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+{"text":"\\section{Introduction}\n\nA \\emph{repetition} of \\emph{size} $t\\geq 1$ in a sequence \nS=s_{1}s_{2}\\ldots s_{n}$ is a subsequence of the form $s_{i+1}s_{i+2}\\ldots\ns_{i+2t}$ satisfying $s_{i+j}=s_{i+t+j}$ for all $j=1,2,\\ldots ,t$. A\nsequence $S$ is \\emph{nonrepetitive} if there is no repetition (of any size)\nin $S$. A celebrated theorem of Thue \\cite{Thue}\\ (cf. \\cite{BerstelThue})\nfrom 1906 asserts that there are arbitrarily long nonrepetitive sequences\nover the set of just $3$ symbols. This result has lots of applications and\ngeneralizations (cf. \\cite{AlloucheShallit}, \\cite{BerstelPerrin}, \\cit\n{GrytczukDM}, \\cite{Lothaire}, \\cite{PezarskiZmarz}). Recently, some game\ntheoretic variants has been introduced leading to new challenges in the area \n\\cite{Pegden}. A basic idea is that a nonrepetitive sequence is created by\ntwo players, Alice and Bob, say, but only Alice cares of avoiding\nrepetitions. For instance, they may be picking alternately symbols from a\nfixed set $A$ and appending them at the end of the existing sequence.\nWhenever a repetition occurs, its second part is being erased immediately.\nIt is proved in \\cite{GKM}\\ that Alice can still create arbitrarily long\nnonrepetitive sequences (no matter what Bob is doing) provided the size of $A\n$ is at least $8$.\n\nIn this paper we introduce another Thue type game, which we call the \\emph\nonline Thue game}. In one round of the game Bob chooses a position in the\nexisting sequence $S$, which is specified by a number $i\\in \\{0,1,\\ldots ,n\\}\n$, and Alice is picking a symbol $x\\in A$ which is inserted right after \ns_{i}$, thereby giving a new sequence $S^{\\prime }=s_{1}\\ldots\ns_{i}xs_{i+1}\\ldots s_{n}$, with $i=0$ meaning that $x$ is placed at the\nbeginning of $S$. The goal of Bob is to force Alice to create a repetition,\nwhile Alice will try to avoid that for as long as possible. For instance, if \n$A=\\{a,b,c\\}$ and $S=acbc$, then Bob catches Alice in one move by choosing \ni=1$. Indeed, picking any $x\\in A$ results in a repetition in $S^{\\prime }$: \n$\\boldsymbol{aa}cbc$, $a\\boldsymbol{bcbc}$, $a\\boldsymbol{cc}bc$. Actually,\nit is easy to check that Alice has no chance for a longer play over $3$\nsymbols. A bit more effort is needed to check that the same is true when $A$\nis of size $4$, and one may start thinking that there is no finite bound at\nall. However we shall prove the following theorem.\n\n\\begin{theorem}\nThere is a strategy guaranteeing Alice arbitrarily long play in the online\nThue game on the set of $12$ symbols.\n\\end{theorem}\n\nThe proof is based on a former result of K\\\"{u}ndegen and Pelsmajer \\cit\n{KundgenPelsmajer}, and Bar\\'{a}t and Varj\\'{u} \\cite{BaratVarju}, on\nnonrepetitive coloring of outerplanar graphs. A vertex coloring of a graph $G\n$ is \\emph{nonrepetitive} if no repetition can be found along any simple\npath of $G$.\n\n\\begin{theorem}\n\\emph{(\\cite{BaratVarju}, \\cite{KundgenPelsmajer}) }Every outerplanar graph\nhas a nonrepetitive coloring using $12$ colors.\n\\end{theorem}\n\nThe proof of Theorem 1, in a slightly different setting, is given in the\nnext section. The last section contains discussion and some open problems.\n\n\\section{Proof of the main result}\n\nWe start with introducing an equivalent setting for the online Thue game\nthat will be more convenient for our purposes. First notice that this game\ncan be played on the real line in such a way that Bob is choosing points of\nthe line and Alice is coloring them using $A$ as the set of colors. Then\nafter $n$ rounds we have a sequence of points $B=b_{1}b_{2}\\ldots b_{n}$, \nb_{i}\\in \\mathbb{R}$, written in increasing order $b_{1}<b_{2}<\\ldots <b_{n}\n, and the corresponding sequence of colors $S=c(b_{1})c(b_{2})\\ldots c(b_{n})\n$, $c(b_{i})\\in A$, assigned to the points $b_{i}$ by Alice. The goal of\nAlice is to avoid repetitions in $S$.\n\n\\begin{proof}\nWithout loss of generality we may assume that Bob starts with $0$, and\nwhenever he wants to extend the sequence $B$ to the left or to the right, he\nalways chooses the closest integer point. So, in the second move he picks\neither $-1$ or $1$. In consequence the extreme points $b_{1}$ and $b_{n}$ of \n$B$ are always integers. Furthermore we assume that when Bob decides to\ninsert a point between $b_{i}$ and $b_{i+1}$, then the new point lies\nexactly in the middle of these two. So, if he has chosen $1$ in the second\nround, then he may now choose $-1,1\/2,$ or $2$. The set of points accessible\nin this way is just the set of \\emph{dyadic rationals} $D$ which consists of\nall numbers of the form $a\/2^{k}$, where $a$ is any integer, and $k$ is any\nnonnegative integer. We assume that $(a,2^{k})=1$, and we call the exponent \nk$ the \\emph{depth} of $a\/2^{k}$. Denote by $D_{k}$ the set of all dyadic\nrationals of depth $k$. Notice that $D_{0}$ is just the set\\ of all integers.\n\nNow, for each integer $k\\geq 0$ consider a graph $P_{k}$ whose vertex set is\nthe union $D_{0}\\cup D_{1}\\cup \\ldots \\cup D_{k}$, with two points joined by\nan edge if and only if there is no other point of the union between them on\nthe line. Clearly $P_{k}$ is a bi-infinite path. Consider finally a graph $G$\non the vertex set $D$ whose set of edges is the union of sets of edges of\nall paths $P_{k}$. It is not hard to see that $G$ is an outerplanar graph.\nIndeed, we may draw the edges of each path $P_{k}$ as half-circles of\ndiameter $1\/2^{k}$ lying above the line. Then no two edges of $G$ can cross,\nand clearly all vertices of $G$ touch the outer face of this embedding.\n\nA simple (but crucial) observation is that any sequence $B=b_{1}b_{2}\\ldots\nb_{n}$ that can be formed by Bob during the game is a path in $G$. Actually, \n$B$ is a path in a subgraph $G_{n}$ of $G$ which is the union of paths \nP_{0},P_{1},\\ldots ,P_{n}$. Indeed, assume inductively that this holds for $n\n$ and consider a sequence $B^{\\prime }$ from the next round. Suppose first\nthat a new point $x$ is placed at the beginning of $B$, that is $B^{\\prime\n}=xB$. Then, accordingly to the rules, $x$ must be the closest integer to \nb_{1}$, which itself is an integer. Hence $xb_{1}$ is an edge in $P_{0}$,\nand consequently $B^{\\prime }$ is a path in $G_{n+1}$. The same holds for \nB^{\\prime }=Bx$. Now, suppose $x$ has been inserted between $b_{i}$ and \nb_{i+1}$. By inductive assumption these two points lie consecutively on some\npath $P_{j}$, $0\\leq j\\leq n$. Hence, $b_{i},x,b_{i+1}$ lie consecutively on\npath $P_{j+1}$. In other words, the edge $b_{i}b_{i+1}$ has been subdivided,\nand therefore $B^{\\prime }$ is a path in $G_{n+1}$.\n\nTo complete the proof we need only to demonstrate that graph $G$ has a\nnonrepetitive coloring using $12$ colors. But this follows easily from\nTheorem 2 by invoking the compactness principle.\n\\end{proof}\n\n\\section{Remarks and open problems}\n\nFirst notice that we have actually proved a stronger result asserting that\nAlice has a strategy to play \\emph{infinitely} long, not just to play \\emph\narbitrarily} long. The strategy seems nonconstructive, as it is based on\nnonrepetitive coloring of an infinite graph obtained via compactness\nprinciple. However, looking into details of the proof of Theorem 2, one has\nan impression that it can be adapted to infinite graphs so that an explicit\nformula for the nonrepetitive coloring of $G$ is perhaps possible. \n\nNotice also that in fact we need a weaker coloring in which only\nforward-directed paths are nonrepetitive. We suspect that this can be\nachieved with smaller number of colors, perhaps $9$ are sufficient.\n\nAnother observation leads to a major open problem of the area of\nnonrepetitive colorings of graphs. Notice that graph $G$ from the proof of\nTheorem 1 has \\emph{page number} $1$. What about nonrepetitive coloring of\ngraphs with higher page number?\n\n\\begin{conjecture}\nThere is a finite constant $N$ such that every graph of page number $2$ has\na nonrepetitive coloring using $N$ colors.\n\\end{conjecture}\n\nThis innocently looking question was propounded some years ago by Idziak.\nHowever, now we know that it is strong enough to imply the following\nstatement.\n\n\\begin{conjecture}\nThere is a finite constant $M$ such that every planar graph has a\nnonrepetitive coloring using $M$ colors.\n\\end{conjecture}\n\nThe best result to date \\cite{Dujmovic2} gives a logarithmic upper bound (in\nthe number of vertices of a graph). Perhaps it would be easier to verify\ndirected versions of the above problems.\n\nFinally, one may consider online versions of other theorems in combinatorics\non words. Particularly interesting seems the case of \\emph{abelian\nrepetitions}, that is subsequences of the form $r_{1}\\ldots r_{t}r_{\\sigma\n(1)}\\ldots r_{\\sigma (t)}$, where $\\sigma $ is any permutation of the set \n\\{1,2,\\ldots ,t\\}$. Answering a question of Erd\\H{o}s \\cite{Erdos}, Ker\\\"{a\nnen \\cite{Keranen} proved that there exist arbitrarily long sequences over $4\n$ symbols avoiding abelian repetitions of any size. Is the online version of\nthis result possible?\n\n\\begin{acknowledgement}\nJ. Grytczuk acknowledges a support of the Polish Ministry of Science and\nHigher Education, grant N206257035.\n\\end{acknowledgement}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the Hamiltonian formulation of classical mechanics, the canonical transformations, given locally by expressions of the form $Q_{i} = Q_{i}(q_{j}, p_{j}, t)$, $P_{i} = P_{i}(q_{j}, p_{j}, t)$, act on the extended phase space (the Cartesian product of the phase space and the time axis) and, therefore, there is a naturally defined action of a canonical transformation on any function defined on the extended phase space (that is, on any function of $q_{i}$, $p_{i}$, and $t$). On the other hand, Hamilton's principal function, usually denoted by an $S$, is a function defined on the extended configuration space (that is, a function of $q_{i}$ and $t$), just like the wave function in the elementary Schr\\\"odinger equation, but there is no obvious way to define the action of a canonical transformation on a function of $q_{i}$ and $t$.\n\nHowever, in Ref.\\ \\cite{DI} it was shown that, under certain conditions, one can define an action of a time-independent canonical transformation [that is, a canonical transformation that does not involve the time, $Q_{i} = Q_{i}(q_{j}, p_{j})$, $P_{i} = P_{i}(q_{j}, p_{j})$] on functions defined on the configuration space, in such a way that if a (time-independent) Hamiltonian is invariant under the canonical transformation, a solution of the corresponding time-independent Hamilton--Jacobi (HJ) equation is mapped into another solution. Furthermore, it was shown that the action of the one-parameter group of canonical transformations generated by an arbitrary function $G(q_{i}, p_{i})$ is determined by an equation similar to the HJ equation.\n\nIn this paper, we consider systems with a Hamiltonian that can depend on the time, we show that one can define the action of the one-parameter group of canonical transformations generated by an arbitrary function $G(q_{i}, p_{i}, t)$ on functions defined on the extended configuration space, in such a way that if $G$ is a constant of motion, then a solution of the corresponding HJ equation is mapped into a one-parameter family of solutions of this equation. In this manner, each constant of motion adds a continuous parameter to a given solution of the HJ equation. We also show that any complete solution of the HJ equation can be obtained from a solution without parameters by means of the action of the groups of canonical transformations generated by $n$ constants of motion in involution.\n\nThroughout this paper, several examples are given in order to illustrate the definitions and results presented here.\n\n\\section{The action of a one-parameter group of canonical transformations on functions defined on the extended configuration space}\nWe start by reviewing the HJ equation, which will give us the pattern to follow in the definition of the action of any one-parameter group of canonical transformations on functions defined on the extended configuration space.\n\n\\subsection{The Hamilton--Jacobi as an evolution equation}\nGiven a Hamiltonian $H(q_{i}, p_{i}, t)$ of a system with $n$ degrees of freedom, the corresponding HJ equation is the first-order partial differential equation\n\\begin{equation}\nH(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial t} = 0. \\label{hj}\n\\end{equation}\nUsually one is interested in {\\em complete solutions}\\\/ of the HJ equation (\\ref{hj}), that is, solutions $S(q_{i}, t, \\alpha_{i})$ of Eq.\\ (\\ref{hj}), containing $n$ arbitrary parameters $\\alpha_{i}$, such that\n\\begin{equation}\n\\det \\left( \\frac{\\partial^{2} S}{\\partial q_{i} \\partial \\alpha_{j}} \\right) \\not= 0, \\label{compl}\n\\end{equation}\nbecause they yield the solution of the Hamilton equations (see, {\\em e.g.}, Refs.\\ \\cite{Wi} and \\cite{Ga}). However, one can consider the HJ equation as an {\\em evolution equation}, which determines the function $S(q_{i}, t)$ that reduces to a given function $f(q_{i})$, for $t = 0$ (or any other initial value, $t_{0}$, of $t$). (This fact does not sound very strange if we consider the relationship between the HJ equation and the Schr\\\"odinger equation.) In fact, if we have the solution of the Hamilton equations, we can use it to find the solution of the HJ equation that satisfies any initial condition $S(q_{i}, t_{0}) = f(q_{i})$. (Similar ideas are often mentioned in the standard textbooks on analytical mechanics, without presenting, however, a clear statement or an explicit procedure.)\n\nFor example, in the case of the Hamiltonian\n\\begin{equation}\nH = \\frac{p_{1}{}^{2} + p_{2}{}^{2}}{2m} + mg q_{2}, \\label{hamex}\n\\end{equation}\nwhere $m$ and $g$ are constants, the solution of the corresponding Hamilton equations can be readily obtained and is given by\n\\begin{equation}\nq_{1} = (q_{1})_{0} + \\frac{(p_{1})_{0} t}{m}, \\qquad q_{2} = (q_{2})_{0} + \\frac{(p_{2})_{0} t}{m} - \\frac{gt^{2}}{2}, \\qquad p_{1} = (p_{1})_{0}, \\qquad p_{2} = (p_{2})_{0} - mgt, \\label{solex}\n\\end{equation}\nwhere $(q_{1})_{0}$ denotes the value of $q_{1}$ at $t = 0$, and so on. As the initial condition we choose\n\\begin{equation}\nS(q_{1}, q_{2}, 0) = \\alpha_{1} q_{1} + \\alpha_{2} q_{2}, \\label{icex}\n\\end{equation}\nwhere $\\alpha_{1}$, $\\alpha_{2}$ are arbitrary constants. We want to find the solution of the HJ equation [see Eqs.\\ (\\ref{hj}) and (\\ref{hamex})]\n\\begin{equation}\n\\frac{\\partial S}{\\partial t} = - \\frac{1}{2m} \\left[ \\left( \\frac{\\partial S}{\\partial q_{1}} \\right)^{2} + \\left( \\frac{\\partial S}{\\partial q_{2}} \\right)^{2} \\right] - mg q_{2}, \\label{hjex}\n\\end{equation}\nthat satisfies the initial condition (\\ref{icex}). Recalling that $\\partial S\/\\partial q_{i} = p_{i}$, making use of Eqs.\\ (\\ref{solex}) and (\\ref{icex}) we have\n\\[\n\\frac{\\partial S}{\\partial q_{1}} = \\left. \\frac{\\partial S}{\\partial q_{1}} \\right|_{t = 0} = \\alpha_{1}, \\qquad \\frac{\\partial S}{\\partial q_{2}} = \\left. \\frac{\\partial S}{\\partial q_{2}} \\right|_{t = 0} - mgt = \\alpha_{2} - mgt.\n\\]\nSubstituting these expressions into the right-hand side of Eq.\\ (\\ref{hjex}) we have\n\\[\n\\frac{\\partial S}{\\partial t} = - \\frac{1}{2m} \\big[ \\alpha_{1}{}^{2} + (\\alpha_{2} - mgt)^{2} \\big] - mg q_{2}.\n\\]\nCombining the last three equations one readily finds that, choosing the integration constant so that Eq.\\ (\\ref{icex}) is satisfied,\n\\begin{equation}\nS(q_{1}, q_{2}, t) = \\alpha_{1} q_{1} + \\alpha_{2} q_{2} - mgtq_{2} - \\frac{\\alpha_{1}{}^{2} t}{2m} + \\frac{(\\alpha_{2} - mgt)^{3} - \\alpha_{2}{}^{3}}{6m^{2}g}. \\label{prinex}\n\\end{equation}\nThe expression (\\ref{prinex}) is a (complete, R-separable) solution of the HJ equation that reduces to the specified function (\\ref{icex}) for $t = 0$. (A solution is R-separable if it is the sum of a function of two or more variables and functions of one variable.)\n\nWe close this subsection with a second example. The solution of the Hamilton equations for the time-dependent Hamiltonian\n\\begin{equation}\nH = \\frac{p^{2}}{2m} - ktq, \\label{n1}\n\\end{equation}\nwhere $m$ and $k$ are constants, is given by\n\\begin{equation}\nq = q_{0} + \\frac{tp_{0}}{m} + \\frac{kt^{3}}{6m}, \\qquad p = p_ {0} + \\frac{1}{2} kt^{2}, \\label{n2}\n\\end{equation}\nwhere $q_{0}$ and $p_{0}$ denote the values of $q$ and $p$ at $t = 0$, respectively. Letting\n\\begin{equation}\nS(q,0,\\alpha) = \\alpha q, \\label{n3}\n\\end{equation}\nwhere $\\alpha$ is an arbitrary constant, with the aid of Eqs.\\ (\\ref{n2}), we have\n\\begin{equation}\n\\frac{\\partial S}{\\partial q} = \\left. \\frac{\\partial S}{\\partial q} \\right|_{t = 0} + \\frac{1}{2} kt^{2} = \\alpha + \\frac{1}{2} kt^{2}. \\label{n4}\n\\end{equation}\nThus, from Eqs.\\ (\\ref{hj}), (\\ref{n1}), and (\\ref{n4}), we have\n\\begin{equation}\n\\frac{\\partial S}{\\partial t} = - \\frac{1}{2m} \\left( \\frac{\\partial S}{\\partial q} \\right)^{2} + ktq = - \\frac{1}{2m} \\left( \\alpha + \\frac{1}{2} kt^{2} \\right)^{2} + ktq. \\label{n5}\n\\end{equation}\nThen, from Eqs.\\ (\\ref{n3})--(\\ref{n5}), we readily obtain\n\\[\nS(q, t, \\alpha) = \\alpha q + \\frac{1}{2} kt^{2} q - \\frac{1}{2m} \\left( \\alpha^{2} t + \\frac{1}{3} \\alpha k t^{3} + \\frac{k^{2} t^{5}}{20} \\right),\n\\]\nwhich is an R-separable complete solution of the HJ equation. (We might start with expressions more complicated than (\\ref{icex}) and (\\ref{n3}), but these simple expressions are enough to obtain complete solutions of the HJ equation.)\n\nIt should be clear that a similar construction can be devised using an arbitrary function of $(q_{i}, p_{i}, t)$ instead of a Hamiltonian.\n\n\\subsection{Families of solutions of the HJ equation and constants of motion}\nAny differentiable function, $G(q_{i}, p_{i}, t)$, defines a (possibly local) one-parameter group of canonical transformations, determined by the (autonomous) system of first-order ordinary differential equations\n\\begin{equation}\n\\frac{{\\rm d} q_{i}}{{\\rm d} \\alpha} = \\frac{\\partial G(q_{j}, p_{j}, t)}{\\partial p_{i}}, \\qquad\n\\frac{{\\rm d} p_{i}}{{\\rm d} \\alpha} = - \\frac{\\partial G(q_{j}, p_{j}, t)}{\\partial q_{i}}, \\label{odes}\n\\end{equation}\nwhich are of the form of the Hamilton equations.\n\nFor example, the function\n\\begin{equation}\nG(q_{1}, q_{2}, p_{1}, p_{2}, t) = \\frac{p_{1} p_{2}}{m} + mgq_{1}, \\label{a}\n\\end{equation}\nwhere $m$ and $g$ are constants, leads to the system of equations\n\\begin{equation}\n\\frac{{\\rm d} q_{1}}{{\\rm d} \\alpha} = \\frac{p_{2}}{m}, \\qquad \\frac{{\\rm d} q_{2}}{{\\rm d} \\alpha} = \\frac{p_{1}}{m}, \\qquad \\frac{{\\rm d} p_{1}}{{\\rm d} \\alpha} = - mg, \\qquad \\frac{{\\rm d} p_{2}}{{\\rm d} \\alpha} = 0. \\label{sim}\n\\end{equation}\nFrom the last two equations we find that\n\\begin{equation}\np_{1} = (p_{1})_{0} - mg \\alpha, \\qquad p_{2} = (p_{2})_{0}, \\label{apes}\n\\end{equation}\nwhere $(p_{i})_{0}$ denotes the value of $p_{i}$ for $\\alpha = 0$. Substituting these expressions into the first pair of equations (\\ref{sim}) we get\n\\[\n\\frac{{\\rm d} q_{1}}{{\\rm d} \\alpha} = \\frac{(p_{2})_{0}}{m}, \\qquad \\frac{{\\rm d} q_{2}}{{\\rm d} \\alpha} = \\frac{(p_{1})_{0} - mg \\alpha}{m}.\n\\]\nHence,\n\\begin{equation}\nq_{1} = (q_{1})_{0} + \\frac{(p_{2})_{0} \\alpha}{m}, \\qquad q_{2} = (q_{2})_{0} + \\frac{(p_{1})_{0} \\alpha}{m} - \\frac{g \\alpha^{2}}{2}. \\label{aqs}\n\\end{equation}\n\nThe action of the canonical transformations defined by a function $G(q_{i}, p_{i}, t)$, on functions of\n$(q_{i}, t)$, will be defined imitating the HJ equation.\n\n\\noindent{\\bf Definition.} The image of a given arbitrary function $f(q_{i}, t)$ under the one-parameter group of canonical transformations generated by $G(q_{i}, p_{i}, t)$ is the function $S(q_{i}, t, \\alpha)$ such that\n\\begin{equation}\nG(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial \\alpha} = 0, \\label{action}\n\\end{equation}\nwith the initial condition $S(q_{i}, t, 0) = f(q_{i}, t)$. ({\\em Cf.}\\ Eq.\\ (\\ref{hj}).)\n\nThe following example shows that this definition produces the expected effect in the case of transformations that only affect the coordinates of the configuration space. Indeed, the one-parameter group of canonical transformations generated by, {\\em e.g.}, $G = p_{1}$ is given by [see Eqs.\\ (\\ref{odes})]\n\\[\nq_{1} = (q_{1})_{0} + \\alpha, \\qquad q_{i} = (q_{i})_{0}, \\quad {\\rm for \\ } i \\geqslant 2, \\qquad p_{i} = (p_{i})_{0}, \\]\nthat is, ``translations'' along the $q_{1}$-axis. Then, Eq.\\ (\\ref{action}) takes the form\n\\[\n\\frac{\\partial S}{\\partial q_{1}} + \\frac{\\partial S}{\\partial \\alpha} = 0,\n\\]\nwhose general solution is\n\\[\nS(q_{i}, t, \\alpha) = F(q_{1} - \\alpha, q_{2}, \\ldots, q_{n}, t),\n\\]\nwhere $F$ is an arbitrary function. By imposing the initial condition $S(q_{i}, t, 0) = f(q_{i}, t)$, we find that\n\\[\nS(q_{i}, t, \\alpha) = f(q_{1} - \\alpha, q_{2}, \\ldots, q_{n}, t),\n\\]\nwhich is the expected effect of a translation along the $q_{1}$-axis.\n\nAs a second example, the images of the function\n\\begin{equation}\nf(q_{i}, t) = (\\alpha_{2} - mgt)q_{2} + \\frac{(\\alpha_{2} - mgt)^{3} - \\alpha_{2}{}^{3}}{6m^{2}g} \\label{icex2}\n\\end{equation}\n(which is the function (\\ref{prinex}) with $\\alpha_{1} = 0$) under the transformations generated by (\\ref{a}) can be obtained proceeding as in the examples of Sec.\\ 2.1. Making use of Eqs.\\ (\\ref{apes}) and (\\ref{icex2}) we have [with $S(q_{i}, t, 0) = f(q_{i}, t)$]\n\\begin{equation}\n\\frac{\\partial S}{\\partial q_{1}} = \\left. \\frac{\\partial S}{\\partial q_{1}} \\right|_{\\alpha = 0} - mg \\alpha = - mg \\alpha, \\qquad \\frac{\\partial S}{\\partial q_{2}} = \\left. \\frac{\\partial S}{\\partial q_{2}} \\right|_{\\alpha = 0} = \\alpha_{2} - mgt. \\label{partial}\n\\end{equation}\nSubstituting these expressions into Eq.\\ (\\ref{action}), taking into account Eq.\\ (\\ref{a}), we obtain\n\\[\n\\frac{\\partial S}{\\partial \\alpha} = - \\frac{1}{m} (- mg \\alpha) (\\alpha_{2} - mgt) - mg q_{1}.\n\\]\nCombining the last equations and the initial condition (\\ref{icex2}) one finds that\n\\begin{equation}\nS(q_{1}, q_{2}, t, \\alpha_{2}, \\alpha) = (\\alpha_{2} - mgt)q_{2} - mg \\alpha q_{1} + \\frac{1}{2} (\\alpha_{2} - mgt)g \\alpha^{2} + \\frac{(\\alpha_{2} - mgt)^{3} - \\alpha_{2}{}^{3}}{6m^{2}g}. \\label{prinex2}\n\\end{equation}\nNote that $t$ is treated here as a constant because the canonical transformations do not affect $t$. Since the function (\\ref{icex2}) is a particular case of (\\ref{prinex}), is a solution of the HJ equation (\\ref{hjex}), and one can readily verify that (\\ref{prinex2}) is {\\em also}\\\/ a solution of Eq.\\ (\\ref{hjex}), for all values of the parameter $\\alpha$ [which is essentially the two-parameter family of solutions (\\ref{prinex})].\n\nThus, we have obtained a two-parameter family of solutions of the HJ equation (\\ref{hjex}), starting from a one-parameter solution of this equation. As we shall show in the following Proposition, this is a consequence of the fact that the function (\\ref{a}) is a constant of motion for the Hamiltonian (\\ref{hamex}).\n\n\\noindent {\\bf Proposition 1.} The image of a solution of the HJ equation, $S_{0}(q_{i}, t)$, under the one-parameter group of canonical transformations generated by a constant of motion $G(q_{i}, p_{i}, t)$ is a one-parameter family of solutions of the same HJ equation.\n\n\\noindent {\\em Proof.} Assuming that $S(q_{i}, t, \\alpha)$ is a solution of Eq.\\ (\\ref{action}), making use of the chain rule repeatedly, we have (here, and in what follows, there is summation over repeated indices)\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial \\alpha} \\left[ H(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial t} \\right] & = & \\frac{\\partial H}{\\partial p_{j}} \\frac{\\partial^{2} S}{\\partial \\alpha \\partial q_{j}} + \\frac{\\partial^{2} S}{\\partial \\alpha \\partial t} \\nonumber \\\\\n& = & - \\frac{\\partial H}{\\partial p_{j}} \\frac{\\partial G(q_{i}, \\partial S\/\\partial q_{i}, t)}{\\partial q_{j}} - \\frac{\\partial G(q_{i}, \\partial S\/\\partial q_{i}, t)}{\\partial t} \\nonumber \\\\\n& = & - \\frac{\\partial H}{\\partial p_{j}} \\left( \\frac{\\partial G}{\\partial q_{j}} + \\frac{\\partial G}{\\partial p_{i}} \\frac{\\partial^{2} S}{\\partial q_{j} \\partial q_{i}} \\right) - \\frac{\\partial G}{\\partial p_{i}} \\frac{\\partial^{2} S}{\\partial t \\partial q_{i}} - \\frac{\\partial G}{\\partial t}. \\label{des1}\n\\end{eqnarray}\nAccording to the hypothesis, $G$ is a constant of motion, that is\n\\begin{equation}\n\\frac{\\partial G}{\\partial t} + \\frac{\\partial G}{\\partial q_{j}} \\frac{\\partial H}{\\partial p_{j}} - \\frac{\\partial G}{\\partial p_{j}} \\frac{\\partial H}{\\partial q_{j}} = 0, \\label{const}\n\\end{equation}\nthus, the last line of Eq.\\ (\\ref{des1}) amounts to\n\\begin{eqnarray*}\n- \\frac{\\partial G}{\\partial p_{i}} \\left( \\frac{\\partial H}{\\partial q_{i}} + \\frac{\\partial H}{\\partial p_{j}} \\frac{\\partial^{2} S}{\\partial q_{j} \\partial q_{i}} + \\frac{\\partial^{2} S}{\\partial t \\partial q_{i}} \\right) & = & - \\frac{\\partial G}{\\partial p_{i}} \\frac{\\partial}{\\partial q_{i}} \\left[ H(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial t} \\right] \\\\\n& = & - \\frac{{\\rm d} q_{i}}{{\\rm d} \\alpha} \\frac{\\partial}{\\partial q_{i}} \\left[ H(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial t} \\right]\n\\end{eqnarray*}\n[see Eqs.\\ (\\ref{odes})]. Hence, we obtain the equation\n\\[\n\\frac{{\\rm d}}{{\\rm d} \\alpha} \\left[ H(q_{i}(\\alpha), \\partial S(q_{i}(\\alpha), t, \\alpha)\/\\partial q_{i}, t) + \\frac{\\partial S(q_{i}(\\alpha), t, \\alpha)}{\\partial t} \\right] = 0,\n\\]\nwhich involves the images of $S$ and the coordinates $q_{i}$ under the transformations generated by $G$. Since the expression inside the brackets in the last equation vanishes for $\\alpha = 0$, it is equal to zero for all values of $\\alpha$, thus proving the assertion.\n\nA partial converse of this Proposition is true. The steps in the proof show that if the images of a solution of the HJ equation under the transformations generated by $G$ are solutions of the same HJ equation, and $S$ is a complete solution of the HJ equation, then $G$ is a constant of motion. The assumption that $S$ is a complete solution of the HJ equation assures that Eq.\\ (\\ref{const}) holds everywhere. In fact, the following stronger result holds.\n\n\\noindent {\\bf Proposition 2.} A complete solution of the HJ equation, $S(q_{i}, t, \\alpha_{i})$ (not necessarily separable or R-separable), is the image of $S(q_{i}, t, 0)$ under the transformations generated by the $n$ constants of motion\n\\begin{equation}\nG_{j}(q_{i}, p_{i}, t) = - \\frac{\\partial S}{\\partial \\alpha_{j}}(q_{i}, t, \\alpha_{i}(q_{k}, p_{k}, t)) \\label{prop3}\n\\end{equation}\n($j = 1, 2, \\ldots, n$), obtained by eliminating the $\\alpha_{i}$ by means of\n\\begin{equation}\np_{j} = \\frac{\\partial S}{\\partial q_{j}} (q_{i}, \\alpha_{i}, t). \\label{pes}\n\\end{equation}\nFurthermore, the functions $G_{i}$ are in involution ({\\em i.e.}, the Poisson bracket of $G_{i}$ and $G_{j}$ is equal to zero). (Note that, according to Eq.\\ (\\ref{compl}), in principle, one can find the $\\alpha_{i}$ from Eqs.\\ (\\ref{pes}).)\n\n\\noindent {\\em Proof.} We note that, by virtue of Eqs.\\ (\\ref{pes}), Eq.\\ (\\ref{prop3}) is equivalent to Eq.\\ (\\ref{action}). Since, by hypothesis, $S(q_{i}, t, \\alpha_{i})$ is a complete solution of the HJ equation, each function $G_{i}$, defined by (\\ref{prop3}), is a constant of motion. In order to prove that the functions $G_{i}$ are in involution, making use of Eq.\\ (\\ref{prop3}), we calculate the mixed partial derivative\n\\[\n\\frac{\\partial^{2} S}{\\partial \\alpha_{i} \\partial \\alpha_{j}} = - \\frac{\\partial}{\\partial \\alpha_{i}} G_{j}(q_{i}, \\partial S\/\\partial q_{i}, t, \\alpha_{k}) = - \\frac{\\partial G_{j}}{\\partial p_{k}} \\frac{\\partial}{\\partial \\alpha_{i}} \\frac{\\partial S}{\\partial q_{k}} = \\frac{\\partial G_{j}}{\\partial p_{k}} \\frac{\\partial G_{i}}{\\partial q_{k}}.\n\\]\nThen we see that the commutativity of the partial derivatives of $S$ implies that the Poisson brackets of the $G_{i}$ are equal to zero. (The assumption that $S(q_{i}, t, \\alpha_{i})$ is a complete solution of the HJ equation implies that the Poisson brackets $\\{ G_{i}, G_{j} \\}$ vanish everywhere.) Since the Poisson brackets $\\{ G_{i}, G_{j} \\}$ are equal to zero, the one-parameter groups of transformations generated by the functions $G_{i}$ commute. (In fact, the groups of canonical transformations generated by two functions $F$ and $G$ commute if and only if $\\{ F, G \\}$ is a trivial constant, not necessarily zero \\cite{DM}.)\n\nLet us consider, for example, the function\n\\begin{equation}\nS(q_{i}, t, \\alpha_{i}) = \\alpha_{1} q_{1} + \\alpha_{2} q_{2} - mgtq_{2} - \\frac{\\alpha_{1}{}^{2} t}{2m} + \\frac{(\\alpha_{2} - mgt)^{3} - \\alpha_{2}{}^{3}}{6m^{2}g}, \\label{prinexo}\n\\end{equation}\nwhich is the complete solution (\\ref{prinex}) of the HJ equation constructed in Sec.\\ 2.1. One finds that\n\\[\n\\frac{\\partial S}{\\partial q_{1}} = \\alpha_{1}, \\qquad \\frac{\\partial S}{\\partial q_{2}} = \\alpha_{2} - mgt,\n\\]\nwhich lead to [see Eqs.\\ (\\ref{pes})]\n\\begin{equation}\n\\alpha_{1} = p_{1}, \\qquad \\alpha_{2} = p_{2} + mgt. \\label{inversi}\n\\end{equation}\nOn the other hand,\n\\[\n\\frac{\\partial S}{\\partial \\alpha_{1}} = q_{1} - \\frac{\\alpha_{1} t}{m}, \\qquad \\frac{\\partial S}{\\partial \\alpha_{2}} = q_{2} + \\frac{(\\alpha_{2} - mgt)^{2} - \\alpha_{2}{}^{2}}{2 m^{2} g},\n\\]\nwhich, taking into account Eqs.\\ (\\ref{inversi}), are of the form (\\ref{prop3}) with\n\\begin{equation}\nG_{1} = - q_{1} + \\frac{p_{1} t}{m}, \\qquad G_{2} = - q_{2} + \\frac{p_{2} t}{m} + \\frac{1}{2} gt^{2}. \\label{ges}\n\\end{equation}\nOne can readily verify that $G_{1}$ and $G_{2}$ are constants of motion in involution, and that the function (\\ref{prinexo}) can be obtained by means of the transformations generated by $G_{1}$ and $G_{2}$ from\n\\begin{equation}\nS(q_{i},t) = - mgt q_{2} - \\frac{mg^{2}t^{3}}{6}, \\label{init}\n\\end{equation}\nwhich is obtained from (\\ref{prinexo}) setting $\\alpha_{1} = \\alpha_{2} = 0$.\n\nIndeed, the function\n\\begin{equation}\nG \\equiv \\alpha_{1} G_{1} + \\alpha_{2} G_{2} = \\alpha_{1} \\left( - q_{1} + \\frac{p_{1} t}{m} \\right) + \\alpha_{2} \\left( - q_{2} + \\frac{p_{2} t}{m} + \\frac{1}{2} gt^{2} \\right) \\label{gee}\n\\end{equation}\nis a constant of motion for all values of the constants $\\alpha_{1}$ and $\\alpha_{2}$. The one-parameter group of canonical transformations generated by $G$ is given by (denoting by $s$ the corresponding parameter)\n\\[\nq_{1} = (q_{1})_{0} + \\frac{\\alpha_{1} t}{m} s, \\quad q_{2} = (q_{2})_{0} + \\frac{\\alpha_{2} t}{m} s, \\quad p_{1} = (p_{1})_{0} + \\alpha_{1} s, \\quad p_{2} = (p_{2})_{0} + \\alpha_{2} s,\n\\]\nhence, making use of the initial condition (\\ref{init}),\n\\[\n\\frac{\\partial S}{\\partial q_{1}} = 0 + \\alpha_{1} s, \\qquad \\frac{\\partial S}{\\partial q_{2}} = - mgt + \\alpha_{2} s,\n\\]\nand substituting these expressions into Eq.\\ (\\ref{action}), with $G$ given by (\\ref{gee}),\n\\[\n\\frac{\\partial S}{\\partial s} = \\alpha_{1} q_{1} - \\frac{\\alpha_{1}t}{m}(\\alpha_{1} s) + \\alpha_{2} q_{2} - \\frac{\\alpha_{2} t}{m} (-mgt + \\alpha_{2} s) - \\frac{1}{2} \\alpha_{2} gt^{2}.\n\\]\nThus,\n\\[\nS(q_{1}, q_{2}, t) = \\alpha_{1} s q_{1} + \\alpha_{2} s q_{2} - mgtq_{2} - \\frac{(\\alpha_{1} s)^{2} t}{2m} + \\frac{(\\alpha_{2} s - mgt)^{3} - (\\alpha_{2} s)^{3}}{6m^{2}g},\n\\]\nwhich reduces to the expression (\\ref{prinexo}) when $s = 1$.\n\n\\section{Concluding remarks}\nAccording to Proposition 2, any complete solution of the HJ equation is, locally, the image of a particular solution (without free parameters) under the action of an Abelian $n$-dimensional group (that is, a group locally isomorphic to the additive Lie group $\\mathbb{R}^{n}$). As pointed out above, these solutions need not be separable or R-separable. It should be noted, however, that some functions are invariant under the transformations generated by a function $G$, even if these transformations are different from the identity. For instance, a function that does not depend on $q_{1}$ is invariant under the translations generated by $p_{1}$.\n\nOne can readily see that, at least in the case of time-independent operators, an analog of Proposition 1 holds in the case of the solutions of the Schr\\\"odinger equation.\n\nIn addition to the R-separable complete solution (\\ref{prinex}), the HJ equation for the Hamiltonian (\\ref{hamex}) also admits complete separable solutions in the same coordinate system (see also Ref.\\ \\cite{Li})\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nA maximally oxidised theory  is a Lagrangian theory of  gravity coupled to forms\nand dilatons defined in the highest possible space-time dimension $D$ which\nupon dimensional reduction  to three dimensions exhibits the symmetry of\na simple Lie group ${\\cal G}$ realised on a coset space\n${\\cal G}\/{\\cal H}$. Here\n$\\cal H$ is the maximal compact subgroup of $\\cal G$. The maximally\noxidised actions  have been constructed for all   $\\cal\nG$ \\cite{cjlp}.  They  comprise in particular pure gravity in $D$\ndimensions and the low energy effective actions of the bosonic string and\nof  M-theory.\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=12cm]{dynk.eps}\n \\caption { \\small The\nnodes labelled 1,2,3 define the Kac-Moody extensions of the  Lie\nalgebras. The horizontal line\nstarting at 1 defines the `gravity line', which is the\nDynkin diagram  of a\n$A_{D-1}$ subalgebra. Note that for some Kac-Moody algebras the choice of a\ngravity line is\n not unique, in which case we take here the one shown in the figure.}\n   \\label{first}\n\\end{figure}\n\n\n\n\nIt\nhas been conjectured that these theories, or some extensions of them,\npossess the much larger very-extended Kac-Moody symmetry\n${\\cal G}^{+++}$. ${\\cal G}^{+++}$ algebras are defined by the Dynkin diagrams depicted\nin Fig.1,  obtained from those of $\\cal G$ by adding three\nnodes~\\cite{ogw}. One first adds the affine node, labelled 3 in the\nfigure, then a second node, 2,  connected to it by a single line and\ndefining the overextended ${\\cal G}^{++}$ algebra,   then  a third\none, 1, connected  by a single line to the overextended node. Such\n${\\cal G}^{+++}$ symmetries were first conjectured in the aforementioned\nparticular cases\n\\cite{west01,lw} and the extension to all\n${\\cal G}^{+++}$ was proposed in\n\\cite{ehtw}. In a different development, the study of the properties\nof cosmological solutions in the vicinity of a space-like singularity,\nknown as cosmological billiards\n\\cite{damourhn00},  revealed an  overextended symmetry ${\\cal G}^{++}$\nfor all maximally oxidised theories\n\\cite{damourh00, damourbhs02}.\n\n\nTo explore the possible fundamental significance of these huge\nsymmetries two Lagrangian formulations \\cite{damourhn02, eh}  {\\it\nexplicitly} invariant under such infinite-dimensional Kac-Moody\nalgebras  have been proposed.\n\nFirst  \\cite{damourhn02} an\n$E_{10}\\equiv E_8^{++}$-invariant action was constructed in the\ncontext of M-theory as a reparametrisation invariant $\\sigma$-model of\nfields depending on one parameter $t$ and living on the coset space\n$E_8^{++}\/K_8^{++}$. Here $K_8^{++}$ is the subalgebra of $E_8^{++}$\ninvariant  under the Chevalley involution. The action is built in a\nrecursive way  by a level expansion of $E_8^{++}$ with respect to its \nsubalgebra $A_9$ whose Dynkin diagram is the `gravity line' defined in Fig. 1,\nwith the node 1 deleted.  The level of an irreducible representation of\n$A_9$ occurring in the decomposition of the adjoint  representation of\n$E_8^{++}$ counts the number of  times the simple root  not\npertaining to the gravity line appears in\nthe decomposition.  The $\\sigma$-model, limited to the\nreal roots up to level 3, reveals  a perfect  match with the bosonic\nequations of motion of 11-dimensional supergravity in the vicinity of\nthe  space-like singularity of the $E_8^{++}$ cosmological billiards,\nwhere the fields depend only on time\\footnote{See also \\cite{damourn04} for an\nanalysis in a different formulation, and\n\\cite{nike} for a discussion of $E_8^{++}$ adapted to $IIB$\nsupergravity.}. In the dictionary relating the\n$\\sigma$-model to supergravity, the  parameter $t$\n is identified with time.  It was conjectured that   space derivatives\nare hidden  in some higher level fields of the $\\sigma$-model.\n This approach is straightforwardly generalised, as seen\nbelow, to   actions $S_{{\\cal G}_C^{++}}$ related to cosmological\nbilliards and invariant under the overextended ${\\cal G}^{++}$. Such\n${\\cal G}^{++}$-invariant formulation  singles out naturally a time coordinate.\n\nThe second  approach \\cite{eh}  yields  a ${\\cal G}^{+++}$-invariant action $S_{{\\cal G}^{+++}}$\nwhich puts  all the space-time coordinates on the same footing. This\nis achieved by   a reparametrisation invariant $\\sigma$-model with\nfields depending on a parameter $\\xi$ spanning a world-line a priori\nunrelated to space-time. One uses a level decomposition of ${\\cal G}^{+++}$ with\nrespect to the subalgebra $A_{D-1}$ of its gravity line\\footnote{ Level expansions of very-extended\nalgebras in terms of the subalgebra $A_{D-1}$ have been\nconsidered in\n\\cite{west02, nifi, weke}.