diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgpia" "b/data_all_eng_slimpj/shuffled/split2/finalzzgpia" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgpia" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n In \\cite{Barrett-Suli}, the authors established the existence of global-in-time weak solutions to the Navier--Stokes--Fokker--Planck equations arising in the kinetic theory of dilute polymer solutions and describing a large class of bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials. For $\\O\\subset \\R^3$ a bounded domain, the solvent density $\\vr$ and the solvent velocity field $\\vu$ satisfy the following equations in the space-time cylinder $(0,T]\\times \\O$, $T>0$:\n\\be\\label{i1}\n\\d_t \\vr + \\Div_x (\\vr \\vu) = 0,\n\\ee\n\\be\\label{i2}\n \\d_t (\\vr\\vu)+ \\Div_x (\\vr \\vu \\otimes \\vu) +\\nabla_x p(\\vr) -\n\\Div_x \\SSS =\\Div_x \\TT +\\vr\\, \\ff,\n\\ee\nwhich we assume here to be supplemented with the no-slip boundary condition\n\\be\\label{i4}\n\\vu=\\mathbf{0}\\quad \\mbox{on}\\ (0,T]\\times \\partial\\Omega.\n\\ee\n\nIgnoring the effect of temperature changes, we consider a barotropic pressure law\n\\be\\label{pressure}\np = p(\\vr), \\ p(\\vr) \\approx \\vr^\\gamma \\ \\mbox{for large values of}\\ \\vr.\n\\ee\nThe {\\em Newtonian stress tensor} $\\SSS$ is defined by\n\\be\\label{i3}\n\\SSS = \\mu^S \\left( \\frac{\\nabla_x \\vu + \\nabla^{\\rm T}_x \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right) + \\mu^B (\\Div_x \\vu) \\II,\n\\ee\nwith the shear and bulk viscosity coefficients $\\mu^S$ and $\\mu^B$ defined below (see \\eqref{mu-eta1}). In contrast with \\cite{Barrett-Suli}, where the shear and bulk viscosity coefficients are taken to be constant, $\\mu^S >0$ and $\\mu^B \\geq 0$, we consider here the case when they are functions of the polymer number density. Such an extension requires nontrivial modifications of the method used in \\cite{Barrett-Suli}, and represents the main contribution of the present paper.\n\nIn \\emph{a bead-spring chain model} consisting of $K+1$ beads coupled with $K$ elastic springs representing a polymer chain, the non-Newtonian elastic extra stress tensor $\\TT$ is defined by a version of the Kramers expression (cf. \\eqref{def-TT0} below), depending on the probability density function $\\psi$, which, in addition to $t$ and $x$, also depends on the conformation vector $(q_1^{\\rm T},\\dots q_K^{\\rm T})^{\\rm T} \\in \\R^{3K}$, with $q_i$ representing the $3$-component \\emph{conformation\/orientation vector} of the $i$th spring in the chain. Let $D:=D_1\\times \\cdots \\times D_K \\subset \\R^{3K}$ be the domain of admissible conformation vectors. Typically $D_i$ is the whole space $\\R^3$ or a bounded open ball centered at the origin $0$ in $\\R^3$, for each $i=1,\\dots,K$. When $K=1$, the model is referred to as the \\emph{dumbbell model}. Here we focus on finitely extensible nonlinear (or, briefly, FENE-type) bead-spring-chain models where $D_i=B(0,b_i^{\\frac12})$, a ball centered at the origin $0$ in $\\mathbb{R}^3$ and of radius $b_i^{\\frac12}$, with $b_i>0$ for each $i \\in \\{1,\\dots, K\\}$. The {\\em extra-stress tensor} $\\TT$ is defined by the formula:\n\\be\\label{def-TT0}\n\\TT (\\psi)(t,x) := \\TT_1 (\\psi) (t,x) -\\left(\\int_{D\\times D} \\gamma(q,q')\\, \\psi(t,x,q)\\,\\psi(t,x,q') \\, \\dq\\,\\dq'\\right)\\II,\n\\ee\nwhere, similarly to \\cite{Barrett-Suli}, the interaction kernel $\\g$ is assumed to be a positive constant $\\gamma(q,q')\\equiv \\de>0.$\nConsequently,\n\\be\\label{def-TT}\n\\TT (\\psi) := \\TT_1 (\\psi) -\\de \\left(\\int_{D} \\psi \\,\\dq\\right)^2\\II.\n\\ee\nThe first part, $\\TT_1(\\psi)$, of $\\TT(\\psi)$ is given by the {\\em Kramers expression}\n\\be\\label{def-TT1}\n\\t_1(\\psi) := k \\left[\\left(\\sum_{i=1}^K \\CC_i(\\psi)\\right)-(K+1) \\left(\\int_{D} \\psi \\ \\dq\\right)\\II\\right],\n\\ee\nwhere $k>0$ is the product of the Boltzmann constant and the absolute temperature and\n\\be\\label{def-Ci}\n\\CC_{i}(\\psi)(t,x) := \\int_D \\psi(t,x,q)\\, U_i'\\bigg(\\frac{|q_i|^2}{2}\\bigg)\\, q_i q_i^{\\rm T}\\,\\dq, \\quad i=1,\\dots,K.\n\\ee\n\nIn the expression \\eqref{def-Ci}, the smooth functions $U_i : [0,\\frac{b_i}{2}) \\to [0,\\infty)$, $i=1,\\dots,K,$ are the spring potentials satisfying\n$U_i(0)=0$, $\\lim_{s\\to \\frac{b_i}{2}-} U_i(s) =+\\infty$.\nWe introduce the {\\em partial Maxwellian} $M_i : D_i \\to [0,\\infty)$ by\n\\be\\label{def-Mi}\nM_i(q_i) := \\frac{1}{Z_i} {\\rm e}^{- U_i\\big(\\frac{|q_i|^2}{2}\\big)},\\quad \\mbox{where } Z_i:=\\int_{D_i} \\mathrm{e}^{- U_i\\big(\\frac{|p_i|^2}{2}\\big)}\\,{\\rm d}p_i.\n\\ee\nThe {\\em Maxwellian} $M : D \\to [0,\\infty)$ is then defined as the product of the $K$ partial Maxwellians: i.e., for any $q=(q_1^{\\rm T},\\dots, q_K^{\\rm T})^{\\rm T}$ in $D = D_1\\times\\cdots \\times D_K$, we have that\n\\[M(q) := \\prod_{i=1}^K M_i(q_i).\\]\nClearly,\n$\n\\int_D M(q)\\ \\dq=1.\n$\nBy direct calculations one verifies that, for any $i \\in \\{1,\\dots,K\\}$,\n\\be\\label{pt2-M}\nM(q)\\, \\nabla_{q_i} (M(q))^{-1} = - M(q)^{-1}\\, \\nabla_{q_i} M(q) = \\nabla_{q_i}\\left(U_i\\bigg(\\frac{|q_i|^2}{2}\\bigg)\\right) = U_i'\\bigg(\\frac{|q_i|^2}{2}\\bigg) q_i\n\\quad \\forall\\,q = (q_1^{\\rm T},\\dots,q_K^{\\rm T})^{\\rm T}\\in D.\n\\ee\n\nAs in \\cite{Barrett-Suli}, we shall suppose that, for any $i\\in\\{1,\\dots, K\\}$, there exist positive constants $c_{ij}, \\ j=1,\\dots,4$, and $\\th_i > 1$ such that\n\\ba\\label{pt1-Mi-Ui}\nc_{i1}\\left({\\rm dist}\\,(q_i,\\d D_i)\\right)^{\\th_i} \\leq M_i(q_i)\\leq c_{i2}\\left({\\rm dist}\\,(q_i,\\d D_i)\\right)^{\\th_i},\\quad\nc_{i3} \\leq \\left({\\rm dist}\\,(q_i,\\d D_i)\\right) U_i'\\left(\\frac{|q_i|^2}{2}\\right)\\leq c_{i4} \\quad \\forall\\,q_i\\in D_i.\n\\ea\nIt is then straightforward to deduce that\n\\be\\label{pt3-Mi-Ui}\n\\int_{D_i} \\left(1+ \\left( U_i\\bigg(\\frac{|q_i|^2}{2}\\bigg) \\right)^2+ \\left( U_i'\\bigg(\\frac{|q_i|^2}{2}\\bigg) \\right)^2 \\right) M_i(q_i)\\,\\dq_i <\\infty,\\quad i=1,\\dots, K.\n\\ee\n\n\n\nThe probability density function $\\psi$ satisfies the following {\\em Fokker--Planck equation} in $(0,T]\\times \\O\\times D$:\n\\ba\\label{eq-psi}\n\\d_t \\psi + \\Div_x (\\vu\\,\\psi) + \\sum_{i=1}^K \\Div_{q_i} \\left( (\\nabla_x \\vu)\\, q_i\\, \\psi \\right)= \\e \\Delta_x \\psi+\\frac{1}{4\\lambda} \\sum_{i=1}^K\\sum_{j=1}^K A_{ij}\\, \\Div_{q_i}\\left( M \\nabla_{q_j} \\left( \\frac{\\psi}{M} \\right)\\right).\n\\ea\n\nThe \\emph{centre-of-mass diffusion} term $\\e \\Delta_x \\psi$ is generally of the form\n$\n\\e \\Delta_x \\left( \\frac{\\psi}{\\zeta(\\vr)} \\right),\n$\nwhich involves the {\\em drag coefficient} $\\zeta(\\cdot)$ depending on the fluid density $\\vr$. Here we assume that $\\zeta$ is a constant function, which is, for simplicity, taken to be identically $1$. The constant parameter $\\e$ is the {\\em centre-of-mass diffusion coefficient}, which is strictly positive.\nThe positive parameter $\\l$ is called the \\emph{Deborah number}; it characterizes the elastic relaxation property of the fluid. The constant matrix $A=(A_{ij})_{1\\leq i,j\\leq K}$, called the \\textit{Rouse matrix}, is symmetric and positive definite. We denote by $A_0$ the smallest eigenvalue of $A$; clearly, $A_0>0$. We refer to Section 1 of Barrett and S\\\"uli \\cite{Barrett-Suli1} for a derivation of the Fokker--Planck equation \\eqref{eq-psi}.\n\nThe Fokker--Planck equation needs to be supplemented by suitable boundary conditions. For any $i=1,\\dots, K$, let $\\d \\overline D_i:= D_1\\times \\cdots \\times D_{i-1} \\times \\d D_i \\times D_{i+1} \\cdots \\times D_K$ and suppose that\n\\ba\\label{boundary-psi}\n\\left(\\frac{1}{4\\lambda} \\sum_{j=1}^K A_{ij}\\, \\Div_{q_i}\\left(M \\nabla_{q_j} \\left( \\frac{\\psi}{M} \\right)\\right) -(\\nabla_x \\vu) \\, q_i\\, \\psi \\right)\\cdot \\frac{q_i}{|q_i|}&=0 \\quad &&\\mbox{on}\\ (0,T]\\times \\O \\times \\d \\overline D_i,\\\\\n \\nabla_x\\psi \\cdot {\\bf n}&=0 \\quad &&\\mbox{on}\\ (0,T]\\times \\d\\O \\times D.\n\\ea\n\nFinally, we introduce the {\\em polymer number density} $\\eta$ defined by\n\\be\\label{def-eta}\n\\eta(t,x) := \\int_D \\psi(t,x,q)\\,\\dq, \\qquad (t,x)\\in [0,T]\\times \\O.\n\\ee\nBy (formally) integrating the partial differential equation \\eqref{eq-psi} over $D$ and using the boundary condition in $\\eqref{boundary-psi}_1$, and by integrating the boundary condition $\\eqref{boundary-psi}_2$ over $D$, we deduce the following partial differential equation and boundary condition for the function $\\eta$:\n\\be\\label{eq-eta}\n\\d_t \\eta + \\Div_x (\\vu \\,\\eta) = \\e \\Delta_x \\eta\\quad \\mbox{in}\\ (0,T] \\times \\O;\\qquad \\nabla_x \\eta \\cdot {\\bf n}=0 \\quad \\mbox{on}\\ (0,T]\\times \\d\\O.\n\\ee\nBy noting \\eqref{def-eta} we see that the expression for the extra-stress tensor in \\eqref{def-TT} and \\eqref{def-TT1} can also be expressed as follows:\n\\be\\label{def-TT-f}\n\\TT (\\psi) := k \\left(\\sum_{i=1}^K \\CC_i(\\psi)\\right)-\\left(k(K+1) \\eta + \\de\\, \\eta^2\\right) \\II.\n\\ee\nAs has already been pointed out above, our main objective is to consider a class of models of this form where the viscosity coefficients $\\mu^S = \\mu^S(\\eta)$ and $\\mu^B = \\mu^B(\\eta)$\ndepend on $\\eta$.\n\n\n\n\\subsection{Dissipative (finite-energy) weak solutions}\n\nWe adopt the following hypotheses on the initial data:\n\\ba\\label{ini-data}\n&\\vr(0,\\cdot) = \\vr_0(\\cdot) \\ \\mbox{with}\\ \\vr_0 \\geq 0 \\ {\\rm a.e.} \\ \\mbox{in} \\ \\O, \\quad \\vr_0 \\in L^\\gamma(\\O) \\ \\mbox{with}\\ \\gamma>\\textstyle{\\frac32};\\\\\n&\\vu(0,\\cdot) = \\vu_0(\\cdot) \\in L^r(\\O;\\R^3) \\ \\mbox{for some $r>1$}\\ \\mbox{such that}\\ \\vr_0|\\vu_0|^2 \\in L^1(\\O);\\\\\n&\\psi(0,\\cdot) = \\psi_0(\\cdot) \\ \\mbox{with}\\ \\psi_0 \\geq 0 \\ \\mbox{a.e. in} \\ \\O\\times D,\\quad \\psi_0 \\left(\\log \\frac{\\psi_0}{M}\\right) \\in L^1(\\O\\times D);\\\\\n&\\eta(0,\\cdot)=\\int_{D}\\psi_0 \\,\\dq =:\\eta_0 \\in L^2 (\\O).\n\\ea\nWe deduce from $\\eqref{ini-data}_1$ and $\\eqref{ini-data}_2$ by using H\\\"older's inequality that $(\\vr\\vu)(0,\\cdot) = \\vr_0 \\vu_0 =\\sqrt{\\vr_0}\\sqrt{\\vr_0} \\vu_0 \\in L^{\\frac{2\\g}{\\g+1}}(\\O;\\R^3)$.\n\n\\begin{definition}\\label{def-weaksl} We say that $(\\vr,\\vu,\\psi,\\eta)$ is a dissipative (finite-energy)\nweak solution in $(0,T]\\times \\O\\times D$ to the system of equations\n\\eqref{i1}--\\eqref{i3}, \\eqref{eq-psi}--\\eqref{def-TT-f}, supplemented by the initial data \\eqref{ini-data}, if:\n\\begin{itemize}\n\n\\item $\\vr \\geq 0 \\ {\\rm a.e.\\ in} \\ (0,T] \\times \\Omega,\\quad \\vr \\in C_w ([0,T]; L^\\gamma(\\Omega)),\\quad \\vu\\in L^{r}(0,T;W_0^{1,r}(\\Omega; \\R^3))\\quad \\mbox{for some $r>1$},$\n\\ba\\label{weak-est}\n&\\vr \\vu \\in C_w([0,T]; L^{\\frac{2 \\gamma}{ \\gamma + 1}}(\\Omega; \\R^3)),\\quad \\vr |\\vu|^2 \\in L^\\infty(0,T; L^{1}(\\Omega));\\\\\n&\\psi \\geq 0 \\ {\\rm a.e.\\ in} \\ (0,T] \\times \\Omega \\times D, \\quad \\psi \\in C_{w}([0,T];L^1(\\Omega\\times D)),\\\\\n& \\nabla_x \\psi\\in L^1((0,T)\\times \\O\\times D;\\R^3), \\quad M \\nabla_{q} \\left(\\frac{\\psi}{M}\\right)\\in L^1((0,T)\\times \\O\\times D;\\R^{3K}),\\\\\n&\\eta =\\int_D \\psi\\ \\dq \\ {\\rm a.e.\\ in} \\ (0,T] \\times \\Omega,\\quad \\eta \\in C_w ([0,T]; L^2(\\Omega)) \\cap L^2 (0,T; W^{1,2}(\\Omega)),\\\\\n&\\TT (\\psi) := k \\left(\\sum_{i=1}^K \\CC_i(\\psi)\\right)-\\left(k(K+1) \\eta + \\de\\, \\eta^2\\right) \\II \\quad {\\rm a.e.\\ in} \\ (0,T] \\times \\Omega,\\quad \\TT\\in L^1((0,T)\\times \\O;\\R^{3\\times 3}).\n\\ea\n\n\n\\item For any $t \\in (0,T]$ and any test function $\\phi \\in C^\\infty([0,T] \\times \\Ov{\\Omega})$, one has\n\n\\be\\label{weak-form1}\n\\int_0^t\\intO{\\big[ \\vr \\partial_t \\phi + \\vr \\vu \\cdot \\Grad \\phi \\big]} \\,\\dt' =\n\\intO{\\vr(t, \\cdot) \\phi (t, \\cdot) } - \\intO{ \\vr_{0} \\phi (0, \\cdot) },\n\\ee\n\\be\\label{weak-form2}\n\\int_0^t \\intO{ \\big[ \\eta \\partial_t \\phi + \\eta \\vu \\cdot \\Grad \\phi - \\e \\nabla_x\\eta \\cdot \\nabla_x \\phi \\big]} \\, \\dt' = \\intO{ \\eta(t, \\cdot) \\phi (t, \\cdot) } - \\intO{ \\eta_{0} \\phi (0, \\cdot) }.\n\\ee\n\n\\item For any $t \\in (0,T]$ and any test function $\\vvarphi \\in \\DC([0,T] \\times {\\Omega};\\R^3)$, one has\n\\ba\\label{weak-form3}\n&\\int_0^t \\intO{ \\big[ \\vr \\vu \\cdot \\partial_t \\vvarphi + (\\vr \\vu \\otimes \\vu) : \\Grad \\vvarphi + p(\\vr)\\, \\Div_x \\vvarphi - \\SSS : \\Grad \\vvarphi \\big] } \\, \\dt'\\\\\n&= \\int_0^t \\intO{ \\TT : \\nabla_x\\vvarphi - \\vr\\, \\ff \\cdot \\vvarphi} \\, \\dt' + \\intO{ \\vr \\vu (t, \\cdot) \\cdot \\vvarphi (t, \\cdot) } - \\intO{ \\vr_{0} \\vu_{0} \\cdot \\vvarphi(0, \\cdot) },\n\\ea\nwhere $\\SSS$ is defined by \\eqref{i3}.\n\n\\item For any $t \\in (0,T]$ and any test function $\\phi \\in \\DC([0,T] \\times \\Ov{\\Omega}\\times D)$, one has\n\\ba\\label{weak-form4}\n&\\int_0^t \\intO{\\int_D \\bigg[ \\psi \\partial_t \\phi + \\psi \\vu \\cdot \\Grad \\phi + \\sum_{i=1}^K (\\nabla_x \\vu)\\, q_i\\, \\psi \\cdot \\nabla_{q_i} \\phi - \\e\\nabla_x \\psi \\cdot \\nabla_x \\phi \\bigg]\\,\\dq}\\,\\dt'\\\\\n&=\\frac{1}{4\\lambda} \\sum_{i=1}^K\\sum_{j=1}^K A_{ij}\\int_0^t \\intO{ \\int_D M\\nabla_{q_j}\\left(\\frac{\\psi}{M}\\right)\\cdot \\nabla_{q_i}\\phi\\ \\dq} \\, \\dt' + \\intO{ \\int_D \\psi \\phi (t, \\cdot)- \\psi_{0} \\phi(0, \\cdot)\\,\\dq }.\n\\ea\n\n\\item The continuity equation holds in the sense of renormalized solutions:\n\\be\\label{weak-renormal}\n\\d_t b(\\vr) +\\Div_x (b(\\vr)\\vu) + (b'(\\vr)\\vr - b(\\vr))\\,\\Div_x \\vu =0 \\quad \\mbox{in} \\ \\mathcal{D}' ((0,T]\\times \\O),\n\\ee\nfor any\n\\be\\label{cond-b-renormal}\nb\\in C^1([0,\\infty)),\\quad |b'(s)s|+| b(s)| \\leq c<\\infty \\qquad \\forall\\, s\\in [0,\\infty).\n\\ee\n\\item Let $\\mathcal{F}(s) := s(\\log s -1)+1$ for $s>0$ and define $\\mathcal{F}(0):=\\lim_{s\\to 0+} \\mathcal{F}(s)=1$. For a.e. $t \\in (0,T]$, the\nfollowing \\emph{energy inequality} holds:\n\\ba\\label{energy}\n&\\int_{\\Omega} \\bigg[\n\\frac{1}{2} \\vr |\\vu|^2 + P(\\vr) +\\de\\, \\eta^2 + k \\int_D M \\mathcal{F} (\\widetilde \\psi)\\ \\dq\\bigg] (t, \\cdot) \\, \\dx\\\\\n&\\quad+ \\int_0^t \\intO{ \\mu^S \\bigg|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu)\\, \\II\\, \\bigg|^2 +\\mu^B |\\Div_x \\vu|^2 } \\, \\dt' + 2\\, \\e\\, \\de \\int_0^t \\intO{ |\\nabla_x \\eta|^2} \\, \\dt'\\\\\n&\\quad+ \\e\\, k \\int_0^t \\intO{ \\int_D M\\bigg|\\nabla_x \\sqrt{\\widetilde \\psi}\\, \\bigg|^2 \\, \\dq} \\,\\dt'\n+ \\frac{k\\,A_0}{4\\lambda} \\int_0^t \\intO{ \\int_D M \\bigg|\\nabla_{q} \\sqrt{\\widetilde \\psi}\\,\\bigg|^2 \\, \\dq} \\, \\dt'\\\\\n&\\leq \\int_{\\Omega} \\bigg[ \\frac{1}{2} \\vr_{0} |\\vu_{0} |^2 + P(\\vr_0) + \\de\\, \\eta_0^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_0}{M}\\bigg)\\, \\dq\\bigg] \\dx + \\int_0^t\\int_\\O \\vr \\,\\ff \\cdot \\vu \\,\\,\\dx\\, \\dt',\n\\ea\n where we have set $P(\\vr) := \\vr \\int_1^\\vr p(z)\/ z^2 \\, {\\rm d}z$ and $\\widetilde \\psi:=\\frac{\\psi}{M}$.\n\n\\end{itemize}\n\n\\end{definition}\n\n\n\n\\begin{remark} Definition \\ref{def-weaksl} is fairly standard. The energy\ninequality (\\ref{energy}) identifies an important class of weak solutions, usually termed \\emph{dissipative (finite-energy)}.\nWe note that, given a \\emph{smooth} solution, by tedious but rather straightforward calculations one can obtain the following a priori bound (see (1.22) in \\cite{Barrett-Suli}):\n\\ba\\label{energy0}\n&\\int_{\\Omega} \\left[\n\\frac{1}{2} \\vr |\\vu|^2 + P(\\vr) +\\de\\, \\eta^2 + k \\int_D M \\mathcal{F} (\\widetilde \\psi)\\, \\dq\\right] (t, \\cdot) \\, \\dx+ \\int_0^t \\intO{ \\SSS : \\Grad \\vu\\, } \\, \\dt' + 2\\, \\e\\, \\de \\int_0^t \\intO{ |\\nabla_x \\eta|^2}\\, \\dt'\\\\\n& \\quad + \\e\\, k \\int_0^t \\intO{ \\int_D M\\left|\\nabla_x \\sqrt{\\widetilde \\psi} \\,\\right|^2 \\dq} \\, \\dt' + \\frac{k}{4\\lambda} \\sum_{i=1}^K\\sum_{j=1}^K A_{ij}\\int_0^t \\intO{ \\int_D M \\nabla_{q_j} \\sqrt{\\widetilde \\psi} \\cdot \\nabla_{q_i} \\sqrt{\\widetilde \\psi} \\, \\dq} \\, \\dt' \\\\\n&\\leq \\int_{\\Omega} \\left[\n\\frac{1}{2} \\vr_{0} |\\vu_{0} |^2 + P(\\vr_0) + \\de\\, \\eta_0^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_0}{M}\\bigg)\\, \\dq\\right]\\!\\dx + \\int_0^t\\int_\\O \\vr \\,\\ff \\cdot \\vu \\, \\dx\\,\\dt',\\nn\n\\ea\nwhich then implies (\\ref{energy}). Indeed, thanks to the form of the Newtonian stress tensor in \\eqref{i3}, direct calculations yield that\n$$\n\\SSS : \\Grad \\vu = \\mu^S \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right|^2 +\\mu^B |\\Div_x \\vu|^2.\n$$\nHence, by the positive definiteness of the Rouse matrix $A=(A_{ij})_{1\\leq i,j\\leq K}$, and recalling that the smallest eigenvalue of $A$ is $A_0>0$, we deduce \\eqref{energy}.\n\\end{remark}\n\n\\subsection{Assumptions and main results}\n\nWe shall suppose that both $\\mu^S$ and $\\mu^B$ are $C^1$ functions of the polymer number density $\\eta$, and we adopt the following assumptions: there exist positive constants $c_j, \\ j=1,\\dots,5$, and an $\\o\\in\\R$ such that\n\\be\\label{mu-eta1}\nc_{1} (1+\\eta)^\\o \\leq \\mu^S(\\eta) \\leq c_2 (1+\\eta)^\\o, \\quad |(\\mu^S)'(\\eta)| \\leq c_3+c_4 (1+\\eta)^{\\o-1}, \\quad 0 \\leq \\mu^B(\\eta) \\leq c_5 (1+\\eta)^\\o\n\\quad \\forall\\,\\eta\\geq 0.\n\\ee\nIn addition, for the sake of simplicity, we shall assume that\n\\be\\label{press}\np(\\vr) = a \\vr^\\gamma, \\quad a > 0, \\ \\gamma > \\frac32.\n\\ee\n\nAs the complete proof of the existence of dissipative weak solutions in the special case of constant viscosity coefficients is already very long and technical (cf. \\cite{Barrett-Suli}), in the more general setting of polymer-number-dependent viscosity coefficients considered here we shall confine ourselves to establishing \\emph{weak sequential stability} of the family of dissipative weak solutions, whose existence we shall assume; we shall however indicate in Section \\ref{End} the main steps of a possible complete existence proof in the case of polymer-number-density-dependent viscosity coefficients.\n\nAccordingly, the main result of the paper reads as follows.\n\n\\begin{theorem}[Weak Sequential Stability]\\label{theorem} Let $\\{(\\vr_n,\\vu_n,\\psi_n,\\eta_n)\\}_{n\\in \\N}$ be a sequence of dissipative (finite-energy) weak solutions in the sense of Definition \\ref{def-weaksl} associated with the initial data $\\{(\\vr_{0,n},\\vu_{0,n},\\psi_{0,n},\\eta_{0,n})\\}_{n\\in\\N}$ satisfying:\n\\ba\\label{cond-ini-n}\n&\\vr_{0,n} \\geq 0 \\ \\mbox{a.e. in} \\ \\O, \\quad \\vr_{0,n} \\to \\vr_0 \\ \\mbox{strongly in}\\ L^\\g(\\O);\\\\\n&\\vu_{0,n} \\to \\vu_0 \\ \\mbox{in}\\ L^r(\\O;\\R^3) \\ \\mbox{for some $r>1$}\\ \\mbox{such that}\\ \\vr_{0,n}|\\vu_{0,n}|^2 \\to \\vr_0|\\vu_0|^2 \\ \\mbox{strongly in}\\ L^1(\\O);\\\\\n& \\psi_{0,n} \\geq 0 \\ \\mbox{a.e. in} \\ \\O\\times D,\\quad \\psi_{0,n} \\to \\psi_{0},\\quad \\psi_{0,n} \\left(\\log \\frac{\\psi_{0,n}}{M}\\right) \\to \\psi_0 \\left(\\log \\frac{\\psi_0}{M}\\right) \\ \\mbox{strongly in}\\ L^1(\\O\\times D);\\\\\n&\\eta_{0,n} = \\int_{D}\\psi_{0,n} \\, \\dq \\to \\eta_0 \\ \\mbox{strongly in} \\ L^2 (\\O).\n\\ea\n\nLet $\\vc{f} \\in L^\\infty((0,T) \\times \\Omega;\\R^3)$.\nSuppose that the exponent $\\g$ in \\eqref{press} and the parameter $\\o$ in \\eqref{mu-eta1} satisfy\n\\be\\label{mu-eta2}\n0\\leq \\o < \\frac{5}{3}, \\quad \\g>\\frac{3}{2}\n\\ee\nor\n\\be\\label{mu-eta3}\n -\\frac{4}{3}<\\o\\leq 0, \\quad \\g>\\frac{6}{4+3\\o}.\n\\ee\nThen, there exists a subsequence (not indicated) such that\n$$\n(\\vr_n,\\vu_n,\\psi_n,\\eta_n) \\to (\\vr,\\vu,\\psi,\\eta) \\quad \\mbox{as $n\\to \\infty$, in the sense of distributions (at least weakly in}\\ L^1),\n$$\nwhere the limit $(\\vr,\\vu,\\psi,\\eta)$ is a dissipative (finite-energy) weak solution in the sense of Definition \\ref{def-weaksl} associated with the initial data $(\\vr_0,\\vu_0,\\psi_0,\\eta_0)$.\n\n\n\\end{theorem}\n\nBefore embarking on the proof of Theorem \\ref{theorem} two remarks are in order.\n\n\\begin{remark}\\label{rem-thm1} The strong convergence assumptions in \\eqref{cond-ini-n} imply that\n\\ba\\label{cond-ini-n1}\n\\vr_{0,n} \\vu_{0,n} \\to \\vr_{0} \\vu_{0} \\quad \\mbox{strongly in}\\ L^{\\frac{2\\g}{1+\\g}}(\\O;\\R^3),\\qquad \\eta_0 = \\int_D \\psi_0\\, \\dq \\quad \\mbox{a.e. in}\\ \\O.\n\\ea\n\n\n\\end{remark}\n\n\\begin{remark}\\label{rem-thm2} It is important to note that we allow the viscosity coefficients $\\mu^B$ and $\\mu^S$ to decay to zero as the polymer number density tends to infinity; this is achieved at the expense of assuming a larger adiabatic exponent $\\g$; cf. \\eqref{mu-eta3}.\n\n\\end{remark}\n\n\n\nThe bulk of the rest of the paper is devoted to the proof of Theorem \\ref{theorem}. Comments on the possibility of carrying out a complete proof of the existence\nof dissipative (finite-energy) weak solutions are given in Section \\ref{End}. Throughout the rest of the paper, if there is no specification, $c$ will denote a positive constant depending only on the length $T$ of the time interval and the following quantity associated with the initial data:\n$$\n\\sup_{n\\in \\N}\\int_{\\Omega} \\bigg[\n\\frac{1}{2} \\vr_{0,n} |\\vu_{0,n} |^2 + P(\\vr_{0,n}) + \\de\\, \\eta_{0,n}^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_{0,n}}{M}\\bigg)\\, \\dq\\bigg] \\dx.\n$$\nWe emphasize, however, that the value of $c$ may vary from line to line.\n\n\\section{Preliminaries}\n\nIn this section, we recall some concepts that will be used systematically throughout the rest of the paper, including Maxwellian-weighted Lebesgue and Sobolev spaces, embeddings of spaces of Banach-space-valued weakly-continuous functions, the Div-Curl lemma, and Riesz operators.\n\n\\subsection{Maxwellian-weighted spaces}\\label{sec:def-LMr-HM1}\nFor any $r \\in [1,\\infty)$, $L_M^r(D)$ denotes the Maxwellian-weighted Lebesgue space over $D$ with norm\n\\be\\label{def-LMD1}\n\\|u\\|_{L^r_M(D)}:=\\left(\\int_{ D} M |u(x)|^r\\, \\dq \\right)^{\\frac{1}{r}}.\n\\ee\nSimilarly, we define $L_M^r(\\O\\times D):=L^r(\\O;L^r_M(D))$ and the Maxwellian-weighted Sobolev spaces\n\\ba\\label{def-SMD2}\nH^1_M(D)&:=\\left\\{ u \\in L_{\\rm loc}^1( D) : \\|u\\|_{H^1_M( D)}^2:=\\int_{ D} M\\big( |u|^2 + |\\nabla_q u|^2 \\big)\\, \\dq<\\infty \\right\\}.\\\\\nH^1_M(\\O\\times D)&:=\\left\\{ u \\in L_{\\rm loc}^1(\\O\\times D) : \\|u\\|_{H^1_M(\\O\\times D)}^2:=\\int_{\\O\\times D} M\\big( |u|^2 + |\\nabla_x u|^2 +|\\nabla_q u|^2\\big)\\, \\dq \\ \\dx<\\infty \\right\\}.\\nonumber\n\\ea\nThe proof of the following lemma can be found in Appendix C and Appendix D in \\cite{Barrett-Suli2}.\n\\begin{lemma}\\label{lem:HM1-emb} The normed spaces $L^r_M (D)$, $L_M^r(\\O\\times D)$, $H^1_M(D)$ and $H^1_M(\\O \\times D)$ are Banach spaces.\nThe embedding $H^1(\\O;L_M^2(D)) \\hookrightarrow L^6(\\O;L_M^2(D))$ is continuous, and the embeddings\n$H^1_M(D) \\hookrightarrow L_M^2(D)$, $H^1_M(\\O\\times D) \\hookrightarrow L^2_M(\\O\\times D)$ are compact.\n\\end{lemma}\n\n\\subsection{On $C_w([0, T];X)$ type spaces}\n\nLet $X$ be a Banach space. We denote by $C_w([0, T];X)$ the set of all functions $u\\in L^\\infty(0, T;X)$ such that the mapping\n$t \\in [0, T] \\mapsto \\langle \\phi,u(t)\\rangle_{X}\\in \\R$ is continuous on $[0, T]$ for all $\\phi \\in X'$. Here and throughout the paper, we use $X'$ to denote the\ndual space of $X$, and $\\langle \\cdot,\\cdot \\rangle_{X}$ to denote the duality pairing between $X'$ and $X$.\n\nWhenever $X$ has a predual $E$, in the sense that $E'=X$, we denote by $C_{w*}([0, T];X)$ the set of all functions $u\\in L^\\infty(0, T;X)$ such that the mapping $t \\in [0, T] \\mapsto \\langle u(t),\\phi\\rangle_{E}\\in \\R$ is continuous on $[0, T]$ for all $\\phi \\in E$. We reproduce Lemma 3.1 from \\cite{Barrett-Suli}.\n\\begin{lemma}\\label{lem-Cw-Cw*}\nSuppose that $X$ and $Y$ are Banach spaces.\n\\begin{itemize}\n\\item[(i)] Assume that the space $X$ is reflexive and is continuously embedded in the space $Y$; then,\n$$ L^\\infty(0, T;X) \\cap C_w([0, T];Y) = C_w([0, T];X).$$\n\\item[(ii)] Assume that $X$ has a separable predual $E$ and $Y$ has a predual $F$ such that $F$ is continuously\nembedded in $E$; then,\n $$ L^\\infty(0, T;X) \\cap C_{w*}([0, T];Y) = C_{w*}([0, T];X).$$\n\\end{itemize}\n\n\\end{lemma}\n\nWe recall an Arzel\\`{a}--Ascoli type result in $C_w([0,T];L^s(\\O))$. We refer to Lemma 6.2 in \\cite{N-book} for the proof.\n\\begin{lemma}\\label{lem-Cw} Let $r, s \\in (1,\\infty)$ and let $\\O$ be a bounded Lipschitz domain in $\\R^d, \\ d\\geq 2$. Suppose $\\{g_n\\}_{n\\in \\N}$ is a sequence of functions in $C_w([0, T]; L^s(\\O))$ such that $\\{g_n\\}_{n\\in \\N}$ is bounded in $C([0, T]; W^{-1,r}(\\O)) \\cap \\ L^\\infty(0, T;L^s(\\O))$. Then, there exists a subsequence (not indicated) such that the following hold:\n\\begin{itemize}\n\\item[(i)] $g_n\\to g$ in $C_w([0, T]; L^s(\\O))$;\n\\item[(ii)] If, in addition, $r\\leq \\frac{d}{d-1}$, or $r>\\frac{d}{d-1}$ and $s>\\frac{d\\,r}{d+r}$, then $g_n\\to g$ strongly in $C([0, T]; W^{-1,r}(\\O))$.\n\\end{itemize}\n\\end{lemma}\n\n\\subsection{Div-Curl lemma}\n\nWe recall the celebrated Div-Curl lemma due to L. Tartar \\cite{L.Tartar}.\n\\begin{lemma}\\label{lem-div-curl2}\nLet $Q\\subset \\R^d$ be a domain and let $\\{({\\bf U}_n,{\\bf V}_n)\\}_{n\\in \\N}$ be a sequence of functions such that\n$$\n{\\bf U}_n \\to {\\bf U} \\ \\mbox{weakly in $L^p(Q;\\R^d)$}, \\quad {\\bf V}_n \\to {\\bf V} \\ \\mbox{weakly in $L^{q}(Q;\\R^d)$}, \\quad \\mbox{as $n\\to \\infty$},\n$$\nwhere\n$\n\\frac{1}{p}+\\frac 1q =\\frac{1}{r}<1.\n$\nSuppose in addition that, for some $s>0$,\n$$\n\\{\\Div \\,{\\bf U}_n\\}_{n\\in \\N} \\ \\mbox{is precompact in $W^{-1,s}(Q)$},\\quad \\{{\\rm curl} \\,{\\bf V}_n\\}_{n\\in \\N} \\ \\mbox{is precompact in $W^{-1,s}(Q;\\R^{d\\times d})$}.\n$$\nThen,\n$\n{\\bf U}_n \\cdot {\\bf V}_n \\to {\\bf U} \\cdot {\\bf V} \\ \\mbox{weakly in}\\ L^r(Q).\n$\n\n\\end{lemma}\n\n\\subsection{On Riesz type operators}\\label{sec:Riesz}\n\nThe Riesz operator $\\RR_{j}$, $1\\leq j \\leq d$, in $\\R^d$ is defined as a Fourier integral operator with symbol $\\frac{\\xi_j}{|\\xi|}$. That is, for any $u\\in \\mathcal{S}'(\\R^d)$, where $\\mathcal{S}'(\\R^d)$ denotes the space of tempered distributions on $\\R^d$, $\\RR_j$ is defined by\n$$\n\\RR_{j}[u] := \\mathfrak{F}^{-1}\\left[\\frac{\\xi_j}{|\\xi|} \\mathfrak{F}[u]\\right],\n$$\nwhere $\\mathfrak{F}$ is the Fourier transform and $\\mathfrak{F}^{-1}$ is the inverse Fourier transform. We then define, for any $u\\in \\mathcal{S}'(\\R^d)$,\n$$\n\\RR_{ij} [u]:=\\RR_i \\circ \\RR_j u= \\mathfrak{F}^{-1}\\left[\\frac{\\xi_i\\xi_j}{|\\xi|^2} \\mathfrak{F}[u]\\right],\\quad 1\\leq i,j\\leq d.\n$$\n\n\nWe define $\\A_j$ by\n$\n\\A_{j} [u] :=- \\mathfrak{F}^{-1}\\left[\\frac{i\\xi_j}{|\\xi|^2} \\mathfrak{F}[u]\\right].\n$\nSince the derivative $\\d_j$ and the Laplacian $\\Delta$ can be seen as Fourier integral operators with symbols $i\\xi_j$ and $-|\\xi|^{-2}$, respectively, we can write\n$$\n\\RR_{ij} = \\d_i\\d_j{\\Delta}^{-1},\\quad \\A_j = -\\d_{j} \\Delta^{-1}.\n$$\n\nLet the matrix-valued operator $\\RR$ and the vector-valued operator $\\A$ be defined by\n\\be\\label{def-Reize}\n\\RR=(\\RR_{ij})_{i,j=1}^d=(\\nabla\\otimes\\nabla) \\Delta^{-1},\\quad \\A:=(A_j)_{j=1}^d = - \\nabla \\Delta^{-1}.\n\\ee\nWe have\n$\n\\RR_{ij}= - \\d_i \\A_j,\\ \\sum_{j=1}^d \\RR_{jj}= -\\sum_{j}\\d_j\\A_j =\\II.\n$ By Theorem 1.55 and Theorem 1.57 in \\cite{N-book} the following result holds.\n\\begin{lemma}\\label{lem-Riesz1} For any $p \\in (1,\\infty)$ the operators $\\RR_j$ and $\\RR_{ij}$, $1\\leq i, j\\leq d$, are bounded from $L^p(\\R^d)$ to $L^p(\\R^d)$. That is,\nthere exists a positive constant $c=c(p,d)$ such that\n$$\n\\|\\RR_j [u] \\|_{L^p(\\R^d)}+ \\|\\RR_{ij} [u] \\|_{L^p(\\R^d)} \\leq c(p,d) \\|u \\|_{L^p(\\R^d)}\\qquad \\forall\\,u\\in L^p(\\R^d).\n$$\nMoreover, for any $u\\in L^p(\\R^d)$, $v\\in L^{p'}(\\R^d)$, $11$ satisfying\n$\n\\frac{1}{r}+\\frac 1p-\\frac 1d < \\frac 1s < \\min\\{1,\\frac{1}{r}+\\frac 1p\\},\n$\nthere exists a constant $c=c(r,p,s,d)$ such that:\n$$\n\\| \\RR_{ij}[w\\,v] - w \\RR_{ij}[v] \\|_{W^{\\b,s}(\\R^d)}\\leq c\\, \\|w\\|_{W^{1,r}(\\R^d)} \\|v\\|_{L^{p}(\\R^d)},\n$$\nfor any $i, j \\in \\{1, \\dots, d\\}$, where $\\b\\in (0,1)$ satisfies\n$\n\\frac{\\b}{d} =\\frac 1d + \\frac 1s - \\frac{1}{r} - \\frac 1p.\n$\n\n\n\\end{lemma}\n\nIn Lemma \\ref{lem-Riesz3}, we used the fractional-order Sobolev space $W^{\\b,s}(\\R^d)$. Let $\\O$ be the whole space $\\R^d$ or a bounded Lipschitz domain in $\\R^d$. For any $\\b \\in (0,1)$ and $s\\in [1,\\infty) $, we define\n\\be\\label{def-frac-Sob}\n W^{\\b,s}(\\O):=\\left\\{u\\in L^s(\\O) : \\| u \\|_{W^{\\b,s}(\\O)}:= \\| u \\|_{L^{s}(\\O)} + \\left(\\int_\\O\\int_\\O \\frac{|u(x)-u(y)|^s}{|x-y|^{d+\\b s}} \\ \\dx \\ {\\rm d}y\\right)^{\\frac{1}{s}} <\\infty\\right\\}.\n\\ee\nWe recall the following classical compact embedding theorem (see Theorem 7.1 in \\cite{NPV}).\n\\begin{lemma}\\label{lem-frac-sob} Let $\\O\\subset \\R^d$ be a bounded Lipschitz domain and suppose that $\\b \\in (0,1)$ and $s\\in [1,\\infty)$; then, the embedding\nof $W^{\\b,s}(\\O)$ into $L^s(\\O)$ is compact, i.e. $W^{\\b,s}(\\O)\\hookrightarrow \\hookrightarrow L^s(\\O)$.\n\n\\end{lemma}\n\n\n\n\\section{Uniform bounds}\\label{sec:estimate}\n\nLet $(\\vr,\\vu,\\psi,\\eta)$ be a dissipative (finite-energy) weak solution in the sense of Definition \\ref{def-weaksl} with initial data $(\\vr_0,\\vu_0,\\psi_0,\\eta_0)$ satisfying \\eqref{ini-data}. This section is devoted to establishing bounds on $(\\vr,\\vu,\\psi,\\eta)$ under the hypotheses \\eqref{mu-eta1}--\\eqref{mu-eta3}.\n\n\n\\subsection{Gronwall's inequality and uniform bounds}\n\nWe begin by noting that\n\\ba\\label{ini-est2}\n\\int_{\\Omega} \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_0}{M}\\bigg)\\,\\dq \\,\\dx =\\int_{\\Omega} \\int_D \\bigg( \\psi_0\\log\\bigg(\\frac{\\psi_0}{M}\\bigg)-\\psi_0 +M\\bigg)\\,\\dq \\,\\dx.\n\\ea\nAs $ s\\log s \\geq s-1$ for all $s\\geq 0$, it follows that $\\psi_0\\log\\big(\\frac{\\psi_0}{M}\\big)\\geq \\psi_0 - M$, which then implies that\n$$\n\\left|\\psi_0\\log\\bigg(\\frac{\\psi_0}{M}\\bigg)-\\psi_0 +M\\right|\\leq 2 \\left|\\psi_0\\log\\bigg(\\frac{\\psi_0}{M}\\bigg)\\right|.\n$$\nThus, by \\eqref{ini-data}, we have that\n\\be\\label{ini-estf}\n\\int_\\O \\bigg[\\frac{1}{2} \\vr_{0} |\\vu_{0} |^2 + P(\\vr_0) + \\de\\, \\eta_0^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_0}{M}\\bigg)\\,\\dq\\bigg]\\dx \\leq c.\n\\ee\n\nSince $\\ff\\in L^\\infty((0,T)\\times \\O;\\R^3)$ and\n$$\n|\\vr \\,\\ff \\cdot \\vu |\\leq |\\ff| \\, |\\sqrt \\vr|\\,| \\sqrt\\vr \\vu| \\leq |\\ff|\\left( \\vr + \\vr|\\vu|^2 \\right) \\leq |\\ff|\\left(1 + \\vr^\\g + \\vr|\\vu|^2 \\right),\n$$\nby using Gronwall's inequality we deduce from \\eqref{energy} and \\eqref{ini-estf} that, for a.e. $t\\in (0,T)$,\n\\ba\\label{energy-f}\n&\\int_{\\Omega} \\bigg[ \\frac{1}{2} \\vr |\\vu|^2 + P(\\vr) +\\de\\, \\eta^2 + k \\int_D M \\mathcal{F} (\\widetilde \\psi)\\, \\dq\\bigg] (t, \\cdot) \\,\\dx\\\\\n&\\quad+ \\int_0^t \\intO{ \\mu^S(\\eta) \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right|^2 +\\mu^B(\\eta) |\\Div_x \\vu|^2 } \\, \\dt' + 2\\, \\e\\, \\de \\int_0^t \\intO{ |\\nabla_x \\eta|^2} \\, \\dt' \\\\\n&\\quad+\\e\\, k \\int_0^t \\intO{ \\int_D M\\left|\\nabla_x \\sqrt{\\widetilde \\psi}\\, \\right|^2\\dq} \\, \\dt+\\frac{k\\,A_0}{4\\lambda} \\int_0^t \\intO{ \\int_D M \\left|\\nabla_{q} \\sqrt{\\widetilde \\psi}\\,\\right|^2\\dq} \\, \\dt' \\\\\n&\\leq c(1+T)\\, {\\rm e}^{ct}.\n\\ea\nIn the rest of this section, we shall establish additional bounds on the unknowns, one by one, by using \\eqref{energy-f}.\n\n\\subsection{Bounds on the fluid density and the polymer number density}\n\nFrom \\eqref{energy-f}, we have\n\\be\\label{est-r-eta1}\n\\vr\\in L^\\infty(0,T;L^\\gamma(\\O)), \\quad \\eta\\in L^\\infty(0,T;L^2(\\O))\\cap L^2(0,T;W^{1,2}(\\O)), \\quad \\vr |\\vu|^2 \\in L^{\\infty}(0,T;L^{1}(\\O)).\n\\ee\nBy Sobolev embedding and interpolation it follows that\n\\be\\label{est-r-eta3}\n \\eta\\in L^\\infty(0,T;L^2(\\O)) \\cap L^2(0,T;L^6(\\O)) \\hookrightarrow L^{a}(0,T;L^{\\frac{6a}{3a-4}}(\\O)),\\quad 2\\leq a\\leq \\infty.\n \\ee\nThe bounds in \\eqref{est-r-eta1} then imply that\n\\be\\label{est-ru2}\n\\vr \\vu = \\sqrt{\\vr} \\sqrt{\\vr}\\vu \\in L^{\\infty}(0,T;L^{\\frac{2\\g}{\\g+1}}(\\O;\\R^3)).\n\\ee\n\n\\subsection{Bounds on the fluid velocity field}\\label{sec:est-u}\n\nBy \\eqref{energy-f} we deduce that\n\\be\\label{est-u1}\ng:= \\sqrt{\\mu^S(\\eta)} \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right| \\in L^2(0,T;L^2(\\O)).\n\\ee\nWe recall that $\\mu^S(\\eta)$ fulfills the hypotheses stated in \\eqref{mu-eta1} with $\\o$ satisfying \\eqref{mu-eta2} or \\eqref{mu-eta3}. We begin by considering the case when $\\o\\geq 0$; \\eqref{mu-eta1} then implies that $\\mu^{-1}$ is uniformly bounded. Thus,\n\\be\\label{est-u2}\n \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right| \\in L^2(0,T;L^2(\\O));\n\\ee\nwhence\nKorn's inequality (see \\cite{Dain}) with the no-slip boundary condition on $\\vu$ implies that\n\\be\\label{est-u4}\n|\\nabla_x \\vu| \\in L^2(0,T;L^2(\\O)), \\qquad \\mbox{and hence}\\qquad \\vu \\in L^2(0,T;W^{1,2}_0(\\O;\\R^3)).\n\\ee\nOn the other hand, when $\\o\\leq 0$ satisfies the constraint \\eqref{mu-eta3}, that is $-4\/3<\\o\\leq 0$, then, by \\eqref{mu-eta1} and \\eqref{est-r-eta3}, we have that\n\\be\\label{est-mu1}\n\\mu(\\eta)^{-1}\\in L^\\infty(0,T;L^{\\frac{2}{|\\o|}}(\\O)) \\cap L^{\\frac{2}{|\\o|}}(0,T;L^{\\frac{6}{|\\o|}}(\\O)) \\cap L^{\\frac{10}{3|\\o|}}((0,T)\\times \\O).\n\\ee\nThus, from \\eqref{est-u1}, we deduce that\n\\be\\label{est-u11}\n \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right| \\in L^2(0,T;L^{\\frac{4}{2+|\\o|}}(\\O)) \\cap L^{\\frac{4}{2+|\\o|}}(0,T;L^{\\frac{12}{6+|\\o|}}(\\O))\\cap L^{\\frac{20}{10+3|\\o|}}((0,T)\\times \\O).\n\\ee\nHence, taking advantage of the no-slip boundary condition, we may use\nKorn's inequality to obtain\n\\be\\label{est-u13}\n\\vu \\in L^2(0,T;W_0^{1,\\frac{4}{2+|\\o|}}(\\O;\\R^3)) \\cap L^{\\frac{4}{2+|\\o|}}(0,T;W_0^{1,\\frac{12}{6+|\\o|}}(\\O;\\R^3))\\cap L^{\\frac{20}{10+3|\\o|}}(0,T;W_0^{1,\\frac{20}{10+3|\\o|}}(\\O;\\R^3)).\n\\ee\n\n\\subsection{Bounds on the Newtonian stress tensor}\n\nConsider first the case when \\eqref{mu-eta2} holds with $\\o\\geq 0$. Let us write\n$$\n\\mu^S(\\eta) \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right| = g \\sqrt{\\mu^S(\\eta)},\n$$\nwhere $g\\in L^2((0,T)\\times \\O)$ is defined by \\eqref{est-u1}. Thanks to \\eqref{mu-eta1}, \\eqref{est-r-eta3}, \\eqref{est-u4}, and by a similar argument as in the derivation of \\eqref{est-u13}, we obtain\n\\be\\label{est-SS1}\n\\SSS\\in L^2(0,T;L^{\\frac{4}{2+|\\o|}}(\\O;\\R^{3\\times 3})) \\cap L^{\\frac{4}{2+|\\o|}}(0,T;L^{\\frac{12}{6+|\\o|}}(\\O;\\R^{3\\times 3}))\\cap L^{\\frac{20}{10+3|\\o|}}((0,T)\\times \\O;\\R^{3\\times 3}).\n\\ee\nOn the other hand, when $\\o\\leq 0$, by observing that $\\sqrt{\\mu^S} + \\sqrt{\\mu^B}\\leq c<\\infty$ we deduce that\n\\be\\label{est-SS2}\n\\SSS\\in L^2((0,T)\\times\\O;\\R^{3\\times 3}).\n\\ee\n\n\n\\subsection{Bounds on the probability density function}\\label{sec:est-psi}\n\nIt follows from \\eqref{energy-f} that\n\\be\\label{est-psi1}\n\\mathcal{F}(\\tpsi) \\in L^\\infty(0,T;L^1_M(\\O\\times D)), \\quad \\sqrt{\\tpsi} \\in L^2(0,T;H_M^1(\\O\\times D)),\n\\ee\nwhere we recall that $\\tpsi := \\psi\/M$. Consequently, by Sobolev embedding and thanks to Lemma \\ref{lem:HM1-emb} we then have that $\\sqrt{\\tpsi} \\in L^2(0,T;L^6(\\O;L^2_M(D)))$.\n\nIf $s>{\\rm e}^2$, then $s\\log s > 2s >2(s-1)$, which then implies that $\\mathcal{F}(s)=s\\log s -(s-1)>\\frac{1}{2}{s\\log s}$ for $s>{\\rm e}^2$. Thus,\n\\ba\\label{est-psi3}\n\\|\\tpsi\\log\\tpsi\\|_{L^1_M(\\O\\times D))}&=\\int_{\\O\\times D} |\\tpsi\\log\\tpsi|\\,M\\,\\dq\\,\\dx = \\int_{0 \\leq \\tpsi \\leq {\\rm e}^2} |\\tpsi\\log\\tpsi|\\,M\\,\\dq\\,\\dx +\\int_{\\tpsi > e^2} |\\tpsi\\log\\tpsi|\\,M\\,\\dq\\,\\dx \\\\\n& \\leq 2 {\\rm e}^2\\int_{\\O\\times D}\\,M\\,\\dq\\,\\dx + 2 \\int_{\\tpsi > {\\rm e}^2} |\\mathcal{F}(\\tpsi)|\\, M\\,\\dq\\,\\dx \\leq c,\n\\ea\nwhere we have used \\eqref{pt1-Mi-Ui} and \\eqref{est-psi1}. This implies that\n\\be\\label{est-psi5}\n\\tpsi\\log\\tpsi \\in L^\\infty(0,T;L^1_M(\\O\\times D)),\\quad \\psi\\log\\psi \\in L^\\infty(0,T;L^1(\\O\\times D)).\n\\ee\n\nClearly,\n\\be\\label{nabla-psi}\n\\nabla_{x}\\tpsi = 2 \\sqrt{\\tpsi} \\ \\nabla_{x}\\sqrt{\\tpsi},\\quad \\nabla_{q}\\tpsi = 2 \\sqrt{\\tpsi} \\ \\nabla_{q}\\sqrt{\\tpsi}.\n\\ee\nBy \\eqref{nabla-psi} and H\\\"older's inequality we therefore have that\n\\ba\\label{est-psi8}\n\\|\\nabla_{q,x} \\tpsi\\|_{L^1_M(D;\\R^{3(K+1)})} & = \\int_{D} | M\\,\\nabla_{q,x}\\tpsi | \\, \\dq = 2\\int_{D} M \\sqrt{\\tpsi} \\, \\left|\\nabla_{q,x}\\sqrt{\\tpsi}\\,\\right|\\dq \\\\\n&\\leq 2 \\left(\\int_{D} M \\left|\\nabla_{q,x}\\sqrt{\\tpsi}\\,\\right|^2 \\, \\dq \\right)^{\\frac 12}\\left(\\int_{D} M \\tpsi \\, \\dq \\right)^{\\frac 12}=2 \\left\\|\\nabla_{q,x}\\sqrt{\\tpsi}\\,\\right\\|_{L^2_M(D;\\R^{3(K+1)})} \\eta^{\\frac 12}.\n\\ea\nThus, we benefit from the estimates in \\eqref{est-r-eta3} for $\\eta$ and deduce that\n\\ba\\label{est-psi9}\n&\\nabla_{q,x} \\tpsi \\in L^2(0,T;L^{\\frac 43}(\\O;L^1_M(D;\\R^{3(K+1)})))\\cap L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;L^1_M(D;\\R^{3(K+1)}))).\n\\ea\nBy observing that $\\|\\nabla_x \\tpsi\\|_{L^1_M(D;\\R^{3})}=\\|\\nabla_x\\psi\\|_{L^1(D;\\R^{3})}$ we obtain\n\\ba\\label{est-psi10}\n\\nabla_x \\psi \\in L^2(0,T;L^{\\frac 43}(\\O;L^1(D;\\R^{3})))\\cap L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;L^1(D;\\R^{3}))).\n\\ea\n\n\n\n\\subsection{Bounds on the extra-stress tensor}\n\nWe recall that the extra-stress tensor can be expressed as (see \\eqref{weak-est})\n$$\\TT := k \\left(\\sum_{i=1}^K \\CC_i(\\psi)\\right)-\\left(k(K+1) \\eta + \\de\\, \\eta^2\\right) \\II. $$\nThanks to the bounds on $\\eta$ stated in \\eqref{est-r-eta1} and \\eqref{est-r-eta3}, we have that\n\\be\\label{est-TT1}\n\\eta^2\\in L^\\infty(0,T;L^1(\\O)) \\cap L^1(0,T;L^3(\\O)).\n\\ee\n\nAccording to \\eqref{pt1-Mi-Ui}, we have $M=0$ on $\\d D$. We then deduce by using \\eqref{def-Ci} and \\eqref{pt2-M} that\n\\ba\\label{est-TT2}\n\\CC_i(\\psi) &= \\CC_i(M\\tpsi) = \\int_D M \\tpsi\\, U_i'\\bigg(\\frac{q_i^2}{2}\\bigg)\\, q_i q_i^{\\rm T} \\, \\dq = - \\int_{D} \\tpsi\\, (\\nabla_{q_i} M)q_i^{\\rm T}\\, \\dq\\\\\n&=\\int_{D} M\\, (\\nabla_{q_i}\\tpsi) q_i^{\\rm T}\\, \\dq + \\left(\\int_{D} M \\tpsi \\, \\dq\\right)\\II = \\int_{D} M (\\nabla_{q_i}\\tpsi) q_i^{\\rm T}\\, \\dq + \\eta \\II.\n\\ea\nThus, by \\eqref{nabla-psi}, H\\\"older's inequality implies that\n\\ba\\label{est-TT4}\n\\left|\\int_{D} M (\\nabla_{q_i}\\tpsi) q_i^{\\rm T}\\, \\dq \\right| \\leq c \\left(\\int_{D} M \\left(\\nabla_{q_i}\\sqrt{\\tpsi}\\right)^2 \\, \\dq \\right)^{\\frac 12}\\left(\\int_{D} M \\tpsi \\, \\dq \\right)^{\\frac 12}= c \\left(\\int_{D} M \\left(\\nabla_{q_i}\\sqrt{\\tpsi}\\right)^2 \\, \\dq \\right)^{\\frac 12} \\eta ^{\\frac 12}.\\nn\n\\ea\nBy \\eqref{est-psi1} we have\n$$\n\\left(\\int_{D} M \\left(\\nabla_{q_i}\\sqrt{\\tpsi}\\right)^2 \\dq \\right)^{\\frac 12} \\in L^2(0,T;L^2(\\O)).\n$$\nThus, by \\eqref{est-r-eta3} and \\eqref{est-TT1}, we obtain\n\\be\\label{est-TTf}\n\\TT \\in L^2(0,T;L^{\\frac 43}(\\O;\\R^{3\\times 3}))\\cap L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;\\R^{3\\times 3})).\n\\ee\n\n\\subsection{Higher integrability of the fluid density and the pressure}\\label{sec:est-higher-vr}\n\nFrom the energy inequality \\eqref{energy} we deduce that\n$\\vr\\in L^\\infty(0,T;L^\\g(\\O))$, which implies that $p(\\vr)\\in L^\\infty(0,T;L^1(\\O))$; unfortunately, $p(\\vr)$ is only in $L^1$ with respect to the spatial variable, $x$. In order to improve the\nintegrability of the pressure with respect to the spatial variable, one may use the so-called Bogovski{\\u{\\i}} operator (see \\cite{bog}), exactly as in \\cite{FNP, F-book, N-book}.\nWe recall the following lemma whose proof can be found in Chapter III of Galdi's book \\cite{Galdi-book}.\n\\begin{lemma}\\label{lem-bog}\n Let $p \\in (1,\\infty)$ and suppose that $\\O\\subset \\R^d$ is a bounded Lipschitz domain. Let $L^{p}_0(\\Omega)$ be the space of $L^p(\\Omega)$ functions with zero mean-value.\nThen, there exists a linear operator $\\mathcal{B}_\\O$ from $L_0^p(\\O)$ to $W_0^{1,p}(\\O;\\R^d)$ such that\n$$\n\\Div_x\\, \\mathcal{B}_\\O (f) =f \\quad \\mbox{in} \\ \\O; \\quad \\|\\mathcal{B}_\\O (f)\\|_{W_0^{1,p}(\\O;\\R^d)} \\leq c (d,p,\\O)\\, \\|f\\|_{L^p(\\O)}\\quad\\, \\forall\\, f\\in L_0^p(\\O),\n$$\nwhere the constant $c$ depends only on $p$, $d$ and the Lipschitz character of $\\O$. If, in addition, $f=\\Div_x \\vg$ for some $\\vg \\in L^{q}, \\ 10$ sufficiently small, that\n\\be\\label{est-bog-p1}\n\\int_0^t\\int_\\O \\phi(t)\\, \\vr^\\g\\, b_{n}(\\vr)\\, \\dx \\, \\dt \\leq c\\qquad \\forall\\,n \\geq 0.\n\\ee\nLetting $n\\to \\infty$ in \\eqref{est-bog-p1} and approximating $1$ by $C_c^\\infty ((0,T))$ functions finally gives\n\\be\\label{est-bog-pf1}\n \\vr \\in L^{\\g+\\th}((0,T)\\times \\O),\\quad p(\\vr) \\in L^{1+\\frac{\\th}{\\g}}((0,T)\\times \\O) \\quad \\mbox{for some $\\th>0$}.\n\\ee\n\n\\medskip\n\n\n\\begin{remark}\\label{rem-bog-th}\n\n\n\nIn order to obtain \\eqref{est-bog-pf1} for some positive $\\th$, the conditions imposed in \\eqref{mu-eta2} and \\eqref{mu-eta3} can be relaxed. By careful analysis the following constraints are found to be sufficient:\n\\be\\label{mu-eta20}\n\\g>\\frac 32,\\ 0\\leq \\o < \\frac{10}{3} \\quad \\mbox{or} \\quad -2<\\o\\leq 0,\\ \\g>\\frac{6}{4+3\\o}.\n\\ee\nThe more restrictive conditions featuring in \\eqref{mu-eta2} and \\eqref{mu-eta3} are needed later on, in Section \\ref{vis-flux}, in order to prove the so-called {\\rm effective viscous flux} equality (see Remark \\ref{rem-mu-eta}).\n\\end{remark}\n\n\\subsection{Bounds on the time derivative and continuity}\\label{sec:est-time}\n\nThis section is devoted to establishing bounds on the time derivatives $(\\d_t \\vr, \\ \\d_t\\eta, \\ \\d_t (\\vr \\vu), \\ \\d_t \\psi)$.\n\nAs $\\d_t \\vr = - \\Div_x(\\vr\\vu)$, the bound in \\eqref{est-ru2} implies that\n$\\d_t \\vr \\in L^\\infty(0,T; W^{-1,\\frac{2\\g}{\\g+1}}(\\O)).$ Then, by \\eqref{est-r-eta1} and Lemma \\ref{lem-Cw-Cw*}, we have that\n$$\n\\vr \\in C_w([0,T]; L^\\g(\\O)).\n$$\n\n\n\\medskip\n\nRecall that $\\d_t \\eta = - \\Div_x(\\eta\\vu) + \\e \\Delta_x \\eta$. For $\\o\\geq 0$ satisfying \\eqref{mu-eta2}, we have $\\vu \\in L^2(0,T; W_0^{1,2}(\\O;\\R^3))$, which is embedded in $L^2(0,T; L^6(\\O;\\R^3))$. Thanks to \\eqref{est-r-eta1}, we then obtain $\\d_t \\eta \\in L^2(0,T; W^{-1,\\frac{3}{2}}(\\O)).$\nIf on the other hand $\\o\\leq 0$ satisfies \\eqref{mu-eta3}, then we have $\\vu \\in L^2(0,T;W_0^{1,\\frac{4}{2+|\\o|}}(\\O;\\R^3)) \\hookrightarrow L^2(0,T; L^\\frac{12}{3|\\o|+2}(\\O;\\R^3))$. Consequently, $\\d_t \\eta \\in L^2(0,T; W^{-1,\\frac{12}{3|\\o|+8}}(\\O))$.\nThus, in both cases, by \\eqref{est-r-eta1} and Lemma \\ref{lem-Cw-Cw*}, we have\n$$\n\\eta \\in C_w([0,T]; L^2(\\O)).\n$$\n\n\\medskip\n\nNext, we recall that $\\d_t(\\vr\\vu)=- \\Div_x (\\vr \\vu \\otimes \\vu) - \\nabla_x p(\\vr) +\n\\Div_x \\SSS +\\Div_x \\TT +\\vr\\, \\ff$. We first consider the case with $\\o\\geq 0$ satisfying \\eqref{mu-eta2}. By the estimates in Section \\ref{sec:est-u} we have $\\vu \\in L^2(0,T; L^6(\\O;\\R^3))$. Since $\\vr\\in L^\\infty(0,T;L^\\g(\\O))$ with $\\g>\\frac{3}{2}$, together with \\eqref{est-r-eta1}, we obtain\n\\be\\label{est-ruu}\n\\vr \\vu \\otimes \\vu \\in L^1(0,T; L^{\\frac{3\\g}{\\g+3}}(\\O;\\R^{3\\times 3}))\\cap L^\\infty(0,T; L^{1}(\\O;\\R^{3\\times 3})) \\hookrightarrow L^r(0,T; L^{r}(\\O;\\R^{3\\times 3})) \\quad \\mbox{for some $r>1$}.\n\\ee\nBy \\eqref{est-SS1}, \\eqref{est-bog-pf1}, \\eqref{est-TTf}, together with \\eqref{est-ruu}, we deduce that\n\\be\\label{est-dt-ru}\n\\d_t (\\vr\\vu) \\in L^r(0,T; W^{-1,r}(\\O;\\R^{3})) \\quad \\mbox{for some $r>1$}.\n\\ee\nFor the case with $\\o\\leq 0$ satisfying \\eqref{mu-eta3}, a similar argument gives the same result as in \\eqref{est-dt-ru}. Then, by \\eqref{est-ru2}, \\eqref{est-dt-ru} and Lemma \\ref{lem-Cw-Cw*}, we have in both cases that\n$$\\vr\\vu \\in C_w([0,T]; L^\\frac{2\\g}{\\g+1}(\\O;\\R^3)).$$\n\n\nBy the Fokker--Planck equation \\eqref{eq-psi} for $\\psi$, the estimates in \\eqref{est-psi9} and \\eqref{est-psi10}, direct calculations yield that\n\\be\\label{est-dt-psi}\n\\d_t \\psi \\in\nL^2(0,T; W^{s,2}(\\O\\times D)') \\quad \\mbox{with $s>1 + \\frac{3}{2}(K+1)$},\n\\ee\nwhere we have used that $W^{s,2}(\\Omega \\times D) \\hookrightarrow W^{1,\\infty}(\\Omega \\times D)$ for $s>1+\\frac{3}{2}(K+1)$.\nMoreover, by the estimates \\eqref{est-psi5} and \\eqref{est-dt-psi}, using Lemma \\ref{lem-Cw-Cw*} and the same argument as in Section 4.5 in \\cite{Barrett-Suli}, we deduce that $$\\psi \\in C_w([0,T];L^1(\\O\\times D)).$$\n\n\n\n\n\\subsection{Summary}\n\nWe summarize the results obtained in this section. Let $(\\vr,\\vu,\\psi,\\eta)$ be a dissipative weak solution in the sense of Definition \\ref{def-weaksl} associated\nwith initial data $(\\vr_0,\\vu_0,\\psi_0,\\eta_0)$ satisfying \\eqref{ini-data}. Then,\n\\ba\\label{est-sum1}\n& \\vr\\in C_w([0,T);L^\\gamma(\\O))\\cap L^{\\g+\\th}((0,T)\\times \\O);\\quad \\eta \\in C_w([0,T];L^2(\\O)) \\cap L^2(0,T;W^{1,2}(\\O));\\\\\n& \\vr |\\vu|^2 \\in L^{\\infty}(0,T;L^{1}(\\O)),\\quad \\vr \\vu \\in C_w([0,T];L^{\\frac{2\\g}{\\g+1}}(\\O;\\R^3));\\quad \\TT \\in L^2(0,T;L^{\\frac 43}(\\O;\\R^{3\\times 3}))\\cap L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;\\R^{3\\times 3}));\\\\\n&\\vu \\in L^2(0,T;W^{1,2}_0(\\O;\\R^3)) \\ \\mbox{if $\\o\\geq 0$ satisfies \\eqref{mu-eta2}};\\\\\n& \\vu \\in L^2(0,T;W_0^{1,\\frac{4}{2+|\\o|}}(\\O;\\R^{3})) \\cap L^{\\frac{4}{2+|\\o|}}(0,T;W_0^{1,\\frac{12}{6+|\\o|}}(\\O;\\R^{3}))\\cap L^{\\frac{20}{10+3|\\o|}}(0,T;W_0^{1,\\frac{20}{10+3|\\o|}}(\\O;\\R^{3})) \\\\\n&\\qquad \\mbox{if $\\o\\leq 0$ satisfies \\eqref{mu-eta3}};\\\\\n&\\mathcal{F}(\\tpsi), \\tpsi\\log\\tpsi, \\ \\tpsi \\in L^\\infty(0,T;L^1_M(\\O\\times D)),\\quad \\sqrt{\\tpsi} \\in L^2(0,T;H_M^1(\\O\\times D)),\\\\\n&\\qquad \\nabla_{q,x} \\tpsi \\in L^2(0,T;L^{\\frac 43}(\\O;L^1_M(D;\\R^{3(K+1)})))\\cap L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;L^1_M(D;\\R^{3(K+1)})));\\\\\n&\\mathcal{F}(\\psi),\\ \\psi\\log\\psi \\ \\in L^\\infty(0,T;L^1(\\O\\times D)),\\quad \\psi \\in C_w([0,T];L^1(\\O\\times D)),\\\\\n &\\qquad \\nabla_x \\psi \\in L^2(0,T;L^{\\frac 43}(\\O;L^1(D;\\R^{3})))\\cap L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;L^1(D;\\R^{3})));\n\\ea\nand\n\n\\begin{equation}\\label{est-sum2}\n\\begin{aligned}\n&\\d_t \\vr \\in L^\\infty(0,T; W^{-1,\\frac{2\\g}{\\g+1}}(\\O)), \\quad \\d_t (\\vr\\vu) \\in L^r(0,T; W^{-1,r}(\\O;\\R^{3})) &&\\quad \\mbox{for some $r>1$};\\\\\n& \\d_t \\eta \\in L^2(0,T; W^{-1,\\frac{3}{2}}(\\O)) &&\\quad \\mbox{if $\\o\\geq 0$ satisfies \\eqref{mu-eta2}};\\\\\n&\\d_t \\eta \\in L^2(0,T; W^{-1,\\frac{12}{3|\\o|+8}}(\\O)) &&\\quad \\mbox{if $\\o\\leq 0$ satisfies \\eqref{mu-eta3}};\\\\\n&\\d_t \\psi \\in L^2(0,T; W^{s,2}(\\O\\times D)') &&\\quad \\mbox{with $s>1 + \\frac{3}{2}(K+1)$}.\n\\end{aligned}\n\\end{equation}\nIt is important to note that all of the above inclusions are consequences of bounds that depend only on the initial energy, the final time $T$, and the structural constants in the hypotheses imposed on the constitutive relations.\n\n\n\n\\section{Passing to the limit}\\label{sec:lim}\n\nLet $(\\vr_n,\\vu_n,\\psi_n,\\eta_n)_{n\\in \\N}$ be a sequence of the dissipative (finite-energy) weak solutions satisfying the assumptions in Theorem \\ref{theorem}. Then, the energy inequality \\eqref{energy} and \\eqref{energy-f} give, respectively,\n\\ba\\label{energy-n}\n&\\int_{\\Omega} \\bigg[\n\\frac{1}{2} \\vr_n |\\vu_n|^2 + P(\\vr_n) +\\de\\, \\eta_n^2 + k \\int_D M \\mathcal{F} (\\psi_n)\\, \\dq\\bigg] (t, \\cdot) \\, \\dx + 2\\, \\e\\, \\de \\int_0^t \\intO{ |\\nabla_x \\eta_n|^2} \\, \\dt'\\\\\n&\\quad+ \\int_0^t \\intO{ \\mu^S(\\eta_n) \\left|\\frac{\\nabla \\vu_n + \\nabla^{\\rm T} \\vu_n}{2} - \\frac{1}{3} (\\Div_x \\vu_n) \\II \\right|^2 +\\mu^B(\\eta_n) |\\Div_x \\vu_n|^2 } \\, \\dt' \\\\\n&\\quad+ \\e\\, k \\int_0^t \\intO{ \\int_D M\\left|\\nabla_x \\sqrt{\\tpsi_n} \\right|^2 \\dq} \\, \\dt+\\frac{k\\,A_0}{4\\lambda} \\int_0^t \\intO{ \\int_D M \\left|\\nabla_{q} \\sqrt{\\tpsi_n}\\right|^2 \\dq} \\, \\dt'\\\\\n&\\leq \\int_{\\Omega} \\bigg[\n\\frac{1}{2} \\vr_{0,n} |\\vu_{0,n} |^2 + P(\\vr_{0,n}) + \\de\\, \\eta_{0,n}^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_{0,n}}{M}\\bigg)\\,\\dq\\bigg]\\dx + \\int_0^t\\int_\\O \\vr_n \\,\\ff \\cdot \\vu_n \\, \\dx\\, \\dt'\n\\ea\n and, therefore,\n \\ba\\label{energy-f-n}\n&\\int_{\\Omega} \\bigg[ \\frac{1}{2} \\vr_n |\\vu_n|^2 + P(\\vr_n) +\\de\\, \\eta_n^2 + k \\int_D M \\mathcal{F} (\\widetilde \\psi_n)\\,\\dq\\bigg] (t, \\cdot) \\, \\dx + 2\\, \\e\\, \\de \\int_0^t \\intO{ |\\nabla_x \\eta_n|^2} \\, \\dt' \\\\\n&\\quad+ \\int_0^t \\intO{ \\mu^S(\\eta_n) \\left|\\frac{\\nabla \\vu_n + \\nabla^{\\rm T} \\vu_n}{2} - \\frac{1}{3} (\\Div_x \\vu_n) \\II \\right|^2 +\\mu^B(\\eta_n) |\\Div_x \\vu_n|^2 } \\ \\dt' \\\\\n&\\quad+\\e\\, k \\int_0^t \\intO{ \\int_D M\\left|\\nabla_x \\sqrt{\\widetilde \\psi_n} \\right|^2\\dq} \\ \\dt'+\\frac{k\\,A_0}{4\\lambda} \\int_0^t \\intO{ \\int_D M \\left|\\nabla_{q} \\sqrt{\\widetilde \\psi_n}\\right|^2\\dq} \\, \\dt' \\\\\n&\\leq c(1+T)\\, {\\rm e}^t.\n\\ea\nThus, from the results in Section \\ref{sec:estimate}, this solution sequence $(\\vr_n,\\vu_n,\\psi_n,\\eta_n)_{n\\in \\N}$ satisfies the uniform bounds \\eqref{est-sum1} and \\eqref{est-sum2}.\n\nThe present section is devoted to studying the limit of this solution sequence. We remark that, throughout this section, the limits are taken up to subtractions of subsequences without identification.\n\n\n\\subsection{Convergence of the fluid density}\\label{sec:vr-lim}\n\nWe have that $\\{\\vr_n\\}_{n\\in \\N}$ is a sequence in $C_w([0,T];L^\\gamma(\\O))$ satisfying\n\\be\\label{vrn-est}\n\\sup_{n\\in \\N} \\left(\\|\\vr_n\\|_{L^\\infty(0,T;L^\\gamma(\\O))} + \\|\\d_t \\vr_n \\|_{L^\\infty(0,T; W^{-1,\\frac{2\\g}{\\g+1}}(\\O))} \\right) \\leq c.\n\\ee\nBy Sobolev embedding one has\n$$\nL^\\g(\\O) \\hookrightarrow W^{-1,\\frac{3\\g}{3-\\g}}(\\Omega) \\quad \\mbox{ if $\\g<3$};\\quad L^\\g(\\O) \\hookrightarrow W^{-1,s}(\\Omega) \\quad \\mbox{ for any $s \\in (1,\\infty)$, \\ if $\\g\\geq3$}.\n$$\nThus, by applying Lemma \\ref{lem-Cw}, we deduce that\n\\ba\\label{vrn-lim}\n& \\vr_n \\to \\vr \\quad \\mbox{in}\\ C_w([0,T];L^\\gamma(\\O));\\\\\n& \\vr_n \\to \\vr \\quad \\mbox{strongly in} \\ C([0,T];W^{-1,r}(\\O)) \\quad \\mbox{for any $r\\geq \\frac{3}{2}$ such that $\\frac{3r}{3+r}<\\g$}.\n\\ea\n\n\\subsection{Convergence of the fluid velocity field}\\label{sec:lim-u}\n\nBy Sobolev embedding, for the case $\\o\\geq 0$, we have\n\\be\\label{vun-limit1}\n\\vu_n \\to \\vu \\quad \\mbox{weakly in}\\ L^2(0,T;W^{1,2}_0(\\O;\\R^3))\\ \\mbox{and in}\\ L^2(0,T;L^{6}(\\O;\\R^3)),\n\\ee\nwhile for the case $\\o\\leq 0$, we have\n\\ba\\label{vun-limit2}\n&\\vu_n \\to \\vu \\quad \\mbox{weakly in}\\ L^2(0,T;W_0^{1,\\frac{4}{2+|\\o|}}(\\O;\\R^3))\\ \\mbox{and weakly in}\\ L^2(0,T;L^{\\frac{12}{2+3|\\o|}}(\\O;\\R^3)),\\\\\n&\\vu_n \\to \\vu \\quad \\mbox{weakly in}\\ L^{\\frac{4}{2+|\\o|}}(0,T;W_0^{1,\\frac{12}{6+|\\o|}}(\\O;\\R^3))\\ \\mbox{and weakly in}\\ L^{\\frac{4}{2+|\\o|}}(0,T;L^{\\frac{12}{2+|\\o|}}(\\O;\\R^3)),\\\\\n&\\vu_n \\to \\vu \\quad \\mbox{weakly in}\\ L^{\\frac{20}{10+3|\\o|}}(0,T;W_0^{1,\\frac{20}{10+3|\\o|}}(\\O;\\R^3)) \\ \\mbox{and weakly in}\\ L^{\\frac{20}{10+3|\\o|}}(0,T;L^{\\frac{60}{10+9|\\o|}}(\\O;\\R^3)).\n\\ea\n\n\\subsection{Convergence of the nonlinear terms $\\vr_n\\vu_n$ and $\\vr_n \\vu_n\\otimes \\vu_n$}\\label{sec:lim-convective}\n\nWe first consider the case with $\\o\\geq 0$ satisfying \\eqref{mu-eta2}. Since $\\g>\\frac 32>\\frac 65$, we have, because of \\eqref{vrn-lim}, that\n\\ba\\label{vrn-lim1}\n \\vr_n \\to \\vr \\quad \\mbox{strongly in} \\ C([0,T];W^{-1,2}(\\O)).\n\\ea\nTogether with \\eqref{vun-limit1}, we have\n\\be\\label{vrvun-lim0}\n\\vr_n\\vu_n \\to \\vr\\vu \\quad \\mbox{in $\\mathcal{D}'((0,T)\\times \\O;\\R^{3})$.}\n\\ee\nBy observing the uniform estimate\n$$\n\\sup_{n \\in \\mathbb{N}}\\left(\\|\\vr_n\\vu_n\\|_{L^\\infty(0,T;L^{\\frac{2\\g}{1+\\g}}(\\O;\\R^{3}))} + \\|\\vr_n\\vu_n\\|_{L^2(0,T;L^{\\frac{6\\g}{6+\\g}}(\\O;\\R^{3}))}\\right)\\leq c\n$$\nwe have\n\\be\\label{vrvun-lim1}\n\\vr_n\\vu_n \\to \\vr\\vu \\quad \\mbox{weakly* in} \\ L^\\infty(0,T;L^{\\frac{2\\g}{1+\\g}}(\\O;\\R^{3})) \\ \\mbox{and weakly in} \\ L^2(0,T;L^{\\frac{6\\g}{6+\\g}}(\\O;\\R^{3})).\n\\ee\n\nMoreover, by \\eqref{est-sum1} and \\eqref{est-sum2}, we have $\\vr_n\\vu_n \\in C_w([0,T];L^{\\frac{2\\g}{\\g+1}}(\\O;\\R^d))$ and $\\{\\d_t (\\vr_n\\vu_n)\\}_{n\\in \\N}$ is uniformly bounded in $L^r(0,T;W^{-1,r}(\\O;\\R^3))$ for some $r>1$. By Lemma \\ref{lem-Cw} and the interpolation argument in Section \\ref{sec:vr-lim} we have\n\\ba\\label{vrvun-lim2}\n&\\vr_n\\vu_n \\to \\vr\\vu \\quad \\mbox{in}\\ C_w([0,T];L^{\\frac{2\\g}{1+\\g}}(\\O;\\R^3));\\\\\n&\\vr_n\\vu_n \\to \\vr\\vu \\quad \\mbox{strongly in} \\ C([0,T];W^{-1,r}(\\O;\\R^3)) \\quad \\mbox{for any $r\\geq \\frac{3}{2}$ such that $\\frac{3r}{3+r}<\\frac{2\\g}{1+\\g}$}.\n\\ea\n\nThe fact that $\\g>\\frac 32$ implies $\\frac{2\\g}{1+\\g} > \\frac{6}{5}$. This gives $\\vr_n\\vu_n \\to \\vr\\vu$ strongly in $C([0,T];W^{-1,2}(\\O;\\R^3))$. Together with \\eqref{vun-limit1}, we have for the sequence of convective terms:\n\\be\\label{vrvuvun-lim0}\n\\vr_n\\vu_n\\otimes \\vu_n \\to \\vr\\vu\\otimes \\vu \\quad \\mbox{ in $\\mathcal{D}'((0,T)\\times \\O;\\R^{3\\times 3})$.}\n\\ee\n\n\nThe case $\\o\\leq 0$ is dealt with similarly to the case $\\o\\geq 0$, and we obtain the results stated in \\eqref{vrvun-lim0} and \\eqref{vrvuvun-lim0}, so we omit the details. We merely remark that when $\\o\\leq 0$, then we have weaker integrability for $\\vu$; this is, however, compensated by supposing stronger integrability for $\\vr$ through hypothesis \\eqref{mu-eta3}.\n\n We also remark that the hypotheses stated in \\eqref{mu-eta2} and \\eqref{mu-eta3} can be relaxed in this part of the analysis: the constraints stated in \\eqref{mu-eta20} in Remark \\ref{rem-bog-th} are sufficient.\n\n\\subsection{Convergence of the polymer number density}\\label{sec:lim-eta}\n\nWe begin by noting that the sequence $\\{\\eta_n\\}_{n\\in \\N}$ is contained in $C_w([0,T];L^2(\\O))$ and satisfies the bound\n\\be\\label{etan-est1}\n\\sup_{n \\in \\mathbb{N}} \\left(\\|\\eta_n\\|_{L^\\infty(0,T;L^2(\\O)) \\cap L^2(0,T;W^{1,2}(\\O))} + \\|\\d_t \\eta_n\\|_{L^2(0,T;W^{-1,\\frac{12}{3|\\o|+8}}(\\O))}\\right) \\leq c.\n\\ee\nThus, by Lemma \\ref{lem-Cw},\n\\be\\label{etan-lim1}\n\\eta_n \\to \\eta \\quad \\mbox{strongly in}\\ C_w([0,T];L^2(\\O)) \\ \\mbox{and weakly in}\\ L^2(0,T;W^{1,2}(\\O)).\n\\ee\nThanks to the compact Sobolev embedding $W^{1,2}(\\O) \\hookrightarrow \\hookrightarrow L^q(\\O)$, $q<6$, the\nAubin--Lions--Simon compactness theorem (see \\cite{Lions-InC} or \\cite{Simon}) implies that\n\\be\\label{etan-lim2}\n\\eta_n \\to \\eta \\quad \\mbox{strongly in}\\ L^2(0,T;L^q(\\O))\\quad \\mbox{for any $q<6$}.\n\\ee\nBy interpolation we also have\n\\be\\label{etan-lim3}\n\\eta_n \\to \\eta \\quad \\mbox{strongly in}\\ L^q((0,T) \\times \\O)\\quad \\mbox{for any $q<\\frac{10}{3}$}.\n\\ee\n\nWe still need to show that $\\eta=\\int_D \\psi \\ \\dq$, where $\\psi$ is the limit of $\\psi_n$. This will be done later on, in Section \\ref{sec:limit-psi}.\n\n\n\\subsection{Convergence of the Newtonian stress tensor}\\label{sec:lim-SS}\nWhen $0\\leq \\o <\\frac{5}{3}$ as in \\eqref{mu-eta2}, by \\eqref{mu-eta1} and \\eqref{etan-lim3}, we have that\n\\be\\label{mu-lim1}\n(\\mu^S(\\eta_n),\\mu^B(\\eta_n)) \\to (\\mu^S(\\eta),\\mu^B(\\eta)) \\quad \\mbox{strongly in}\\ L^{\\frac{10}{3\\o}}((0,T)\\times \\O).\n\\ee\nTogether with \\eqref{vun-limit1}, we deduce that\n\\be\\label{Newtonian-lim1}\n\\SSS_n \\to \\SSS :=\\mu^S(\\eta) \\left( \\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right) + \\mu^B(\\eta) (\\Div_x \\vu) \\II\\quad {\\rm weakly\\ in} \\ L^{\\frac{10}{3\\o+5}}((0,T)\\times \\O;\\R^{3\\times 3}).\n\\ee\n\nOn the other hand, when $-\\frac{4}{3}<\\o\\leq 0$ as in \\eqref{mu-eta2}, then we have\n\\be\\label{mu-lim2}\n(\\mu^S(\\eta_n),\\mu^B(\\eta_n)) \\to (\\mu^S(\\eta),\\mu^B(\\eta)) \\quad \\mbox{strongly in}\\ L^q(0,T;L^q(\\O)) \\quad \\mbox{for any $q<\\infty$}.\n\\ee\nThus, by \\eqref{vun-limit2}, we have, for any $r<\\frac{20}{3\\o+10}$,\n\\ba\\label{Newtonian-lim10}\n\\SSS_n \\to \\SSS :=\\mu^S(\\eta) \\left( \\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right) + \\mu^B(\\eta) (\\Div_x \\vu) \\II\\quad\n{\\rm weakly~in} \\ L^{r}((0,T)\\times \\O;\\R^{3\\times 3}).\n\\ea\n\n\nWe remark that, by rewriting\n\\ba\\label{Newtonian-est2}\n\\SSS_n = \\sqrt{\\mu^S(\\eta_n)} \\sqrt{\\mu^S(\\eta_n)} \\left( \\frac{\\nabla \\vu_n + \\nabla^{\\rm T} \\vu_n}{2} - \\frac{1}{3} (\\Div_x \\vu_n) \\II \\right) + \\sqrt{\\mu^B(\\eta_n)}\\sqrt{\\mu^B(\\eta_n)} (\\Div_x \\vu_n) \\II,\n\\ea\nwe can relax the constraints in \\eqref{mu-eta2} and \\eqref{mu-eta3} as in \\eqref{mu-eta20} in Remark \\ref{rem-bog-th}.\n\n\\subsection{Convergence of the probability density function}\\label{sec:limit-psi}\n\nFirst of all, by the energy inequality \\eqref{energy-f-n} we have\n\\be\\label{est-psin1}\n\\sup_{n\\in\\N}\\left( \\left\\|\\mathcal{F}(\\tpsi_n)\\right\\|_{L^\\infty(0,T;L^1_M(\\O\\times D))} + \\left\\|\\sqrt{\\tpsi_n}\\,\\right\\|_{L^2(0,T;H_M^1(\\O\\times D))} \\right)\\leq c.\n\\ee\n\n\nAs in Section 4 in \\cite{Barrett-Suli} and Section 5 in \\cite{Barrett-Suli4}, we use Dubinski{\\u\\i}'s compactness theorem (cf. \\cite{Dubin}; see also \\cite{Barrett-Suli3}, Theorem 3.1 or \\cite{Barrett-Suli}) by setting\n\\ba\\label{dubinskii-spaces}\n&X:= L^1_M(\\O\\times D),\\quad X_0:=\\{ \\vp\\in X : \\vp\\geq 0,\\ \\sqrt{\\vp} \\in H^1_M(\\O\\times D) \\},\\\\\n&X_1: = M^{-1} W^{s,2}(\\O\\times D)':=\\{ M^{-1} \\vp : \\vp \\in W^{s,2}(\\O\\times D)'\\} \\quad \\mbox{with $s>1+\\frac{3}{2}(K+1)$},\n\\ea\nwhere $X_0$ is a seminomed space (in the sense of Dubinski{\\u\\i}) with seminorm defined by\n$$\n[\\vp]_{X_0}: = \\|\\vp\\|_{X} + \\int_{\\O\\times D} M\\bigg( \\left|\\nabla_x \\sqrt{\\vp}\\,\\right|^2 + \\left|\\nabla_q \\sqrt{\\vp}\\,\\right|^2 \\bigg) \\, \\dq \\, \\dx.\n$$\n\nBy \\eqref{est-sum1} and \\eqref{est-sum2} we have that\n\\be\\label{psin-est1}\n\\mbox{$\\{\\widetilde \\psi_n\\}_{n\\in \\N}$ is uniformly bounded in $L^1(0,T;X_0)$\\quad and \\quad $\\{\\d_t \\widetilde \\psi_n\\}_{n\\in \\N}$ is uniformly bounded in $L^2(0,T;X_1)$}.\n\\ee\nThe continuity of the embedding $X\\hookrightarrow X_1$ and the compactness of the embedding $X_0 \\hookrightarrow X$ are shown in Section 5 in \\cite{Barrett-Suli4}. Then, by virtue of Dubinski{\\u\\i}'s compact embedding theorem, we have that\n\\be\\label{psin-lim1}\n\\tpsi_n \\to \\tpsi \\quad \\mbox{strongly in} \\ L^1(0,T;L_M^1(\\Omega\\times D)),\n\\ee\nwhich is equivalent to\n\\be\\label{psin-lim2}\n\\psi_n \\to \\psi \\quad \\mbox{strongly in} \\ L^1(0,T;L^1(\\Omega\\times D)).