}  and\n$D$ is identified to  the space-time dimension. The $\\xi$-dependent\nfields live in a coset space ${\\cal G}^{+++} \/K^{+++}$ where the subalgebra\n$K^{+++}$ is invariant under a temporal involution\n\\cite{eh} which preserve the Lorentz algebra $SO(1,D-1)$ and ensures\nthat the action $S_{{\\cal G}^{+++}}$ is Lorentz invariant\nat each level.\n\nIn this paper, we analyse the ${\\cal G}^{++}$ content of the\n${\\cal G}^{+++}$-invariant action $S_{{\\cal G}^{+++}}$. We find that the ${\\cal G}^{++}$-invariant action\n$S_{{\\cal G}_C^{++}}$  is obtained by putting to zero in $S_{{\\cal G}^{+++}}$,\nconsistently with all its equations of motion, all the fields\nwhich do not appear in $S_{{\\cal G}_C^{++}}$. This consistent\ntruncation of the ${\\cal G}^{+++}$ theory implies that all the solutions of the equations of\nmotion of $S_{{\\cal G}_C^{++}}$ are also solutions of the\nequations of motion of $S_{{\\cal G}^{+++}}$. We then find that $S_{{\\cal G}^{+++}}$ contains another\n action $S_{{\\cal G}_B^{++}}$ obtained by performing the\nconsistent truncation {\\em after}  conjugation by a Weyl reflection in\n${\\cal G}^{+++}$. The action $S_{{\\cal G}_B^{++}}$ is also invariant under a ${\\cal G}^{++}$\nalgebra but is nonequivalent to $S_{{\\cal\nG}_C^{++}}$ and singles out naturally a space direction. Through Weyl\ntransformations in ${\\cal G}^{++}$ expressed in terms of fields,\n$S_{{\\cal G}_B^{++}}$ can be formulated\n in various equivalent forms endowed with\nvarious Lorentzian signatures $(-\\dots -, +\\dots +)$,  a consequence of\nthe non-commutativity of the temporal involution with Weyl reflections\n\\cite{keu1,keu2}. Exact solutions of $S_{{\\cal G}_B^{++}}$ are obtained in\naccordance with references \\cite{eh,eh2}.  These solutions   are   {\\em\nidentical} to those of the covariant Einstein and field equations describing,\ndepending on the signature, conventional or exotic intersecting extremal branes\nsmeared in all directions but one\n\\cite{aeh, ah}. They  transform into each other by\n`duality' Weyl transformations. As all these brane solutions of\n$S_{{\\cal G}_B^{++}}$  are also solutions of the\n${\\cal G}^{+++}$-invariant action $S_{{\\cal G}^{+++}}$, the latter necessarily contains solutions\nof exotic counterparts of the original maximally oxidised theories. In\nparticular the $E_{11}\\equiv E_8^{+++}$-invariant action contains\nin addition to the M-theory solutions, the exotic branes of the\nrelated M' and M* theories \\cite{hull1,hull2,ah, exop}.\nOf course $S_{{\\cal G}^{+++}}$ can, as $S_{{\\cal\nG}_B^{++}}$, be formulated with different signatures related\nby field transformations.\n\nThe fact that $S_{{\\cal G}_B^{++}}$, and hence $S_{{\\cal G}^{+++}}$, contains\n{\\em exact} intersecting extremal  brane solutions of space-time\ncovariant theories compactified to all dimensions but one provides a\nlaboratory to analyse the significance of at least some subset of the\ninfinite many fields appearing in the Kac-Moody invariant theories.\nExtremal branes in   more non-compact dimensions differ from the\none dimensional ones only by the dependance\nof a harmonic function on the number of non-compact dimensions. For\nsuch  decompactified solutions to exist in\nthe Kac-Moody theory,   higher level fields must provide the\nderivatives needed to obtain  higher dimensional harmonic\nfunctions. This test is crucial to settle the issue of whether or not\nthe Kac-Moody theories discussed here can really describe uncompactified\nspace-time covariant theories.\n\nThe paper is organised as follows. In section 2, we recall  the\nconstruction of the ${\\cal G}^{+++}$-invariant actions $S_{{\\cal G}^{+++}}$.  In section 3, we\nshow how the `cosmological' ${\\cal G}^{++}$-invariant action $S_{{\\cal G}_C^{++}}$\nis obtained from $S_{{\\cal G}^{+++}}$ by a consistent truncation.  In section 4, we\nobtain the\n`brane' action $S_{{\\cal G}_B^{++}}$ which arises from a Weyl\ntransformation in ${\\cal G}^{+++}$ and a consistent truncation.\nWe show how to formulate $S_{{\\cal G}_B^{++}}$ in various equivalent\nways to exhibit various Lorentzian signatures. Differential\nequations for the Weyl  transformations of fields ensuring the\nequivalence of the different formulations are obtained. The equivalence \nproves that, in addition to \nconventional ones, exotic dualities are present in  ${\\cal G}^{+++}$. This is made\nexplicit in the particular case of $E_{11}$ and is  in agreement\nwith the results of reference \\cite{keu1, keu2}, linking in\n${\\cal G}^{+++}$ M-theory to M' and M*-theories. In Section 5, we derive the exact solutions\nof $S_{{\\cal G}_B^{++}}$, and hence of $S_{{\\cal G}^{+++}}$, which are identical to the\nspace-time solutions describing intersecting extremal branes, conventional and\nexotic, smeared in all directions but one. We illustrate by a specific\nnon-trivial example how dualities transforming these solutions into one\nanother operate. In Section 6 we stress  perspectives suggested by our\nresults.\n\n\n\n\\section{${\\cal G}^{+++}$-invariant action}\n\n\n\nActions $S_{{\\cal G}^{+++}}$ invariant under non-linear\ntransformations of\n${\\cal G}^{+++}$ are constructed recursively from a level\ndecomposition with\nrespect to a subalgebra\n$A_{D-1}$ where $D$ is interpreted as the space-time dimension. The\naction is defined in a reparametrisation invariant way  on a\nworld-line, a priori unrelated to space-time, in terms of fields\n$\\varphi(\\xi)$ where $\\xi$ spans the world-line. The fields $\\varphi(\\xi)$\nlive in a coset space\n${\\cal G}^{+++}\/K^{+++}$ where the subalgebra $K^{+++}$ is invariant under a\n`temporal involution'\n preserving at each level a Lorentz algebra\\footnote{In Section 4, we shall see\nthat $G^{+++}$  contains other $A_{D-1}$ subalgebras intersecting\ndifferently with\n$K^{+++}$.}\n$SO(1,D-1)= A_{D-1} \\cap K^{+++}$.\n\nWe now recall in more detail the construction of these\n${\\cal G}^{+++}$-invariant theories\n\\cite{eh}.\n\n${\\cal G}^{+++}$ contains a  subalgebra $GL(D)$ such that $SL(D) (=A_{D-1})\n\\subset GL(D) \\subset\n{\\cal G}^{+++}$. The generators of the $GL(D)$ subalgebra are taken to be\n$K^a_{~b}\\ (a,b=1,2,\\ldots ,D)$   with commutation relations\n\\begin{equation}\n\\label{Kcom} [K^a_{~b},K^c_{~d}]   =\\delta^c_b\nK^a_{~d}-\\delta^a_dK^c_{~b}\\,  .\n\\end{equation}  The $K^a_{~b}$ along with the  abelian generator $R$,\nwhich is present  when the corresponding maximally oxidised action\n$S_{\\cal G}$ has one\ndilaton\\footnote{All the maximally oxidised  theories have at most one dilaton except\nthe\n$C_{q+1}$-series.  The maximally oxidised theory\n$C_{q+1}$ is a four dimensional theory   containing $q$ dilatons and\n $C_{q+1}^{+++}$  is constructed with\n$q$ abelian generators $R_i \\quad i=1 \\dots q$ (see for instance ref\n\\cite{eh} appendix A3). To avoid crowding of indices, we consider\nhere  only the theories with at most one dilaton.}, are the level zero\ngenerators. The step operators of level greater than zero are tensors\n$R^{\\quad c_1\\dots c_r}_{ d_1\\dots d_s}$ of the\n$A_{D-1}$ subalgebra. The lowest levels contain antisymmetric tensor\nstep  operators\n$R^{a_1a_2 \\dots a_r}$ associated with  electric and magnetic\nroots arising from the dimensional reduction of field strength forms in $S_{\\cal\nG}$. They satisfy the tensor and scaling relations\n\\begin{eqnarray}\n\\label{root} &&[K^a_{~b},R^{a_1\\dots a_r}]   =\\delta^{a_1}_b R^{aa_2\n\\dots a_r} +\\dots +\n\\delta^{a_r}_b R^{a_1 \\dots a_{r-1}a}\\, ,\\\\\n\\label{root2} && [R, R^{a_1\\dots a_r}\n] =   -\\frac{\\varepsilon_A\na_A}{2}\\,  R^{a_1\\dots a_r}\\, ,\n\\end{eqnarray} where $a_A$ is the dilaton coupling constant to the\n field strength form   and\n$\\varepsilon_A$ is $+1\\, (-1)$ for an\nelectric (magnetic) root \\cite{ehtw}. The generators obey the invariant\nscalar product relations\n\\begin{eqnarray}\n\\label{killing}\n&&\\langle K_{~a}^a K_{~b}^b\\rangle =G_{ab}\\, ,\\quad \\langle\nK^b_{~a}K_{~c}^d\\rangle=\n\\delta_c^b\\delta_a^d \\ a\\neq b\\, , \\quad\\langle R R\\rangle\n=\\frac{1}{2}\n\\,,\\\\\n\\label{step}\n&&\\langle\nR^{\\quad a_1\\dots a_r}_{ b_1\\dots b_s} , \\bar  R_{ d_1\\dots   d_r}^{\\quad\nc_1\\dots c_s}\\rangle\n=\\delta^{c_1}_{b_1}\\dots\\delta^{c_s}_{b_s}\\delta^{a_1}_{d_1}\\dots\\delta^\n{a_r}_{d_r}\\,.\n\\end{eqnarray}\nHere $G= I_D -\n\\frac{1}{2}\\Xi_D$ where $\\Xi_D$ is a D-dimensional matrix with all\nentries   equal to unity and   $\\bar  R_{ d_1\\dots   d_r}^{\\quad\nc_1\\dots c_s}$ designates the negative step operator conjugate to $ R^{\\quad\nd_1\\dots   d_r}_{ c_1\\dots c_s}$.\n\nThe temporal involution $\\Omega_1$ generalises the Chevalley\ninvolution to allow identification of the index 1 to a time coordinate in\n$SO(1,D-1)$. It is  defined by\n\\begin{equation}\n\\label{map} K^a_{~b}\\stackrel{\\Omega_1}{\\mapsto}\n-\\epsilon_a\\epsilon_b K^b_{~a}\\quad R\\stackrel{\\Omega_1}{\\mapsto} -R\\quad\n, \\quad R^{\\quad c_1\\dots c_r}_{ d_1\\dots d_s}\n\\stackrel{\\Omega_1}{\\mapsto}\n-\\epsilon_{c_1}\\dots\\epsilon_{c_r}\\epsilon_{d_1}\\dots\\epsilon_{d_s}\n  \\bar R_{ c_1\\dots c_r}^{\\quad d_1\\dots d_s}\\, ,\n\\end{equation}\n with $\\epsilon_a =-1$ if $a=1$ and\n$\\epsilon_a=+1$ otherwise. It leaves invariant a subalgebra $K^{+++}$ of\n${\\cal G}^{+++}$.\n\n The fields $\\varphi(\\xi)$ living in the coset space ${{\\cal G}^{+++}}\/K^{+++}\n$ parametrise the Borel group  built out of Cartan and positive\nstep operators in\n${\\cal G}^{+++}$. Its elements $\\cal V$  are written as\n\\begin{equation}\n\\label{positive} {\\cal V(\\xi)}= \\exp (\\sum_{a\\ge b}\nh_b^{~a}(\\xi)K^b_{~a} -\n\\phi(\\xi) R) \\exp (\\sum\n\\frac{1}{r!s!} A^{\\quad a_1\\dots a_r}_{ b_1\\dots b_s}(\\xi) R_{\na_1\\dots   a_r}^{\\quad b_1\\dots b_s} +\\cdots)\\, ,\n\\end{equation}\n where the first exponential contains only  level zero  operators and\nthe second one the positive step operators of levels strictly greater\nthan zero. Defining\n\\begin{equation}\n\\label{sym}\ndv(\\xi)= d{\\cal V} {\\cal V}^{-1}\\quad d\\tilde\nv(\\xi)=   -\\Omega_1\n\\, dv(\\xi)\n\\qquad\\quad dv_{sym}=\\frac{1}{2} (dv+d\\tilde v)\\, ,\n\\end{equation}\none obtains, in terms of the\n$\\xi$-dependent fields, an  action $S_{\\cal {\\cal G}^{+++}}$ invariant under\nglobal ${\\cal G}^{+++}$ transformations, defined on  the\ncoset ${{\\cal G}^{+++}}\/K^{+++}$\n\\begin{equation}\n\\label{actionG} S_{\\cal {\\cal G}^{+++}}=\\int d\\xi  \\frac{1}{n(\\xi)}\\langle\n(\\frac{dv_{sym}(\\xi)}{d\\xi})^2\\rangle\\, ,\n\\end{equation} where\n$n(\\xi)$ is an arbitrary lapse function ensuring reparametrisation\ninvariance on the world-line. One has\n\\begin{equation}\n\\label{sumsym}\ndv_{sym}= dv_{sym}^0+\\sum_A dv_{sym}^{(A)}\\, ,\n\\end{equation}\nwhere $dv_{sym}^0$ contains all the level zero contributions. One gets\n\\begin{equation}\n\\label{v0} dv_{sym}^0=-\\frac{1}{2}\\sum_{a\\ge b} [e^  h (de^{-\nh})]_b^{~a} (K^b_{~a}- \\Omega_1 K^{b}_{~a}) -\nd\\phi  R\\, ,\n\\end{equation}\n\\begin{equation}\n\\label{vform} dv^{(A)}_{sym}=\\frac{1}{2r!s!}DA_{\\mu_1\\dots\n\\mu_r}^{\\quad\n\\nu_1\\dots\n\\nu_s} \\, \\exp (-\n\\lambda \\phi)\\, e^{~\\mu_1}_{{a}_1}...\ne^{~\\mu_r}_{{a}_r}e_{\\nu_1}^{~{b}_1}\n... e_{\\nu_s}^{~{b}_s}\\, (R^{\\quad a_1\\dots a_r}_{ b_1\\dots\nb_s}-\\Omega_1 R^{\\quad a_1\\dots a_r}_{ b_1\\dots\nb_s})\\,  ,\n\\end{equation}\nwhere $e_\\mu^{~a}=(e^{-h(\\xi)})_\\mu^{~a}$, $\\lambda$ is the\ngeneralisation of the scale parameter\n$-\\varepsilon_A a_A\/2$ to all roots\nand $D\/D\\xi$ is a  covariant derivative generalising\n$d\/d\\xi$ through  non-linear terms arising from  non-vanishing\ncommutators  between  positive  step operators.\n\nWriting\n\\begin{equation}\n\\label{full} S_{{\\cal G}^{+++}} =S_{{\\cal G}^{+++}}^{(0)}+\\sum_A\nS_{{\\cal G}^{+++}}^{(A)}\\, ,\n\\end{equation} where $S_{{\\cal G}^{+++}}^{(0)}$ contains all level\nzero contributions, one obtains\n\\begin{equation}\n\\label{fullzero} S_{{\\cal G}^{+++}}^{(0)} =\\frac{1}{2}\\int d\\xi\n\\frac{1}{n(\\xi)}\\left[\\frac{1}{2}(g^{\\mu\\nu}g^{\\sigma\\tau}-\n\\frac{1}{2}g^{\\mu\\sigma}g^{\\nu\\tau})\\frac{dg_{\\mu\\sigma}}{d\\xi}\n\\frac{dg_{\\nu\\tau}}{d\\xi}+\n\\frac{d\\phi}{d\\xi}\\frac{d\\phi}{d\\xi}\\right],\n\\end{equation}\n\\begin{equation}\n\\label{fulla} S_{{\\cal G}^{+++}}^{(A)}=\\frac{1}{2 r! s!}\\int d\\xi\n\\frac{ e^{- 2\\lambda\n\\phi}}{n(\\xi)}\\left[\n\\frac{DA_{\\mu_1\\dots \\mu_r}^{\\quad \\nu_1\\dots\n\\nu_s}}{d\\xi} g^{\\mu_1{\\mu}^\\prime_1}...\\,\ng^{\\mu_r{\\mu}^\\prime_r}g_{\\nu_1{\\nu}^\\prime_1}...\\,\ng_{\\nu_s{\\nu}^\\prime_s}\n\\frac{DA_{{\\mu}^\\prime_1\\dots {\\mu}^\\prime_r}^{\\quad\n{\\nu}^\\prime_1\\dots {\\nu}^\\prime_s}}{d\\xi}\\right].\n\\end{equation}  The $\\xi$-dependent fields $g_{\\mu\\nu}$ are defined as\n$g_{\\mu\\nu} =e_\\mu^{~a}e_\\nu^{~b}\\eta_{ab}$. The appearance of the\nLorentz metric $\\eta_{ab}$ with $\\eta_{11}=-1$ is a consequence of the\ntemporal involution $\\Omega_1$. The metric $g_{\\mu\\nu}$ allows a\nswitch from  the Lorentz  indices  $(a,b)$ of the fields appearing  in\nEq.(\\ref{positive}) to\n$GL(D)$ indices $(\\mu,\\nu)$.\n\nExact solutions have been found for all ${\\cal G}^{+++}$ theories. In \\cite{eh}\nsolutions describing the algebraic properties of the BPS extremal\nbranes have been discovered.  Each single extremal p-brane is\ncharacterised by  {\\it one\\\/} non-zero field multiplying a  component\nof an antisymmetric tensor step operator of low level.  Each extremal\nbrane is related in this way to a  {\\em real positive root} in ${\\cal G}^{+++}$. In\n\\cite{eh2} these results have been extended and exact solutions of\n$S_{{\\cal G}^{+++}}$ for all  the extremal intersecting brane solutions\n\\cite{aeh,ah} of the maximally oxidised theory  were found. The\nintersection rules determining such  configurations are neatly encoded\nin the ${\\cal G}^{+++}$ algebra. They are expressed as\northogonality  conditions on the real roots characterising  the intersecting\nbranes.\n\n\n\\setcounter{equation}{0}\n\\section{From ${\\cal G}^{+++}$ to  the cosmological ${\\cal G}^{++}_C$-invariant action }\n\nConsider the overextended algebra ${\\cal G}^{++}_C$ obtained from the very-extended\nalgebra ${\\cal G}^{+++}$  by deleting the  node labelled 1 from the Dynkin\ndiagrams of ${\\cal G}^{+++}$ depicted in Fig.1. This algebra is realised in the space of\nKasner solutions of the action $S_{{\\cal G}^{+++}}$ and more generally in the\ncosmological billiards solutions\\footnote{In what follows, we use the notation\n${\\cal G}^{++}$ when referring to the multiplication table of the overextended algebra and\nadd an index to ${\\cal G}^{++}$  when we want to keep track of the\nembedding in ${\\cal G}^{+++}$.}. In this\nsection we show how  an action invariant under\n${\\cal G}^{++}_C$  emerges from the\n${\\cal G}^{+++}$-invariant action $S_{{\\cal G}^{+++}}$   Eq.(\\ref{full}) through a {\\it\nconsistent} truncation. This `cosmological' action $S_{{\\cal G}^{++}_C}$ generalises to\nall ${\\cal G}^{++}$  the\n$E_{10}\\, (\\equiv E_8^{++})$ action of reference \\cite{damourhn02}. It is \nobtained below from\n$S_{\\cal\n{\\cal G}^{+++}}$ by putting  to zero  in the coset representative Eq.(\\ref{positive}) the\nfield multiplying the Chevalley generator\n$H_1=  K_{~1}^1- K_{~2}^2$ and all the\nfields multiplying the positive step operators associated to  roots whose\ndecomposition in terms of simple roots contains  the deleted root\n$\\alpha_1$.  Such a truncation of the action\n$S_{\\cal {\\cal G}^{+++}}$ will be shown to be consistent with all its equations of motion.\n\n\nConsider first the fields $A^{\\quad a_1\\dots a_r}_{ b_1\\dots\nb_s}(\\xi)$ multiplying the positive step\noperators $R_{\na_1\\dots   a_r}^{\\quad b_1\\dots b_s}$.  The $A^{\\quad a_1\\dots a_r}_{\nb_1\\dots b_s}(\\xi)$ equated to zero fall into two classes.\nFirst the set of all  fields appearing in  irreducible\nrepresentations of $A_{D-1}$ whose lowest weight contains $\\alpha_1$.\nSecond, in the other representations of\n$A_{D-1}$, the set of  tensor components with at least one time index. The\nconsistency of the truncation of the fields from the first class is obvious as\nall  terms in the action Eq.(\\ref{full}) either contain no such field or\ncontain at least two of them. Fields of the second class may occur linearly but\nare then multiplied by non diagonal metrics\n$g_{1\\hat{\\mu}}$ where the hatted indices  $\\hat{\\mu}$ run from\n$2$ to $D$. In the triangular gauge for $h_\\mu^{~a}$ stemming from the Borel\ngroup defined in Eq.(\\ref{positive}), the linear terms in the second class\n$A$-fields are necessarily multiplied by at least\none $h_1^{~a} $-field $(a>1)$ associated through $K^1_{~a}$ to\na root $\\alpha_1$. Taking\n\\begin{equation}\n\\label{ofdiag} g_{1\\hat{\\mu}}=0\\, ,\n\\end{equation}\nwe ensure the consistency of putting to zero the fields of the second\nclass and in fact of all the fields multiplying step operators associated\nto a root\n$\\alpha_1$, including those of level zero. Indeed, the equation of\nmotions of the second class $A$-fields are now satisfied. Consider now\nthe equations of motion of the $g_{1\\hat{\\mu}}$. All the terms\ncontaining $g_{1\\hat{\\mu}}$  contain necessarily at least\none other\n$g_{1\\hat{\\nu}}$ in Eq.(\\ref{fullzero}) or at least one \n$A$-field component put to zero in\nEq.(\\ref{fulla}). The equations of motion of the $g_{1\\hat{\\mu}}$\nare thus also consistent with the truncation.\n\nWe now turn to the  fields multiplying Cartan generators. Labelling  $q_1$\n the field multiplying $H_1$ in the\nChevalley basis we express the equation $q_1=0$ in terms of the metric\n$g_{\\mu\\nu}$. We define\n\\begin{equation}\n\\label{pdef}\np_a =  - h^{~a}_a\\, ,\n\\end{equation}\nwhere $h^{~a}_a$ is the\ncoefficient of $K^a_{~a}$ in Eq.(\\ref{positive}) . One gets for all ${\\cal G}^{+++}$\n\\cite{ehtw} the equivalence\n\\begin{equation}\n\\label{embed}\n q_1=0\\qquad\\Longleftrightarrow\\qquad\n p_1 = \\sum_{a=2}^{D} p_a\\, .\n\\end{equation}  Using the fact that the coset model Eq.(\\ref{actionG})\nwas computed in  the triangular Borel  gauge and using Eq.(\\ref{ofdiag}),\nwe  rewrite Eq.(\\ref{embed}) as\n\\begin{eqnarray}\n\\label{det1} g_{11}&=&{\\bf g}\\, ,\\\\\n\\label{det2} {\\bf g}&\\equiv&\\det{g_{\\hat{\\mu}\\hat{\\nu}}}\\, .\n\\end{eqnarray}\n\n\nConsistency of the truncation from ${\\cal G}^{+++}$ to ${\\cal G}^{++}$ requires\nthat Eq.(\\ref{det1}) satisfy the equation of motion of\n$g_{11}$. Only variations with respect to $g_{11}$ in the action\nEq.(\\ref{fullzero}) have to be considered, as the\nvariations in Eq.(\\ref{fulla}) are automatically satisfied because\n$g_{11}$ multiplies  $A$-fields already equated to zero.\nUsing Eq.(\\ref{ofdiag})  we get\n\\begin{equation}\n\\label{eomzero}\n\\frac{d^2}{d\\xi^2}\\ln |g_{11}|-\\frac{d^2}{d\\xi^2}\\ln{\\bf g} = 0\\, ,\n\\end{equation} which admits  as solution Eq.(\\ref{det1}).\n\nTo obtain the  actions $S_{{\\cal G}^{++}_C}$  whose\nequations of motion are all contained in the equations of motion\nobtained from\n$S_{{\\cal G}^{+++}}$ given by Eq.(\\ref{actionG}), it thus suffices to substitute in\nEq.(\\ref{fullzero}) $g_{11}$ by its value given in Eq.(\\ref{det1}) and\nequate to zero in Eq.(\\ref{fulla}) all first and second\nclass $A$-fields. Denoting the remaining $A$-fields by $B$ and using the\nhatted indices $\\hat\\mu , (\\mu=2,\\dots D)$, we obtain\n\n\\begin{equation}\n\\label{fullp} S_{{\\cal G}^{++}_C} = S_{{\\cal G}^{++}_C}^{(0)}+\\sum_B\nS_{{\\cal G}^{++}_C}^{(B)}\\, ,\n\\end{equation} where\n\\begin{eqnarray}\n\\label{fullop} S_{{\\cal G}^{++}_C}^{(0)}& =&\\frac{1}{2}\\int dt\n\\frac{1}{n(t)}\\left[\\frac{1}{2}(g^{\\hat{\\mu}\\hat{\\nu}}g^{\\hat{\\sigma}\\hat{\\tau}}-\ng^{\\hat{\\mu}\\hat{\\sigma}}g^{\\hat{\\nu}\\hat{\\tau}})\\frac{dg_{\\hat{\\mu}\n\\hat{\\sigma}}}{d\\tau}\n\\frac{dg_{\\hat{\\nu}\\hat{t}}}{dt}+\n\\frac{d\\phi}{dt}\\frac{d\\phi}{dt}\\right] ,\\\\\n\\label{fullc} S_{{\\cal G}^{++}_C}^{(B)}&=&\\frac{1}{2 r! s!}\\int dt\n\\frac{ e^{- 2\\lambda\n\\phi}}{n(t)}\\left[\n\\frac{DB_{\\hat{\\mu}_1\\dots \\hat{\\mu}_r}^{\\quad \\hat{\\nu}_1 \\dots\n\\hat{\\nu}_s}}{dt} g^{\\hat{\\mu}_1{\\hat{\\mu}}^\\prime_1}...\\,\ng^{\\hat{\\mu}_r{\\hat{\\mu}}^\\prime_r}g_{\\hat{\\nu}_1{\\hat{\\nu}}^\\prime_1}\n...\\ ,    g_{\\hat{\\nu}_s{\\hat{\\nu}}^\\prime_s}\n\\frac{DB_{{\\hat{\\mu}}^\\prime_1\\dots {\\hat{\\mu}}^\\prime_r}^{\\quad\n{\\hat{\\nu}}^\\prime_1\\dots {\\hat{\\nu}}^\\prime_s}}{dt}\\right] .\n\\end{eqnarray} Here we  renamed $\\xi$ as $t$.\nEqs.(\\ref{fullp}),(\\ref{fullop}) and(\\ref{fullc}) follow from \nEq.(\\ref{actionG}) by replacing\n$dv_{sym}$ by\n$dv{_{sym}}_{(+)}$,   which describes a motion on the coset\n${\\cal G}^{++}\/K^{++}_{(+)}$ where  $K^{++}_{(+)}$ is the subalgebra of ${\\cal G}^{++}$\ninvariant under the Chevalley involution. In particular, we\nsee that for ${\\cal G}^{++} = E_8^{++}$, we recover as a consistent truncation of\n$S_{E_8^{+++}}$  given by Eq.(\\ref{actionG}) the action for the\n$E_{10}\\equiv E_8^{++}$-invariant theory proposed in reference\n\\cite{damourhn02}.\n\nThe fact that the solutions of the ${\\cal G}^{++}$-equations of motion are\nalso solutions of the ${\\cal G}^{+++}$-equations of motion corresponding to a\nparticular choice of the initial conditions can be viewed in a\ndifferent way. Consider the general solution of the ${\\cal G}^{+++}\/K^{+++}$\nnon linear sigma-model. In the notations of \\cite{damourhn00}, it\nreads\n\\begin{equation} {\\cal M} (\\xi) = {\\cal M} (0) \\cdot \\exp (\\xi J)\\, , \\label{solu}\n\\end{equation}\nwhere ${\\cal M}$ is defined as ${\\cal M} = {\\cal V}^{T'} {\\cal V}$ in terms\nof the ${\\cal G}^{+++}$-group element ${\\cal V}$.  Here, the generalised\ntransposition is defined in terms of the involution $\\Omega_1$ as\n$E^{T'} = - \\Omega_1(E)$ for any Lie algebra element $E$ (and\nextended by the rule $(AB)^{T'} = B^{T'} A^{T'}$). In\n(\\ref{solu}), the $J$'s are the conserved currents and belong\nto ${\\cal G}^{+++}$.  If we choose the initial conditions in such a way\nthat ${\\cal M} (0)$ belongs to the ${\\cal G}^{++}$-subgroup and the current $J$\nis in the subalgebra ${\\cal G}^{++}$, then we see from (\\ref{solu}) that\n${\\cal M} (\\xi)$ is in the ${\\cal G}^{++}$-subgroup for all ``times\" $\\xi$.\nThis proves consistency of the truncation at the level of the\nequations of motion.\n\n\n\\setcounter{equation}{0}\n\\section{From ${\\cal G}^{+++}$ to  the brane ${\\cal G}^{++}_B$-invariant action}\n\n\\subsection{Weyl reflections of the gravity line}\nThe Weyl reflection $W_{\\alpha_1}$ in the hyperplane perpendicular to\n$\\alpha_1$ generates a subalgebra ${\\cal G}^{++}_2$ conjugate to ${\\cal G}^{++}_C$ in\n${\\cal G}^{+++}$. More generally, performing  Weyl reflections from roots of the\ngravity line, we generate in this way $(D-1)$  subalgebras ${\\cal G}^{++}_a,\n(a = 2,\\dots, D)$, which are all conjugate in ${\\cal G}^{++}$.  While the generators of\n${\\cal G}^{++}_C$ were labelled by  spatial indices only, the\nindex\n$a$ in\n${\\cal G}^{++}_a$ must   be interpreted as  a time coordinate. The fact\nthat we have $D-1$ different identifications of the  time\ncoordinate follows from the non-commutativity of  the time\ninvolution with the Weyl reflections \\cite{keu1} as seen\nbelow. This non-commutativity will be exploited in the \nmore general context of Section 4.2 but is well illustrated by\nits effect on the gravity line.\n\nExpressing a Weyl transformation $W$  as a conjugation by a group\nelement\n$U_W$ of ${\\cal G}^{+++}$, we define the involution $\\Omega^\\prime$ operating on\nthe conjugate elements by\n\\begin{equation}\n\\label{newinvolve}\nU_W\\, \\Omega T\\, U^{-1}_W=\\Omega^\\prime \\, U_W  T U^{-1}_W\\, ,\n\\end{equation}\nwhere $T$ is any generator of ${\\cal G}^{+++}$.  Applying Eq.(\\ref{newinvolve}) to\nthe Weyl reflection $W_{\\alpha_1}$  generates the subalgebra\n${\\cal G}^{++}_2$ conjugate to\n${\\cal G}^{++}_C$. One gets\n\\begin{eqnarray}\n\\label{permute}\n&&U_1\\, \\Omega K^2_{\\ 1} \\, U^{-1}_1= \\rho K^2_{\\ 1}= \\rho\\Omega^\\prime\n K^1_{\\ 2}\\nonumber\\, ,\\\\\n&&U_1\\, \\Omega K^1_{\\ 3} \\, U^{-1}_1= \\sigma K^3_{\\ 2}=\n\\sigma\\Omega^\\prime\n K^2_{\\ 3} \\, ,\\\\\n&&U_1\\, \\Omega K^i_{\\ i +1} \\, U^{-1}_1= -\\tau K^{i+1}_{\\ \\, i}=\n\\tau\\Omega^\\prime  K^i_{\\ i +1}\\quad i >2\\, .\\nonumber\n\\end{eqnarray}\nHere $\\rho,\\sigma,\\tau$ are plus or minus signs which may arise\nas step operators are representations of the Weyl group \nup to signs. Eq.(\\ref{permute}) illustrate the general result\nthat such signs always cancel in the determination of\n$\\Omega^\\prime$ because they are identical in the Weyl transform of\ncorresponding positive and negative roots, as\ntheir commutator is in the Cartan subalgebra which  forms a true\nrepresentation of the Weyl group. The content of Eq.(\\ref{permute}) is\nrepresented in Table 1. The signs below the generators of the gravity\nline indicate the sign in front of the\n negative step operator obtained by the involution: a\nminus sign is in agreement with the conventional Chevalley involution\nand indicates that the indices in\n$K^a_{\\ a +1}$ are both either space or time indices while a plus sign\nindicates that one index must be time and the other  space.\n\\begin{table}[h]\n\\caption{\\small Involution switch from $\\Omega$ to\n$\\Omega^\\prime$ due to the Weyl reflection $W_{\\alpha_1}$}\n\\begin{center}\n\\begin{tabular}{|c|ccccc|c|}\n\\hline\ngravity line&$K^1_{\\ 2}$&$K^2_{\\ 3}$&$K^3_{\\ 4}$&$\\cdots$&$K^{D-1}_{\\\nD}$&time coordinate\\\\\n\\hline\\hline\n$\\Omega$&$+$&$-$&$-$&$-$&$-$&1\\\\\n\\hline$\\,\\Omega^\\prime$&$+$&$+$&$-$&$-$&$-$&2\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{gravity}\n\\end{table}\n\n\\noindent\nTable 1 show that\nthe  time coordinates in\n${\\cal G}^{++}_2$ must now be identified either with 2, or with all indices\n$\\neq 2$. We choose the first description, which leaves\nunaffected coordinates attached to planes invariant under the Weyl\ntransformation. Similarly the time coordinate in\n${\\cal G}^{++}_a$ is taken to be\n$a$.\n\nOne may now apply the analysis of Section 3 to obtain  actions\n$S_{{\\cal G}^{++}_a}$ invariant under ${\\cal G}^{++}_a$, namely we equate as before to zero\nall the fields in the ${\\cal G}^{+++}$-invariant action Eq.(\\ref{full}) which\nmultiply generators not involving the root $\\alpha_1$, but this\ntruncation is performed {\\em after} the Weyl transformation which\ntransmutes the time index 1 to a space index. The actions $S_{{\\cal G}^{++}_a}$ are then\nformally identical to the one given by Eqs.(\\ref{fullp}),\n(\\ref{fullop}) and (\\ref{fullc}) but with a Lorentz signature for the\nmetric, which in the flat coordinates amounts to a negative sign for the\nLorentz metric component $\\eta_{aa}$, and with $\\xi$ identified to the missing\nspace coordinate instead of $t$.\n\nThe $D-1$ actions $S_{{\\cal G}^{++}_a}$ for $a = 2,\\dots, D$  differ only by the\nindex identifying the time coordinate and the concomitant identification of\n$\\xi$ with a space coordinate. They are thus equal up to a trivial redefinition\nof all tensor fields by an interchange of indices. This redefinition may be\nviewed as the Weyl transformations generated by the roots of the subalgebra\n$A_{D-2}$ on the space of fields. Equivalence under Weyl transformations from\nroots which do not belong to the gravity line are far less trivial and will now\nbe examined.\n\n\n\n\n\\subsection{General Weyl reflections}\n\nThe $D-1$  actions $S_{{\\cal G}^{++}_a}$ differ by the labelling of the\ntime coordinate and have all the same global signature $(1,9)$.   They\nare related through Weyl transformations of ${\\cal G}^{++}$ from roots of\nthe gravity line. Consider now the Weyl transformations of ${\\cal G}^{++}$\ngenerated in addition by the simple roots not belonging to $A_{D-2}$.\nThese will yield  actions with different global signatures.\nAll such actions will be shown to be equal and related by field\nredefinitions. This equivalence realises in the action\nformalism  the general analysis of Weyl transformations by Keurentjes\n\\cite{keu1,keu2}. It will be related to the existence, in\naddition to conventional duality symmetries, to exotic dualities, which\nin the particular case of M-theory, are relating it to M* and\nM'-theories \\cite{hull1,hull2}. To illustrate how Weyl transformation change\nglobal signatures, we first consider explicitly  the signature changes for\n$E_8^{++}\\equiv E_{10}$.\n\nThe Weyl reflection from the root\nassociated to any generator $R^{abc}$ multiplying the three-form\npotential $A_{abc}$ can always be mapped to the Weyl reflection\n$W_{\\alpha_{11}}$ from the simple root $\\alpha_{11}$ by products of\nWeyl reflections of the gravity line, thereby changing the  time\ncoordinate, originally positioned at 1, to any position.\n\nLet us choose the time coordinate to be 9 and consider the Weyl\nreflection $W_{\\alpha_{11}}$. The only generator of the gravity line\naffected by the Weyl transformation is $K^8_{\\ 9}$. One has from\nEq.(\\ref{newinvolve})\n\\begin{equation}\n\\label{weyla}\nU_{11}\\, \\Omega R^{8\\,10\\,11} \\, U^{-1}_{11}= -\\rho K^9_{\\ 8}=\n\\rho\\Omega^\\prime\n K^8_{\\ 9}\\, .\n\\end{equation}\nImposing that the coordinate 2 remains unaffected by the transformation,\nwe see that both 10 and 11 become time coordinates, as\nillustrated in Table 2. The transformation of the  generator\n$R^{9\\,10\\,11}$ yields\n\\begin{equation}\n\\label{weylb}\nU_{11}\\, \\Omega R_{9\\,10\\,11} \\, U^{-1}_{11}= \\sigma R_{9\\,10\\,11}=\n\\sigma\\Omega^\\prime\n R^{9\\,10\\,11}.\n\\end{equation}\nThe action of the involution\n$\\Omega^\\prime$  on the simple root  not\npertaining to the gravity line, $\\Omega^\\prime R^{9\\,10\\,11}=\nR_{9\\,10\\,11}$, differs by a sign from the action of a temporal\ninvolution  defined according to Eq.(\\ref{map}) when the times are\n10 and\n11.   This shift of sign is shown\nin the last column of Table~\\ref{e10}. It will lead to negative kinetic\nenergy terms in the corresponding actions\n$S_{{\\cal G}{}_{(10\\,11)}^{++}}^{(2,8,-)}$ below. Table~\\ref{e10}\nshows how by repeated use of Eqs.(\\ref{weyla}) and (\\ref{weylb}) one\nreaches the signatures $(5,5,+)$,\n$(6,4,-)$ and\n$(9,1,+)$.\n\n\\begin{table}[h]\n\n\\caption{\\small Involution switches from $\\Omega$ to\n$\\Omega^\\prime$ due to the Weyl reflection $W_{\\alpha_{11}}$}\n\n\\begin{center}\n\\begin{tabular}{|ccccccccc|c|c|}\n\\hline\n$K^2_{\\ 3}$&$K^3_{\\ 4}$&$K^4_{\\ 5}$&$K^5_{\\ 6}$&$K^6_{\\\n7}$&$K^7_{\\ 8}$&$K^8_{\\ 9}$&$K^9_{\\ 10}$&$K^{10}_{\\ 11}$&\ntimes & $(t,s,\\pm)$ \\\\\n\\hline\\hline\n$-$&$-$&$-$&$-$&$-$&$-$&$+$&$+$&$-$&9&$(1,9,+) $\\\\\n\\hline $-$&$-$&$-$&$-$&$-$&$-$&$-$&$+$&$-$&10,11&$(2,8,-)\n$\\\\\n\\hline\\hline\n$-$&$-$&$-$&$-$&$+$&$-$&$+$&$-$&$-$&7,8&$(2,8,-) $\\\\\n\\hline $-$&$-$&$-$&$-$&$+$&$-$&$-$&$-$&$-$&7,8,9,10,11&$(5,5,+)\n$\\\\\n\\hline\\hline$-$&$-$&$-$&$+$&$-$&$-$&$+$&$+$&$-$&2,3,4,5,9&$(5,5,+)\n$\\\\\n\\hline$-$&$-$&$-$&$+$&$-$&$-$&$-$&$+$&$-$&2,3,4,5,10,11&$(6,4,-)\n$\\\\\n\\hline\\hline$-$&$-$&$-$&$-$&$-$&$+$&$-$&$-$&$-$&2,3,4,5,6,7&$(6,4,-)\n$\\\\\n\\hline$-$&$-$&$-$&$-$&$-$&$+$&$+$&$-$&$-$&2,3,4,5,6,7,9,10,11&$(9,1,+)\n$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{e10}\n\\end{table}\n\nIt is interesting to illustrate the string interpretation of the\nWeyl transformations of Table 2  in terms of signature changing\ndualities discussed in \\cite{hull1,hull2}. Consider the type IIA interpretation\nof $E_8^{+++}$. In the Dynkin diagram of\n$E_8^{+++}$ in Fig. 1 the node labelled 10 is no longer on\nthe gravity line and should be redrawn as perpendicular to it at the node\nlabelled 9. Recall that in the embedding of  $E_8^{++}$ in\n$E_8^{+++}$ the coordinate 1 is  a space coordinate and that the string\ninterpretation of the Weyl reflection $W_{\\alpha_{11}}$ is a double\n$T$-duality in the directions 9 and 10 plus an exchange\nof these  directions \\cite{weylt,banks,ehtw}.\n\nThe first Weyl\nreflection of Table 2  maps  the\n$E_8^{++}$ signature\n$(1,9,+)$ to $(2,8,-)$. To get the corresponding string theories we  ignore the\ndirection $11$ and take into account the direction\n$1$ which is spacelike. The $(1,9,+)$ signature corresponds to type\n$IIA$ theory. \nThe action of $W_{\\alpha_{11}}$ renders 11  timelike. Consequently the\nstring interpretation of the $(2,8,-)$ signature is a string theory with\nsignature $(1,9)$ and wrong signs for the kinetic terms in the $RR$ sector.\nThis defines the type $IIA^*$ theory consistently  with \nstring dualities. Indeed a double T-duality with respect to one time\ndirection, 9, and one space direction, 10, maps type $IIA$ onto\ntype $IIA^*$ \\cite{hull1,hull2}. The second Weyl\nreflection of Table 2  maps  the\n$E_8^{++}$ signature\n$(2,8,-)$ with  time directions $7,8$ to $(5,5,+)$ with time directions\n$7,8,9,10,11$. The string interpretation is obtained by the same argument\nas above and one gets in this way a mapping of  $IIA_{8+2}$ to\n$IIA_{6+4}$ theories, in the notation of\n\\cite{hull2}, where the index $s+t$ designates a theory with $s$ space\n and $t$ time directions. This result is in agreement with the string\ndualities. Indeed a double spacelike\n$T$-duality in the direction $9,10$ maps $IIA_{8+2}$ to\n$IIA_{6+4}$ (see Figure 5 of \\cite{hull2}). The third Weyl\nreflection of Table 2  maps  the\n$E_8^{++}$ signature\n$(5,5,+)$ with  time directions $2,3,4,5,9$ on $(6,4,-)$ with time\ndirections\n$2,3,4,5,10,11$. The string interpretation maps accordingly type $IIA_{5+5}$\nto  type $IIA^*_{5+5}$. These two theories are indeed related by a double\n$T$-duality in the time direction 9 and in the space direction\n10. The last Weyl reflection of Table 2 maps the $E_8^{++}$ signature\n$(6,4,-)$ with  time directions $2,3,4,5,6,7$ to $(9,1,+)$ with time\ndirections\n$2,3,4,5,6,7,9,10,11$, in accordance with     the string interpretation\nwhere type $IIA_{4+6}$ is mapped to type $IIA_{2+8}$ by\ndouble spacelike T-duality.\n\n\\subsection{Weyl equivalence of different space-time\nsignatures}\n\nThe action $S_{{\\cal G}^{++}_2}$ is, as $S_{{\\cal G}^{++}_C}$, given by\nEqs.(\\ref{fullp}),(\\ref{fullop}) and (\\ref{fullc}) but with\n$\\eta_{22}=-1$. It can be expressed in the general form \nEq.(\\ref{actionG}) by replacing\n$dv_{sym}$ by $dv{_{sym}}_{(-)}$   which describes a motion on the coset\n${\\cal G}^{++}\/K^{++}_{(-)}$.  $K^{++}_{(-)}$ is the subalgebra of ${\\cal G}^{++}$\ninvariant under the  time involution $\\Omega_2$ defined as in\nEq.(\\ref{map}) with 2 as the time coordinate and restricted to ${\\cal G}^{++}$. To\ncompute\n$dv{_{sym}}_{(-)}$ one specifies a\ncoset  representative of ${\\cal G}^{++}\/K^{++}_{(-)}$ and a level expansion\nabout a gravity line endowed with the involution\n$\\Omega_2$. We shall refer to this embedding of ${\\cal G}^{++}$ in ${\\cal G}^{+++}$ by ${\\cal G}^{++}_B$\nto distinguish it from ${\\cal G}^{++}_C$.\n\n\n\nThe action $S_{{\\cal G}^{++}_2}$  is characterised by a global signature $(1,9, +)$ and\ntime coordinate~2. We may reparametrise the coset space ${\\cal G}^{++}\/K^{++}_{(-)}$ by\nsubjecting its elements to  a Weyl conjugation by an element $U$ of ${\\cal G}^{++}$.\nThis selects a new gravity line in ${\\cal G}^{++}$ which may be endowed with a\ndifferent involution than $\\Omega_2$, as the temporal involution does\nnot in general commute with Weyl transformations. The Borel group and the\nlevel expansion accommodating the new involution will then not coincide with\nthe old one, although the new $\\xi$-dependant fields could\nalways in principle be expressed in terms of the old ones. One can\nconstruct in this way apparently many distinct actions in terms of these\ndifferent set of fields.  Weyl\ntransformations from the gravity line roots permute the tensor indices and\nhence may change  time indices while Weyl reflections from roots which do\nnot belong to the gravity line may change  the global signature, as\nexemplified for\n$E_8^{++}$ in Table 2.  We denote these actions by\n$S_{{\\cal G}{}_{(i{}_1i{}_2\\dots i{}_t)}^{++}}^{(t,s,\\varepsilon)}$, where\nthe global signature is $(t,s,\\varepsilon)$ with\n$\\varepsilon$ denoting  a set of $+1$ or $-1$ signs associated to each\nsimple root which does not pertain to the gravity line, and\n$i{}_1i{}_2\\dots i{}_t$ are the time indices. We shall verify that all these\nactions are equivalent and that their corresponding Lagrangians are equal\nfor all $\\xi$. We show the equivalence by deriving the differential equations\n relating the fields parametrising the different coset representatives.\n\n\n\nWe write  $S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)}$  in terms of\n$dv_{sym \\, (i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}$ as\n\\begin{equation}\n\\label{newaction}\n S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)} =\\int d\\xi  \\frac{1}{n(\\xi)}\\langle\n(\\frac{dv_{sym \\, (i{}_1i{}_2\\dots\ni{}_t)}^{(t,s,\\varepsilon)}(\\xi)}{d\\xi})^2\\rangle\\, ,\n\\end{equation}\nwhere $dv_{sym \\, (i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}$ is\ncomputed  as in Eqs.(\\ref{positive}) and (\\ref{sym}), but using a\nBorel  representative  of\n${\\cal G}^{++}\/K^{++}_{(-)}$,  built  from a gravity line endowed\nwith the involution\n$\\widetilde\n\\Omega$ defining the signature $(t,s,\\varepsilon)$ with\n$\\{i{}_1i{}_2\\dots i{}_t\\}$ as time coordinates.\nThis yields the same expressions as in Eqs.(\\ref{fullp}),\n(\\ref{fullop}) and (\\ref{fullc}) but with a different space-time\nsignature and a change of sign for all terms in levels generated by an\nodd number of simple roots not pertaining to the gravity line and\ncarrying a negative contribution to $\\varepsilon$.\n\n\nWe denote the Cartan\ngenerators  $K^a_{~a}$ and $R$  as $H^{\n i}$ and the tensor positive step operators as $R^{\nj}$. In terms of the differential forms $\\{X_i, Y_j\\}$ built from\nthe Borel group fields , one has\n\\begin{equation}\n\\label {new}\n dv_{sym \\, (i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}=\n\\sum_i X_i H^i + \\sum_j  Y_j\n (R^j -\n\\widetilde\\Omega R^j)\\, .\n\\end{equation}\nOn the other hand $S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)}$ can be obtained by\nconjugation from $S_{{\\cal G}^{++}_2}$ in the following way. Write\n$dv{_{sym}}_{(-)}$ as\n\\begin{equation}\n\\label{sym-}\ndv{_{sym}}_{(-)} = \\sum_i X_i^\\prime H^{\\prime\\, i} + \\sum_j Y_j^\\prime\n(R^{\\prime\\, j} -\n\\Omega_2 R^{\\prime\\, j})\\, ,\n\\end{equation}\nwhere $\\{X_i^\\prime, Y_j^\\prime\\}$ are the differential forms built   \n   from the Borel fields defined by a gravity line endowed with\nthe involution\n$\\Omega_2$. Select\n generators $R^{\\prime\\prime\\, j}$  whose  Weyl transform\n$R^i=\\widetilde U R^{\\prime\\prime\\, j} \\widetilde U^{-1},\\,\n\\widetilde U\n\\subset{\\cal G}^{++}$,\n spans the\npositive step generators in the level expansion defining $dv_{sym \\,\n(i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}$ and  maps the\ninvolution\n$\\Omega_2$ to\n$\\widetilde \\Omega$. We have\n\\begin{equation}\n\\label{correspond}\n\\widetilde U (R^{\\prime\\prime\\, j} -\n\\Omega_2 R^{\\prime\\prime\\, j}) \\widetilde U^{-1}=\\rho^j (R^j -\n\\widetilde\\Omega R^j)\\, ,\n\\end{equation}\nand $\\rho^j$ is a\nsign. Note that\n$R^{\\prime\\prime\\, j} $ need not be a positive step operator\n$R^{\\prime\\, j}$ in Eq.(\\ref{sym-})  but one  always has\n\\begin{equation}\n\\label{negative}\n (R^{\\prime\\prime\\, j}  -\n\\Omega_2 R^{\\prime\\prime\\, j} ) =\\lambda^j ( R^{\\prime\\, j} -\n\\Omega_2  R^{\\prime\\, j})\\, ,\n\\end{equation}\n  where $\\lambda^j$ is a\nsign. Performing the conjugation  on $dv{_{sym}}_{(-)}$ one gets\n\\begin{equation}\n\\label {conjugate}\n\\widetilde U dv{_{sym}}_{(-)}\\widetilde\nU^{-1}= \\sum_i X_i^\\prime H^i + \\sum_j  Y_j^\\prime\n\\lambda^j\\rho^j (R^j -\n\\widetilde\\Omega R^j)\\, .\n\\end{equation}\nComparing Eq.(\\ref{new}) with Eq.(\\ref{conjugate}) we see that the\nintegrable differential equations relating the different Borel fields\n\\begin{equation}\n\\label{field}\n\\hbox{\\framebox{$ X_i^\\prime= X_i$\\, ,\\, $  Y_j^\\prime \\lambda^j\\rho^j=\nY_j$}}\\,\n\\end{equation}\n ensures that $dv_{sym \\, (i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}$\nand $dv{_{sym}}_{(-)}$ define the same Lagrangian for all $\\xi$ and\nprove the equivalence of\n $S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)}$ and  $S_{{\\cal G}^{++}_2}$.\nThe resulting field transformations realise\nthe Weyl transformation $\\widetilde U$ in field space. We denote the\nset of equivalent actions  by $S_{{\\cal G}^{++}_B}$.\nThe two distinct  actions $S_{{\\cal G}^{++}_B}$ and $S_{{\\cal G}^{++}_C}$ are both contained\nas consistent distinct truncations of the unique action $S_{{\\cal G}^{+++}}$, which\nencompasses different signatures and can be formulated, as was\ndone for $S_{{\\cal G}^{++}_B}$, as separate actions identified through field\nredefinitions.\n\n\nWe illustrate the field transformations obtained from Eq.(\\ref{field}) by\nconsidering for ${\\cal G}^{++} =E_8^{++}$ the equivalence of $S_{{\\cal\nG}{}_{(9)}^{++}}^{(1,9,+)}$ and $ S_{{\\cal G}{}_{(10\\,\n11)}^{++}}^{(2,8,-)} ,$ under the Weyl transformation $W_{\\alpha_{11}}$\ngiven in the first block of Table 2. First consider the non-trivial\nrelation $ Y_j^\\prime \\lambda^j\\rho^j= Y_j$ at the lowest level. Taking into\naccount Eq.(\\ref{weyla}) and Eqs.(\\ref{v0})-(\\ref{vform}) we get the following\nnon-linear relations (up to the sign $\\lambda^j\\rho^j$ not taken into account here)\n\\begin{eqnarray}\n\\label{relaa}\n\\frac{1}{3!}dA^\\prime_{\\mu \\nu\n\\rho}\\,\n (e^{h^\\prime})^{~\\mu}_{8} (e^{h^\\prime})^{~\\nu}_{10}\n(e^{h^\\prime})^{~\\rho}_{11}   &=& -[e^  h\n(de^{-h})]_8^{~9}\\, ,\\\\\n\\label{relab}-[e^{h^\\prime}\n(de^{-{h^\\prime}})]_8^{~9}\n &=&\\frac{1}{3!}dA_{\\mu \\nu\n\\rho}\\, (e^h)^{~\\mu}_{8} (e^h)^{~\\nu}_{10}\n(e^h)^{~\\rho}_{11}\\, .\n\\end{eqnarray}\nWe now turn to the relation $X_j^\\prime= X_j$ between the Cartan\ngenerators. From $\\sum_{i=2}^{11} X_i^\\prime H^{\\prime\\, i}\n\\equiv -\\sum_{a=2}^{11} dp^{\\prime a} K^a_{~a}~(p^{\\prime a}=- h^{~\\prime a}_a)$ \none expresses the Weyl reflection\n$W_{\\alpha_{11}}$ as\n\\cite{ehtw}\n\\begin{eqnarray}\\nonumber K^{\\prime a}{}_a&=&K^a_{~a}\\quad a=2,\\ldots ,8\n\\\\ \\nonumber K^{\\prime a}{}_{a}&=&K^a_{~a}+{1\\over\n3}(K^2_{~2}+\\dots+K^8_{~8})-{2\\over\n3}(K^{9}_{~9}+K^{10}_{~10}+K^{11}_{~11})~~ a=9,10,11 \\,\n\\end{eqnarray} to recover the known result\n\\begin{eqnarray}\n\\label{WeylMe2} p^{\\prime a}+\\frac{1}{3}(p^{\\prime\n9}+p^{\\prime 10}+p^{\\prime 11})&=&p^a\n\\qquad\na=2,\\dots, 8\\\\\n\\label{WeylMe1} p^{\\prime a} -\\frac{2}{3}(p^{\\prime\n9}+p^{\\prime 10}+p^{\\prime 11}) &=&p^a\n\\qquad\na=9,10,11\\, .\n\\end{eqnarray}\nWe shall apply in the next section Eqs. (\\ref{relaa}),(\\ref{relab}),\n(\\ref{WeylMe2}), and (\\ref{WeylMe1}) to exact \nextremal brane solutions of ${\\cal G}^{++}_B$.\n\n\\setcounter{equation}{0}\n\\section{Intersecting extremal branes in ${\\cal G}^{++}$ theories}\nExact solutions of  $S_{{\\cal G}^{+++}}$  giving the algebraic properties\nof  intersecting  brane configurations have been constructed in\n\\cite{eh2}. We  show in this section that in $S_{{\\cal G}^{++}_B}$,\nthese coincide with the solutions of the Einstein and field equations of\nthe maximally oxidised theories describing  the intersecting branes\n smeared in all directions but one. $\\xi$ is  identified to the\nnon-compact coordinate.\n\n\n\nWe consider the level decomposition in ${\\cal G}^{++}_B$ for a  signature\n$(t,s,\\varepsilon)$ with $D-1=t+s$ connected by  Weyl\nreflections  to the `phase' $(1,D-2,+)$ with time coordinate 2 of the\n${\\cal G}^{++}_B$ theory. This is the phase described by the fields in the action\n$S_{{\\cal G}^{++}_2}\\equiv S_{{\\cal G}{}_{(2)}^{++}}^{(1,9,+)}$.\n\nThe equations of motion yielding the intersecting brane\nconfiguration solutions from $S_{{\\cal G}^{+++}}$ in \\cite{eh2} can immediately be\nread in $S_{{\\cal G}^{++}_2}$ provided the embedding relation\nEq.(\\ref{embed}) is satisfied with  $p_1 = \\sum_{a=2}^{D} p_a$ \nlabelling  a space direction orthogonal to the branes. The solutions are\nunaltered for\n$a=2,3\\dots D$. As pointed out in\n$\\cite{eh2}$, these solutions can be extended to exotic branes with\n$t_A$ longitudinal timelike and $s_A$ spacelike directions connected in\n${\\cal G}^{+++}$ by Weyl reflections. The results of Section 4 yield therefore the\nintersecting brane configuration solutions, exotic or not, from the fields\ndefining \n$S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)}$. For each of the $\\cal N$ branes\npresent in the configuration and characterised by\n$\\tau_1 \\dots \\tau_{t_A}$ longitudinal timelike directions and\n$\\lambda_1\\dots\n\\lambda_{s_A}$ longitudinal spacelike directions, one has\n\\begin{equation} A_{\\tau_1 \\dots \\tau_{t_A}\\lambda_1\\dots \\lambda_{s_A}}=\n\\epsilon_{\\tau_1 \\dots \\tau_{t_A}\\lambda_1\\dots \\lambda_{s_A}}\n[\\frac{2(D-2)}{\\Delta_A}]^{1\/2}H_A^{-1}(\\xi)\n\\qquad A=1 \\dots {\\cal N}\\, ,\n\\label{aequ}\n\\end{equation} and\n\\begin{eqnarray}\n &&p^a= \\sum_{A=1}^{\\cal N} p_A^a=\\sum_{A=1}^{\\cal N}\n\\frac{\\eta_A^a}{\\Delta_A}\n\\ln H_A(\\xi)\\qquad a=2,3,\\dots,D  \\label{pequ}\\\\\n\\label{phiequ} &&\\phi =\\sum_{A=1}^{\\cal N} \\, \\phi_A = \\sum_{A=1}^{\\cal\nN}\\frac{D-2}{\\Delta_A}\\varepsilon_A a_A\n\\ln H_A(\\xi) \\, .\n\\end{eqnarray} Here $\\eta_A^a=s_A+t_A$ or $-(D-2-s_A-t_A)$ depending on\nwhether  the direction $a$ is perpendicular or parallel to the\n$q_A$-brane and\n$\\Delta_A= (s_A+t_A)(D-2-s_A-t_A)+\\frac{1}{2}a_A^2(D-2)$.  The factor\n$\\varepsilon_A$ is $+1$ for an electric brane and $-1$ for  a magnetic one. Each\nof the  branes in the configuration is thus described as electrically charged and\nis characterised by one positive harmonic function  in\n$\\xi$-space, namely one has\n\\begin{equation}\n\\frac{d^2H_A(\\xi)}{d\\xi^2}=0 \\qquad A=1 \\dots {\\cal N}\\, .\n\\label{harmo}\n\\end{equation}\nFrom Eq.(\\ref{pequ}), the embedding relation Eq.(\\ref{embed}) yields for\nthe spatial  direction 1 the result\n\\begin{equation}\n\\label{dir1}\np^1= \\sum_{A=1}^{\\cal N}\n\\frac{s_A+t_A}{\\Delta_A}\\,\n\\ln H_A(\\xi)\\, ,\n\\end{equation}\nidentifying it to a direction transverse to all branes.\nThe Eqs. (\\ref{pequ}), (\\ref{phiequ}) and (\\ref{dir1})\nare solutions provided the generalised intersection rules\n\\cite{ah}\n\\begin{equation}\n\\bar{s}+\\bar{t}=\\frac{(s_A+t_A)(s_B+t_B)}{D-2}-\\frac{1}{2}\n\\varepsilon_A a_A \\varepsilon_B a_B \\label{exorule}\n\\end{equation}\nare satisfied. The intersection rules Eq.(\\ref{exorule}) can be expressed\n as an orthogonality condition between the real positive roots of\n${\\cal G}^{++}_B$ (and\n${\\cal G}^{+++}$) for all  branes present in the configuration\n\\cite{eh2}. This orthogonality condition is in fact the input that\npermits the derivation of the exact solutions  Eqs.\n(\\ref{aequ}), (\\ref{pequ}),  (\\ref{phiequ}) and (\\ref{harmo}) by\nallowing a reduction of $S_{{\\cal G}^{++}_B}$ to  quadratic\nterms.\n\nWe may   verify from the above equations  that the  lapse constraint in\n$S_{{\\cal G}^{++}_B}$ is satisfied and takes the form\n\\begin{equation}\n\\label{xiextremal}\n\\sum_{a=2}^D ( d p^{a})^2 -(\\sum_{a=2}^D\ndp^{a})^2 +\n\\frac{1}{2} (d\\phi)^2-  \\sum_{A=1}^{\\cal N}\\frac{D-2}{\\Delta_A} (d\\ln\nH_A)^2=0\\, ,\n\\end{equation}\nwhere the differentials are taken in $\\xi$-space.\nEq.(\\ref{xiextremal}) expresses the vanishing of the action for\nsolutions involving fields associated to orthogonal roots. This\ncondition is preserved under a Weyl reflection, whether or not the\nlatter induces a signature change. It thus follows from\nthe invariance of $S_{{\\cal G}^{++}_B}$, as expressed by Eq.(\\ref{field}), that the lapse\nequation is invariant under  Weyl reflections. In addition the term $\\Lambda= -\n\\sum_{A=1}^{\\cal N}(D-2\/\\Delta_A) (d\\ln H_A)^2$ is separately invariant because\nthe other terms in Eq.(\\ref{xiextremal}) are the  quadratic form of ${\\cal G}^{++}$\nrestricted to its Cartan subalgebra, which is Weyl invariant. This\ninvariance relates different intersecting branes, Kaluza-Klein momenta\nand monopoles \\cite{eh,eh2} of different phases, conventional or exotic.\n\n\nThese exact solutions of $S_{{\\cal G}^{++}_B}$ describe  not only the\nalgebraic properties of all corresponding solutions, exotic or not,\nof all the maximally oxidised theories and of their exotic counterparts\nrelated to them by `dualities' encoded in the Weyl group of ${\\cal G}^{++}_B$.\nThey are now  identical to the solutions of the Einstein and field\nequations describing  these\nintersecting branes smeared in all directions but the spatial\ndimension 1. This follows on the one hand from the fact that the\nEq.(\\ref{harmo}) describes a harmonic function in one dimension as\nwould be required by the latter solutions when $\\xi$ is identified with\nthe single non-compact space coordinate $x^1$. On the other hand such\nidentification is consistent with Eq.(\\ref{dir1}) which shows that this\nnon-compact direction is indeed transverse to all branes.\n\n\nIt is instructive to  illustrate by a specific example how the\naforementioned  solutions of the Einstein and field\nequations are transformed into themselves by the Weyl transformations\nensuring the uniqueness of $S_{{\\cal G}^{++}_B}$ for conventional and exotic\nsolutions. We consider the mapping of the Kaluza-Klein momentum in\n11-dimensional supergravity, smeared in all transverse directions but\n$x^1$ to the exotic smeared membrane with 2 longitudinal times. We\ntake, as in Table 2, the time to be 9 and the momentum in the direction\n8.  The metric is\n\\begin{equation} ds^2=- \\widetilde H^{-1}\\, (dx^9)^2+ \\widetilde H\n\\left[ dx^8 +(\\widetilde H^{-1}-1)dx^9 \\right]^2 +(dx^1)^2\n+\\sum_{\\mu=2\\dots7, 10, 11}   (dx^\\mu)^2,\n  \\label{kkmetric}\n\\end{equation} where $\\widetilde H(x^1)$  is\na positive   harmonic function in one dimension\n$\\widetilde H(x^1)= A+Q |x^1|\n$.  In the triangular gauge the relevant vielbein are given by\n\\begin{eqnarray}\n\\label{vielkk}e_8{}^{8} &=&  (2-\\widetilde H)^{-{1\\over 2}}= H^{-{1\\over\n2}}\\, ,\\nonumber\\\\ e_9{}^9 &=& (2-\\widetilde H)^{1\\over 2}=H^{1\\over\n2}\\, ,\\nonumber\\\\ e_8{}^9 &=&  (2-\\widetilde H)^{-{1\\over\n2}}-(2-\\widetilde H)^{1\\over 2} =H^{-{1\\over 2}}-H^{1\\over 2}\\, .\n\\end{eqnarray}\nHere we have defined $H=2-\\widetilde H$ and the range of $x^1$ is restricted\nto $H > 0$. These results, when expressed in terms of $H$, differs from\nthe vielbein computed in Appendix B1 of\n\\cite{eh}, where the time direction was 1, by a  sign in the\nexponents of the diagonal time and space vielbein. This difference, which is a\nconsequence of the triangular gauge when the time index is bigger than\nthe space one, is crucial in Eq.(\\ref{weylbrane}) below.\nThe $p^{\\prime a}$ parametrising the Cartan subalgebra  are from\nEq (\\ref{vielkk})\n\\begin{equation}\n\\label{cartanmkk} p^{\\prime 8}=-{1 \\over 2} \\ln H\\, , \\qquad   p^{\\prime\n9}={1\n\\over 2} \\ln H\\, ,\n\\end{equation} while, using Eq.(\\ref{vielkk}), one may compute the non-zero\nnon-Cartan field (see Appendix B1  of\n\\cite{eh})\n\\begin{equation}\n\\label{stepmkk} h_8{}^{\\prime 9}= \\ln H\\, .\n\\end{equation}\nThe  vielbein equation Eq.(\\ref{vielkk}) also yields\n\\begin{equation}\n\\label{vielk}\n-[e^{h^\\prime}\n(de^{-{h^\\prime}})]_8^{~9} =d\\ln H\\, .\n\\end{equation}\nWe perform the Weyl reflection $W_{\\alpha_{11}}$ using Eq.(\\ref{relab}),\n(\\ref{WeylMe2}) and (\\ref{WeylMe1}) . For the Cartan we find  \n\\begin{eqnarray}\n\\nonumber\np^a&=&\\frac{1}{6}\\ln H \\qquad a=2,3,4,5,6,7,9\\\\\n\\label{weylbrane}\np^a&=&-\\frac{1}{3}\\ln H \\qquad a=8,10,11\\, .\n\\end{eqnarray}\nFor the non-Cartan fields we find from Eqs.(\\ref{relab}), (\\ref{vielk}) and\n(\\ref{weylbrane}), up to a sign,\n\\begin{equation}\n\\label{Afield}\ndA_{8 \\, 10\n\\, 11} = dH^{-1}.\n\\end{equation}\nTaking into account the relation Eq.(\\ref{dir1}) this solution describes\nan exotic `membrane' in the directions 8, 10 and 11,  in a phase with\nsignature $(2,8,-)$ with time directions 10 and 11,  characterised by\nthe harmonic function\n$H=2-\\widetilde H$.\nUnder this Weyl transformation the smeared $KK$-momentum of M-theory is\nthus mapped onto an exotic membrane with two longitudinal times of\n$M^*$-theory. The exotic membrane obtained is in perfect agreement\nwith the T-duality interpretation as it can be checked by applying\nthe Buscher transformations \\cite{buscher}\non the KK-momentum solution in 10 dimensions\nobtained by reducing  Eq.(\\ref{kkmetric}) along the direction 11 and\nthen uplifting back to  $M^*$.\n\n\nWhile Eq.(\\ref{weylbrane}) illustrate the invariance of the  quadratic\nform of ${\\cal G}^{++}$ restricted to its Cartan subalgebra, \nEq.(\\ref{Afield})  confirms the invariance of the lapse constraint,\nyielding\n\\begin{equation}\n\\label{branex}\n\\Lambda= -\\frac{1}{2} (d\\ln H)^2\\, ,\n\\end{equation}\nwhere, despite the fact that the membrane has two longitudinal times 10\nand 11, a minus sign does arise in the quadratic lapse constraint from\nthe negative kinetic term in the\n$(2,8,-)$ signature.\n\n\nWe point out that the Einstein solutions required  the additional\nconditions\n\\begin{equation}\n\\Theta_E (-1)^{t_A+1}=1\\qquad \\Theta_E=(-1)^{T+1} \\Theta_M\\, ,\n\\label{theta}\n\\end{equation}\nwhere $\\Theta_E$ and $\\Theta_M$ are the signs of the kinetic\nterms for the electric and magnetic potentials  in the exotic actions.\nThese conditions are trivially satisfied in the conventional phase\n$S_{{\\cal G}^{++}_2}\\equiv S_{{\\cal G}{}_{(2)}^{++}}^{(1,9,+)}$ where $\\Theta_E =1$. This\nphase  is characterised by one time which is always longitudinal to the\nbranes. Therefore they are automatically satisfied in all phases as the\nsolutions in these phases can be obtained from Weyl transformations and\nhence do exist with $\\Theta_E$ identified with the corresponding sign\nencoded in $\\varepsilon$.\n\n\\setcounter{equation}{0}\n\\section{Perspectives}\n\nWe have shown that all solutions of the cosmological and the brane\noverextended invariant actions $S_{{\\cal G}^{++}_C}$ and $S_{{\\cal G}^{++}_B}$ are solutions\nof the very-extended invariant actions $S_{{\\cal G}^{+++}}$. The variable $\\xi$\nused to parametrise the motion of the fields on the coset space\n${\\cal G}^{+++}\/K^{+++}$ in the non-linear realisation $S_{{\\cal G}^{+++}}$ of ${\\cal G}^{+++}$ is then\nidentified respectively to a time  or to a space coordinate and a\nset of fields are consistently made to\nvanish. This suggests that the generators of ${\\cal G}^{+++}$ contain a huge gauge\nredundancy. These generators are indeed\nassociated to fields which must be interpreted as potentials rather\nthan field strengths and equivalent descriptions may be possible with\nboth space and time components of gauge fields. The\nconventional intersecting extremal branes,  easily described by the\n$S_{{\\cal G}^{++}_B}$, could perhaps also be obtained from $S_{{\\cal G}^{++}_C}$ in a\ncomplicated way, while the latter action describes trivially   the\nKasner-like cosmological solutions.\n\nThus the fact that ${\\cal G}^{+++}$ contains on equal footing time and space allows\na selection of the best gauge potentials to describe a\nparticular solution but this `covariant' approach need not  in\nprinciple comprehend a larger gauge invariant content than a\nnon-covariant ${\\cal G}^{++}$,  as least as long as only conventional `phases' of\ntheories are considered. The existence of exotic solutions is a specific\nfeature of ${\\cal G}^{+++}$ which has no counterpart in the overextended action\n$S_{{\\cal G}^{++}_C}$ and is a necessary concomitant of ${\\cal G}^{+++}$-invariance and\nthe temporal involution, as suggested in reference \\cite{keu1}. The only way\nexotic phases could decouple in  very-extended algebra would be by restricting\nthe signature to be Euclidean, a rather problematic alternative.\n\nAn essential result of this work is the existence of solutions of\n$S_{{\\cal G}^{+++}}$ and $S_{{\\cal G}^{++}_B}$ which are identical to the space-time covariant\nsolutions of intersecting extremal branes smeared in all directions but\none. This was made possible by the interpretation of  $\\xi$ as a\nspatial coordinate in the context of $S_{{\\cal G}^{++}_B}$. In this way  we do not\nhave to restrict the interpretation of  $S_{{\\cal G}^{+++}}$ to  algebraic\nconsiderations as in reference \\cite{eh}. The reason of the\nimportance of this result is that the exact solutions of the space-time\ncovariant theories exist with the same algebraic structure but\ndifferent, although simple, functional dependance in more uncompactified\ndimensions. Thus we have a laboratory to check whether or not the\nKac-Moody theories can reach, at least in these simple cases,\nuncompactified gravity from the information contained in higher levels.\nIf this turns out not to be the case, then the content of these\napproaches would probably only be a consistent dimensional reduction of\nthe covariant space time theories down to one dimension through an\ninfinity of equivalent fields. If uncompactified theories are reached,\neven at this elementary level, one has attained the first step of a\nformulation of gravity coupled to some matter fields, which is\nconceptually completely different from the Einstein approach, and which\npossibly includes new degrees of freedom which are hinted at by the\nstring perturbative approaches. We hope to be able to provide the\nanswer to this question in future work.\n\n\\section*{Acknowledgments}\n\n\nWe thank Riccardo Argurio, Hermann Nicolai and Philippe Spindel  for\nfruitful discussions. \n\nThis work was supported in part  by the NATO grant PST.CLG.979008, by\nthe ``Interuniversity Attraction Poles Programme -- Belgian Science\nPolicy '', by IISN-Belgium (convention 4.4505.86),   by Proyectos\nFONDECYT 1020629, 1020832 and 7020832 (Chile) and by the   European\nCommission FP6 programme MRTN-CT-2004-005104, in which F.~E., M.~H. and\nL.~H. are associated to the V.U.Brussel (Belgium).\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Notes}\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\n\n\n\n\\subsection{The Multiple Point Principle}\n\\label{sec:MPP}\n\n\nThe origin of the values for the 19 parameters of the Standard Model is unknown. These parameters are the three lepton masses, six quark masses,\nthe three coupling strengths, the QCD vacuum angle, the CKM mixing angles along with a phase, \nthe electroweak vacuum-expectation value and the Higgs-boson mass.\nAs an underlying mechanism to generate parameter values, the so-called Mulitple Point Principle (MPP) ~\\cite{Bennett:1993pj,Bennett:1996hx,Bennett:1996vy} has been proposed.\nThis principle may explain how parameters of the Standard Model, like the masses of the Higgs boson and the top quark, come about.\nThe basic idea of this principle may be illustrated by the triple point of water (as pointed out for \ninstance in~\\cite{Bennett:2003yr}): At the triple point of water we observe\n three coexisting phases, that is, its solid, liquid, and gaseous form.  \nThe triple point occurs at specific, that is, fine-tuned values for\nthe intensive parameters temperature and pressure of about 273.16 K and 0.612 kPa. \nSince the transitions between the three phases are all of first order,\nenergy and volume can be varied in a certain range without changing neither\ntemperature nor pressure of the triple point. \nIn an analogous way coexisting phases of the Higgs potential may determine\nthe fine-tuned values of the masses of the bosons and fermions.\nBesides one minimum at the electroweak scale of  ${\\cal O}(100) \\text{ GeV}$ there\nshould be a degenerate minimum or degenerate minima at a scale $\\Lambda$ far above the electroweak scale up to the Planck scale. \nBased on this principle\nthe Higgs boson mass has been predicted more than 20 years ago to be \n$135 \\pm 9 \\text{ GeV}$~\\cite{Froggatt:1995rt}!\nA more refined analysis yielded a mass $129.4 \\pm 2$ GeV \\cite{Degrassi:2012ry}\nto be compared to the observed mass $125.10 \\pm 0.14$ GeV \\cite{Tanabashi:2018oca} of the Higgs boson\nwhich was discovered by the CMS and ATLAS collaborations in 2012 \\cite{Chatrchyan:2012xdj,Aad:2012tfa}.\n\nLet us briefly sketch the derivation of this remarkable result following closely~\\cite{Froggatt:1995rt}:\nThe MPP states that\n\\begin{enumerate}\n\\item The Higgs-boson doublet $\\varphi$ has at least two coexisting vacua 1 and 2: with the same potential value, that is,\n\\begin{equation} \\label{Cond1}\nV_{\\text{eff}}(\\langle \\varphi \\rangle_1 )= V_{\\text{eff}}(\\langle \\varphi \\rangle_2 ).\n\\end{equation}\n\\item The additional minimum or minima should appear at the high scale $\\Lambda$ with $100 \\text{ GeV}\\ll \\Lambda <  M_{\\text{Planck}}$, \n\\begin{equation} \\label{Cond2}\n\\langle \\varphi \\rangle_2 = {\\cal O}(\\Lambda).\n\\end{equation}\n\\end{enumerate}\nIn the Standard Model with only one Higgs-boson doublet, the\neffective gauge-invariant potential is written, where we define  $\\phi = \\sqrt{ \\varphi^\\dagger \\varphi}$,\n\\begin{equation}\nV_{\\text{eff}} = \\mu^2(\\phi) \\phi^2 + \\frac{\\lambda(\\phi)}{8} \\phi^4 \\;,\n\\end{equation}\nand where the dependence of the parameters on the scale is written explicitly.\nClose to the second vacuum the quartic term is dominant,\n\\begin{equation}\nV_{\\text{eff}} \\approx \\frac{\\lambda(\\phi)}{8} \\phi^4 \\;.\n\\end{equation}\nThe two conditions (1) and (2) above then give, using the fact that \nthe degeneracy of the potential values requires that $\\lambda(\\Lambda) \\approx 0$,\n\\begin{equation}\n\\frac{d V_{\\text{eff}} }{d \\phi} \\bigg |_{\\text{vac 2}} \n= 0 \n = \\frac{1}{2} \\lambda (\\Lambda) \\Lambda^3\n + \\frac{1}{8} \\frac{d \\lambda(\\phi)}{d \\phi}  \\bigg |_{\\text{vac 2}}  \\Lambda^4 \n = \\frac{1}{8} \\frac{d \\lambda(\\phi)}{d \\phi}  \\bigg |_{\\text{vac 2}}  \\Lambda^4 \n = \\frac{1}{8} \\beta_\\lambda(\\Lambda)  \\Lambda^3\\;.\n\\end{equation}\nThis means that in addition to the quartic parameter $\\lambda$ also\nits $\\beta_\\lambda$ function has to vanish at the scale $\\Lambda$. \nThe $\\beta_\\lambda$ function depends in the following way on the Higgs-boson field $\\phi$:\n\\begin{equation}\n\\beta_\\lambda (\\phi) = \\frac{d \\lambda(\\phi)}{d \\ln (\\phi)}\n=\\phi \\frac{d \\lambda(\\phi)}{d (\\phi)}\n =\\beta_\\lambda ( \\lambda (\\phi), g_t(\\phi), g_1(\\phi), g_2(\\phi), g_3(\\phi) )\n\\end{equation}\nwith $g_t(\\phi)$ the top-Yukawa coupling and $g_{1\/2\/3}(\\phi)$ the scale-dependent gauge couplings. From\nthe explicit form of the $\\beta_\\lambda$ function in the Standard Model, Nielsen and Froggatt evaluate the renormalization group equation numerically, using two loop beta functions and plot $\\lambda(\\phi)$. \nThe evolution depends on the masses of the top-quark and the Higgs boson.\nRequiring a vanishing quartic parameter $\\lambda(\\phi)$  as well as a vanishing $\\beta_\\lambda$ function  at the \nhigh scale $\\Lambda$, the masses of the top quark and the Higgs boson are predicted.\\\\\n\nThis prediction for two of the Standard Model parameters is clearly remarkable and raises the question whether the particular form of the Higgs effective potential \nat the high scale is just an accident or whether there is a deeper law of nature represented by the MPP. \nFor this reason, it is worthwhile to have a closer look how the MPP has been motivated in the literature. \nOriginally, the MPP was justified using thermodynamical arguments \\cite{Bennett:1993pj,Bennett:1996hx,Bennett:1996vy} considering a micro-canonical\nensemble in which an extensive quantity (energy, volume, number of particles) is fixed to a given value. It is then argued that this leads to the MPP.\nThe system is described by a set of intensive quantitiies (temperature, pressure, chemical potential) and it is assumed that more than one\nphase exists with a strong (first order) phase transition between different phases.\n\nIt is then argued that in such a system with a fixed extensive variable\nthe probability is high that intensive quantities (temperature, pressure, chemical potential) take on critical values corresponding to a state with two or more coexisting phases.\n\nMotivated by this result, the question arises  what are the consequences of the MPP in the Two-Higgs-Doublet Model (THDM)?\nThe original motivation by T. D.~Lee~\\cite{Lee:1973iz} to study the two-Higgs-doublet extension\nhas been to have another source for CP violation -- one of the shortcomings of the Standard Model, where violation of CP only arises from the CKM matrix (and the PMNS matrix) and is too small to explain the observed baryon asymmetry dynamically. Another motivation has been given by supersymmetric models which require to have more than one Higgs-boson doublet in order to give masses to up- and down type fermions. A more pragmatic reason is that there is nothing which prevents the introduction of more copies of Higgs-boson doublets. \nIn particular, the $\\rho$ parameter, relating the masses of the electroweak gauge bosons with the weak mixing angle (see~\\cite{Bernreuther:1998rx} for details) is measured close to one in agreement with the Standard Model.\nThe $\\rho$ parameter is known to keep unchanged at tree level with respect to additional copies of Higgs-boson doublets.\nEventually, let us mention that compared to the two real parameters of the Higgs potential of the Standard Model, the potential of the THDM has a much richer structure allowing for different phases.\nFor a review of the THDM we refer to~\\cite{Branco:2011iw}.\n\nThe most general gauge-invariant potential with two Higgs-boson doublets\n\\begin{equation} \\label{doublets}\n\\varphi_1 = \\begin{pmatrix} \\varphi_1^{(+)}\\\\ \\varphi_1^{(0)} \\end{pmatrix},\n\\qquad\n\\varphi_2 = \\begin{pmatrix} \\varphi_2^{(+)}\\\\ \\varphi_2^{(0)} \\end{pmatrix},\n\\end{equation}\nin the convention with both Higgs doublets carrying hypercharge $y=+1\/2$, \nreads~\\cite{Gunion:1989we}\n\\begin{equation}\n\\label{Vconv}\n\\begin{split}\nV^{\\text{THDM}} (\\varphi_1, \\varphi_2) =~& \nm_{11}^2 (\\varphi_1^\\dagger \\varphi_1) +\nm_{22}^2 (\\varphi_2^\\dagger \\varphi_2) -\nm_{12}^2 (\\varphi_1^\\dagger \\varphi_2) -\n(m_{12}^2)^* (\\varphi_2^\\dagger \\varphi_1)\\\\\n& +\\frac{1}{2} \\lambda_1 (\\varphi_1^\\dagger \\varphi_1)^2 \n+ \\frac{1}{2} \\lambda_2 (\\varphi_2^\\dagger \\varphi_2)^2 \n+ \\lambda_3 (\\varphi_1^\\dagger \\varphi_1)(\\varphi_2^\\dagger \\varphi_2) \\\\ \n&+ \\lambda_4 (\\varphi_1^\\dagger \\varphi_2)(\\varphi_2^\\dagger \\varphi_1)\n+ \\frac{1}{2} [\\lambda_5 (\\varphi_1^\\dagger \\varphi_2)^2 + \\lambda_5^* \n(\\varphi_2^\\dagger \\varphi_1)^2] \\\\ \n&+ [\\lambda_6 (\\varphi_1^\\dagger \\varphi_2) + \\lambda_6^* \n(\\varphi_2^\\dagger \\varphi_1)] (\\varphi_1^\\dagger \\varphi_1) + [\\lambda_7 (\\varphi_1^\\dagger \n\\varphi_2) + \\lambda_7^* (\\varphi_2^\\dagger \\varphi_1)] (\\varphi_2^\\dagger \\varphi_2).\n\\end{split}\n\\end{equation}\nThe parameters $m_{12}^2$, $\\lambda_{5\/6\/7}$ are complex, whereas all other parameters have to be real \nin order to yield a real potential. Therefore we count in total 14 real parameters in contrast to two real parameters of the Standard Model.\\\\\n\nIn the work~\\cite{Froggatt:2004st} the MPP has been studied with respect to the general THDM. \nThe argumentation in this work has been developed in the following way:\nSupposing that the potential\nhas a second minimum at a high scale $\\Lambda$, after an appropriate \n\\mbox{$SU(2)_L \\times U(1)_Y$}  transformation the vacuum expectation values of the two Higgs-boson doublets are parametrized as \\cite{Froggatt:2004st}\n\\begin{equation} \\label{Holgpara}\n\\langle \\varphi_1 \\rangle = \\phi_1 \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}, \n\\qquad\n\\langle \\varphi_2 \\rangle = \\phi_2 \\begin{pmatrix} \\sin(\\theta) \\\\ \\cos (\\theta) e^{i \\omega} \\end{pmatrix},\n\\end{equation}\nwhere $\\Lambda^2= \\phi_1^2+\\phi_2^2$.\nThen the conditions are studied for the potential \\eqref{Vconv} and its derivatives with respect to $\\phi_{1\/2}$  to be independent of the phase $\\omega$ at the scale $\\Lambda$. This leads to conditions for the quartic couplings as well as their derivatives, that is, the $\\beta$ functions, evaluated at the scale $\\Lambda$.\nIn particular it is shown that these conditions originating from the MPP yield a CP conserving potential, obeying\nin addition a softly broken $Z_2$ symmetry giving an argument for the absence of flavor-changing neutral currents.\nTherefore, they continue their analysis in the framework of models with natural flavor conservation, e.g.\\ the THDM type II.\n\n{In \\cite{McDowall:2018ulq} a detailed phenomenological study of the MPP is carried out, starting from the results of \\cite{Froggatt:2004st}, and applying them to the THDM type II as well as the Inert Doublet Model. It is found that in both cases, the MPP is incompatible with the requirement of providing simultaneously the experimental value of the top quark mass, electroweak symmetry breaking and stability.\n}\n\nHere we show that the study of the MPP in the THDM can be implemented \nconcisely in the bilinear formalism~\\cite{Nagel:2004sw,Maniatis:2006fs,Nishi:2006tg}.\nThe advantage of the bilinear formalism is that all unphysical gauge degrees of freedom are eliminated systematically and the potential and all parameters are real. Basis transformations, that is, \na unitary mixing of the two doublets, for instance, are given by simple rotations.\nSimilar to the case of the Standard Model, where we have to have a vanishing quartic coupling together with its $\\beta$ function at the high scale $\\Lambda$, we find conditions among the potential parameters and its derivatives in order to satisfy the MPP.\nWe present a classification of all possible realizations of the MPP in the THDM.  \nThe conditions for these classes of realizations are given in a basis-invariant way and can be checked easily for any THDM. \n\nIn order to arrive at the conditions for the MPP we present the $\\beta$ functions\nof the potential parameters in the bilinear formalism (see also~\\cite{Ma:2009ax}).\nWe demonstrate the conditions of different realizations of the MPP in examples\nand we show in particular that the results of \\cite{Froggatt:2004st} can be recovered in the presently developed formalism as one possible realization of the MPP.\nIt should be noted that other MPP solutions have been discussed in \\cite{Froggatt:2008am} in the conventional formalism.\nHere, we will present a complete classification of the MPP solutions in a transparent way using the bilinear formalism.\n\n\\subsection{Brief review of bilinears in the THDM}\n\\label{sec:bilinears}\n\nHere we briefly review the bilinears in the THDM~\\cite{Nagel:2004sw, Nishi:2006tg, Maniatis:2006fs} in order to make this article \nself contained. We will also discuss briefly basis transformations.\nBilinears systematically avoid unphysical gauge degrees of freedom and are defined in the following way: \nAll possible gauge-invariant scalar products of the two doublets $\\varphi_1$ and $\\varphi_2$ which may appear\n in the potential can be arranged in one matrix \n\\begin{equation}\n\\twomat{K} = \n\\begin{pmatrix}\n \\varphi_1^\\dagger \\varphi_1 &  \\varphi_2^\\dagger \\varphi_1\\\\\n \\varphi_1^\\dagger \\varphi_2 &  \\varphi_2^\\dagger \\varphi_2\n \\end{pmatrix}.\n \\end{equation}\n This  hermitian matrix can be decomposed into a basis of \n the unit matrix and the Pauli matrices,\n \\begin{equation}\n \\twomat{K} = \\frac{1}{2} \\left(\nK_0 \\mathbbm{1}_2 + K_a \\sigma_a \\right), \\qquad a=1,2,3,\n\\end{equation}\nwith four real coefficients $K_0$, $K_a$, called bilinears.\nBuilding traces on both sides of this equation (also with products of Pauli matrices) \nwe get the four real bilinears explicitly,\n\\begin{align}\n\\label{eqKphi}\n&K_0 = \\varphi_1^\\dagger \\varphi_1 + \\varphi_2^\\dagger \\varphi_2,\n&&K_1 = \\varphi_1^\\dagger \\varphi_2 + \\varphi_2^\\dagger \\varphi_1, \\nonumber \\\\\n&K_2 = i\\varphi_2^\\dagger \\varphi_1 - i\\varphi_1^\\dagger \\varphi_2, \n&&K_3 = \\varphi_1^\\dagger \\varphi_1 - \\varphi_2^\\dagger \\varphi_2. \n\\end{align}\nThe matrix $\\twomat{K}$ is positive semi-definite. \nFrom $K_0 = \\tr (\\twomat{K})$  and $\\det(\\twomat{K}) = \\tfrac{1}{4} (K_0^2 -K_a K_a)$\nwe get \n\\begin{equation} \\label{Kdom}\nK_0 \\ge 0, \\qquad K_0^2 - K_a K_a \\ge 0.\n\\end{equation}\nAs has been shown in~\\cite{Maniatis:2006fs} there is a one-to-one correspondence between the original\ndoublet fields and the bilinears apart from unphysical gauge-degrees of freedom. \nIn terms of bilinears\nwe can write any THDM (a constant term can always be dropped),\n\\begin{equation}\n\\label{pot}\nV^{\\text{THDM}} (K_0, K_a)=\n \\xi_0 K_0 + \\xi_a K_a + \\eta_{00} K_0^2 + 2 K_0 \\eta_a K_a + K_a E_{ab} K_b,\n\\end{equation}\nwith real parameters $\\xi_0$, $\\xi_a$, $\\eta_{00}$, $\\eta_a$, $E_{ab} = E_{ba}$, $a,b \\in \\{1,2,3\\}$.\nExpressed in terms of the conventional parameters~\\eqref{Vconv}, these new parameters read\n\\begin{align}\n\\xi_0 &= \\frac{1}{2}\\left(m_{11}^2+m_{22}^2\\right) ,\n\\quad\n\\tvec{\\xi} = (\\xi_\\alpha)=\\frac{1}{2}\n\\begin{pmatrix}\n- 2 \\re(m_{12}^2), &\n 2 \\im(m_{12}^2), &\n m_{11}^2-m_{22}^2\n\\end{pmatrix}^\\trans,\n\\label{eq:para1}\n\\\\\n\\eta_{00} &= \\frac{1}{8}\n(\\lambda_1 + \\lambda_2) + \\frac{1}{4} \\lambda_3  ,\n\\quad\n\\tvec{\\eta} = (\\eta_{a})=\\frac{1}{4}\n\\begin{pmatrix}\n\\re(\\lambda_6+\\lambda_7), & \n-\\im(\\lambda_6+\\lambda_7), & \n\\frac{1}{2}(\\lambda_1 - \\lambda_2)\n\\end{pmatrix}^\\trans, \n\\label{eq:para2}\n\\\\\nE &= (E_{ab})= \\frac{1}{4}\n\\begin{pmatrix}\n\\lambda_4 + \\re(\\lambda_5) & \n-\\im(\\lambda_5) & \\re(\\lambda_6-\\lambda_7) \n\\\\ \n-\\im(\\lambda_5) & \\lambda_4 - \\re(\\lambda_5) & \n-\\im(\\lambda_6-\\lambda_7) \\\\ \n\\re(\\lambda_6-\\lambda_7) & \n-\\im(\\lambda_6 -\\lambda_7) & \n\\frac{1}{2}(\\lambda_1 + \\lambda_2) - \\lambda_3\n\\end{pmatrix}.\n\\label{eq:para3}\n\\end{align}\n\nA unitary basis transformations of the doublets,\n\\begin{equation}\n\\varphi'_i = U_{ij} \\varphi_j, \\quad \\text{with } U = (U_{ij}),  \\quad U^\\dagger U = \\mathbbm{1}_2\\, ,\n\\end{equation}\ncorresponds to a transformation of the bilinears,\n\\begin{equation} \\label{b1}\nK_0' = K_0, \\quad K_a' = R_{ab}(U) K_b \\;,\n\\end{equation}\nwith $R_{ab}(U)$ defined by\n\\begin{equation}\nU^\\dagger \\sigma^a U = R_{ab}(U) \\sigma^b \\;.\n\\end{equation}\nIt follows that $R(U) \\in SO(3)$, that is, $R(U)$ is a proper rotation in three dimensions. \nWe see that the potential~\\eqref{pot} stays invariant under a \nchange of basis of the bilinears \\eqref{b1}\nif we simultaneously transform the parameters~\\cite{Maniatis:2006fs}\n\\begin{equation} \\label{b2}\n\\xi'_0 = \\xi_0, \\quad\n\\xi_a' = R_{ab} \\xi_b, \\quad\n\\eta_{00}' = \\eta_{00}, \\quad\n\\eta_{a}' = R_{ab} \\eta_{b}, \\quad\nE_{cd}' = R_{ca}  E_{ab} R_{bd}^\\trans \\, .