\n\\ee\nThis implies that\n\\be\\label{psin-lim3}\n\\eta_n = \\int_D \\psi_n \\,\\dq \\to \\int_D \\psi \\, \\dq \\quad \\mbox{strongly in} \\ L^1((0,T)\\times \\Omega);\n\\ee\nin addition, by the uniqueness of the limit, the function $\\eta$ obtained in Section \\ref{sec:lim-eta} satisfies $\\eta = \\int_D \\psi \\, \\dq$.\n\n\n\nSince $\\mathcal{F}(\\tpsi)$ is nonnegative on $(0,T)\\times \\Omega \\times D$, by applying Fatou's lemma we have that\n\\be\\label{psin-est3}\n \\left\\|\\mathcal{F}(\\tpsi)\\right\\|_{L^\\infty(0,T;L^1_M(\\O\\times D))} \\leq \\liminf_{n\\to \\infty} \\left\\|\\mathcal{F}(\\tpsi_n)\\right\\|_{L^\\infty(0,T;L^1_M(\\O\\times D))} \\leq c.\n\\ee\nBy the same technique as in Section \\ref{sec:est-psi} we deduce from \\eqref{psin-est3} that\n\\be\\label{psin-est4}\n\\tpsi\\log\\tpsi, \\ \\tpsi \\in L^\\infty(0,T;L^1_M(\\O\\times D)),\\quad \\psi\\log\\psi, \\ \\psi \\in L^\\infty(0,T;L^1(\\O\\times D)).\n\\ee\n\nBy \\eqref{psin-est1} we have\n\\be\\label{psin-est5}\n\\d_t \\psi \\in L^2(0,T;W^{s,2}(\\O\\times D)') \\quad \\mbox{for any $s>1 + \\frac{3}{2}(K+1)$}.\n\\ee\n\nFrom the estimates in \\eqref{psin-est4} and \\eqref{psin-est5}, by using the second part of Lemma \\ref{lem-Cw-Cw*} and the same argument as in Section 4.5 in \\cite{Barrett-Suli}, we deduce that\n$\n \\psi \\in C_w([0,T];L^1(\\O\\times D)).\n$\n\n\nAgain, we write $\\nabla_{q_i}\\tpsi_n = 2 \\sqrt{\\tpsi_n}\\nabla_{q_i}\\sqrt{\\tpsi_n}$. By \\eqref{psin-lim1} we have\n$$\n\\sqrt{\\tpsi_n} \\to \\sqrt{\\tpsi} \\quad \\mbox{strongly in} \\ L^2(0,T;L_M^2(\\Omega\\times D)).\n$$\nFurther, by \\eqref{est-psin1}, we have that\n$$\n\\nabla_q \\sqrt{\\tpsi_n} \\to \\nabla_q \\sqrt{\\tpsi} \\quad \\mbox{weakly in} \\ L^2(0,T;L_M^2(\\Omega\\times D;\\R^{3K})).\n$$\nThus,\n\\be\\label{psin-limf1}\n\\nabla_{q}\\tpsi_n \\to \\nabla_{q}\\tpsi \\quad \\mbox{weakly in} \\ L^1(0,T;L_M^1(\\Omega\\times D;\\R^{3K})).\n\\ee\n\n\nSimilarly, we also have that\n\\be\\label{psin-limf2}\n\\nabla_{x}\\tpsi_n \\to \\nabla_{x}\\tpsi \\quad \\mbox{weakly in} \\ L^1(0,T;L_M^1(\\Omega\\times D;\\R^{3})),\\quad\n\\nabla_{x}\\psi_n \\to \\nabla_{x}\\psi \\quad \\mbox{weakly in} \\ L^1(0,T;L^1(\\Omega\\times D;\\R^{3})).\n\\ee\n\n\n\\subsection{Convergence of the extra-stress tensor}\\label{sec:lim-TT}\n\nWe recall the formula for the extra-stress tensor. By \\eqref{weak-est} and \\eqref{est-TT2} we have that\n\\be\\label{TTn-est1}\n\\TT_n := k \\left(\\sum_{i=1}^K \\int_{D} M (\\nabla_{q_i}\\tpsi_n) q_i^{\\rm T}\\, \\dq \\right) - \\left(k \\,\\eta_n + \\de\\, \\eta_n^2\\right) \\II,\n \\ee\nwhich, because of \\eqref{est-TTf}, satisfies\n\\be\\label{TTn-est2}\n\\sup_{n\\in\\N}\\left(\\|\\TT_n\\|_{L^2(0,T;L^{\\frac 43}(\\O;\\R^{3\\times 3}))} + \\|\\TT_n\\|_{L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;\\R^{3\\times 3}))}\\right) -\\frac{4}{3}$, appearing in \\eqref{mu-eta3}, is really needed. Indeed, for any test function $\\vvarphi$, just as in \\eqref{nonlin-FP-lim3}, we have that\n\\be\\label{nonlin-FP-lim5}\n\\int_D (\\psi_n-\\psi) \\, \\vvarphi \\,q_i \\ \\dq \\to \\mathbf{0} \\quad \\mbox{in}\\ L^q(0,T;L^q(\\O;\\R^3)) \\quad \\mbox{for any $q<\\frac{10}{3}$}.\n\\ee\nAccording to the bounds established in Section \\ref{sec:lim-u} for the case $\\o\\leq 0$, the sequence $\\left\\{\\nabla_x \\vu_n\\right\\}_{n\\in \\N}$ is uniformly bounded in $L^{\\frac{20}{10+3|\\o|}}((0,T)\\times\\O;\\R^{3\\times 3})$. To deduce the desired convergence result, we therefore need that\n\\be\\label{mu-eta30}\n\\frac{10+3|\\o|}{20} + \\frac{3}{10}<1 \\Longleftrightarrow |\\o| < \\frac{4}{3}.\n\\ee\n\n\n\\subsection{Convergence of the fluid pressure}\nAs was shown in Section \\ref{sec:est-higher-vr} concerning higher integrability of the fluid density and pressure, we have that\n\\be\\label{pn-lim1}\np(\\vr_n) \\to \\overline{p(\\vr)} \\quad \\mbox{weakly in} \\ L^{1+\\frac{\\th}{\\g}}((0,T)\\times \\O).\n\\ee\n\nBy applying this, together with the convergence results from the previous sections, in particular the convergence results for the nonlinear terms stated in Section \\ref{sec:lim-convective}, Section \\ref{sec:lim-SS} and Section \\ref{sec:lim-TT}, we deduce that\n\\be\\label{i1-0}\n\\d_t \\vr + \\Div_x (\\vr \\vu) = 0 \\quad \\mbox{in}\\ \\mathcal{D}'((0,T)\\times \\O),\n\\ee\n\\be\\label{i2-0}\n \\d_t (\\vr\\vu)+ \\Div_x (\\vr \\vu \\otimes \\vu) +\\nabla_x \\overline{p(\\vr)} -\n\\Div_x \\SSS =\\Div_x \\TT +\\vr\\, \\ff\\quad \\mbox{in}\\ \\mathcal{D}'((0,T)\\times \\O;\\R^3),\n\\ee\nwhere the Newtonian stress tensor $\\SSS$ and the extra-stress tensor $\\TT$ are defined by \\eqref{Newtonian-lim1} and \\eqref{TTn-lim2}, respectively.\n\n\nConcerning the convergence of the nonlinear terms in the Navier--Stokes--Fokker--Planck system with variable viscosity coefficients considered here, it remains to show that $p(\\vr) = \\overline{p(\\vr)},$\nwhich, because of the strict convexity of $p(\\cdot)$, is equivalent to the strong convergence of $\\vr_n$:\n\\be\\label{pn-limf2}\n\\vr_n \\to \\vr \\quad \\mbox{a.e. in} \\ (0,T)\\times \\O.\n\\ee\n\nThis is also one of the main difficulties in the study of global existence of weak solutions to the compressible Navier--Stokes equations (see \\cite{Lions-C, FNP,F-book,N-book}), where the so called \\emph{effective viscous flux} introduced by P. L. Lions \\cite{Lions-C} plays a crucial role. It turns out that the effective viscous flux as a whole is more regular than its components.\n We will prove the following lemma, which is in the spirit of Proposition 5.1 in \\cite{Feireisl-2001}.\n\\begin{lemma}\\label{lem-viscous-flux}\nFor any $\\phi\\in \\mathcal{D}(0,T)$, $\\vp \\in \\mathcal{D}(\\O)$, we have that\n\\ba\\label{vis-flux}\\lim_{n\\to \\infty}\\int_0^T \\intO{\\phi(t)\\vp(x) \\left[p(\\vr_n) - \\left(\\frac{2\\mu^S(\\eta_n)}{3}-\\mu^B(\\eta_n)\\right) \\Div_x \\vu_n \\right] T_k(\\vr_n) }\\, \\dt\\\\\n=\\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\overline{p(\\vr)} - \\left(\\frac{2\\mu^S(\\eta)}{3}-\\mu^B(\\eta)\\right) \\Div_x \\vu\\right] \\overline{T_k(\\vr)} }\\, \\dt,\n\\ea\nwhere $T_k$ is a cut-off function defined by $T_k(\\cdot) := k \\,T(\\frac{\\cdot}{k})$ for some concave function $T\\in C^\\infty([0,\\infty))$ such that $T(s)=s$ for $0\\leq s\\leq 1$ and $T(s)=2$ for $s\\geq 3$. Here $\\overline{T_k(\\vr)}$ is the weak* limit of the sequence $\\{T_k(\\vr_n)\\}_{n\\in \\N}$ in $L^\\infty((0,T)\\times \\O)$ as $n$ goes to infinity.\n\\end{lemma}\n\n\nThe viscosity coefficients $\\mu^B$ and $\\mu^S$ in our case are not constant, which gives rise to additional complications in the proof of this lemma. We shall employ the commutator estimates stated in Lemma \\ref{lem-Riesz3} and the techniques used in Section 3.6.5 in \\cite{F-N-book} to overcome the resulting difficulties.\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem-viscous-flux}] As in the proof of the effective viscous flux lemma for the compressible Navier--Stokes equations with constant viscosity coefficients (see Proposition 5.1 in \\cite{Feireisl-2001} or Proposition 7.36 in \\cite{N-book}), we introduce the following test functions:\n\\be\\label{def-test-flux}\n\\vv_n(t,x) = \\phi(t) \\vp(x) \\A \\left[T_k(\\vr_n) \\right],\\quad \\vv(t,x) = \\phi(t) \\vp(x) \\A \\left[ \\overline{T_k(\\vr)}\\right],\n\\ee\nwhere $\\phi\\in C_c^\\infty(0,T), \\ \\vp \\in C_c^\\infty(\\O)$, and $\\A$ is the Fourier integral operator introduced in \\eqref{def-Reize}. We remark that $\\vr_n$ and $\\vr$ are extended by zero outside of $\\O$ in \\eqref{def-test-flux}.\n\n\nTaking these $\\vv_n$ and $\\vv$ as test functions in the weak formulation \\eqref{weak-form3} and the weak formulation of \\eqref{i2-0}, respectively, results in\n\\begin{align}\n\\label{weak-vn}\n&\\int_0^T \\intO{ \\big[ \\vr_n \\vu_n \\cdot \\partial_t \\vv_n + (\\vr_n \\vu_n \\otimes \\vu_n) : \\Grad \\vv_n + p(\\vr_n)\\, \\Div_x \\vv_n - \\SSS_n : \\Grad \\vv_n \\big]} \\,\\dt\n\\nonumber\\\\\n&\\hspace{3in}= \\int_0^T \\intO{ \\TT_n : \\nabla_x \\vv_n - \\vr_n\\, \\ff \\cdot \\vv_n} \\, \\dt,\\\\\n\\label{weak-v}\n&\\int_0^T \\intO{ \\left[ \\vr \\vu \\cdot \\partial_t \\vv + (\\vr \\vu \\otimes \\vu) : \\Grad \\vv + \\overline{p(\\vr)}\\, \\Div_x \\vv - \\SSS : \\Grad \\vv \\right]}\\,\\dt\n\\nonumber\\\\\n&\\hspace{3in}= \\int_0^T \\intO{ \\TT : \\nabla_x \\vv - \\vr\\, \\ff \\cdot \\vv} \\, \\dt.\n\\end{align}\n\nThe main idea is to pass to the limit $n\\to \\infty$ in \\eqref{weak-vn}, and compare the resulting limit with \\eqref{weak-v}, which will ultimately imply our desired result \\eqref{vis-flux}. Since, following the contributions of Lions \\cite{Lions-C} and Feireisl \\cite{F-book}, this type of argument is by now well understood, instead of including all of the details here we shall focus on the terms that do not appear in the case of the compressible Navier--Stokes equations with constant viscosity coefficients. These `new' terms are the following:\n\\ba\\label{flux-new-terms1}\n&\\int_0^T \\intO{ \\TT_n : \\nabla_x \\vv_n} \\, \\dt,\\quad \\int_0^T \\intO{ \\TT : \\nabla_x \\vv } \\, \\dt,\\quad \\int_0^T \\intO{ \\SSS_n : \\Grad \\vv_n } \\, \\dt,\\quad \\int_0^T \\intO{ \\SSS : \\Grad \\vv } \\, \\dt,\\nn\n\\ea\nand the goal is to prove that\n\\ba\\label{flux-new-terms2}\n\\lim_{n\\infty}\\int_0^T \\intO{ \\TT_n : \\nabla_x \\vv_n} \\, \\dt = \\int_0^T \\intO{ \\TT : \\nabla_x \\vv } \\, \\dt,\n\\ea\nand\n\\ba\\label{flux-new-terms3}\n&\\lim_{n\\to\\infty}\\int_0^T \\intO{ \\SSS_n : \\Grad \\vv_n } \\, \\dt - \\int_0^T \\intO{ \\SSS : \\Grad \\vv } \\, \\dt \\\\\n&= \\lim_{n\\to\\infty} \\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\left(\\frac{2\\mu^S(\\eta_n)}{3}-\\mu^B(\\eta_n)\\right) \\Div_x \\vu_n \\right] T_k(\\vr_n) }\\, \\dt \\\\\n&\\quad - \\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\left(\\frac{2\\mu^S(\\eta)}{3}-\\mu^B(\\eta)\\right) \\Div_x \\vu\\right] \\overline{T_k(\\vr)} }\\, \\dt.\n\\ea\n\nBy the strong convergence of $\\TT_n$ stated in \\eqref{TTn-lim-strff} in Section \\ref{sec:lim-TT} it is straightforward to show the convergence result \\eqref{flux-new-terms2}, so we only focus on proving \\eqref{flux-new-terms3}. We begin by noting that\n\\ba\\label{flux-new31}\n\\int_0^T \\intO{ \\SSS_n : \\Grad \\vv_n } \\, \\dt = \\int_0^T \\phi(t)\\intO{ \\SSS_n :\\, \\left(\\nabla_x \\vp \\otimes \\A[ T_k(\\vr_n)]\\right)} \\, \\dt + \\int_0^T \\phi(t)\\intO{ \\vp \\, \\SSS_n : \\RR [ T_k(\\vr_n)]} \\, \\dt,\\nn\n\\ea\nwhere $\\RR$ is the Riesz operator defined in \\eqref{def-Reize} in Section \\ref{sec:Riesz}.\n\n\nFor any fixed $k\\in\\N$, the sequence $\\{ T_k(\\vr_n)\\}_{n\\in \\N}$ is uniformly bounded in $ L^\\infty(0,T;L^r(\\R^3))$ for all $r \\in [1, \\infty]$. Thus, by Lemma \\ref{lem-Riesz1}, we have that\n\\be\\label{A-Tk-vrn-1}\n\\sup_{n\\in \\N} \\left\\| \\A[T_k(\\vr_n)]\\right\\|_{L^\\infty(0,T;W^{1,r}(\\O;\\R^3))} \\leq c\\quad \\mbox{for any $r \\in (1,\\infty)$}.\n\\ee\n\nObserving that $T_k(\\cdot)$ fulfills the properties in \\eqref{cond-b-renormal}, it follows from \\eqref{weak-renormal1} that\n\\be\\label{renomal-Tk}\n\\d_t T_k(\\vr_n) + \\Div_x\\left(T_k(\\vr_n)\\vu_n\\right) + \\left(T_k'(\\vr_n)\\vr_n-T_k(\\vr_n)\\right)\\Div_x \\vu_n =0\\quad \\mbox{in $ \\mathcal{D}'((0,T)\\times \\R^3)$}.\n\\ee\nThis implies that\n\\ba\\label{A-Tk-vrn-2}\n\\d_t \\A[ T_k(\\vr_n)]= \\A[\\d_t T_k(\\vr_n)] = - \\A\\left[ \\Div_x\\left(T_k(\\vr_n)\\vu_n\\right) \\right] - \\A\\left[ \\left(T_k'(\\vr_n)\\vr_n-T_k(\\vr_n)\\right)\\Div_x \\vu_n \\right].\n\\ea\nSince\n$\n\\A_j \\Div_x = -\\sum_{i=1}^3\\d_j \\d_i \\Delta^{-1} = - \\sum_{i=1}^3\\RR_{ij}\n$\nare Riesz type operators, by Lemma \\ref{lem-Riesz1} we have that\n$$\n\\|\\A_j \\left[ \\Div_x \\vvarphi \\right]\\|_{L^r(\\R^3:\\R^3)}\\leq c \\, \\|\\vvarphi\\|_{L^r(\\R^3;\\R^3)} \\quad \\mbox{for any $\\vvarphi\\in L^r(\\R^3;\\R^3)$ and any $r \\in (1,\\infty)$}.\n$$\nBy using \\eqref{vun-limit1} and \\eqref{vun-limit2} we deduce that\n\\ba\\label{A-Tk-vrn-3}\n\\left\\|\\A\\left[ \\Div_x\\left(T_k(\\vr_n)\\vu_n\\right) \\right]\\right\\|_{L^2(0,T;L^6(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\geq 0$}; \\\\\n\\left\\|\\A\\left[ \\Div_x\\left(T_k(\\vr_n)\\vu_n\\right) \\right]\\right\\|_{L^2(0,T;L^{\\frac{12}{2+6|\\o|}}(\\O;\\R^3))} &\\leq c &&\\quad\\mbox{when $\\o\\leq 0$}.\n\\ea\nFurthermore, by Lemma \\ref{lem-Riesz1} in conjunction with \\eqref{vun-limit1} and \\eqref{vun-limit2} we deduce that\n\\ba\\label{A-Tk-vrn-4}\n\\left\\| \\A\\left[ \\left(T_k'(\\vr_n)\\vr_n-T_k(\\vr_n)\\right)\\Div_x \\vu \\right] \\right\\|_{L^2(0,T;L^6(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\geq 0$}; \\\\\n\\left\\| \\A\\left[ \\left(T_k'(\\vr_n)\\vr_n-T_k(\\vr_n)\\right)\\Div_x \\vu \\right] \\right\\|_{L^2(0,T;L^{\\frac{12}{2+6|\\o|}}(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\leq 0$}.\n\\ea\nThus,\n\\ba\\label{A-Tk-vrn-5}\n\\left\\| \\d_t \\A[ T_k(\\vr_n)] \\right\\|_{L^2(0,T;L^6(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\geq 0$};\\\\\n\\left\\| \\d_t \\A[ T_k(\\vr_n)] \\right\\|_{L^2(0,T;L^{\\frac{12}{2+6|\\o|}}(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\leq 0$}.\n\\ea\nIt follows from the uniform estimates in \\eqref{A-Tk-vrn-1} and \\eqref{A-Tk-vrn-5} and the Aubin--Lions--Simon compactness theorem that\n\\ba\\label{A-Tk-vrn-f}\n\\A[T_k(\\vr_n)] \\to \\overline{\\A[T_k(\\vr)]} = \\A[\\overline{T_k(\\vr)}] \\quad \\mbox{strongly in $L^\\infty(0,T;L^r(\\O;\\R^3))$ \\quad for any $r\\in (1,\\infty)$}.\n\\ea\nTogether with the weak convergence of $\\SSS_n$ stated in \\eqref{Newtonian-lim1} and \\eqref{Newtonian-lim10}, we deduce that\n\\ba\\label{flux-new32}\n \\lim_{n\\to\\infty}\\int_0^T \\phi(t)\\intO{ \\SSS_n : (\\nabla_x \\vp \\otimes \\A[ T_k(\\vr_n)])} \\, \\dt = \\int_0^T \\phi(t)\\intO{ \\SSS : (\\nabla_x \\vp \\otimes \\A[ \\overline{T_k(\\vr)}])} \\, \\dt.\n\\ea\nTherefore, to show \\eqref{flux-new-terms3}, it suffices to prove that\n\\ba\\label{flux-new33}\n& \\lim_{n\\to \\infty} \\int_0^T \\phi(t)\\intO{ \\vp \\, \\SSS_n : \\RR [ T_k(\\vr_n)]} \\, \\dt - \\int_0^T \\phi(t)\\intO{ \\vp \\, \\SSS : \\RR [ \\overline{T_k(\\vr)}]} \\, \\dt\\\\\n &= \\lim_{n\\to\\infty} \\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\left(\\frac{2\\mu^S(\\eta_n)}{3}-\\mu^B(\\eta_n)\\right) \\Div_x \\vu_n \\right] T_k(\\vr_n) }\\, \\dt \\\\\n&\\quad - \\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\left(\\frac{2\\mu^S(\\eta)}{3}-\\mu^B(\\eta)\\right) \\Div_x \\vu\\right] \\overline{T_k(\\vr)} }\\, \\dt.\n\\ea\n\nBy \\eqref{i3} we have that\n\\ba\\label{flux-new34}\n&\\int_0^T \\phi(t)\\intO{ \\vp \\, \\SSS_n : \\RR [ T_k(\\vr_n)]} \\ \\dt= \\int_0^T \\phi(t)\\intO{ \\vp \\, \\mu^S(\\eta_n) \\left( \\frac{\\nabla \\vu_n + \\nabla^{\\rm T} \\vu_n}{2} \\right) : \\RR [ T_k(\\vr_n)]} \\, \\dt\\\\\n&\\qquad + \\int_0^T \\phi(t)\\intO{ \\vp \\, \\left(\\mu^B(\\eta_n)-\\frac{\\mu^S(\\eta_n)}{3}\\right) (\\Div_x \\vu_n) \\II: \\RR [ T_k(\\vr_n)]} \\, \\dt\\\\\n& = \\int_0^T \\phi(t)\\intO{ \\vp \\, \\mu^S(\\eta_n) \\nabla \\vu_n : \\RR [ T_k(\\vr_n)]} \\, \\dt + \\int_0^T \\phi(t)\\intO{ \\vp \\, \\left(\\mu^B(\\eta_n)-\\frac{\\mu^S(\\eta_n)}{3}\\right) (\\Div_x \\vu_n) \\, [ T_k(\\vr_n)]} \\, \\dt,\n\\ea\nwhere have we used the fact that $\\RR$ is symmetric (see Lemma \\ref{lem-Riesz1}) and that $\\II : \\RR = \\sum_{i=1}^3 \\RR_{ii}=\\II$.\nFurther, by Lemma \\ref{lem-Riesz1}, and noting that $\\sum_{i,j=1}^3 \\RR_{ij}\\partial_j \\vu_n^i = \\Div_x \\vu_n$, we have that\n\\ba\\label{flux-new35}\n& \\int_0^T \\phi(t)\\intO{ \\vp \\, \\mu^S(\\eta_n) \\nabla \\vu_n : \\RR [ T_k(\\vr_n)]} \\, \\dt = \\int_0^T \\phi(t)\\intO{ \\sum_{i,j=1}^3 \\RR_{ij} \\left[\\vp \\, \\mu^S(\\eta_n)\\, \\d_j \\vu_n^i \\right] T_k(\\vr_n)} \\, \\dt\\\\\n&\\quad =\\int_0^T \\phi(t)\\intO{ \\sum_{i,j=1}^3 \\left( \\vp \\, \\mu^S(\\eta_n)\\, \\RR_{ij} \\left[ \\d_j \\vu_n^i \\right] T_k(\\vr_n) + R_n^{ij} \\, T_k(\\vr_n) \\right)\\!} \\, \\dt \\\\\n&\\quad = \\int_0^T \\phi(t)\\intO{ \\left(\\vp \\, \\mu^S(\\eta_n)\\, (\\Div_x \\vu_n) \\, T_k(\\vr_n) + \\sum_{i,j=1}^3 R_n^{ij} \\, T_k(\\vr_n) \\right)\\!} \\, \\dt,\n\\ea\nwhere $R_n^{ij}$ is the commutator defined by $R_n^{ij}:= \\RR_{ij} \\left[\\vp \\, \\mu^S(\\eta_n) \\d_j \\vu_n^i \\right]- \\vp \\, \\mu^S(\\eta_n) \\RR_{ij} \\left[ \\d_j \\vu_n^i \\right].$ We will use Lemma \\ref{lem-Riesz3} to derive uniform bounds on $R_n^{ij}$. We first consider the case $0\\leq \\o < \\frac 53$. Then, by \\eqref{mu-eta1} and \\eqref{etan-est1}, the sequence\n$\\{\\nabla_x \\mu^S(\\eta_n)\\}_{n\\in\\N} = \\{(\\mu^S)'(\\eta_n) \\nabla_x \\eta_n\\}_{n\\in\\N}$ is uniformly bounded in $L^2(0,T;L^2(\\O;\\R^3))$ when $0\\leq \\o\\leq 1$, and in $L^{2}(0,T;L^\\frac{2}{\\o}(\\O;\\R^3))$ when $1\\leq \\o < \\frac 53$.\nThus,\n$$\n\\sup_{n\\in \\N}\\| \\mu^S(\\eta_n) \\|_{L^2(0,T;W^{1,2}(\\O))} \\leq c \\quad \\mbox{when}\\ 0\\leq \\o\\leq 1;\\qquad \\sup_{n\\in \\N}\\| \\mu^S(\\eta_n) \\|_{L^2(0,T;W^{1,\\frac{2}{\\o}}(\\O))} \\leq c \\quad \\mbox{when}\\ 1 \\leq \\o < \\frac 53.\n$$\n\nBy \\eqref{vun-limit1} the sequence $\\{\\nabla_x \\vu_n\\}_{n\\in\\N}$ is uniformly bounded in $ L^2(0,T; L^{2}(\\O;\\R^{3 \\times 3}))$. Furthermore, we note that\n\\ba\\label{mu-eta10}\n\\frac{1}{2} + \\frac{1}{2} -\\frac{1}{3} = \\frac{2}{3}< 1;\\quad \\frac{1}{2}+\\frac{\\o}{2} - \\frac{1}{3} =\\frac{1+3\\o}{6} < 1, \\quad \\mbox{as long as}\\ 0\\leq \\o < \\frac{5}{3}.\n\\ea\nHence, by Lemma \\ref{lem-Riesz3}, for any $s>1$ such that\n\\ba\\label{est-mun-2}\n \\frac{1}{s} >\\frac{1}{2}+\\frac{1}{2} - \\frac{1}{3}=\\frac{2}{3} \\quad \\mbox{when}\\ 0\\leq \\o\\leq 1;\\quad \\frac{1}{s} >\\frac{1}{2}+\\frac{\\o}{2} - \\frac{1}{3} =\\frac{1+3\\o}{6} \\quad \\mbox{when}\\ 1\\leq \\o < \\frac{5}{3},\n\\ea\nwe have, for $r_\\o=2$ when $0\\leq \\o\\leq 1$ and for $r_\\o=\\frac{2}{\\o}$ when $1\\leq \\o < \\frac{5}{3}$, that\n\\ba\\label{flux-commu-2}\n&\\left\\|R_n^{ij}\\right\\|_{W^{\\b,s}(\\R^3)} \\leq c\\, \\| \\vp \\mu^S(\\eta_n) \\|_{W^{1,r_\\o}(\\O)} \\|\\d_j \\vu_n^i\\|_{L^2(\\O)},\n\\ea\nwhere $\\b\\in (0,1)$ is such that\n\\ba\\label{est-mun-3}\n\\frac{\\b}{3} =\\frac 13 + \\frac 1s - 1 \\quad \\mbox{when}\\ 0\\leq \\o\\leq 1;\\quad \\frac{\\b}{3} =\\frac 13 + \\frac 1s - \\frac{1+\\o}{2}\\quad \\mbox{when}\\ 1\\leq \\o < \\frac 53.\n\\ea\nTherefore, for some $s>1$ and some $0<\\b<1$, we have the uniform estimate\n\\ba\\label{flux-commu-3}\n\\sup_{n\\in \\N} \\left\\|R_n^{ij} \\right\\|_{L^1(0,T;W^{\\b,s}(\\R^3))} \\leq c\\, \\sup_{n\\in \\N} \\| \\vp \\mu^S(\\eta_n) \\|_{L^2(0,T;W^{1,r_\\o}(\\O))}\\, \\sup_{n\\in \\N} \\|\\d_j \\vu_n^i\\|_{L^2(0,T;L^2(\\O))}\\leq c.\n\\ea\nHence, by the strong convergence of $\\eta_n$ shown in Section \\ref{sec:lim-eta} we have that\n\\ba\\label{flux-commu-4}\nR_n^{ij} \\to R^{ij}:= \\RR_{ij} \\left[\\vp \\, \\mu^S(\\eta) \\d_j \\vu^i \\right]- \\vp \\, \\mu^S(\\eta) \\RR_{ij} \\left[ \\d_j \\vu^i \\right] \\quad \\mbox{weakly in} \\ L^{\\frac{10}{3\\o+5}}((0,T)\\times \\O).\n\\ea\n\nBy the boundedness of $T_k$ we have\n\\be\\label{Tk-weak}\nT_k(\\vr_n)\\to \\overline{T_k(\\vr)} \\quad \\mbox{weakly in $L^r((0,T)\\times \\O)$\\quad for any $11$ such that\n\\ba\\label{est-mun-2-w<0}\n& \\frac{1}{s} > \\frac{1}{2}+\\frac{2+|\\o|}{4} - \\frac{1}{3},\n\\ea\nwe have that\n\\ba\\label{flux-commu-2-w<0}\n&\\left\\|R_n^{ij}\\right\\|_{W^{\\b,s}(\\R^3)} \\leq c\\, \\| \\vp \\mu^S(\\eta_n) \\|_{W^{1,2}(\\O)} \\|\\d_j \\vu_n^i\\|_{L^\\frac{4}{2+|\\o|}(\\O)},\n\\ea\nwhere $\\b\\in (0,1)$ is such that\n\\ba\\label{est-mun-3-w<0}\n\\frac{\\b}{3} =\\frac 13 + \\frac 1s - \\frac{1}{2}-\\frac{2+|\\o|}{4}.\n\\ea\nTherefore, for some $s>1$ and some $\\b>0$, the following uniform estimate holds:\n\\ba\\label{flux-commu-3-w<0}\n\\sup_{n\\in \\N} \\left\\|R_n^{ij} \\right\\|_{L^1(0,T;W^{\\b,s}(\\R^3))} \\leq c\\, \\sup_{n\\in \\N} \\| \\vp \\mu^S(\\eta_n) \\|_{L^2(0,T;W^{1,2}(\\O))}\\,\\sup_{n\\in \\N} \\|\\d_j \\vu_n^i\\|_{L^2(0,T;L^\\frac{4}{2+|\\o|}(\\O))}\\leq c.\n\\ea\nThanks to the strong convergence of $\\eta_n$ shown in Section \\ref{sec:lim-eta}, we have that\n\\ba\\label{flux-commu-4-w<0}\n R_n^{ij} \\to R^{ij}:= \\RR_{ij} \\left[\\vp \\, \\mu^S(\\eta) \\d_j \\vu^i \\right]- \\vp \\, \\mu^S(\\eta) \\RR_{ij} \\left[ \\d_j \\vu^i \\right], \\\\\n \\mbox{weakly in} \\ L^{r}((0,T)\\times \\O),\\ \\mbox{for any $r<\\frac{20}{10+3|\\o|}$}.\n\\ea\nThus, the Div-Curl lemma implies that\n\\ba\\label{flux-commu-5-w<0}\nR_n^{ij}\\, T_k(\\vr_n) \\to R^{ij} \\overline{T_k(\\vr_n)} \\quad \\mbox{weakly in} \\ L^{r}(0,T;L^{r}(\\O)) \\quad \\mbox{for any $r<\\frac{20}{10+3|\\o|}$},\n\\ea\nwhich, once again, implies \\eqref{flux-new36}.\n\n\\medskip\n\n\n\n\nFinally, from \\eqref{flux-new34}, \\eqref{flux-new35} and \\eqref{flux-new36}, we deduce the desired result \\eqref{flux-new33}. This implies \\eqref{flux-new-terms3} by using \\eqref{flux-new32}. At the same time, by tedious but, by now, standard calculations, as in the proof of the effective viscous flux lemma in \\cite{Lions-C, F-book, N-book}, we have that\n\\ba\\label{flux-new-terms3f}\n\\lim_{n\\to\\infty}\\int_0^T \\intO{ \\SSS_n : \\Grad \\vv_n } \\ \\dt - \\int_0^T \\intO{ \\SSS : \\Grad \\vv } \\ \\dt =0.\n\\ea\nCombining \\eqref{flux-new-terms3f} with \\eqref{flux-new-terms3}, we obtain \\eqref{vis-flux} and complete the proof of Lemma \\ref{lem-viscous-flux}.\n\\end{proof}\n\n\n\\medskip\n\nWith Lemma \\ref{lem-viscous-flux} in hand, the proof of the strong convergence result \\eqref{pn-limf2} then proceeds along a well-understood route (see for example \\cite{Feireisl-2001}, from Section 6 to Section 8), so we shall not dwell on the details here. In particular, as in Proposition 7.1 in \\cite{Feireisl-2001}, the limit $(\\vr, \\vu)$ satisfies \\eqref{weak-renormal1} in the sense of renormalized weak solutions.\n\n\\medskip\n\nIt remains to show that the limit $(\\vr,\\vu,\\psi,\\eta)$ satisfies the energy inequality \\eqref{energy}. This is easily seen by noting the strong convergence assumption on the initial data in \\eqref{cond-ini-n} and passage to the limit $n\\to \\infty$ in \\eqref{energy-n}, which directly imply the energy\ninequality \\eqref{energy} \nby the application of Tonelli's sequential weak (weak*) lower semicontinuity theorem (cf. Theorem 10.16 in \\cite{RR-book}, for example,) to the terms appearing on the left-hand side of \\eqref{energy-n}; in particular, the sequential weak lower semicontinuity of the $L^p$ norm, $1< p<\\infty$, the sequential weak* lower semicontinuty of the $L^\\infty$ norm and inequality \\eqref{psin-est3} are used.\nThus we have shown that the limit $(\\vr,\\vu,\\psi,\\eta)$ is a dissipative (finite-energy) weak solution in the sense of Definition \\ref{def-weaksl}. That completes the proof of Theorem \\ref{theorem}.\n\n\\begin{remark}\\label{rem-mu-eta}\nThe constraint $\\o<\\frac{5}{3}$ for the case $\\o\\geq 0$ is crucially determined by \\eqref{mu-eta10} and the condition $\\frac{10}{3\\o+5}>1$, which appears in \\eqref{flux-commu-4}, while the constraint $\\o>-\\frac{4}{3}$ for the case $\\o\\leq 0$ is crucially determined by \\eqref{mu-eta10-w<0}. It is unclear whether,\nwith our present techniques at least, these restrictions on $\\o$ can be relaxed.\n\\end{remark}\n\n\n\n\\section{Conclusion: existence of dissipative weak solutions}\\label{sec:end}\n\n\\label{End}\n\nThe conclusions of Theorem \\ref{theorem} \\emph{do not}, of course, imply the existence of dissipative (finite-energy) weak solutions to the Navier--Stokes--Fokker--Planck system with polymer-number-density-dependent viscosity coefficients. A rigorous proof of the existence of dissipative weak solutions would require the following:\n\\begin{itemize}\n\n\\item a suitable approximation scheme, compatible with the energy inequality and the compactness arguments presented in this paper;\n\n\\item a rigorous proof of the existence of a solution to the approximation scheme;\n\n\\item proof of the convergence of the sequence of approximate solutions to a dissipative weak solution, mimicking the arguments presented in this paper.\n\n\\end{itemize}\n\nGiven the formal similarity of the present model to the one studied in \\cite{Barrett-Suli}, a natural approach would be to adjust the approximation scheme used\nin \\cite{Barrett-Suli}, based on time-discretization, in the case of the Navier--Stokes--Fokker--Planck system with constant viscosity coefficients (or, alternatively, to use a Galerkin approximation scheme in the spatial variables, similar to the one in \\cite{FNP}). The added technical difficulties, caused by the presence of the variable viscosity coefficients, can be handled exactly as in \\cite{AbFei}, where a similar scheme, based on time-discretization, was applied to a diffuse interface model with viscosity coefficients that depended on the concentration.\n\n\n\n\\section*{Acknowledgements}\n\\thispagestyle{empty}\n\nThe authors acknowledge the support of the project LL1202 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe discovery of the standard model (SM) Higgs boson~\\cite{Chatrchyan:2012xdj,Aad:2012gk}, \nthe only known elementary scalar particle, has been the main result of the Large Hadron Collider\nphysics program. Remarkably, the measured Higgs boson mass was actually predicted with a good accuracy already in 1995~\\cite{Froggatt:1995rt} by\napplying the principle of multiple point criticality (PMPC)~\\cite{Bennett:1993pj,Bennett:1996vy,Bennett:1996hx} to the Higgs boson effective potential. \nToday the SM Higgs criticality has been studied to three-loop accuracy, indicating that the present data prefers a meta-stable Higgs vacuum \nsufficiently close to criticality~\\cite{Buttazzo:2013uya}.\\footnote{This statement depends most strongly on the precise value of the top quark Yukawa coupling, as the critical point remains within $3\\sigma$ uncertainty of $y_t$. Since $y_t$ is not a directly measured quantity, the systematic uncertainty related to its derivation from collider top mass measurements may be presently underestimated~\\cite{Corcella:2019tgt}.}\nThis fact is the most interesting unexplained experimental outcome of collider physics so far.\n\n\nWithin the SM, there is no explanation as to why the SM parameters -- the gauge, Yukawa and scalar quartic couplings -- must take values leading to the Higgs criticality.\nTherefore, in the original proposal~\\cite{Froggatt:1995rt}, the existence of multiple degenerate vacua was argued to be analogous to the \nsimultaneous existence of multiple phases in thermodynamics. This statement is purely empirical and cannot follow solely from the SM physics, nevertheless\nit may have some support from statistical systems in solid state physics~\\cite{Volovik:2003hn}.\nTo derive the PMPC from fundamental physical principles, Higgs criticality has been attributed to the proposed asymptotic safety of quantum gravity~\\cite{Shaposhnikov:2009pv}.\nMore recently Dvali~\\cite{Dvali:2011wk} revived the old argument by Zeldovich~\\cite{Zeldovich:1974py,Kobzarev:1974cp} that physical systems with multiple vacua must all be degenerate because of the consistency of quantum field theory -- tunneling from a Minkowski vacuum to a lower one must be exponentially fast as in any theory possessing ghosts. \nThe Zeldovich-Dvali argument contradicts Coleman's semi-classical computation in the Euclidian space that the formation of vacuum bubbles is exponentially slow~\\cite{Coleman:1977py}, and has also been challenged by studies of different physical systems~\\cite{Garriga:2011we}. However, attempts to generalize Coleman's computation to the realistic case with zero-mass vacuum bubbles, which is the origin of the divergence, have failed and the Zeldovich-Dvali argument has not been disproved~\\cite{Gross:2020tph}. Altogether, the underlying fundamental physics behind the PMPC remains presently unclear: we accept it on empirical grounds.\n\nNevertheless, the phenomenological success of PMPC has motivated different studies of scalar potentials in extended models. Already the very first papers\nafter the Higgs boson discovery argued that if the SM is extended by a singlet scalar, the Higgs vacuum stability as well as criticality can be achieved due to the radiative corrections of the singlet~\\cite{Kadastik:2011aa}. Later the PMPC has been applied to the singlet~\\cite{Haba:2014sia,Kawana:2014zxa,Haba:2016gqx,McDowall:2019knq} and to the two Higgs doublet models (2HDM)~\\cite{Froggatt:2004st,Froggatt:2006zc,Froggatt:2008am,McDowall:2018ulq,Maniatis:2020wfz}.\nThe Higgs criticality may play an important role in understanding Higgs inflation~\\cite{Bezrukov:2007ep}. We do not make an attempt to form a comprehensive list of all possible applications of PMPC and limit ourselves to the above examples.\n\nThe aim of this work is to point out that so far all the applications of PMPC to scalar potentials, including the examples given above, deal with radiatively generated\nmultiple minima. However, if the degeneracy of multiple vacua is a fundamental property of Nature, all vacua of multi-scalar models -- including the ones \ndominated by {\\it tree-level} potential terms -- should be degenerate. We first argue that this formulation of PMPC should be correct, and after that exemplify its physics potential, analyzing the vacuum solutions of the scalar singlet dark matter (DM) model~\\cite{Silveira:1985rk,McDonald:1993ex,Burgess:2000yq,Cline:2013gha} (see~\\cite{Arcadi:2019lka} for a review).\nWe present three simple scalar DM models utilizing the PMPC: I) $\\mathbb{Z}_2$-symmetric real scalar DM model, II) $\\mathbb{Z}_2$-symmetric pseudo-Goldstone DM (pGDM) model and III) $\\mathbb{Z}_3$-symmetric pseudo-Goldstone DM model.\\footnote{In a more complicated pGDM model, all these global symmetries can be broken, save for the CP-like $S \\to S^{*}$ that stabilizes DM \\cite{Alanne:2020jwx}.