\n\\end{equation}\nNote that by a change of basis we \ncan always diagonalize the real symmetric matrix~$E$. \n\n{As an illustration, the following parametrization for a unitary transformation~\\cite{Maniatis:2006fs}}\n\\begin{align}\\label{EWbasisChange}\n    \\begin{pmatrix} \\varphi_1' \\\\ \\varphi_2' \\end{pmatrix} &= \n    \\begin{pmatrix} \\cos({\\beta}) && \\sin({\\beta})\\,e^{- i \\zeta} \\\\ -\\sin({\\beta})\\, e^{i \\zeta} && \\cos({\\beta}) \\end{pmatrix}  \n    \\begin{pmatrix} \\varphi_1 \\\\ \\varphi_2 \\end{pmatrix} \\equiv U \\begin{pmatrix} \\varphi_1 \\\\ \\varphi_2 \\end{pmatrix}\n\\end{align}\n{corresponds, in terms of bilinears, to the rotation matrix}\n\\begin{equation} \\label{unitR}\nR(U) = \\begin{pmatrix}\n \\cos ^2(\\beta )-\\sin ^2(\\beta ) \\cos (2 \\zeta ) & -\\sin ^2(\\beta ) \\sin (2 \\zeta ) & -\\sin (2 \\beta ) \\cos (\\zeta ) \\\\\n -\\sin ^2(\\beta ) \\sin (2 \\zeta ) & \\sin ^2(\\beta ) \\cos (2 \\zeta )+\\cos ^2(\\beta ) & -\\sin (2 \\beta ) \\sin (\\zeta ) \\\\\n \\sin (2 \\beta ) \\cos (\\zeta ) & \\sin (2 \\beta ) \\sin (\\zeta ) & \\cos (2 \\beta )\n\\end{pmatrix}\n\\end{equation}\n{\nand is useful to relate a given general basis to the so-called Higgs basis (see appendix~\\ref{EWbreak}) if the doublets acquire non-zero \nvacuum-expectation values. In this case, the angle $\\beta$ fulfils $|v_1^0| \\sin{\\beta} = |v_2^0| \\cos{\\beta}$ (or $\\tan{\\beta} = |v_2^0|\/|v_1^0|$). \\\\\n}\n\nLet us also briefly recall that (standard) CP transformations, that is, $\\varphi_i \\to \\varphi_i^*$, $i=1,2$, have a simple geometric picture in terms of bilinears~\\cite{Maniatis:2007vn}. With view on~\\eqref{eqKphi} we see that a (standard) CP transformation corresponds to $K_2 \\to - K_2$ keeping all other bilinears invariant in addition to the\nparity transformation which flips the sign of the arguments not written explicitly.\nNow let us assume for simplicity, that by a change of basis, the parameter matrix $E$ is diagonal. \nFor the general case of arbitrary matrices $E$ we refer to~\\cite{Maniatis:2007vn}. \nWith $E$ diagonal we see that\nthe potential~\\eqref{pot}  is invariant under the (standard) \nCP transformation if the parameters $\\xi_2$ and  $\\eta_2$ vanish. \nEventually we note that by a basis change \\eqref{b1}, \\eqref{b2} this is equivalent to any commonly vanishing\nentries of the parameter vectors  $\\tvec{\\xi}=(\\xi_1, \\xi_2, \\xi_3)^\\trans$ and \n$\\tvec{\\eta}=(\\eta_1, \\eta_2, \\eta_3)^\\trans$ at the same position.\n\nLet us also prepare the analysis of the THDM for the case of large $K_0$.\nFirst we note that $K_0 \\ge 0$ and for $K_0=0$ the potential is trivially vanishing.\nWe define for $K_0 > 0$~\\cite{Maniatis:2006fs}\n\\begin{equation}\n\\label{eq-ksde}\nk_a = \\frac{K_a}{K_0}, \\quad a \\in \\{1,2,3\\} \\quad \\text{and} \\quad \n\\tvec{k} = \n\\begin{pmatrix} k_1, & k_2, & k_3 \\end{pmatrix}^\\trans \\;.\n\\end{equation}\nWith \\eqref{eq-ksde}\nwe can write the potential~\\eqref{pot}\nin the form\n\\begin{equation}\n\\label{eq-vk}\nV^{\\text{THDM}} = K_0\\, J_2(\\tvec{k}) + K_0^2\\, J_4(\\tvec{k})\n\\end{equation}\nwith the functions\n\\begin{equation}\n\\label{J2J4}\nJ_2(\\tvec{k}) = \\xi_0 + \\tvec{\\xi}^\\trans \\tvec{k},\\qquad\nJ_4(\\tvec{k}) = \\eta_{00} \n  + 2 \\tvec{\\eta}^\\trans \\tvec{k} + \\tvec{k}^\\trans E \\tvec{k},\n\\end{equation}\ndefined \non the compact domain, as follows from~\\eqref{Kdom},\n\\begin{equation}\n|\\tvec{k}| \\leq 1 \\;.\n\\end{equation}\n\nIn appendix \\ref{EWbreak} we recap some parts of\n electroweak symmetry breaking in the THDM in terms of bilinears.\n \n\n\n\n\\section{Classification of the vacua}\n\\label{class}\nWe now apply the MPP to the THDM potential, that is, we study \n the two conditions (1) and (2) as mentioned in section~\\ref{sec:MPP} for the case of the THDM.\nAnalogously to the Standard Model case we consider the parameters as scale dependent. \nIn the bilinear formalism, advantageous in the \ndescription of the THDM potential, large field configurations correspond to a\n large bilinear\n$K_0$ which itself is bilinear in the Higgs-doublet fields, see \\eqref{eqKphi}.\nTherefore the \nbilinears depend quadratically on the mass scale $\\Lambda$.\nThe THDM potential is \nconsidered as an effective parametrization $V_{\\text{eff}}^{\\text{THDM}}$ of the form \\eqref{eq-vk}, \\eqref{J2J4}.\nHigher-dimensional order operators are neglected since \nwe consider a  \nscale $\\Lambda$ much larger than the electroweak scale but also sufficiently below\nthe Planck mass.\nFor large fields close to the high scale the quartic terms are dominant,\nso we have\n\\begin{equation}\n\\label{Veff}\nV_{\\text{eff}}^{\\text{THDM}} \\approx K_0^2\\, J_4, \\quad \\text{ for large $K_0$}\n\\end{equation}\nwith \n\\begin{equation}\nJ_4 = \\eta_{00} (K_0)\n  + 2 \\eta_a (K_0) k_a + k_a E_{ab}(K_0) k_b,\n\\end{equation}\nwhere we write explicitly the dependence of the parameters on the\nscale $K_0$. In appendix~\\ref{appendixJ4} we briefly discuss the condition of \na vanishing function $J_4$ at the high scale.\n Note that the bilinear dimensionless field $\\tvec{k}$ is \ndefined on the domain $|\\tvec{k}| \\le 1$. \nWith respect to the MPP we are looking for a potential\n which has a second minimum at the high scale, that is, \n $\\langle K_0 \\rangle_2 = \\Lambda^2$ at a \ncorresponding ``direction'' of the second minimum, see \\eqref{eq-ksde}-\\eqref{J2J4},\n\\begin{equation}\n\\langle \\tvec{k} \\rangle_2 \\;.\n\\end{equation}\n\nIn order to have a degenerate vacuum at the high scale with the same\npotential value we find with respect to \\eqref{Veff} for large $K_0$ the condition\n\\begin{equation}   \\label{cond1THDM}\nJ_4 ( K_0 = \\Lambda^2, \\langle \\tvec{k} \\rangle_2 ) = \n\\eta_{00} (\\Lambda^2)\n  + 2 \\eta_a (\\Lambda^2) \\langle k_a \\rangle \n  + \\langle k_a \\rangle E_{ab}(\\Lambda^2) \\langle k_b \\rangle\n= 0 \\;.\n\\end{equation}\nAs a function of $K_0$, $J_4( K_0, \\langle \\tvec{k}\\rangle_2 )$\nshould provide a minimum with $J_4 = 0$. \nThis requires also\n\\begin{equation}\n\\left .\\frac{ \\partial V_{\\text{eff}}^{\\text{THDM}} } { \\partial K_0} \\right |_{ \\text{vac 2}}\n= 0 \n= \n 2 \\Lambda^2 J_4 \\bigg |_{ \\text{vac 2}}\n+\n\\Lambda^4\n \\frac{ \\partial J_4 } { \\partial K_0} \\bigg |_{ \\text{vac 2}}\n=\n\\Lambda^4 \n\\left . \\frac{ \\partial J_4 } { \\partial K_0} \\right |_{ \\text{vac 2}} .\n\\end{equation}\nWith \n\\begin{equation}\n2 K_0\n\\frac{ \\partial J_4 } { \\partial K_0}\n= \n2 K_0\n\\left(\n\\frac{ \\partial \\eta_{00}(K_0) } { \\partial K_0}\n+\n2 \\frac{ \\partial \\eta_a (K_0) } { \\partial K_0} \nk_a\n+\nk_a\n\\frac{ \\partial E_{ab}(K_0) } { \\partial K_0} \nk_b\n\\right)\n=\n\\beta_{\\eta_{00}} + 2 \\beta_{\\eta_{a}} \nk_a\n+ \nk_a\n \\beta_{E_{ab}} \nk_b\n \\, ,\n\\end{equation}\nwe find the condition for the beta functions at the Planck scale:\n\\begin{equation} \\label{cond2THDM}\n\\beta_{\\eta_{00}} (\\Lambda^2) + 2 \\beta_{\\eta_{a}}(\\Lambda^2)   \\langle {k}_a  \\rangle_2 + \n\\langle {k}_a \\rangle_2 \\beta_{E_{ab}}(\\Lambda^2)  \\langle {k}_b \\rangle_2 = 0.\n\\end{equation}\nFor the THDM the conditions \\eqref{cond1THDM} and \\eqref{cond2THDM}  \nreplace the ones for the SM. \n\n\\subsection{Stationary points at the high scale $\\Lambda$}\n\\label{statPoints}\n\nSo far we have found that a minimum $\\langle \\tvec{k} \\rangle_2$ should satisfy\nthe conditions \\eqref{cond1THDM} and \\eqref{cond2THDM}.\nWe now study the stationarity structure of the dominant quartic terms.\nThis study is quite analogous to the stability study in~\\cite{Maniatis:2006fs}. \nWe have $|\\tvec{k} | \\le 1$ and we  consider the cases\n $|\\tvec{k} | < 1$ and  $|\\tvec{k} | = 1$ separately.\n\nFor $|\\tvec{k} | < 1$, stationarity of the potential requires,\nexpressed in terms of $J_4$ (since $K_0 >0$),\n\\begin{equation}\n\\nabla_{\\tvec{k}} J_4 (\\tvec{k})\n=\n 2 \\tvec{\\eta}^\\trans + 2 \\tvec{k}^\\trans E  = 0, \n\\end{equation}\nthat is, since $E$ is symmetric\n\\begin{equation} \\label{con}\n\\tvec{\\eta} + E  \\tvec{k} =0\\;.\n\\end{equation}\nNote that we do not write explicitly the scale dependence of the parameters which \nimplicitly is given by $\\Lambda^2$.\nWith the condition \\eqref{con} for a vanishing gradient we can \nin the case $|\\tvec{k} | < 1$ write the condition for vanishing $J_4$ \\eqref{cond1THDM} \nnow in the form\n\\begin{equation} \\label{cond12}\n\\tvec{\\eta}^\\trans \\langle \\tvec{k} \\rangle_2 = - \\eta_{00} \\;.\n\\end{equation}\n\nFor $\\det (E) \\neq 0$ the regular solution of \\eqref{con} is\n\\begin{equation} \\label{solreg}\n\\langle \\tvec{k} \\rangle_2 = - E^{-1} \\tvec{\\eta}\n\\end{equation}\nor we have for $\\det (E) = 0$ exceptional solutions. \nWe check that for the regular solutions we have with $1-\\tvec{k}^\\trans \\tvec{k} > 1$ indeed\n\\begin{equation} \\label{J4reg}\nJ_4 (\\langle \\tvec{k} \\rangle_2)\n=\n \\eta_{00} - \\tvec{\\eta}^\\trans E^{-1} \\tvec{\\eta}  = 0, \n \\quad\n \\text{if }\n 1 - \\tvec{\\eta}^\\trans E^{-2} \\tvec{\\eta} >0 .\n\\end{equation}\n\nFor $|\\tvec{k} | = 1$ we impose a Lagrange multiplier,\nand the stationary solutions follow from \n\\begin{equation}\n\\nabla_{\\tvec{k}, u} (J_4 (\\tvec{k}) + u (1-\\tvec{k}^2) )\n= 0 \\;,\n\\end{equation}\nthat is,\n\\begin{equation} \\label{eqbord}\n(E-  \\mathbbm{1}_3 u) \\tvec{k} + \\tvec{\\eta} = 0, \\quad  \n(1-\\tvec{k}^\\trans \\tvec{k})=0 \\;.\n\\end{equation}\nWith these conditions for a vanishing gradient we can \nin the case $|\\tvec{k} | = 1$ write the condition for a vanishing $J_4$ \\eqref{cond1THDM} \nnow in the form\n\\begin{equation} \\label{cond12u}\n\\tvec{\\eta}^\\trans \\langle \\tvec{k} \\rangle_2 = - \\eta_{00} -u \\;.\n\\end{equation}\n\nThe regular solution of \\eqref{eqbord}, that is, a solution \nwith $\\det(E- \\mathbbm{1}_3 u) \\neq 0$ is\n\\begin{equation} \\label{solbord}\n\\langle \\tvec{k} \\rangle_2 = - (E- \\mathbbm{1}_3 u)^{-1} \\tvec{\\eta}\\;,\n\\end{equation}\nwhere $u$ follows from $(1-\\tvec{k}^\\trans \\tvec{k})=0$ with \\eqref{solbord} from\n\\begin{equation} \\label{solu}\n 1 - \\tvec{\\eta}^\\trans (E- \\mathbbm{1}_3 u)^{-2} \\tvec{\\eta} = 0 \\;.\n\\end{equation}\n We check that for the regular solution we have indeed\n \\begin{equation}\n J_4 (\\langle \\tvec{k} \\rangle_2)\n=\n u + \\eta_{00} - \\tvec{\\eta}^\\trans E^{-1} \\tvec{\\eta}  = 0, \n\\end{equation}\nwith $u$ the solution of \\eqref{solu}.\nAlternatively, we may have for $\\det(E- \\mathbbm{1}_3 u) = 0$\nexceptional solutions of \\eqref{eqbord}.\n\n\nEventually we note that we have to ensure that the \nstationary solutions of \\eqref{con}\nfor $|\\tvec{k}| < 1$, respectively, \\eqref{eqbord} for $|\\tvec{k}| = 1$ are local minima. \nAs usual this can be \ndone by considering the (bordered) Hessian matrix. Alternatively, in \ncase of a stable potential, that is, a potential which is bounded from below, we \nmay look for the deepest stationary solution or in the degenerate case, solutions,\nwhich are then of course minima. \n\n\n\n\n\\subsection{Classification of the MPP in the THDM}\n\\label{classification}\n\nLet us now study the vacuum structure with respect to the MPP in detail.\nEspecially, we derive the conditions to have isolated points, respectively, continuous\nstationarity regions, corresponding to the MPP in a weaker or \na stronger sense. \nFirst we recall that we can, by a basis change, \\eqref{b1}, \\eqref{b2}, diagonalize the real symmetric matrix $E$ and therefore we suppose to have\n\\begin{equation}\nE' = \\diag (E_{11}', E_{22}', E_{33}' ) .\n\\end{equation}\nWe emphasize that $E'$ diagonal is assumed to hold at the scale $\\Lambda^2$. This in particular means that in principle the matrix $E'$ may be non-diagonal at a different scale. We discuss the running\nof the parameters of the THDM in the next section.\n\nIn order to distinguish the parameters in the new basis, where the matrix $E'$ is diagonal, from the\noriginal ones, we denote them with a prime symbol.\nIn conventional notation the potential with a diagonal matrix $E$ corresponds\nto arbitrary parameters with  \n$\\lambda_6=\\lambda_7$ general complex and $\\lambda_5$ real.\n\nFor $K_0 \\neq 0$ the bilinear space is defined on the domain $|\\tvec{k}| \\le 1$. \nLet us first consider the case $|\\tvec{k}| < 1$. \n\nAs pointed out above, for $\\det (E) = \\det (E') \\neq 0$ the regular solution of\nthe gradient equation \\eqref{con}\nis a single point \\eqref{solreg} and requires that $\\eta_{00}$ also satisfies \\eqref{cond12},\nproviding a degenerate value of the potential. \nWe emphasize that the condition $\\det (E) \\neq 0$ as well as the parameter $\\eta_{00}$ \nare invariant under a change of basis.\n\nIf one of the eigenvalues of $E$ is zero, say in the \ndiagonalized matrix $E'$ its upper component, \nwe get from \\eqref{con} \n\\begin{multline}\nE'_{11} = 0,\\; E'_{22} \\neq 0,\\; E'_{33} \\neq 0  :\\\\\n\\quad \\text{solution } \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} x, &  -\\frac{\\eta_2'}{E_{22}'}, & - \\frac{\\eta_3'}{E_{33}'} \\end{pmatrix}^\\trans\n\\text{ with } \nx^2 < 1 -  \\frac{\\eta_2'^2}{E_{22}'^2}  -  \\frac{\\eta_3'^2}{E_{33}'^2}  \\text{ for } \\eta_1' =0.\n\\end{multline}\n This is a line segment. \nIn the case that $\\eta_1'$ together with $E_{11}'$ are vanishing, that is,\n the two zero components of $\\tvec{\\eta'}$ and $E'$  \n are aligned, \n we may have a degenerate line of solutions satisfying \\eqref{con}. \nFor $\\eta_1' \\neq 0$  there is no solution of \\eqref{con}.\n Besides, the\n $\\eta_{00}$ parameter has in any case to satisfy \\eqref{cond12}.\n We want to derive these conditions in a basis-invariant way. \nFirstly, we remark that one vanishing eigenvalue of the matrix $E$ (note that $E$ is \nthe original parameter matrix and not necessarily diagonal) corresponds to \n$\\rank(E) = 2$ and this in turn gives the basis-invariant conditions\n\\begin{equation} \\label{Erank2}\n \\rank(E) = 2: \\qquad  \\det(E) = 0 \\text{ and } \\tr^2(E) - \\tr(E^2)  \\neq 0 \\;.\n\\end{equation}\nNow we can construct the conditions to have one zero in the parameter vector $\\tvec{\\eta}'$\naligned with $E'$ in a basis-invariant way:\n\\begin{equation} \\label{1align}\n(\\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans E^2 \\tvec{\\eta} = 0,\n\\quad \\text{and} \\quad\n(E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans (E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta})) \\neq 0 \\;.\n\\end{equation}\nWe get the statement, that for a matrix $E$ of rank 2, that is, a matrix $E$ fulfilling the conditions \\eqref{Erank2},\nthere is a line of second degenerate vacua possible if in addition the two conditions\n\\eqref{1align} hold and the basis-invariant parameter \n$\\eta_{00}$ satisfies \\eqref{cond12}. This corresponds to the MPP in the stronger sense.\n In case that only the rank conditions \\eqref{Erank2} hold but not the conditions \\eqref{1align} there is no realization \n of the MPP possible. \n\nSimilarly we can treat the case that two eigenvalues of $E$ vanish. Going to a basis where $E'$ is diagonal, we suppose\nthat the two upper components of the diagonal matrix $E'$ vanish,  \nthen we find from \\eqref{con} the solutions\n\\begin{multline} \\label{2zero}\nE'_{11} =  E'_{22} = 0,\\; E'_{33} \\neq 0:\\\\\n\\text{solution } \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} x, & y , &  -\\frac{\\eta_3'}{E_{33}'} \\end{pmatrix}^\\trans\n\\text{ with } x^2 + y^2  < 1 -  \\frac{\\eta_3'^2}{E_{33}'^2},\\text{ for }\n\\eta_1'=\\eta_2'=0.\n\\end{multline}\nThis solution is a disk. In case \nthat not both components of $\\tvec{\\eta'}$ aligned with\nthe vanishing diagonal entries of  $E'$ vanish,  there is no solution.\nWe note that for a solution also the parameter $\\eta_{00}$ has to satisfy \\eqref{cond12}.\n\nThe formulation in a basis-invariant way is as follows:\nTwo vanishing eigenvalues correspond to a matrix $E$ of $\\rank(E) =1$, that is,\n\\begin{equation} \\label{Erank1}\n \\rank(E) = 1:\\qquad  \\det(E) = 0 ,\n \\text{ and } \\tr^2(E) - \\tr(E^2)  = 0 ,\n \\text{ and }\n\\tr(E) \\neq 0\\;.\n\\end{equation}\nSince two eigenvalues vanish, we can by a basis change always achieve \nthat also one of the components of $\\tvec{\\eta}$ vanish, aligned with one of the vanishing\nentries of $E$. This can be written in a basis-invariant way:\n\\begin{equation} \\label{cond2}\n\\tvec{\\eta}^\\trans E \\tvec{\\eta} \\neq 0, \n\\quad\n\\text{and}\n\\quad\n(E \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (E \\tvec{\\eta} \\times \\tvec{\\eta}) = 0 \\;.\n\\end{equation}\nOnly in case that in addition to \\eqref{Erank1} \nthe conditions \\eqref{cond2} are satisfied,\nwe can have the MPP in form of a disk in bilinear space, that is, the MPP in the stronger sense.\nWe note, that in this case it is required that the parameter \n$\\eta_{00}$ satisfies \\eqref{cond12}.\nOtherwise, if the rank 1 conditions are fulfilled, but \\eqref{cond2} do not both hold, there is no solution as a multiple point.\n\nEventually, we consider the case $E=0$. Note that a vanishing\nmatrix $E$ does not depend on the chosen basis. Now, we find from \\eqref{cond12},\nthat is, the condition of a vanishing potential at the second minimum,\n that there is for $\\tvec{\\eta} \\neq 0$ no solution with respect to the MPP.\n However, if we have in addition to $E=0$ also $\\tvec{\\eta} = 0$ and $\\eta_{00}=0$ \nwe have a sphere of solutions, that is, the MPP realized in the stronger sense.\n\n\n\n\nThe case $|\\tvec{k}| =1$ can be treated analogously to the previous one. \nWe look for solutions of the gradient equation \\eqref{eqbord} instead of \n\\eqref{con}.\nThis system of four equations has in general solutions for the indeterminants\n$\\langle \\tvec{k} \\rangle_2$ as well as the Lagrange multiplier $u$.\nThe solutions of \\eqref{eqbord} and in particular the degeneracy of the solutions depend on the determinant of the matrix\n\\begin{equation}\nM=E - \\mathbbm{1}_3 u \\;.\n\\end{equation}\nAgain we get for  $\\det{M} \\neq 0$\n from \\eqref{eqbord} solution points \\eqref{solbord}. \nThe determinant is of course invariant under basis changes.\n \nLet us now turn to the exceptional cases, with at least one  \neigenvalue of the matrix $M$ vanishing.\nThe argumentation is quite analogously to the previous study where\nwe have to replace the matrix $E$ by $M$ and have to\ntake into account the condition $\\tvec{k}^\\trans \\tvec{k} =1$. \n\nIf one of the eigenvalues of $M$ is zero, say, without loss of generality in the \ndiagonalized matrix $M'$ its upper component, \nwe get from \\eqref{con} \n\\begin{multline}\nM'_{11} = 0,\\; M'_{22} \\neq 0,\\; M'_{33} \\neq 0  :\\\\\n\\text{solution } \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} x, &  -\\frac{\\eta_2'}{M_{22}'}, & - \\frac{\\eta_3'}{M_{33}'} \\end{pmatrix}^\\trans\n\\quad \\text{with } \nx^2 = 1 -  \\frac{\\eta_2'^2}{M_{22}'^2}  -  \\frac{\\eta_3'^2}{M_{33}'^2},\\text{ for }\n\\eta_1' =0 .\n \\end{multline}\n This gives at most two points, supposed that $\\eta_1' =0$, and\n otherwise there is no solution. \n In addition, the $\\eta_{00}$ parameter has to satisfy \\eqref{cond12u} in order to \n give a degenerate second vacuum.\n We want to find the conditions independent of the chosen basis. \nOne vanishing eigenvalue of the matrix $M$ corresponds to \n$\\rank(M) = 2$, hence, basis-invariantly written,\n\\begin{equation} \\label{Erank2u}\n \\rank(M) = 2: \\qquad  \\det(M) = 0 \\text{ and } \\tr^2(M) - \\tr(M^2)  \\neq 0 \\;.\n\\end{equation}\nThe conditions to have one zero in the parameter vector $\\tvec{\\eta'}$\naligned with the vanishing eigenvalue in $M$ are\n\\begin{equation} \\label{1alignM}\n(\\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans M^2 \\tvec{\\eta} = 0,\n\\quad \\text{and} \\quad\n(M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans (M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta})) \\neq 0 \\;.\n\\end{equation}\nIn case there is a solution of  a vacuum with $u$ from \\eqref{eqbord} satisfying the conditions \\eqref{Erank2u}\nand also the conditions \\eqref{1alignM},\nthere are points as a second degenerate vacuum possible\nsupposed that $\\eta_{00}$ satisfies \\eqref{cond12u}.\n\nSuppose now that the two components, say, the upper components of the diagonalized matrix $M'$ vanish, then we find from \\eqref{eqbord} the solutions\n\\begin{multline} \\label{2zerou}\nM'_{11} =  M'_{22} = 0,\\; M'_{33} \\neq 0:\\\\\n\\text{solution } \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} x, & y , &  -\\frac{\\eta_3'}{M_{33}'} \\end{pmatrix}^\\trans\n\\quad \\text{with } x^2 + y^2  = 1 -  \\frac{\\eta_3'^2}{M_{33}'^2},\\text{ for }\n\\eta_1' = \\eta_2' =0.\n\\end{multline}\nOnly in case that we have also the two vanishing components of $\\tvec{\\eta'}$ \naligned with the vanishing eigenvalues of $M'$, we \nget a circle of degenerate solutions; otherwise there\nis no solution.\n\nThe formulation in a basis-invariant way is as follows:\nTwo zero eigenvalues correspond to a matrix $M$ of rank one, that is,\n\\begin{equation} \\label{Erank1u}\n \\rank(M) = 1:\\qquad  \\det(M) = 0 ,\n \\text{ and } \\tr^2(M) - \\tr(M^2)  \\neq 0 ,\n \\text{ and }\n\\tr(M) \\neq 0\\;.\n\\end{equation}\nSince two eigenvalues vanish, we can by a further change of basis always achieve \nthat one of the components of $\\tvec{\\eta}'$ vanishes, aligned with one of the vanishing\nentries of $M'$. \nTo this purpose we construct two additional conditions, manifestly basis invariant,\n\\begin{equation} \\label{cond2u}\n\\tvec{\\eta}^\\trans M \\tvec{\\eta} \\neq 0, \n\\quad\n\\text{and}\n\\quad\n(M \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (M \\tvec{\\eta} \\times \\tvec{\\eta}) =0.\n\\end{equation}\nOnly in the case that both conditions \\eqref{Erank1u} and \\eqref{cond2u} hold,\nwith $\\eta_{00}$ satisfying \\eqref{cond12u} we can have the MPP realized in form of a \ncircle in bilinear space.\nIn all other cases there is no second vacuum possible.\n\nEventually, we consider the case $M=0$. Then, we find from \\eqref{cond12u},\nthat is, a vanishing potential at the second minimum,\n that there is for $\\tvec{\\eta} \\neq 0$ no solution with respect to the MPP and\nin case of $\\tvec{\\eta} = 0$ we have a surface of a sphere of solutions, that is, \nthe MPP in the stronger sense.\nNote that these conditions are already basis invariant. \n\n\nLet us mention that we have seen, that\nin addition to a vanishing eigenvalue of the matrix $E$, respectively, \n$M=E - u \\mathbbm{1}_3$, also the corresponding component \nof $\\tvec{\\eta'}$ has to vanish (in a basis where $E'$, respectively $M'$, is diagonal). \nThis in turn means that we do have CP conservation in\nthis case~\\cite{Maniatis:2007vn}.\nWe thus confirm the result~\\cite{Froggatt:2004st}\nthat the MPP in the THDM in the stronger sense corresponds\nto a CP conserving potential.\nIn the strongest case where the second vacuum is a degenerate sphere we have \nto have\n$E = 0$ together with $\\tvec{\\eta}$ and $\\eta_{00}$ vanishing. \nThis means that the potential has $J_4 =0$ for all $\\tvec{k}$. \nIn conventional notation this\ngives $\\lambda_i = 0$, $i=1,\\ldots,7$.\n\nMoreover, let us note the interesting aspect of solutions corresponding \nto $|\\tvec{k}| < 1$ (see \\cite{Maniatis:2006fs} for details),\nwhich give charge-breaking minima and solutions corresponding to\n$|\\tvec{k}| = 1$ \nwhich give  electroweak symmetry breaking \\mbox{$SU(2)_L \\times U(1)_Y$}  $\\to$ \\mbox{$U(1)_{em}$} , \nhowever, for a second vacuum at\na high vacuum expectation scale $\\Lambda$, there is no reason to discard the possibility of charge-breaking minima.\n\nWe summarize our findings in table \\ref{tabMPP}.\n\n\\begin{table}[h!]\n\\begin{tabular}{lllll}\n\\hline\n\\hline\n$\\det(E) \\neq 0$ & & & & point\\\\\n\\hline\n\\multirow{12}{*}{$\\det(E) = 0$}\n\t & \\multirow{6}{*}{$\\tr^2(E)-\\tr(E^2) \\neq 0$}\n\t\t\t& & \\multicolumn{2}{l}{\\hspace{-24pt}$(\\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans E^2 \\tvec{\\eta} = 0$}\n\\\\\n& & & and & line\\\\\n& & & \n\\multicolumn{2}{l}{\\hspace{-24pt}$(E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans (E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta})) \\neq 0$}\n\\\\\n\t \t\t\\cline{4-5}\n& & & \\multicolumn{2}{l}{\\hspace{-24pt}$(\\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans E^2 \\tvec{\\eta} \\neq 0$}\\\\\n& & &  or \t &  no \\\\\n& & & \\multicolumn{2}{l}{\\hspace{-24pt}$(E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans (E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta})) = 0$}\t\n\\\\\n\t \\cline{2-5}\n\t &  \\multirow{8}{*}{$\\tr^2(E)-\\tr(E^2) = 0$}\n\t \t& \\multirow{2}{*}{$\\tr(E) \\neq 0$}\n\t \t\t&\n\t \t\t$\\tvec{\\eta}^\\trans E \\tvec{\\eta} \\neq 0$\\\\\n& & &\t\\text{ and } & disk\\\\\n& & & $(E \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (E \\tvec{\\eta} \\times \\tvec{\\eta}) = 0$ \\\\\n\t \t\t\\cline{4-5}\n\t \t\t& & \t& \n \t\t$\\tvec{\\eta}^\\trans E \\tvec{\\eta} = 0$ \\\\\n& & &\t\t\t\\text{ or } & no\\\\\n& & &\t\t\t$(E \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (E \\tvec{\\eta} \\times \\tvec{\\eta}) \\neq 0$\\\\ \n\t \t\\cline{3-5}\n\t\t& &  \\multirow{2}{*}{$E=0$} & $\\tvec{\\eta} \\neq 0$  & no\\\\\n\t\t\\cline{4-5}\n \t\t& &  & $\\tvec{\\eta} = 0$ & sphere\\\\\n\\hline\n\\hline\n$\\det(M) \\neq 0$ & & & & point\\\\\n\\hline\n\\multirow{5}{*}{$\\det(M) = 0$}\n\t & \\multirow{6}{*}{$\\tr^2(M)-\\tr(M^2) \\neq 0$}\n\t\t\t& & \\multicolumn{2}{l}{\\hspace{-24pt}$(\\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans M^2 \\tvec{\\eta} = 0$}\n\\\\\n& & & and & point\\\\\n& & & \n\\multicolumn{2}{l}{\\hspace{-24pt}$(M^2 \\tvec{\\eta} \\times (M\\tvec{\\eta}))^\\trans (M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta})) \\neq 0$}\n\\\\\n\t \t\t\\cline{4-5}\n& & & \\multicolumn{2}{l}{\\hspace{-24pt}$(\\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans M^2 \\tvec{\\eta} \\neq 0$}\\\\\n& & &  or \t &  no \\\\\n& & & \\multicolumn{2}{l}{\\hspace{-24pt}$(M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans (M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta})) = 0$}\t\\\\\n\t \\cline{2-5}\n\t &  \\multirow{8}{*}{$\\tr^2(M)-\\tr(M^2) = 0$}\n\t \t& \\multirow{2}{*}{$\\tr(M) \\neq 0$}\n\t \t\t&\n\t \t\t$\\tvec{\\eta}^\\trans M \\tvec{\\eta} \\neq 0$\\\\\n& & &\t\t\t\\text{ and } & circle\\\\\n& & &\t\t\t$(M \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (M \\tvec{\\eta} \\times \\tvec{\\eta}) = 0$\\\\\n\t \t\t\\cline{4-5}\n\t \t\t& & \t& \n\t \t\t$\\tvec{\\eta}^\\trans M \\tvec{\\eta} = 0$\\\\\n& & &\t\t or & no\\\\\n& & &\t\t$(M \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (M \\tvec{\\eta} \\times \\tvec{\\eta}) \\neq 0$\n\t\t  \\\\\n\t\t \t\\cline{3-5}\n\t\t& &  \\multirow{2}{*}{$M=0$} & $\\tvec{\\eta} \\neq 0$  & no\\\\\n\t\t\\cline{4-5}\n \t\t& &  & $\\tvec{\\eta} = 0$ & surface\\\\\n \\hline\n \\hline\n\\end{tabular}\n\\caption{\n\\label{tabMPP} \nClassification of possible realizations of the MPP in the THDM. \nThe last column gives the kind of realization of the MPP or ``no'' in \ncase the MPP is not realized. The upper part gives solutions for \nthe case $|\\tvec{k}|<1$ and the lower part for the case $|\\tvec{k}|=1$.\nIn all cases where the MPP is realizable, the parameter $\\eta_{00}$ has\nto fulfill the condition \\eqref{cond12}, respectively, \\eqref{cond12u}. The solutions\nof the vacuum vector $\\langle \\tvec{k} \\rangle$ follow from \\eqref{con}, respectively,\nthe solutions of $\\langle \\tvec{k} \\rangle$ and $u$ from \\eqref{eqbord}.\nThe conditions are given in a basis-invariant way and\nare therefore directly applicable to any THDM.\n}\n\\end{table}\n\n\n\n\\subsection{Constraints from the quantum potential}\n\\label{quantum}\n\nThanks to the bilinear formalism, the 1-loop $\\beta$-functions of the general THDM can be put in a concise tensor form (see appendix~\\ref{rges}), allowing one to perform an analytical study of the renormalization group. In the case where $|\\tvec{k}| < 1$, using~\\eqref{con} and~\\eqref{cond12}, the constraint~\\eqref{cond2THDM} can be put into \nthe following form:\n\\begin{align}\\label{nullDK0}\n    \\begin{split}\n    0 ={}& 8\\left[\\left( \\tvec{k}^\\trans E \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans E^2 \\tvec{k} + \\tr \\left( E^2 \\right) \\right] + g \\tvec{k}^\\trans \\tvec{k} + G\\\\\n    &-\\frac{1}{2}\\Big\\{ \\tensor{\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right)}{^a_b}\\tensor{\\left(\\sigma_0 + \\tvec{k}^\\trans\\tvec{\\sigma}\\right)}{^c_d} \\Big\\}\\,\\tensor{\\mathcal{T}}{_a^b_c^d} + \\left(1 - \\tvec{k}^\\trans \\tvec{k} \\right) \\mathcal{T}_{\\mathcal{U}\\mathcal{D}}\\,,\n    \\end{split}\n\\end{align}\nwhere we defined for convenience the strictly positive quantities \n\\begin{equation}\n    g \\equiv \\frac{9}{20} g_1^2 g_2^2 \\quad\\text{and}\\quad G \\equiv \\frac{9}{8}\\left(\\frac{3 g_1^4}{25}+g_2^4\\right)\n\\end{equation}\nand where the definition of the $\\mathcal{T}$-tensors is given in~\\eqref{tTrace}.\nIf $|\\tvec{k}| = 1$, we must use \\eqref{eqbord} and~\\eqref{cond12u} and the constraint reads \n\\begin{align}\\label{nullDK0u}\n    \\begin{split}\n    0 ={}& 8\\left[ 2 u^2 + \\left( \\tvec{k}^\\trans M \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans M^2 \\tvec{k} + \\mathrm{Tr}\\left( M^2 \\right) \\right] + g + G \\\\\n    & -\\frac{1}{2}\\Big\\{ \\tensor{\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right)}{^a_b}\\tensor{\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right)}{^c_d} \\Big\\}\\,\\tensor{\\mathcal{T}}{_a^b_c^d} \\,.\n    \\end{split}\n\\end{align}\n\nLet us first consider a simplified situation where the theory does not contain Yukawa couplings. In that case, eq.~\\eqref{nullDK0} becomes\n\\begin{align}\n    0 ={}& 8\\left[\\left( \\tvec{k}^\\trans E \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans E^2 \\tvec{k} + \\mathrm{Tr}\\left( E^2 \\right) \\right] + g \\tvec{k}^\\trans \\tvec{k} + G \\,,\n\\end{align}\nand has in fact no solutions. This can be seen by working in a basis where $\\tvec{k} = (k_1,0,0)^\\trans$, and where the term in the brackets now reads\n\\begin{align}\n\\begin{split}\n    \\left( \\tvec{k}^\\trans E \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans E^2 \\tvec{k} + \\mathrm{Tr}\\left( E^2 \\right) ={}& k_1^4 E_{11}^2 -2 k_1^2 \\left(E_{11}^2+ E_{12}^2+E_{13}^2\\right) \\\\\n   &+E_{11}^2+E_{22}^2+E_{33}^2 + 2 (E_{12}^2+ E_{13}^2 + E_{23}^2) \\\\\n   ={}&(1-k_1^2)^2 E_{11}^2 + E_{22}^2 + E_{33}^2 + 2(1-k_1^2)\\big[ E_{12}^2 + E_{13}^2 + E_{23}^2 \\big]\\,.\n\\end{split}\n\\end{align}\nClearly, since $\\tvec{k}^\\trans \\tvec{k} <1$ the right-hand side is a positive quantity. This holds in any basis, and implies in turn that \n\\begin{align}\n    8\\left[\\left( \\tvec{k}^\\trans E \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans E^2 \\tvec{k} + \\tr \\left( E^2 \\right) \\right] + g \\tvec{k}^\\trans \\tvec{k} + G > 0\\,.\n\\end{align}\n\nWe therefore have proven that in absence of Yukawa couplings, and if $\\tvec{k}^\\trans \\tvec{k} < 1$, eq.~\\eqref{cond2THDM} cannot be satisfied, which means that the MPP cannot be realized. Applying the above reasoning to the case $\\tvec{k}^\\trans \\tvec{k}=1$ we get the same conclusion.\\\\\n\nThe situation changes if we consider Yukawa couplings. For the known fermions we will see that only the dominant \ncontribution from the top quark allows for a realization of the MPP. Alternatively, lower bounds on the Yukawa couplings\ncould be given in order to have the MPP realized.\n\n\\subsection{Comparison with previous work on the MPP in the THDM}\n\\label{comparisonPreviousWorks}\n\nAs a concrete application of the results derived in sections~\\ref{classification} and~\\ref{quantum}, let us take the example of a \nTHDM type II\\footnote{In fact, the following discussion remains valid for any THDM where $\\lambda_6 = \\lambda_7 = 0$ and where \nnone of the fermions couple simultaneously to both doublets.}. \nThus we will be able to compare our results with the ones from \\cite{Froggatt:2004st} and \\cite{McDowall:2018ulq}.\\\\\n\nThe scalar potential \nconsidered in \\cite{Froggatt:2004st,McDowall:2018ulq} is the general potential but with the restriction $\\lambda_6 = \\lambda_7 = 0$. \nWith these assumptions the quartic couplings \\eqref{eq:para2}, \\eqref{eq:para3} simplify as:\n\\begin{align}\n\\eta_{00} &= \\frac{1}{8}\n(\\lambda_1 + \\lambda_2) + \\frac{1}{4} \\lambda_3  ,\n\\quad\n\\tvec{\\eta} = (\\eta_{a})=\\frac{1}{4}\n\\begin{pmatrix}\n0, & \n0, & \n\\frac{1}{2}(\\lambda_1 - \\lambda_2)\n\\end{pmatrix}^\\trans, \n\\\\\nE &= (E_{ab})= \\frac{1}{4}\n\\begin{pmatrix}\n\\lambda_4 + \\re(\\lambda_5) & \n-\\im(\\lambda_5) & 0\n\\\\ \n-\\im(\\lambda_5) & \\lambda_4 - \\re(\\lambda_5) & \n0 \\\\ \n0 & \n0 & \n\\frac{1}{2}(\\lambda_1 + \\lambda_2) - \\lambda_3\n\\end{pmatrix}.\n\\end{align}\n\n\nBy a basis change we can diagonalize the matrix $E$ without changing $\\eta_{00}$, $\\tvec{\\eta}$. In the new basis the parameter $\\lambda_5$ is real,\n\\begin{align}\nE &= (E_{ab})= \\frac{1}{4}\n\\begin{pmatrix}\n\\lambda_4 + \\lambda_5 & \n0 & 0\n\\\\ \n0 & \\lambda_4 - \\lambda_5 & \n0 \\\\ \n0 & \n0 & \n\\frac{1}{2}(\\lambda_1 + \\lambda_2) - \\lambda_3\n\\end{pmatrix}.\n\\end{align}\n\nThe most general form for the vacuum expectation values of the doublets may be parametrized as \\cite{Froggatt:2004st}\n\\begin{equation}\n\\langle \\varphi_1 \\rangle = \\phi_1 \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}, \n\\qquad\n\\langle \\varphi_2 \\rangle = \\phi_2 \\begin{pmatrix} \\sin(\\theta) \\\\ \\cos (\\theta)\\, e^{i \\omega} \\end{pmatrix},\n\\end{equation}\nwhere $\\Lambda^2= \\phi_1^2+\\phi_2^2$. In terms of the bilinear fields, this corresponds to \n\\begin{align}\n\t\\begin{split}\\label{Ks}\n\t\tK_0 &= \\phi_1^2 + \\phi_2^2 = \\Lambda^2,\\\\\n\t\tK_1 &= 2 \\phi_1 \\phi_2  \\cos(\\theta) \\cos(\\omega),\\\\\n\t\tK_2 &= 2 \\phi_1 \\phi_2  \\cos(\\theta) \\sin(\\omega),\\\\\n\t\tK_3 &= \\phi_1^2 - \\phi_2^2\\,,\n\t\\end{split}\n\\end{align}\nso there is a direct mapping between $(\\phi_1, \\phi_2, \\theta, \\omega)$ and $(K_0, K_1, K_2, K_3)$. Since this will be useful in the following, we note that\n\\begin{equation}\n\t\\tvec{k}^\\trans \\tvec{k} \n\t= \\frac{\\tvec{K}^\\trans \\tvec{K}}{K_0^2} \n\t= \\frac{1}{\\left(\\phi_1^2 + \\phi_2^2\\right)^2}\\left( \\phi_1^4 + 2 \\cos(2\\theta) \\phi_1^2 \\phi_2^2 + \\phi_2^4 \\right )\\,.\n\\end{equation}\nTherefore we find that for $\\cos(2\\theta) = 1$ or, equivalently, $\\cos(\\theta) = \\pm 1$ we have \n$\\tvec{k}^\\trans \\tvec{k} =1$.\\\\\n\nWe can now begin the comparison between the present work and the previous analysis of the MPP by Froggatt and Nielsen. The study in \\cite{Froggatt:2004st} results in two possible vacuum configurations, namely\n\\begin{subequations}\n\t\\begin{equation}\\label{chargeCons}\n\t\t\\langle \\varphi_1 \\rangle = \\begin{pmatrix} 0 \\\\  \\phi_1 \\end{pmatrix}, \n\\qquad\n\\langle \\varphi_2 \\rangle = \\begin{pmatrix} 0 \\\\  e^{i \\omega} \\phi_2 \\end{pmatrix},\n\t\\end{equation}\n\t\\begin{equation}\\label{chargeBr}\n\t\t\\langle \\varphi_1 \\rangle =\\begin{pmatrix} 0 \\\\  \\phi_1  \\end{pmatrix}, \n\\qquad\n\\langle \\varphi_2 \\rangle = \\begin{pmatrix}  \\phi_2 \\\\ 0 \\end{pmatrix},\n\t\\end{equation}\n\\end{subequations}\n\nwhich respectively correspond to a charge-conserving, CP-violating and a charge-breaking CP-conserving minimum.\\\\\n\nIn the first case, \\eqref{chargeCons}, that is, $\\cos(\\theta) = \\pm 1$, we have $\\tvec{k}^\\trans \\tvec{k} =1$ at the vacuum. \nAlso Froggatt and Nielsen require that the value of the potential at the minimum is independent of $\\omega$, meaning that, with view on \\eqref{Ks}, in the approach developed in the present work, this corresponds to a circle-shaped vacuum.\nThis in turn requires, as can be seen from table \\ref{tabMPP}, that there is a basis where the parameters must satisfy\n\\begin{equation}\n\tM = E - u \\mathbbm{1}_3 = \\diag(0, 0, E_{33}-u) \\quad\n\\text{yielding} \\quad \\lambda_5 = 0 \\quad\\text{and}\\quad u = \\frac{1}{4}\\lambda_4\\, .\n\\end{equation}\n\nThe value of $u$ being fixed, we may solve equations~\\eqref{eqbord} and \\eqref{cond12u}, giving\n\\begin{gather}\n\t\\eta_3^2 = (\\eta_{00} + u)(E_{33}-u)\\,,\n\\end{gather}\nwhich in terms of the conventional parameters gives\n\\begin{equation}\n\t\\pm \\sqrt{\\lambda_1 \\lambda_2} + \\lambda_3 + \\lambda_4 = 0\\,,\n\\end{equation}\nRequiring that $|k_3| < 1$, it can be shown that the choice of the solution depends on the sign of $\\lambda_1$ and $\\lambda_2$, thus giving\n\\begin{gather}\n\t\\lambda_1 > 0\\quad,\\quad\\lambda_2 > 0 \\quad,\\quad +\\sqrt{\\lambda_1 \\lambda_2} + \\lambda_3 + \\lambda_4 = 0\\quad,\\quad \\lambda_5 = 0 \\label{minimum1}\\\\[.1cm]\n\t\\textbf{or} \\nonumber\\\\[.1cm]\n\t\\lambda_1 < 0\\quad,\\quad\\lambda_2 < 0 \\quad,\\quad -\\sqrt{\\lambda_1 \\lambda_2} + \\lambda_3 + \\lambda_4 = 0\\quad,\\quad \\lambda_5 = 0\\,. \n\t\\label{minimum2}\n\\end{gather}\nIn addition, ensuring that the extremum is a minimum rules out the second solution \\eqref{minimum2}. \nThe only remaining set of constraints \\eqref{minimum1} corresponds to the one derived in \\cite{Froggatt:2004st}.\\\\\n\nWe now turn to the second case~\\eqref{chargeBr} where $\\cos(\\theta) = 0$. Making use of~\\eqref{Ks}, this means that $k_1 = k_2 = 0$ and, in the case \nwhere neither $\\phi_1$ nor $\\phi_2$ vanish, $|k_3| < 1$.\\footnote{Note that for a vacuum at a large scale we cannot have $\\phi_1 = \\phi_2 = 0$.}\nIn any case, from a geometric point of view, the solution is a point, meaning that none of the eigenvalues of the matrix $E$ should vanish. \nApplying the constraints~\\eqref{con} and~\\eqref{cond12} gives:\n\\begin{equation}\n\t\\eta_3^2 = E_{33} \\eta_{00}\n\\end{equation}\nor, in conventional parameters,\n\\begin{equation}\n\t\\lambda_3 = \\pm \\sqrt{\\lambda_1 \\lambda_2}\\,,\n\\end{equation}\nwhere obviously the quantity $\\lambda_1 \\lambda_2$ must be positive. In addition to the above constraint, we can work out the condition $|\\tvec{k}| < 1$ to give in the conventional formalism:\n\\begin{equation}\n\t\\lambda_1 < 0\\;,\\; \\lambda_2 < 0\\;,\\; \\lambda_3 = \\sqrt{\\lambda_1 \\lambda_2} \\quad\\textbf{or}\\quad \\lambda_1 > 0\\;,\\; \\lambda_2 > 0\\;,\\; \n\t\\lambda_3 = -\\sqrt{\\lambda_1 \\lambda_2} \\, .\n\\end{equation}\n\nOnce again we want to ensure that the solution is a minimum, meaning here that $E_{33}$ must be positive. This rules out the first solution while the second set of conditions corresponds to the one in \\cite{Froggatt:2004st}. Thus the present formalism agrees with the results of Froggatt and Nielsen in both cases.\\\\\n\nFinally, relation~\\eqref{cond2THDM} must hold if the MPP is to be realized. This results in an additional constraint among the beta-functions, \nthat is\\footnote{The fact that $\\beta_{\\eta_1}$ and $\\beta_{\\eta_2}$ vanish is due to the property of the Yukawa sector that each fermion couples to only one doublet.}:\n\\begin{equation}\n\t\\beta_{\\eta_{00}} + 2 k_3 \\beta_{\\eta_3} + k_1^2 \\beta_{E_{11}} + k_2^2 \\beta_{E_{22}} + k_3^2 \\beta_{E_{33}} = 0\\,.\n\\end{equation}\nInjecting the expression of the bilinear couplings in terms of the conventional parameters gives:\n\\begin{equation}\\label{betaFN}\n\t(1 + k_3)^2 \\beta_{\\lambda_1} + (1 - k_3)^2 \\beta_{\\lambda_2} + 2(1-k_3^2) \\beta_{\\lambda_3} + 2(k_1^2 + k_2^2)\\beta_{\\lambda_4} + 2(k_1^2 - k_2^2)\\beta_{\\lambda_5} = 0\\,.\n\\end{equation}\n\nIn the case of the charge-conserving vacuum~\\eqref{chargeCons} we have $\\lambda_5 = \\beta_{\\lambda_5} = 0$ and $k_1^2 + k_2^2 + k_3^2 = 1$. \nUsing these relations as well as the expression of $k_3$ in \\eqref{cond12u} in terms of the conventional parameters we find\n\\begin{equation}\n\t\\frac{1}{2}\\sqrt{\\frac{\\lambda_2}{\\lambda_1}} \\beta_{\\lambda_1} + \\frac{1}{2} \\sqrt{\\frac{\\lambda_1}{\\lambda_2}} \\beta_{\\lambda_2} + \\beta_{\\lambda_3}  + \\beta_{\\lambda_4} = 0 \\, .\n\\end{equation}\nIn the other case \\eqref{chargeBr}, we have $k_1 = k_2 = 0$ and with \\eqref{cond12} we find\n\\begin{equation}\n\t\\frac{1}{2}\\sqrt{\\frac{\\lambda_2}{\\lambda_1}} \\beta_{\\lambda_1} + \\frac{1}{2} \\sqrt{\\frac{\\lambda_1}{\\lambda_2}} \\beta_{\\lambda_2} + \\beta_{\\lambda_3} = 0 \\, .\n\\end{equation}\n\nThese two expressions exactly match the condition (26) from ref.\\ \\cite{Froggatt:2004st}, with the only difference that here we did not need to consider the sign of $\\lambda_4$.\n Instead, what distinguishes the two cases is the shape of the MPP vacuum.\\\\\n\nTo summarize this section, we have shown that the constraints derived by Froggatt and Nielsen \\cite{Froggatt:2004st, Froggatt:2006zc} and \nlater reused by McDowall and Miller \\cite{McDowall:2018ulq} in the framework of a THDM type II constitute in fact a very particular case of the \napplication of the formalism developed in this paper and can be easily recovered. \nWe stress that numerous different realizations of the MPP may be derived in the same manner using the classification from table~\\ref{tabMPP}, which \nmight lead to various phenomenological implications of this principle at the EW scale.\nLet us emphasize that the conclusion in the works \\cite{Froggatt:2004st, Froggatt:2006zc, McDowall:2018ulq}, that the MPP in the THDM cannot be realized \nfor a SM-like Higgs-boson mass and the observed top-quark mass is based on a special case of the MPP.\nHere we have seen in the geometric approach in terms of bilinears that the THDM may develop many more\ndifferent kinds of realizations of the MPP.\n\n\n\\subsection{Example potential}\n\\label{examplePotential}\n\nAs an additional example we study a CP conserving THDM potential \nin which the parameters in conventional notation \\eqref{Vconv}\nsatisfy\n\\begin{equation}\n\\lambda_1 = \\lambda_2 = \\lambda_3, \\quad \n\\lambda_4 = \\lambda_5 = \\lambda_5^*, \\quad\n\\lambda_6 = \\lambda_6^* = \\lambda_7, \\quad \n\\text{ with } \\lambda_4 \\neq - \\lambda_5.\n\\end{equation}\nLet us recall that these relations are assumed to hold at the scale $\\Lambda$.\nIn bilinear space \\eqref{pot} this corresponds\nto the quartic parameters in the form\n\\begin{equation}\nE = \\diag (E_{11}, 0, 0 ), \\text{ with } E_{11} \\neq 0, \n\\qquad\n\\tvec{\\eta} = \\begin{pmatrix} \\eta_1, & 0, & 0 \\end{pmatrix}^\\trans, \\text{ with } \\eta_1 \\neq 0.\n\\end{equation}\nWith view on table \\ref{tabMPP} we have the case $\\det(E)=0$, \n$\\tr^2(E)-\\tr(E^2)=0$, but with $\\tr(E) \\neq 0$.\ntogether with $\\tvec{\\eta}^\\trans E \\tvec{\\eta} = E_{11} \\eta_1^2 \\neq 0$\nand also \n$(E \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (E \\tvec{\\eta} \\times \\tvec{\\eta}) = 0$,\nthat is the MPP is realizable as a disk supposed $\\eta_{00}$ satisfies \nthe condition \\eqref{cond12}. \n\nLet us look into the solutions in detail. \nFirst we note that we can, by a change of basis \\eqref{b2}, with the rotation matrix \n\\begin{equation}\nR = \n\\begin{pmatrix}\n0 & \\phantom{+}0 & - 1\\\\\n0 & \\phantom{+}1 & \\phantom{+}0\\\\\n1 & \\phantom{+}0 & \\phantom{+}0\n\\end{pmatrix}\n\\end{equation}\nshift both,\nthe diagonal entry as well as the corresponding entry of $\\tvec{\\eta}$.\nThis case corresponds therefore to the case \\eqref{2zero} with \ntwo vanishing eigenvalues of $E$.\n\nIn order to study the MPP we consider first the case $|\\tvec{k}|<1$. \nThe second vacuum follows from \\eqref{con} \n\\begin{equation} \\label{vac21}\n\\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} -\\frac{\\eta_1}{E_{11}}, & y, & z \\end{pmatrix}^\\trans\n\\quad \\text{with } y^2 + z^2  < 1 -  \\frac{\\eta_1^2}{E_{11}^2} \n\\text{ for } \\eta_1^2 < E_{11}^2.\n\\end{equation}\nThis is indeed an open disk in the $y-z$ direction with \nborder radius $\\sqrt{  1 - \\eta_1^2\/E_{11}^2}$.\nThe parameter $\\eta_{00}$ has thereby to fulfill the \ncondition \\eqref{cond12},\n\\begin{equation} \\label{sol1eta00}\n\\eta_{00} =  \\frac{\\eta_1^2}{E_{11}}\\;.\n\\end{equation}\nMoreover, the $\\beta$ functions have to satisfy \\eqref{cond2THDM}.\nIn the case that we only consider the potential without any Yukawa couplings \nthese conditions read\n\\begin{equation}  \\label{exabeta}\n\\beta_{\\eta_{00}} =  2 \\frac{\\eta_1}{E_{11}} \\beta_{\\eta_1}  - \\left( \\frac{\\eta_1}{E_{11}}\\right)^2 \\beta_{E_{11}}\n\\end{equation}\nagain at the scale $\\Lambda^2$. \n\nFor $|\\tvec{k}|=1$ the stationarity condition is \ngiven by \\eqref{eqbord} with a Lagrange multiplier $u$. \nFor $u=0$ we get an exceptional solution at the border of\nthe solution \\eqref{vac21},\n\\begin{equation} \\label{vac21u}\n\\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} -\\frac{\\eta_1}{E_{11}}, & y, & z \\end{pmatrix}^\\trans\n\\quad \\text{with } y^2 + z^2  = 1 -  \\frac{\\eta_1^2}{E_{11}^2}\n\\text{ for } \\eta_1^2 \\le E_{11}^2.\n\\end{equation}\nwhich is, as to be expected from table \\ref{tabMPP} a circle\nin case that $\\eta_{00}$ fulfills \\eqref{cond12u} which equals \\eqref{sol1eta00}\nin this case. Also the $\\beta$ functions have to fulfill \\eqref{cond2THDM}\nwhich give, neglecting Yukawa interactions \\eqref{exabeta}.\n\nFor $u \\neq 0$ we immediately get the solution from \\eqref{eqbord}\n\\begin{equation} \\label{vac22}\n \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix}  \\pm 1  & 0, & 0 \\end{pmatrix}^\\trans ,\n\\end{equation}\nthat is, two possible points \nwith corresponding values for the Lagrange multiplier \n\\begin{equation}\nu = E_{11} \\pm \\eta_1 \\;.\n\\end{equation}\nThe condition \\eqref{cond12u} restricts the parameter $\\eta_{00}$, \n\\begin{equation} \\label{soleta002}\n\\eta_{00} = - E_{11} \\mp 2 \\eta_1 \\;.\n\\end{equation}\ncorresponding to one of the two discrete vacua in \\eqref{vac22}.\nBesides, we have to satisfy the condition for the beta functions \\eqref{cond2THDM},\nthat is,\n\\begin{equation} \\label{ex1beta}\n\\beta_{\\eta_{00}}  \\pm  2 \\beta_{\\eta_{1}}   + \n\\beta_{E_{11}}  = 0.\n\\end{equation}\n\nWe note that the potential is CP conserving \\cite{Maniatis:2007vn} (see section \\ref{sec:bilinears})\nand we conclude that the MPP is realizable \nin the stronger sense with a continuous disk of degenerate stationary points with \nthe parameters and its $\\beta$ functions satisfying the discussed constraints. \nIf these constraints are not fulfilled we may get at most an isolated point, that is,\nthe MPP in the weaker sense, where the parameters and $\\beta$ functions\n have to have in particular\nto satisfy \\eqref{soleta002} and \\eqref{cond2THDM}.\n\n\n\\section{Application of the MPP: From the high scale to the EW scale}\n\\label{analysisMPP}\n\nHaving classified the possible types of vacua at the high scale $\\Lambda$, we now want to study the MPP and its low-energy phenomenological implications. \nThe method we use in this analysis can be summarized by the following steps:\n\\begin{itemize}\n\t\\item At the high scale $\\Lambda$ we encounter 7 real parameters from the quartic part of the potential besides\n\tthe parameters of the Yukawa couplings. The three gauge couplings can be run up from their known values\n\tat the electroweak scale. \n\t\n\t\\item We consider the constraints provided by the MPP at the high scale $\\Lambda$, as given by Tab.~\\ref{tabMPP}.\n\t\n\t\\item We run all couplings down to the electroweak scale by the evolution equations at one-loop accuracy. \n\tAt the electroweak scale we have to consider in addition the \n\tquadratic parameters of the potential. These quadratic parameters are constrained since the model should provide\n\t the observed spontaneous electroweak symmetry breaking.\n\t\n\t\\item\n\tWe scan over all remaining free parameters.\n\t\n\t\\item For every parameter set, we compute the masses of the physical Higgs bosons of the THDM and the masses of the fermions.\n\\end{itemize}\n\n\nThe purpose of this analysis is to study the implications of the MPP on the masses of the physical states.\nThe main goal is to determine whether the application of the MPP to the THDM may yield correct (\\textit{i.e.} observed) masses of a Standard-Model-like Higgs boson and the top-quark. \nIn appendix \\ref{EWbreak} we recall the mechanism of spontaneous symmetry breaking in the THDM in\nthe bilinear formalism. In particular we present in this appendix the mass matrices of the Higgs bosons and\nthe mass of the pair of charged Higgs bosons.\n\n\n\\subsection{First example: MPP as a spherical vacuum}\n\nAs an application of our methods, \nwe first study a THDM potential with the MPP realized as \na spherical vacuum characterized by the potential parameter matrix $E=0$ (respectively $M=0$ for $|\\tvec{k}|=1$) (see table~\\ref{tabMPP}). We consider for simplicity only the top Yukawa coupling and we stay in the framework of a THDM type III to keep the discussion general. Note however that other types of THDM may always be obtained as special cases when either $y_t$ or $\\epsilon_t$ is set to 0 in some well-chosen basis. In appendix~\\ref{top} we show the Lagrangian of the top quark Yukawa coupling and its behavior under Higgs-basis changes.\n\nFollowing the discussion in section~\\ref{class} we have to consider the cases $|\\tvec{k}| < 1$\nand $|\\tvec{k}| = 1$. In the first case, equation~\\eqref{nullDK0} reads\n\\begin{align}\n\\begin{split}\n\tg \\tvec{k}^\\trans \\tvec{k} + G &= \\frac{3}{2}\\left[ Y^\\dagger\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right) Y \\right]^2\\\\\n\t&= \\frac{3}{2}\\left[(1+k_3) \\abs{y_t}^2 + (1-k_3)\\abs{\\epsilon_t}^2 + k_1(\\epsilon_t^* y_t + y_t^*\\epsilon_t) + i k_2(\\epsilon_t^* y_t - y_t^*\\epsilon_t)\\right]^2\\,. \\label{sphereCase1}\n\\end{split}\n\\end{align}\n\nIn order to simplify the evaluation of this relation, we may first perform a change of basis making the top quark Yukawa couplings real. Defining $y_t = \\abs{y_t} e^{i \\theta_y}$ and $\\epsilon_t = \\abs{\\epsilon_t} e^{i \\theta_\\epsilon}$, the associated unitary transformation is (see appendix \\ref{top})\n\\begin{equation}\n\\label{Ureal}\n\tU = \\begin{pmatrix}\n\t\te^{-i \\theta_y} & 0 \\\\\n\t\t0 & e^{-i \\theta_\\epsilon}\n\t\\end{pmatrix}\\,,\n\\end{equation}\nand corresponds, in terms of bilinears, to a rotation matrix $R(U)$ \\eqref{unitR}. The latter is in fact a rotation around the $z$-axis which in our case can be performed without loss of generality. Equation \\eqref{sphereCase1} can be further simplified after rotation to a basis where \\mbox{$Y = (y'_t, \\epsilon'_t)^\\trans = (y'_t,  0)^\\trans$}. The associated transformation $U$ is a 2D rotation matrix and corresponds, in the bilinear space, to a rotation around the $y$-axis. In this new basis, \\eqref{sphereCase1} now reads \n\\begin{equation}\n\tg \\tvec{k}^\\trans \\tvec{k} + G = \\frac{3}{2} (1+k_3)^2 y_t^4\\,,\n\\end{equation}\nwhere we dropped the primes for clarity. We see that choosing a point $\\tvec{k}$ within the sphere fixes the value of the top Yukawa coupling. Furthermore, it can be shown that a necessary condition for the above equation to be satisfied is\\footnote{To obtain this inequality, we used the fact that $g < G$ at all energy scales.} \n\\begin{equation}\n\t\\abs{y_t} \\geq \\left(\\frac{g + G}{6} \\right)^{1\/4}\\, ,\n\\end{equation}\nwhich can be reformulated in a basis-invariant way:\n\\begin{equation}\n\t\\label{yukInequality}\n\t\\sqrt{\\abs{y_t}^2 + \\abs{\\epsilon_t}^2} \\geq \\left(\\frac{g + G}{6}  \\right)^{1\/4} \\approx 0.38 \\quad \\text{for} \\quad \\Lambda=\\SI[parse-numbers=false]{10^{18}}{\\GeV}\\, .\n\\end{equation}\n\nGiven the above inequality, and under the assumption that the left hand-side is roughly of order $\\abs{y_t^{SM}}$, we expect the top quark to be the only fermion that couples strongly enough to the Higgs doublets to satisfy it. However the situation may not be as clear in some limiting cases, e.g. the THDM type II with a high value of $\\tan \\beta$.\\\\\n\n\nWe now turn to the case $|\\tvec{k}| = 1$, where the evaluation of~\\eqref{nullDK0u} in the same basis as above gives a constraint involving the Lagrange multiplier $u$, namely \n\\begin{equation}\n\t16 u^2 + g + G = \\frac{3}{2} (1+k_3)^2 y_t^4\\, .\n\\end{equation}\nWe note that in this case, condition \\eqref{yukInequality} must be satisfied as well.\n\n\nChoosing a specific value for $u$ and $k_3$ we can fix all the relevant parameters at the high scale: In case $|\\tvec{k}| < 1$ all the quartic couplings vanish whereas in case $|\\tvec{k}| = 1$ the non-zero parameters are related to the Lagrange multiplier $u$ via\n\\begin{subequations}\n\t\\begin{align}\n\t\t\\eta_{00} &= -u\\,,\\\\\n\t\tE_{ii} &= u\\:\\:,\\:\\: i = 1,2,3\\,.\n \t\\end{align}\n\\end{subequations}\n\nIt is remarkable that in both cases this set of couplings implies CP conservation at the level of the quartic part of the scalar potential \\cite{Maniatis:2006fs}. The only remaining possible source of CP violation is a non-zero value for the scalar mass coupling $\\xi_2$ in the basis chosen above.\\\\\n\nThe next step is to perform the running of the couplings down to the electroweak scale, where the study of spontaneous symmetry breaking will eventually allow for the determination of the masses of the Higgs bosons and the fermions. \nAt the electroweak scale we have to consider the quadratic parameters $\\xi_0$, $\\xi_{i,\\ i=1,2,3}$  of the Higgs potential.\nThese couplings are however subject to constraints in order to give a proper $SU(2)_L\\times U(1)_Y\\rightarrow U(1)_{em}$ symmetry breaking pattern.\n Using~\\eqref{EWbasisChange}, we can trade these four parameters for $u_\\text{EW}$, $\\beta$, $\\zeta$ and $v_0$. The latter is known since it corresponds to the Standard Model vacuum-expectation value $v_0 \\approx \\SI{246}{\\GeV}$.\\\\\n\nThe angles $\\beta$ and $\\zeta$ \nparametrize the basis transformation \\eqref{EWbasisChange} which allows to achieve the form of the Higgs basis \\eqref{higgsBasis}.\nWe note that a non-zero value for $\\zeta$ will in our case generate CP violation at the level of the scalar potential, and accordingly imply a mixing between the scalar and pseudo-scalar physical states. For simplicity reasons\nwe consider the case $\\zeta = 0$, in which we will identify the lightest neutral scalar\nwith the CP-even Standard-Model like Higgs boson.\\\\\n\n\\subsection{Second example: MPP as a disk-shaped vacuum}\n\nWe now briefly discuss the next-to-maximal symmetric vacuum, namely the disk-shaped one. This includes in particular the CP-conserving potential discussed in section~\\ref{examplePotential}.\n\nIn the following, we will work in a basis where $E = \\diag (0, 0, E_{33})$, respectively $M = \\diag (0, 0, M_{33})$ if $\\abs{\\tvec{k}} = 1$. In the latter case, the MPP vacuum reduces to a circle. Using the results from section~\\ref{statPoints}, the quartic parameters at the high scale satisfy the following constraints in order to provide stationary points:\n\\begin{align}\n\t\\abs{\\tvec{k}} < 1 &: \\begin{cases}\n\t\t\\eta_3 + E_{33} k_3 = 0 \\, ,\\\\\n\t\t\\eta_{00} + \\eta_3 k_3 = 0\\, ,\\\\\n\t\tk_1^2 + k_2^2 < 1 - k_3^2\\, ,\n\t\\end{cases}\\\\[.15cm]\n\t\\abs{\\tvec{k}} = 1 &: \\begin{cases}\n\t\t\\eta_3 + M_{33} k_3 = 0\\, , \\\\\n\t\tu + \\eta_{00} + \\eta_3 k_3 = 0\\, ,\\\\\n\t\tk_1^2 + k_2^2 = 1 - k_3^2\\, .\n\t\\end{cases}\n\\end{align}\nNote that, as discussed earlier, all other quartic parameters have to vanish at the high scale. The constraints from the quantum potential \\eqref{nullDK0}, \\eqref{nullDK0u} respectively simplify as\n\\begin{gather}\n\\begin{split}\n\\label{diskQpot1}\n\t8 E_{33}^2 \\left(1 - k_3^2 \\right)^2 + g\\tvec{k}^\\trans \\tvec{k} + G &= \\frac{3}{2}\\left[ Y^\\dagger\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right) Y \\right]^2 \\, ,\n\\end{split}\\\\\n\\begin{split}\n\\label{diskQpot2}\n\t16 u^2 + 8 M_{33}^2 \\left(1 - k_3^2 \\right)^2 + g + G &= \\frac{3}{2}\\left[ Y^\\dagger\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right) Y \\right]^2 \\, .\n\\end{split}\n\\end{gather}\n\nAlthough it is still possible to make the Yukawa couplings real using transformation~\\eqref{Ureal}, we cannot rotate to a basis where either $y_t$ or $\\epsilon_t$ vanishes without introducing a mixing between $k_1$ and $k_3$. In this context, equations~\\eqref{diskQpot1} and~\\eqref{diskQpot2} can be respectively rewritten as\n\\begin{gather}\n\\begin{split}\n\t8 E_{33}^2 \\left(1 - k_3^2 \\right)^2 + g\\tvec{k}^\\trans \\tvec{k} + G &= \\frac{3}{2}\\left[(1+k_3) y_t^2 + (1-k_3)\\epsilon_t^2 + 2 k_1 \\epsilon_t y_t\\right]^2\\, ,\n\\end{split}\\\\\n\\begin{split}\n\t16 u^2 + 8 M_{33}^2 \\left(1 - k_3^2 \\right)^2 + g + G &= \\frac{3}{2}\\left[(1+k_3) y_t^2 + (1-k_3)\\epsilon_t^2 + 2 k_1 \\epsilon_t y_t\\right]^2\\, .\n\\end{split}\n\\end{gather}\n\nNote that the number of free parameters at the high scale is increased here compared to the spherical case. We also emphasize that the previous comments about CP conservation are still valid in this case.\n\n\n\\subsection{Results and discussion}\n\nWe now present the results of the numerical analyses for the MPP for \na Higgs potential providing a spherical and disk-shaped vacuum. \nIn order to detect the minima at the electroweak scale we have to solve the \ncorresponding gradient equation (see appendix~\\ref{EWbreak} and \\cite{Maniatis:2006fs}). In principle, we may encounter regular and\nirregular solutions of these equations. In the spherical case, the irregular solutions provide very low values of $u_{EW}$. Consequently, either the scalar mass matrix \\eqref{scalarMasses} develops negative eigenvalues or leads to situations where the lightest scalar is much lighter than the SM-like Higgs boson. On the other hand, in the disk and circle cases, these solutions may provide some acceptable values for the masses of the physical states. In any case, for simplicity, we consider here only the regular solutions of eq.~\\eqref{eqEW}.\n\nFor these solutions, we show in  Fig.~\\ref{BManalysis} the results of the numerical analysis in the $(m_h, m_t)$ plane for $\\Lambda = \\SI[parse-numbers=false]{10^{18}}{\\GeV}$. The list of free parameters at the high scale along with their allowed range is presented in Table~\\ref{freeParams}. We systematically excluded from the parameter scan the points violating perturbativity and unitarity bounds \\cite{Branco:2011iw}. We also studied the conditions of a bounded from below potential \\cite{Branco:2011iw, McDowall:2018ulq} at all energy scales as illustrated in Fig.~\\ref{BManalysis}. We finally note that the ranges of $u$, $y_t$ and $\\epsilon_t$ at the high scale were chosen based on the observations that $\\abs{u} > 0.5$ sytematically violate perturbativity and unitarity bounds and that for Yukawa couplings greater than $1$ the RG flow tends to develop Landau poles.\n\nAt the EW scale, we have to scan over two remaining free parameters, namely $\\beta$ and $u_{EW}$. The former is taken in the range $[\\shortminus\\frac{\\pi}{2}, \\frac{\\pi}{2}]$ and the latter\\footnote{In order to give a physical meaning to $u_{EW}$, we note that $m_{H^\\pm} =v_0\\sqrt{ 2 u_{EW} }$ takes values between $0$ and $\\SI{1.1}{\\TeV}$ when $0 < u_{EW} < 10$.} in the range $[0, 10]$. We emphasize that the classical potential is ensured to develop a global minimum at the EW scale since we require Theorem 3 of \\cite{Maniatis:2006fs} to be satisfied.\n\n\\begin{table}[h]\n{\n\\hfill\n\\begin{tabular}{|C{1cm}|C{1.5cm}|C{1.5cm}|}\n\\hline\n\\multicolumn{3}{|c|}{\\large Spherical vacuum}                                       \\\\ \\hline\n $\\abs{\\tvec{k}}$ & $\\left[0, 1\\right[$ & $1$                                 \\\\ \\hline\n $u$              & \/                   & $\\left[\\shortminus 0.5, 0.5\\right]$ \\\\ \\hline\n $k_3$            & $\\left]\\shortminus1, 1\\right[$ & $\\left[\\shortminus1, 1\\right]$                 \\\\ \\hline\n\\end{tabular}\n\\hfill\n\\begin{tabular}{|C{1cm}|C{1.5cm}|C{1.5cm}|}\n\\hline\n\\multicolumn{3}{|c|}{Disk-shaped vacuum}                                                \\\\ \\hline\n$\\abs{\\tvec{k}}$ & $\\left[0, 1\\right[$            & $1$                                 \\\\ \\hline\n$u$              & \/                              & $\\left[\\shortminus 0.5, 0.5\\right]$ \\\\ \\hline\n$k_3$            & $\\left]\\shortminus1, 1\\right[$ & $\\left[\\shortminus1, 1\\right]$      \\\\ \\hline\n$k_1$            & $\\left]\\shortminus1, 1\\right[$ & $\\left[\\shortminus1, 1\\right]$      \\\\ \\hline\n$y_t$            & \\multicolumn{2}{c|}{$\\left[0, 1\\right]$}                             \\\\ \\hline\n$\\epsilon_t$     & \\multicolumn{2}{c|}{$\\left[0, 1\\right]$}                             \\\\ \\hline\n\\end{tabular}\n\\hfill\n}\n\\caption{Ranges of the free parameters at the high scale, in the spherical and disk case. The value of the $k_i$'s are always chosen such that $\\abs{\\tvec{k}} \\leq 1$.}\n\\label{freeParams}\n\\end{table}\n\n\n\\begin{figure}[h]\n\t\\centering\n\t\\begin{subfigure}[h]{0.49\\linewidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Figures\/case1.png}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[h]{0.49\\linewidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Figures\/case2.png}\n\t\\end{subfigure}\\\\\n\t\\begin{subfigure}[h]{0.49\\linewidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Figures\/case3.png}\n         \\caption{}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[h]{0.49\\linewidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Figures\/case4.png}\n         \\caption{}\n\t\\end{subfigure}\n\t\\caption{Results of the parameter scans in the $(m_h, m_t)$ plane in cases (a) $|\\tvec{k}| < 1$ and (b) $|\\tvec{k}| = 1$ for $\\Lambda = \\SI[parse-numbers=false]{10^{18}}{\\GeV}$. The two upper plots are obtained in the case of a spherical vacuum, and the lower ones in the case of a disk\/circle-shaped vacuum. For the points shown in red, the scalar potential is bounded from below at all energy scales between $M_Z$ and $\\Lambda$. The red stars indicate the position of $(m_h, m_t^{\\overline{\\mathrm{MS}}}) \\approx (\\SI{125}{\\GeV}, \\SI{163}{\\GeV})$. \\label{BManalysis}}\n\\end{figure}\n\n\nIn all four considered cases, there are regions of the parameter space for which the MPP constraints are compatible with the observed values of $m_h$ and $m_t$. However, if we also take the constraint into account that the potential has to be bounded from below, the spherical vacua are ruled out. In contrast, the disk- and circle-shaped vacua provide acceptable values for the masses along with the requirement of a stable potential. Although the question of identifying the precise regions of the parameter space yielding correct $(m_h, m_t)$ values goes beyond the scope of this work, the conclusion of this simple analysis is that the realization of the MPP is \\textit{a priori} not incompatible with a SM-like Higgs boson and top quark in the context of the general THDM\n\n\n\nEventually, we would like to mention that we varied in our numerical analysis the value of the high scale $\\Lambda$ in the range $10^{15} - \\SI[parse-numbers=false]{10^{20}}{\\GeV}$ and found that our predictions on the masses show a negligible dependence on this parameter.\n\n\\section{Conclusion and outlook}\n\nThe MPP \\cite{Froggatt:1995rt} forces the Higgs potential to provide degenerate vacua with the same potential value.\nA long time before the discovery of the Higgs boson, its mass has been predicted to be $135 \\pm 9 \\text{ GeV}$\nbased on this principle applied to the Standard Model -- a quite remarkable result.\\\\\n \nSome effort \\cite{Froggatt:2004st, Froggatt:2006zc, McDowall:2018ulq}\nhas been spent to apply the MPP to the TDHM. \nThis has been done in the conventional \nformalism, where the gauge degrees of freedom appear explicitly. Here we have studied the MPP\nin the THDM applying the bilinear formalism \\cite{Nagel:2004sw, Maniatis:2006fs, Nishi:2006tg} . \nThis formalism turns out to be quite powerful to \nstudy models with additional Higgs-boson doublets.\\\\\n\nWe have classified all different types of degenerate vacua \n in the THDM. In particular, we find degenerate vacua which realize the MPP in a weaker sense, providing\nadditional isolated points, but also realizations in  a stronger sense of line segments, circles, \nsurfaces of spheres, as well as spheres.\nWe have presented the classification in a basis-invariant way.\nFor any THDM the corresponding conditions for the different types of realizable MPP's can be easily checked. \\\\\n\nWe have studied the $\\beta$-functions of the THDM in detail in terms of bilinears and have shown that the MPP, considering only the THDM potential, is not realizable.\nThis changes if we consider the Yukawa couplings.\\\\\n\nWe recover the MPP cases studied in the literature but in addition can identify\ndifferent realizations of the MPP in the THDM. \nWe explored in section~\\ref{analysisMPP} four different realizations of the MPP in the context of a general THDM with a simplified matter content. Two of them (the disk- and circle-shaped vacua) are compatible with the measured values of the Higgs and top quark masses, satisfying simultaneously the constraints of a correct electroweak symmetry breaking, perturbativity and unitarity as well as the requirement of a stable scalar potential at all energy scales.\\\\\n\n\n\n\nOur analysis is done at the one-loop order of the $\\beta$-functions and using the tree-level RG-improved potential. In the future we would like to extend this analysis to higher orders of the effective potential, the RGEs, as well as the matching conditions between the $\\overline{\\mathrm{MS}}$ and pole masses. Also we would like to consider the experimental constraints from negative searches of additional Higgs bosons. Eventually, we plan to extend this study by considering all three families of fermions also in the context of THDMs with natural flavour conservation.\n\n\n\\acknowledgments\nThe project was supported in part by the UBB projects ''Materia Obscura y los bosones de Higgs''\nwith No. DIUBB 193209 1\/R and Fondecyt with No. 1200641.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{The isentropic compressible Navier-Stokes system and the inviscid limit problem}\n\nWe consider the motion of a compressible fluid in the half-space $\\Omega=\\mathbb{R}^{2}\\times\\mathbb{R}^{+}$. The density of the fluid is a scalar function $\\rho(t,x)$ with $t$ being the time variable and $x \\in\\Omega$, its velocity is $u(t):\\Omega \\rightarrow \\mathbb{R}^{3}$: these will be the unknowns of the system. \nThe temperature of the fluid, $\\theta$, is constant throughout the paper, and the pressure $P$ will follow a barotropic law: $P=P(\\rho)=\\frac{k}{\\gamma}\\rho^{\\gamma}$, with $k>0$ and the adiabatic constant $\\gamma> 1$. \nThe motion of the fluid is governed by the isentropic compressible Navier-Stokes system, which consists of two conservation laws:\n\\begin{itemize}\n\\item conservation of mass\n\\begin{equation} \\derp{t}\\rho + \\div(\\rho u) = 0, \\label{NSdiv} \\end{equation}\n\\item and conservation of momentum\n\\begin{equation} \\derp{t}(\\rho u) + \\div(\\rho u\\otimes u) = \\div \\Sigma + \\rho F, \\label{NS} \\end{equation}\n\\end{itemize}\nwith $F(t,x)$ a force term. To avoid technical complications with compatibility conditions, we consider that the force is smooth and that fluid is at rest for negative times: $F(t,x)=0$ and $(\\rho,u)(t,x)=(1,0)$ for $t<0$.\n\nIn these equations, $\\Sigma$ is the internal stress tensor,\n$$ \\Sigma = 2\\varepsilon\\mu Su + (\\varepsilon\\lambda_0 \\div u - P(\\rho)) I_{3}, $$\nin which $\\varepsilon>0$ will be arbitrarily small, $Su=\\frac{1}{2}(\\nabla u+\\nabla u^{T})$ is the symmetric gradient, and $I_{3}$ is the $3\\times 3$ identity matrix. The parameters $\\mu>0$ (dynamic viscosity) and $\\lambda:=\\lambda_0 + \\mu >0$ (bulk viscosity) will be given regular functions of $\\rho$. In equation (\\ref{NS}), we will write\n$$ \\div \\Sigma = \\varepsilon(\\mu \\Delta u +\\lambda \\nabla\\div u) - \\nabla (P(\\rho)) + \\varepsilon\\sigma(\\nabla \\rho,\\nabla u). $$\nThe final term $\\sigma$ is non-zero only when $\\lambda'(\\rho)$ and $\\mu'(\\rho)$ are non-zero ($\\lambda$, $\\mu$ not constant).\n\\newline\n\nWe shall not go into detailed historics about the isentropic system; a survey of existence and regularity results up to 1998 was put together by B.~Desjardins and C-K.~Lin, \\cite{DL}. \nWe shall quickly cite the emblematic result on the subject: the global existence theorem for weak solutions, as proved by P-L.~Lions in the 1990s (\\cite{Lpl} or \\cite{Lplbook2}), a result that has since been improved, for instance, by E.~Feireisl, A.~Novotn\\'y and H.~Petzeltov\\'a, \\cite{FNP}, \nand extended to some cases with density-dependent (variable, therefore) viscosity coefficients by D.~Bresch, B.~Desjardins and D.~G\\'erard-Varet \\cite{BDG}. \nIn the latter paper, a nonlinear drag force, written as $F=F(u)=r_0 |u| u$, is considered. \nWe also refer to the local strong existence theory, initiated in the 1960s and 70s by J.~Nash and then V.A.~Solonnikov \\cite{Nj, Sva} and improved on by R.~Danchin (for instance, see survey \\cite{Drsurv}). In this framework, blowup can occur \\cite{Xz}.\n\\newline\n\nIn this article, we will aim to obtain local-in-time existence of strong solutions to the isentropic Navier-Stokes system, with a lower bound for the time of existence that does not depend on the viscosity parameters when these are small, and work on the inviscid limit problem. \nAs we are on the half-space, we will have some boundary conditions.\n\n\\begin{nota} Throughout the paper, we will use the notation $x=(y,z)$, with $y\\in\\mathbb{R}^{2}$ and $z\\in\\mathbb{R}^{+}$; the boundary is therefore the set $\\{x=(y,z)\\in\\mathbb{R}^{3}~|~z=0\\}$.\n\nMoreover, for a vector field $v(x)$, the tangential part to the boundary is, for $x$ on the boundary, $v_\\tau(x)=v(x)+(v(x)\\cdot\\vec{n}(x))\\vec{n}(x)$, where $\\vec{n}(x)$ is the outer normal vector to $\\partial\\Omega$ at point $x$. As $\\vec{n}(x)\\equiv(0,-1)$, we extend the notation to all $\\Omega$: $v_\\tau(x)=(v_{1}(x),v_{2}(x))$. \\end{nota}\n\nThe boundary conditions on $u$ are the standard non-penetration condition on the boundary,\n\\begin{equation} u\\cdot\\vec{n}|_{z=0}=u_{3}|_{z=0}=0 , \\label{NPB} \\end{equation}\nand the Navier (slip) boundary condition $\\left[\\left(\\frac{1}{\\varepsilon}\\Sigma\\vec{n}+ au\\right)_\\tau\\right]|_{z=0} = 0$ in which $a>0$. In our case, with a flat boundary, the Navier condition can be rewritten as\n\\begin{equation} [\\mu(\\rho) \\derp{z}u_\\tau]|_{z=0} = 2a u_\\tau |_{z=0} \\label{NBC} \\end{equation}\nfor $t>0$ and $y\\in\\mathbb{R}^{2}$. As opposed to the Dirichlet or no-slip condition, which, in our setting, would be $u|_{z=0}=0$, the Navier condition, proposed by H.~Navier himself in the XIX\\textsuperscript{th} century \\cite{Nh}, allows the fluid to slip along the boundary, and this occurs wherever interaction at the boundary is non negligeable. \nFor instance, the slip phenomenon can be observed on the contact line of two immiscible flows \\cite{QWS}, and in capillary blood vessels, which are the microscopic, tissue-irrigating vessels where molecular exchanges with the neighbouring cells take place \\cite{PRD}. \nIt also appears when homogenising rough and porous boundaries (\\cite{JM} and \\cite{GVM}), and can be derived mathematically from a Boltzmann microscopic model with a Maxwell reflection boundary condition \\cite{MSR}. \nTo be physically pertinent, the slip coefficient $a$ should be chosen positive, but our results do not technically require $a$ to have a specific sign, so we take $a\\in \\mathbb{R}$.\n\nWe also impose the limit condition\n\\begin{equation} U(t,x):=(\\rho(t,x)-1,u(t,x))\\stackrel{|x|\\rightarrow+\\infty}{\\rightarrow} 0 \\label{limcond} \\end{equation}\nso that $U(t) \\in L^{2}(\\Omega)$.\n\\newline\n\nFormally, taking $\\varepsilon=0$ leads to the compressible Euler equations\n\\begin{equation} \\left\\{ \\begin{array}{rcl} \\derp{t}\\rho + \\div(\\rho u) & = & 0 \\\\\n\\rho \\derp{t}u + \\rho u\\cdot\\nabla u + \\nabla P(\\rho) & = & \\rho F . \\end{array} \\right. \\label{Euler} \\end{equation}\nThe Euler equation is of order one, so it only requires one boundary condition equation, which is (\\ref{NPB}). \nThis leads to the appearance of boundary layers: if the solutions of low-viscosity Navier-Stokes equations are expected to behave like a solution of the Euler equation far away enough from the boundary, solutions of Navier-Stokes are still required to satisfy a second boundary condition, whereas the reference solution of the Euler equation is not. \nA typical boundary layer expansion for solutions to Navier-Stokes will read\n\\begin{equation} u^\\varepsilon(t,y,z) = u^E(t,y,z) + V \\left(t,y,\\frac{z}{\\sqrt{\\varepsilon}}\\right) , \\label{blayer} \\end{equation}\nwith $u^E$ solving the Euler equation, and $V$, acting on a shorter scale, picking up the boundary condition.\n\nThe inviscid limit problem is a major challenge for mathematicians, whether one considers compressible or incompressible fluids. We remind the reader of the (non-forced) incompressible system:\n$$ \\left\\{ \\begin{array}{rcl} \\div u & = & 0 \\\\\n\\derp{t}u + u\\cdot\\nabla u - \\varepsilon\\nu\\Delta u + \\nabla q & = & 0 \\\\ u|_{t=0} & = & u_0 , \\end{array} \\right. $$\nin which $\\nu=\\frac{\\mu}{\\rho}$, with $\\rho$ constant, is the kinematic viscosity and $q$ is the kinematic pressure. \nRegarding the inviscid limit results on the incompressible system with Navier boundary conditions, the problem is solved in $L^2$ framework in 2D (see \\cite{Bc}, \\cite{CMR}, \\cite{Kjp}), and convergence of weak solutions of the Navier-Stokes equation towards a strong solution of the Euler equation, when the limit initial condition is regular enough, has been obtained for a range of Navier slip coefficients of the form $a=a'\\varepsilon^{-\\beta}$: \nstarting with D.~Iftimie and G.~Planas \\cite{IP} ($\\beta=0$), \\cite{WWX} ($\\beta<1\/2$) and \\cite{Pm11} ($\\beta<1$ for positive slip coefficients and $\\beta\\leq 1\/2$ regardless of sign) have extended the range of numbers $\\beta$ for which convergence occurs. \nC.~Bardos, F.~Golse and L.~Paillard recently obtained weak convergence results for Leray solutions of Navier-Stokes towards dissipative solutions of the Euler equation \\cite{BGP}.\n\nThe solutions to the 3D incompressible Navier-Stokes equations with a Navier boundary condition that does not depend on $\\varepsilon$ have a better asymptotic expansion than (\\ref{blayer}); D.~Iftimie and F.~Sueur showed in \\cite{IS} that the boundary layer $V$ has a smaller amplitude:\n\\begin{equation} u^\\varepsilon(t,y,z)=u^E(t,y,z) + \\sqrt\\varepsilon V(t,y,\\varepsilon^{-1\/2}z) . \\label{blayern} \\end{equation}\nThe problems of local existence of strong solutions on a time interval that does not depend on $\\varepsilon$, and of the corresponding inviscid limit, showing behaviour in agreement with this ansatz, have been solved by N.~Masmoudi and F.~Rousset \\cite{MR}. \nTheir approach is based on energy estimates in conormal Sobolev spaces, and the same technique has allowed them to prove similar results for the corresponding free-boundary system \\cite{MRfs}, and we will see that a similar approach for the isentropic system is valid. We refer the reader to \\cite{XX} and \\cite{BC2} for other studies in 3D.\n\nIn the incompressible case also, a variable viscosity coefficient, $\\nu=\\nu(q,|Su|^2)$, can be considered. This appears in elastohydrodynamics or the mechanics of granular materials for example. \nExistence of weak solutions for such equations on a bounded domain with the Navier boundary condition has been obtained by M.