} Our results will show that the allowed parameter spaces of those models are constrained and the PMPC \nwill, in principle, be testable if the DM of Universe is the singlet scalar. Generalizing this result, our work demonstrates that the PMPC might be a useful tool to discriminate\nbetween different models of new physics beyond the SM.\n\nThis work is organized as follows. In Section~\\ref{sec:PMPC} we generalize the concept of PMPC to the case of multiple scalar fields. In Sections~\\ref{sec:model1},~\\ref{sec:model2} and~\\ref{sec:model3} we analyze our example models. We conclude and discuss perspectives in Section~\\ref{sec:conc}.\n\n\n\n\\section{Tree-level PMPC}\n\\label{sec:PMPC}\n\nSo far the PMPC has been used to constrain model parameters by requiring that radiatively generated minimum of the\nscalar potential is degenerate with the electroweak vacuum. This was the case with the SM as well as with the scalar singlet and doublet\nextensions of the SM. Clearly, in multi-scalar models there can be several minima of the \nscalar potential already at tree level. If the PMPC follows from some underlying fundamental physics principle, all those vacua must be degenerate as well.\nThis is the concept we adopt and study in this work. Although we do not know which physics principle is behind the PMPC, we exemplify with the following discussion\nthat such principles may exist.\n\nAs Zeldovich pointed out in early 70ies~\\cite{Zeldovich:1974py,Kobzarev:1974cp}, if there exist two minima of the scalar potential with $V=0$ and $V<0$,\nthe vacuum decay bubble with mass $m = 0$ can appear with any initial velocity, giving rise to a divergent boost integral. It should be noticed that \nthere is no such a problem with the pair creation of ordinary particles. Instead, this problem is intrinsic to the theories with ghosts, {\\it i.e.}, to particles with negative energy. \nWhile the theories with ghosts are widely believed to be pathological precisely because of the exponentially fast vacuum decay, similar phenomenon\nshould appear in the Lorentz-invariant decay of the Minkowski vacuum, rendering this configuration unphysical. Dvali, who apparently was not aware of the \nZeldovich paper, independently make this claim in Ref.~\\cite{Dvali:2011wk}. More recently this setup was studied in theories of ghosts~\\cite{Gross:2020tph}, finding the same divergence.\n\n\nNeedless to say, this claim contradicts Coleman's result that the vacuum decay rate is finite and exponentially suppressed~\\cite{Coleman:1977py}. \nAs Coleman computed one bubble decay rate in Euclidean space, the authors of Ref.~\\cite{Gross:2020tph} wanted to generalize Coleman's computation to\nthe case of ghosts, but did not succeed. The Zeldovich-Dvali claim was also criticized in Ref.~\\cite{Garriga:2011we} which,\nhowever, studies a completely different physical situation -- particle pair production in uniform electric field. Whether this set-up is analogous to the case of \nvacuum decay is far from being obvious. It seems that the main source of different opinions is whether the divergent boost integral has to be taken or not,\nand there is no definitive answer at the moment. \n\nNevertheless, this discussion demonstrates that there exist attempts to derive PMPC from general physics principles.\nThe PMPC in this, broader formulation has been used to criticize certain supersymmetry breaking scenarios~\\cite{Bajc:2011iu}.\nIn this work we perform phenomenological analyses of implications of tree-level PMPC on multi-scalar DM models.\n\n\n\\section{Model I: $\\mathbb{Z}_2$ real singlet}\n\\label{sec:model1}\n\nThe real scalar singlet DM model extends the SM with one real scalar singlet which is stabilized by a $\\mathbb{Z}_2$ symmetry. \nThis is among the most popular DM models whose phenomenology has been thoroughly studied~\\cite{Davoudiasl:2004be,Ham:2004cf,OConnell:2006rsp,Patt:2006fw,Profumo:2007wc,Barger:2007im,He:2007tt,He:2008qm,Yaguna:2008hd,Lerner:2009xg,Farina:2009ez,Goudelis:2009zz,Profumo:2010kp,Guo:2010hq,Barger:2010mc,Arina:2010rb,Bandyopadhyay:2010cc,Drozd:2011aa,Djouadi:2011aa,Low:2011kp,Mambrini:2011ik,Espinosa:2011ax,Mambrini:2012ue,Djouadi:2012zc,Cheung:2012xb,Cline:2013gha,Urbano:2014hda,Endo:2014cca,Feng:2014vea,Duerr:2015bea,Duerr:2015mva,Beniwal:2015sdl,Cuoco:2016jqt,Escudero:2016gzx,Han:2016gyy,He:2016mls,Ko:2016xwd,Athron:2017kgt,Ghorbani:2018yfr}. Nevertheless, we shall show that applying the PMPC to this model scalar potential will open new perspectives in\nunderstanding the allowed parameter space of the model.\n\nThe scalar potential of the model is given by\n\\begin{equation}\nV=\\mu_h^2 \\lvert H \\rvert^{2} + \\frac{\\mu_s^2}{2}s^2+\\lambda_h \\lvert H \\rvert^{4}+\\frac{\\lambda_s}{4}s^4+\\frac{\\lambda_{hs}}{2} \\lvert H \\rvert^{2} s^2,\n\\end{equation}\nwhere $H$ is the SM Higgs doublet in unitary gauge:\n\\begin{equation}\nH=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}\n0\\\\\nh+v_h\n\\end{array}\n\\right).\n\\end{equation}\nThe Higgs vacuum expectation value is $v_h=246$ GeV.\nThe scalar $s$, as the dark matter candidate, must not acquire a non-zero VEV: then the scalar $s$ does not mix with the Higgs and remains stable due to the $\\mathbb{Z}_2$ symmetry. The freeze-out of the $s$ is independent of the self-coupling of the scalar singlet $\\lambda_s$ and its relic abundance is therefore set only by the Higgs portal coupling $\\lambda_{hs}$. \n\nThe potential may have three types of minima in the field space $(h,s)$: the Higgs vacuum $(v_h,0)$, singlet vacuum $(0,v_s)$ and a general or mixed vacuum $(v'_h,v'_s)$. Presently, the Universe is in the Higgs vacuum. According to the PMPC these minima are to be degenerate, if they exist simultaneously. Let us examine the three vacuum configurations more closely. \n\n\\subsection{The Higgs vacuum configuration $(v_h,0)$}\nThe Higgs vacuum $(v_h,0)$, in which we live, satisfies the following minimization condition:\n\\begin{equation}\n\\mu_h^2=-\\lambda_h v_h^2.\\label{Our vacuum minimization condition}\n\\end{equation} \nThe scalar mass matrix is given by\n\\begin{equation}\n\\mathcal{M}^2=\\left(\\begin{array}{cc}\n2\\lambda_h v_h^2 & 0\\\\\n0 & \\frac{1}{2}\\lambda_{hs} v_h^2+\\mu_s^2\n\\end{array}\n\\right).\n\\end{equation}\n\n\n\\subsection{The singlet vacuum configuration $(0,v_s)$}\nThe minimization condition for the singlet vacuum $(0,v_s)$ is given by\n\\begin{equation}\n\\mu_s^2=-\\lambda_s v_s^2.\\label{singlet vacuum minimization condition}\n\\end{equation}\n\nIn this case, the scalar mass matrix is given by\n\\begin{equation}\n\\mathcal{M}^2=\\left(\\begin{array}{cc}\n\\frac{1}{2}\\lambda_{hs} v_s^2+\\mu_h^2 & 0\\\\\n0 & 2\\lambda_s v_s^2\n\\end{array}\n\\right).\n\\end{equation}\n\n\n\n\\subsection{The mixed vacuum configuration $(v'_h,v'_s)$}\nThe minimization conditions for the mixed vacuum $(v'_h,v'_s)$ are given by\n\\begin{eqnarray}\n\\mu_h^2 & = & -\\lambda_h {v'_h}^2-\\frac{1}{2}\\lambda_{hs}{v'_s}^2\\\\\n\\mu_s^2 & = & -\\lambda_{s} {v'_s}^2-\\frac{1}{2}\\lambda_{hs} {v'_h}^2\n\\end{eqnarray}\nThe scalar mass matrix is given by\n\\begin{equation}\n\\mathcal{M}^2=\\left(\\begin{array}{cc}\n2\\lambda_{h} {v'_h}^2 & \\lambda_{hs} {v'_h}v'_s\\\\\n\\lambda_{hs} {v'_h}v'_s & 2\\lambda_{s} {v'_s}^2\n\\end{array}\n\\right),\n\\end{equation}\nleading to mixing of the Higgs doublet with the real singlet.\n\n\\subsection{Comparison of minima}\nThe current minimum configuration is the Higgs vacuum $(v_h,0)$, which allows dark matter and correctly breaks the electroweak symmetry. If the other vacuum configurations are to exist simultaneously with the $(v_h,0)$, they must be degenerate. Imposing that the $(0,v_s)$ vacuum configuration is degenerate to $(v_h,0)$,\n\\begin{equation}\nV(h=v_h,s=0)= V(h=0,s=v_s),\n\\end{equation}\nleads to the condition\n\\begin{equation}\\label{the constraint for our vacuum VS singlet vacuum}\nm_s^4-\\lambda_{hs} m_s^2 v_h^2 -\\lambda_s\\lambda_h v_h^4+\\frac{1}{4}\\lambda_{hs}^2 v_h^4=0.\n\\end{equation}\n\nDemanding that the $(v'_h,v'_s)$ vacuum configuration be degenerate to the present Higgs vacuum $(v_h,0)$ leads to a contradiction. Therefore, according to the PMPC, there can at most only be two degenerate vacua: the Higgs vacuum $(v_h,0)$ and the singlet vacuum $(0,v_s)$. \n\nThe collider experiments impose constraints on the singlet $s$ when it is lighter than $m_h\/2$. Then the Higgs decay width to the $ss$ final state is \\cite{Cline:2013gha}\n\\begin{equation}\n\\Gamma (h\\to ss) =\\frac{\\sqrt{m_h^2-4m_s^2}}{32\\pi m_h^2}\\lambda_{hs}^2 v_h^2. \n\\end{equation}\nIn addition, the Higgs invisible branching ratio is then given by\n\\begin{equation}\n\\text{BR}_\\text{inv} = \\frac{\\Gamma_{h \\to ss}}{\\Gamma_{h \\to \\text{SM}} + \\Gamma_{h \\to ss}},\n\\end{equation}\nwhich is constrained to be below $0.24$ at $95\\%$ confidence level \\cite{Khachatryan:2016whc,ATLAS-CONF-2018-031} by direct measurements and below about $0.17$ by statistical fits of Higgs couplings \\cite{Giardino:2013bma,Belanger:2013xza}, which excludes singlet masses below $53$ GeV.\n\nThe portal coupling $\\lambda_{hs}$ is determined by demanding that the correct relic abundance $\\Omega_{c} h^{2} = 0.120$ \\cite{Aghanim:2018eyx} is produced in the freeze-out of the singlet. The PMPC in Eq.~(\\ref{the constraint for our vacuum VS singlet vacuum}) allows us to determine the singlet self-coupling $\\lambda_s$ as a function of the dark matter mass, as is presented in the Fig. \\ref{dark matter self-coupling as a function of dark matter mass}. The self-coupling $\\lambda_s$ becomes non-perturbative as the dark matter mass reaches electroweak scale. \n\nDirect detection also limits \nthe dark matter mass so that the only allowed mass region is near the Higgs resonance. The PMPC thus predicts that the dark matter mass must be in the resonance region. \n\n\\begin{figure*}[tb]\n\t\\begin{center}\n\t\\includegraphics[width=0.45\\linewidth]{RealSingletLowMassPlot.pdf}\n\t\\includegraphics[width=0.45\\linewidth]{RealSingletHighMassPlot.pdf}\n\t\\end{center}\n\t\\caption{\\label{dark matter self-coupling as a function of dark matter mass} The value of $\\mathbb{Z}_2$ real singlet DM self-coupling $\\lambda_s$ as a function of DM mass according to PMPC in Eq. \\eqref{the constraint for our vacuum VS singlet vacuum}. The portal coupling is fixed so that the correct relic abundance is obtained. Left panel: the self-coupling $\\lambda_s$ near the Higgs resonance, $m_s\\in [53,70]$ GeV. Right panel: the self-coupling $\\lambda_s$ away from the resonance, $m_s\\in [65,1000]$ GeV. }\n\\end{figure*}\n\n\n\n\n\n\\section{Model II: $\\mathbb{Z}_2$ pseudo-Goldstone DM} \n\\label{sec:model2}\n\nThis model is an extension of SM with a complex scalar $S$ carrying a global $U(1)$ charge. The most general scalar potential invariant under global $U(1)$ transformation is given by\n\\begin{equation}\\label{}\n\\begin{split}\n V_0 &= \\mu_h^2 \\lvert H \\rvert^{2} + \\mu_s^2 \\lvert S \\rvert^{2} +\\lambda_h \\lvert H \\rvert^{4}\n +\\lambda_s \\lvert S \\rvert^{4} \\\\\n &+\\lambda_{hs} \\lvert H \\rvert^{2} \\lvert S \\rvert^{2},\n\\end{split}\n\\end{equation}\nwhere $H$ is the SM Higgs doublet. \nThe global $U(1)$ symmetry is spontaneously broken as the scalar $S$ acquires a non-zero vacuum expectation value. This would result in a massless Goldstone boson in the physical spectrum. In order to avoid this, the global $U(1)$ symmetry is explicitly broken into its discrete subgroup $\\mathbb{Z}_2$ by the following soft term:\n\\begin{equation}\nV_{\\mathbb{Z}_2}=-\\frac{\\mu^{\\prime 2}}{4}S^2+\\text{h.c},\n\\end{equation}\nso the full potential becomes\n\\begin{equation}\\label{Z2 scalar potential}\nV=V_0+V_{\\mathbb{Z}_2}.\n\\end{equation}\n\nThe model was first considered in \\cite{Barger:2008jx,Chiang:2017nmu}, but only in the Higgs resonance region. The suppression of the direct detection cross section of the pseudo-Goldstone DM was noted in \\cite{Gross:2017dan}; there has been large interest in the phenomenology of the model \\cite{Huitu:2018gbc,Azevedo:2018oxv,Azevedo:2018exj,Alanne:2018zjm,Cline:2019okt,Kannike:2019wsn}.\n\nThe parameter $\\mu^{\\prime 2}$ can be made real and positive by absorbing its phase into the field $S$. When $\\mu^{\\prime 2}$ is positive, then the VEV of $S$ is real. The reality of the VEV can be easily seen by writing the singlet in polar coordinates:\n\\begin{equation}\nS=\\frac{1}{\\sqrt{2}}r e^{i\\theta}.\n\\end{equation}\nThe soft term now becomes\n\\begin{equation}\nV_{\\mathbb{Z}_2}=-\\frac{\\mu^{\\prime 2}}{4}r^2\\cos (2\\theta).\n\\end{equation}\nThe soft term is the only part of the potential that depends on the phase of the singlet $S$. Therefore the global minimum of the potential is the minimum of the soft term. Because $\\mu^{\\prime 2}>0$, the minimum is obtained when $\\cos(2\\theta)=1$. It follows that\n\\begin{equation}\n2\\theta =\\pm 2\\pi n\\Rightarrow \\theta=0,\\pi,\n\\end{equation}\nand therefore the VEV is real.\n\n\nWe parametrize the fields in the unitary gauge as:\n\\begin{equation}\nH=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}\n0\\\\\nh+v_h\n\\end{array}\n\\right),\n\\end{equation}\nand\n\\begin{equation}\nS=\\frac{1}{\\sqrt{2}}(v_s+s+i\\chi).\n\\end{equation}\nThe Higgs vacuum expectation value is $v_h=246$ GeV and $v_s$ is real.\n\nThe scalar potential in Eq. (\\ref{Z2 scalar potential}) is invariant under two different $\\mathbb{Z}_2$ symmetries,\n\\begin{equation}\nS\\to -S,\n\\end{equation}\nand the CP-like\n\\begin{equation} \nS\\to S^\\ast.\n\\end{equation} \nAfter the symmetry breaking the latter symmetry becomes equivalent to $\\chi\\to -\\chi$, which stabilizes $\\chi$, making it a dark matter candidate. \n\nWe will now move on to study the vacuum structure of the potential in Eq. (\\ref{Z2 scalar potential}).\n\n \n\\subsection{Different vacua}\nThe potential in \\eqref{Z2 scalar potential} is bounded from below if\n\\begin{equation} \n\\lambda_h > 0,\\quad \\lambda_s > 0, \\quad 2\\sqrt{\\lambda_h \\lambda_s}+\\lambda_{hs} > 0, \n\\end{equation}\n There are three possible vacuum configurations in field space $(h,s,\\chi)$: the mixed vacuum $(v_h, v_s,0)$, the Higgs vacuum $(v'_h, 0, 0)$ and the singlet vacuum $(0, v'_s, 0)$. Let us study if there can be degenerate minima.\n\n\n\\subsubsection{Our vacuum $(v_h,v_s,0)$}\n\nIn order for the pseudo-Goldstone to be the DM candidate, our Universe has to reside in the mixed vacuum. This vacuum $(v_h,v_s,0)$ satisfies the following minimization conditions:\n\\begin{align}\n\\mu_h^2 \n& = -\\lambda_h v_h^2-\\frac{1}{2}\\lambda_{hs} v_s^2\\label{Our vacuum minimization condition h},\\\\\n\\mu_s^2 & = -\\lambda_s v_s^2-\\frac{1}{2}\\lambda_{hs} v_h^2+\\frac{1}{2}\\mu^{\\prime 2},\n\\label{Our vacuum minimization condition s}\n\\end{align}\nwhich can be inverted in favor of the VEVs:\n\\begin{align}\nv_h^2 & = \\frac{-4\\lambda_s \\mu_h^2 + 2\\lambda_{hs}\\mu_s^2 -\\lambda_{hs}\\mu^{\\prime 2}}{4\\lambda_h\\lambda_s-\\lambda_{hs}^2}>0\\label{solved VEVs 1},\\\\\nv_s^2 & = \\frac{-4\\lambda_h \\mu_s^2 + 2\\lambda_{hs}\\mu_h^2 -2\\lambda_{h}\\mu^{\\prime 2}}{4\\lambda_h\\lambda_s-\\lambda_{hs}^2}>0.\\label{solved VEVs 2}\n\\end{align}\n\nThe mass matrix in the $(h,s,\\chi)$-basis is given by\n\\begin{equation}\n\\mathcal{M}^2=\\begin{pmatrix}\n2\\lambda_h v_h^2 & \\lambda_{hs} v_h v_s & 0\\\\\n\\lambda_{hs} v_h v_s & 2\\lambda_s v_s^2 & 0\\\\\n0 & 0 & \\mu^{\\prime 2}\n\\end{pmatrix}.\n\\end{equation}\n\nThe determinant of the $2\\times 2$ CP-even block matrix has to be strictly positive in order for the mixed configuration to be a physical minimum, yielding the condition\n\\begin{equation}\\label{positive CP-even masses}\n4\\lambda_h\\lambda_s-\\lambda_{hs}^2 >0.\n\\end{equation}\nWe have chosen the parameter $\\mu^{\\prime 2}$ to be positive and therefore all the masses will be positive. The mixed vacuum configuration $(v_h, v_s,0)$ is therefore the minimum to which the other configurations are compared.\n\nWhen one combines Eq. \\eqref{positive CP-even masses} with Eqs. \\eqref{solved VEVs 1} and \\eqref{solved VEVs 2}, the following inequalities are obtained:\n\\begin{align}\n-4\\lambda_s\\mu_h^2+2\\lambda_{hs}\\mu_s^2-\\lambda_{hs}\\mu^{\\prime 2} &>0,\\label{VEV positivity condition 1}\\\\\n-4\\lambda_h\\mu_s^2+2\\lambda_{hs}\\mu_h^2+2\\lambda_h\\mu^{\\prime 2} &>0.\\label{VEV positivity condition 2}\n\\end{align}\n\n\n\\subsubsection{The Higgs vacuum $(v'_h,0,0)$}\n\nWhile the Higgs vacuum $(v'_h,0,0)$ gives rise to correct electroweak symmetry breaking, in this case it is not the pseudo-Goldstone boson but the complex singlet $S$ that is the DM candidate, contrary to the present assumptions. The mass matrix in this vacuum is given by\n\\begin{equation}\n\\mathcal{M}^2=\\begin{pmatrix}\n-2\\mu_h^2 & 0 & 0\\\\\n0 & -\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}+\\mu_s^2-\\frac{\\mu^{\\prime 2}}{2} & 0\\\\\n0 & 0 & -\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}+\\mu_s^2+\\frac{\\mu^{\\prime 2}}{2}\n\\end{pmatrix}.\n\\end{equation}\nBy using Eq. (\\ref{VEV positivity condition 2}), one sees that the mass in the middle is negative. Therefore this vacuum configuration is not a minimum.\n\n\\subsubsection{The singlet vacuum $(0,v'_s,0)$}\n\nThe singlet vacuum $(0,v'_s,0)$ can in principle be a minimum. Let us compare, however the potential energy of this minimum to that of our mixed minimum $(v_h,v_s,0)$. Their difference is\n\\begin{equation}\n\\begin{split}\n V(v_h,v_s,0)&-V(0,v'_s,0) =-\\frac{1}{4}\\frac{1}{4\\lambda_h\\lambda_s-\\lambda_{hs}^2} \\frac{1}{\\lambda_s}\n \\\\\n &\\times \\left[\\frac{\\lambda_{hs}}{2}(2\\mu_s^2-\\mu^{\\prime 2})-2\\lambda_s\\mu_s^2\\right]^2<0,\n\\end{split}\n\\end{equation}\naccording to Eq.~(\\ref{positive CP-even masses}). Therefore, we have that\n\\begin{equation}\nV(v_h,v_s,0)0$, the VEV of $S$ is negative.\n\n\nWe now parametrize the scalar fields in unitary gauge as\n\\begin{equation}\nH=\\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n0\\\\\nh+v_h\n\\end{pmatrix},\n\\end{equation}\nand\n\\begin{equation}\nS=\\frac{1}{\\sqrt{2}}(v_s+s+i\\chi).\n\\end{equation}\nThe Higgs vacuum expectation value is $v_h=246$ GeV.\nThe pseudo-scalar $\\chi$ can be dark matter \\cite{Kannike:2019mzk}. The $\\mathbb{Z}_3$-symmetric complex singlet model has also been studied in the phase where the $\\mathbb{Z}_3$ remains unbroken \\cite{Belanger:2012zr,Hektor:2019ote}. \n\nThe scalar potential in Eq. (\\ref{Z3 scalar potential}) is invariant under the CP-like $\\mathbb{Z}_2$-symmetry\n\\begin{equation}\nS\\to S^\\ast.\n\\end{equation}\nThis symmetry is becomes \n\\begin{equation} \n\\chi\\to -\\chi,\n\\end{equation}\nwhen $S$ acquires VEV. This symmetry stabilizes $\\chi$, making it a dark matter candidate. \n\nWe next study the vacuum structure of the model.\n \n\\subsection{Different vacua}\n\nThe potential in Eq. (\\ref{Z3 scalar potential}) is bounded from below, if\n\\begin{equation} \n\\lambda_h > 0,\\quad \\lambda_s > 0, \\quad 2\\sqrt{\\lambda_h \\lambda_s}+\\lambda_{hs} > 0. \n\\end{equation}\n \nThere are three possible vacuum configurations in the field space $(h,s,\\chi)$: the mixed vacuum $(v_h,v_s,0)$, the Higgs vacuum $(v'_h,0,0)$ and the singlet vacuum $(0,v'_s,0)$.\n\n\n\n\n\\subsection{Our vacuum $(v_h,v_s,0)$}\n\nIn order for the pseudo-Goldstone to be the DM candidate, our Universe has to reside in the mixed vacuum. \nThe vacuum $(v_h,v_s,0)$ satisfies the following minimization conditions:\n\\begin{eqnarray}\n\\mu_h^2 \n& = & -\\lambda_h v_h^2-\\frac{1}{2}\\lambda_{hs} v_s^2,\n\\label{Our vacuum minimization condition h}\\\\\n\\mu_s^2 & = &-\\lambda_s v_s^2-\\frac{1}{2}\\lambda_{hs} v_h^2-\\frac{3}{2\\sqrt{2}}v_s{\\mu'_s}.\\label{Our vacuum minimization condition s}\n\\end{eqnarray} \nThe minimization conditions are now hard to invert in favor of VEVs as minimization condition of $\\mu_s^2$ contains a linear term in $v_s$, due to the soft cubic self-coupling of $S$. The VEVs are solved to be:\n\\begin{align}\nv_s & = \\frac{-\\frac{3\\mu'_s}{\\sqrt{2}}\\pm 2\\sqrt{\\frac{1}{\\lambda_h}\\left(4\\lambda_h\\lambda_s-\\lambda_{hs}^2\\right)\\left(\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}-\\mu_s^2\\right)+\\frac{9\\mu_{s}^{\\prime 2}}{8}}}{\\frac{1}{\\lambda_h}\\left(4\\lambda_h\\lambda_s-\\lambda_{hs}^2\\right)},\n\\label{vs our vacuum minimum}\\\\\nv_h & = \\pm\\frac{\\sqrt{-\\lambda_{hs} v_s^2-2\\mu_h^2}}{\\sqrt{2}\\sqrt{\\lambda_h}}.\n\\end{align}\nWe choose the positive sign $v_s$-solution to be the VEV in our minimum. The negative sign $v_s$-solution is not a minimum as one of the scalar masses is necessarily negative in this case.\n\nThe mass matrix in the $(h,s,\\chi)$-basis is given by\n\\begin{equation}\\label{Z3 our minimum masses}\n\\mathcal{M}^2=\\left(\\begin{array}{ccc}\n2\\lambda_h v_h^2 & \\lambda_{hs} v_h v_s & 0\\\\\n\\lambda_{hs} v_h v_s & 2\\lambda_s v_s^2+\\frac{3}{2\\sqrt{2}}v_{s} \\mu'_s & 0\\\\\n0 & 0 & -\\frac{9v_s \\mu'_s}{2\\sqrt{2}}\n\\end{array}\n\\right).\n\\end{equation}\n\nThe $2 \\times 2$ block in the upper-left corner of Eq. \\eqref{Z3 our minimum masses} can be diagonalized by an orthogonal transformation\n\\begin{equation}\nU^T M^2 U = \\textrm{diag}(m_h^2,m_H^2),\n\\end{equation}\nwhere\n\\begin{equation}\nM^2=\\left(\\begin{array}{cc}\n2\\lambda_h v_h^2 & \\lambda_{hs} v_h v_s \\\\\n\\lambda_{hs} v_h v_s & 2\\lambda_s v_s^2+\\frac{3}{2\\sqrt{2}}vs \\mu'_s \n\\end{array}\n\\right),\n\\end{equation}\nand\n\\begin{equation}\nU=\\left(\\begin{array}{cc}\n\\cos\\theta & \\sin\\theta \\\\\n-\\sin\\theta & \\cos\\theta \n\\end{array}\n\\right).\n\\end{equation}\nThe mixing angle $\\theta$ is defined as\n\\begin{equation}\n\\tan 2\\theta =-\\frac{\\lambda_{hs}v_h v_s}{\\lambda_h v_h^2-\\lambda_s v_s^2-\\frac{3}{4\\sqrt{2}}v_s \\mu'_s}.\n\\end{equation}\nWith the knowledge that we must choose the positive sign solution of $v_s$, the parameters in the potential can be interchanged to more suitable ones. The potential parameters $\\mu_h^2$, $\\mu_s^2$, $\\lambda_h$, $\\lambda_s$, $\\lambda_{hs}$ and $\\mu'_s$ are replaced by $v_h$, $v_s$, $m_h$, $m_H$, $m_\\chi$ and the mixing angle $\\theta$:\n\\begin{align}\n\\mu_h^2 & = -\\frac{1}{4}(m_h^2+m_H^2)+\\frac{1}{4v_h}(m_H^2-m_h^2)\n \\notag\n \\\\\n &\\times (v_h\\cos 2\\theta -v_s\\sin 2\\theta),\\\\\n\\mu_s^2 & = -\\frac{1}{4}(m_h^2+m_H^2)+\\frac{1}{4v_s}(m_h^2-m_H^2)\n\\notag\n\\\\\n&\\times(v_s\\cos 2\\theta +v_h\\sin 2\\theta)+\\frac{1}{6}m_\\chi^2,\\\\\n\\lambda_h & = \\frac{m_h^2+m_H^2+(m_h^2-m_H^2)\\cos 2\\theta}{4 v_h^2},\\\\\n\\lambda_s & = \\frac{3(m_h^2+m_H^2)+3(m_H^2-m_h^2)\\cos 2\\theta+2m_\\chi^2}{12 v_s^2},\\\\\n\\lambda_{hs} & = -\\frac{(m_h^2-m_H^2)\\sin 2\\theta}{2 v_h v_s},\\\\\n\\mu'_s & = -\\frac{2\\sqrt{2}}{9}\\frac{m_\\chi^2}{v_s}.\n\\end{align}\n\n\\subsection{The Higgs vacuum $(v'_h,0,0)$}\n\nWhile the Higgs vacuum $(v'_h,0,0)$ gives rise to correct electroweak symmetry breaking, in this case it is not the pseudo-Goldstone boson but the complex singlet $S$ that is the DM candidate, contrary to the present assumptions.\nThe mass matrix in this vacuum is given by\n\\begin{equation}\n\\mathcal{M}^2=\\begin{pmatrix}\n-2\\mu_h^2 & 0 & 0\\\\\n0 & -\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}+\\mu_s^2 & 0\\\\\n0 & 0 & -\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}+\\mu_s^2\n\\end{pmatrix}.\n\\end{equation}\nAll the masses can be positive simultaneously. This minimum can be degenerate with our mixed vacuum, leading into the condition\n\\begin{equation}\n\\mu^{\\prime 2}=\\frac{1}{\\lambda_h}\\left(4\\lambda_h\\lambda_s-\\lambda_{hs}^2\\right)\\left(-\\frac{\\lambda_{hs}}{2\\lambda_h}\\mu_h^2+\\mu^2_s\\right),\n\\end{equation}\nor equivalently,\n\\begin{equation}\\label{higgs only vacuum degeneracy}\nm_\\chi^2 = \\frac{9m_h^2 m_H^2}{m_h^2\\cos^2\\theta+m_H^2\\sin^2\\theta}.\n\\end{equation}\n\n\n\\subsection{The singlet vacuum $(0,v'_s,0)$}\nIn the singlet vacuum $(0,v'_s,0)$, only the complex singlet $S$ gets a VEV and electroweak symmetry is not broken.\nThe minimization condition is given by\n\\begin{equation}\n\\mu_s^2=-\\lambda_s {v'_s}^2-\\frac{3}{2\\sqrt{2}}v'_s \\mu'_s.\n\\end{equation}\nThe VEV $v'_s$ can be found to be\n\\begin{equation}\nv'_s=\\frac{-3\\mu'_s\\pm \\sqrt{-32\\lambda_s\\mu_s^2+9\\mu^{\\prime 2}}}{4\\sqrt{2}\\lambda_s}.\n\\end{equation}\n\nThe mass matrix in this vacuum is given by\n\\begin{equation}\n\\mathcal{M}^2=\\left(\\begin{array}{ccc}\n\\mu_h^2-\\frac{\\lambda_{hs}}{2 \\lambda_s}\\left[\\mu_s^2+\\frac{3\\mu'_s v'_s}{2\\sqrt{2}}\\right] & 0 & 0\\\\\n0 & -2\\mu_s^2-\\frac{3\\mu'_s v'_s}{2\\sqrt{2}} & 0\\\\\n0 & 0 & -\\frac{9\\mu'_s v'_s}{2\\sqrt{2}}\n\\end{array}\n\\right).\n\\end{equation}\n\nThis vacuum configuration does not seem to be able to be a minimum when the mixed vacuum $(v_h,v_s,0)$ and the Higgs vacuum $(v'_h,0,0)$ are degenerate. It is hard to show analytically, but all numerical evidence points to that.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{direct_detection}\n\\end{center}\n\\caption{Direct detection cross section for $\\mathbb{Z}_{3}$ pGDM. The XENON1T result \\cite{Aprile:2018dbl} is shown in solid line and prediction \\cite{Ni2017} for the XENONnT in dashed line. The red points are excluded by the Higgs invisible width or a too large Higgs total width.}\n\\label{fig:Z3:direct:detection}\n\\end{figure}\n\n\n\n\\subsection{Final verdict on $\\mathbb{Z}_3$ pGDM from the PMPC}\nThe two vacuum configurations $(v_h,v_s,0)$ and $(v'_h,0,0)$ can be minima and degenerate at the same time. This degeneracy of these two minima will impose the condition in Eq. (\\ref{higgs only vacuum degeneracy}), which aproximately reads:\n\\begin{equation}\nm_\\chi\\approx 3 m_H,\n\\end{equation}\nso strictly speaking, $\\chi$ is on the borderline of not being a true pseudo-Goldstone boson.\nThe third vacuum configuration $(0,v'_s,0)$ does not seem to be a minimum when the $(v_h,v_s,0)$ and $(v'_h,0,0)$ are degenerate. This is also checked only numerically. \n\n\nWe also impose constraints from collider experiments. The fit of the Higgs signal strengths constrains the value of the mixing angle to $\\cos^2\\theta\\ge 0.9$~\\cite{Beacham:2019nyx}. If the mass of $x \\equiv \\chi \\text{ or }h_2$ is less than $m_h\/2$, then the Higgs decay width into this particle is given by \\cite{Kannike:2019mzk}\n\\begin{equation}\n \\Gamma_{h \\to x x} = \\frac{g_{hxx}}{8 \\pi} \\sqrt{1 - 4 \\frac{m_x^2}{m_h^2}}\n\\end{equation}\nwith\n\\begin{align}\n g_{h1 \\chi \\chi} &= \\frac{m_h^2 + m_\\chi^2}{v_S},\n \\\\\n g_{h1 h_2 h_2} &= \\frac{1}{v v_S} \\left[ \\left(\\frac{1}{2} m_h^2 + m_2^2\\right) (v \\cos \\theta + v_S \\sin \\theta) \\right.\n \\notag\n \\\\\n & \\left. \n + \\frac{1}{6} v m_\\chi^2 \\cos \\theta \\right] \\sin 2 \\theta.\n\\end{align}\nThe latest measurements of the width of an SM-like Higgs boson\ngives $\\Gamma^{h_1}_\\text{tot}\\,=3.2^{+2.8}_{-2.2}$ MeV, with 95\\% CL limit on\n$\\Gamma_\\text{tot} \\leq 9.16$ MeV \\cite{Sirunyan:2019twz}.\nIn addition, the Higgs invisible branching ratio is then given by\n\\begin{equation}\n\\text{BR}_\\text{inv} = \\frac{\\Gamma_{h \\to \\chi\\chi}}{\\Gamma_{h_1 \\to \\text{SM}} + \\Gamma_{h \\to \\chi\\chi} + \\Gamma_{h \\to h_2 h_2}},\n\\end{equation}\nwhich is constrained to be below $0.24$ at $95\\%$ confidence level \\cite{Khachatryan:2016whc,ATLAS-CONF-2018-031} by direct measurements and below about $0.17$ by statistical fits of Higgs couplings \\cite{Giardino:2013bma,Belanger:2013xza}. Decays of the resulting $h_{2}$ into the SM may be visible \\cite{Huitu:2018gbc}.\n\nA scan of the parameter space with the condition \\eqref{higgs only vacuum degeneracy} imposed is shown in Fig.~\\ref{fig:Z3:direct:detection}. We take the mass of the other CP-even scalar $m_{2} \\in [10, 1500]$ and the mixing angle $\\sin \\theta \\in \\pm [0.01, 0.3]$. The VEV $v_{s}$ is fitted from the DM relic density $\\Omega_{c} h^{2} = 0.120$ \\cite{Aghanim:2018eyx}. The points in red are excluded by the Higgs invisible width and\/or by an overly large Higgs total width.\n\n\\section{Discussion and Conclusion}\n\\label{sec:conc}\n\n\nIn this work we have argued that the principle of multiple point criticality must be generalized to all vacuum states, implying that all minima of a scalar potential \nmust be degenerate. Indeed, if the PMPC follows from some fundamental underlying principle, its validity must not depend on the particulars of the model.\nFor example, in the most studied case, one minimum is taken to be the SM EW Higgs minimum and another one the Planck-scale minimum generated by the running of Higgs quartic coupling. We argue that the principle must apply to all minima in multi-scalar models, whether they are generated radiatively or at tree level.\n\nPhenomenological implications of this formulation of the PMPC are straightforward but profound -- the PMPC becomes a tool to constrain the parameter space of \nmulti-scalar models. Since the scalar singlet models represent one of the most popular class of dark matter models, we exemplified the usefulness of the PMPC by studying how different realizations of scalar DM models are constrained by imposing the PMPC requirements. \n\nIn the simplest possible model, studied in Section~\\ref{sec:model1}, the DM is a real scalar singlet stabilized by a $\\mathbb{Z}_2$ symmetry. The PMPC predicts that the singlet self-coupling hits a Landau pole at DM mass of about $300$~GeV, invalidating the model above that scale. Due to the stringent direct detection constraints, the DM mass in this model is predicted to be near the Higgs resonance.