~Bul\\'i\\v{c}ek, J.~M\\'alek and K.~Rajagopal \\cite{BMR}.\n\\newline\n\nOn the 3D isentropic Navier-Stokes system with Navier boundary conditions that we are interested in, F.~Sueur recently showed the convergence of weak solutions to a strong solution of the Euler equation when the limit initial condition is smooth, and for slip coefficients that can depend on $\\varepsilon$, such as $a\/\\varepsilon^\\beta$ with $\\beta<1$ \\cite{Sf13}. \nIn the case where the slip coefficient does not depend on $\\varepsilon$, D.~Hoff obtained global solutions with intermediate regularity (more regularity than weak solutions \\textit{\\`a-la-}Lions, but not classical solutions) in 2005 \\cite{Hoff}, while Y-G.~Wang and M.~Williams justified a WKB expansion for strong solutions \\cite{WW} in 2012. \nIn particular, Wang and Williams show that solutions to the isentropic Navier-Stokes system behave similarly to their incompressible counterparts in the inviscid limit, in the sense that we have the asymptotic expansion (\\ref{blayern}). \nAs we aim to get existence of strong solutions of the Navier-Stokes equation through uniform \\textit{a priori} estimates, we will make use of conormal derivatives that we introduce below.\n\\newline\n\nTo complete the references on the inviscid limit problem for compressible fluids, we cite F.~Huang, Y.~Wang and T.~Yang (ideal gas, \\cite{HWY}), and Feireisl and Novotn\\'y (weak-strong convergence in the Navier-Stokes-Fourier setting, \\cite{FN13}), for advances on the full compressible Navier-Stokes equations, with an extra equation on the internal energy, temperature or entropy. \nWe also refer to \\cite{XY} for results on the linearised 2D system, \\cite{Rf05} and \\cite{GMWZ} for boundary layer analysis with characteristic and non-characteristic boundary conditions respectively, and \\cite{MZ}, \\cite{AB} for results on more general parabolic-hyperbolic systems.\n\n\\subsection{The conormal functional setting}\n\nIf we consider a boundary-layer expansion of the form (\\ref{blayern}), which the solutions of the equation we will study satisfy, we see that we can expect uniform control of $u^\\varepsilon$, its derivatives, but not its second derivatives: a factor $\\varepsilon^{-1\/2}$ results from the differentiation of the boundary layer $V$. \nThus, the functional setting used in this paper will be that of conormal Sobolev spaces. Introduced in the mid-60s \\cite{Hl}, these spaces have been used to work on hyperbolic systems with characteristic boundaries (see, for example, \\cite{Rj}, \\cite{Go}, \\cite{Sp95}). \nSuch spaces on a domain $\\Omega$, which has a boundary, are constructed by differentiating functions following a finite set of generators of vector fields that are tangent to the boundary of $\\Omega$. Namely, in the case of the half-space, we can choose\n$$ Z_{1,2} = \\derp{y_{1},y_{2}},~Z_{3} = \\phi(z)\\derp{z} , $$\nwith $\\phi$ a smooth, positive, bounded function of $\\mathbb{R}^{+}$ such that $\\phi(0)=0$ and $\\phi'(0)\\neq 0$ - typically, consider $\\phi(z)=\\frac{z}{1+z}$. \n\nConormal derivatives will allow us to get high-order uniform-in-$\\varepsilon$ estimates. Considering an expansion like (\\ref{blayern}), if we look at the conormal derivatives of $\\derp{z}u^\\varepsilon$, the boundary-layer term is written as\n$$ Z^3 (\\partial V(\\varepsilon^{-1\/2}z)) = \\varepsilon^{-1\/2}\\phi(z)\\partial^2 V(\\varepsilon^{-1\/2}z) , $$\nand this is of amplitude ${\\cal O}(1)$ in a neighbourhood of the boundary of size $\\sqrt\\varepsilon$ thanks to the factor $\\phi(z)$. Thus the conormal setting is the only one in which we can expect uniform bounds on a large number of derivatives.\n\nThe conormal Sobolev space on $\\Omega$, $W^{m,p}_{co}(\\Omega)$, is then naturally defined as the set of functions $f(x) \\in L^p(\\Omega)$ such that the conormal derivatives of order at most $m$ of $f$ are also in $L^p(\\Omega)$. \n\\newline\n\nAs part of their estimation process in \\cite{MR}, Masmoudi and Rousset used the incompressibility equation to express $\\derp{z}u_3$ as a combination of conormal derivatives:\n$$ \\derp{z}u_3 = -\\derp{y_1}u_1 -\\derp{y_2}u_2 = -Z_1 u_1 - Z_2 u_2. $$\nWe will be able to use a similar trick, but with the equation of conservation of mass (\\ref{NSdiv}), in which $\\derp{t}\\rho$ intervenes. Also, we will regularly use equation (\\ref{NS}) to replace terms with two normal derivatives ($\\derp{zz}u$), and there, $\\derp{t}u$ is involved. \nFor these reasons, we add $Z_0 = \\derp{t}$ for functions that depend on $(t,x)$, and introduce the conormal Sobolev spaces on $[0,T]\\times \\Omega$ in the sense of O.Gu\\`es \\cite{Go}, for a set time $T$. \nFor $\\alpha\\in\\mathbb{N}^{4}$, we write $Z^{\\alpha}=Z_{0}^{\\alpha_{0}}Z_{1}^{\\alpha_{1}}Z_{2}^{\\alpha_{2}}Z_{3}^{\\alpha_{3}}$, and $|\\alpha|=\\sum_{i=0}^{3} \\alpha_i$: the conormal Sobolev space $W^{m,p}_{co}([0,T]\\times \\Omega)$ is the set of functions $f:\n[0,T]\\times \\Omega\\rightarrow\\mathbb{R}^{d}$ such that $Z^{\\alpha}f\\in L^{p}([0,T]\\times \\Omega)$, for every $\\alpha$ with $|\\alpha|\\leq m$. We will only use $p=+\\infty$ and $p=2$, with the notation $H^m_{co}=W^{m,2}_{co}$. We therefore have\n$$ H^m_{co}([0,T]\\times\\Omega) = \\{ f(t,x)~|~ \\forall ~0\\leq k\\leq m, ~ \\derp{t}^k f \\in L^2([0,T],H^{m-k}_{co}(\\Omega))\\} . $$\nCompared to $W^{m,p}_{co}(\\Omega)$, the notation is slightly abusive, in that $W^{m,p}_{co}([0,T]\\times\\Omega)$ is not a space whose conormal derivatives are tangent to the boundary ($Z_0=\\derp{t}$ is not tangent to the boundary of $[0,T]$).\n\nIn our \\textit{a priori} estimation process, we will be interested in the following space:\n$$ X^m_T(\\Omega) = \\{ f(t,x) ~|~ \\forall ~0\\leq k\\leq m,~ \\derp{t}^k f\\in L^\\infty([0,T],H^{m-k}_{co}(\\Omega)\\}. $$\nThis is more restrictive than asking for $f\\in H^{m}_{co}([0,T]\\times\\Omega)$. For a set $t\\geq 0$, we introduce the semi-norms\n$$ \\norme{m}{f(t)}^2 = \\sum_{|\\alpha|\\leq m} \\norme{L^2(\\Omega)}{Z^\\alpha f(t)}^2 ~~\\mathrm{and}~~ \\norme{m,\\infty}{f(t)} = \\sum_{|\\alpha|\\leq m} \\norme{\\infty}{Z^\\alpha f(t)}. $$\nNote that these semi-norms coincide with the $H^m_{co}(\\Omega)$ and $W^{m,\\infty}_{co}(\\Omega)$ norms if $f$ is stationary. Based on these semi-norms, we construct two norms on $X^m_T$ and $W^{m,\\infty}_{co}([0,T]\\times\\Omega)$ respectively, which are essentially $L^\\infty$-in-time norms,\n$$ \\trinorme{m,T}{f} := \\sup_{t\\in[0,T]} \\norme{m}{f(t)} ~~\\mathrm{and}~~ \\trinorme{m,\\infty,T}{f} := \\sup_{t\\in[0,T]} \\norme{m,\\infty}{f(t)} , $$\nthe latter of which coincides with the $W^{m,\\infty}_{co}([0,T]\\times\\Omega)$ norm, and $L^2$-in-time norms\n$$ \\int_0^T \\norme{m}{f(t)}^2~dt ~~\\mathrm{and}~~ \\int_0^T \\norme{m,\\infty}{f(t)}^2~dt , $$\nthe former of which is the natural norm for $H^m_{co}([0,T]\\times\\Omega)$. We will prefer not to abbreviate these last norms, as the forms we have given will make the \\textit{a priori} estimation process clearer.\n\nWe add the following abbreviations: $\\norme{\\infty}{f(t)}:=\\norme{0,\\infty}{f(t)}$, $\\trinorme{\\infty,T}{f}:=\\trinorme{0,\\infty,T}{f}$, and we denote by $\\trinorme{\\mathrm{Lip},T}{f}$ the standard Lipschitz norm on $[0,T]\\times\\Omega$.\n\n\\subsection{Results and proof strategy}\n\nWe introduce the notation $U(t,x)=(\\rho-1,u)(t,x)$, and will consider the class of solutions satisfying the following property:\n$$ {\\cal E}_m(T,U) := \\trinorme{m,T}{U}^2 + \\trinorme{m-1,T}{\\derp{z}u_{\\tau}}^2+\\int_0^T \\norme{m-1}{\\derp{z}u_{3}(s)}^2+\\norme{m-1}{\\derp{z}\\rho(s)}^2~ds \\hspace{50pt} $$\n\\begin{equation} \\hspace{130pt} +\\trinorme{1,\\infty,T}{\\derp{z}u_{\\tau}}^2+\\int_0^T \\norme{1,\\infty}{\\derp{z}\\rho(s)}^2+\\norme{1,\\infty}{\\derp{t}\\derp{z}\\rho(s)}^2~ds <+\\infty . \\label{emtu} \\end{equation}\nIn terms of our functional setting in the previous section, if we have $U\\in X^m_T$ and $\\derp{z} U\\in X^{m-1}_T \\cap W^{2,\\infty}_{co}([0,T]\\times\\Omega)$ (with $L^\\infty$ in time norms only), then ${\\cal E}_m(T,U)$ is also finite. \n\nNote that in ${\\cal E}_m(T,U)$ we only have control of $L^2$-in-time norms on the derivatives of $u_3$ and $\\rho$. Simply using $|f(t,x)|^2 = |f(0,x)|^2 + \\int_0^t \\derp{t}(|f(s,x)|^2)~ds$, we get that\n\\begin{equation} \\trinorme{1,\\infty,T}{\\derp{z}\\rho}^2 \\leq \\norme{1,\\infty}{\\derp{z}\\rho(0)}^2 + C\\int_0^t \\norme{1,\\infty}{\\derp{z}\\rho(s)}^2+\\norme{1,\\infty}{\\derp{t}\\derp{z}\\rho(s)}^2~ds , \\label{l2timeinf} \\end{equation}\nthus the final integral in ${\\cal E}_m(T,U)$ acts to control the $W^{1,\\infty}_{co}$ norm of $\\derp{z}\\rho$. \nRequiring control of the $W^{1,\\infty}_{co}$ norm of $\\nabla U$ is typical of characteristic hyperbolic problems, see \\cite{Go}.\n\nThe force must be smooth on $\\mathbb{R}\\times \\Omega$, hence we introduce the notation:\n$$ {\\cal N}_m(T,F) = \\sup_{t \\in [-T,T]} \\norme{m}{F(t)}^2 + \\norme{m-1}{\\nabla F(t)}^2 + \\norme{2,\\infty}{\\nabla F(t)}^2 . $$\n\n\\begin{theo} \\underline{Uniform existence of solutions to the Navier-Stokes system.}\n\n\\noindent Let $m\\geq 7$, $F=(0,F)$ be such that ${\\cal N}_m(t,F)<+\\infty$ for any $t>0$, and $\\mu$ and $\\lambda$ be positive and bounded ${\\cal C}^m$ functions of $\\rho$. \nThen, for $\\varepsilon_0>0$, there exists $T^*>0$ such that, for every $0<\\varepsilon<\\varepsilon_0$, there is a unique $U^\\varepsilon$ satisfying ${\\cal E}_m(T^*,\\cdot)<+\\infty$, solution to (\\ref{NSdiv})-(\\ref{NS})-(\\ref{NPB})-(\\ref{NBC})-(\\ref{limcond}), the isentropic compressible Navier-Stokes system on $(0,T^*)\\times\\Omega$ with Navier boundary conditions. \nMoreover, there is no vacuum on this time interval: there exists $c>0$ such that $\\rho(t,x)\\geq c$ for $t\\in [0,T^*]$ and $x\\in\\Omega$. \\label{unifexist} \\end{theo}\n\n\\begin{theo} \\underline{Inviscid limit.}\n\nUnder the same conditions as above, the family $(U^\\varepsilon=(\\rho^\\varepsilon-1,u^\\varepsilon))_{0<\\varepsilon<\\varepsilon_0}$ of solutions to the Navier-Stokes system converges in $L^2([0,T^*]\\times\\Omega)$ and $L^\\infty([0,T^*]\\times\\Omega)$, towards $V=(\\rho-1,u)$, the unique solution to the isentropic compressible Euler system, (\\ref{Euler})-(\\ref{NPB})-(\\ref{limcond}), that satisfies ${\\cal E}_m(T^*,V)<+\\infty$. \\label{inviscid} \\end{theo}\n\nNote that there are no restrictions on the viscosity parameters other than positiveness and sufficient regularity (${\\cal C}^m$). It seems physically justified to ask $\\mu(\\rho)$ and $\\lambda(\\rho)$ to be increasing with the density, but, like the sign of the slip coefficient, the signs of $\\mu'$ and $\\lambda'$ do no intervene technically. \nAlso, the results are shown for a barotropic pressure law, but we can extend them to any positive, increasing ${\\cal C}^1$ pressure law.\n\nFinally, we expect that our results are also valid in any domain of $\\mathbb{R}^3$ with a ${\\cal C}^{m'}$ boundary, for $m'$ large enough, locally characterised by equations of type $z=\\psi(y)$. As shown in \\cite{MR}, the differences are technical, and extra difficulties arise only because the normal vector to the boundary is no longer constant.\n\\newline\n\nOur results are obtained by classical arguments once a uniform estimate is shown. The key bound is the following.\n\\begin{theo} \\underline{Uniform energy bound.}\n\nLet $m\\geq 7$ and $M_0>0$. We assume that ${\\cal N}_m(t,F)<+\\infty$ for every $t>0$, and that the initial value of $U$ satisfies ${\\cal N}_m(0,U) \\leq M_0$. Then, there exist $\\varepsilon_0>0$, $T^*>0$ and a positive increasing function $Q: \\mathbb{R}^{+}\\rightarrow \\mathbb{R}^{+}$, with $Q(z)\\geq z$, such that, for $\\varepsilon\\leq \\varepsilon_0$ and $0\\leq t\\leq T^*$,\n\\begin{equation} {\\cal E}_m(t,U)+\\trinorme{1,\\infty,t}{\\derp{z}u_3}^2 + \\varepsilon\\int_0^t \\norme{m}{\\nabla u(s)}^2 + \\norme{m-1}{\\nabla^2 u_\\tau(s)}^2~ds \\leq Q(2M_0) \\label{th13} \\end{equation} \\label{thEE} \\end{theo}\n\nLet us outline the proof of \\thref{thEE}. It is proved by showing that the left-hand side of (\\ref{th13}) is bounded by\n$$ Q(M_0) + (t+\\varepsilon) Q({\\cal E}_m(t,U)+{\\cal N}_m(t,F)). $$\nThe energy function we consider, ${\\cal E}_m$, contains $L^2$ in space norms with many derivatives, on the first line of (\\ref{emtu}), and conormal-Lipschitz norms on the second line. The first type of term is dealt with by performing energy estimates, in which we will have to control the commutators between the conormal derivatives and the operators that appear in the equation. \nIn particular, we will make use of the symmetrisable hyperbolic-parabolic structure of the compressible model to get the energy estimates on $U$. For the $L^\\infty$ norms, we will widely use an anisotropic Sobolev embedding theorem (\\thref{sobo}), which is the main contributor to the restriction on $m$. \nFor the terms in the second line of (\\ref{emtu}), a maximum principle will provide us with bounds on $\\trinorme{1,\\infty,t}{\\derp{z}u_\\tau}$, while the Duhamel formula for the equation satisfied by $\\derp{z}\\rho$, which is obtained by combining $\\varepsilon\\times(\\ref{NSdiv})$ with the third component of (\\ref{NS}), will give us the bounds for the $W^{1,\\infty}_{co}$ norms on $\\derp{z}\\rho$. \nA bootstrap argument closes the proof.\n\nNote that, in the context of \\tworef{unifexist}{inviscid}{Theorems}, $M_0$ can be arbitrarily small, as ${\\cal N}_m(0,U)=0$. We will be able to prove the energy estimate taking initial conditions into account; this allows one to extend our results to less regular force terms or different initial values, providing the compatibility conditions yield uniform bounds on the norms of $U$ at $t=0$.\n\\newline\n\n\\textbf{\\underline{Organisation of the paper.}} In the next section, we prove \\tworef{unifexist}{inviscid}{Theorems}, assuming \\thref{thEE}. The remaining sections will all be dedicated to proving this uniform estimate. \nStarting with some important commutator estimates in section 3, we then proceed to prove the bound \\textit{a priori}, looking at each component of $(U,\\derp{z}U)$ separately, and getting the required conormal and $L^\\infty$ bounds on each of them: $U$ in section 4, the normal derivative of $u_\\tau$ in section 5, $\\derp{z}u_3$ in section 6, and the normal derivative of the density in section 7. \nWe conclude the proof of \\thref{thEE} in section 8.\n\n\\section{Proof of \\tworef{unifexist}{inviscid}{Theorems}}\n\nIn this section, we assume the uniform estimate in \\thref{thEE}. To obtain \\thref{unifexist}, we can use a classical fixed-point iteration method to get existence of solutions for any fixed $\\varepsilon>0$ on a time interval $[0,T^\\varepsilon]$ depending on the viscosity (see \\cite{WW}, section 4.1). A bootstrap argument with the uniform bounds then yields a uniform existence time.\n\\newline\n\nWe will further detail the proof of \\thref{inviscid}. We first get local convergence by using a standard compactness argument.\n\n\\begin{propo} \\underline{Conormal compact embedding theorem.}\n\nLet $T>0$ and $(U_n)_{n\\in\\mathbb{N}}$ be a bounded sequence of $H^m_{co}([0,T]\\times\\Omega)$, such that the sequence $(\\nabla U_n)_n$ is bounded in $H^{m-1}_{co}([0,T]\\times\\Omega)$. \nThen we can extract a sub-sequence $(U_{n_j})_{j\\in\\mathbb{N}}$ such that, for every $\\alpha\\in\\mathbb{N}^4$ with $|\\alpha|\\leq m-1$, $Z^\\alpha U_{n_j}$ converges in $L^2_{loc}([0,T]\\times\\Omega)$ - we will say that $(U_n)$ is \\underline{locally compact} in $H^{m-1}_{co}([0,T]\\times\\Omega)$.\n\\label{rellich} \\end{propo}\n\nThe bound on the gradient of $U_n$ is crucial here. In other contexts in which conormal Sobolev spaces have been used, such as in \\cite{Go} and \\cite{Sp}, one normal derivation costs two conormal derivations, thus their $H^m_{co}$ spaces locally and compactly embed in $H^{m-2}_{co}$. \nBut in our energy estimates on the isentropic Navier-Stokes system, we find that one normal derivative can be controlled by the same number of conormal derivatives. The proof of \\propref{rellich} is similar to that in \\cite{Sp}.\n\\newline\n\nFor a given $\\varepsilon>0$, let $U^\\varepsilon=(\\rho^\\varepsilon-1,u^\\varepsilon)$ be the solution to (\\ref{NSdiv})-(\\ref{NS})-(\\ref{NPB})-(\\ref{NBC})-(\\ref{limcond}) with viscosity coefficient $\\varepsilon$, given by \\thref{unifexist}. \nThe energy estimate in \\thref{thEE} tells us that the family $({\\cal E}_m(T^*,U^{\\varepsilon}))_{0<\\varepsilon<\\varepsilon_{0}}$ is bounded, so we can immediately state that the sequence $(U^\\varepsilon)_{0<\\varepsilon<\\varepsilon_0}$ is locally compact in $H^{m-1}_{co}([0,T^*]\\times \\Omega)$ by \\propref{rellich}. \nAs $H^{m-1}_{co}([0,T^*]\\times \\Omega) \\hookrightarrow {\\cal C}([0,T^*],L^2(\\Omega))$, we can consider a sequence $\\varepsilon_{n}\\stackrel{n\\rightarrow+\\infty}{\\rightarrow}0$ such that $U^{\\varepsilon_{n}}$ converges locally in $H^{m-1}_{co}([0,T^*]\\times \\Omega)$ and in ${\\cal C}([0,T^*],L^2(\\Omega))$ towards a function $V=(\\rho-1,u)$, which is easily seen to be a weak solution to the compressible Euler system. \nThanks to the uniform bounds, we see that $V$ has the same regularity as $U^{\\varepsilon}$, and in particular $V$ is Lipschitz-class, which yields uniqueness of solutions for the Euler equation in the space of functions satisfying ${\\cal E}_m(T^*,V)<+\\infty$. \nThe whole sequence then converges towards $V$, strongly and locally in $H^{m-1}_{co}([0,T^*]\\times\\Omega)$, and $U^\\varepsilon(t)$ converges weakly in $L^2([0,T^*]\\times\\Omega)$ towards $V(t)$.\n\\newline\n\nWe now prove strong convergence in $L^2$. For $t\\leq T^*$, we start with the classical energy inequality for the Navier-Stokes system, as in \\cite{LM},\n\\begin{equation} \\left| E(t,U^\\varepsilon) - E(0,U^\\varepsilon) - \\int_0^t \\int_\\Omega \\rho^\\varepsilon F(s)\\cdot u^\\varepsilon (s) ~dx~ds \\right| \\leq C\\varepsilon\\int_0^t \\norme{H^1(\\Omega)}{u^\\varepsilon(s)}^2~ds , \\label{clasen} \\end{equation}\n$$ \\mathrm{where}\\hspace{10pt} E(t,U^\\varepsilon) = \\int_\\Omega \\frac{1}{2}\\rho^\\varepsilon(t)|u^\\varepsilon(t)|^2 + \\frac{k}{\\gamma(\\gamma-1)}\\int_\\Omega (\\rho^\\varepsilon)^\\gamma(t) - 1 - \\gamma(\\rho^\\varepsilon(t)-1)~dx. $$\nNote that $E(0,U^\\varepsilon) = 0$ for every $\\varepsilon$. As $\\norme{H^1([0,T^*]\\times\\Omega)}{u^\\varepsilon}$ is bounded by \\thref{thEE} and $\\rho^\\varepsilon u^\\varepsilon$ converges weakly towards $\\rho u$ in $L^2$ (as that is the case in the sense of distributions by local strong convergence), we get that, for $t\\in [0,T^*]$,\n$$ E(t,U^\\varepsilon) \\stackrel{\\varepsilon\\rightarrow 0}{\\longrightarrow} \\int_0^t \\int_\\Omega \\rho(s,x) F(s,x)\\cdot u(s,x)~dx~ds . $$\nBut $\\int_{[0,t]\\times\\Omega} \\rho F\\cdot u$ is equal to $E(t,V)$, as (\\ref{clasen}) with $\\varepsilon=0$ yields the energy equality for the compressible Euler equation. So, we have $E(t,U^\\varepsilon) \\stackrel{\\varepsilon\\rightarrow 0}{\\longrightarrow} E(t,V)$ for every $t\\leq T^*$.\n\nWe get $L^2$ convergence of the density by using the strict convexity of $z\\mapsto z^\\gamma$ for $z>0$ and $\\gamma>1$. On one hand, we have\n$$ \\rho^\\varepsilon|u^\\varepsilon|^2-\\rho|u|^2 - 2 \\sqrt{\\rho}u \\cdot (\\sqrt{\\rho^\\varepsilon}u^\\varepsilon - \\sqrt{\\rho}u) = |\\sqrt{\\rho^\\varepsilon}u^\\varepsilon - \\sqrt{\\rho}u|^2 , $$\nand on the other hand, there exists $c>0$ such that $\\rho(t,x)>c>0$, thus the Taylor expansion of $z\\mapsto z^\\gamma$ yields\n$$ (\\rho^\\varepsilon)^\\gamma - \\rho^\\gamma - \\gamma \\rho^{\\gamma-1}(\\rho^\\varepsilon-\\rho) \\geq c\\gamma(\\gamma-1) (\\rho^\\varepsilon-\\rho)^2 . $$\nAdding the two together with the coefficients that appear in $E$, and taking the integral on $\\Omega$, we get\n\\begin{equation} E(t,U^\\varepsilon)-E(t,V) - R(t,U^\\varepsilon,V) \\geq c_0 \\left(\\norme{L^2(\\Omega)}{(\\sqrt{\\rho^\\varepsilon}u^\\varepsilon - \\sqrt{\\rho}u)(t)}^2 + \\norme{L^2(\\Omega)}{(\\rho^\\varepsilon-\\rho)(t)}^2\\right) \\label{strconv} \\end{equation}\nwhere $R(t,U^\\varepsilon,V) = \\int_\\Omega \\sqrt{\\rho}u \\cdot (\\sqrt{\\rho^\\varepsilon}u^\\varepsilon - \\sqrt{\\rho}u) + \\frac{k}{\\gamma-1}(\\rho^\\varepsilon-\\rho)(\\rho^{\\gamma-1}-1)~dx$, which converges to zero. \nIndeed, the uniform boundedness of the energy means that $\\norme{L^2}{\\sqrt{\\rho^\\varepsilon(t)}u^\\varepsilon(t)}$ is bounded, so we can extract a weakly converging sub-sequence in $L^2(\\Omega)$, $\\sqrt{\\rho^{\\varepsilon_n}(t)}u^{\\varepsilon_n}(t)$, whose limit is necessarily $\\sqrt{\\rho(t)} u(t)$, thus the whole sequence converges weakly in $L^2(\\Omega)$. \nSo the first term of $R(t,U^\\varepsilon,V)$ goes to zero. Likewise, we have $\\rho^\\varepsilon(t) \\rightharpoonup \\rho(t)$ in $L^2(\\Omega)$, and $\\rho^{\\gamma-1}-1$ is seen to be in $L^2(\\Omega)$ by use of the order-one Taylor expansion of $z\\mapsto z^{\\gamma-1}$ at $z=1$, so the second term also converges to zero.\n\nMoreover, $E(t,U^\\varepsilon)-E(t,V) \\stackrel{\\varepsilon\\rightarrow 0}{\\longrightarrow} 0$, so (\\ref{strconv}) gives us the global, strong $L^2$ convergence of $\\rho^\\varepsilon$ towards $\\rho$. Now, by remarking that\n\\begin{equation} \\left|\\norme{L^2(\\Omega)}{(\\sqrt{\\rho^\\varepsilon}u^\\varepsilon)(t)}^2 - \\norme{L^2(\\Omega)}{(\\sqrt{\\rho}u^\\varepsilon)(t)}^2\\right| \\leq \\norme{L^\\infty}{u^\\varepsilon(t)}\\norme{L^2}{u^\\varepsilon(t)}\\norme{L^2}{\\rho^\\varepsilon(t)-\\rho(t)} , \\label{normu} \\end{equation}\nwe get that $\\displaystyle \\lim_{\\varepsilon\\rightarrow 0} \\norme{L^2(\\Omega)}{(\\sqrt{\\rho} u^\\varepsilon)(t)} = \\lim_{\\varepsilon\\rightarrow 0} \\norme{L^2(\\Omega)}{(\\sqrt{\\rho^\\varepsilon}u^\\varepsilon)(t)} = \\norme{L^2(\\Omega)}{\\sqrt{\\rho}(t) u(t)}$, as $u^\\varepsilon$ is uniformly bounded in $L^2$ and $L^\\infty$. \nSo $u^\\varepsilon$ converges towards $u$ in $L^2([0,T^*]\\times \\Omega,dt~\\rho dx)$, and $\\rho dx$ is an equivalent measure to the Lebesgue measure, hence we conclude that $U^\\varepsilon$ converges towards $V$ in $L^2([0,T^*]\\times\\Omega)$. \n$L^\\infty$ convergence is obtained by using the Sobolev embedding inequality and the uniform bounds. $\\square$\n\n\\section{Preliminary properties of conormal derivatives}\n\nWe begin this section by reminding the reader of some important properties of our functional setting, and that will be used throughout the \\textit{a priori} estimation process. Set $T>0$.\n\n\\begin{propo} \\underline{Generalised Sobolev-Gagliardo-Nirenberg inequality (or tame estimate).} \\cite{Go}\n\nThere exists a constant $C(T)$, which does not blow up as $T\\rightarrow 0$, such that, for $f,g\\in L^{\\infty}([0,T]\\times\\Omega)\\cap H^{m}_{co}([0,T]\\times\\Omega)$, and $\\alpha_{1},~\\alpha_{2}\\in\\mathbb{N}^{4}$ such that $|\\alpha_{1}|+|\\alpha_{2}|=m$,\n$$ \\int_0^T \\norme{0}{(Z^{\\alpha_{1}}f Z^{\\alpha_{2}}g)(s)}^{2}~ds \\leq C(T)\\left[\\trinorme{\\infty,T}{f}^{2}\\int_0^T \\norme{m}{g(s)}^{2}~ds +\\trinorme{\\infty,T}{g}^{2}\\int_0^T \\norme{m}{f(s)}^{2}~ds \\right].$$ \\label{tame} \\end{propo}\n\n\\begin{propo} \\underline{Trace inequality.}\n\nIf $f\\in L^2([0,T]\\times\\Omega)$ and $\\nabla f\\in L^2([0,T]\\times\\Omega)$, then\n$$ \\int_0^T \\int_{\\partial\\Omega} |f(t,y,0)|^{2}~dt~dy \\leq C \\int_0^T \\norme{0}{f(t)}\\norme{0}{\\derp{z}f(t)}~dt $$ \\label{trace} \\end{propo}\n\n\\begin{propo} \\underline{Anisotropic Sobolev embedding theorem.}\n\nIf $f\\in H^3_{co}([0,T]\\times\\Omega)$, $\\nabla f\\in H^2_{co}([0,T]\\times\\Omega)$, then $f\\in L^\\infty([0,T]\\times\\Omega)$ and\n$$ \\trinorme{\\infty,T}{f}^{2} \\leq C\\left(\\norme{2}{f(0)}^2+\\norme{1}{\\derp{z}f(0)}^2 + \\int_0^T \\norme{3}{f(t)}^{2}+\\norme{2}{\\derp{z}f(t)}^2 ~dt\\right) $$ \\label{sobo} \\end{propo}\n\nThis last theorem is a direct application of the $H^m_{co}(\\Omega)$ Sobolev embedding, used in \\cite{MR} and \\cite{MRfs}: for a given $t$, we have\n$$ \\norme{\\infty}{f(t)}^2 \\leq C(\\norme{H^2_{co}(\\Omega)}{f(t)}^2 + \\norme{H^1_{co}(\\Omega)}{\\nabla f(t)}^2) \\leq C(\\norme{2}{f(t)}^2+\\norme{1}{\\derp{z}f(t)}^2). $$\nWe combine this with the following property: for $f\\in H^{m+1}_{co}([0,T]\\times\\Omega)$,\n\\begin{equation} \\trinorme{m,t}{f}^2 \\leq \\norme{m}{f(0)}^2 + C \\int_0^t \\norme{m+1}{f(s)}^2~ds . \\label{l2time} \\end{equation}\nThis is shown by writing $n(t)=n(0)+\\int_0^t n'(s)~ds$ for $n(s)=\\norme{m}{f(s)}^2$; given that\n$$ n'(s) = 2\\sum_{|\\alpha|\\leq m} \\int_\\Omega (\\derp{t}Z^\\alpha f(s,x))(Z^\\alpha f(s,x))~dx, $$\nwe easily see that $n'(s)\\leq 2\\norme{m+1}{f(s)}^2$. \nMorally, the bound (\\ref{l2time}) means that we can exchange an $L^\\infty$-in-time norm for an $L^2$-in-time norm for the cost of one conormal derivative, similarly to (\\ref{l2timeinf}).\n\nWe will now show the important commutator properties we will need.\n\n\\subsection{Commuting with $\\derp{z}$}\n\nThe first technical key to the proof in the subsequent parts of the paper is how to estimate the commutators that will appear when applying $Z^{\\alpha}$ to equations (\\ref{NSdiv}), (\\ref{NS}). \nA lot of the commutators are trivial, since all $Z_{i}$'s commute with $\\derp{t}$, $\\derp{y_{1}}$ and $\\derp{y_{2}}$, but $Z_{3}$ does not commute with $\\derp{z}$. \nSpecifically, we have\n\\begin{equation} [Z_{3},\\derp{z}] = \\phi(z)\\derp{zz}-\\derp{z}(\\phi(z)\\derp{z}) = -\\phi'(z)\\derp{z} \\label{commu1} \\end{equation}\nLikewise, we can observe the commutator of $Z_3$ with $\\derp{zz}$, which will come from the $\\div\\Sigma$ term of (\\ref{NS}). This time, we have\n\\begin{equation} [Z_{3},\\derp{zz}] = -2\\phi'\\derp{zz} - \\phi''\\derp{z} . \\label{commul} \\end{equation}\n\nWhen commutating with a higher order operator, $Z_3^m$ for $m>1$, we show the following:\n\\begin{propo} (a) For $m\\geq 1$, there exist two families of bounded functions $(\\varphi_{\\beta,m})_{0\\leq \\beta <m}$ and $(\\varphi^{\\beta,m})_{0\\leq\\beta<m}$, such that\n\\begin{equation} [Z_{3}^{m},\\derp{z}] = \\sum_{\\beta=0}^{m-1} \\varphi_{\\beta,m}(z)Z_{3}^{\\beta}\\derp{z} = \\sum_{\\beta=0}^{m-1} \\varphi^{\\beta,m}(z)\\derp{z}Z_{3}^{\\beta} \\label{commum} \\end{equation}\n\n(b) For $m\\geq 1$, there exist four families of bounded functions $(\\psi_{1,\\beta,m})$, $(\\psi_{2,\\beta,m})$, $(\\psi^{1,\\beta,m})$ and $(\\psi^{2,\\beta,m})$, for $0\\leq \\beta <m$, such that\n\\begin{eqnarray*} [Z_{3}^{m},\\derp{zz}] & = & \\sum_{\\beta=0}^{m-1} \\psi_{1,\\beta,m}(z) Z_{3}^{\\beta}\\derp{z} + \\psi_{2,\\beta,m}(z)Z_{3}^{\\beta}\\derp{zz} \\\\\n & = & \\sum_{\\beta=0}^{m-1} \\psi^{1,\\beta,m}(z)\\derp{z}Z_{3}^{\\beta} + \\psi^{2,\\beta,m}(z)\\derp{zz}Z_{3}^{\\beta}. \\end{eqnarray*}\n\\label{comprop} \\end{propo}\nIn practice, we can therefore choose to place the normal derivatives as the first or last derivative to be applied in all the terms of the commutator. We can deduce from the proposition the basic estimate for the commutators with $\\derp{z}$:\n\\begin{coro} For any $f\\in H^{m}_{co}([0,T]\\times \\Omega)$ such that $\\derp{z}f\\in H^{m-1}_{co}([0,T]\\times \\Omega)$, and for any $|\\alpha|\\leq m$,\n$$ \\int_0^T \\norme{0}{[Z^\\alpha,\\derp{z}]f(t)}^2~dt \\leq C\\int_0^T \\norme{m-1}{\\derp{z}f(t)}^2~dt $$ \\label{comcoro} \\end{coro}\nWe will deal with the commutators with $\\derp{zz}$ directly in context: they often appear with a factor $\\varepsilon$, and equation (\\ref{NS}) will allow us to substitute the difficult terms.\n\\newline\n\n\\preu{Proof of \\propref{comprop}}: equation (\\ref{commu1}) shows the case $m=1$, and we continue by induction. Let us just explain the case $m=2$ to show the mechanism; the rest is left to the reader. We have\n\\begin{eqnarray*} [Z_{3}^{2},\\derp{z}] & = & Z_{3}[Z_{3},\\derp{z}]+[Z_{3},\\derp{z}]Z_{3} \\\\\n & = & -\\phi\\phi''\\derp{z}-\\phi'(Z_{3}\\derp{z}+\\derp{z}Z_{3}). \\end{eqnarray*}\nUsing (\\ref{commu1}), we can write the second part of the last line as either $\\phi'(z)(2Z_{3}\\derp{z}+\\phi'\\derp{z})$ or $\\phi'(z)(2\\derp{z}Z_{3}-\\phi'\\derp{z})$, which proves the proposition for $m=2$.\n\nThe proof of (b) is also an elementary induction. $\\square$\n\n\\subsection{Commuting with a function}\n\n\\begin{nota} For $\\alpha,~\\beta\\in\\mathbb{N}^{4}$, we write $\\beta \\leq \\alpha$ if, for every $i$, $\\beta_{i}\\leq\\alpha_{i}$. \\end{nota}\n\n\\begin{propo} Let $\\alpha\\in\\mathbb{N}^{4}$, $|\\alpha|=m>0$, be fixed, and $f\\in H^m_{co}([0,T]\\times\\Omega)\\cap L^\\infty([0,T]\\times\\Omega)$ and $g\\in L^\\infty$ such that $\\derp{t}g,~\\nabla g\\in H^{m-1}_{co}([0,T]\\times\\Omega)\\cap L^\\infty([0,T]\\times\\Omega)$. Then we have the following inequality:\n\\begin{eqnarray} \\int_{0}^{T}\\norme{0}{[Z^{\\alpha},g]f}^{2}~dt & \\leq & C\\sum_{j=0}^3\\int_{0}^{T}\\trinorme{\\infty,T}{Z_j g}^{2}\\norme{m-1}{f}^{2}+\\trinorme{\\infty,T}{f}^{2}\\norme{m-1}{Z_{j}g}^{2}~dt, \\label{funcom} \\\\ & \\leq & C\\int_{0}^{T}\\trinorme{1,\\infty,T}{g}^{2}\\norme{m-1}{f(s)}^{2}+\\trinorme{\\infty,T}{f}^{2}\\norme{m}{g(s)}^{2}~dt, \\nonumber \\end{eqnarray}\nif, moreover, $g\\in L^2([0,T]\\times\\Omega)$. \\label{funcprop} \\end{propo}\n\nThis proposition is easily proved, using the Leibniz formula and \\propref{tame}. We will require formulation (\\ref{funcom}) whenever $g\\notin L^2$ (for example when $g=\\rho$).\n\\newline\n\nWe prove one more commutator estimate, which we will need when estimating the normal derivatives of $u_\\tau$ in the conormal spaces (section 5), as directly using the above would lead to $\\norme{m-2}{\\derp{zz} u}$ appearing, which we cannot bound uniformly in $\\varepsilon$ by using the equation.\n\\begin{propo} Let $f,~g,~\\alpha$ be as in \\propref{funcprop}, with $g$ scalar such that $g|_{z=0}=0$. Then there exists $C$ which does not depend on $T$ such that\n$$ \\int_{0}^{T}\\norme{0}{[Z^{\\alpha},g\\derp{z}]f}^{2}~dt \\leq C\\int_{0}^{T}(\\trinorme{\\mathrm{Lip},T}{g}^{2}+\\trinorme{1,\\infty,T}{\\nabla g}^{2})\\norme{m}{f(s)}^{2}+\\trinorme{1,\\infty,T}{f}^{2}\\norme{m-1}{\\nabla g(s)}^{2}~dt $$ \\label{comlem} \\end{propo}\n\n\\preu{Proof:} we decompose the commutator as $[Z^{\\alpha},g\\cdot\\nabla]f = [Z^{\\alpha},g]\\derp{z} f + g [Z^{\\alpha},\\derp{z}]f$, and start by taking a closer look at the second term. \nBy \\propref{comprop}, it is equal to a sum of terms of the form $\\varphi^{\\beta}(z)g~\\derp{z}Z^{\\beta}f$, with $\\beta\\leq\\alpha$, $\\beta\\neq\\alpha$. As $g|_{z=0}=0$, we have\n\\begin{equation} |g(t,x)|\\leq \\phi(z)\\trinorme{\\infty,T}{\\derp{z}g}, \\label{u3phi} \\end{equation}\nso, as $Z_{3}=\\phi(z)\\derp{z}$,\n\\begin{equation} \\norme{0}{g[Z^{\\alpha},\\derp{z}]f} \\leq C\\trinorme{\\infty,T}{\\nabla g}\\norme{m-1}{\\phi(z)\\derp{z}f} \\leq C\\trinorme{\\mathrm{Lip},T}{g}\\norme{m}{f}. \\label{hmcom3} \\end{equation}\n\nNow we look at the first term of the decomposition. We prove that\n\\begin{equation} \\int_{0}^{T}\\norme{0}{[Z^{\\alpha},g]\\derp{z}f}^{2} \\leq C\\int_{0}^{T}(\\trinorme{\\mathrm{Lip},T}{g}^{2}+\\trinorme{1,\\infty,T}{\\nabla g}^{2})\\norme{m}{f}^{2} + \\trinorme{1,\\infty,T}{f}^{2}\\norme{m-1}{\\nabla g}^{2}~dt, \\label{comfinal} \\end{equation}\nwhich would end the proof of the proposition.\n\nWe can write $[Z^{\\alpha},g]\\derp{z}f$ as a sum, on $\\beta$ and $\\delta$, of terms of the form $\\varphi^{\\delta}(z) Z^{\\beta}g~\\derp{z}Z^{\\delta} f$, where $\\beta\\leq\\alpha$ and $|\\beta|>0$, and $\\delta\\leq(\\alpha-\\beta)$. The idea is to insert $\\frac{1}{\\phi(z)}\\times\\phi(z)$, thus we have to estimate\n$$ \\norme{0}{\\frac{1}{\\phi(z)}(Z^{\\beta}g)(Z_{3}Z^{\\delta}f)}. $$\nThis will obviously be done by using the tame estimate, but we also have to deal with the first factor. We have $\\phi^{-1}Z^{\\beta}g = Z^{\\beta}(\\phi^{-1}g) - [Z^{\\beta},\\phi^{-1}]g$, and we need to write this last term more explicitly.\n\\begin{lemma} For every $b\\in\\mathbb{N}$, $\\phi Z_{3}^{b}(\\phi^{-1})$ is smooth and bounded, then there exists a family of smooth bounded functions $(\\sigma_{p})_{0\\leq p\\leq b}$ such that for every $f\\in H^{m}_{co}([0,T]\\times\\Omega)$,\n$$ [Z^{m}_{3},\\phi^{-1}]f = \\sum_{p=0}^{m} \\sigma_{p}(z) Z^{p}_{3}(\\phi^{-1}f) $$ \\label{phicom} \\end{lemma}\n\n\\preu{Proof:} as $\\phi$ only depends on $z$, we are only interested in $[Z_3^m,\\phi^{-1}]$ for $m\\geq 1$. The Leibniz formula yields that the commutator is a linear combination of terms written as $(Z_3^b(\\phi^{-1}))(Z^{m-b}f)$, for $0< b\\leq m$. \nIn the case of $\\phi(z)=\\frac{z}{1+z}$, we notice that $Z_3^b (\\phi^{-1})$ has the same following properties as $\\phi^{-1}$: it is smooth on $]0,+\\infty[$, bounded at infinity and has the same blow-up rate at $z=0$ as $\\phi^{-1}$ (blows up like $z^{-1}$). So, for each $b$, $\\phi Z^b_3(\\phi^{-1})$ is a bounded function on $[0,+\\infty[$, and if we write\n$$ (Z^b_{3}(\\phi^{-1}))(Z^{m-b}f) = \\phi Z^b_{3}(\\phi^{-1})\\left[Z^{m-b}\\left(\\frac{1}{\\phi}f\\right) - [Z^{m-b},\\phi^{-1}]f\\right], $$\nwe have $m-b<m$, and we can reiterate the process. As $m$ is fixed, we conclude the proof with a finite number of iterations. $\\square$\n\\newline\n\nApplying the lemma, we have that, in any norm,\n\\begin{equation} \\norme{}{\\phi^{-1}Z^{\\beta}g} \\leq C\\sum_{\\beta'\\leq\\beta} \\norme{}{Z^{\\beta'}(\\phi^{-1}g)} \\label{phicom2} \\end{equation}\nWith that, the tame estimate gives us\n$$ \\int_{0}^{T}\\norme{0}{\\phi^{-1}(Z^{\\beta}g)(Z_{3}Z^{\\delta}f)}^{2}~dt \\leq C \\int_{0}^{T} \\trinorme{\\infty,T}{Z_{3}f}^{2} \\sum_{j=0}^{3} \\norme{m-2}{Z_j\\left(\\frac{g(s)}{\\phi}\\right)}^{2} $$\n$$ \\hspace{70pt} + \\left(\\trinorme{\\infty,T}{\\phi^{-1}g}^2 +\\sum_{j=0}^{3} \\trinorme{\\infty,T}{Z_{j}(\\phi^{-1}g)}^2\\right)\\norme{m-1}{Z_3 f(s)}^{2}~dt $$\nand the $\\trinorme{\\infty,T}{\\phi^{-1}g_3}$ comes from the terms in (\\ref{phicom2}) with $\\beta'=0$.\n\nIt remains to deal with the terms involving $\\phi^{-1}g$, and the key fact here is that $Z^{\\beta}g|_{z=0}=0$. We easily have by (\\ref{u3phi}),\n\\begin{equation} \\norme{\\infty}{\\phi^{-1}g} \\leq \\norme{\\mathrm{Lip}}{g} ~~\\mathrm{and}~~ \\norme{\\infty}{Z_{j}(\\phi^{-1}g)}\\leq C\\norme{1,\\infty}{\\nabla g}, \\label{hardyinf} \\end{equation}\nas $Z_{3}(\\phi^{-1}g) = \\phi^{-1}Z_{3}g+\\phi'\\phi^{-1}g$, with $\\phi'$ bounded, and likewise we can use the Hardy inequality to get\n$$ \\norme{m-2}{Z_{j}(\\phi^{-1}g)} \\leq C\\norme{m-1}{\\derp{z}g}, $$\nwhich ends the proof of (\\ref{comfinal}). $\\square$\n\n\\section[Proof of \\thref{thEE}, part I: \\textit{a priori} estimates on $U$]{Proof of \\thref{thEE}, part I \\\\ \\textit{A priori} estimates on $U$\n \\sectionmark{Proof of \\thref{thEE} part I: a priori estimates on $U$}}\n\\sectionmark{Proof of \\thref{thEE} part I: a priori estimates on $U$}\n\n\\begin{assu} Throughout the \\textit{a priori} estimation process, we will assume that there is no vacuum on the time of study: there exists $0<c_{0}<1$ such that $c_{0}\\leq\\rho(t,x)$ for $x\\in\\Omega$. \nAlso, as $u_3|_{z=0}=0$, for any $\\delta>0$, we assume that there exists $z_\\delta>0$, independent of $\\varepsilon$, such that $|u_3(t,x)|\\leq \\delta$ for $x\\in \\sdel{F} := \\mathbb{R}^2\\times [0,\\sdel{z}]$.\n\nAfter getting the estimates, we will show in section 8 that the final bounds actually imply these two properties on $[0,T^*]$, with $T^*$ such that ${\\cal E}_m(T^*,U)\\leq M$, and prove by a bootstrap argument that $T^*$ does not depend on $\\varepsilon$.\n\\label{assu1} \\end{assu}\n\n\\begin{nota} As of now, $0<c<1$ will designate a small constant, $C>1$ a large constant, and $Q(z)$ a positive increasing function of $\\mathbb{R}^{+}$, with polynomial growth and $Q(z)\\geq z$. \nAll three can change from one line to the next, and can depend on any of the system's parameters (constants $a$, $k$ and $\\gamma$, or bounds of the viscosity functions $\\mu(\\rho)$ and $\\lambda(\\rho)$ - we will often omit the dependence on $\\rho$), on the order of derivation $m$ or on $\\varepsilon_0$. \\end{nota}\n\n\\subsection{Conormal energy estimates}\n\nWe start with energy estimates on $U=(\\rho-1,u)$ and its conormal derivatives: this will estimate the $\\trinorme{m}{U}$ part of ${\\cal E}_{m}(t,U)$, and it will uncover other terms of ${\\cal E}_{m}(t,U)$ that we will need to estimate later.\n\nWe will be able to estimate $\\rho-1$ and $u$ simultaneously by taking full advantage of the symmetrisable hyperbolic structure of the order-one part of the isentropic Navier-Stokes system $(\\ref{NSdiv}),~ (\\ref{NS})$. We rewrite it as\n\\begin{equation} A_{0}(\\rho)\\derp{t}U + \\sum_{j=1}^{3} A_{j}(U)\\derp{x_{j}}U - \\varepsilon(0,\\mu(\\rho) \\Delta u + \\lambda(\\rho) \\nabla\\div u)^t = (0,\\rho F-\\varepsilon\\sigma(\\nabla U))^t , \\label{matrixform} \\end{equation}\nwhere $\\lambda=\\mu+\\lambda_0>0$, $\\sigma$ has the following expression\n$$ \\sigma(\\nabla \\rho,\\nabla u) = 2Su\\cdot\\nabla (\\mu(\\rho)) + \\div u \\nabla (\\lambda_0(\\rho)) = 2\\mu'(\\rho) Su\\cdot \\nabla \\rho + \\lambda'(\\rho)\\div u \\nabla \\rho. $$\nThe matrices $A_{j}(U)$ are: $A_{0}(\\rho)=\\mathrm{diag}(1,\\rho,\\rho,\\rho)$,\n$$ A_{1}(U)=\\left(\\begin{array}{cccc} u_{1} & \\rho & 0 & 0 \\\\ k\\rho^{\\gamma-1} & \\rho u_{1} & 0 & 0 \\\\ 0 & 0 & \\rho u_{1} & 0 \\\\ 0 & 0 & 0 & \\rho u_{1} \\end{array} \\right) ~,~ A_{2}(U)=\\left(\\begin{array}{cccc} u_{2} & 0 & \\rho & 0 \\\\ 0 & \\rho u_{2} & 0 & 0 \\\\ k\\rho^{\\gamma-1} & 0 & \\rho u_{2} & 0 \\\\ 0 & 0 & 0 & \\rho u_{2} \\end{array} \\right) $$\n$$ \\mathrm{and} ~ A_{3}(U)=\\left(\\begin{array}{cccc} u_{3} & 0 & 0 & \\rho \\\\ 0 & \\rho u_{3} & 0 & 0 \\\\ 0 & 0 & \\rho u_{3} & 0 \\\\ k\\rho^{\\gamma-1} & 0 & 0 & \\rho u_{3} \\end{array} \\right). $$\nThis system is symmetrisable: multiplying these matrices on the left by the positive diagonal matrix $D(\\rho)=\\mathrm{diag}(k\\rho^{\\gamma-2},1,1,1)$, we get:\n\\begin{equation} DA_{0}\\derp{t}U + \\sum_{j=1}^{3} DA_{j}\\derp{x_{j}}U -\\varepsilon(0,\\mu\\Delta u +\\lambda\\nabla\\div u)^t = (0,\\rho F-\\varepsilon\\sigma(\\nabla U))^t \\label{sym} \\end{equation}\n\n$DA_{0}(\\rho)$ is symmetric, so\n$$ \\frac{1}{2} \\frac{d}{dt}\\left(\\int_{\\Omega} DA_{0}U\\cdot U\\right) = \\int_{\\Omega} DA_{0}\\derp{t}U\\cdot U + \\frac{1}{2}\\int_{\\Omega} \\derp{t}(DA_{0})U\\cdot U. $$\nIntegrating this in time between $0$ and $t$, and since $\\rho$ is uniformly bounded from below by $c_{0}$ and from above by $c_{1}$, there exist $0<c<C$ such that\n\\begin{equation} c \\norme{0}{U(t)}^{2} \\leq C\\norme{0}{U(0)}^{2}+C\\trinorme{\\infty,t}{\\derp{t}DA_0}\\int_{0}^{t}\\norme{0}{U(s)}^{2} + \\int_{0}^{t}\\int_{\\Omega} DA_{0}\\derp{t}U(s)\\cdot U(s)~ds \\label{l2ee1} \\end{equation}\n\nWe replace $DA_{0}\\derp{t}U$ by its expression in (\\ref{sym}), and we use integration by parts on the integrals with order-one derivatives of $U$:\n\\begin{equation} \\int_{\\Omega} DA_{j}\\derp{x_{j}}U \\cdot U = -\\frac{1}{2} \\int_{\\Omega} \\derp{x_{j}}(DA_{j}) U\\cdot U . \\label{l2ee2} \\end{equation}\nIndeed, $DA_{j}(U)$ is symmetric, so $DA_{j}U\\cdot \\derp{x_{j}}U = U\\cdot DA_{j}\\derp{x_{j}}U$, and we notice that, for each $j$, $DA_{j}U\\cdot U = (3\\rho P'(\\rho)+\\rho|u|^2)u_j$, which means that there is no boundary term (at $z=0$ for $j=3$) when we integrate by parts, which gives us\n\\begin{eqnarray} c\\norme{0}{U(t)}^2 & \\leq & C\\norme{0}{U(0)}^2 + C\\left(\\sum_{j=0}^3 \\trinorme{\\mathrm{Lip},t}{DA_j}\\right)\\int_0^t \\norme{0}{U(s)}^2 ~ds \\nonumber \\\\\n & & \\hspace{10pt} + \\varepsilon\\int_0^t \\int_\\Omega \\mu\\Delta u\\cdot u + \\lambda \\nabla \\div u \\cdot u + \\sigma\\cdot u~ds + \\int_0^t \\int_\\Omega \\rho F\\cdot u~ds \\nonumber \\\\\n & \\leq & C\\norme{0}{U(0)}^2 + C(1+\\trinorme{\\mathrm{Lip},t}{U}^2)\\int_0^t \\norme{0}{U(s)}^2 ~ ds \\nonumber \\\\\n & & \\hspace{10pt} + \\varepsilon \\int_0^t \\int_\\Omega \\mu\\Delta u \\cdot u + \\lambda \\nabla\\div u\\cdot u + \\sigma\\cdot u ~ds + \\int_0^t \\int_\\Omega \\rho F\\cdot u~ds \\label{l2ee2b} \\end{eqnarray}\n\nWe use integration by parts on the order-two derivatives and the Navier boundary condition (\\ref{NBC}) to deal with the boundary term:\n\\begin{eqnarray} \\int_\\Omega \\mu\\Delta u\\cdot u + \\lambda \\nabla\\div u\\cdot u & = & -\\int_\\Omega (\\mu|\\nabla u|^2 + \\lambda|\\div u|^2) -\\int_{z=0} 2a |u_\\tau|^2 \\hspace{50pt} \\nonumber \\\\\n & & \\hspace{20pt} - \\int_\\Omega \\left(\\nabla (\\mu(\\rho)) \\cdot \\nabla u\\cdot u + \\div u~ \\nabla (\\lambda(\\rho))\\cdot u\\right) , \\label{l2ee3} \\end{eqnarray}\nwith the notation $\\int_{z=0}f = \\int_{\\mathbb{R}^{2}} f(y,0)~dy$. The first term of (\\ref{l2ee3}) is moved the left-hand side of (\\ref{l2ee2b}), as is the second if $a>0$. \nWhen $a<0$, we can use the trace theorem, \\propref{trace}, and absorb the norm of $\\nabla u$ in the left-hand side by using Young's inequality, $\\nu\\zeta \\leq \\frac{\\eta}{2} \\nu^{2} + \\frac{1}{2\\eta} \\zeta^{2}$ for any $(\\nu,\\zeta)\\in\\mathbb{R}^2$ and $\\eta>0$, with an adequate parameter $\\eta$ (same as in \\cite{Pm11}). \nYoung's inequality is also used on the term containing the derivatives of $\\mu$ and $\\lambda$, which turns (\\ref{l2ee3}) into\n\\begin{equation} \\int_{\\Omega} (\\mu\\Delta u\\cdot u+\\lambda\\nabla \\div u\\cdot u) \\leq \\frac{c(1+\\trinorme{\\mathrm{Lip},t}{\\rho})}{\\eta}\\norme{0}{u}^2 - (c-(|a|+1)\\eta) \\norme{0}{\\nabla u}^2 , \\label{l2ee3b} \\end{equation}\nso we choose $\\eta$ so that $c-(|a|+1)\\eta = c\/2$, which allows us to move the $\\norme{0}{\\nabla u}^2$ term to the left-hand side and absorb it with the first term of (\\ref{l2ee3}) that we moved there earlier. The remainder $\\int_\\Omega \\sigma\\cdot u$ is bounded the same way.\n\nCombining (\\ref{l2ee2b}) and (\\ref{l2ee3b}), and the assumption that $\\rho$ is uniformly bounded, there exist $0<c<C$ such that\n$$ c\\left[\\norme{0}{U(t)}^{2} + \\varepsilon\\int_{0}^{t}\\norme{0}{\\nabla u}^{2} \\right] \\leq C\\left(\\norme{0}{U(0)}^{2}+(1+\\trinorme{\\mathrm{Lip},t}{U}^2) \\int_{0}^{t}\\norme{0}{U(s)}^{2} + \\norme{0}{F(s)}^2~ds\\right), $$\nas $\\norme{\\infty}{\\nabla(DA_{j})}\\leq C\\norme{\\infty}{U}\\norme{\\infty}{\\nabla U}$.\n\\newline\n\nWe have shall now show the following, higher order estimate.\n\n\\begin{esti} For every $m\\geq 0$,\n$$ c\\left[\\norme{m}{U(t)}^{2} + \\varepsilon\\int_{0}^{t}\\norme{m}{\\nabla u(s)}^{2} ~ds\\right] \\hspace{250pt} $$\n$$ \\hspace{10pt} \\leq C\\left[\\norme{m}{U(0)}^{2}+(1+\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{\\infty}{F}^2)\\int_{0}^{t} \\norme{m}{U}^{2} + \\norme{m-1}{\\nabla U}^{2} + \\norme{m}{F}^2 ~ds\\right] $$\n(the negative index that appears when $m=0$ is ignored). \\label{hmee} \\end{esti}\n\n\\preu{Proof:} higher-order estimates work exactly the same way as above, only we will have to estimate the commutators between $Z^{\\alpha}$, with $|\\alpha|\\leq m$, and the operators in the equation. \nWe apply $Z^{\\alpha}$ to equation (\\ref{matrixform}), and isolate the highest-order terms as follows:\n\\begin{equation} A_0 \\derp{t}Z^{\\alpha}U + \\sum_{j=1}^{3} A_{j}\\derp{x_{j}}(Z^{\\alpha}U) - \\varepsilon(0,\\div Z^\\alpha(\\mu\\nabla u)+\\lambda\\nabla\\div (Z^{\\alpha}u))^t = (0,Z^\\alpha (\\rho F+\\varepsilon\\tilde{\\sigma}))^t-{\\cal C}^{\\alpha}. \\label{matrixformalpha} \\end{equation}\n${\\cal C}^{\\alpha}$ contains the commutators:\n$$ {\\cal C}^{\\alpha} = [Z^\\alpha,A_0\\derp{t}]U + \\sum_{j=1}^{3} [Z^{\\alpha},A_{j}\\derp{x_{j}}]U -\\varepsilon(0, [Z^{\\alpha},\\div] (\\mu \\nabla u)+[Z^{\\alpha},\\lambda(\\rho)\\nabla \\div]u)^t. $$\nNotice the peculiar formulation for the laplacian term. \nThis allows us to avoid difficult commutator terms on the boundary when $\\mu$ is not constant; indeed, the Navier boundary condition for $Z^\\alpha u$ is $Z^\\alpha(\\mu\\derp{z}u_\\tau) = 2a Z^\\alpha u_\\tau$, thus, when we multiply equation (\\ref{matrixformalpha}) by $Z^\\alpha u$ and integrate by parts, we can use this boundary condition immediately. \nThe remainder of the order-two part of the equation $\\tilde{\\sigma}$ is the equivalent of $\\sigma$ for this formulation.\n\nWe then multiply by the matrix $D$, which is uniformly bounded in $L^\\infty$, and can repeat the above to obtain:\n$$ c\\left[\\norme{0}{Z^{\\alpha}U(t)}^{2} + \\varepsilon\\int_{0}^{t}\\left(\\norme{0}{Z^\\alpha \\nabla u}^{2} + \\int_{z=0}|Z^{\\alpha}u|^{2}\\right)\\right] \\leq C\\norme{0}{Z^{\\alpha}U(0)}^{2} \\hspace{100pt} $$\n$$ \\hspace{80pt} +C\\int_0^t (1+\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{\\infty,t}{F})\\norme{m}{U}^{2} + \\varepsilon \\norme{m-1}{\\nabla U}^2 + \\norme{m}{F}^2 ~ds $$\n\\begin{equation} \\hspace{120pt} + C\\int_0^t\\int_{\\Omega} |(\\varepsilon Z^\\alpha \\sigma+{\\cal C}^{\\alpha})\\cdot Z^{\\alpha}U| + |{\\cal I}^\\alpha_\\nabla| ~ds. \\label{hmee1} \\end{equation}\nThe term ${\\cal I}^\\alpha_\\nabla$ contains extra commutators arising from the following integration by parts:\n\\begin{eqnarray*} -\\int_\\Omega \\div Z^\\alpha(\\mu\\nabla u)\\cdot Z^\\alpha u & = & \\int_{\\partial\\Omega} Z^\\alpha(\\mu\\derp{z}u)\\cdot Z^\\alpha u + \\int_\\Omega Z^\\alpha(\\mu \\nabla u):\\nabla Z^\\alpha u \\\\\n & = & \\int_{\\partial\\Omega} 2a |Z^\\alpha u_\\tau|^2 + \\int_\\Omega \\mu|Z^\\alpha \\nabla u|^2 \\\\\n & & \\hspace{40pt} +\\int_\\Omega ([Z^\\alpha,\\mu]\\nabla u) : Z^\\alpha \\nabla u - Z^\\alpha(\\mu\\nabla u):([Z^\\alpha,\\nabla] u), \\end{eqnarray*}\nso ${\\cal I}^\\alpha_\\nabla$ is the final integral multiplied by $\\varepsilon$. Here, we have used the contracted matrix product: $A:B = \\sum_{i,j} a_{i,j} b_{i,j}$.\n\nTo deal with the commutator terms, we will use the tools shown in section 3. Let us first look at ${\\cal I}^\\alpha_\\nabla$. In the first term, we have\n$$ \\varepsilon \\left| \\int_\\Omega ([Z^\\alpha,\\mu]\\nabla u) : Z^\\alpha \\nabla u \\right| \\leq \\norme{0}{[Z^\\alpha,\\mu]\\nabla u} \\times \\varepsilon\\norme{0}{Z^\\alpha \\nabla u}. $$\nBy estimate (\\ref{funcom}) in \\propref{funcprop}, the integral in time of the square of the first norm is easily bounded, whereas the second can be moved to the left-hand side (absorbed) by using Young's inequality with an adequate parameter $\\eta$ to have\n\\begin{equation} \\left|\\int_0^t \\varepsilon \\norme{m}{\\nabla u} \\norme{0}{[Z^\\alpha,\\mu]\\nabla u}\\right | \\leq C\\varepsilon (1+\\trinorme{\\mathrm{Lip},t}{U}^2)\\int_0^t (\\norme{m}{\\rho-1}^2+\\norme{m-1}{\\nabla u}^{2}) + \\frac{c}{4}\\varepsilon\\int_0^t \\norme{0}{Z^\\alpha \\nabla u}^{2} . \\label{hmeeb} \\end{equation}\nThe second term of ${\\cal I}^\\alpha_\\nabla$ is dealt with in exactly the same fashion, with \\cororef{comcoro} controlling the side of the product containing the commutator, and Young's inequality allowing to absorb the other.\n\\newline\n\nNow to the commutators in ${\\cal C}^\\alpha$ relative to the order-one part of the equation. We notice that, for $j\\in\\{0,1,2,3\\}$,\n$$ [Z^{\\alpha},A_{j}\\derp{x_j}]U = [Z^{\\alpha},A_{j}]\\derp{x_j} U+A_{j}[Z^{\\alpha},\\derp{x_j}]U, $$\nwhere $x_0$ is understood to be $t$. The first term is estimated using inequality (\\ref{funcom}) of \\propref{funcprop} (we cannot write $\\norme{m}{A_{j}}$ because $\\rho \\notin L^{2}$), and the second, which is either 0, if $j\\neq 3$, or $A_{3}[Z^{\\alpha},\\derp{z}]U$, is estimated using \\cororef{comcoro}, so estimating this commutator yields\n\\begin{equation} \\int_{\\Omega} |A_{3}[Z^{\\alpha},\\derp{z}]U\\cdot Z^{\\alpha}U | \\leq C(1+\\trinorme{\\mathrm{Lip},t}{U}^{2})\\norme{m-1}{\\nabla U}^{2}+\\norme{m}{U}^{2}. \\label{hmee2} \\end{equation}\n\nThe commutators on the order-two part of the equation can be split into terms of two types as follows: commutators on the viscosity parameters (functions of $\\rho$) and commutators on the differential operators, for instance\n\\begin{equation} [Z^\\alpha,\\lambda\\nabla\\div] = [Z^\\alpha,\\lambda]\\nabla \\div + \\lambda[Z^\\alpha,\\nabla\\div] . \\label{lapcomsplit} \\end{equation}\n\nWe begin with the first term. Notice that we can write this commutator as a sum of the following type of integral,\n$$ I_{i,j} = \\int_0^t \\int_\\Omega \\varepsilon [Z^\\alpha,\\lambda]\\derp{x_i,x_j}u \\cdot Z^\\alpha u, $$\nwith $i,~j \\in \\{1,2,3\\}$. Using \\tworef{funcprop}{comprop}{Propositions}, we write\n\\begin{equation} I_{i,j} = \\varepsilon \\sum_{|\\beta|+|\\delta|\\leq m, |\\delta|< m} \\int_0^t \\int_\\Omega \\psi_{\\beta,\\delta}(z) Z^\\beta(\\lambda(\\rho))  ~ \\derp{x_i}Z^\\delta\\derp{x_j}u\\cdot Z^\\alpha u ~ ds , \\label{801} \\end{equation}\nwhere the $\\psi_{\\beta,\\delta}$ are ${\\cal C}^\\infty$ bounded functions of $z$. If $|\\beta|< m$, we can integrate by parts on the $x_i$ variable:\n\\begin{equation} I_{i,j} \\leq \\sum_{|\\beta|+|\\delta|\\leq m} c_{\\beta,\\delta} \\varepsilon \\int_0^t \\int_\\Omega |\\derp{x_i}Z^\\beta(\\lambda(\\rho))~Z^\\delta\\derp{x_j}u\\cdot Z^\\alpha u + Z^\\beta(\\lambda(\\rho))~Z^\\delta\\derp{x_j}u\\cdot \\derp{x_i}Z^\\alpha u|~ds . \\label{802} \\end{equation}\nWe use Young's inequality, then we use the tame estimate on the left of the scalar products to get\n$$ I_{i,j} \\leq \\frac{c\\varepsilon}{2}\\int_0^t \\norme{m}{\\nabla u}^2~ds + \\varepsilon Q(\\trinorme{\\mathrm{Lip},t}{\\rho}^2) \\int_0^t \\norme{m-1}{\\nabla u}^2 + \\norme{m}{u}^2~ds \\hspace{50pt} $$\n$$ \\hspace{100pt} + \\varepsilon\\trinorme{\\infty,t}{\\nabla u}^2 Q\\left(\\int_0^t \\norme{m-1}{\\nabla \\rho}^2~ds\\right) . $$\nThis allows us to absorb the first term. Note that we have used the tame estimate on the pair $(Z \\rho,\\derp{x_j}u)$ rather than $(\\rho,\\derp{x_j}u)$ on the second term of (\\ref{802}) to get $H^{m-1}_{co}$ norms of $\\nabla u$ with a factor $\\varepsilon$, which can be controlled by induction on $m$.\n\nThere is a slight difference when $\\beta=\\alpha$ in (\\ref{801}). In this case, $\\delta=0$, but after integrating by parts, we would need to control $\\norme{m}{\\nabla \\rho}$. \nSo we cannot integrate by parts and must estimate $J_{i,j}:=\\varepsilon\\int_0^t \\int_\\Omega Z^\\alpha(\\lambda(\\rho)) ~\\derp{x_i,x_j}u\\cdot u$ directly. \nThis is not a problem if $(i,j)\\neq (3,3)$: we bound $J_{i,j}$ by\n$$ J_{i,j}\\leq \\varepsilon \\int_0^t \\trinorme{\\infty}{u}^2 \\norme{1}{\\nabla u}^2 Q\\left(\\norme{m}{\\rho-1}^2\\right)~ds . $$\nAt this stage, if $m=1$, we use Young's inequality to absorb $\\varepsilon\\int_0^t \\norme{1}{\\nabla u}^2$, and if $m\\geq 2$, we can use \\propref{hmee} with $m=1$. When $(i,j)=(3,3)$, we instead replace $\\varepsilon\\derp{zz}u$ by using the equation. We have\n$$ \\varepsilon(\\mu \\derp{zz}u_\\tau, (\\mu+\\lambda)\\derp{zz}u_3) = \\rho\\derp{t}u + (\\rho u\\cdot\\nabla u) + \\nabla P - \\rho F -\\varepsilon v_2, $$\nwhere order-two derivatives appear in $v_2$. The terms that appear are therefore either controlled in $L^\\infty$ by $\\norme{\\mathrm{Lip}}{U}$, or, in the case of $v_2$, dealt with using Young's inequality as above. \nThe term $\\int_\\Omega \\varepsilon Z^\\alpha\\sigma\\cdot Z^\\alpha u$ in (\\ref{hmee1}) is dealt with the same techniques as these commutators (the terms it contains ressemble those in (\\ref{802})).\n\\newline\n\nFor the remaining commutator in (\\ref{lapcomsplit}) (the estimation of the last term $[Z^\\alpha,\\div] (\\mu\\nabla u)$ follows the same lines), we assume that $\\alpha_3>0$, as these commutators are zero otherwise. \nThe terms of the commutator $[Z^{\\alpha},\\nabla\\div]u$ of the form $[Z^{\\alpha},\\nabla_\\tau\\derp{z}]u_{3}$ (the others are either trivial or contain $\\derp{zz}$), can then be estimated with \\cororef{comcoro}:\n$$ \\int_{\\Omega} \\varepsilon\\lambda|[Z^{\\alpha},\\derp{z}]\\nabla_\\tau u_{3}\\cdot Z^{\\alpha}u_\\tau| \\leq C\\varepsilon\\norme{m}{\\derp{z}u}\\norme{m}{u} \\leq C\\varepsilon\\norme{m}{u}^{2}+\\frac{c}{4}\\varepsilon\\norme{m}{\\derp{z}u}^{2}, $$\nagain using Young's inequality to allow us to absorb $\\varepsilon\\int_0^t \\norme{m}{\\derp{z}u}^{2}$. On to the remaining term $\\varepsilon[Z^{\\alpha},\\derp{zz}]u$. By \\propref{comprop}, there exist two families of functions, $(\\varphi_{\\beta})_{\\beta\\leq\\alpha}$ and $(\\psi^\\beta)_{\\beta\\leq\\alpha}$, such that\n$$ [Z^{\\alpha},\\derp{zz}]u = \\sum_{\\beta\\leq\\alpha,~\\beta\\neq\\alpha} \\varphi_\\beta(z) Z^{\\beta}\\derp{z}u+\\psi^\\beta(z) \\derp{zz}Z^{\\beta}u $$\nThe first part of the right-hand side is simply bounded by $\\norme{m-1}{\\derp{z}u}$, so it only remains to control $J:=\\varepsilon\\int_{\\Omega} |\\derp{zz}Z^{\\beta}u\\cdot Z^{\\alpha}u|$. As $\\alpha_{3}>0$, we have $Z^{\\alpha}u=0$ on the boundary, so integrating by parts and again using Young's inequality provides us with\n$$ J = \\varepsilon\\int_{\\Omega} |\\derp{z}Z^{\\beta}u\\cdot\\derp{z}Z^{\\alpha}u| \\leq C\\varepsilon\\norme{m-1}{\\derp{z}u}\\norme{m}{\\derp{z}u} \\leq C\\varepsilon\\norme{m-1}{\\derp{z}u}^{2}+\\frac{c}{4}\\varepsilon\\norme{m}{\\derp{z}u}^{2}. $$\n\nCombining this last inequality with (\\ref{hmee1}), (\\ref{hmeeb}) and (\\ref{hmee2}), then summing for all $|\\alpha|\\leq m$, we get the result. $\\square$\n\n\\subsection{$L^\\infty$ estimates}\n\nWith conormal energy estimates, we obtain inequalities for any order of derivation $m$, and these inequalities contain $L^\\infty$ or Lipschitz norms of $U$, and, as we shall see soon, $W^{1,\\infty}_{co}$ norms of $\\nabla U$. \nThese need to be controlled, and the goal of this control is to get the conormal energy estimates to be closed for $m$ large enough, by using the Sobolev embedding inequality in \\propref{sobo}. \nPutting the $L^\\infty$ norm of the normal derivative $\\derp{z}U$ to one side for now (this will be dealt with in the next sections), we can apply \\propref{sobo} to $\\trinorme{1,\\infty,t}{U}$, and (\\ref{l2time}) again on the terms involving the derivatives of $\\rho$ and $u_3$.\n\n\\begin{esti} We have the following bound for $\\trinorme{1,\\infty,t}{U}$:\n\\begin{eqnarray*} \\trinorme{1,\\infty,t}{U}^2 & \\leq & C(\\norme{4}{U(0)}^2+\\norme{4}{\\derp{z}U(0)}^2) \\\\\n & & \\hspace{15pt} + Ct\\left(\\trinorme{5,t}{U}^2+\\trinorme{4,t}{\\derp{z}u_\\tau}^2+\\int_0^t \\norme{5}{\\derp{z}u_3}^2+\\norme{5}{\\derp{z}\\rho}^2~ds\\right) \\end{eqnarray*} \\label{linfu} \\end{esti}\n\nWe can inject this into \\propref{hmee} and partially close the energy estimate for $m-1\\geq 5$. We widely use the Young inequality to separate product terms, and we get the following.\n\n\\begin{esti} For every $m\\geq 6$, there exists a constant $C>1$ and a positive increasing function $Q:\\mathbb{R}^{+}\\rightarrow \\mathbb{R}^{+}$ such that\n\\begin{eqnarray*} \\norme{m}{U(t)}^2  & \\leq & C\\norme{m}{U(0)}^2 + t Q(\\trinorme{m,t}{U}^2+\\trinorme{m-1,t}{\\derp{z}u_\\tau}^2+\\trinorme{m,t}{F}^2+\\trinorme{m-1,t}{\\nabla F}^2) \\\\\n & & \\hspace{15pt} +C \\trinorme{\\infty,t}{\\derp{z}U}^2\\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2 + \\norme{m-1}{\\derp{z}u_3(s)}^2 ~ds. \\end{eqnarray*}\nThus, under the conditions of \\assuref{assu1}, we have, for $t\\leq T^*$,\n$$ \\trinorme{m,t}{U}^2 + \\varepsilon \\int_0^t \\norme{m}{\\nabla u}^2 \\leq M_0 + Q(M+M_F)\\left(t + \\trinorme{\\infty,t}{\\derp{z}U}^2\\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2+\\norme{m-1}{\\derp{z}u_3(s)}^2~ds\\right). $$ \\label{hmeeinf} \\end{esti}\n\n\\section[Proof of \\thref{thEE}, part II: \\textit{a priori} estimates on $\\derp{z}u_\\tau$]{Proof of \\thref{thEE}, part II \\\\ \\textit{A priori} estimates on $\\derp{z}u_\\tau$\n \\sectionmark{Proof of \\thref{thEE} part II: a priori estimates on $\\derp{z}u_\\tau$}}\n\\sectionmark{Proof of \\thref{thEE} part II: a priori estimates on $\\derp{z}u_\\tau$}\n\n\\subsection{Conormal energy estimates}\n\nIn this section, we get estimates on the tangential components of $\\derp{z}u$. We will perform conormal energy estimates on the first two components of the equation of the vorticity\n$$ \\omega=\\overrightarrow{\\mathrm{rot}~} u = \\left( \\begin{array}{c} \\derp{y_{2}}u_{3}-\\derp{z}u_{2} \\\\ \\derp{z}u_{1}-\\derp{y_{1}}u_{3} \\\\ \\derp{y_{1}}u_{2}-\\derp{y_{2}}u_{1} \\end{array} \\right). $$\nBy applying derivations to equation (\\ref{NS}), we get that $w=\\omega_\\tau$ solves the following equation:\n\\begin{equation} \\rho\\derp{t}w + \\rho u\\cdot\\nabla w - \\varepsilon\\mu\\Delta w = {\\cal M} , \\label{roteqn} \\end{equation}\n$$ \\mathrm{where} \\hspace{20pt} {\\cal M}=-\\rho\\omega\\cdot\\nabla u + \\rho(\\div u)\\omega + \\overrightarrow{\\mathrm{rot}~} (\\rho F) + {\\cal M}^I + \\varepsilon {\\cal M}^{II}. $$\nThe remainder ${\\cal M}^I$ is a sum of terms written as $\\derp{x_i}\\rho(\\derp{t}u_j+u\\cdot\\nabla u_j)e_j$, and ${\\cal M}^{II}$ is a sum of terms written as $\\kappa'(\\rho) \\derp{x_j}\\rho (\\derp{x_k,x_l}^2 u_j)e_j$, with $i,j,k,l\\in\\{1,2,3\\}$, $i\\neq j$, $(e_1,e_2,e_3)$ is the canonical basis of $\\mathbb{R}^3$, and $\\kappa$ is either $\\lambda$ or $\\mu$. \nThe terms in ${\\cal M}^{II}$ come from the derivation of the laplacian terms, but also from $\\overrightarrow{\\mathrm{rot}~} \\sigma$, in which there are no terms with two derivatives on $\\rho$ thanks to $\\overrightarrow{\\mathrm{rot}~}\\nabla=0$. \nBut the boundary condition on $\\omega_\\tau$, according to (\\ref{NBC}) and the non-penetration condition $u_{3}|_{z=0}=0$, is\n$$ \\mu(\\rho)\\omega_\\tau|_{z=0} = 2a u_\\tau^{\\bot}|_{z=0}, $$\nwhere for $v=(v_{1},v_{2})\\in\\mathbb{R}^{2}$, $v^{\\bot} = (-v_{2},v_{1})$, which makes integrations by parts difficult. So we introduce a modified vorticity:\n$$ W = \\omega_\\tau - \\frac{2a}{\\mu(\\rho)} u_\\tau^{\\bot} $$\nBy the tame estimate, we can harmlessly identify conormal Sobolev norms of $W$ with those of $\\omega_\\tau$ and $\\derp{z}u_\\tau$. The modified vorticity satisfies $W=0$ on the boundary, and solves the equation\n\\begin{equation} \\rho\\derp{t}W + \\rho u\\cdot\\nabla W - \\varepsilon\\mu\\Delta W = H, \\label{vorteqn} \\end{equation}\n$$ \\mathrm{with} \\hspace{10pt} H=2a[(\\rho\\derp{t}+\\rho u\\cdot\\nabla - \\varepsilon\\mu\\Delta),\\mu^{-1}] u_\\tau^\\bot-2a\\varepsilon\\mu^{-1}\\lambda\\nabla^{\\bot}_\\tau\\div u \\hspace{70pt}  $$\n$$ \\hspace{70pt} + 2ak\\mu^{-1}\\rho^{\\gamma-1}\\nabla^{\\bot}_\\tau\\rho -2a\\mu^{-1}\\rho F_\\tau^\\bot + {\\cal M}_\\tau . $$\nWe now prove the following.\n\n\\begin{esti} The modified vorticity $W$ satisfies the following conormal energy estimate for every $m\\geq 1$:\n$$ c\\left[\\norme{m-1}{W(t)}^{2} + \\varepsilon\\int_{0}^{t} \\norme{m-1}{\\nabla W(s)}^{2}~ds\\right] \\leq C\\left[\\norme{m-1}{W(0)}^{2} + \\norme{m}{U(0)}^{2}\\right] \\hspace{110pt} $$\n\\begin{equation} \\hspace{20pt} + Q(\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{1,\\infty,t}{\\nabla U}^2+\\trinorme{\\infty,t}{\\nabla F}^2) \\int_0^t (\\norme{m}{U}^{2} + \\norme{m-1}{\\derp{z}U}^{2} + \\norme{m}{F}^2 + \\norme{m-1}{\\nabla F_\\tau}^2)~ds, \\label{hmvfinal} \\end{equation}\nfor some positive increasing function $Q(z)$. \\label{hmvort} \\end{esti}\n\n\\preu{Proof:} we repeat the reasoning of section 4.1 on the above equation. We start by taking the $L^{2}$ scalar product of (\\ref{vorteqn}) with $W$ and integrate by parts to get\n$$ c\\norme{0}{W(t)}^{2} + \\varepsilon\\mu\\int_{0}^{t}\\norme{0}{\\nabla W(s)}^{2}~ds \\leq C\\norme{0}{W(0)}^{2} \\hspace{125pt} $$\n\\begin{equation} \\hspace{80pt} +C\\int_{0}^{t} \\left[\\trinorme{\\infty,t}{\\div(\\rho u)}\\norme{0}{W(s)}^{2} + \\int_\\Omega H(s)\\cdot W(s) \\right]~ ds. \\label{l2vort} \\end{equation}\nWe notice that, thanks to the compressibility equation (\\ref{NSdiv}), $\\trinorme{\\infty,t}{\\div(\\rho u)}$ can be replaced by $\\trinorme{\\infty,t}{\\derp{t}\\rho}\\leq \\trinorme{1,\\infty,t}{\\rho}$. It remains to estimate $\\int_\\Omega H\\cdot W$. \nThe only terms we need this scalar product form for are the order-two terms in ${\\cal M}^{II}$, and $\\varepsilon \\int_\\Omega \\mu^{-1} (\\Delta \\mu) u_\\tau\\cdot W$, which comes from the commutator $\\varepsilon \\mu [\\Delta,\\mu^{-1}]u_\\tau$. After integrating $\\int_\\Omega \\mu^{-1} (\\Delta \\mu) u_\\tau\\cdot W$ by parts, we can simply bound $\\mu$ and $\\nabla \\mu$ in $L^\\infty$ and use Young's inequality on the terms involving $u$ and $W$. \nThe term $\\int_\\Omega {\\cal M}^{II} \\cdot W$ is bounded by integration by parts on the variable $x_k$ with $k\\neq 3$ and the use of Young's inequality with an adequate to absorb $\\norme{0}{\\nabla W}$ whenever it appears, while the terms containing $\\derp{x_k,x_i}^2\\rho$ are controlled by bounding the order-two part on $\\rho$ in $L^\\infty$ (thus getting $\\\nnorme{1,\\infty}{\\nabla \\rho}$). When $(k,l)=(3,3)$, we use (\\ref{NS}) to replace $\\varepsilon\\derp{zz}u$. For example,\n$$ \\varepsilon\\mu\\derp{zz}u_1 = \\rho\\derp{t}u_1 + (\\rho u \\cdot\\nabla) u_1 + \\derp{y_1}P(\\rho) - \\rho F_1 - \\varepsilon [\\mu (\\derp{y_1 y_1}+\\derp{y_2 y_2})u_1 + \\lambda \\derp{y_1}\\div u] , $$\nand the only difficulty is to control $\\varepsilon\\int_0^t \\norme{1}{\\derp{z}u_1}^2$, but this is given by \\propref{hmee}.\n\nWe can use the Cauchy-Schwarz inequality and handle the norm of the rest of $H$ as follows:\n\\begin{itemize}\n\\item the other commutators with $\\mu^{-1}$ are controlled by $Q(1+\\norme{\\mathrm{Lip}}{U})(\\norme{1}{U}+\\norme{0}{W})$,\n\\item $\\norme{0}{\\rho^{\\gamma-1}\\nabla_\\tau^{\\bot}\\rho} \\leq C\\norme{1}{U}$ as $\\rho$ is assumed to be uniformly bounded,\n\\item $\\norme{0}{\\rho\\omega\\cdot\\nabla u_\\tau + \\rho(\\div u)\\omega} \\leq C\\norme{\\infty}{\\nabla u}(\\norme{0}{W}+\\norme{0}{u})$,\n\\item ${\\cal M}^I$ is bounded using $\\norme{0}{\\derp{x_{i}}\\rho(\\derp{t}u_j+u\\cdot\\nabla u_j)} \\leq \\trinorme{\\infty,t}{\\nabla \\rho}(\\norme{1}{u}+\\trinorme{\\infty,t}{\\nabla u}\\norme{0}{u})$,\n\\item and finally, we have $\\varepsilon\\norme{0}{\\nabla_\\tau \\div u}\\norme{0}{W} \\leq \\varepsilon(\\norme{1}{\\nabla u}^{2}+\\norme{0}{W}^{2})$ and use \\estiref{hmee} to control $\\varepsilon\\norme{1}{\\nabla u}$,\n$$ \\varepsilon\\int_{0}^{t}\\norme{1}{\\nabla u(s)}^{2}~ds \\leq C\\left(\\norme{1}{U(0)}^{2}+(1+\\trinorme{\\mathrm{Lip},t}{U}^{2})\\int_{0}^{t}(\\norme{1}{U(s)}^{2} + \\norme{0}{\\nabla U(s)}^{2})~ds\\right). $$\n\\end{itemize}\n\nWe thus get the final $L^{2}$ estimate on $W$:\n$$ c\\left[\\norme{0}{W(t)}^{2}+\\varepsilon\\int_{0}^{t}\\norme{0}{\\nabla W(s)}^{2}~ds\\right] \\leq C(\\norme{0}{W(0)}^{2} + \\norme{1}{U(0)}^{2}) \\hspace{120pt} $$\n$$ \\hspace{60pt} +Q(1+\\trinorme{\\mathrm{Lip},t}{U}^{2}+\\trinorme{1,\\infty,t}{\\nabla U}^2)\\int_{0}^{t} \\norme{1}{U}^{2} + \\norme{0}{\\derp{z}U}^{2}+\\norme{1}{F}^2 + \\norme{0}{\\nabla F}^2~ds . $$\n\nNow we move on to the $H^{m-1}_{co}$ estimate with $m>1$, which will also follow the same pattern as the previous section. We apply $Z^{\\alpha}$ to (\\ref{vorteqn}), for $|\\alpha|\\leq m-1$ and isolate the maximum order terms:\n$$ \\rho\\derp{t}Z^{\\alpha}W + \\rho u\\cdot \\nabla Z^{\\alpha} W - \\varepsilon\\mu\\Delta Z^{\\alpha}W = Z^{\\alpha}H - {\\cal C}^{\\alpha}_{W}, $$\nwhere ${\\cal C}^{\\alpha}_{W}= [Z^{\\alpha},\\rho]\\derp{t}W+[Z^{\\alpha},\\rho u\\cdot\\nabla]W - \\varepsilon[Z^\\alpha,\\mu\\Delta]W$ contains the commutators. Multiplying the above equation by $Z^\\alpha W$ in $L^{2}$, we have\n$$ c\\left[\\norme{0}{Z^{\\alpha}W(t)}^{2} + \\varepsilon\\int_{0}^{t}\\norme{0}{Z^{\\alpha}\\nabla W}^{2}~ds\\right] \\leq C\\left[\\norme{0}{Z^\\alpha W(0)}^{2} + \\trinorme{1,\\infty,t}{\\rho}\\int_0^t \\norme{m-1}{W}^2~ds\\right] \\hspace{50pt} $$\n\\begin{equation} \\hspace{60pt} + C\\varepsilon\\int_{0}^{t} \\norme{m-2}{\\nabla W}^2~ds + C\\int_0^t \\left[\\norme{m-1}{H}^2+\\int_\\Omega |{\\cal C}^{\\alpha}_{W}\\cdot Z^\\alpha W|\\right] ~ds . \\label{hmvort1} \\end{equation}\n\nThe terms on the second line of this inequality are the ones we need to control. \nFirst, $\\varepsilon\\int_0^t \\norme{m-2}{\\nabla W}^2$, which comes from changing $\\nabla Z^\\alpha W$ into $Z^\\alpha \\nabla W$ (using \\propref{comprop}) in the integration by parts on the laplacian, is dealt with by induction, by using (\\ref{hmvfinal}) at a lower order. \nThen, $\\norme{m-1}{H}$ is easily estimated as above, also using the tame estimate in \\propref{tame}:\n\\begin{eqnarray*} \\int_0^t \\norme{m-1}{H(s)}^2~ds & \\leq & \\varepsilon\\int_0^t \\norme{m}{\\nabla u}^{2}~ds + Q(\\trinorme{\\infty,t}{U}^2+ \\trinorme{1,\\infty,t}{\\nabla U}^2+\\trinorme{\\infty,t}{\\nabla F}^2) \\\\\n & & \\hspace{10pt} \\times \\left(\\int_0^t \\norme{m}{U}^{2}+\\norme{m-1}{\\derp{z}U}^{2} + \\norme{m}{F}^2+ \\norme{m-1}{\\nabla F_\\tau}^2~ds\\right) . \\end{eqnarray*}\nUsing \\estiref{hmee} to cover the worst term $\\varepsilon\\int_{0}^{t}\\norme{m}{\\nabla u(s)}^{2}~ds$, we have\n\\begin{eqnarray} \\int_0^t \\norme{m-1}{H(s)}^2~ds & \\leq & C\\norme{m}{U(0)}^2 + Q(\\trinorme{\\infty,t}{U}^2+\\trinorme{1,\\infty,t}{\\nabla U}^2+\\trinorme{\\infty,t}{\\nabla F}^2) \\nonumber \\\\\n & & \\hspace{10pt} \\times \\int_0^t \\norme{m}{U}^2+\\norme{m-1}{\\derp{z}U}^2 + \\norme{m}{F}^2 + \\norme{m-1}{\\nabla F}^2~ds . \\label{hmvrhs} \\end{eqnarray}\n\nWe finally need to estimate the commutators in $\\norme{0}{{\\cal C}^{\\alpha}_{W}}$. The first and last terms of ${\\cal C}^{\\alpha}_{W}$ are easily estimated as in the previous section,\n\\begin{equation} \\int_0^t \\norme{0}{[Z^{\\alpha},\\rho]\\derp{t}W}^2~ds \\leq C\\int_0^t (\\trinorme{\\infty,t}{\\rho}^2+\\trinorme{\\infty,t}{\\derp{t}W}^2)(\\norme{m-1}{W}^2+\\norme{m-1}{U}^2)~ds \\label{hmcom1} \\end{equation}\nby \\propref{funcprop}, and again using the decomposition in \\propref{comprop} (b) and integrating by parts, we have\n\\begin{equation} \\varepsilon \\int_{\\Omega} |[Z^\\alpha,\\mu\\Delta]W\\cdot Z^{\\alpha} W| \\leq \\frac{1+\\trinorme{\\mathrm{Lip},t}{\\rho}}{c}\\varepsilon\\norme{m-2}{\\derp{z}W}^{2}+\\frac{c}{4}\\varepsilon\\norme{m-1}{\\derp{z}W}^{2}, \\label{hmcom2} \\end{equation}\nin which the first term is estimated by induction (using (\\ref{hmvfinal}) at order $m-2$) and the second is absorbed by the left-hand side of (\\ref{hmvort1}). \nFinally, instead of applying the commutator results we have used so far to $\\int_0^t \\norme{0}{[Z^\\alpha, \\rho u\\cdot\\nabla] W}^2$, which would yield a $\\norme{m-2}{\\derp{zz}^2 U}$ term that we do not expect to control uniformly in $\\varepsilon$, we use \\propref{comlem}:\n$$ \\int_{0}^{t}\\norme{0}{[Z^{\\alpha},(\\rho u)(s)\\cdot\\nabla]W(s)}^{2}~ds \\leq \\int_{0}^{t}Q(\\trinorme{\\mathrm{Lip},t}{U}^{2}+\\trinorme{1,\\infty,t}{\\nabla U}^{2})\\norme{m-1}{W(s)}^{2}~ds $$\n$$ \\hspace{135pt} +\\int_{0}^{t}Q(\\trinorme{1,\\infty,t}{\\nabla U}^{2})\\norme{m-1}{\\nabla U(s)}^{2}~ds. $$\nThis finishes off the proof of \\estiref{hmvort}. $\\square$\n\n\\subsection{$L^\\infty$ estimates}\n\nTo deal with $\\norme{1,\\infty}{\\derp{z}u_\\tau}$, we examine $\\omega_\\tau$. The result is:\n\\begin{esti} There exists a positive, increasing function on $\\mathbb{R}^{+}$, $Q$, such that\n$$ \\norme{1,\\infty}{\\omega_\\tau(t)}^2 \\leq Q(\\norme{6}{U(0)}^2+\\norme{5}{\\omega_\\tau(0)}^2+\\norme{1,\\infty}{\\omega_\\tau(0)}^2) \\hspace{170pt} $$\n$$ \\hspace{30pt} + Q(\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{1,\\infty,t}{\\nabla U}^2+{\\cal N}_6(t,F))\\int_0^t 1+\\norme{6}{U(s)}^2+\\norme{6}{\\omega_\\tau(s)}^2+\\norme{5}{\\derp{z}\\rho(s)}^2~ds. $$  \\label{linfvort} \\end{esti}\n\nWe can now deduce the following update of \\propref{hmeeinf}.\n\\begin{coro} Under the conditions of \\assuref{assu1}, for $m\\geq 7$ and $t\\leq T^*$,\n$$ \\trinorme{m,t}{U}^2+\\trinorme{m-1,t}{\\nabla u_\\tau}^2+\\trinorme{1,\\infty,t}{\\nabla u_\\tau}^2 + \\varepsilon\\int_0^t \\norme{m}{\\nabla u}^2 + \\norme{m-1}{\\nabla W}^2~ds \\hspace{70pt} $$\n$$ \\hspace{80pt} \\leq Q(M_0) + Q(M+M_F)\\left(t+ \\trinorme{1,\\infty,t}{\\derp{z}(\\rho,u_3)}^2 \\int_0^t \\norme{m-1}{\\derp{z}\\rho}^2+\\norme{m-1}{\\derp{z}u_3}^2~ds\\right). $$\n\\label{uvortbd} \\end{coro}\n\nThe first tool to prove \\propref{linfvort} is the following version of the maximum principle.\n\\begin{propo} Consider $X$ a Lipschitz-class solution to the following hyperbolic-parabolic system on the half-space $\\Omega\\subset\\mathbb{R}^3$:\n\\begin{equation} \\left\\{ \\begin{array}{rcll} a\\derp{t}X + b\\cdot\\nabla X - \\varepsilon\\mu(a)\\Delta X & = & G & \\mathrm{in}~\\Omega \\\\\nX & = & h & \\mathrm{on}~\\partial\\Omega \\\\ X|_{t=0} & = & X_0\\in L^\\infty(\\Omega), & \\end{array} \\right. \\label{mpsys} \\end{equation}\nin which $X:\\mathbb{R}^{+}\\times\\Omega\\rightarrow \\mathbb{R}^d$, $a$ is a scalar function, bounded and positive uniformly in time ($\\inf_{(t,x)} a(t,x)>c>0$), $b:\\mathbb{R}^{+}\\times\\Omega\\rightarrow\\mathbb{R}^3$ is tangent to the boundary, $G:\\mathbb{R}^{+}\\times\\Omega\\rightarrow\\mathbb{R}^d$ and $h:\\mathbb{R}^{+}\\times \\mathbb{R}^{2} \\rightarrow \\mathbb{R}^d$ are in $L^\\infty$ and $\\mu$ is a positive regular function of $a$. \nWe assume that $a$ and $b$ satisfy $\\derp{t}a+\\div b = 0$. Then we have, for $t\\in [0,T]$,\n$$ \\norme{\\infty}{X(t)} \\leq \\norme{\\infty}{X(0)} + \\trinorme{\\infty,T}{h} + \\frac{1}{\\inf a}\\int_0^t \\norme{\\infty}{G(s)} + \\varepsilon\\norme{\\infty}{\\mu'(a(s))\\nabla a(s) \\cdot \\nabla X(s)} ~ds. $$ \\label{mp1} \\end{propo}\n\nNote that when $\\mu$ is constant, the term involving $\\nabla(\\mu(a))$ vanishes on the right-hand side. \nIn fact, in that case we can directly apply the result to the modified vorticity $W$ instead of $\\omega$, which has the further advantage of satisfying a homogeneous boundary condition ($h=0$). \nIn the non-constant case however, if we consider $W$, we cannot deal with the term $\\derp{zz}\\rho$ that emerges from the commutator $[\\Delta,\\mu^{-1}]$, so we will use the proposition on $\\omega_\\tau$. \nHaving $\\nabla X$ in the right-hand side looks disastrous, but in the context of the estimation process, the factor $\\varepsilon$ intervenes crucially.\n\\newline\n\n\\preu{Proof:} we look for a function $g(t)$ that will control $\\norme{\\infty}{X(t)}$. To do so, we perform energy estimates on the functions $(X-g)_+$ and $(X+g)_-$ ($g$ will indifferently designate the scalar function $g$ and the vector $g\\mathbf{1}\\in\\mathbb{R}^d$), with the convention that, for a given scalar function $f$, $f=f_+ + f_-$ (so $f_-\\leq 0$).\n\nWe concentrate on $(X-g)_+$, the procedure on $(X+g)_-$ being identical. We want to find a function $g(t)$ that satisfies\n$$ \\norme{0}{(X(0)-g(0))_+} = 0 ~~\\mathrm{and}~~ \\frac{d}{dt}\\norme{0}{(X(t)-g(t))_+}^2 \\leq 0,~t> 0; $$\nsuch a function $g$ is an upper bound for $\\norme{\\infty}{X_+}$. Choosing $g(0)=\\norme{\\infty}{X(0)}$, we examine\n\\begin{eqnarray} \\frac{1}{2}\\frac{d}{dt}\\left(\\int_\\Omega a|(X(t)-g(t))_+|^2\\right) & = & \\int_\\Omega a\\derp{t}X\\cdot(X-g)_+ - \\int_\\Omega a\\derp{t}g\\cdot(X-g)_+ \\nonumber \\\\\n & & \\hspace{10pt} + \\frac{1}{2} \\int_\\Omega \\derp{t}a |(X-g)_+|^2 \\nonumber \\\\\n & = & \\int_\\Omega (-b\\cdot\\nabla X + \\varepsilon\\mu\\Delta X + G)\\cdot(X-g)_+ \\nonumber \\\\\n & & \\hspace{10pt} + \\int_\\Omega -a\\derp{t}g\\cdot(X-g)_+ + \\frac{1}{2}\\derp{t}a|(X-g)_+|^2 \\label{mpee} \\end{eqnarray}\nBecause of the scalar product with $(X-g)_+$, which is zero wherever $X-g\\leq 0$, we can replace $\\nabla X$ by $\\nabla ((X-g)_+)$ and $\\Delta X$ by $\\Delta((X-g)_+)$. Then, integration by parts gives us\n$$ -\\int_\\Omega b\\cdot\\nabla X\\cdot(X-g)_+ = \\frac{1}{2}\\int_\\Omega \\div b |(X-g)_+|^2 = -\\frac{1}{2} \\int_\\Omega \\derp{t}a|(X-g)_+|^2, $$\nby the first line of (\\ref{mpsys}), so this term cancels out with the final term of (\\ref{mpee}), and it remains to look at the term involving the laplacian. When integrating by parts, we need to guarantee that $(X-g)_+$ vanishes on the boundary. We therefore impose the condition\n\\begin{equation} g(t) \\geq \\trinorme{L^\\infty(\\partial\\Omega),T}{h} \\label{gbc} \\end{equation}\nand write\n\\begin{eqnarray*} \\varepsilon\\mu\\int_\\Omega \\Delta X\\cdot(X-g)_+ & = & -\\varepsilon\\mu\\norme{0}{\\nabla ((X-g)_+)}^2 - \\varepsilon \\int_\\Omega \\nabla (\\mu(a))\\cdot\\nabla X\\cdot (X-g)_+ \\\\\n & \\leq & -\\varepsilon \\int_\\Omega \\mu'(a) \\nabla a\\cdot \\nabla X\\cdot (X-g)_+ \\end{eqnarray*}\nTherefore, setting $\\til{G} = G-\\varepsilon \\mu'(a) \\nabla X\\cdot\\nabla a$, of (\\ref{mpee}) there only remains\n$$ \\frac{d}{dt}\\left(\\int_\\Omega a|(X(t)-g(t))_+|^2\\right) \\leq 2\\int_\\Omega (\\til{G}-a\\derp{t}g)\\cdot(X-g)_+, $$\nwhich is negative if, for each $j\\in\\{1,\\cdots,p\\}$, $\\til{G}_j-a\\derp{t}g\\leq 0$ on $\\Omega$. Integrating this leads to $g(t)\\geq g(0)+\\int_0^t \\frac{\\til{G}_j(s,x)}{a(s,x)}~ds$ for every $x\\in\\Omega$.\n\nIdentical estimates on $(X+g)_-$ lead to $g(t)\\geq g(0)-\\int_0^t \\frac{\\til{G}_j(s,x)}{a(s,x)}~ds$, so, also taking into account (\\ref{gbc}), we choose\n$$ g(t) = \\norme{\\infty}{X(0)} + \\trinorme{L^{\\infty}(\\partial\\Omega),T}{h} + \\int_0^t \\norme{\\infty}{\\frac{\\til{G}(s)}{a(s)}}~ds, $$\nand this controls $\\norme{\\infty}{X(t)}$ as desired. $\\square$\n\\newline\n\n\\preu{Proof of \\propref{linfvort}:} we immediately apply \\propref{mp1} to $w:=\\omega_\\tau$ and $Z_j w$, to get\n$$ \\norme{1,\\infty}{w(t)} \\leq \\norme{1,\\infty}{w(0)} + 2a\\trinorme{1,\\infty,t}{\\mu^{-1} u_\\tau} + \\int_0^t \\norme{1,\\infty}{\\frac{{\\cal M}_\\tau(s)}{\\rho(s)}} + \\sum_{j=0}^3\\norme{\\infty}{{\\cal C}^j_w(s)}~ds, $$\nwhere ${\\cal C}_w^j=[Z_j,\\rho]\\derp{t}w+[Z_j,\\rho u\\cdot\\nabla] w - \\varepsilon[Z_j,\\mu\\Delta]w$ are the commutator terms that appear in the equations on $Z_j w$. Thus, by using the Cauchy-Schwarz and Young inequalities,\n\\begin{equation} \\norme{1,\\infty}{w(t)}^2 \\leq \\norme{1,\\infty}{w(0)}^2 + 4a^2\\trinorme{1,\\infty,t}{\\mu u_\\tau}^2 + t\\int_0^t \\norme{1,\\infty}{\\frac{{\\cal M}_\\tau(s)}{\\rho(s)}}^2+\\sum_{j=0}^3 \\norme{\\infty}{{\\cal C}^j_w(s)}^2~ds . \\label{inftyv1} \\end{equation}\nNote that a factor $t$ has been extracted in front of the source terms and commutators, so we can be satisfied with fairly crude bounds on these. The $W^{1,\\infty}_{co}$ term stemming from the boundary is easily dealt with using \\propref{sobo}.