\n\nThe second model under investigation, the $\\mathbb{Z}_2$ symmetric pseudo-Goldstone DM model, studied in Section~\\ref{sec:model2}, cannot have degenerate minima at all. Therefore it cannot be consistent with the PMPC requirements. \n\nThe third model under investigation, the $\\mathbb{Z}_3$ symmetric pseudo-Goldstone DM model, studied in Section~\\ref{sec:model3}, can have degenerate minima. Applying the PMPC in this case leads to a simple relation between the DM and the second scalar masses and excludes DM masses below about $190$~GeV via the measured Higgs width. This model represents an example how the PMPC can constrain the phenomenology of multi-scalar models.\n\nIn conclusion, if the PMPC is indeed realized in Nature, which we have assumed in this work, it may lead to significant constraints and simplifications of the phenomenology of multi-scalar models. Further studies of degenerate tree-level minima in those models is worth of pursuing. \n\n\n\\vspace{5mm}\n\\noindent \\textbf{Acknowledgement.} We thank Gia Dvali, Luca Marzola, \nAlessandro Strumia, Daniele Teresi and Hardi Veerm{\\\"a}e for discussions on the origin of PMPC. \nThis work was supported by the Estonian Research Council grants PRG434, PRG803, PRG803, MOBTT5 and MOBTT86, and by the EU through the European Regional Development Fund CoE program TK133 ``The Dark Side of the Universe.\" \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\bf 1. Introduction.}\n\nIn two previous works, [\\putref{DandK2}], [\\putref{DandK1}], we have\ninvestigated the thermal quantities for scalar and spinor fields on the\nspace--time T$\\times$S$^d$, where $d$ is odd. The conformal scalar\ninternal energy, $E$, for example, is a linear combination of `partial'\nenergies, $\\epsilon_t,\\, t=1,2,\\ldots$, (see below). These quantities possess\na known behaviour under modular transformations and are central to\nelliptic function theory. In particular, a temperature inversion symmetry\ncan be exhibited.\n\nFrom basic elliptic properties, it was shown that $E$ could be expressed\nas a polynomial in $\\epsilon_2$ and $\\epsilon_3$, while, for the specific heat, and\nall higher derivatives, one has to include $\\epsilon_1$, preferably via a\nmodular covariant derivative. The free energy involves an integration and\nthings are not so simple as obstructions arise to simple modular\nbehaviour. It is this aspect that is explored in the present work. I use\nthis thermal angle just to motivate the introduction of various\nquantities, the analysis of which allows us to make contact with various\ntopics in the theory of modular forms. Some of the procedures can be\ngiven a physical terminology, which might be suggestive.\n\n\\section{\\bf 2. Mellin transforms and the period polynomial.}\n\nIt is advantageous to begin from the basic Mellin transform, used {\\it e.g. } by\nMalurkar and Hardy, for the Eisenstein series (or partial internal\nenergy, [\\putref{DandK1}] [\\putref{DandK2}]),\n $$\\eqalign{\n \\epsilon_t({\\mbox{{\\ssb\\char98}}})&=-{B_{2t}\\over 4t}+\n \\sum_{n=1}^\\infty {n^{2t-1}q^{2n}\\over1-q^{2n}}\\cr\n &={1\\over2}\\,\\zeta_R(1-2t)+{1\\over2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,\\Gamma(s)\\,\n \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,(2\\pi{\\mbox{{\\ssb\\char98}}})^{-s}\\,,}\n \\eql{mell1}$$\nwhere $q=e^{i\\pi\\tau}$ and $c$ lies above $2t$. I am using\n${\\mbox{{\\ssb\\char98}}}=-i\\tau= \\beta\/2\\pi=1\/\\xi$ as yet another convenient variable.\nAlthough, physically, $\\mbox{{\\ssb\\char98}}$ is real, if we wish to consider general\nmodular transformations then we must allow it to become complex.\n\nThe Mellin transform has been used systematically in finite temperature\nfield theory and elsewhere in discussions of asymptotic limits. The\nsurvey by Elizalde {\\it et al}, [\\putref{Elizalde2}], is useful and I\nfurther mention, as being somewhat relevant here, Cardy [\\putref{Cardy}],\nKutasov and Larsen, [\\putref{KandL}]. Terras, [\\putref{Terras}] pp.55,229,\ncan be consulted for aspects of the mathematical side. \\marginnote{OTHERS} In\nnumber theory see Hardy and Littlewood, [\\putref{HandL}], and Landau,\n[\\putref{Landau,Landau2}].\n\nEquation (\\puteqn{mell1}) can be looked upon as a continuous expansion in\n${\\mbox{{\\ssb\\char98}}}$. Various limits in ${\\mbox{{\\ssb\\char98}}}$ are uncovered by displacing the\nintegration contour. The pole in the integrand at $s=2t$ corresponds to\nthe Planck term, while the remaining one at $s=0$ corresponds to minus\nthe Casimir value, which is the first term in (\\puteqn{mell1}). The high\ntemperature limit (${\\mbox{{\\ssb\\char98}}}\\to0$) follows by pushing the contour to the\nleft, beyond the Casimir pole, the contribution from which cancels the\nfirst term in (\\puteqn{mell1}) leaving just the effect of the Planck pole\nand a correction that tends to zero exponentially. Conversely, on moving\nthe contour all the way to the right only the zero temperature Casimir\nterm remains.\n\nThis cosmetic asymmetry can be avoided (if desired) by also extracting\nthe Planck term, defining,\n $$\n \\epsilon_t^{\\rm sub}({\\mbox{{\\ssb\\char98}}})=\\epsilon_t(\\mbox{{\\ssb\\char98}})-{1\\over2}\\,\\zeta_R(1-2t)\n \\big(1+({i\\mbox{{\\ssb\\char98}}})^{-2t}\\big)\\,,\n \\eql{rege}$$\nand writing\n $$\\eqalign{\n\\epsilon_t^{\\rm sub}({\\mbox{{\\ssb\\char98}}})={1\\over2\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty}ds\\,\\Gamma(s)\\,\n \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,(2\\pi{\\mbox{{\\ssb\\char98}}})^{-s}\\,,}\n \\eql{mell2}$$\nwhere now $c$, at the moment, lies anywhere {\\it between} $0$ and $2t$.\n\nThe quantity in the integrand is the Hecke $L$--series of the normalised\nEisenstein modular form,\n $$\n L(G_t,s)=\\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,,\n \\eql{eisl1}$$\nsatisfying the (typical) functional relation\n $$\n (2\\pi)^{-s}\\Gamma(s)\\,L(G_t,s)=(-1)^t(2\\pi)^{s-2t}\\Gamma(2t-s)\\,L(G_t,2t-s)\\,.\n \\eql{gfr}$$\n\nI now consider a multiple integration with respect to ${\\mbox{{\\ssb\\char98}}}$ of the\ninternal energy. Actually for the free energy only one integration is\nneeded but the general case is mathematically important. So integrate\n$\\epsilon_t^{\\rm sub}$, $h$ times to give, using (\\puteqn{mell2}),\n $$\\eqalign{\n{1\\over\\Gamma(h)}&\\int_\\mbox{{\\ssb\\char120}}^\\infty d{\\mbox{{\\ssb\\char98}}}\\,({\\mbox{{\\ssb\\char98}}}-\\mbox{{\\ssb\\char120}})^{h-1}\n\\,\\epsilon_t^{\\rm sub}({\\mbox{{\\ssb\\char98}}})\\cr\n & ={1\\over2\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty}ds\\,{\\Gamma(s-h)\\over(2\\pi)^s}\\,\n \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,\\mbox{{\\ssb\\char120}}^{-s+h}\\,,}\n \\eql{mint}$$\nwith $h2t-2$ and, continued in $s$, satisfies the\nfunctional relation,\n $$\n (2\\pi)^{-s}\\,\\Gamma(s)\\,L({\\cal F},s)=(-1)^t(2\\pi)^{s-2t}\\,\\Gamma(2t-s)\\,L({\\cal F},2t-s)\\,,\n \\eql{lfr}$$\nby virtue of the (inversion) modularity of ${\\cal F}$, and conversely. This\nis usually attributed to Hecke, [\\putref{Hecke}] (but see the Appendix).\nTerras, [\\putref{Terras}] p.229, gives a useful pedagogical treatment of\nthis topic.\n\nThen the periods could be {\\it defined}, in general, by,\n[\\putref{KandZ}],[\\putref{Zagier}],\n $$\\eqalign{\n r_n({\\cal F})&=\\lim_{s\\to n+1}\\,{\\Gamma(s)\\over(2\\pi)^{s}}\\,L({\\cal F},s)\n \\cr\n &\\equiv\\lim_{s\\to n+1}L^*({\\cal F},s)\\,,\n }\n \\eql{periods2}$$\nbecause this coincides with the integral expression, (\\puteqn{periods}), for\ncusp forms.\n\nThe explicit values of $r_n$ ($n=0,\\ldots 2t-2$) for Eisenstein series\n(which is all that is required because of a decomposition theorem for\nmodular form space) are easily computed from (\\puteqn{eisl1}) and are given\nin [\\putref{KandZ}]. They agree with (\\puteqn{even}) and (\\puteqn{moments}), as\nalready noted.\n\nHowever, the Dirichlet series, $L^*({\\cal F},s)$ has poles at $s=0$ and\n$s=2t$. A means of incorporating these values is given by Zagier,\n[\\putref{Zagier}], using the period polynomial. One notes that the\nsummation in the cusp period polynomials can be extended,\n $$\\eqalign{\n r_{\\cal F}(\\mbox{{\\ssb\\char120}})&=\\sum_{n\\in\\,\\mbox{\\open\\char90}}(-1)^n\\comb{2t-2}{n}\\,r_n({\\cal F})\\,\n \\mbox{{\\ssb\\char120}}^{2t-2-n} \\cr\n&=\\sum_{n\\in\\,\\mbox{\\open\\char90}}(-1)^n\\comb{2t-2}{n}\\,i^{n+1}\\,L^*({\\cal F},n+1)\\,\n\\mbox{{\\ssb\\char120}}^{2t-2-n}\\,,}\n \\eql{ppol6}$$\nbecause the binomial coefficient vanishes outside the range\n$n=0,\\ldots,2t-2$. For {\\it non}--cusp forms, however, the sum extends\nfrom $-1$ to $n=2t-1$, the nonzero end values arising from the\nsingularities of $L^*({\\cal F},s)$ mentioned above. The resulting expression\ncan then be taken as a {\\it definition} of the period polynomial of a\nnon--cusp form.\n\nAgain, for the Eisenstein series, $G_{2t},\\,=\\epsilon_{t}$, the computation is\nstraightforward, [\\putref{Zagier}], and (\\puteqn{ppol6}) with (\\puteqn{eisl1})\nyields, ($n=2j-1$),\n $$\\eqalign{\n r_G(x)={(2t-2)!\\over2(2\\pi)^{2t}}\\bigg(2\\pi\n \\zeta_R&(2t-1)\\,\\big((-1)^{t-1}\\mbox{{\\ssb\\char120}}^{2t-2}-1\\big)-\\cr\n &(2\\pi)^t\\sum_{j=0}^t\\,(-1)^{t-j}{B_{2j}\\,B_{2t-2j}\\over(2j)!(2t-2j)!}\n \\,\\mbox{{\\ssb\\char120}}^{2j-1}\\bigg)\\,,\n }\n \\eql{eperiodp}\n $$\nwhich is $1\/\\mbox{{\\ssb\\char120}}$ times a polynomial. For comparison with\n[\\putref{Zagier}], Zagier's $X$ equals my $i\\mbox{{\\ssb\\char120}}$.\n\nI now comment on this construction. The periods, and period polynomials,\nfor non--cusp forms are defined by analogy to those of cusp forms, the\njustification being that the expressions, for ${\\cal F}(i\\infty)$ equal to\nzero, reduce to those for cusp forms. The means to attain this is not\nunique. Instead of the `zeta--function' regularisation, as in the\ndefinition of $L$, (\\puteqn{ell}), I employ the {\\it subtracted} form, as in\n(\\puteqn{rege}),\n $$\n {\\cal F}^{\\rm sub}\\equiv {\\cal F}-a_0-{a_0\\over\\tau^{2t}}\\,,\\quad\n a_0={\\cal F}(i\\infty)\\,,\n $$\nwhose moments are non--infinite and provide an alternative definition of\nthe periods of a non--cusp form which equally reduces to that for cusp\nforms when $a_0=0$. This subtraction maintains inversion modularity, but\nviolates translational, whereas the subtraction of just $a_0$ does the\nreverse. Making use of my earlier considerations of Eisenstein series,\nsee (\\puteqn{ppol}), I define the subtracted period polynomial as an\nintegral,\n $$\n r_{\\cal F}^{\\rm sub}(\\mbox{{\\ssb\\char120}})=\\int_0^{\\infty}d\\mbox{{\\ssb\\char98}}\\,\n (\\mbox{{\\ssb\\char98}}-\\mbox{{\\ssb\\char120}})^{2t-2}{\\cal F}^{\\rm sub}(\\mbox{{\\ssb\\char98}})\\,.\n $$\n\nFor Eisenstein series, (\\puteqn{ppol}) shows that $r_G^{\\rm sub}(\\mbox{{\\ssb\\char120}})$ is\nproportional to $\\overline P_t(\\mbox{{\\ssb\\char120}})$ whose explicit form is given in\n(\\puteqn{ppoly2}).\n\nIt is necessary to compare this definition with that of Zagier,\n(\\puteqn{eperiodp}). The difference is the absence of the `end point' terms\nproportional to $1\/\\mbox{{\\ssb\\char120}}$ and $\\mbox{{\\ssb\\char120}}^{2t-1}$. This can be remedied by\nincluding, in the terminology of this paper, the regularised\ncontributions from the subtracted terms, the Casimir and Planck terms,\nwhich can be accomplished simply by expanding the closed contour in\n(\\puteqn{ppoly2}) to include the corresponding poles at $s=0$ and $s=2t$.\nThis just extends the sum by the two end points giving the well known\nquantity,\n $$\\eqalign{\n\\overline R_t(\\mbox{{\\ssb\\char120}})=2\\pi\\,\\zeta_R(2t-1)\\big((i \\mbox{{\\ssb\\char120}})^{2t-2}-1\\big)-4i\n \\sum_{j=0}^t\\zeta_R(2j)\\,\\zeta_R(2t-2j)(i \\mbox{{\\ssb\\char120}})^{2t-2j-1}\n \\,,}\n \\eql{ppoly3}\n $$\nwhich agrees, up to a factor, with Zagier's expression given in\n(\\puteqn{eperiodp}).\n\nFurthermore, the use of the subtracted form, ${\\cal F}^{\\rm sub}$, allows one\nto give a cleaner integral expression for the enlarged period polynomial\nthan that in [\\putref{Zagier}], {\\it i.e. },\n $$\n r_{\\cal F}^{\\rm enl}(\\mbox{{\\ssb\\char120}})=\\int_0^{\\infty}d\\mbox{{\\ssb\\char98}}\\,\n (\\mbox{{\\ssb\\char98}}-\\mbox{{\\ssb\\char120}})^{2t-2}{\\cal F}^{\\rm sub}(\\mbox{{\\ssb\\char98}})+\n {a_0\\over2t-1}\\big(\\mbox{{\\ssb\\char120}}^{2t-1}+\\mbox{{\\ssb\\char120}}^{-1}\\big)\\,.\n $$\n\nWhile equivalent to the method in [\\putref{Zagier}], I believe the contour\napproach is neater and is capable of being extended to the general theory\nof periods. Zagier, [\\putref{Zagier}], shows that logistic simplifications\noccur in this theory when it is extended to include non--cusp forms such\nas the Eisenstein series.\n\nAs an historical point, it will be noticed that the above expressions\nhave been available since the time of Ramanujan, with contour derivations\nby Malurkar, [\\putref{Malurkar}], and Guinand, [\\putref{Guinand}]. In order\nto investigate this point further, I return to the mode sum expressions\nfor the thermodynamic related quantities.\n\n\\section{ \\bf 3. Back to summations.}\nI start with the Eisenstein series forms and work the development to\nparallel the quantities of the previous section. This allows me to\nintroduce the useful calculations of Smart, [\\putref{Smart}], on the\nEpstein function (which will arise later) using its relation with\nEisenstein series.\n\nThe subtracted partial energy $\\epsilon_t^{\\rm sub}$ in (\\puteqn{rege})\ncorresponds to the the `regularised' Eisenstein series,\n $$\\eqalign{\n G_t^{\\,\\rm sub}(\\tau)&=\\sumdasht{m=-\\infty}{\\infty}\\,\\sumdasht{n=-\\infty}{\\infty}\n {1\\over (m\\tau+n)^{2t}}\\cr\n &=\\,G_t(\\tau)-2\\zeta_R(2t)(1+\\tau^{-2t})\\,,}\n \\eql{geesub}$$\nThe relation is, [\\putref{DandK1}],\n $$\n \\epsilon^{\\rm sub}_t(\\tau)={(2t-1)!\\over2(2\\pi i)^{2t}}\\, \\overline\n G^{\\,\\rm sub}_t(\\tau)\\,.\n \\eql{inten5}$$\n\nNow, corresponding to (\\puteqn{mint2}), integrate (\\puteqn{geesub}) with\nrespect to $\\tau$ from $\\tau$ to $i\\infty$, $2t-1$ times to give,\nformally,\n $$\n {1\\over(2t-1)!}\\,\\phi_{2t}(\\tau)\\,,\n \\eql{geeint}$$\nwhere $\\phi$ is defined by, [\\putref{Smart}], eqn.(2.11),\n $$\n \\phi_{2t}(\\tau)=\\sumdasht{m=-\\infty}{\\infty}\\,\\sumdasht{n=-\\infty}{\\infty}\n {1\\over m^{2t-1}}{1\\over m\\tau+n}\\,.\n \\eql{phis}$$\n\nThe behaviour of the series $\\phi_{2k}(\\tau)$ under modular\ntransformations, in particular $\\tau\\to-1\/\\tau$, has been especially\nconsidered by Smart, [\\putref{Smart}], who refers to the $\\phi$'s as `{\\it\nmodular forms of weight $2-2t$ with rational period functions}'.\n\nUsing the partial fraction identity (see also Hurwitz, [\\putref{Hurwitz}],\nEisenstein, [\\putref{Eisenstein}]),\n $$\n {1\\over m^{2t-1}}{1\\over m\\tau+n}=\n -{\\tau^{2t-1}\\over n^{2t-1}}{1\\over m\\tau+n}+\\sum_{j=1}^{2t-1}(-1)^{j+1}\n {\\tau^{2t-j-1}\\over m^j\\,n^{2t-j}}\\,,\n \\eql{pf}$$\none finds easily the inversion relation, [\\putref{Smart}], eqn.(1.10b),\n $$\n \\phi_{2t}(\\tau)-\\tau^{2t-2}\\phi_{2t}(-1\/\\tau)=-4\\sum_{j=1}^{t-1}\n \\tau^{2t-2j-1}\\zeta_R(2j)\\,\\zeta_R(2t-2j)\\,.\n \\eql{sm1}$$\nAlso, under translations,\n $$\n \\phi_{2t}(\\tau+1)-\\phi_{2t}(\\tau)=2{\\zeta_R(2t)\\over\\tau(\\tau+1)}\\,.\n \\eql{sm2}$$\n\nUp to a simple factor, one might expect $\\phi_{2t}$ to be the same as\n${\\overline\\phi}_{2t}$ in (\\puteqn{mint2}). However the cocycle functions in\n(\\puteqn{sm1}) and (\\puteqn{ppoly2}) differ by the two {\\it even} powers. I\nnote that these are related to the $j=1$ and $j=2t-1$ terms in the sum in\n(\\puteqn{pf}) which have gone out in the passage to the difference,\n(\\puteqn{sm1}), by antisymmetry. However the summation over $m$,\nrespectively $n$, for these values of $j$ is conditionally convergent and\nis therefore subject to ambiguity, that is to say, a choice has to be\nmade, as in the discussion of the Eisenstein series, $G_2$. The\nregularised Mellin transform approach results in a different definition\nof the series (\\puteqn{phis}) to that used by Smart, [\\putref{Smart}],p.3. The\nrelation is easily found by taking the even powers in (\\puteqn{ppoly2}) over\nto the left and then we see that one has the relation, (further remarks\nare given later),\n $$\n {\\overline\\phi}_{2t}(\\mbox{{\\ssb\\char120}})=\ni\\phi_{2t}(\\tau)-2\\pi \\zeta_R(2t-1)\\,,\\quad \\mbox{{\\ssb\\char120}}=-i\\tau\\,,\n \\eql{connec}\n $$\nalso agreeing with (\\puteqn{trans}) and (\\puteqn{sm2}).\n\nSmart, [\\putref{Smart}], also gives the relation with a known Lambert\nseries as\n $$\\eqalign{\n S_t(\\tau)\\equiv\\sum_{m=1}^\\infty{1\\over m^{2t-1}}{q^{2m}\\over1-q^{2m}}\n &=-{1\\over4\\pi i}\\,\\phi_{2t}(\\tau)-{1\\over2\\pi i\\tau}\\,\n\\zeta_R(2t)-{1\\over2} \\zeta_R(2t-1)\\cr\n&={1\\over4\\pi}\\,\\big(\\overline\\phi_{2t}(\\mbox{{\\ssb\\char120}})+{2\\over\\mbox{{\\ssb\\char120}}}\\zeta_R(2t)\\big)\\cr\n&\\equiv{1\\over4\\pi}\\,\\overline\\psi_{2t}(\\mbox{{\\ssb\\char120}})\\,,\n}\n \\eql{ls}$$\nwhere $\\overline\\psi_{2t}$ is the function having $\\overline R_{2t}(\\mbox{{\\ssb\\char120}})$,\n(\\puteqn{ppoly3}), as its inversion cocycle function,\n $$\n {\\overline\\psi}_{2t}(\\mbox{{\\ssb\\char120}})-(i\\mbox{{\\ssb\\char120}})^{2t-2}\\,{\\overline\\psi}_{2t}(1\/\\mbox{{\\ssb\\char120}})\n= \\overline R_{2t}(\\mbox{{\\ssb\\char120}})\\,,\n \\eql{are1}$$\nas is easily confirmed.\n\nThe introduction of $\\overline\\psi$ has restored translational\ninvariance, (the Planck term has been put back in (\\puteqn{ls})),\n $$\n \\overline\\psi_{2t}(\\mbox{{\\ssb\\char120}}-i)- \\overline\\psi_{2t}(\\mbox{{\\ssb\\char120}})=0\\,.\n \\eql{per}$$\nIt is almost convention to consider modular integrals, for example, to be\nperiodic ({\\it e.g. } Knopp, [\\putref{Knopp}]). See some later comments.\n\nEquations (\\puteqn{are1}) and (\\puteqn{ls}) imply an inversion relation for\nthe series $S_t$ which has a certain history, some of which is detailed\nby Berndt, [\\putref{Berndt}], p.153, who says that it was first written\ndown by Ramanujan. As is clear, it is quite the same as the expression\nfor the Eisenstein period polynomials.\n\nThe Lambert series, $S_t(\\tau)$, has been considered by Apostol,\n[\\putref{Apostol}], Berndt, [\\putref{Berndt}], Grosswald, [\\putref{Grosswald}],\nand Smart [\\putref{Smart}], for example. It can be related to statistical\nmechanics in the following way.\n\nOne can resum the statistical expression for the partial free energy in a\nstandard manner, beginning, ($q=e^{\\pi i\\tau}=e^{-\\pi\/\\xi}$),\n $$\n f_t=\\epsilon_{t,0}-{\\xi\\over2\\pi}\\sum_{n=1}^\\infty n^{2t-2}\\log(1-q^{2n})\n =\\epsilon_{t,0}-{\\xi\\over2\\pi}\\sum_{m,n=1}^\\infty{n^{2t-2}\\over m}q^{2mn}\\,.\n \\eql{ssum}$$\n\nThe scaled entropy is defined by,\n $$\n s_t(\\xi)={\\partial f_t(\\xi)\\over\\partial \\xi}=-{1\\over2\\pi}\\bigg(1-{2\\pi\\over\\xi}D\n\\bigg) \\sum_{m,n=1}^\\infty{n^{2t-2}\\over m}q^{2mn}\\,.\n \\eql{scent}$$\n\nConcentrate now on the summation and rewrite it,\n $$\n \\sum_{m,n=1}^\\infty{n^{2t-2}\\over m}q^{2mn}=D^{2t-2}\n\\sum_{m=1}^\\infty{1\\over m^{2t-1}}{q^{2m}\\over1-q^{2m}}\\,,\n \\eql{sum9}$$\nin terms of $S_t(\\tau)$, (\\puteqn{ls}).\n\nThis Lambert series has been discussed in connection with generalised\nDedekind $\\eta$--functions by Berndt, [\\putref{Berndt}], and it is worth\nnoting that, in a certain sense, $\\log \\eta$ is a modular form of zero\nweight, in agreement with the results $\\epsilon_1=-D\\log\\eta$,\n[\\putref{DandK2}], which could therefore be written,\n $\n \\epsilon_1=-{\\cal D}\\log\\eta\\,,\n $\nin terms of the covariant derivative, ${\\cal D}$.\n\nThe question, not answered here, is whether there is an `expansion' or\nrepresentation theorem for modular {\\it integrals} combining that for\nmodular forms (of {\\it any} real weight, see [\\putref{Rankin}]). It is not\nlikely that one can find a pure elliptic formula. As evidence I cite the\nparticular evaluations of Smart who gives at the lemniscate point\n($\\tau=i,\\mbox{{\\ssb\\char120}}=1$),\n $$\n \\overline\\psi_{4}(1)={7\\pi^4\\over90}-2\\pi\\zeta_R(3)\\,,\n $$\nand, more generally, if $2t=0({\\rm mod} 4)$,\n $$\n S_t(i)={\\zeta_R(2t)\\over2\\pi}-{\\zeta_R(2t-1)\\over2}+{1\\over2\\pi}\n \\sum_{j=1}^{t-1}(-1)^{j+1}\\zeta_R(2t-2j)\\zeta_R(2j)\\,,\n $$\na result actually due to Lerch which follows from (\\puteqn{are1}) with\n(\\puteqn{ppoly3}).\n\nAs a final general comment in this section, it is always possible to\nobtain the required quantities as (infinite) $q$-series by direct\nintegration, but these do not count as closed forms. For example, one\nintegration yields the $q$-series for the partial free energy in terms of\nthe arithmetic functions, $\\sigma_k(m)$, (the sum of the $k$-th powers of\nthe divisors of $m$),\n $$\n f_t=\\epsilon_{t,0}+{1\\over\\beta}\\sum_{m=1}^\\infty\\sigma_{2t-1}(m)\\,{q^{2m}\\over m}\\,,\n $$\nwhich amounts only to doing the $n$-summation in ({\\puteqn{ssum}), or an\nintegration of the standard relation, (\\puteqn{mell1}),\n $$\n \\epsilon_t=-{B_{2t}\\over4t}+\\sum_{m=1}^\\infty \\sigma_{2t-1}(m)\\,q^{2m}\\,.\n $$\n\nOne also has the formula, {\\it e.g. } [\\putref{Rad,Glaisher}],\n $$\n S_t(\\tau)=\\sum_{m=1}^\\infty \\sigma_{-2t+1}(m)\\,q^{2m}=\n \\sum_{m=1}^\\infty {\\sigma_{2t-1}(m)\\over m^{2t-1}}\\,q^{2m}\\,,\\quad\\forall\\, t\\,,\n \\eql{Ssi}$$\nin agreement with the result of a $(2t-1)$-fold integration.\n\\section{\\bf 4. Another approach via modular integrals.}\nThe period functions are cocycle functions associated with Eichler\ncohomology and modular integrals. These notions put a different slant on\nthe preceding formulae. In effect one begins again.\n\nThe best place to start is the important (but particular) analytical result\nof Bol, [\\putref{Bol}], which states that, for {\\it any} reasonable function,\n$\\varphi(\\tau)$,\n $$\n \\big(D^{r+1}\\varphi\\big)(\\gamma\\tau)=\n (c\\tau+d)^{r+2}D^{r+1}\\big((c\\tau+d)^r\\,\\varphi(\\gamma\\tau)\\big)\\,,\n \\eql{bol}$$\nwhere $r$ is a nonnegative number, and I use,\n $$(D^{r+1}\\varphi)(\\tau)\\equiv \\bigg(q{d\\over dq}\\bigg)^{r+1}\\varphi(\\tau)\\,,\n $$\nwith,\n $$\n \\gamma=\\left(\\matrix{*&*\\cr\n c&d}\\right)\\in {\\rm SL}(2,\\mbox{\\open\\char82})\\,.\n $$\nThis shows that, {\\it if} $\\varphi$ is an automorphic form of weight\n$-r$, then $D^{r+1}\\,\\varphi$ is an automorphic form of weight $2+r$, see\nalso Petersson, [\\putref{Petersson}]. Note that it is the ordinary\nderivative that enters here. The proof of (\\puteqn{bol}) follows by\ninduction.\n\nThe classic discussion of {\\it modular integrals} can proceed by asking\nfor the inverse of Bol's result. That is, given a form of weight $2+r$,\n$\\rho(\\tau)$, as a source, can one integrate the differential equation,\n $$\n D^{r+1}\\varphi=\\rho\n \\eql{de}$$\nto obtain $\\varphi$, a form of weight $-r$? Clearly this will not be so\nin general because of the freedom introduced by constants of integration.\nThat is, one would expect $\\varphi$ to behave as a form of weight $-r$ up\nto an additional piece, the general form of which can be determined from\nthe differential equation (\\puteqn{de}) and Bol's theorem, (\\puteqn{bol}).\n\nGenerally we would write (ignoring any multiplier systems for\nsimplicity),\n $$\n \\varphi(\\gamma\\tau)=(c\\tau+d)^{-r}\\big(\\varphi(\\tau)+P(\\gamma,\\tau)\\big)\\,,\n \\eql{co2}$$\nwhere $P(\\gamma,\\tau)$ has to be a polynomial in $\\tau$ of order $\\le r$,\nThe inverse to Bol's result is that, if any $\\varphi$ satisfies\n(\\puteqn{co2}), then $\\rho$, of (\\puteqn{de}), is a form of weight $(2+r)$.\nThis follows most simply by substituting (\\puteqn{co2}) into (\\puteqn{bol}).\n\n$P$ must satisfy the 1--cocycle consistency condition, [\\putref{Eichler}],\n $$\\eqalign{\n P(\\gamma_1\\gamma_2,\\tau)&=(c_2\\tau+d_2)^r\\,P(\\gamma_1,\\gamma_2\\tau)+P(\\gamma_\n 2,\\tau)\\cr\n &\\equiv P(\\gamma_1,\\tau)\\big|\\gamma_2+P(\\gamma_2,\\tau)\\,.\n }\n \\eql{cocycle}$$\n\n$\\varphi$ is known as an {\\it automorphic} or {\\it Eichler integral}, of\nweight $-r$, and $P$ is the {\\it associated period polynomial},\n[\\putref{Eichler,HandK}]. I have also introduced the useful, standard\n`stroke' operator, $\\big|\\gamma$, sometimes denoted $\\big|[\\gamma]$.\n\nThe {\\it general} solution (equivalently the indefinite integral) of the\ndifferential equation (\\puteqn{de}) is a particular integral plus the\ncomplementary function (a zero mode) {\\it i.e. }\n $$\n \\varphi(\\tau)={(2\\pi i)^{r+1}\\over\\Gamma(r+1)}\n \\int_{\\tau_0}^\\tau d\\sigma\\, (\\tau-\\sigma)^{r}\\,\\rho(\\sigma)+\\Theta(\\tau)\\,,\n \\eql{gensol}\n $$\nwhere $\\Theta(\\tau)$ is a polynomial of degree $r$. The integral, in the\nupper half plane, is path independent if $\\rho$ has weakish analytic\nproperties and, if desired, the solution can be made independent of the\nlower limit by suitable adjustment of $\\Theta(\\tau)$. In elementary fashion,\n$\\Theta(\\tau)$ is determined by the first $r$ derivatives of $\\varphi(\\tau)$\nat $\\tau_0$.\n\nAll this is general analysis. In modular applications to SL(2,$\\mbox{\\open\\char90}$),\n$\\tau_0$ is chosen as the cusp $i\\infty$ and the limits in (\\puteqn{gensol})\nreversed. In terms of Fourier expansions, a holomorphic cusp form source\nis\n $$\n \\rho(\\tau)=\\sum_{m=1}^\\infty \\rho_m\\,q^{2m}\\,,\n \\eql{fe1}$$\nand the particular Eichler integral of $\\rho$, is, from (\\puteqn{gensol}),\n $$\n \\varphi(\\tau)=\\sum_{m=1}^\\infty {\\rho_m\\over m^{r+1}}\\,q^{2m}\\,,\n \\eql{fe2}$$\nwhich has the same periodicity as $\\rho$, a result of choosing\n$\\tau_0=i\\infty$.\n\nThe problem in the present Eisenstein case has already been noted. The\nEisenstein series, $G_t(\\tau)$ does not vanish at $i\\infty$ (or $0$). It\nis a `non--cusp' form and the direct integration of (\\puteqn{de}) meets\nobstacles at an elementary level. This can be appreciated simply from the\nappearance of $1\/m$ factors during integration, and $m$ can be zero in\n$G_t$. This fits in because the $m=0$ terms in $G_t$ are those making\n$G_t$ nonzero at $i\\infty$, the zero temperature value. Saying it yet\nagain, the Fourier transform of $G_t$ begins with an $m=0$ term, the\nCasimir contribution, so (\\puteqn{fe2}) makes no immediate sense.\n\nOne way of avoiding these obstructions, is to regularise $G_t$ by\ndropping the offending term(s) as in (\\puteqn{geesub}) and consider the\ndifferential equation,\n $$\\eqalign{\n D^{2t-1}\\,\\varphi_{2t}=-4\\pi\\epsilon_t^{\\rm sub}\\,,}\n \\eql{gee}$$\nwhich has the particular integral, $\\varphi_{2t}=\\overline\\phi_{2t}$\nwhere,\n $$\n \\overline\\phi_{2t}(\\tau)=-4\\pi{(2\\pi i)^{2t-1}\\over\\Gamma(2t-1)}\n \\int_\\tau^{i\\infty}d\\tau'\\,(\\tau'-\\tau)^{2t-2}\n \\,\\epsilon_t^{\\rm sub}(\\tau')\\,,\n $$\nthe constants and boundary conditions being chosen so as to reproduce\n(\\puteqn{mint2}). The upper limit is a cusp. Smart's function, $\\phi_{2t}$,\nresults if one adds a complementary constant (and multiplies by $i$).\n(See (\\puteqn{connec}).)\n\nBecause of the subtraction, the quantity $\\epsilon_t^{\\rm sub}(\\tau)$ is not\nexactly a modular form, rather one has, expressed slightly generally,\n($r=2t-2$),\n $$\n \\epsilon(\\gamma\\tau)=(c\\tau+d)^{r+2}\\big(\\epsilon(\\tau)+E(\\gamma,\\tau)\\big)\n \\eql{submod}\n $$\nand also\n $$\n \\phi(\\gamma\\tau)=(c\\tau+d)^{-r}\\big(\\phi(\\tau)+P(\\gamma,\\tau)\\big)\\,,\n \\eql{co3}\n $$\nwhich are related by the same differential equation, (\\puteqn{de}),\n $$\n D^{r+1}\\phi=-4\\pi\\epsilon\\,.\n $$\nBol's formula, (\\puteqn{bol}), relates $P$ and $E$,\n $$\n D^{r+1}P=-4\\pi E\\,,\n \\eql{bol2}\n $$\nand these relations generalise the usual ones applicable to cusp forms.\nThe cocycle formula (\\puteqn{cocycle}) still holds of course, as (\\puteqn{co3})\nis the same as (\\puteqn{co2}).\n\nI remark that $\\overline\\phi$ and $\\epsilon^{\\rm sub}$, not being periodic, do\nnot quite possess Fourier expansions, as in (\\puteqn{fe1}) and (\\puteqn{fe2}).\nHowever $\\overline\\psi$, in view of (\\puteqn{per}), does,\n $$\n \\overline\\psi(\\mbox{{\\ssb\\char120}})=\\sum_{m=1}^\\infty \\psi_m\\,q^{2m}\\,,\n $$\n\nBecause $P$, this time, satisfies (\\puteqn{bol2}), it is not required to be\na polynomial. Generally, since the modular group is generated \\footnote{\nBeware that some authors, {\\it e.g. } Knopp, swap the usage of the symbols $S$\nand $T$.} by $S$ and $T$, $P(\\gamma,\\tau)$ is a {\\it rational period\nfunction}. In the present case, however, $P(S,\\tau)$ {\\it is} a\npolynomial, given by (\\puteqn{ppoly2}),\n $$\n P(S,\\tau)=\\overline P_t(\\tau)\\,,\n \\eql{pt}$$\nwhile the (non--zero) $P(T,\\tau)$, which follows from (\\puteqn{trans}),\n $$\n P(T,\\tau)=2\\,\\zeta_R(2t)\\bigg({1\\over \\tau}-{1\\over\\tau+1}\\bigg)\\,,\n \\eql{ps}$$\nis a ratio of polynomials. Of course, $P(\\pm{\\bf 1},\\tau)=0$ and I remark\nthat the polynomial $P(S,\\tau)$ runs from $\\tau^0$ to $\\tau^{2t-2}$.\n\nAs a simple check of the algebra, set $\\gamma_1=\\gamma_2=S$ in the cocycle\nrelation, (\\puteqn{cocycle}). This gives,\n $$\n P(S,\\tau)\\big|(1+S)\\equiv P(S,\\tau)+\\tau^{r}P(S,S\\tau)=\n P(S,\\tau)+\\tau^{r}P(S,-1\/\\tau)=0\n $$\nwhich, for $r=2t-2$, agrees with the explicit form, (\\puteqn{ppoly2}).\n\nAlso, as examples,\n $$\\eqalign{\n P(TS,\\tau)=&2\\,\\zeta_R(2t){\\tau^{2t}\\over1-\\tau}+\\overline P_t(\\tau)\\cr\n P(ST,\\tau)=&{2\\,\\zeta_R(2t)\\over\\tau(1+\\tau)}+\\overline P_t(1+\\tau)\\,.}\n $$\nI note that the period function appears to have poles at $-1$, $0$, $+1$\nand $i\\infty$.\n\nFrom (\\puteqn{submod}) and the form of $\\epsilon^{\\rm sub}$, (\\puteqn{rege}), one\nfinds\n $$\n E(S,\\tau)=0\\,,\\quad E(T,\\tau)=\\bigg({1\\over(\\tau+1)^{2t}}\n -{1\\over\\tau^{2t}}\\bigg)\n $$\nwhich are consistent with (\\puteqn{ps}) and (\\puteqn{pt}).