\n\\newline\n\nLet us start with the control of the commutators. Given that, by (\\ref{commul}),\n$$ \\varepsilon\\mu[Z_3,\\derp{zz}]w=-2\\varepsilon\\mu\\phi'\\derp{zz}w-\\varepsilon\\mu\\phi''\\derp{z}w , $$\nwe replace $\\varepsilon\\mu\\derp{zz}w$ by its expression in the equation. So we write, using the notation $\\Delta_\\tau = \\derp{y_1 y_1}+\\derp{y_2 y_2}$,\n\\begin{eqnarray*} \\varepsilon^2 \\mu^2\\norme{\\infty}{[Z_3,\\derp{zz}]w}^2 & \\leq & C\\varepsilon^2\\norme{\\infty}{\\derp{z}w}^2 + C\\norme{\\infty}{\\rho\\derp{t}w+\\rho u\\cdot\\nabla w - \\varepsilon\\mu\\Delta_\\tau w - {\\cal M}_\\tau}^2 \\\\\n & \\leq & C\\varepsilon^2\\norme{\\infty}{\\derp{z}w}^2 + C(\\norme{1,\\infty}{\\rho}^2+\\norme{\\mathrm{Lip}}{u}^2\\norme{1,\\infty}{w}^2) \\\\\n & & \\hspace{30pt} + C\\varepsilon^2\\norme{2,\\infty}{w}+C\\norme{\\infty}{{\\cal M}_\\tau}^2, \\end{eqnarray*}\nby using the fact that $\\norme{\\infty}{u_3\\derp{z}w} \\leq C\\norme{\\infty}{\\derp{z}u_3}\\norme{\\infty}{Z_3 w}$. \nWe see that $\\norme{\\infty}{{\\cal C}_w^3}$ is bounded, among other terms, by $\\norme{\\infty}{{\\cal M}_\\tau}$; we'll examine the source term last. \nIn this commutator, it remains to control\n$$ J:=\\int_0^t \\varepsilon^2 (\\norme{\\infty}{\\derp{z}w(s)}^2+\\norme{2,\\infty}{w(s)}^2)~ds . $$\nOn both parts of this term, we start by applying \\propref{sobo}, the Sobolev embedding theorem, to get\n$$ J \\leq C\\int_0^t \\varepsilon^{2}(\\norme{3}{\\derp{zz}w(s)}^2+\\norme{5}{\\derp{z}w(s)}^2+\\norme{5}{w(s)}^2)~ds. $$\nThe second term, $\\int_0^t \\varepsilon^2\\norme{5}{\\derp{z}w}^2$, is controlled by using \\estiref{hmvort}, and, like above, the first term, $\\int_0^t \\norme{3}{\\varepsilon\\derp{zz}w}^2$, is controlled by replacing $\\varepsilon\\derp{zz}w$ by its expression in (\\ref{roteqn}), and by using the tame estimate:\n\\begin{eqnarray*} \\int_0^t \\norme{3}{\\varepsilon\\derp{zz}w(s)}^2~ds & \\leq & C \\int_0^t (\\trinorme{\\infty,t}{\\rho}^2+\\trinorme{\\infty,t}{\\rho u}^2)\\norme{4}{w}^2+\\trinorme{1,\\infty,t}{w}^2\\norme{3}{U}^2 \\\\\n& & \\hspace{50pt} + \\norme{3}{\\rho u_3 \\derp{z} w} + \\norme{5}{w}^2 + \\norme{3}{{\\cal M}_\\tau}^2 ~ ds \\end{eqnarray*}\nThe source term norm $\\int_0^t \\norme{3}{{\\cal M}_\\tau(s)}^2~ds$ is given by (\\ref{hmvrhs}), and $\\int_0^t \\norme{3}{\\rho u_3 \\derp{z} w}^2~ds$ is controlled in the same way as is done in \\propref{comlem}:\n\\begin{eqnarray*} \\int_0^t \\norme{3}{(\\rho u_3 \\derp{z} w)(s)}^2~ds & = & \\int_0^t \\norme{3}{\\frac{(\\rho u_3)(s)}{\\phi} Z_3 w(s)}^2~ds \\\\\n & \\leq & C \\int_0^t \\trinorme{1,\\infty,t}{w}^2 \\norme{4}{\\derp{z}(\\rho u)}^2+(\\trinorme{\\mathrm{Lip},t}{\\rho u}^2+\\trinorme{1,\\infty,t}{\\nabla (\\rho u)}^2)\\norme{4}{w}^2~ds \\end{eqnarray*}\nThe commutator $\\varepsilon[Z_3,\\mu]\\Delta w=Z_3(\\mu(\\rho)) \\Delta w$ is controlled by replacing $\\varepsilon\\derp{zz}w$ by its expression. Therefore, in total, we have\n$$ \\varepsilon^2\\int_0^t \\norme{\\infty}{[Z_3,\\mu\\Delta]w}^2~ds \\leq C[\\norme{6}{U(0)}^2+\\norme{5}{w(0)}^2] + C\\int_0^t \\norme{\\infty}{{\\cal M}_\\tau(s)}^2~ds \\hspace{50pt} $$\n\\begin{equation} \\hspace{60pt} + C\\int_0^t Q(\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{1,\\infty,t}{\\nabla U}^2)(1+\\norme{6}{U}^2+\\norme{5}{w}^2+\\norme{5}{\\derp{z}\\rho}^2)~ds+{\\cal N}_6(t,F) \\label{lapcomwinf} \\end{equation}\nIt is now straightforward to notice that the whole commutator satisfies (\\ref{lapcomwinf}), the main tool to bound $\\norme{\\infty}{[Z_j,\\rho]\\derp{t}w+[Z_j,\\rho u \\cdot\\nabla]w}$ being property (\\ref{hardyinf}), as used in \\propref{comlem}.\n\\newline\n\nMoving on to the source term, we now focus on the terms in $\\varepsilon {\\cal M}^{II}$ which arise when the viscosity coefficients are not constant. First, we examine terms involving one normal derivative on $u_1$ or $u_2$. These are linked to $\\nabla w$, which we split into two parts:\n\\begin{eqnarray*} \\varepsilon \\nabla w & = & \\varepsilon\\nabla ((\\overrightarrow{\\mathrm{rot}~} u)_\\tau) = \\varepsilon \\nabla (\\derp{z} u_\\tau^\\bot - \\nabla_\\tau^\\bot u_3) \\\\\n & = & \\varepsilon \\left(\\begin{array}{c} \\nabla_\\tau \\\\ \\derp{z} \\end{array} \\right) \\derp{z}u_\\tau^\\bot - \\varepsilon \\nabla (\\nabla_\\tau^\\bot u_3) \\end{eqnarray*}\nWe are interested in the $L^2$-in-time $W^{1,\\infty}_{co}$ norm of this. On the terms with only one normal derivative, which are the second and the $\\nabla_\\tau$ components of the first, we apply the Sobolev embedding, \\propref{sobo}, leading to having to bound $\\norme{5}{\\varepsilon \\derp{zz}u_j}^2$ for a certain $j$; once again we can do so by replacing $\\derp{zz}u_j$ by its expression in (\\ref{NS}). \nThe remains of the first term, $\\derp{zz}u_\\tau$, are no more difficult: having replaced $\\varepsilon\\derp{zz}u_\\tau$ using (\\ref{NS}), all the subsequent terms are dealt with easily using the Sobolev embedding in most places, including on $\\varepsilon\\lambda \\nabla_\\tau \\derp{z}u_3$, which comes from the $\\nabla\\div$term, and another replacement of $\\varepsilon\\derp{zz}u_3$ completes the estimate.\n\nIn the remaining terms of $\\varepsilon {\\cal M}^{II}$, terms with two normal derivatives on $u$ can appear: we then replace them using the equation. Terms with two conormal derivatives are dealt with by using the Sobolev embedding inequality.\n\\newline\n\nEstimating the $L^2$-in-time conormal-Lipschitz norm of the rest of the source term, ${\\cal M}_\\tau$, which, we remind the reader, is the tangential component of $-\\rho\\omega\\cdot\\nabla u + \\rho(\\div u)\\omega + \\overrightarrow{\\mathrm{rot}~} (\\rho F) + {\\cal M}^I$, is straight-forward. $\\square$\n\n\\section[Proof of \\thref{thEE}, part III: \\textit{a priori} estimates on $\\derp{z}u_3$]{Proof of \\thref{thEE}, part III \\\\ \\textit{A priori} estimates on $\\derp{z}u_3$\n \\sectionmark{Proof of \\thref{thEE} part III: a priori estimates on $\\derp{z}u_3$}}\n\\sectionmark{Proof of \\thref{thEE} part III: a priori estimates on $\\derp{z}u_3$}\n\nTo begin with, we bound $\\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds$, which will essentially only need the tame estimate in \\propref{tame}. The compressibility equation (\\ref{NSdiv}) gives us\n$$ \\rho Z^\\alpha \\derp{z} u_3 = - Z^\\alpha \\derp{t}\\rho - Z^\\alpha(u\\cdot\\nabla \\rho) - Z^\\alpha(\\rho \\mathrm{div}_\\tau u_\\tau) - [Z^\\alpha,\\rho]\\derp{z}u_3, $$\nwith $|\\alpha|\\leq m-1$ and introducing the notation $\\mathrm{div}_\\tau u_\\tau = \\derp{y_1}u_1 + \\derp{y_2}u_2$. Integrating the square of this equality in time and space, we quickly get\n$$ c\\int_0^t \\norme{0}{Z^\\alpha \\derp{z}u_3(s)}^2~ds \\leq C (1+\\trinorme{\\mathrm{Lip},t}{U}^2) \\int_0^t \\norme{m}{U(s)}^2~ds \\hspace{75pt} $$\n$$ \\hspace{50pt} + C\\trinorme{\\infty,t}{u_3}^2 \\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds + C\\trinorme{1,\\infty,t}{\\rho}^2 \\int_0^t \\norme{m-2}{\\derp{z}u_3(s)}^2~ds, $$\nusing \\propref{funcprop} on the commutator term. With this, we can perform an induction on $m>0$ and conclude that there exists a positive increasing polynomial function $Q$ such that\n$$ \\int_0^t \\norme{m-1}{\\derp{z}u_{3}(s)}^2~ds \\leq Q(1+\\trinorme{\\mathrm{Lip},t}{U}^2) \\int_0^t \\norme{m}{U(s)}^2 + \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds . $$\nEssentially, this means that $H^{m-1}_{co}([0,T]\\times\\Omega)$ norms of $\\derp{z}u_3$ can be replaced by the same norms of $\\derp{z}\\rho$ and $W^{1,\\infty}_{co}$ norms of $\\derp{z}u_3$, which, in turn, will be bounded by $H^{m_0}_{co}$ norms for a certain $m_0$, so we need to extract a small parameter, in this case $t$, to close the estimate. We already have\n\\begin{equation} \\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds \\leq Q(1+\\trinorme{\\mathrm{Lip},t}{U}^2)\\left(t\\trinorme{m,t}{U} + \\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds \\right) , \\label{hmdzu3} \\end{equation}\nso we need the estimates on $\\norme{1,\\infty}{\\derp{z}u_3}$ and $\\norme{1,\\infty}{\\derp{z}\\rho}$ to yield a small parameter, $t$ or $\\varepsilon$, as we cannot use (\\ref{l2time}) on the $L^2$-in-time norm present here (by \\estiref{linfu}, $\\norme{1,\\infty}{U}$ already yields the desired parameter).\n\\newline\n\nFor now, we focus on $\\norme{1,\\infty}{\\derp{z}u_3}$. Simply reading (\\ref{NSdiv}), we have\n$$ \\trinorme{\\infty,t}{\\derp{z}u_3}^2 \\leq \\trinorme{\\infty,t}{\\derp{t}\\rho + \\mathrm{div}_\\tau (\\rho u_\\tau)}^2 + \\trinorme{\\infty,t}{u_3}^2\\trinorme{\\infty,t}{\\derp{z}\\rho}^2, $$\nto which we can apply (\\ref{l2timeinf}), and we get\n\\begin{eqnarray} \\trinorme{\\infty,t}{\\derp{z}u_3}^2 & \\leq & Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2) + \\int_0^t Q(1+\\norme{2,\\infty}{U(s)}^2+\\norme{1,\\infty}{\\derp{z}\\rho(s)}^2)~ds \\nonumber \\\\\n & \\leq & Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2) + tQ(1+\\trinorme{5,t}{U}^2+\\trinorme{4,t}{\\derp{z}U}^2 + \\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2) \\label{linfnd1} \\end{eqnarray}\nWe can now do the same for $\\norme{1,\\infty}{\\derp{z}u_3}$, bearing in mind that there is a commutator, $[Z_j,\\rho]\\derp{z}u_3 = (Z_j\\rho)\\derp{z}u_3$, and that we do not want to lose derivatives on $\\norme{1,\\infty}{\\derp{z}\\rho}$ this time:\n\\begin{eqnarray*} \\trinorme{1,\\infty,t}{\\derp{z}u_3}^2 & \\leq & \\trinorme{1,\\infty,t}{\\derp{t}\\rho + \\mathrm{div}_\\tau(\\rho u_\\tau )}^2 + \\trinorme{1,\\infty,t}{u_3}^2\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2 + \\trinorme{1,\\infty}{\\rho}^2\\trinorme{\\infty}{\\derp{z}u_3}^2 \\\\\n & \\leq & Q(1+\\norme{2,\\infty}{U(0)}^2) + \\int_0^t \\norme{2,\\infty}{\\derp{t}\\rho+ \\mathrm{div}_\\tau (\\rho u_\\tau )}^2 \\\\\n & & \\hspace{30pt} +\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2\\left(\\norme{1,\\infty}{u_3(0)}^2+\\int_0^t \\norme{2,\\infty}{u_3(s)}~ds\\right) \\\\\n & & \\hspace{30pt} + \\trinorme{\\infty,t}{\\derp{z}u_3}^2\\left(\\norme{1,\\infty}{\\rho(0)}^2+\\int_0^t \\norme{2,\\infty}{\\rho(s)}^2~ds\\right) \\\\\n & \\leq & Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2+\\norme{1,\\infty}{\\nabla U(0)}^2) + C\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^4 \\\\\n & & \\hspace{30pt}  + tQ(\\trinorme{6,t}{U}^2+\\trinorme{5,t}{\\derp{z}U}^2+\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2) , \\end{eqnarray*}\nby (\\ref{linfnd1}), which, by applying (\\ref{l2time}) provides us with the following.\n\n\\begin{esti} (a) There exists an increasing, positive polynomial function $Q$ on $\\mathbb{R}^{+}$ such that\n$$ \\trinorme{1,\\infty,t}{\\derp{z}u_3}^2 \\leq Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2+\\norme{1,\\infty}{\\nabla U(0)}^2) + Q(\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2) \\hspace{110pt} $$\n\\begin{equation} \\hspace{40pt} + t Q\\left( 1+\\trinorme{6,t}{U}^2+\\trinorme{5,t}{\\nabla u_\\tau}^2+\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2+\\int_0^t \\norme{6}{\\derp{z}u_3(s)}^2+\\norme{6}{\\derp{z}\\rho(s)}^2~ds \\right) \\label{linfnd} \\end{equation}\n\n(b) Combining (\\ref{hmdzu3}) and (\\ref{linfnd}), we get that, for $m\\geq 7$,\n$$ \\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds \\leq Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2+\\norme{1,\\infty}{\\nabla U(0)}^2) \\hspace{150pt} $$\n$$ \\hspace{35pt} + t Q\\left(1+\\trinorme{m,t}{U}^2+\\trinorme{m-1,t}{\\nabla u_\\tau}^2 + \\trinorme{1,\\infty,t}{\\nabla u_\\tau}^2 + \\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds \\right) $$\n\\begin{equation} + Q\\left(\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2 + \\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds \\right) \\hspace{130pt} \\label{hmnddef} \\end{equation}\n\\label{hmnd} \\end{esti}\n\nAs a result, we can update \\cororef{uvortbd}. Set\n\\begin{eqnarray*} \\til{E}^\\varepsilon _{m}(t,U)^2 & = & \\trinorme{m}{U}^2+\\trinorme{m-1,t}{\\nabla u_\\tau}^2 + \\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds+\\trinorme{1,\\infty,t}{\\nabla u}^2 \\\\\n & & \\hspace{40pt} + \\varepsilon\\int_0^t \\norme{m}{\\nabla u}^2 + \\norme{m-1}{\\nabla W}^2~ds . \\end{eqnarray*}\nThis contains ${\\cal E}_m(t,U)$ except the terms involving $\\derp{z}\\rho$, plus the $W^{1,\\infty}_{co}$ norm of $\\derp{z}u_3$ and the gradient terms from the energy estimates. The combination of \\cororef{uvortbd} and \\propref{hmnd} (b) gives us the following.\n\\begin{coro} Under the conditions of \\assuref{assu1}, for $m\\geq 7$ and $t\\leq T^*$,\n$$ \\til{E}^\\varepsilon_m(t,U) \\leq Q(M_0) + Q(M+M_F)\\left(t + Q\\left(\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2 + \\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds\\right)\\right) $$ \\label{endp3} \\end{coro}\nWe see on this estimate that it remains to look at the normal derivative of the density.\n\n\\section[Proof of \\thref{thEE}, part IV: \\textit{a priori} estimates on $\\derp{z}\\rho$]{Proof of \\thref{thEE}, part IV \\\\ \\textit{A priori} estimates on $\\derp{z}\\rho$\n \\sectionmark{Proof of \\thref{thEE} part IV: a priori estimates on $\\derp{z}\\rho$}}\n\\sectionmark{Proof of \\thref{thEE} part IV: a priori estimates on $\\derp{z}\\rho$}\n\n\\subsection{Conormal energy estimates}\n\nIn this section, we examine $R:=\\derp{z}\\rho$. The equation satisfied by $R$ is given by the differentiation of (\\ref{NSdiv}):\n$$ \\derp{t}R + u\\cdot\\nabla R + R(\\div u+\\derp{z}u_3) + \\rho\\derp{zz}u_3 = - \\rho\\derp{z}\\mathrm{div}_\\tau u_\\tau - \\derp{z}u_\\tau \\cdot\\nabla_\\tau \\rho. $$\nA very problematic term appears in this equation: $\\derp{zz}u_3$. The idea is to multiply the equation by $l\\varepsilon:=(\\mu+\\lambda)\\varepsilon$, in order to replace $l\\varepsilon\\derp{zz}u_3$ by its expression in the equation,\n\\begin{equation} l\\varepsilon\\derp{zz} u_3 = \\rho \\derp{t}u_3 + \\rho u\\cdot\\nabla u_3 - \\mu\\varepsilon\\Delta_\\tau u_3 - \\lambda\\varepsilon\\derp{z}\\mathrm{div}_\\tau u_\\tau + R. \\label{NSrwR} \\end{equation}\nThis brings us to\n\\begin{equation} l\\varepsilon(\\derp{t}R + u\\cdot\\nabla R) + \\rho P'(\\rho) R = h, \\label{eqnR} \\end{equation}\nwhere we remind the reader that $\\rho P'(\\rho) = k\\rho^\\gamma$ (and $P'$ is positive; we can thus expect our results to extend to other, strictly increasing and positive pressure laws), and $h$ will be treated as a source term:\n\\begin{eqnarray*} h & = & \\varepsilon[\\rho(\\mu\\Delta_\\tau u_3+\\lambda\\derp{z}\\mathrm{div}_\\tau u_\\tau+\\sigma(\\nabla U)_3) \\\\\n & & \\hspace{20pt} -l\\derp{z}u_\\tau\\cdot\\nabla_\\tau\\rho-l(\\div u+\\derp{z}u_3)R]-\\rho^2(\\derp{t}u_3+u\\cdot\\nabla u_3)+\\rho^2 F_3. \\end{eqnarray*}\n\nIn these estimates, we will strongly use \\assuref{assu1}: for $t\\leq T^*$, we have $\\rho(t,x)\\geq c_0>0$, $|u_3(t,x)|$ is uniformly bounded near the boundary on the same time interval, and ${\\cal E}_m(t,U)\\leq M$ for some $m\\geq 7$. We use the latter assumption to simplify the presentation of what follows.\n\n\\begin{esti} Under the conditions of \\assuref{assu1}, for $m\\geq 1$, there exists a positive, increasing function on $\\mathbb{R}^{+}$, $Q$, such that, for $t\\leq T^*$ and $\\varepsilon\\leq \\varepsilon_0$,\n$$ \\int_0^t \\norme{m-1}{R(s)}^2~ds \\leq Q(\\norme{m-1}{R(0)}^2 +\\norme{m}{U(0)}^2) + (t+\\varepsilon)Q(M+M_F) + Q(\\trinorme{1,\\infty,t}{R}^2). $$\n\\label{hmeeR} \\end{esti}\n\n\\preu{Proof:} we start with the $L^2$ estimate. We multiply (\\ref{eqnR}) by $R$ and integrate in space, and, as usual, we integrate the term containing $u\\cdot\\nabla R$ by parts and get\n\\begin{equation} \\frac{l\\varepsilon}{2}\\frac{d}{dt}(\\norme{0}{R}^2) + \\gamma \\int_\\Omega \\rho P'(\\rho) R^2 = \\int_\\Omega \\left(\\frac{l\\varepsilon}{2} R^2 \\div u + \\frac{l'(\\rho)\\varepsilon}{2} R^2 u\\cdot\\nabla \\rho + hR\\right). \\label{l2eeR} \\end{equation}\nThe coefficients $l$ and $\\rho P'$ are bounded from below, so there exist $0<c<1<C$ such that the left-hand side, integrated in time, is greater than\n$$ c\\left(\\varepsilon \\norme{0}{R(t)}^2 + \\int_0^t \\norme{0}{R(s)}^2~ds\\right) -C\\norme{0}{R(0)}^2 . $$\nThen, using the Young inequality on the right-hand side of (\\ref{l2eeR}) with the parameter $\\eta=\\frac{c}{2}$, we get\n$$ c\\varepsilon \\norme{0}{R(t)}^2 + \\frac{c}{2}\\int_0^t \\norme{0}{R(s)}^2~ds \\leq C\\norme{0}{R(0)}^2 + C\\int_0^t \\varepsilon \\trinorme{\\mathrm{Lip},t}{U}\\norme{0}{R(s)}^2 + \\norme{0}{h(s)}^2~ds. $$\n\nIt only remains to bound $\\int_0^t \\norme{0}{h}^2$, and this is simple:\n$$ \\int_0^t \\norme{0}{h(s)}^2~ds \\leq C\\varepsilon^2\\int_0^t \\trinorme{\\infty,t}{\\nabla u}^2\\norme{0}{R(s)}^2 + \\norme{1}{\\nabla u(s)}^2+\\norme{1}{\\rho(s)-1}^2~ds \\hspace{50pt} $$\n$$ \\hspace{150pt} + C\\int_0^t (1+\\norme{\\infty}{\\derp{z}u_3}^2)\\norme{1}{u_3}^2 + \\norme{0}{F}^2~ds, $$\nin which $\\varepsilon\\int_0^t \\norme{1}{\\nabla u(s)}$ is bounded using \\propref{hmee}, so we have proved the energy estimate on $R$ for $m=1$.\n\\newline\n\nWe now move to the $H^{m-1}_{co}$ estimates with $m\\geq 2$, and once again, the main focus will be the commutators. Given $\\alpha$ of length $m-1$, the equation on $Z^\\alpha R$ is\n$$ l\\varepsilon(\\derp{t}Z^\\alpha R + u\\cdot\\nabla Z^\\alpha R) + \\rho P'(\\rho) Z^\\alpha R = Z^\\alpha h - {\\cal C}^\\alpha_R, $$\nwith ${\\cal C}^\\alpha_R = l\\varepsilon[Z^\\alpha, u\\cdot\\nabla] R + [Z^\\alpha, \\rho P'(\\rho)]R$. Repeating the above procedure, we reach the estimate\n$$ c\\varepsilon\\norme{0}{Z^\\alpha R(t)}^2 + c\\int_0^t \\norme{0}{Z^\\alpha R(s)}^2~ds \\leq C\\norme{m-1}{R(0)}^2 \\hspace{100pt} $$\n\\begin{equation} \\hspace{100pt} + C\\int_0^t \\varepsilon\\trinorme{\\mathrm{Lip},t}{U}\\norme{m-1}{R(s)}^2+\\norme{m-1}{h(s)}^2 + \\norme{0}{{\\cal C}^\\alpha_R(s)}^2~ds. \\label{hmndr1} \\end{equation}\n\nControlling $\\int_0^t \\norme{m-1}{h}^2$ is immediate, except for the term $\\rho^2 u_3 \\derp{z}u_3$, which figures in $\\rho^2 u\\cdot\\nabla u_3$. We start by replacing $\\rho\\derp{z}u_3$ by using (\\ref{NSdiv}),\n$$ \\rho^2 u_3 \\derp{z}u_3 = -\\rho u_3(\\derp{t}\\rho + \\rho \\mathrm{div}_\\tau u_\\tau + u_\\tau\\cdot\\nabla_\\tau \\rho + u_3\\derp{z}\\rho), $$\nand the problematic term here is clearly $\\rho u_3^2 \\derp{z}\\rho$. Directly using the tame estimate basically yields $\\int_0^t \\norme{m-1}{R}^2$, but there is no factor $\\varepsilon$ in this term to allow us to hope to absorb it. \nHowever, we have a factor $u_3$, and we use the assumption that, for a given $\\delta>0$, there exist $\\sdel{z}>0$ which does not depend on $\\varepsilon$, such that $|u_3(x)|<\\delta$ for $x$ in the strip $\\sdel{\\omega} = \\mathbb{R}^2\\times [0,\\sdel{z}]$. \nNow, we split the $H^{m-1}_{co}$ norm of $\\rho u_3^2 R$ into two parts,\n$$ \\int_0^t \\norme{m-1}{\\rho u_3^2 R(s)}^2 ~ds = \\int_0^t \\norme{m-1,\\sdel{\\omega}}{\\rho u_3^2 R(s)}^2 + \\norme{m-1,\\Omega\\backslash \\sdel{\\omega}}{\\rho u_3^2 R(s)}^2~ds. $$\nHere, for $\\omega\\subset \\Omega$, we set $\\norme{m-1,\\omega}{f(s)}$ as a sort of semi-norm of $f(s)$ restricted to $\\omega$:\n$$ \\norme{m-1,\\omega}{f(s)}^2 = \\sum_{|\\beta|\\leq m-1} \\norme{0}{(Z^\\beta f)(s) |_{\\omega}}^2 $$\nWe apply the tame estimate to both norms:\n\\begin{eqnarray*} \\int_0^t \\norme{m-1}{\\rho u_3^2 R}^2 & \\leq & C\\int_0^t \\trinorme{L^\\infty(\\sdel{\\omega}),t}{\\rho u_3^2}^2\\norme{m-1}{R}^2 + \\trinorme{\\infty,t}{R}^2\\norme{m-1}{\\rho u_3^2}^2 \\\\\n& & \\hspace{50pt} + \\trinorme{\\infty,t}{\\rho u_3^2}^2\\norme{m-1,\\Omega\\backslash\\sdel{\\omega}}{\\derp{z}\\rho}^2~ds. \\end{eqnarray*}\nThe two key terms are the first and the last. To deal with the last term, we use \\propref{comprop} to write\n$$ \\norme{m-1,\\Omega\\backslash\\sdel{\\omega}}{\\derp{z}\\rho}^2 \\leq C\\sum_{|\\beta|< m} \\norme{0}{\\derp{z} Z^\\beta \\rho}^2, $$\nand we note that, for $z\\geq \\sdel{z}$, $\\phi(z)\\geq \\phi(\\sdel{z})$, therefore\n$$ |\\derp{z} Z^\\beta \\rho(x)|\\leq \\frac{1}{\\phi(\\sdel{z})}|Z_3 Z^\\beta \\rho(x)| $$\nfor $x\\neq \\sdel{\\omega}$. This means that conormal derivatives are equivalent to standard derivatives away from the boundary, thus, $\\norme{m-1,\\Omega\\backslash\\sdel{\\omega}}{\\derp{z}\\rho}^2 \\leq \\norme{H^m_{co}(\\Omega)}{\\rho-1}^2$. \nThe first term is led by $\\norme{L^\\infty(\\sdel{\\omega})}{\\rho u_3^2}^2$, which, given the boundedness of $\\rho$ and the properties of $u_3$ on $\\sdel{\\omega}$, is bounded by $c_1^2\\delta^4$, which is smaller that $\\frac{c}{2}$ if $\\delta$ is small enough, where $c$ is the coefficient on the left-hand side of (\\ref{hmndr1}): we therefore choose $\\delta$ so that this term can be absorbed.\n\nThe other terms in $h$ are straight-forward, as we can use \\estiref{hmee} on the $\\varepsilon^2\\int_0^t \\norme{m}{\\nabla u}^2$ term that comes from the order-two terms of $h$ - and the factor $\\varepsilon^2$ is essential, as it leaves a factor $\\varepsilon$ which will allow us to close the complete estimate for $\\varepsilon$ and $t$ small. In total, we get\n$$ \\int_0^t \\norme{m-1}{h(s)}^2~ds \\leq C\\norme{m}{U(0)}^2 + \\left(\\varepsilon Q(1+\\trinorme{\\mathrm{Lip},t}{U}^2) + \\frac{c}{2}\\right)\\int_0^t \\norme{m-1}{R(s)}^2~ds $$\n\\begin{equation} \\hspace{50pt} + C\\int_0^t Q(1+\\trinorme{\\mathrm{Lip},t}{U}^2 + \\trinorme{1,\\infty,t}{\\nabla u}^2) (\\norme{m}{U(s)}^2+\\norme{m-1}{\\nabla u_\\tau(s)}^2+\\norme{m-1}{F(s)}^2)~ds \\label{hmndr2} \\end{equation}\n\nThe estimation of the commutators is also mostly straight-forward. $[Z^\\alpha,u\\cdot\\nabla]R$ can be controlled by using \\propref{comlem}, bearing in mind that there is a factor $\\varepsilon$ in front of it:\n$$ \\varepsilon^2\\int_0^t \\norme{0}{[Z^\\alpha,u\\cdot\\nabla]R}^2~ds \\leq C\\varepsilon \\int_{0}^{T}(\\trinorme{\\mathrm{Lip},t}{u}^{2}+\\trinorme{1,\\infty,t}{\\nabla u}^{2})\\norme{m-1}{R(s)}^{2}~ds \\hspace{30pt} $$\n$$ \\hspace{110pt} +C\\varepsilon \\int_{0}^{t}\\trinorme{1,\\infty,t}{R}^{2}\\norme{m-1}{\\nabla u(s)}^{2}+\\trinorme{\\infty,t}{R}^{2}\\norme{m}{u(s)}^{2}~ds. $$\nThe other commutator, $[Z^\\alpha,\\rho P'(\\rho)]$ does not have a factor $\\varepsilon$, so we need to gain a derivative on $R$ by using \\propref{funcprop},\n$$ \\int_{0}^{t}\\norme{0}{[Z^{\\alpha},\\rho^\\gamma]R}^{2}~ds \\leq C\\trinorme{1,\\infty,t}{\\rho}^{2\\gamma}\\int_{0}^{t}\\norme{m-2}{R(s)}^{2}~ds+\\trinorme{\\infty,t}{R}^{2}\\int_0^t \\norme{m-1}{(\\rho-1)(s)}^{2}~ds , $$\nand use (\\ref{l2time}) to extract a factor $t$ in the first term. This leaves us with an isolated $t\\trinorme{1,\\infty,t}{\\rho}$, to which we apply the anistropic Sobolev embedding, \\propref{sobo}. Thus,\n$$ t\\trinorme{1,\\infty,t}{\\rho}^2 \\leq Ct\\left(1+\\norme{4}{R(0)}^2 +\\trinorme{4,t}{U}^2+ \\int_0^t \\norme{5}{R(s)}^2~ds\\right) $$\nNote that it is this last inequality that restricts us to $m-1\\geq 5$. We conclude the proof of \\estiref{hmeeR} by combining (\\ref{hmndr1}), (\\ref{hmndr2}) and these bounds on the commutators. $\\square$\n\n\\subsection{$L^\\infty$ estimates}\n\nAs stated in the introduction, we will control $L^\\infty$ norms of $R$ with $L^2$-in-time bounds by virtue of (\\ref{l2timeinf}),\n\\begin{equation} \\trinorme{1,\\infty,t}{R}^2 \\leq \\norme{1,\\infty}{R(0)}^2 + C\\int_0^t \\norme{1,\\infty}{\\derp{t}R(s)}^2+\\norme{1,\\infty}{R(s)}^2~ds . \\label{l2timeinfr} \\end{equation}\nLet us define $Y_t$ the space of functions $f$ satisfying $\\norme{Y_t}{f}^2 := \\int_0^t \\norme{1,\\infty}{f(s)}^2+\\norme{1,\\infty}{\\derp{t}f(s)}^2~ds$ finite.\n\n\\begin{esti} Under the conditions of \\assuref{assu1}, there exists an increasing, positive function $Q$ on $\\mathbb{R}^{+}$ such that, for $t\\leq T^*$ and $\\varepsilon\\leq \\varepsilon_0$,\n$$ \\norme{Y_t}{R}^2 \\leq Q(M_0) + (t+\\varepsilon)Q(M+M_F) $$ \\label{infR} \\end{esti}\nThis ends the proof of \\thref{thEE} providing we can pick up \\assuref{assu1}.\n\\newline\n\n\\preu{Proof:} the main tool in this proof will be the Duhamel formula for the ordinary differential equation $\\varepsilon f' + \\til{p}f = \\til{h}$. We reach this ODE by considering $R$ along the characteristics of the transport equation $\\derp{t}R+u\\cdot\\nabla R = 0$, in other words\n$$ f(t,x) = R^X(t,x) := R(t,X(t,x)), $$\nwhere $X(t,x)$ satisfies $\\derp{t}X(t,x) = u(t,X(t,x))$ and $X(0,x)=x$. We extend the notation $g^X$ to any function $g$ as above. \nThus, we have the identity $\\derp{t}(R^X) = (\\derp{t}R+u\\cdot\\nabla R)^X$, so (\\ref{eqnR}) becomes,\n$$ \\varepsilon l(\\rho^X) \\varepsilon\\derp{t}(R^X) + \\rho^X P'(\\rho^X) R^X = h^X. $$\n\nFor the higher-order estimates, it is important to apply the conormal derivatives first, then follow the flow of $u$. So, the equation we are interested in is\n$$ \\varepsilon l(\\rho^X)\\varepsilon\\derp{t}((Z^\\alpha R)^X) + \\rho^X P'(\\rho^X) (Z^\\alpha R)^X = (Z^\\alpha h)^X - ({\\cal C}_{R}^\\alpha)^X, $$\nwith ${\\cal C}_R^\\alpha=l\\varepsilon[Z^\\alpha,u\\cdot\\nabla]R+ [Z^\\alpha,\\rho P'(\\rho)]R$ as in the previous paragraph, and $\\alpha$ is either of length $\\leq 1$, or of length 2 with $\\alpha_0\\geq 1$ (these are the $\\alpha$ that intervene in $Y_t$).\n\nTo lighten the load, we introduce the following notations: $g_\\alpha = Z^\\alpha h - {\\cal C}^\\alpha_R$,\n$$ j(s',s,x) = \\int_s^{s'} \\frac{[\\rho^X P'(\\rho^X)](\\sigma,x)}{\\varepsilon l(\\rho^X)(\\sigma,x)} ~d\\sigma ~,~\\mathrm{and}~ J(s,x) = \\int_0^s \\frac{g_\\alpha^X (\\sigma,x)}{\\varepsilon l(\\rho^X)(\\sigma,x)} e^{-j(s,\\sigma,x)}~d\\sigma. $$\nThe Duhamel formula for this equation then reads:\n$$ (Z^\\alpha R)^X(s,x)=Z^\\alpha R(0,x)e^{-j(s,0,x)} + J(s,x) . $$\nWe integrate the square of this equality in time between $0$ and $t$, which yields\n\\begin{equation} \\int_0^t [(Z^\\alpha R)^X(s,x)]^2~ds \\leq C\\left[\\norme{\\infty}{Z^\\alpha R(0)}^2\\int_0^t e^{-2j(s,0,x)}~ds + \\int_0^t J^{2}(s,x)~ds\\right] . \\label{duhamel2} \\end{equation}\nAgain, we use the uniform bounds on $\\rho$ to get, for any $s,~s' \\in \\mathbb{R}^{+}$, $x\\in\\Omega$,\n\\begin{equation} j(s',s,x) \\geq \\int_s^{s'} \\frac{c}{\\varepsilon} = \\frac{c(s'-s)}{\\varepsilon}. \\label{jbelow} \\end{equation}\nThus,\n\\begin{equation} \\int_0^t e^{-cj(s,0,x)}~ds \\leq C\\varepsilon \\label{jest} , \\end{equation}\nwhich deals with the first term of the right-hand side of (\\ref{duhamel2}). For the second term, we use (\\ref{jbelow}) again and write the integral to reveal a convolution in the time variable:\n\\begin{eqnarray*} \\int_0^t J^2(s,x)~ds & \\leq & \\int_0^t \\left[\\int_\\mathbb{R} |g_\\alpha^X|(\\sigma,x)\\mathbf{1}_{(0,t)}(\\sigma) \\frac{e^{-c\\varepsilon^{-1}(s-\\sigma)}}{\\varepsilon}\\mathbf{1}_{(0,t)}(s-\\sigma)~d\\sigma\\right]^2 ~ds \\\\\n & \\leq & \\norme{L^2(0,t)}{\\left(g_\\alpha^X(\\cdot,x)\\mathbf{1}_{(0,t)}\\right) \\ast \\left(\\varepsilon^{-1}e^{-c\\varepsilon^{-1}(\\cdot)}\\mathbf{1}_{(0,t)}\\right)}^2 \\\\\n & \\leq & \\norme{L^2(0,t)}{g_\\alpha^X(\\cdot,x)}^2 \\norme{L^1(0,t)}{\\varepsilon^{-1}e^{-c\\varepsilon^{-1}(\\cdot)}}^2 \\\\\n & \\leq & C\\int_0^t \\norme{\\infty}{g_\\alpha (s)}^2~ds, \\end{eqnarray*}\nby (\\ref{jest}) and standard convolution inequalities.\n\\newline\n\nSo now we only need to control the $Y_t$ norm of $h$ and the $L^\\infty$ norm of the commutators. We remind the reader that\n\\begin{eqnarray*} h & = & \\varepsilon[\\rho(\\mu\\Delta_\\tau u_3+\\lambda\\derp{z}\\mathrm{div}_\\tau u_\\tau+\\sigma(\\nabla U)_3) \\\\\n & & \\hspace{20pt} -l\\derp{z}u_\\tau\\cdot\\nabla_\\tau\\rho-l(\\div u+\\derp{z}u_3)R]-\\rho^2(\\derp{t}u_3+u\\cdot\\nabla u_3)+\\rho F_3. \\end{eqnarray*}\nThe starting point here is to notice that, for two functions $f$ and $g$,\n\\begin{equation} \\norme{2,\\infty}{fg} \\leq C(\\norme{2,\\infty}{f}\\norme{\\infty}{g} + \\norme{1,\\infty}{f}\\norme{2,\\infty}{g}). \\label{fg2inf} \\end{equation}\nIn the case of $h_1:=\\rho(\\mu\\Delta_\\tau u_3 + \\lambda\\derp{z}\\mathrm{div}_\\tau u_\\tau)$, $f=\\rho$, so\n$$ \\norme{2,\\infty}{h_1}^2 \\leq C(\\norme{2,\\infty}{\\rho}^2(\\norme{2,\\infty}{u}^2+\\norme{1,\\infty}{\\nabla u_\\tau}^2) + \\norme{1,\\infty}{\\rho}(\\norme{4,\\infty}{u}^2 + \\norme{3,\\infty}{\\nabla u_\\tau}^2)), $$\nin which we apply the Sobolev inequality to all terms except $\\norme{1,\\infty}{\\nabla u_\\tau}^2$ (it is part of the total quantity we wish to bound), and use the Young inequality to split the products up. \nIn the process, we obtain $(\\varepsilon^2 \\int_0^t \\norme{4}{\\derp{z}\\nabla u_\\tau(s)}^2~ds)^2$, which is controlled by using \\estiref{hmvort}. Therefore, in total, using all the estimates available to us,\n$$ \\int_0^t \\varepsilon^2\\norme{2,\\infty}{h_1}^2 ~ds \\leq \\varepsilon Q(M_0) + \\varepsilon \\int_0^t Q(1+\\norme{7}{U}^2+\\norme{6}{\\nabla u_\\tau}^2+\\norme{6}{R}^2 +\\norme{1,\\infty}{\\nabla u_\\tau}^2 + \\norme{1,\\infty}{R}^2)~ds, $$\n\\begin{equation} \\mathrm{thus}~~\\int_0^t \\varepsilon^2\\norme{2,\\infty}{h_1}^2 ~ds \\leq Q(M_0) + C\\varepsilon(t+1)Q(M+M_F) + C\\varepsilon t \\trinorme{1,\\infty,t}{R}^2. \\label{densinfh1} \\end{equation}\nLikewise, setting $h_2:= \\derp{z}u_\\tau\\cdot\\nabla_\\tau\\rho+(\\div u + \\derp{z}u_3)R + \\sigma_3 = \\derp{z}u\\cdot\\nabla \\rho + R\\div u + \\sigma_3$, we use (\\ref{fg2inf}) again, applied to the space $Y_t$ instead of $W^{2,\\infty}_{co}$, and we get, after applying the Sobolev embedding to all the terms except $\\norme{1,\\infty}{\\nabla u_\\tau}^2$, and $\\norme{Y_t}{R}^2$, to get\n$$ \\varepsilon^2\\norme{Y_t}{h_2}^2 \\leq Q(M_0) + \\varepsilon(t+1)Q(M+M_F) + C\\varepsilon^2 \\norme{Y_t}{R}^2 + C\\varepsilon^2 \\int_0^t \\norme{6}{\\derp{zz}u_3}^2~ds . $$\nWe have once again used \\estiref{hmvort} to control the $H^4_{co}([0,T]\\times\\Omega)$ norm of $\\varepsilon \\derp{z}\\nabla u_\\tau$, and we have used (\\ref{l2time}) to get an $L^2$-in-time norm of $\\derp{z}u_3$ (which can then be controlled using \\propref{hmee}) and $\\derp{zz}u_3$. \nThus, we replace $\\varepsilon\\derp{zz}u_3$ by its expression in (\\ref{NSrwR}), use the tame estimate, take the supremum inside the integral and extract a factor $t$, and this yields\n\\begin{equation} \\varepsilon^2\\norme{Y_t}{h_2}^2 \\leq Q(M_0) + \\varepsilon(t+1)Q(M+M_F) + C\\varepsilon \\norme{Y_t}{R}^2. \\label{densinfh2} \\end{equation}\n\nIn the source term $h$, it remains to examine $h_3:= \\rho^2(\\derp{t}u_3+u\\cdot\\nabla u_3+F_3)$. The difference here is that there is no factor $\\varepsilon$ ready to provide us with a small parameter. Instead, we once again extract $t$ from the integral to use as the small parameter. \nBy simply using the Sobolev embedding inequality and breaking the result down almost completely with the tame estimate, we get\n\\begin{eqnarray} \\int_0^t \\norme{2,\\infty}{h_3(s)}^2~ds & \\leq & Q(M_0)+ C\\int_0^t \\norme{5}{h_3(s)}^2+\\norme{4}{\\derp{z}h_3(s)}^2~ds \\nonumber \\\\\n & \\leq & Q(M_0) + \\int_0^t Q(1+\\trinorme{1,\\infty,t}{U}^2+\\trinorme{1,\\infty,t}{\\derp{z}U}^2+M_F) \\nonumber \\\\\n & & \\hspace{35pt} \\times \\left(1+\\norme{6}{U}^2+\\norme{5}{\\derp{z}U}^2+\\norme{4}{u_3\\derp{zz}u_3}^2~ds\\right) . \\label{h31} \\end{eqnarray}\nThe only product we do not split using the tame estimates and the Young inequality is $u_3\\derp{zz}u_3$, which appears in $\\derp{z}h_3$. Indeed, we need the factor $u_3$ to be able to compensate for the two $z$-derivatives. We have already used in (\\ref{h31}) the fact that\n$$ \\trinorme{\\infty,t}{u_3\\derp{zz}u_3}^2 \\leq \\trinorme{1,\\infty,t}{\\derp{z}u_3}^4 , $$\ndue to (\\ref{u3phi}), and $\\norme{3}{u_3\\derp{zz}u_3}$ is dealt with in same way as in the proof of (\\ref{comfinal}) in \\propref{comlem}: multiply and divide by $\\phi$, which means we are actually looking at $\\norme{3}{\\phi^{-1}u_3 Z_3\\derp{z}u_3}$, and use the Hardy inequality to get that this quantity satisfies (\\ref{comfinal}), with $g=u_3$ and $f=\\derp{z}u_3$, so\n$$ \\int_0^t \\norme{2,\\infty}{h_3(s)}^2 ~ds \\leq Q(M_0) + Q(1+\\trinorme{1,\\infty,t}{U}^2+\\trinorme{1,\\infty,t}{\\derp{z}U}^2)\\int_0^t \\norme{6}{U}^2+\\norme{5}{\\derp{z}U}^2~ds . $$\nNow we extract $t$ and use (\\ref{l2time}) to get $L^2$-in-time norms on $R$ and $\\derp{z}u_3$, so\n\\begin{equation} \\int_0^t \\norme{2,\\infty}{h_3(s)}^2~ds \\leq Q(M_0) + tQ(M+M_F) . \\label{densinfh3} \\end{equation}\n\nFinally, we examine the $L^{\\infty}$ norms of the commutators, ${\\cal C}^{\\alpha}_R$, with $|\\alpha|= 1$, or $|\\alpha|=2$ and $\\alpha_0\\geq 1$. We begin with the case $|\\alpha|=1$, so $Z^\\alpha=Z_j$ for a certain $j$, and\n$$ {\\cal C}^{\\alpha}_{R} = l\\varepsilon[Z_j,u\\cdot\\nabla] R + [Z_j,\\rho P'(\\rho)]R = l\\varepsilon(Z_j u)\\cdot\\nabla R + l\\varepsilon u_3[Z_j,\\derp{z}]R + \\gamma(Z_j P)R , $$\nin which the second term is either 0 ($j\\neq 3$) or $-l\\varepsilon\\phi' u_3\\derp{z}R$. Bounding this is straight-forward, using (\\ref{hardyinf}) along the way:\n\\begin{equation} \\int_0^t \\norme{\\infty}{{\\cal C}^\\alpha_R(s)}^2~ds \\leq Ct(1+\\varepsilon^2)\\trinorme{1,\\infty,t}{\\nabla u}^2\\trinorme{1,\\infty,t}{R}^2 . \\label{densinfc1} \\end{equation}\nThe second case, $|\\alpha|=2$ and $\\alpha_0>0$ is also simple. We can write $Z^\\alpha = Z_j \\derp{t}$ for a certain $j$, so\n\\begin{eqnarray*} {\\cal C}^{\\alpha}_R & = & l\\varepsilon\\left((Z_j\\derp{t} u)\\cdot\\nabla R + (Z_ju)\\cdot\\nabla(\\derp{t}R) + (\\derp{t}u)\\cdot\\nabla (Z_j R) + (\\derp{t}u_3)[Z_j,\\derp{z}]R \\right. \\\\\n & & \\hspace{25pt} \\left. + u_3[Z_j,\\derp{z}]\\derp{t}R\\right) + \\gamma((Z_j\\derp{t}P)R+(Z_j P)(\\derp{t}R)+(\\derp{t} P)(Z_jR)) \\end{eqnarray*}\nThe $L^\\infty$ norm of this is, for most terms, bounded using only the Sobolev embedding and\/or (\\ref{hardyinf}), wherever $u_3\\derp{z}$ or conormal derivatives of $u_3 \\derp{z}$ appear. One can then take the supremum in time inside the integral and integrate to get a factor $t$. For instance, the last term satisfies\n\\begin{eqnarray*} \\norme{\\infty}{(\\derp{t}r)(Z_jR)}^2 & \\leq & C(\\norme{2,\\infty}{u_\\tau}^2+\\norme{1,\\infty}{\\derp{z}u_3}^2+\\norme{1,\\infty}{\\rho}^2)\\norme{1,\\infty}{R}^2 \\\\\n & \\leq & C(1+\\norme{5}{U}^2+\\norme{4}{\\nabla u_\\tau}^2+\\norme{3}{R}^2+\\norme{1,\\infty}{\\derp{z}u_3}^2)\\norme{1,\\infty}{R}^2 , \\end{eqnarray*}\nthen we apply \\estiref{hmnd} (a) on $\\norme{1,\\infty}{\\derp{z}u_3}$. Two specificities do appear though in the terms containing $u_3 \\derp{z}R$ and the like.\n\\begin{itemize}\n\\item The term $l \\varepsilon u_3[Z_3,\\derp{z}]\\derp{t}R = -l\\varepsilon\\phi' u_3\\derp{zt}R$ leads to two conormal derivatives on $R$. We cannot extract a factor $t$ from the integral here, since we want $L^2$-in-time norms of $\\norme{1,\\infty}{\\derp{t}R}$. However, we do have a factor $l\\varepsilon$, which will act as the small parameter. So,\n$$ \\int_0^t \\norme{\\infty}{l\\varepsilon\\phi' u_3\\derp{zt}R}^2~ds \\leq C\\varepsilon^2\\trinorme{\\infty,t}{\\derp{z}u_3}^2\\int_0^t \\norme{1,\\infty}{\\derp{t}R(s)}^2~ds . $$\n\\item The term $l \\varepsilon(Z_j\\derp{t}u)\\cdot\\nabla R$ contains $\\varepsilon(Z_j\\derp{t}u_3)\\derp{z}R$, which, after using (\\ref{hardyinf}), leads to a term with two conormal derivatives on $\\varepsilon\\derp{z}u_3$ (and no more than one on $R$, which is therefore not problematic), so we re-use the trick we used when estimating $\\varepsilon h_2$: we use the Sobolev inequality on the term with two conormal derivatives, to get\n$$ \\int_0^t \\varepsilon^2\\norme{2,\\infty}{\\derp{z}u_3(s)}^2~ds \\leq M_0 + C\\int_0^t \\varepsilon^2\\norme{5}{\\derp{z}u_3(s)}^2 + \\norme{5}{\\varepsilon\\derp{zz}u_3(s)}^2~ds , $$\nand we replace $\\varepsilon\\derp{zz}u_3$ by its expression in (\\ref{NS}), use the tame estimate, take the supremum inside the integral and extract a factor $t$.\n\\end{itemize}\n\nIn total, combining the above consideration with inequalities (\\ref{densinfh1}) to (\\ref{densinfc1}), we conclude that\n$$ \\int_0^t \\norme{\\infty}{g_\\alpha(s)}^2~ds \\leq Q(M_0) + (t+\\varepsilon) Q(M+M_F) , $$\nwhich ends the proof of the estimate. $\\square$\n\n\\section[Proof of \\thref{thEE}, part V: conclusion]{Proof of \\thref{thEE}, part V \\\\ Conclusion\n \\sectionmark{Proof of \\thref{thEE} part V: conclusion}}\n\\sectionmark{Proof of \\thref{thEE} part V: conclusion}\n\nLet us consider times $t\\leq T'$, where\n$$ T' = \\sup\\{ t\\leq T_0 ~|~ {\\cal E}_m(t,U) + \\trinorme{1,\\infty,t}{\\derp{z}u_3}^2 \\leq Q(2M_0)\\} . $$\nThe time $T'$ depends \\textit{a priori} on $\\varepsilon$. With this, we pick up the uniform boundedness from below of $\\rho$ and the uniform smallness of $u_3$ near the boundary that we have so far assumed.\n\\begin{itemize}\n\\item Let $\\rho(0,x)\\geq c_{0}$. Equation (\\ref{NSdiv}) provides us with the differential equation\n$$ \\derp{t}\\rho^{X}(t,x) = - (\\rho\\div u)^{X}(t,x), $$\nwhere $f^{X}$ is once again $f$ following the flow of $u$. Thus,\n$$ \\rho^{X}(t,x) = \\rho(0,x)\\exp\\left(-\\int_{0}^{t}\\div u(s,x)~ds\\right), $$\nand $\\div u$ is uniformly bounded on $[0,T']\\times \\Omega$, so\n$$ |\\rho(t,x)|\\geq c_{0}e^{-Q(2M_0)T'} := c_0', $$\nfor $t\\leq T'$, and $c_0'>0$ can be chosen independent of $\\varepsilon$ (if $T'=+\\infty$ for some $\\varepsilon$, we replace with some finite $T'_0$ independent of $\\varepsilon$). The density $\\rho$ is therefore uniformly bounded from below on $[0,T']$.\n\\item For $t\\leq T'$, we can repeat the proof of (\\ref{linfnd1}),\n$$ \\norme{\\infty}{\\derp{z}u_3}^2 \\leq Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2) + tQ(1+\\trinorme{4,t}{U}^2+\\trinorme{3,t}{\\derp{z}U}^2 + \\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2), $$\nwhich is only a reading of (\\ref{NSdiv}) combined with the Sobolev embedding and property (\\ref{l2timeinf}), hence $\\derp{z}u_3(t)$ has a uniform bound for $t\\leq T'$, thus $|u_3(t,z)|\\leq M'z$ for some $M'$ when $t\\leq T'$. \nAs a result, we get uniform smallness of $u_3$ near the boundary: $|u_3(t,z)|\\leq M'\\delta$ when $z<\\delta$. \\end{itemize}\n\nFor $t\\leq T'$, we have the bounds required to make the \\textit{a priori} estimation process valid, so we have\n$$ {\\cal E}_m(t,U) \\leq Q(M_0)+(t+\\varepsilon)Q(Q(2M_0)+M_F). $$\nFor $\\varepsilon\\leq \\varepsilon_0$ small enough, we see that the right-hand side is smaller than $Q(2M_0)$ for $t\\leq T^*\\leq T'$, with $T^*$ depending on $\\varepsilon_0$ but not on $\\varepsilon$. Thus $T^*$ can be chosen independent of $\\varepsilon$, and the theorem is proved. $\\square$\n\\vspace{12pt}\n\n\\textbf{\\underline{Acknowledgements.}} This is one of the results from my Ph.D thesis, prepared at the University of Rennes 1. Therefore, I am deeply grateful to my supervisor Fr\\'ed\\'eric Rousset for the opportunity to work on this topic, for his guidance and patience throughout the preparation of the present article. \nI would also like to thank Mark Williams for the discussions that have helped improve the clarity of the paper.\n\n\\begin{small}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}