\n\nFor completeness, and to aid comparison, this is a convenient place to\noutline some standard elementary things about period polynomials, mostly\nfor SL$(2,\\mbox{\\open\\char90})$. Two basic `polynomials' are the inversion one,\n$P(S,\\tau)$, and the translation one, $P(T,\\tau)$. For cusp forms, the\nlatter vanishes because of periodicity, $P(T,\\tau)=0$, [\\putref{Knopp}],\ncorresponding to the periodicity of the modular integrals, {\\it cf }\n(\\puteqn{fe1}), (\\puteqn{fe2}). In this case, $P(S,\\tau)$ satisfies the\nEichler--Shimura relations\n $$\\eqalign{\n P(S,\\tau)\\big|(1+S)&=0 \\cr\n P(S,\\tau)\\big|(1+TS+(TS)^2)&=0\n }\n \\eql{ES}\n $$\nwith $S^2=1$ and $(TS)^3=1$. A standard way of proving these uses the\ncomplex integral form for $P$, {\\it e.g. } [\\putref{Zagier}], but they can also\nfollow, algebraically from the cocycle relation as I now show. (See also\nKnopp, [\\putref{Knopp3}], for slightly different details.)\n\nFor the first identity, just set, as above, $\\gamma_1=1$ and $\\gamma_2=S$ in\n(\\puteqn{cocycle}) and it falls straight out. The second identity is\nslightly more work. Using (\\puteqn{cocycle}) twice, one has, dropping the\n$\\tau$ argument temporarily and noting $(TS)^2=(TS)^{-1}$,\n $$\\eqalign{\n P(S)\\big|\\big(1+&TS+(TS)^2\\big)=\\cr\n &P(S)+P(STS)-P(TS)+P(SS^{-1}T^{-1})-P(TSTS)\\,.\n }\n \\eql{ES2}\n $$\n\nFurther, setting $\\gamma_1=T$ in the cocycle condition, and using the\nassumption that $P(T)=0$, yields\n $$\n P(T\\gamma)=P(\\gamma)\\,.\n \\eql{pt0}\n $$\nOne then sees that the first and third terms on the right--hand side of\n(\\puteqn{ES2}) cancel, as do the second and fifth. The fourth term vanishes\non its own because (\\puteqn{pt0}) shows that $P(T^{-1})=0$, and (\\puteqn{ES})\nfollows.\n\nIn one development, the notion of modular integral has been somewhat\ndivorced from that of integration. One says that a {\\it modular integral}\nis any function (with appropriate analyticity properties) that obeys the\nquasi--modular relation (\\puteqn{co2}), where the {\\it period function} $P$\nmust still, of course, satisfy the cocycle condition (\\puteqn{cocycle}).\nMotivation is then provided to look for modular integrals with {\\it\nrational} period functions, {\\it i.e. } ratios of polynomials. In this case it\nseems to be conventional to {\\it assume} that the translational period\nfunction, $P(T,\\tau)$, vanishes, [\\putref{Knopp}], so that both\nEichler--Shimura relations, ({\\puteqn{ES}), still hold. There is a certain\ninterest in computing such rational period functions and investigating\ntheir zeros and poles. The easiest introduction to this topic is by\nKnopp, [\\putref{Knopp3}]. It is, however, somewhat peripheral to our\nconsiderations which are more concerned with non--cusp forms {\\it i.e. } the\nEisenstein series, {\\it e.g. }\\ [\\putref{Zagier}].\n\n\\section{\\bf 5. Green function distributional description.}\n\nDeveloping the formalism a little further, it is possible to give the\ndiscussion of Eichler integrals a distributional or Green function look.\nFor this purpose, it is better to use the `real' variable form and consider\nthe differential equation,\n $$\n {d^{2t-1}\\over d{\\mbox{{\\ssb\\char120}}}^{2t-1}}\\,\\psi(\\mbox{{\\ssb\\char120}})=\\rho(\\mbox{{\\ssb\\char120}})\\,,\n \\eql{rde}$$\nfor $\\mbox{{\\ssb\\char120}}>0$ with solution vanishing at $\\infty$, {\\it cf } (\\puteqn{mint2}),\n $$\n \\psi(\\mbox{{\\ssb\\char120}})=\n\\int_{0^-}^\\infty d{\\mbox{{\\ssb\\char120}}'}\\,\\Phi_{2t-1}(\\mbox{{\\ssb\\char120}}'-\\mbox{{\\ssb\\char120}})\\,\\rho({\\mbox{{\\ssb\\char120}}'})\n \\eql{soln}$$\nwhere the generalised function, $\\Phi_\\alpha(\\mbox{{\\ssb\\char120}})$, is\n $$\n\\Phi_\\alpha(\\mbox{{\\ssb\\char120}})={\\mbox{{\\ssb\\char120}}^{\\alpha-1}_+\\over\\Gamma(\\alpha)}\\,.\n \\eql{genp}$$\nThe {\\it generalised} function $\\mbox{{\\ssb\\char120}}^\\alpha_+$ is concentrated on the positive\n$\\mbox{{\\ssb\\char120}}$-axis, {\\it i.e. } $\\mbox{{\\ssb\\char120}}^\\alpha_+$ is equal to $\\mbox{{\\ssb\\char120}}^\\alpha$ for $\\mbox{{\\ssb\\char120}}\\ge0$ and is\nzero for $\\mbox{{\\ssb\\char120}}<0$, {\\it e.g. } Gelfand and Shilov, [\\putref{GandS}] I,\\S5.5.\n$\\Phi_{2t-1}$ acts as a Green function satisfying,\n $$\n {d^{2t-1}\\over d{\\mbox{{\\ssb\\char120}}}^{2t-1}}\\,\\Phi_{2t-1}(\\mbox{{\\ssb\\char120}}'-\\mbox{{\\ssb\\char120}})=\\delta(\\mbox{{\\ssb\\char120}}'-\\mbox{{\\ssb\\char120}})\\,.\n $$\nThe formal, distributional equivalent is obtained from the convolution, valid\nfor all $\\alpha$ and $\\beta$,\n $$\n \\Phi_\\alpha*\\Phi_\\beta=\\Phi_{\\alpha+\\beta}\\,,\n \\eql{conv}$$\nby setting $\\alpha=-\\beta=2t-1$ and noting\n $$\n\\Phi_{-k}(\\mbox{{\\ssb\\char120}})=\\delta^{(k)}(\\mbox{{\\ssb\\char120}})\\,,\\quad k=0,1,2,\\ldots.\n \\eql{deriv}$$\n\nAn important formula is the Fourier transform of $\\Phi$, [\\putref{GandS}]\nI.p.360,\n $$\n \\int_0^\\infty d\\mbox{{\\ssb\\char120}}\\, \\Phi_\\alpha(\\mbox{{\\ssb\\char120}})\\,e^{i\\sigma \\mbox{{\\ssb\\char120}}}\n ={1\\over (\\sigma+i0)^\\alpha}\\,e^{i\\pi\\alpha\/2}\\,,\n \\eql{ftphi}$$\nand the inverse, which is a (continuous) eigenfunction expansion,\n $$\n \\Phi_\\alpha(\\mbox{{\\ssb\\char120}})=e^{i\\pi\\alpha\/2}{1\\over2\\pi}\\int_{-\\infty}^\\infty d\\sigma\\,\n {e^{-i\\sigma \\mbox{{\\ssb\\char120}}}\\over(\\sigma+i0)^\\alpha}\\,.\n \\eql{invft}$$\n\nIn the present case, the source $\\rho$ is assumed to be periodic under\n$\\mbox{{\\ssb\\char120}}\\to\\mbox{{\\ssb\\char120}}-i$ (the Planck term is not removed) and to converge as\n$\\mbox{{\\ssb\\char120}}\\to\\infty$. So one has the `Fourier' series,\n $$\n \\rho(\\mbox{{\\ssb\\char120}})=\\sum_{m=1}^\\infty \\rho_m\\, e^{-2m\\pi\\mbox{{\\ssb\\char120}}}\\,,\\quad \\mbox{{\\ssb\\char120}}>0\\,.\n \\eql{ftrho}$$\nSubstitution of this and (\\puteqn{invft}) into (\\puteqn{soln}) gives,\n $$\n \\psi(\\mbox{{\\ssb\\char120}})=(-1)^{t+1}{1\\over2\\pi}\\sum_{m=1}^\\infty\\rho_m\n \\int_{-\\infty}^\\infty d\\sigma\\,\n {e^{i\\sigma\\mbox{{\\ssb\\char120}}}\\over(\\sigma+i0)^{2t-1}(\\sigma-2m\\pi i)}\\,.\n \\eql{psi1}\n $$\nClosing the contour in the upper half $\\sigma$--plane I obtain, $\\mbox{{\\ssb\\char120}}>0$,\n $$\n \\psi(\\mbox{{\\ssb\\char120}})={1\\over(2\\pi)^{2t-1}}\\sum_{m=1}^\\infty\n {\\rho_m\\over m^{2t-1}}\\,e^{-2m\\pi\\mbox{{\\ssb\\char120}}}\\,,\n \\eql{soln2}$$\nwhich, it is no surprise to see, is just (\\puteqn{fe2}) derived by a much\nlonger route but one on which we encounter some handy information. It is\nseen also that the source periodicity has been transmitted to the solution.\n\nIncluding an $m=0$ term in the Fourier series, (\\puteqn{ftrho}), still gives\nconvergence at infinity, but of course the solution, (\\puteqn{soln2}),\nbreaks down, as has already been mentioned. As with all Green function\napproaches, when this happens the corresponding eigenfunction has to be\nremoved from the source. This can be done formally by redefining the\nGreen function to exclude this function which is, in the present case, a\nconstant zero mode. Altering the contour in the Fourier integral\n(\\puteqn{invft}) gives a modified Green function,\n $$\n \\Phi^{\\rm sub}_{2t-1}(\\mbox{{\\ssb\\char120}})\\equiv(-1)^{t}{1\\over2\\pi i}\n \\int_{-\\infty+i\\delta}^{\\infty+i\\delta} d\\sigma\\,\n {e^{-i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^{2t-1}}\\,,\\quad 0<\\delta<2\\,,\n \\eql{phisub}$$\nwhich automatically takes out any constant part of the source. This can\nbe seen by going back to (\\puteqn{psi1}) and replacing the integral by one\nfrom $-\\infty+i\\delta$ to $\\infty+i\\delta$,\n $$\n \\psi(\\mbox{{\\ssb\\char120}})=(-1)^{t+1}{1\\over2\\pi}\\sum_{m=1}^\\infty\\rho_m\n \\int_{-\\infty+i\\delta}^{\\infty+i\\delta} d\\sigma\\,\n {e^{i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^{2t-1}(\\sigma-2m\\pi i)}\\,.\n \\eql{psi2}\n $$\nFor $ 0<\\delta<2$, if $\\mbox{{\\ssb\\char120}}>0$, closing the contour in the upper half plane\nstill yields (\\puteqn{soln2}). However now, even if the source Fourier\nseries, (\\puteqn{ftrho}), is extended to the constant term, $m=0$, the\nresult is again (\\puteqn{soln2}). This should be denoted by $\\psi^{\\rm\nsub}(\\mbox{{\\ssb\\char120}})$ and one can write\n $$\n \\psi^{\\rm sub}(\\mbox{{\\ssb\\char120}})=\n\\int_{0^-}^\\infty d{\\mbox{{\\ssb\\char120}}'}\\,\\Phi^{\\rm\nsub}_{2t-1}(\\mbox{{\\ssb\\char120}}'-\\mbox{{\\ssb\\char120}})\\,\\rho({\\mbox{{\\ssb\\char120}}'})\\,.\n \\eql{soln3}$$\n\nRetaining the general structure, (\\puteqn{invft}), for any $\\alpha$, one can\ndefine for any $\\alpha$,\n $$\n \\Phi^{\\rm sub}_\\alpha(\\mbox{{\\ssb\\char120}})\\equiv e^{i\\pi\\alpha\/2}{1\\over2\\pi}\n\\int_{-\\infty+i\\delta}^{\\infty+i\\delta}d\\sigma\\,\n {e^{-i\\sigma \\mbox{{\\ssb\\char120}}}\\over\\sigma^\\alpha}\\,.\n \\eql{invft2}$$\nThen $\\Phi^{\\rm sub}_\\alpha(\\mbox{{\\ssb\\char120}})$ is still concentrated on the positive\n$\\mbox{{\\ssb\\char120}}$--axis and the convolution (\\puteqn{conv}) also remains valid,\n $$\n \\Phi^{\\rm sub}_\\alpha* \\Phi^{\\rm sub}_\\beta =\\Phi^{\\rm sub}_{\\alpha+\\beta}\n \\eql{conv2}\n $$\ntogether with (\\puteqn{deriv}),\n $$\n \\Phi^{\\rm sub}_{-k}=\\delta^{(k)}\\,.\n \\eql{deriv2}\n $$\n\nTo check these statements, firstly, if $\\mbox{{\\ssb\\char120}}<0$, closing the contour in\n(\\puteqn{invft2}) in the upper half plane yields zero as the only\nsingularity is at $\\sigma=0$. Then\n $$\\eqalign{\n \\big(\\Phi^{\\rm sub}_\\alpha* \\Phi^{\\rm sub}_\\beta\\big) (\\mbox{{\\ssb\\char120}})=\n {e^{i\\pi(\\alpha+\\beta)\/2}\\over(2\\pi)^2}\\int_{-\\infty}^\\infty d\\mbox{{\\ssb\\char120}}'\n\\int_{-\\infty+i\\delta}^{\\infty+i\\delta}d\\sigma\\,\\int_{-\\infty+i\\delta}^{\\infty+i\\delta}d\\sigma'\\,\n {e^{-i\\sigma\\mbox{{\\ssb\\char120}}-i\\mbox{{\\ssb\\char120}}'(\\sigma'-\\sigma)}\\over\\sigma^\\alpha\\sigma'^\\beta}\n }\n $$\nwhere the lower limit on the $\\mbox{{\\ssb\\char120}}'$ integral is allowed because\n$\\Phi_\\beta^{\\rm sub}(\\mbox{{\\ssb\\char120}}')$ is concentrated on the positive $\\mbox{{\\ssb\\char120}}'$ axis.\n\nSetting $\\sigma=s+i\\delta$, $\\sigma'=s'+i\\delta$, I get, in detail,\n $$\\eqalign{\n \\big(\\Phi^{\\rm sub}_\\alpha* \\Phi^{\\rm sub}_\\beta\\big) (\\mbox{{\\ssb\\char120}})&= e^{i\\pi(\\alpha+\\beta)\/2}{1\\over(2\\pi)^2}\\int_{-\\infty}^\\infty d\\mbox{{\\ssb\\char120}}'\n\\int_{-\\infty}^{\\infty}ds\\,\\int_{-\\infty}^{\\infty}ds'\\,\n {e^{-i\\sigma\\mbox{{\\ssb\\char120}}-i\\mbox{{\\ssb\\char120}}'(s'-s)}\\over\\sigma^\\alpha\\sigma'^\\beta}\\cr\n &= e^{i\\pi(\\alpha+\\beta)\/2}{1\\over2\\pi}\n\\int_{-\\infty}^{\\infty}ds\\,\\int_{-\\infty}^{\\infty}ds'\\,\n {e^{-i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^\\alpha\\sigma'^\\beta}\\,\\delta(s-s')\\cr\n &=e^{i\\pi(\\alpha+\\beta)\/2}{1\\over2\\pi}\n\\int_{-\\infty}^{\\infty}ds\\,\n {e^{-i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^\\alpha\\sigma^\\beta}=e^{i\\pi(\\alpha+\\beta)\/2}{1\\over2\\pi}\n\\int_{-\\infty+i\\delta}^{\\infty+i\\delta}d\\sigma\\,\n {e^{-i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^{\\alpha+\\beta}}\\cr\n &=\\Phi^{\\rm sub}_{\\alpha+\\beta}(\\mbox{{\\ssb\\char120}})\\,.\n }\n $$\n\n Equation (\\puteqn{deriv2}) follows either from (\\puteqn{conv2}) or\n (\\puteqn{invft2}) since the integrand has no singularity, and so one can\n set $\\delta=0$.\n\n\\subsection{\\it Eichler cohomology.}\nAs a side remark, the general solution to (\\puteqn{rde}) consists of the\nparticular integral, (\\puteqn{soln}), plus a solution of the homogeneous\nequation, {\\it i.e. } a zero mode, which in this case is a polynomial,\n$\\Theta(\\mbox{{\\ssb\\char120}})$, of degree at most $2t-2$. The contribution of this to the\ncocycle polynomial, $P$ of (\\puteqn{co2}), constitutes an element of the set\nof {\\it coboundaries} and a cohomology of polynomials can be set up\n(Eichler cohomology). Working with the cohomology classes removes the\nconstants of integration ambiguity in the solution of (\\puteqn{rde}). That\nis, there is a one-to-one correspondence between $\\rho$ and an element of\nthe cohomology group denoted $H^1(\\Gamma,\\Pi_r)$, where $\\Pi_r$ is the\nvector space of polynomials of degree $\\le r=2t-2$ (for cusp forms) and\n$\\Gamma$ is (here) the modular group. The development of these ideas, while\nsignificant generally, is not required here.\n\n\n\\section{\\bf 6. Epstein approach to thermal quantities.}\n\nAn `earlier' form of the free energy, or effective action, follows from\nthe thermal $\\zeta$--function, which, in this case is related to the Epstein $\\zeta$--function.\nStarting from this I will be able to make contact with the Eisenstein\nformulation and derive useful relations. Using the Epstein function is\nequivalent to the non-holomorphic Eisenstein series. As a rule the\nanalysis is harder. In reality one is doing twice the necessary work.\n\nIn some ways we are going backwards. The use of holomorphic forms, {\\it e.g. }\n(\\puteqn{inten5}), should be considered a simplification available through\nworking with an explicit square root of the Laplacian.\n\nAgain splitting up the degeneracy, one is led to define a `partial'\n$\\zeta$--function,\n $$\n \\zeta_t(s,\\beta)={i\\over2\\beta}\\sumdasht{{m,n=\\atop-\\infty}}\n{\\infty}{n^{2t-2}\\over\n \\big(4\\pi^2m^2\/\\beta^2+n^2\/a^2\\big)^s}\\,,\n \\eql{epthzf}$$\nand the related free energy,\n $$\n \\overline F_t=-{\\xi\\over8\\pi}\\lim_{s\\to0}{1\\over s}\\,\n \\sumdasht{{m,n=\\atop-\\infty}}{\\infty}{n^{2t-2}\\over\n \\big(m^2+n^2\\xi^{-2}\\big)^s}\\,.\n \\eql{eff}$$\n\nI have rescaled the free energy by defining, as above and as in\n[\\putref{DandK1}], $\\overline F_t=a F_t$ and have also taken a preliminary\nlimit of $s\\to0$ in an extracted factor of $(2\\pi\/\\beta)^{2s}$. This will\nbe justified in a moment.\n $$\n \\overline F_t=(-1)^{t-1}{\\xi\\over8\\pi}\\bigg({d\\over d(1\/\\xi^2)}\\bigg)^{t-1}\n \\lim_{s\\to0}{\\Gamma(s-t+1)\\over\\Gamma( s+1)}\\,\n \\sumdasht{{m,n=\\atop-\\infty}}{\\infty}{1\\over\n \\big(m^2+n^2\\xi^{-2}\\big)^{s-t+1}}\\,.\n \\eql{eff2}$$\n\nNot surprisingly, the degeneracy factor is replaced by a derivative and the\nresulting sum is precisely an Epstein $\\zeta$--function. This has been used before {\\it e.g. }\n[\\putref{Kennedy}]. The point is that there is no pole at $s=0$, which\njustifies the limit. In terms of the Epstein function, $Z_2$,\n $$\n \\overline F_t=(-1)^{t-1}{\\xi\\over8\\pi}\\bigg({d\\over d(1\/\\xi^2)}\\bigg)^{t-1}\n \\lim_{s\\to0}\\,{\\Gamma(s-t+1)\\over\\Gamma( s+1)}\\,Z_2(2s-2t+2,A)\\,,\n \\eql{eff3}$$\nwhere $A$ is the matrix (the `modulus'),\n $$\n A=\\left(\\matrix{1&0\\cr\n 0&\\xi^{-2}}\\right)\\,.\n $$\nThe idea now is to use the standard functional relation satisfied by\n$Z_2(s,A)$, repeated here for convenience,\n $$\n Z_2(2s,A)={\\pi^{2s-1}\\over\\sqrt{{\\rm det\\,} A}}{\\Gamma(1-s)\\over\\Gamma(s)}\n \\,Z_2(2-2s,A^{-1})\\,,\n \\eql{freln}$$\nto give\n $$\n \\overline F_t={\\xi\\over8\\pi}\\bigg(-{d\\over d(1\/\\xi^2)}\\bigg)^{t-1}\\xi\n \\lim_{s\\to0}\\pi^{2s-2t+1}\\,{\\Gamma(t-s)\\over\\Gamma( s+1)}\\,Z_2(2t-2s,A^{-1})\\,.\n $$\n$Z_2(2s,A)$ has a pole at $s=1$ only, so working at $t>1$ to begin with,\none has\n $$\n \\overline F_t=\\xi{(-1)^{t-1}\\Gamma(t)\\over8\\pi^{2t}}\n \\bigg({d\\over d(1\/\\xi^2)}\\bigg)^{t-1}\\xi\n \\,\\,Z_2(2t,A^{-1})\\,.\n \\eql{epder}$$\nAs the simplest case, set $t=2$ giving the three-sphere. Then\n $$\n \\overline F_3={\\xi\\over8\\pi^4}{d\\over d(1\/\\xi^2)}\\,\\xi\\,Z_2(4,A^{-1})=\n -{\\xi^4\\over16\\pi^4}{d\\over d\\xi}\\,\\xi\\,Z_2(4,A^{-1})\\,,\n \\eql{3free1}$$\nagreeing with the result in [\\putref{Kennedy}], allowing for the changes in\nnotation.\n\nThis formula can be taken further as shown by Epstein [\\putref{Epstein}]\np.633 who gives,\n $$\n Z_2(4, A^{-1})={\\pi^4\\over45}+\\pi\\xi^{-3}\\zeta_R(3)+2\\pi\\xi^{-3}\\chi(q')\n +4\\pi^2\\xi^{-2}\\,D'\\,\\chi(q')\\,,\n \\eql{epz2}$$\nwhere $q'=e^{-\\pi\\xi}$ and\n $$\n \\chi(q')=S_2(-1\/\\tau)=\\sum_{m=1}^\\infty {q'^{2m}\\over m^3(1-q'^{2m})}\\,.\n $$\n\nEquation (\\puteqn{3free1}) with (\\puteqn{epz2}) is appropriate for the high\ntemperature limit since $q'$ is the `inverse' temperature Boltzmann factor.\nThe low temperature form follows by simple homogeneous scaling of the\nEpstein $\\zeta$--function, which gives,\n $$\n \\,Z_2(2s,A^{-1})=\\xi^{-2s}Z_2(2s,A),\n \\eql{scale}$$\nand therefore the equivalent form,\n $$\n Z_2(4, A^{-1})={\\pi^4\\over45}\\xi^{-4}+\\pi\\xi^{-1}\\zeta_R(3)+2\\pi\\xi^{-1}\\chi(q)\n +4\\pi^2\\xi^{-2}\\,D\\,\\chi(q)\\,,\n \\eql{epz21}$$\nwhere $q=e^{-\\pi\/\\xi}$.\n\nThe low temperature form (\\puteqn{epz21}) yields a somewhat simpler expression\nfor the free energy, (\\puteqn{3free1}), as the $\\zeta_R(3)$ term goes out and\nthere is some cancellation that produces,\n $$\n \\overline F_3={1\\over240}+{\\xi\\over2\\pi}D^2\\,\\chi(q)\\,.\n $$\nI have thus regained, at some length, the summation form (\\puteqn{ssum})\nwith (\\puteqn{sum9}).\n\nAll that has been done is to rederive, in a special case and in a\ndetailed way, the standard statistical mode sum, the general form of\nwhich was obtained, in this fashion, in [\\putref{DandK,Dow2}] and we are no\nfurther forward in finding a closed form for the free energy, which is\none of my aims.\n\nThe equivalence of (\\puteqn{epz2}) and (\\puteqn{epz21}), derived here rather\ntrivially (granted Epstein's calculation), is a known inversion identity,\nand can also be derived from (\\puteqn{sm1}). The exact identity is written\nout in Katayama [\\putref{Kata}]. His derivation seems to be similar to that\nvia the Epstein function. Clearly the higher odd sphere expressions will\ninvolve the Riemann $\\zeta$--function\\ at positive odd integers. Katayama writes out\nthe one appropriate for the (partial) five-sphere.\n\nSmart, [\\putref{Smart}], also derives these identities, which is not\nsurprising since his work is concerned with the evaluation of the Epstein\n$\\zeta$--function\\ at integer arguments, through which he is led to the forms\n$\\phi_{2k}$. We see that Epstein had already arrived at the same\ndevelopment. Consult also Bodendiek and Halbritter, [\\putref{BandH}], for a\nrelated treatment.\n\n\\section{\\bf 7. The Epstein--Kober--Selberg--Chowla formula.}\nSmart makes use of the, oft quoted, paper of Selberg and Chowla,\n[\\putref{SandC}], which is concerned, partly, with the evaluation of the\nEpstein function via the Kronecker limit formula \\footnote{Landau,\n[\\putref{Landau2}], gives some interesting history of this famous formula.}\nand elegantly yields some of the finite forms for the complete elliptic\nintegral used in [\\putref{DandK1}]. The important expression for the\nEpstein function quoted by Selberg and Chowla, [\\putref{SandC2}], and\nderived by them in [\\putref{SandC}], (equ.(6)), in terms of $K$--Bessel\nfunctions, is essentially the same as that already given by Epstein,\n[\\putref{Epstein}], p.631, equn.(12), p.622, equn.(16). Rankin\n[\\putref{Rankin3}] obtained the same result and it also occurs in Bateman\nand Grosswald, [\\putref{BandG}]. These are the standard references. See\nalso Terras, [\\putref{Terras}] p.209. Smart, [\\putref{Smart}], employs this\nformula to split the Epstein function into (simpler) holomorphic and\nanti--holomorphic parts using the explicit form of the Bessel function,\n$K_{t-1\/2}$, which henceforth disappears from his analysis. Other\nexpressions were given by Deuring [\\putref{Deuring}] and Mordell,\n[\\putref{Mordell}].\n\nAn early, and little known, derivation is that by Kober, [\\putref{Kober}].\nFollowing, and generalising, Watson, [\\putref{Watson}], he obtains, {\\it\nstarting} from the Bessel function end, the formula (I retain his\nnotation),\n $$\\eqalign{\n {1\\over\\sqrt u}\\,&\\Delta^{(2w+1)\/4}{\\Gamma(w+1\/2)\\over8\\pi^{w+1\/2}}Z_2(a,b,c;w+1\/2)\\cr\n&={u^{-w}\\over4}{\\Gamma(w)\\over\\pi^{w}}\\,\\zeta_R(2w)\n+{u^{w}\\over4}{\\Gamma(w+1\/2)\\over\\pi^{w+1\/2}}\\,\\zeta_R(2w+1)+\\,\\cr\n &\\hspace{************}\n \\sum_{n=1}^\\infty \\sigma_{2w}(n)n^{-w}\\cos(2\\pi vn)K_{w}(2\\pi un)\\,,}\n \\eql{kob1}$$\nwhere the quadratic form in the Epstein function, $Z_2$, is\n$am_1^2+2bm_1m_2+cm_2^2$ and\n $$\n \\Delta=ac-b^2\\equiv a^2\\,u^2\\,,\\quad v\\equiv{b\\over a}\\,.\n $$\nEquation (\\puteqn{kob1}) is identical to the corresponding one in\n[\\putref{Epstein}] and [\\putref{SandC}] and therefore, with justice, should\nbe called the Epstein--Kober formula. It is the {\\it Fourier expansion}\nof the Epstein function in $v$. For the sum of squares case, $v=0$, the\nresult is rederived by Guinand [\\putref{Guinand2}] in the same way.\n\nKober also discusses behaviour under `reciprocal' transformations,\n$u\\to1\/u$. In particular the diagonal case, $v=0$, when one can write\n$u=\\mbox{{\\ssb\\char120}}$.\n\nIn this way of doing things, the Bessel function can be eliminated using\nthe standard formula\n $$\n \\sum_{n=1}^\\infty {\\sigma_{2w}(n)\\over n^{s+w}}=\\zeta_R(s+w)\\,\\zeta_R(s-w)\\,,\n \\eql{sig}$$\nand the Mellin transform (Heaviside's integral),\n $$\n {1\\over4}\\pi^{-s}\\Gamma\\big({s+w\\over2}\\big)\\,\\Gamma\\big({s-w\\over2}\\big)=\n \\int_0^\\infty dy \\,y^{s-1}\\,K_w(2\\pi y)\\,,\\quad {\\rm Re\\,} s>1+|{\\rm Re\\,} w|\\,,\n \\eql{hi}$$\nso that,\n $$\n \\sum_{n=1}^\\infty {\\sigma_{2w}(n)\\over n^w} K_{w}(2\\pi un)\n=\n{1\\over2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,u^{-s}\\xi(s+w)\\,\\xi(s-w)\\,,\n \\eql{epmell}$$\nwith $c>1+w$ and where,\n $$\\xi(2w)\\equiv{1\\over2}\\pi^{-w}\\,\\Gamma(w)\\,\\zeta_R(2w)\\,.\n $$\nThis is the same as Deuring's expression, [\\putref{Deuring}]\np.589.\n\nThe integrand possesses a functional equation which follows most easily\nfrom that for the Riemann $\\zeta$--function, $\\xi(2w)=\\xi(1-2w)$, and is,\n $$\n \\xi(s+w)\\,\\xi(s-w)\\equiv f_w(s)=f_w(1-s)\\,.\n \\eql{ffrel}$$\n\nUsing (\\puteqn{ffrel}) one could {\\it derive} the functional equation for the\nEpstein $\\zeta$--function\\ in (\\puteqn{kob1}). The usual derivation employs the inversion\nrelation for generalised $\\theta$--functions, [\\putref{Epstein,Krazer}].\n\nFurthermore, it is possible to reverse the argument and use (\\puteqn{epmell})\nto obtain (\\puteqn{kob1}) (for $v=0$). The diagonal Epstein function is\nparticularly easy to deal with being the $\\zeta$--function\\ on the torus,\nS$^1\\times$S$^1$. It can be expressed very simply in terms of the $\\zeta$--functions\\ on\nthe circle factors (these are Riemann $\\zeta$--functions\\ ) and the result is exactly\n(\\puteqn{kob1}) with (\\puteqn{epmell}) {\\it without} the intervention of the\nBessel function. The individual Riemann $\\zeta$--functions\\ in (\\puteqn{kob1}) arise from\nadjusting the zero modes. All this is standard and can easily be\ngeneralised to higher dimensions.\n\n\nA natural step is to proceed as with the holomorphic modular forms and ask\nfor the corresponding period polynomials which will follow directly from\n(\\puteqn{epmell}) and (\\puteqn{ffrel}) in the standard manner.\n\nThere are four poles in the relevant strip, $-c\\le s\\le c$, and\na straightforward contour calculation produces the result,\n $$\\eqalign{\n \\sum_{n=1}^\\infty {\\sigma_{2w}(n)\\over n^w}K_{w}(2\\pi n u)-&\n {1\\over u}\\sum_{n=1}^\\infty {\\sigma_{2w}(n)\\over n^w}K_{w}(2\\pi n\/u)\\cr\n &={1\\over2}\\xi(2w)(u^{w-1}-u^{-w})+{1\\over2}\\xi(-2w)(u^{-w-1}-u^{w})\\,,}\n \\eql{ppk}$$\nThe right hand side of (\\puteqn{ppk}) might be termed a period function.\n\nThis formula was obtained by Guinand, [\\putref{Guinand2}], but using the\ncomplete equation (\\puteqn{kob1}), with $v=0$, and properties of the Epstein\nfunction. This is unnecessary as I have just shown.\n\nIn order to make a connection with the period polynomials encountered\nearlier,{\\it e.g. } (\\puteqn{are1}), one introduces the formula for the $K$-Bessel\nfunction, written as,\n $$\n K_{t-1\/2}(x)=\\sqrt{2\\over \\pi}\\,x^{t-1\/2}\\,(-1)^{t-1}\\,\\bigg({1\\over x}\n{d\\over dx}\\bigg)^{t-1}{e^{-x}\\over x}\\,,\\quad t\\in\\mbox{\\open\\char90}\\,.\n \\eql{kbess}$$\nThen, somewhat similarly to [\\putref{Smart}],\n $$\\eqalign{\n \\sum_{n=1}^\\infty {\\sigma_{2t-1}(n)\\over n^{t-1\/2}}K_{t-1\/2}(2\\pi n u)&=\n \\sqrt{2\\over\\pi}{(-1)^{t-1}\\over(2\\pi)^{t-1\/2}}\\bigg({1\\over u}\n {d\\over du}\\bigg)^{t-1}\n \\sum_{n=1}^\\infty{\\sigma_{2t-1}(n)\\over n^{2t-1}}\\,q^{2n}\\cr\n &=\\sqrt{2\\over\\pi}{(-1)^{t-1}\\over(2\\pi)^{t-1\/2}}\n \\bigg({1\\over u}{d\\over du}\\bigg)^{t-1}\n S_t(iu)\\,,}\n $$\nso that, for the left--hand side of (\\puteqn{ppk}), one finds,\n $$\n (-1)^{t-1}{1\\over{\\pi}^t}\\bigg(\\bigg({d\\over du^2}\\bigg)^{t-1}\n S_t(iu)-{1\\over u}\\bigg({d\\over d(1\/u^2)}\\bigg)^{t-1}S_t(i\/u)\\bigg)\\,.\n \\eql{krelp} $$\nIt is not obvious from (\\puteqn{are1}) that this reduces so nicely to\n(\\puteqn{ppk}), and I leave this as a question mark.\n\nFurther expressions involving Bessel functions appear in the Appendix.\n\\begin{ignore}\n\nThis can be turned around in the following way. Write (\\puteqn{kbess})\nexplicitly in the form,\\marginnote{THis is wrong}\n $$\n e^{-x}\\sum_{j=0}^{t-1}\n {(t-1+j)!\\over j!(t-1-j)!}\\,(2x)^{-j-1\/2}=\\pi^{-1\/2}\\,K_{t-1\/2}(x)\\,,\n $$\nand solve this for $(2x)^{-j-1\/2}$, $0\\le j\\le t-1$,\n $$\n e^{-x}\\,(2x)^{-j-1\/2}={1\\over\\sqrt\\pi}\\sum_{t=0}^{j-1}{()!\\over()!}\n \\,K_{t-1\/2}(x)\\,.\n $$\nNow set $x=2\\pi n u$ and get\n $$\n {e^{-2\\pi nu}\\over n^{j+1\/2}}=(2\\pi)^{j+1\/2}\n {1\\over\\sqrt\\pi}\\sum_{t=0}^{j-1}{()!\\over()!}\\,K_{t-1\/2}(2\\pi nu)\n $$\n\\end{ignore}\n\\begin{ignore}\n\\section{\\bf More hyperbolic sums.}\nIn terms of the hyperbolic expression, (\\puteqn{speen}), the quantity\ncorresponding to the $c_t$ is,\n $$\n {\\partial\\over\\partial\\beta}\\,E(\\beta)=-{1\\over2^{d+1}}\\sum_{m=1}^\\infty m\\bigg({(d-1)^2\n\\cosh(m\\beta\/2)\\over\\sinh^{d-1}(m\\beta\/2)}+{d(d+1)\\cosh(m\\beta\/2)\n\\over\\sinh^{d+1}(m\\beta\/2)}\\bigg)\\,,\n \\eql{speen2}$$\nwhich means that it is also possible to evaluate the sums,\n $$\n \\sum_{m=1}^\\infty {m^{2p+1}\\cosh(mx)\\over\\sinh^d(mx)}\n $$\nand\n $$\n \\sum_{m=1}^\\infty {m^{2p}\\over\\sinh^d(mx)}\\,,\n $$\nat special values of $x$. Cases with $d=1$ have been considered by Zucker,\n[\\putref{zucker2}] and Ling, [\\putref{Ling2}] and can follow from expansion of\nthe dn function.\n\\section{\\bf The fermion case. More to be done.}\n\nIn [\\putref{DandK1}] it was shown that just as for bosons, the fermion\ninternal energies on all spheres can be expressed in terms of just two\nquantities which could be taken to be the first two partial energies,\n$\\eta_1$ and $\\eta_2$. This was proved without recourse to the boson result\nand in a different manner. The recursion is not similar to (\\puteqn{recurs})\nbut has a cubic form. One might therefore reasonably expect an equation\ncorresponding to (\\puteqn{shr}) to exist, but in a different form and derived\nby a different route, possibly from the Jacobi function ds $u$.\n\nA preliminary fact, that it is useful to confirm, starts from the\nexpression for the partial spinor energy on the circle given in\n[\\putref{DandK1}],\n $$\n \\eta_1(\\beta)=-{1\\over24} \\mbox{{\\goth\\char75}}^2\\,(1-2k^2)\\,,\n $$\nwhich, using (\\puteqn{diffrel}), can be integrated to give,\n $$\n \\int d\\beta\\, \\eta_1(\\beta)={1\\over12}\\int d\\kappa {1-2\\kappa\\over\\kappa(1-\\kappa)}=\n {1\\over12}\\log(\\kappa\\ka'\/16)\\,,\n $$\nwhere $\\kappa=k^2$ and a constant of integration has been chosen to make the\nfree energy and internal energies agree at absolute zero.\n\nI have, in this roundabout way, derived another of Jacobi's formulae,\n $$\n q^{1\/24}\\prod_{j=0}^\\infty(1+q^{2j+1})=2^{1\/6}(kk')^{-1\/12}\\,,\n $$\nthe logarithm of which gives the mode sum form of the circle fermion\nfree energy, as is well known in various contexts.\n\nWe can also see the difficulty if we attempt the same approach for $\\eta_2$\n $$\\eqalign{\n -\\int d\\beta\\,\\eta_2(\\beta)&={1\\over120}\\int d\\kappa\\,\\mbox{{\\goth\\char75}}^2(\\kappa)\\,\n{(7+8\\kappa-8\\kappa^2)\\over\\kappa(1-\\kappa)}\\cr\n&={1\\over120}\\int d\\kappa\\,\\mbox{{\\goth\\char75}}^2(\\kappa)\\,\\bigg({7\\over\\kappa}\n+{7\\over\\kappa'}-8\\bigg)\\,,}\n $$\nand the integral is not obvious. I merely note the hypergeometric\nrepresentation,\n $$\n \\mbox{{\\goth\\char75}}(\\kappa)=F\\big({1\\over2},{1\\over2},1; \\kappa\\big)={1\\over\\pi\\kappa^{1\/4}}\\,\n Q_{-1\/2}\\big((\\kappa+1)\/2\\sqrt\\kappa\\big)\\,.\n $$\nA similar thing holds in the boson case of course.\n\\end{ignore}\n\\section{\\bf 8. Conclusion}\n\nIt does not seem possible to ellipticise integrals of the internal\nenergy, in particular the free energy (except for the circle), because\nthe modular properties are more complicated possessing nonzero cocycle\nfunctions. These are related to a known Lambert series connected with the\nEisenstein series and Zagier's expression for the period functions of\nthese is more neatly re--computed by a contour method. The use of the\nfully subtracted series $\\epsilon_t^{\\rm sub}$ allows a more compact\ntreatment.\n\nSufficient historical material has been exhibited to indicate that the\nSelberg--Chowla formula should be renamed the Epstein--Kober formula.\n\\newpage\n\\section{\\bf Appendix, Dirichlet Series.}\n\nIt is possible to give a fairly broad context to the inversion behaviour\n({\\it e.g. } modular properties), without too much restriction, in terms of\nDirichlet series. Although these ideas, by now classic, are associated\nwith Hecke, and have been expounded by Ogg [\\putref{Ogg,Ogg2}], for\nexample, the simplest case occurs earlier in other places. I mention\nKoshliakov, [\\putref{Kosh}], whose discussion I now paraphrase both for\nhistorical justice and interest.\n\nDefine two ordinary Dirichlet series (`zeta functions'),\n $$\n \\phi(s)=\\sum_{n=1}^\\infty {a_n\\over n^s}\\,,\\,\\,({\\rm Re\\,} s>\\nu_a)\\,,\\quad\n \\psi(s)=\\sum_{n=1}^\\infty {b_n\\over n^s}\\,,\\,\\,({\\rm Re\\,} s>\\nu_b>\\nu_a)\\,,\n $$\nfor some $\\nu$'s depending on the asymptotics of $a_n$ and $b_n$, and\n{\\it assume} that the $a_n$ and $b_n$ are such that the corresponding\n`cylinder kernels' satisfy the `formula of transformation',\n $$\n \\sum_{n=0}^\\infty a_n\\,e^{-nb\\rho}={a\\over\\rho^\\nu}\n \\sum_{n=0}^\\infty b_n\\,e^{-nb\/\\rho}\\,,\\quad \\nu=\\nu_a\\,,\\quad \\rho>0\\,,\n \\eql{trans2}$$\nfor fixed $a$ and $b$, where the numbers (not necessarily integers)\nof `zero modes' are taken to be,\n $$\n a_0=-\\phi(0)\\,\\quad b_0=-\\psi(0)\\,.\n $$\nThen Koshliakov, [\\putref{Kosh}], shows the equivalence of (\\puteqn{trans2})\nwith\n $$\n a\\,{\\Gamma(\\nu-s)\\,\\psi(\\nu-s)\\over b^{\\nu-s}}={\\Gamma(s)\\,\\phi(s)\\over b^s}\\,,\n \\eql{frel}$$\nby brute force using the summation function, $\\sigma(z)$, given by the\nBessel function expression,\n $$\n \\sigma(z)=-2ab\\,z^{(\\nu-1)\/2}\\,\\sum_{n=1}^\\infty {b_n\\over n^{(\\nu-1)\/2}}\\,\n K_{\\nu-1}\\big(2b\\sqrt{nz}\\big)\\,,\n $$\nand Heaviside's formula, (\\puteqn{hi}). $\\sigma(z)$ has the important property\nof possessing poles at $z=1,2,\\ldots$ with residues $a_1,a_2,\\ldots$ and\nalso has a cut along the negative $x$--axis with discontinuity\n$ab^\\nu2\\pi i\\psi(0)(-x)^{\\nu-1}\/\\Gamma(\\nu)$, $z=x+iy$.\n\nKoshliakov shows that $\\phi(s)$ is a single--valued holomorphic function\nhaving a first-order pole at $s=\\nu$,\n $$\n \\phi(s)\\sim-\\psi(0){ab^\\nu\\over\\Gamma(\\nu)}{1\\over s-\\nu}\\,,\n \\eql{phipole}$$\nand it is then a theorem that (\\puteqn{frel}) together with (\\puteqn{phipole}) is\nequivalent to (\\puteqn{trans2}), as can also be established easily, and\ninstructively, by Mellin transform. It follows, symmetrically, that\n$\\psi(s)$ is a single--valued holomorphic function having a first-order\npole at $s=\\nu$,\n $$\n \\psi(s)\\sim-\\phi(0){b^\\nu\\over a\\Gamma(\\nu)}{1\\over s-\\nu}\\,.\n $$\n\nKoshliakov also gives an integral form of $\\phi(s)$,\n $$\n \\phi(s)=-{\\sin\\pi s\\over\\pi}\\int_0^\\infty dx\\, {\\sigma(x)\\over x^s}\n \\,,\\quad {\\rm Re\\,} s<0\\,,\n \\eql{intphi}$$\nshowing that $\\phi(s)$ vanishes at negative integers in agreement with\n(\\puteqn{frel}).\n\nHecke, [\\putref{Hecke}], derives the same equivalence, but only for\n$a_n=b_n$, and eight years later. The more general case can also be found\nin the standard references Ogg [\\putref{Ogg,Ogg2}] and Weil,\n[\\putref{Weil2}], and no doubt elsewhere. A related formula involving\nRamanujan's `reciprocal functions' occurs in Hardy and Littlewood,\n[\\putref{HandL}].\n\nDefining cylinder--kernels, or theta--series, including the zero modes,\nby,\n $$\n \\Phi(\\beta)=\\sum_{n=0}^\\infty a_n\\,e^{-n\\beta}\\,,\\quad\n \\Psi(\\beta)=\\sum_{n=0}^\\infty b_n\\,e^{-n\\beta}\\,,\n $$\nthe reciprocal relation, (\\puteqn{trans2}), reads,\n $$\n \\Phi(b\\rho)={a\\over\\rho^\\nu}\\,\\Psi(b\/\\rho)\\,,\n \\eql{recip}\n $$\n(continued from the regions of convergence in $s$) and one has, in\nstandard fashion ({\\it cf } (\\puteqn{ell})),\n $$\n \\phi(s)={1\\over\\Gamma(s)}\\int_0^\\infty d\\beta\\,\\beta^{s-1}\\,\\big(\\Phi(\\beta)-a_0\\big)\\,,\n \\quad\n \\psi(s)={1\\over\\Gamma(s)}\\int_0^\\infty d\\beta\\,\\beta^{s-1}\n \\,\\big(\\Psi(\\beta)-b_0\\big)\\,.\n $$\n\nAs an example, the Dirichlet series corresponding to the classical\nEisenstein series is, from (\\puteqn{mell1}) or (\\puteqn{sig}),\n $$\n \\sum_{m=1}^\\infty \\sigma_{2t-1}(m)\\,m^{-s}=\\zeta_R(s)\\,\\zeta_R(s+1-2t)\\,,\n $$\nand this is a case discussed by Koshliakov, [\\putref{Kosh}] p.19, with\n$a_n=b_n=\\sigma_{2t-1}(n), \\nu=2t,a=(-1)^t,b=2\\pi,\\phi(0)=\\psi(0)=-B_{2t}\/4t$.\n\nKoshliakov also considers Jacobi elliptic cases that yield reciprocal\nidentities covered earlier by Glaisher [\\putref{Glaisher}].\n\n\\subsection{\\it Generalised Dirichlet series.}\nI have described above the classic Dirichlet set up. A certain, and\nsometimes only apparent, generalisation is obtained by replacing $n$ by\narbitrary sequences, $\\lambda_m$ and $\\mu_n$, and defining new $\\phi$ and\n$\\psi$ by\n $$\n \\phi(s)=\\sum_{m=1}^\\infty {a_m\\over \\lambda_m^s}\\,,\\quad\\quad\n \\psi(s)=\\sum_{n=1}^\\infty {b_n\\over \\mu_n^s}\\,,\n \\eql{newdir}$$\nwhich satisfy, {\\it by definition}, the functional equation\n(one of many possible),\n $$\n \\Gamma(\\delta-s)\\,\\psi(\\delta-s)=\\Gamma(s)\\,\\phi(s)\\,,\\quad \\delta\\in\\mbox{\\open\\char82}\\,,\n \\eql{frel2}$$\nwith corresponding heat--kernels,\n $$\n \\Phi(\\beta)=\\sum_{m=1}^\\infty a_m\\,e^{-\\lambda_m\\beta}\\,,\\quad\n \\Psi(\\beta)=\\sum_{n=1}^\\infty b_n\\,e^{-\\mu_n\\beta}\\,.\n \\eql{hkz}$$\n\nBochner, [\\putref{Bochner}], (see also Chandrasekharan and Narasimhan,\n[\\putref{CandN}], Knopp, [\\putref{Knopp2}] ), derives, by Mellin transforms,\nthe `modular relation',\n $$\n \\Phi(\\beta)-\\beta^{-\\delta}\\,\\Psi\\big({1\\over\\beta}\\big)=B(\\beta)\\,,\n \\eql{modrel}$$\nwhere the `residual function' $B(\\beta)$ is\n $$\n B(\\beta)={1\\over2\\pi i}\\int_{C({\\cal S})} ds\\,\\chi(s)\\,\\beta^{-s}=\\sum_{s\\in{\\cal S}}\\beta^{-s}\n {\\rm Res}\\,\\chi(s)\\,.\n $$\n$\\chi(s)$ is the joint continuation of the two sides of (\\puteqn{frel2}) and\nis assumed to have only simple poles for singularities confined to a\ncompact region, ${\\cal S}$, of the $s$--plane, usually on the real axis.\n$C({\\cal S})$ is a loop surrounding ${\\cal S}$. I especially note that if there\nexist higher--order poles then integer powers of $\\log \\beta$ occur.\n\nIn certain situations, {\\it e.g. } Koshliakov's, $B(\\beta)$ can be naturally\namalgamated with\nthe left--hand side to give an exact modular relation like (\\puteqn{trans2}),\n{\\it e.g. } Bochner,\n[\\putref{Bochner}] p.342. Typically one would include zero modes $\\lambda_0=0$\nand $\\mu_0=0$ in (\\puteqn{hkz}), but not in (\\puteqn{newdir}). (There seems to be\na misprint on p.342 of [\\putref{Bochner}] which has $\\lambda_0=\\mu_0=1$.)\n\nThere is another formula connected with this analysis that is useful. It\nis concerned with the representation of the {\\it modified} series,\n $$\n \\phi(s,w)=\\sum_{n=1}^\\infty {a_n\\over (\\lambda_n+w^2)^s}\\,\\quad{\\rm Re\\,} w>0.\n \\eql{asser}$$\n\nThere are many reasons why we should wish to add the number $w^2$. For\nexample it might correspond to a mass or simply be added for extra\nflexibility as when calculating heat--kernel expansion coefficients via\nresolvents or it might be another part of the denominator as in Epstein's\nderivation of (\\puteqn{kob1}); see [\\putref{DandKi}].\n\nThe representation alluded to is, (Berndt [\\putref{Berndt4}]),\n $$\n \\Gamma(s)\\,\\phi(s,w)=R(s,w)+2\\sum_{n=1}^\\infty b_n\\,\\bigg({\\mu_n\\over w^2}\n \\bigg)^{(s-\\delta)\/2}\\,K_{s-\\delta}\\big(2w\\sqrt\\mu_n\\big)\\,,\n \\eql{berndt}$$\nwhere $s$ is such that the summation converges absolutely and $R(s,w)$ is\ngiven by\n $$\n R(s,w)=\\sum_{s'\\in\\,{\\cal S}} \\Gamma(s-s')\\,w^{2s'-2s}{\\rm Res}\\,\\, \\chi(s')\\,.\n \\eql{resi}$$\n\nBerndt's second proof of (\\puteqn{berndt}) proceeds via the\ncylinder--kernels, $\\Phi$ and $\\Psi$. I prefer his first, Mellin, proof,\n[\\putref{Berndt6}]. Of course, the information employed is the same. I give\nthis proof out of interest.\n\nThe Mellin transform relation between $\\phi(s)$ and the $\\phi(s,w)$ of\n(\\puteqn{asser}), is\n $$\n \\Gamma(s)\\,\\phi(s,w)={1\\over2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds'\\,w^{2s'-2s}\n \\Gamma(s-s')\\Gamma(s')\\,\\phi(s')\n $$\nwhere $c$ is chosen sufficiently positive to make both $\\phi(s)$ and\n$\\psi(s)$ converge. The vertical contour is now moved to the left so as\nto run from $\\delta-c-i\\infty$ to $\\delta-c+i\\infty$ picking up the residues,\n$R(s,w)$, (\\puteqn{resi}), on the way. The region, ${\\cal S}$, lies between\nthese vertical lines and also the horizontal pieces contribute zero.\n\nOn the shifted vertical contour make the coordinate change $s'\\to\\delta-s'$\nand use (\\puteqn{frel2}) {\\it i.e. } $\\chi(\\delta-s')=\\Gamma(s')\\psi(s')$, valid in this\nrange, and so for this part I find,\n $$\\eqalign{\n \\sum_n b_n {1\\over2\\pi i}&\\int_{c-i\\infty}^{c+i\\infty}ds'\\,\\Gamma(s'-\\delta+s)\n \\Gamma(s')\\mu_n^{-s'} w^{2\\delta-2s'-2s}\\cr\n&=2\\sum_{n=1}^\\infty b_n\\,\\bigg({\\mu_n\\over w^2}\n \\bigg)^{(s-\\delta)\/2}\\,K_{s-\\delta}\\big(2w\\sqrt\\mu_n\\big)\\,,}\n $$\non using (\\puteqn{hi}). Hence (\\puteqn{berndt}) has been established.\n\nIn specific situations (\\puteqn{berndt}) is quite familiar in the $\\zeta$--function\\ area\nand in background and finite--temperature quantum field theory.\n\nAn illustrative example is the simplest, diagonal Epstein function,\n $$\n \\phi(s)=Z_p(s)\\equiv\\sumdasht{{\\bf m}=-\\infty}{\\infty}\n {1\\over({\\bf m.m})^s}\\,,\\quad {\\bf m}=(m_1,m_2,\\ldots,m_p)\\,,\n \\eql{diagep}$$\nwhich obeys the functional equation (Epstein [\\putref{Epstein}]),\n $$\n \\Gamma(s)\\,Z_p(s)=\\pi^{2s-p\/2}\\,\\Gamma(p\/2-s)\\,Z_p(p\/2-s)\\,,\n \\eql{epfunc}$$\nso that, see (\\puteqn{frel2}), $\\delta=p\/2$, $a_n=1$, $\\lambda_m = {\\bf m.m}$,\n$b_n=\\pi^{p\/2}$, $\\mu_n=\\pi^2\\,{\\bf m.m}$.\n\n$\\Gamma(s)\\,Z_p(s)$ has poles at $s=0$ and $s=p\/2$ and (\\puteqn{berndt}) gives\nfor the modified, or inhomogeneous, or `massive' Epstein function,\n$Z_p(s,w)$,\n $$\\eqalign{\n \\sumdasht{{\\bf m}=-\\infty}{\\infty}\n {1\\over({\\bf m.m}+w^2)^s}=-w^{-2s}&+{\\Gamma(p\/2)\\Gamma(s-p\/2)\\over\\Gamma(s)}\n \\,w^{p-2s}\\cr\n & +{2\\pi^s\\over\\Gamma(s)}\\sumdasht{{\\bf m}=-\\infty}{\\infty}\n \\bigg({|{\\bf m}|\\over w}\\bigg)^{s-p\/2}\\,\n K_{s-p\/2}\\big(2w\\pi|{\\bf m}|\\big)\\,,}\n \\eql{modep}$$\nwhere the dash means to omit the ${\\bf m=0}$ term. Note that the $w^{-2s}$ could be\nincluded in the left--hand sum if this were extended to include ${\\bf m=0}$.\n\nThe right--hand side can be taken as the continuation of the left--hand\nside showing the single pole at $s=p\/2$, as is correct and may be proved in\nother ways.\n\nActually (\\puteqn{modep}) is shown much more directly using the Jacobi\ninversion relation, which is, of course, how Epstein obtained\n(\\puteqn{epfunc}). Equivalent to Jacobi inversion is Poisson summation.\n\nEquations like (\\puteqn{modep}) also occur in lattice summations. Some references,\nbut none earlier than those already quoted, are given in the review by Glasser\nand Zucker [\\putref{GandZ}] p.109.\n\nA similar formula holds for the general Epstein function which may, or may\nnot, have a pole at $s=p\/2$ and may, or may not, vanish at $s=0$. The works\n[\\putref{Kirsten1,Elizalde2,Elizalde3}], for example, can be consulted for details,\nsome further references and physical applications.\n\nWhen $p=1$, $\\phi(s)=2\\zeta_R(2s)$ and I recover an earlier, well--known\nformula, [\\putref{Watson,Kober}], which yields, after putting $w=u\\,m_2$\nand summing over $m_2$, the formula for the Epstein $\\zeta$--function\\ mentioned\nearlier; see (\\puteqn{kob1}).\n\nIt is left as an exercise to show that the non-diagonal formula,\n(\\puteqn{kob1}), can be obtained in the same way using reciprocal relations\nfor the Hurwitz $\\zeta$--function, or, equivalently, the Lipshitz formula ({\\it cf } Epstein,\n[\\putref{Epstein}], Kober [\\putref{Kober}] p.622).\n\nAs an application of (\\puteqn{modep}) it might be helpful if I present a\nstandard derivation of the usual statistical free energy mode sum from\nthe thermal $\\zeta$--function\\ expression, for a reasonably general system. The\nstarting point is the determinant form,\n $$\n F={i\\over2}\\lim_{s\\to0}{\\zeta(s,\\beta)\\over s}\n =-{1\\over2\\beta}\\lim_{s\\to0}{1\\over s}\n \\sumdasht{{m=-\\infty\\atop n}}\n{\\infty}{d_n\\over\n \\big(\\omega_n^2+4\\pi^2m^2\/\\beta^2\\big)^s}\\,,\n \\eql{eff2}$$\nwhere $\\omega_n^2$ and $d_n$ are the eigenvalues and degeneracies of the\nappropriate operator (related to the Laplacian) on the spatial section,\n${\\cal M}$. The $\\omega_n$ are the single--particle energies. The dash here means\nthat the denominator should never be zero ({\\it cf } (\\puteqn{eff})) but the sum\nincludes $m=0$. For simplicity it is assumed that there is no spatial zero\nmode and so the dash can be removed.\n\nNow consider (\\puteqn{modep}) for $p=1$. Include the $w^{-2s}$ with the\nleft--hand sum, set $w=\\beta\\omega_n\/2\\pi$, multiply by the degeneracy, $d_n$\nand sum over $n$. The limit (\\puteqn{eff2}) is then easily taken since\n$s\\Gamma(s)=\\Gamma(s+1)$ and only the $K_{1\/2}$ Bessel appears. There is\nnothing deep about this calculation and the result is a standard form,\n $$\\eqalign{\n F=&{1\\over2}\\,\\zeta_{\\cal M}(-1\/2)+{1\\over\\beta}\\sum_n d_n\n \\sum_{m=1}^\\infty{e^{-m\\omega_n\\beta}\\over m}\\cr\n =&{1\\over2}\\,\\zeta_{\\cal M}(-1\/2)+{1\\over\\beta}\\sum_n d_n\\log(1-e^{-\\omega_n\\beta})\\,.}\n \\eql{modesum}$$\nTo avoid specifically field theoretic problems, it has been assumed that\n$\\zeta_{\\cal M}(-1\/2)$ exists. This is true for the cases discussed in this\npaper.\n\nThe result, (\\puteqn{modesum}), is an analogue of the (first) Kronecker\nlimit--formula applied to the generalised torus S$^1\\times{\\cal M}$; see\n[\\putref{DandA}]. The original formula relates to ${\\cal M}=$S$^1$ and, in this\ncase, there exists a functional relation, (\\puteqn{frel2}), so that the\nlimit--formula is often expressed in terms of the remainder about the\npole in $\\zeta(s,\\beta)$ at $s=1$. In the general case this is not possible.\n\nFinally, if one introduces the standard arithmetic quantity $r_p(n)$,\nequal to the number of representations of the integer $n$ as the sum of\n$p$ squares, then the diagonal Epstein function, (\\puteqn{diagep}), is\nobtained by writing,\n $$\n \\phi(s)=Z_p(s)=\\sum_n{r_{p}(n)\\over n^s}\\,,\n $$\nand is treated as such by Koshliakov who also gives the `exact', product\nforms of $Z_p(s)$ for $p=2$ and $p=4$.\n\n\\begin{ignore}\nThe other useful formula is a generalisation of one occurring in the\ntheory of Riesz means which has already been encountered, essentially in\n(\\puteqn{mint}) and (\\puteqn{soln}).\n\nWith the same notation as above, one has, for $h\\ge0$,\n $$\n {1\\over\\Gamma(h+1)}\\sum_{\\lambda_n}a_n\\,\\theta(x-\\lambda_n)\\,(x-\\lambda_n)^h\n ={1\\over2\\pi i}\\int_{c-i\\infty}\n ^{c+i\\infty}ds\\,{\\Gamma(s)\\,\\phi(s)\\,x^{s+h}\\over\\Gamma(h+1+s)}\\,,\n $$\nwhere $\\theta$ is Heaviside's step function.\n\\end{ignore}\n\\newpage\n\\section{\\bf References.}\n\n\\begin{putreferences}\n \\reference{RandA}{Rao,M.B. and Ayyar,M.V. \\jims{15}{1923\/24}{150}.}\n \\reference{DandK2}{Dowker,J.S. and Kirsten,K. {\\it Elliptic aspects of\n statistical mechanics on spheres} ArXiv:0807.1995.}\n \\reference{DandK1}{Dowker,J.S. and Kirsten,K. \\np {638}{2002}{405}.}\n \\reference{GandS}{Gel'fand, I.M. and Shilov, G.E. {\\it Generalised Functions}, vol.1\n (Academic Press, New York, 1964). }\n \\reference{DandA}{Dowker,J.S. and Apps, J.S. \\cqg{12}{1995}{1363}.}\n \\reference{Weil}{Weil,A., {\\it Elliptic functions according to Eisenstein and\n Kronecker}, Springer, Berlin, 1976.}\n \\reference{Ling}{Ling,C-H. {\\it SIAM J.Math.Anal.} {\\bf5} (1974) 551.}\n \\reference{Ling2}{Ling,C-H. {\\it J.Math.Anal.Appl.}(1988).}\n \\reference{BMO}{Brevik,I., Milton,K.A. and Odintsov, S.D. {\\it Entropy bounds in\n $R\\times S^3$ geometries}. hep-th\/0202048.}\n \\reference{KandL}{Kutasov,D. and Larsen,F. {\\it JHEP} 0101 (2001) 1.}\n \\reference{KPS}{Klemm,D., Petkou,A.C. and Siopsis {\\it Entropy\n bounds, monoticity properties and scaling in CFT's}. hep-th\/0101076.}\n \\reference{DandC}{Dowker,J.S. and Critchley,R. \\prD{15}{1976}{1484}.}\n \\reference{AandD}{Al'taie, M.B. and Dowker, J.S. \\prD{18}{1978}{3557}.}\n \\reference{Dow1}{Dowker,J.S. \\prD{37}{1988}{558}.}\n \\reference{Dow3}{Dowker,J.S.\\prD{28}{1983}{3013}.}\n \\reference{DandK}{Dowker,J.S. and Kennedy,G. \\jpa{11}{1978}{895}.}\n \\reference{Dow2}{Dowker,J.S. \\cqg{1}{1984}{359}.}\n \\reference{DandKi}{Dowker,J.S. and Kirsten, K. {\\it Comm. in Anal. and Geom.\n }{\\bf7}(1999) 641.}\n \\reference{DandKe}{Dowker,J.S. and Kennedy,G.\n \\jpa{11}{1978}{895}.} \\reference{Gibbons}{Gibbons,G.W. \\pl{60A}{1977}{385}.}\n \\reference{Cardy}{Cardy,J.L. \\np{366}{1991}{403}.}\n \\reference{ChandD}{Chang,P. and\n Dowker,J.S. \\np{395}{1993}{407}.}\n \\reference{DandC2}{Dowker,J.S. and\n Critchley,R. \\prD{13}{1976}{224}.}\n \\reference{Camporesi}{Camporesi,R.\n \\prp{196}{1990}{1}.}\n \\reference{BandM}{Brown,L.S. and Maclay,G.J.\n \\pr{184}{1969}{1272}.}\n \\reference{CandD}{Candelas,P. and Dowker,J.S.\n \\prD{19}{1979}{2902}.}\n \\reference{Unwin1}{Unwin,S.D. 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1$ for any algebraic real number $\\alpha$.\nRoth \\cite{Roth55} improved Liouville Theorem, that is, he showed that $w_1 (\\alpha) = 1$ for all algebraic irrational real numbers $\\alpha$.\nFor higher Diophantine exponents, from the Schmidt Subspace Theorem, it is known that\n\\begin{equation*}\n\tw_n (\\alpha ) = w_n ^ {*} (\\alpha ) = \\min \\{ n, d-1 \\},\n\\end{equation*}\nwhere $n \\geq 1$ is an integer and $\\alpha$ is an algebraic real number of degree $d$ (see \\cite[Section 3]{Bugeaud04}).\n\nLet $p$ be a prime, $q=p^s$, where $s \\geq 1$ is an integer and $\\lF_q$ be the finite field containing $q$ elements.\nWe denote by $\\lF_q [T], \\lF_q (T), $ and $\\lF_q ((T ^ {- 1}))$ respectively the ring of polynomials, the field of rational functions, the field of Laurent series in $T ^ {- 1}$ over $\\lF_q$.\nWe consider analogues of Diophantine exponents $w_n$ and $w_n ^ {*}$ for Laurent series (see Section \\ref{sec:notation} for the precisely definition).\nAs an analogue of Liouville Theorem, Mahler \\cite{Mahler49} showed $w_1 (\\alpha) \\leq \\deg \\alpha - 1$ for any algebraic Laurent series $\\alpha \\in \\lF_q((T^{-1}))$.\nHowever, there exist counter examples of analogue of Roth Theorem for Laurent series over a finite field.\nLet $r_1 = p ^ {t_1}$, where $t_1 \\geq 1$ is an integer.\nHe proved that the Laurent series $\\alpha_1 := \\sum_{n = 0}^{\\infty}T ^ {- r_1 ^ n}$ is algebraic of degree $r_1$ with $w_1 (\\alpha_1) = r_1 - 1$.\nAfter that many authors investigated rational approximations (e.g.\\ \\cite{Firicel13, Mathan92, Schmidt00, Thakur99}) and algebraic approximations (e.g.\\ \\cite{Ooto18, Thakur11, Thakur13}) for algebraic Laurent series.\nIn particular, the second author (\\cite{Ooto18}) proved that, for any rational number $w > 2n-1$, there exists an algebraic Laurent series $\\alpha \\in \\lF_q((T^{-1}))$ such that\n\\begin{equation*}\n\tw_1(\\alpha) = w_1^{*}(\\alpha) = \\ldots = w_n(\\alpha) = w_n^{*}(\\alpha) = w.\n\\end{equation*}\nThe first purpose of this paper is to consider Problem 2.2 in \\cite{Ooto18}.\n\n\n\n\\begin{prob}\\label{prob:main1}\n\tIs it true that\n\t\\begin{equation*}\n\t\tw_n(\\alpha) = w_n^{*}(\\alpha)\n\t\\end{equation*}\n\tfor an integer $n \\geq 1$ and an algebraic Laurent series $\\alpha \\in \\lF_q((T^{-1}))$?\n\\end{prob}\n\n\n\nNote that it is well-known and easy to see that $w_1(\\xi) = w_1^{*}(\\xi)$ for all $\\xi \\in \\lF_q((T^{-1}))$.\nWe give a negative answer of Problem \\ref{prob:main1} for any $n \\geq 2$ and $q \\geq 4$.\n\n\n\n\\begin{thm}\\label{thm:neq_alg}\n\tLet $n \\geq 2$ be an integer, $q = p^s$, and $r = q^t$ with $s, t \\geq 1$ be integers and \n\t\\begin{equation*}\n\t\tr \\geq \\frac{3n + 2 + \\sqrt{9n^2 + 4n + 4}}{2}.\n\t\\end{equation*}\n\tIf $q \\geq 4$, then there exist algebraic Laurent series $\\alpha \\in \\lF_q ((T ^ {- 1}))$ with $\\deg \\alpha = r+1$ such that\n\t\\begin{equation*}\n\t\tw_n (\\alpha) \\neq w_n ^ {*} (\\alpha)\n\t\\end{equation*}\n\\end{thm}\n\n\n\nLet $\\alpha$ be in $\\lF_q ((T ^ {- 1}))$ and $r = p ^ t$, where $t \\geq 0$ is an integer.\nWe call $\\alpha$ \\textit{hyperquadratic} if $\\alpha$ is irrational and there exists $(A, B, C, D) \\in \\lF_q [T] ^ 4 \\setminus \\{ {\\bf 0} \\}$ such that\n\\begin{equation}\\label{eq:hyperquad}\n\tA \\alpha^{r+1} + B \\alpha ^ r + C \\alpha + D = 0.\n\\end{equation}\nFor example, quadratic Laurent series are hyperquadratic, $\\alpha_1$ is hyperquadratic and satisfies $T \\alpha_1^r - T \\alpha_1 + 1 = 0$.\nBaum and Sweet \\cite{Baum76} started a study of hyperquadratic continued fractions in characteristic two.\nIn 1986, Mills and Robbins \\cite{Mills86} showed the existence of hyperquadratic continued fractions with all partial quotient of degree one in odd characteristic with a prime field.\nWe refer the reader to \\cite{Lasjaunias172} for a survey of recent works of hyperquadratic continued fractions.\nWe observe that the degree of hyperquadratic Laurent series which satisfies \\eqref{eq:hyperquad} is less than or equal to $r+1$.\nHowever, the exact degree of these is open.\nIn this paper, we discuss the following problem.\n\n\n\n\\begin{prob}\n\tGiven a hyperquadratic Laurent series $\\alpha \\in \\lF_q((T^{-1}))$, determine the exact degree of $\\alpha$.\n\\end{prob}\n\n\n\nIn order to determine the exact degree, we approach to quadratic approximations.\nSince $w_2(\\alpha) \\leq \\deg \\alpha - 1$ for $\\alpha$ as in \\eqref{eq:hyperquad} (see Lemma \\ref{lem:alg}), if $w_2(\\alpha) > r-1$, then $\\deg \\alpha = r+1$.\nIn this paper, we deal with two families for hyperquadratic continued fractions introduced in \\cite{Lasjaunias02, Lasjaunias17}.\nWe determine the exact degree of these families in particular cases.\n\nThis paper is organized as follows.\nIn Section \\ref{sec:notation}, we recall notations which are used in this paper.\nIn Section \\ref{sec:main}, we state main results of quadratic approximation for hyperquadratic continued fractions and applications of the results.\nIn Section \\ref{sec:lemma}, we give lemmas for the proof of main results.\nIn Section \\ref{sec:proof}, we prove the main results.\n\n\n\n\n\n\n\\section{Notations}\\label{sec:notation}\n\n\n\nA nonzero Laurent series $\\xi \\in \\lF_q((T ^ {- 1}))$ is represented by $\\xi = \\sum_{n = N}^{\\infty}a_n T ^ {- n}$, where $N \\in \\lZ, a_n \\in \\lF_q,$ and $a_N \\neq 0$.\nThe field $\\lF_q ((T ^ {- 1}))$ has a non-Archimedean absolute value $\\abs{\\xi} = q ^ {- N}$ and $\\abs{0} = 0$.\nThe absolute value can be uniquely extended to the algebraic closure of $\\lF_q ((T ^ {- 1}))$ and we also write $\\abs{\\cdot}$ for the extended absolute value.\n\nThe {\\itshape height} of $P(X) \\in (\\lF_q [T])[X]$, denoted by $H(P)$, is defined to be the maximum of the absolute values of the coefficients of $P(X)$.\nFor $\\alpha \\in \\overline{\\lF_q (T)}$, there exists a unique non-constant, irreducible, primitive polynomial $P(X) \\in (\\lF _q [T]) [X]$ whose leading coefficients are monic polynomials in $T$ such that $P(\\alpha) = 0$. \nThe polynomial $P(X)$ is called the {\\itshape minimal polynomial} of $\\alpha$.\nThe {\\itshape height} (resp.\\ the {\\itshape degree}, the {\\itshape inseparable degree}) of $\\alpha$, denoted by $H(\\alpha)$ (resp.\\ $\\deg \\alpha$, $\\insep \\alpha$), is defined to be the height of $P(X)$ (resp.\\ the degree of $P(X)$, the inseparable degree of $P(X)$).\nLet $n \\geq 1$ be an integer and $\\xi$ be in $\\lF_q (( T ^ {- 1}))$.\nWe denote by $w_n (\\xi)$ (resp.\\ $w_n ^ {*} (\\xi)$) the supremum of the real numbers $w$ (resp.\\ $w ^ {*}$) which satisfy \n\\begin{equation*}\n\t0 < \\abs{P(\\xi )} \\leq H(P) ^ {- w} \\quad (\\text{resp.\\ } 0 < \\abs{\\xi - \\alpha } \\leq H(\\alpha) ^ {- w ^ {*} - 1})\n\\end{equation*}\nfor infinitely many $P(X) \\in (\\lF_q [T]) [X]$ of degree at most $n$ (resp.\\ $\\alpha \\in \\overline{\\lF_q (T)}$ of degree at most $n$).\n\nLet $\\alpha \\in \\overline{\\lF_q (T)}$ be a quadratic number.\nIf $\\insep \\alpha = 1$, let $\\alpha ' \\neq \\alpha$ be the Galois conjugate of $\\alpha$.\nIf $\\insep \\alpha = 2$, let $\\alpha ' = \\alpha $.\n\nA Laurent series $\\xi \\in \\lF_q ((T ^ {- 1}))$ can be expressed as a continued fraction:\n\\begin{equation*}\n\t\\xi = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{\\cdots }}},\n\\end{equation*}\nwhere $a_n \\in \\lF_q [T]$ for all $n \\geq 0$ and $\\deg _T a_m \\geq 1$ for all $m \\geq 1$.\nWe write the expansion $\\xi = [a_0, a_1, a_2, \\ldots ]$.\nThe continued fraction expansion of $\\xi$ is finite if and only if $\\xi$ is in $\\lF_q (T)$.\nIt is known that the continued fraction expansion of $\\xi$ is ultimately periodic if and only if $\\xi$ is quadratic (see Th\\'{e}or\\`{e}me 4 in \\cite[CHAPITRE IV]{Mathan70}).\nWe define sequences $(p_n)_{n \\geq -1}$ and $(q_n)_{n \\geq -1}$ by\n\\begin{equation*}\n\t\\begin{cases}\n\t\tp_{- 1} = 1, \\ p_0 = a_0, \\ p_n = a_n p_{n - 1} + p_{n - 2},\\ n \\geq 1,\\\\\n\t\tq_{- 1} = 0,\\ q_0 = 1,\\ q_n = a_n q_{n - 1} + q_{n - 2},\\ n \\geq 1.\n\t\\end{cases}\n\\end{equation*}\nWe call $(p_n \/ q_n)_{n \\geq 0}$ the {\\itshape convergent sequence} of $\\xi$ and $p_n \/ q_n$ the \\textit{$n$-th convergent} of $\\xi$.\n\nLet $n \\geq 1$ be an integer and $W, V$ be finite words.\nWe denote by $W \\bigoplus V$ the word of concatenation of $W$ and $V$.\nWe put $W ^ {[n]} := W \\bigoplus \\cdots \\bigoplus W$ ($n$ times) and $\\overline{W} := W\\bigoplus W\\bigoplus \\cdots \\bigoplus W\\bigoplus \\cdots $ (infinitely many times).\nWe write $W ^ {[0]}$ the empty word.\n\nLet $A, B$ be real numbers with $B \\neq 0$.\nWe write $A \\ll B$ (resp.\\ $A \\ll_a B$) if $\\abs{A} \\leq C \\abs{B}$ for some constant (resp.\\ some constant depending at most on $a$) $C > 0$.\nWe write $A \\asymp B$ (resp.\\ $A \\asymp_a B$) if $A \\ll B$ and $B \\ll A$ (resp.\\ $A \\ll_a B$ and $B \\ll_a A$) hold.\n\n\n\n\n\n\n\\section{Main results}\\label{sec:main}\n\n\n\nWe recall a family of hyperquadratic continued fractions.\nLet $s, t \\geq 1$ be integers and we put $q := p ^ s$ and $r := q ^ t$.\nLet $k \\geq 0$ be an integer and $\\lambda$ be in $\\lF_q ^ {*}$.\nIf $p \\neq 2$, then we assume that $\\lambda \\neq 2$ and put $\\mu := 2 - \\lambda$.\nWe define $\\Theta_k ^ {t} (\\lambda) \\in \\lF_q ((T ^ {- 1}))$ by\n\\begin{equation}\\label{eq:firstfamily}\n\t\\Theta_k ^ {t} (\\lambda)\n\t=\n\t\\begin{cases}\n\t\t[0, T ^ {[k]}, \\bigoplus_{i \\geq 1} (T, (\\lambda T, \\mu T) ^ {[(r ^ i - 1) \/ 2]}) ^ {[k+1]}] & \\text{if}\\ p \\neq 2,\\\\\n\t\t[0, T ^ {[k]} \\bigoplus_{i \\geq 1} (T, \\lambda T ^ {[r ^ i - 1]}) ^ {[k + 1]}] & \\text{if}\\ p = 2.\n\t\\end{cases}\n\\end{equation}\nLasjaunias and Ruch \\cite{Lasjaunias02} proved that $\\Theta_k ^ {t} (\\lambda)$ is hyperquadratic and satisfies the algebraic equation\n\\begin{equation}\\label{eq:firstfamilyequation}\n\t\\begin{cases}\n\t\tq_k X ^ {r + 1} - p_k X ^ r + (\\lambda \\mu ) ^ {(r - 1) \/ 2} q_{k + r} X - (\\lambda \\mu) ^ {(r - 1) \/ 2} p_{k + r} = 0 \\quad \\text{if}\\ p \\neq 2,\\\\\n\t\tq_k X ^ {r + 1} - p_k X ^ r + q_{k + r} X - p_{k + r} = 0 \\quad \\text{if}\\ p = 2,\n\t\\end{cases}\n\\end{equation}\nwhere $(p_n \/ q_n)_{n \\geq 0}$ is the convergent sequence of $\\Theta_k ^ {t} (\\lambda)$.\nNote that if $\\lambda = 1$, then $\\Theta_k ^ {t} (\\lambda)$ is ultimately periodic continued fraction, that is, $\\Theta_k ^ {t} (\\lambda)$ is quadratic.\n\nOur first main result is the following.\n\n\n\n\\begin{thm}\\label{thm:main1}\n\tLet $d \\geq 2$ be an integer and $\\Theta_k ^ {t} (\\lambda) \\in \\lF_q ((T ^ {- 1}))$ be as in \\eqref{eq:firstfamily}.\n\tAssume that $\\lambda \\neq 1$.\n\tThen we have\n\t\\begin{gather}\\label{eq:w2*}\n\t\tw_2 ^ {*} (\\Theta_k ^{t} (\\lambda)) \\geq \\max \\left\\lbrace \\frac{r - 1}{k + 1}, r - 1 - \\frac{r(r - 1)(r - 4)}{2r(k + 1) + (r - 1)(r - 2)}\\right\\rbrace ,\\\\\n\t\t\\label{eq:w2}\n\t\tw_2 (\\Theta_k ^ {t} (\\lambda)) \\geq \\max \\left\\lbrace \\frac{r - 1}{k + 1} + 1, r - \\frac{r(r - 1)(r - 3)}{2r(k + 1)+(r - 1)(r - 2)}\\right\\rbrace .\n\t\\end{gather}\n\tIf\n\t\\begin{equation}\\label{eq:maincond1}\n\t\tk \\leq \\frac{2(r - 1)}{d + \\sqrt{d ^ 2 + 4(2r - 1)d + 4}} - 1,\n\t\\end{equation}\n\tthen we have\n\t\\begin{equation}\\label{eq:mainequal1}\n\t\tw_n ^ {*} ( \\Theta_k ^ {t} (\\lambda)) = \\frac{r - 1}{k + 1}, \\quad w_n (\\Theta_k ^ {t} (\\lambda)) = \\frac{r - 1}{k + 1} + 1\n\t\\end{equation}\n\tfor all integers $2 \\leq n \\leq d$.\n\tIf\n\t\\begin{equation}\\label{eq:maincond2}\n\t\t2rd \\leq \\left( r - 2 - \\frac{r(r - 1)(r - 4)}{2r(k + 1) + (r - 1)(r - 2)}\\right) \\left( r - d - \\frac{r(r - 1)(r - 4)}{2r(k + 1) + (r - 1)(r - 2)}\\right) ,\n\t\\end{equation}\n\tthen we have\n\t\\begin{gather}\\label{eq:mainequal2}\n\t\tw_n ^ {*} (\\Theta_k ^ {t} (\\lambda)) = r - 1 - \\frac{r(r - 1)(r - 4)}{2r(k + 1) + (r - 1)(r - 2)},\\\\\n\t\t\\label{eq:mainequal22}\n\t\tw_n (\\Theta_k ^ {t} (\\lambda )) = r - \\frac{r(r - 1)(r - 3)}{2r(k + 1) + (r - 1)(r - 2)}\n\t\\end{gather}\n\tfor all integers $2 \\leq n \\leq d$.\n\\end{thm}\n\n\n\n\\begin{rem}\n\tIt is well-known and easy to see that for any $\\xi \\in \\lF_q ((T ^ {- 1}))$,\n\t\\begin{equation}\\label{eq:w1_determine}\n\t\tw_1 (\\xi) = w_1 ^ {*} (\\xi) = \\limsup_{n \\longrightarrow \\infty } \\frac{\\deg q_{n + 1}}{\\deg q_n},\n\t\\end{equation}\n\twhere $(p_n \/ q_n)_{n \\geq 0}$ is the convergent sequence of $\\xi$.\n\tSince $\\Theta_k ^ {t}(\\lambda)$ has a bounded partial quotient, we have $w_1 (\\Theta_k ^ {t}(\\lambda)) = w_1 ^ {*} (\\Theta_k ^ {t} (\\lambda)) = 1$.\n\tIf\n\t\\begin{equation*}\n\t\tr > \\frac{3d + 2 + \\sqrt{9d ^ 2 + 4d + 4}}{2},\n\t\\end{equation*}\n\tthere exists an effectively computable positive constant $C_1 (r, d)$, depending only on $r$ and $d$, such that we have \\eqref{eq:mainequal2} and \\eqref{eq:mainequal22} for all $k \\geq C_1 (r, d)$.\n\\end{rem}\n\n\n\nThe key point of the proof of Theorem \\ref{thm:main1} is to find two sequences of very good quadratic approximation, that is, ultimately periodic continued fractions.\n\nAs an application of Theorem \\ref{thm:main1}, we give a sufficient condition to determine the exact degree of $\\Theta_k ^ {t} (\\lambda)$.\n\n\n\n\\begin{cor}\\label{cor:main1}\n\tLet $\\Theta_k ^ {t} (\\lambda) \\in \\lF_q ((T ^ {- 1}))$ be as in \\eqref{eq:firstfamily}.\n\tAssume that $\\lambda \\neq 1$.\n\tIf\n\t\\begin{equation}\\label{eq:main1_cor_cond}\n\t\tk = 0 \\ \\text{or}\\ k > \\frac{(r ^ 2 - 4r + 2)(r - 1)}{2r} - 1,\n\t\\end{equation}\n\tthen we have $\\deg \\Theta_k ^ {t} (\\lambda) = r + 1$.\n\\end{cor}\n\n\n\nWe recall another family of hyperquadratic continued fractions.\nWe define a sequence $(F_n)_{n \\geq 0}$ of polynomials in $\\lF_q [T]$ by\n\\begin{equation}\\label{eq:fib}\n\tF_0 = 1, \\quad F_1 = T, \\quad F_{n + 1} = TF_n + F_{n - 1}\\quad \\text{for}\\ n \\geq 1.\n\\end{equation}\nNote that the sequence can be regarded as the analogue of the Fibonacci sequence.\nLet $s, t \\geq 1$ be integers and we put $q := 2 ^ s, r := 2 ^ t$.\nLet $\\ell \\geq 1$ be an integer, ${\\bm \\lambda} = (\\lambda_1, \\ldots , \\lambda_{\\ell}) \\in (\\lF_q ^ {*}) ^ {\\ell}, {\\bm \\varepsilon} = (\\varepsilon_1, \\varepsilon_2) \\in (\\lF_q ^ {*}) ^ 2$, and $(p_n \/ q_n)_{0 \\leq n \\leq \\ell - 1}$ be the convergent sequence of the continued fraction $[\\lambda_1 T, \\ldots , \\lambda_{\\ell}T]$.\nWe consider the algebraic equation\n\\begin{equation}\\label{eq:char2eq}\n\tq_{\\ell - 1} X ^ {r + 1} + p_{\\ell - 1} X ^ r + (\\varepsilon_1 q_{\\ell - 2} F_{r-1} + \\varepsilon_2 q_{\\ell - 1} F_{r - 2}) X + \\varepsilon_1 p_{\\ell - 2} F_{r - 1} + \\varepsilon_2 p_{\\ell - 1} F_{r - 2} = 0.\n\\end{equation}\nLasjaunias \\cite{Lasjaunias17} proved that \\eqref{eq:char2eq} has a unique root $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ in $\\lF_q ((T ^ {- 1}))$ with $\\abs{\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})} \\geq q$, $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ is hyperquadratic and its continued fraction expansion is the following:\n\\begin{equation}\\label{eq:secondfamily}\n\t\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon}) = [\\lambda_1 T, \\lambda_2 T, \\ldots ],\n\\end{equation}\nwhere a sequence $(\\lambda_n)_{n \\geq 1}$ satisfy\n\\begin{gather*}\n\t\\lambda_{\\ell + r m + 1} = (\\varepsilon_2 \/ \\varepsilon_1) \\varepsilon_2 ^ {(- 1) ^ {m + 1}} \\lambda_{m + 1} ^ r,\\\\\n\t\\lambda_{\\ell + r m + i} = (\\varepsilon_1 \/ \\varepsilon_2 ) ^ {(- 1) ^ i} \\quad \\text{for}\\ m \\geq 0 \\ \\text{and}\\ 2 \\leq i \\leq r.\n\\end{gather*}\nNote that if $\\varepsilon_1 = \\varepsilon_2 = 1$ and $\\lambda_j = 1$ for all $1\\leq j \\leq \\ell$, then $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ is ultimately periodic continued fraction, that is, $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ is quadratic.\n\nIn this paper, we consider the continued fractions $\\Phi_{\\ell } ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ in particular cases.\n\n\n\\begin{thm}\\label{thm:main2}\n\tLet $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ be as in \\eqref{eq:secondfamily}, $m \\geq 1, d \\geq 2$ be integers with $m \\leq \\ell$, and $\\lambda$ be in $\\lF_q ^ {*}$ with $\\lambda \\neq 1$.\n\t\\begin{enumerate}\n\t\t\\item[(1)] Assume that $\\lambda _1 = \\lambda , \\lambda _i = 1$ for all $2 \\leq i \\leq m, \\varepsilon_1 = \\varepsilon_2 = 1$ and if $m < \\ell $, then $\\lambda_{m + 1} \\neq 1$.\n\t\tThen we have\n\t\t\\begin{equation}\\label{eq:main22}\n\t\t\tw_2 ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon}))\\geq \\frac{m}{\\ell}(r - 1),\\quad \n\t\t\tw_2 (\\Phi_{\\ell} ^ t ({\\bm \\lambda }; {\\bm \\varepsilon})) \\geq 1 + \\frac{m}{\\ell} (r - 1).\n\t\t\\end{equation}\n\t\tIf\n\t\t\\begin{equation}\\label{eq:maincond3}\n\t\t\t\\frac{\\ell}{m} \\leq \\frac{2(r - 1)}{d + \\sqrt{d ^ 2 + 4(2r - 1)d + 4}},\n\t\t\\end{equation}\n\t\tthen we have\n\t\t\\begin{equation}\\label{eq:mainequal3}\n\t\t\tw_n ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = \\frac{m}{\\ell}(r - 1), \\quad \n\t\t\tw_n (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = 1 + \\frac{m}{\\ell}(r - 1)\n\t\t\\end{equation}\n\t\tfor all integers $2 \\leq n \\leq d$.\n\t\t\n\t\t\\item[(2)] Assume that $r$ is a power of $q, \\varepsilon_1 = \\varepsilon_2 = 1$ and $\\lambda_i = \\lambda$ for all $1 \\leq i \\leq \\ell$.\n\t\tThen we have\n\t\t\\begin{gather}\\label{eq:mainequal6}\n\t\t\tw_2 ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) \\geq r - 1 - \\frac{r(r - 1)(r - 4)}{2\\ell r + (r - 1)(r - 2)},\\\\\n\t\t\t\\label{eq:mainequal66}\n\t\t\tw_2 (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) \\geq r - \\frac{r(r - 1)(r - 3)}{2\\ell r + (r - 1)(r - 2)}.\n\t\t\\end{gather}\n\t\tIf\n\t\t\\begin{equation}\\label{eq:maincond7}\n\t\t\t2 r d \\leq \\left( r - 2 - \\frac{r(r - 1)(r - 4)}{2 \\ell r + (r - 1)(r - 2)}\\right) \\left( r - d - \\frac{r(r - 1)(r - 4)}{2 \\ell r + (r - 1)(r - 2)}\\right) ,\n\t\t\\end{equation}\n\t\tthen we have\n\t\t\\begin{gather}\\label{eq:mainequal7}\n\t\t\tw_n ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = r - 1 - \\frac{r(r - 1)(r - 4)}{2 \\ell r + (r - 1)(r - 2)},\\\\\n\t\t\t\\label{eq:mainequal77}\n\t\t\tw_n (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = r - \\frac{r (r - 1)(r - 3)}{2 \\ell r + (r - 1)(r - 2)}\n\t\t\\end{gather}\n\t\tfor all integers $2 \\leq n \\leq d$.\n\t\\end{enumerate}\n\\end{thm}\n\n\n\\begin{rem}\n\tBy \\eqref{eq:w1_determine}, we have $w_1 (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = w_1 ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = 1$.\n\tIf\n\t\\begin{equation*}\n\t\tr > \\frac{3d + 2 + \\sqrt{9d ^ 2 + 4d + 4}}{2},\n\t\\end{equation*}\n\tthere exists an effectively computable positive constant $C_2 (r, d)$, depending only on $r$ and $d$, such that we have \\eqref{eq:mainequal7} and \\eqref{eq:mainequal77} for all $\\ell \\geq C_2 (r, d)$.\n\\end{rem}\n\n\n\n\\begin{cor}\\label{cor:main2}\n\tLet $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ be as in \\eqref{eq:secondfamily}, $m \\geq 1, d \\geq 2$ be integers with $m \\leq \\ell$, and $\\lambda$ be in $\\lF_q ^ {*}$ with $\\lambda \\neq 1$.\n\t\\begin{enumerate}\n\t\t\\item[(1)] Assume that $\\lambda _1 = \\lambda , \\lambda _i = 1$ for all $2 \\leq i \\leq m, \\varepsilon_1 = \\varepsilon_2 = 1$ and if $m < \\ell $, then $\\lambda_{m + 1} \\neq 1$.\n\t\tIf\n\t\t\\begin{equation*}\n\t\t\t\\frac{m}{\\ell} > \\frac{r - 2}{r - 1},\n\t\t\\end{equation*}\n\t\tthen we have $\\deg \\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon}) = r + 1$.\n\t\t\n\t\t\\item[(2)] Assume that $r$ is a power of $q, \\varepsilon_1 = \\varepsilon_2 = 1$ and $\\lambda_i = \\lambda$ for all $1 \\leq i \\leq \\ell$.\n\t\tIf\n\t\t\\begin{equation*}\n\t\t\t\\ell > \\frac{(r ^ 2 - 4r + 2)(r - 1)}{2r},\n\t\t\\end{equation*}\n\t\tthen we have $\\deg \\Phi_{\\ell} ^ t({\\bm \\lambda}; {\\bm \\varepsilon}) = r + 1$.\n\t\\end{enumerate}\n\\end{cor}\n\n\n\nIn the last part of this section, we mention a problem associated to Problem \\ref{prob:main1}, Corollary \\ref{cor:main1} and \\ref{cor:main2}.\n\n\n\n\\begin{prob}\n\tLet $n\\geq 2$ be an integer.\n\tDetermine the set of all values taken by $w_n - w_n ^ {*}$ over the set of algebraic Laurent series.\n\\end{prob}\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec:lemma}\n\n\n\nIn this section, we gather lemmas for the proof of main results.\n\n\n\n\\begin{lem}\\label{lem:alg}{\\rm (\\cite[Theorem 5.2]{Ooto17}).}\n\tLet $n \\geq 1$ be an integer and $\\alpha \\in \\lF_q ((T ^ {- 1}))$ be an algebraic Laurent series.\n\tThen we have \n\t\\begin{equation*}\n\t\tw_n ^ {*} (\\alpha), w_n (\\alpha) \\leq \\deg \\alpha - 1.\n\t\\end{equation*}\n\\end{lem}\n\n\n\nLemma \\ref{lem:contidif} and \\ref{lem:conti.lem} are immediately seen.\n\n\n\n\\begin{lem}\\label{lem:contidif}\n\tLet $\\xi = [0, a_1, a_2, \\ldots ], \\zeta =[0, b_1, b_2, \\ldots ]$ be in $\\lF_q ((T ^ {- 1}))$.\n\tAssume that there exists an integer $k \\geq 1$ such that $a_n = b_n$ for all $1 \\leq n \\leq k$ and $a_{k + 1} \\neq b_{k + 1}$.\n\tThen we have\n\t\\begin{equation*}\n\t|\\xi - \\zeta| = \\frac{\\abs{a_{k + 1} - b_{k + 1}}}{\\abs{a_{k + 1} b_{k + 1}}\\abs{q_k} ^ 2},\n\t\\end{equation*}\n\twhere $(p_n \/ q_n)_{n \\geq 0}$ is the convergent sequence of $\\xi$.\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:conti.lem}\n\tLet $\\xi = [a_0, a_1, \\ldots ]$ be in $\\lF_q ((T ^ {- 1}))$ and $(p_n \/ q_n)_{n \\geq 0}$ be the convergent sequence of $\\xi$.\n\tThen, for any $n \\geq 1$, we have\n\t\\begin{equation}\\label{enum:q}\n\t\t|q_n| = |a_1| |a_2| \\cdots |a_n|.\n\t\\end{equation}\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:conj2}{\\rm (\\cite[Lemma 4.6]{Ooto17}).}\n\tLet $r, s \\geq 1$ be integers and $\\alpha = [0, a_1,\\ldots ,a_r, \\overline{a_{r + 1}, \\ldots , a_{r + s}}] \\in \\lF_q ((T ^ {- 1}))$ be an ultimately periodic continued fraction with $a_r \\neq a_{r + s}$.\n\tLet $(p_n \/ q_n)_{n \\geq 0}$ be the convergent sequence of $\\alpha$.\n\tThen we have\n\t\\begin{equation*}\n\t\t\\frac{\\min (|a_r|, |a_{r + s}| )}{|q_r| ^ 2} \\leq |\\alpha - \\alpha '| \\leq \\frac{|a_r a_{r + s}|}{|q_r| ^ 2}.\n\t\\end{equation*}\n\\end{lem}\n\n\n\nThe following lemma is well-known and easy to see.\n\n\n\n\\begin{lem}\\label{lem:quadheightupper}\n\tLet $r \\geq 0, s \\geq 1$ be integers and $\\alpha = [0, a_1, \\ldots , a_r, \\overline{a_{r + 1}, \\ldots , a_{r + s}}] \\in \\lF_q ((T ^ {- 1}))$ be an ultimately periodic continued fraction.\n\tLet $(p_n \/ q_n)_{n \\geq 0}$ be the convergent sequence of $\\xi$.\n\tThen we have $H(\\alpha) \\leq |q_r q_{r + s}|$.\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:quadheight}\n\tLet $b, c, d \\in \\lF_q [T]$ be distinct polynomials with $\\deg_T b, \\deg_T c, \\deg_T d \\geq 1$.\n\tLet $n, m, \\ell \\geq 1$ be integers and $a_1, \\ldots , a_{n - 1} \\in \\lF_q [T]$ be polynomials with $\\deg_T a_i \\geq 1$ for all $1 \\leq i \\leq n - 1$.\n\tWe put\n\t\\begin{gather*}\n\t\t\\alpha_1 := [0, a_1, \\ldots , a_{n - 1}, c, \\overline{b}], \\quad \n\t\t\\alpha_2 := [0, a_1, \\ldots , a_{n - 1}, c, \\overline{b ^ {[m]}, d}],\\\\\n\t\t\\alpha_3 := [0, a_1, \\ldots , a_{n - 1}, c, \\overline{b ^ {[m]}, c, b ^ {[\\ell]}}], \\quad\n\t\t\\alpha_4 := [0, a_1, \\ldots , a_{n - 1}, c, \\overline{(b, d) ^ {[m]}, c, (b, d) ^ {[\\ell]}}].\n\t\\end{gather*}\n\tLet $(p_{i, k} \/ q_{i, k})_{k \\geq 0}$ be the convergent sequence of $\\xi_i$ for $i = 1, 2, 3, 4$.\n\tThen we have\n\t\\begin{gather}\\label{eq:quadheight12}\n\t\tH(\\alpha_1) \\asymp_{b, c} \\abs{q_{1, n}} ^ 2, \\quad\n\t\tH(\\alpha_2) \\asymp _{b, c, d} \\abs{q_{2, n} q_{2, n + m + 1}},\\\\\n\t\t\\label{eq:quadheight3}\n\t\tH(\\alpha_3) \\asymp_{b, c} \\abs{q_{3, n} q_{3, n + m + \\ell + 1}}, \\quad\n\t\tH(\\alpha_4) \\asymp_{b, c, d} \\abs{q_{4, n} q_{4, n + 2m + 2\\ell + 1}}.\n\t\\end{gather}\n\\end{lem}\n\n\n\n\\begin{proof}\n\tSee Lemma 4.7 in \\cite{Ooto17} for the proof of \\eqref{eq:quadheight12}.\n\t\n\tIt follows from Lemma \\ref{lem:quadheightupper} that $H(\\alpha_3) \\leq |q_{3, n} q_{3, n + m + \\ell + 1}|$.\n\tWe put \n\t\\begin{equation*}\n\t\t\\beta_3 :=[0, a_1, \\ldots , a_{n - 1}, c, b ^ {[m]}, c, \\overline{b}].\n\t\\end{equation*}\n\t\n\tLet $P_3(X)$ and $Q_3(X)$ be the minimal polynomial of $\\alpha_3$ and $\\beta_3$, respectively.\n\tSince $P_3$ and $Q_3$ do not have a common root, we have\n\t\\begin{equation*}\n\t\t1 \\leq |\\Res(P_3, Q_3)| \\leq H(\\beta_3) ^ 2 H(\\alpha_3) ^ 2 |\\alpha_3 - \\beta_3| |\\alpha_3' - \\beta_3| |\\alpha_3 - \\beta_3'| |\\alpha_3' - \\beta_3'|.\n\t\\end{equation*}\n\tBy Lemma \\ref{lem:contidif}, \\ref{lem:conti.lem} and \\ref{lem:conj2}, we obtain\n\t\\begin{gather*}\n\t\t|\\alpha_3 - \\beta_3| \\ll_{b, c} |q_{3, n + 2m + \\ell + 1}| ^ {- 2}, \\quad \n\t\t|\\alpha_3 - \\beta_3'| \\ll_{b, c} |q_{3, n + m + 1}| ^ {- 2},\\\\\n\t\t|\\alpha_3' - \\beta_3|, |\\alpha_3' - \\beta_3'| \\ll_{b, c} |q_{3, n}| ^ {- 2}.\n\t\\end{gather*}\n\tTherefore, by Lemma \\ref{lem:conti.lem} and \\ref{lem:quadheightupper}, we have $H(\\alpha_3) \\gg_{b, c} |q_{3, n} q_{3, n + m + \\ell + 1}|$.\n\t\n\tIn the same way to the above proof, we obtain $H(\\alpha_4) \\asymp_{b, c, d} \\abs{q_{4, n} q_{4, n + 2m + 2\\ell + 1}}$.\n\\end{proof}\n\n\n\n\\begin{lem}\\label{lem:wnwn*}{\\rm (\\cite[Proposition 5.6]{Ooto17}).}\n\tLet $n \\geq 1$ be an integer and $\\xi \\in \\lF_q ((T ^ {- 1}))$.\n\tThen we have \n\t\\begin{equation*}\n\t\tw_n ^ {*} (\\xi) \\leq w_n (\\xi).\n\t\\end{equation*}\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:wnlower}\n\tLet $n, m \\geq 1$ be integers with $n \\geq m$ and $\\xi \\in \\lF_q ((T ^ {- 1}))$ be not algebraic of degree at most $m$.\n\tThen we have $w_n (\\xi) \\geq m$.\n\\end{lem}\n\n\n\n\\begin{proof}\n\tSince the sequence $(w_k (\\xi))_{k \\geq 1}$ is increasing, we have $w_n (\\xi) \\geq m$ (see e.g.\\ \\cite[page~200]{Bugeaud04}).\n\\end{proof}\n\n\n\nThe following lemma is well-known and immediately seen.\n\n\n\n\\begin{lem}\\label{lem:height}\n\tLet $P(X)$ be in $(\\lF_q [T]) [X]$.\n\tAssume that $P(X)$ can be factorized as\n\t\\begin{equation*}\n\t\tP(X) = A \\prod_{i = 1}^{n} (X - \\alpha_i),\n\t\\end{equation*}\n\twhere $A \\in \\lF_q [T]$ and $\\alpha_i \\in \\overline{\\lF_q (T)}$ for $1 \\leq i \\leq n$.\n\tThen we have\n\t\\begin{equation*}\n\t\tH(P) = |A| \\prod_{i = 1}^{n} \\max (1, |\\alpha_i|).\n\t\\end{equation*}\n\\end{lem}\n\n\n\nThe following lemma is immediately seen by the definition of discriminant.\n\n\n\n\\begin{lem}\\label{lem:Galois}\n\tLet $\\alpha \\in \\overline{\\lF_q (T)}$ be a quadratic number.\n\tIf $\\alpha \\neq \\alpha'$, then we have\n\t\\begin{equation*}\n\t\t|\\alpha - \\alpha'| \\geq H(\\alpha) ^ {- 1}.\n\t\\end{equation*}\n\\end{lem}\n\n\n\nThe following two lemmas are key lemmas for proving Theorem \\ref{thm:main1} and \\ref{thm:main2}.\n\n\n\n\\begin{lem}\\label{lem:bestquad}{\\rm (\\cite[Lemma 3.19]{Ooto18}).}\n\tLet $d \\geq 2$ be an integer.\n\tLet $\\xi$ be in $\\lF_q ((T ^ {- 1}))$, $\\theta , \\delta$ be positive numbers, and $\\varepsilon$ be a non-negative number.\n\tAssume that there exist a sequence $(\\alpha_j)_{j \\geq 1}$ and positive number $c$ such that for any $j \\geq 1$, $\\alpha_j \\in \\overline{\\lF_q (T)}$ is quadratic with $0 < |\\alpha_j - \\alpha_j'| \\leq c$ and $\\xi \\neq \\alpha_j$, $(H(\\alpha_j))_{j \\geq 1}$ is a divergent increasing sequence, and\n\t\\begin{gather*}\n\t\t\\limsup_{k \\rightarrow \\infty } \\frac{\\log H(\\alpha_{k + 1})}{\\log H(\\alpha_k)} \\leq \\theta ,\\\\\n\t\t\\lim_{k \\rightarrow \\infty } \\frac{- \\log |\\xi - \\alpha_k|}{\\log H(\\alpha_k)} = d + \\delta , \\quad \\lim_{k \\rightarrow \\infty } \\frac{- \\log |\\alpha_k - \\alpha_k'|}{\\log H(\\alpha_k)} = \\varepsilon .\n\t\\end{gather*}\n\tIf $2 d \\theta \\leq (d - 2 + \\delta ) \\delta $, then we have for all $2 \\leq n \\leq d$,\n\t\\begin{equation}\\label{eq:bestquad_eq}\n\t\tw_n ^ {*} (\\xi) = d - 1 + \\delta , \\quad w_n(\\xi) = d - 1 + \\delta + \\varepsilon .\n\t\\end{equation}\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:bestquad2}\n\tLet $\\xi$ be in $\\lF_q ((T ^ {- 1})) \\setminus \\lF_q (T)$.\n\tAssume that there exists a sequence $(\\alpha_j)_{j \\geq 1}$ such that for any $j \\geq 1$, $\\alpha_j \\in \\overline{\\lF_q (T)}$ is quadratic with $\\alpha_j \\neq \\alpha_j'$ and $\\xi \\neq \\alpha_j$, and $(H(\\alpha_j))_{j \\geq 1}$ is a divergent increasing sequence.\n\tIf there exist limits of the sequences \n\t\\begin{equation*}\n\t\t\\left( \\frac{- \\log |\\xi - \\alpha_j|}{\\log H(\\alpha_j)}\\right) _{j \\geq 1}\\ \\text{and}\\ \\left( \\frac{- \\log |\\alpha_j - \\alpha_j'|}{\\log H(\\alpha_j)}\\right) _{j \\geq 1},\n\t\\end{equation*}\n\tthen we have\n\t\\begin{gather}\\label{eq:quad1}\n\t\tw_2 ^ {*} (\\xi) \\geq \\lim_{j \\rightarrow \\infty } \\frac{- \\log |\\xi - \\alpha_j|}{\\log H(\\alpha_j)} - 1,\\\\\n\t\t\\label{eq:quad2}\n\t\tw_2 (\\xi) \\geq \\lim_{j \\rightarrow \\infty } \\frac{- \\log |\\xi - \\alpha_j|}{\\log H(\\alpha_j)} + \\lim_{j \\rightarrow \\infty }\\frac{- \\log |\\alpha_j - \\alpha_j'|}{\\log H(\\alpha_j)} - 1.\n\t\\end{gather}\n\\end{lem}\n\n\n\n\\begin{proof}\n\tThe inequality \\eqref{eq:quad1} is immediately seen.\n\tIn what follows, we show \\eqref{eq:quad2}.\n\tWe put\n\t\\begin{equation*}\n\t\t\\gamma_1 := \\lim_{j \\rightarrow \\infty } \\frac{- \\log |\\xi - \\alpha_j|}{\\log H(\\alpha_j)}, \\quad \\gamma_2 := \\lim_{j \\rightarrow \\infty } \\frac{- \\log |\\alpha_j - \\alpha_j'|}{\\log H(\\alpha_j)}.\n\t\\end{equation*}\n\t\n\tWe may assume that $\\gamma_1 > 1 $ and $\\gamma_2 > 0$.\n\tIn fact, if $\\gamma_2 \\leq 0$, then we have \\eqref{eq:quad2} by Lemma \\ref{lem:wnwn*}.\n\tBy Lemma \\ref{lem:Galois}, we have $\\gamma_2 \\leq 1$, which implies $\\gamma_2 < \\gamma_1$.\n\tIf $\\gamma_1 \\leq 1$, then we obtain \\eqref{eq:quad2} by Lemma \\ref{lem:wnlower}.\n\t\n\tThen we obtain\n\t\\begin{equation*}\n\t\t|\\xi - \\alpha_j| < |\\alpha_j - \\alpha_j'| \\leq 1\n\t\\end{equation*}\n\tfor all sufficiently large $j$.\n\tTherefore, we have\n\t\\begin{gather*}\n\t\t\\max (1, |\\xi|) = \\max (1, |\\alpha_j|) = \\max (1, |\\alpha_j'|),\\\\\n\t\t|\\xi - \\alpha_j'| = |\\alpha_j - \\alpha_j'|\n\t\\end{gather*}\n\tfor all sufficiently large $j$.\n\tIt follows from Lemma \\ref{lem:height} that\n\t\\begin{equation*}\n\t\tH(P_j) = |A_j| \\max (1, |\\xi|) ^ 2\n\t\\end{equation*}\n\tfor all sufficiently large $j$, where $P_j (X) = A_j (X - \\alpha_j) (X - \\alpha_j ')$ is the minimal polynomial of $\\alpha_j$.\n\tHence, we have\n\t\\begin{equation*}\n\t\t|P_j (\\xi)| = H(P_j) |\\xi - \\alpha_j| |\\alpha_j - \\alpha_j'| \\max (1, |\\xi|) ^ {- 2}\n\t\\end{equation*}\n\tfor all sufficiently large $j$.\n\tThus, we obtain\n\t\\begin{equation*}\n\t\t\\lim_{j\\rightarrow \\infty }\\frac{- \\log |P_j (\\xi)|}{\\log H(P_j)} = \\gamma_1 + \\gamma_2 - 1,\n\t\\end{equation*}\n\twhich implies \\eqref{eq:quad2}.\n\\end{proof}\n\n\n\n\n\n\n\\section{Proof of Main results}\\label{sec:proof}\n\n\n\nIn this section, we prove the main results, that is, Theorem \\ref{thm:neq_alg}, \\ref{thm:main1}, \\ref{thm:main2}, Corollary \\ref{cor:main1} and \\ref{cor:main2}.\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main1}]\n\tFor an integer $n \\geq 3$, we put\n\t\\begin{gather*}\n\t\t\\alpha_n :=\n\t\t\\begin{cases}\n\t\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 1}(T, (\\lambda T, \\mu T) ^ {[(r ^ i - 1) \/ 2]}) ^ {[k + 1]}, T, \\overline{\\lambda T, \\mu T}] & \\text{if}\\ p \\neq 2,\\\\\n\t\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 1}(T, \\lambda T ^ {[r ^ i - 1]}) ^ {[k + 1]}, T, \\overline{\\lambda T}] & \\text{if}\\ p = 2,\n\t\t\\end{cases}\\\\\n\t\t\\beta_n :=\n\t\t\\begin{cases}\n\t\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 1} (T, (\\lambda T, \\mu T) ^ {[(r ^ i - 1) \/ 2]}) ^ {[k + 1]}, \\overline{T, (\\lambda T, \\mu T) ^ {[(r ^ n - 1) \/ 2]}}] & \\text{if}\\ p \\neq 2,\\\\\n\t\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 1} (T, \\lambda T ^ {[r ^ i - 1]}) ^ {[k + 1]}, \\overline{T, \\lambda T ^ {[r ^ n - 1]}}] & \\text{if}\\ p = 2.\n\t\t\\end{cases}\n\t\\end{gather*}\n\tThen we have\n\t\\begin{align*}\n\t\t\\beta_n =\n\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 2}(T, (\\lambda T, & \\mu T) ^ {[(r ^ i - 1) \/ 2]}) ^ {[k + 1]}, (T, (\\lambda T, \\mu T) ^ {[(r ^ {n - 1} - 1) \/ 2]})^{[k]},\\\\\n\t\t& T, \\overline{(\\lambda T, \\mu T) ^ {[(r ^ {n - 1} - 1) \/ 2]}, T, (\\lambda T, \\mu T) ^ {[(r ^ n - r ^ {n - 1}) \/ 2]}}]\t\n\t\\end{align*}\n\tif $p \\neq 2$, and\n\t\\begin{align*}\n\t\t\\beta_n\t=\n\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 2} (T, \\lambda T ^ {[r ^ i - 1]}) ^ {[k + 1]}, (T, \\lambda T ^ {[r ^ {n - 1} - 1]}) ^ {[k]}, T, \\overline{\\lambda T ^ {[r ^ {n - 1} - 1]}, T, \\lambda T ^ {[r ^ n - r ^ {n - 1}]}}]\n\t\\end{align*}\n\tif $p = 2$.\n\tIt follows from Lemma \\ref{lem:conti.lem} and \\ref{lem:quadheight} that\n\t\\begin{equation*}\n\t\tH(\\alpha_n) \\asymp q ^ {2 (k + 1) (r ^ n - 1) \/ (r - 1)}, \\quad\n\t\tH(\\beta_n) \\asymp q ^ {2 (k + 1) (r ^ n - 1) \/ (r - 1) + r ^ n - 2 r ^ {n - 1}}.\n\t\\end{equation*}\n\tSince $\\Theta_k ^ {t} (\\lambda)$ and $\\alpha_n$ have the same first $((k + 1) \\sum_{i = 0} ^ {n - 1} r ^ i + r ^ n - 1)$-th partial quotients, while the next partial quotient are different, we have\n\t\\begin{equation*}\n\t\t\\abs{\\Theta_k ^ {t} (\\lambda) - \\alpha_n} \\asymp q ^ {- 2 (k + 1) (r ^ n - 1) \/ (r - 1) - 2 r ^ n}\n\t\\end{equation*}\n\tby Lemma \\ref{lem:contidif} and \\ref{lem:conti.lem}.\n\tSimilary, since $\\Theta_k ^ {t} (\\lambda)$ and $\\beta_n$ have the same first $((k + 1) \\sum_{i = 0} ^ {n} r ^ i + r ^ n - 1)$-th partial quotients, while the next partial quotient are different, we get\n\t\\begin{equation*}\n\t\t\\abs{\\Theta_k ^ {t} (\\lambda) - \\beta_n} \\asymp q ^ {- 2 (k + 1) (r ^ {n + 1} - 1) \/ (r - 1) - 2 r ^ n}.\n\t\\end{equation*}\n\tBy Lemma \\ref{lem:conti.lem} and \\ref{lem:conj2}, we obtain\n\t\\begin{equation*}\n\t\t\\abs{\\alpha_n - \\alpha_n'} \\asymp q ^ {- 2 (k + 1) (r ^ n - 1) \/ (r - 1)},\\quad\n\t\t\\abs{\\beta_n - \\beta_n'} \\asymp q ^ {- 2 (k + 1) (r ^ n - 1) \/ (r - 1) + 2 r ^ {n - 1}}.\n\t\\end{equation*}\n\tTherefore, we deduce that\n\t\\begin{gather*}\n\t\t\\lim_{n \\rightarrow \\infty } \\frac{\\log H(\\alpha_{n + 1})}{\\log H(\\alpha_n)} = \\lim_{n \\rightarrow \\infty } \\frac{\\log H(\\beta_{n + 1})}{\\log H(\\beta_n)} = r,\\\\\n\t\t\\lim_{n \\rightarrow \\infty } \\frac{- \\log \\abs{\\Theta_k ^ {t} (\\lambda) - \\alpha_n}}{\\log H(\\alpha_n)} = 1 + \\frac{r - 1}{k + 1},\\quad \n\t\t\\lim_{n \\rightarrow \\infty } \\frac{- \\log \\abs{\\alpha_n - \\alpha_n'}}{\\log H(\\alpha_n)} = 1,\\\\\n\t\t\\lim_{n \\rightarrow \\infty } \\frac{- \\log \\abs{\\Theta_k ^ {t} (\\lambda) - \\beta_n}}{\\log H(\\beta_n)} = r - \\frac{r (r - 1)(r - 4)}{2 r (k + 1) + (r - 1) (r - 2)},\\\\\n\t\t\\lim_{n \\rightarrow \\infty } \\frac{- \\log \\abs{\\beta_n - \\beta_n'}}{\\log H(\\beta_n)} = 1 - \\frac{r (r - 1)}{2 r (k + 1) + (r - 1)(r - 2)}.\n\t\\end{gather*}\n\tHence, we obtain \\eqref{eq:w2*} and \\eqref{eq:w2} by Lemma \\ref{lem:bestquad2}.\n\t\n\tIf \\eqref{eq:maincond1} holds, then we heve\n\t\\begin{equation*}\n\t\t2 r d \\leq \\left( \\frac{r - 1}{k + 1} - 1\\right) \\left( \\frac{r - 1}{k + 1} - d + 1 \\right) .\n\t\\end{equation*}\n\tTherefore, by Lemma \\ref{lem:bestquad}, we obtain \\eqref{eq:mainequal1}.\n\t\n\tSimilaly, if \\eqref{eq:maincond2} holds, then we heve \\eqref{eq:mainequal2} and \\eqref{eq:mainequal22} by Lemma \\ref{lem:bestquad}.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Corollary \\ref{cor:main1}]\n\tIt follows from \\eqref{eq:hyperquad} that $\\deg \\Theta_k ^ {t} (\\lambda) \\leq r+1$.\n\tIf \\eqref{eq:main1_cor_cond} holds, then we have $w_2(\\Theta_k ^ {t} (\\lambda)) > r-1$ by Theorem \\ref{thm:main1}.\n\tTherefore, by Lemma \\ref{lem:alg}, we obtain $\\deg \\Theta_k ^ {t} (\\lambda) = r+1$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:neq_alg}]\n\tIt follows from $q \\geq 4$ that we can take $\\lambda \\in \\lF_q ^ {*}$ with $\\lambda \\neq 1$ and $\\lambda \\neq 2$.\n\tLet $n \\geq 2$ be integers.\n\tSince $r \\geq (3n + 2 + \\sqrt{9n^2 + 4n + 4})\/2$, we take $r = p^t$, where $t \\geq 0$ is an integer with\n\t\\begin{equation*}\n\t\t0 \\leq \\frac{2(r - 1)}{n + \\sqrt{n ^ 2 + 4(2r - 1)n + 4}} - 1.\n\t\\end{equation*}\n\tThen, by Theorem \\ref{thm:main1} and Corollary \\ref{cor:main1}, we have $w_n(\\Theta_0 ^ {t} (\\lambda)) \\neq w_n^{*}(\\Theta_0 ^ {t} (\\lambda))$ and $\\deg \\Theta_0 ^ {t} (\\lambda) = r + 1$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main2}]\n\tSince $w_n (\\xi) = w_n (\\xi^{- 1})$ and $w_n ^ {*} (\\xi) = w_n ^ {*} (\\xi ^{- 1})$ for all $0 \\neq \\xi \\in \\lF_q ((T ^ {- 1}))$ and $n \\geq 1$, we consider $\\Phi := \\Phi_{\\ell} ^ t({\\bm \\lambda}; {\\bm \\varepsilon}) ^ {- 1} = [0, \\lambda_1T, \\lambda_2T, \\ldots ]$ instead of $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$.\n\tBy the assumption, we obtain\n\t\\begin{equation*}\n\t\t\\lambda_n\n\t\t=\n\t\t\\begin{cases}\n\t\t\t\\lambda_{i + 1} ^ {r ^ j} & \\text{if}\\ n = 1 + \\ell \\sum_{h = 0} ^ {j - 1} r ^ h + i r ^ j \\ \\text{for some}\\ j \\geq 0, 0 \\leq i < \\ell ,\\\\\n\t\t\t1 & \\text{otherwise}.\n\t\t\\end{cases}\n\t\\end{equation*}\n\t\n\tFirst, we prove (1).\n\tFor an integer $n \\geq 1$, we put\n\t\\begin{equation*}\n\t\t\\alpha_n := [0, \\lambda_1T, \\lambda_2T, \\ldots ,\\lambda_{i(n)}T, \\overline{T}],\n\t\\end{equation*}\n\twhere $i(n) = 1 + \\ell \\sum_{j = 0} ^ {n - 1} r ^ j$.\n\tThen we have $\\lambda_{i(n)} = \\lambda ^ {r ^ n}$.\n\tIn a similar way to the proof of Theorem \\ref{thm:main1}, we obtain\n\t\\begin{gather*}\n\t\\abs{\\Phi - \\alpha_n} \\asymp q ^ {- 2 \\ell (r ^ n - 1) \/ (r - 1) - 2 m r ^ n},\\quad \n\t\\abs{\\alpha_n - \\alpha_n'} \\asymp q ^ {- 2 \\ell (r ^ n - 1) \/ (r - 1)},\\\\\n\tH(\\alpha_n) \\asymp q ^ {2 \\ell (r ^ n - 1) \/ (r - 1)}.\n\t\\end{gather*}\n\tTherefore, we have \\eqref{eq:main22} by Lemma \\ref{lem:bestquad2}.\n\tIf \\eqref{eq:maincond3} holds, then we obtain \\eqref{eq:mainequal3} by Lemma \\ref{lem:bestquad}.\n\t\n\tNext, we prove (2).\n\tFor an integer $n \\geq 1$, we put\n\t\\begin{equation*}\n\t\\beta_n := [0, \\lambda_1T, \\lambda_2T, \\ldots ,\\lambda_{i(n) - 1}T, \\overline{\\lambda ^ {r ^ n}T, T ^ {[r ^ n - 1]}}],\n\t\\end{equation*}\n\tThen we have\n\t\\begin{equation*}\n\t\\beta_n = [0, \\lambda_1T, \\lambda_2T, \\ldots ,\\lambda_{i(n) - r ^ {n - 1}}T, \\overline{T ^ {[r ^ {n - 1} - 1]},\\lambda T, T ^ {[r ^ n - r ^ {n - 1}]}}],\n\t\\end{equation*}\n\tand $\\lambda_{i(n) - r ^ {n - 1}}=\\lambda$.\n\tIn a similar way to the proof of Theorem \\ref{thm:main1}, we obtain\n\t\\begin{gather*}\n\t\\abs{\\Phi - \\beta_n} \\asymp q ^ {- 2 \\ell (r ^ n - 1) \/ (r - 1) - 2 m r ^ n},\\quad \n\t\\abs{\\beta_n - \\beta_n'} \\asymp q ^ {- 2 \\ell (r ^ n - 1) \/ (r - 1) + 2 r ^ {n - 1}},\\\\\n\tH(\\beta_n) \\asymp q ^{2 \\ell (r ^ n - 1) \/ (r - 1) + r ^ n - 2 r ^ {n - 1}}.\n\t\\end{gather*}\n\tTherefore, we have \\eqref{eq:mainequal6} and \\eqref{eq:mainequal66} by Lemma \\ref{lem:bestquad2}.\n\tIf \\eqref{eq:maincond7} holds, then we obtain \\eqref{eq:mainequal7} and \\eqref{eq:mainequal77} by Lemma \\ref{lem:bestquad}.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Corollary \\ref{cor:main2}]\n\tIn the same way to the proof Corollary \\ref{cor:main1}, we prove Corollary \\ref{cor:main2}.\n\\end{proof}\n\n\n\n\n\n\\subsection*{Acknowledgements}\nThe author would like to thank the referee for helpful comments.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn the eighteenth century, the famous astronomer William Herschel showed us the\npowerful method of star count to understand our own Milky Way \\citep{Hers1785}.\nThe technique has been used since then by generations of astronomers. With\ngreat improvement on data collection over the years, more and more details of\nour Milky Way were unfolded. However, the characteristic scales of smooth\nGalactic structures (i.e., disks and halo) obtained by previous studies does\nnot converge to a common value as an outcome of improving data collection\n(see Table~\\ref{table_ref}). The spread of values is attributed to the\ndegeneracy of Galactic model parameters (i.e., same star count data could be\nfitted equally well by different Galactic models) \\citep{Chen2001,\nSiegel2002, Juric2008, Bilir2008}.\nThis is due to the different sky regions\nand limiting magnitudes (i.e., limiting volumes) used in these studies\n\\citep{Siegel2002, Karaali2004, Bilir2006a, Bilir2006b, Juric2008}. On the\ncontrary, \\citet{Bilir2008, Yaz2010} did not show such degeneracy in determining\nthe Galactic model parameters. This controversy is still\nactively debated.\nTherefore, systematic all sky surveys with deeper limiting magnitude and wider sky\nregion, such as the Two Micron All Sky Survey \\citep[2MASS;][]{Skrutskie2006},\nthe Sloan Digital Sky Survey \\citep[SDSS;][]{York2000}, the Panoramic Survey\nTelescope \\& Rapid Response System \\citep[Pan-Starrs;][]{Kaiser2002} and the\nGAIA mission \\citep{Perryman2001} could provide a good opportunity for us to\nstudy our Galaxy from a global\nperspective. Free from limited sky\nfields, astronomers can acquire many more information from the stellar distribution\nof these surveys.\n\nOn Galactic structure study, besides the simple and smooth\ntwo-component model \\citep{Bahcall1980} or the three-component model\n\\citep{Gilmore1985}, many more structures have been discovered, such as\ninner bars in the Galactic center\n\\citep{Alves2000, Hammersley2000, vanLoon2003, Nishiyama2005, Cabrera2008},\nand flares and warps \\citep{Lopez2002, Robin2003, Momany2006, Reyle2009},\nwhich has been contributed the variation of disk model parameters with\nGalactic longitude \\citep{Cabrera2007, Bilir2008, Yaz2010}.\nMoreover, the overdensities in the halo, such as Sagittarius\n\\citep{Majewski2003}, Triangulum-Andromeda \\citep{Rocha2004, Majewski2004},\nVirgo \\citep{Juric2008}, and in the outer disk, such as Canis Major\n\\citep{Martin2004}, Monoceros \\citep{Newberg2002,Rocha2003}, show the\ncomplexity of the Milky Way. The formation history of our Galaxy is more\ncomplicated than what we thought previously.\n\nOn stellar luminosity function\nstudy, a recent study by \\citet{Chang2010} using the $K_s$ band star count of\n2MASS point source catalog \\citep[2MASS PSC;][]{Cutri2003} verified for\nthe first time the universality hypothesis of the luminosity function (i.e.,\nthe common practice of assuming one luminosity function for the entire Milky\nWay).\n\nWe are interested in the global smooth structure of our Galaxy. In view of the\nexisting fine structures (e.g., flares, warps, overdensities\\dots etc.), the\nstructure of the smooth components can be determined either by including these\nfine structures in a grand Galaxy model or by avoiding the sky area\n``contaminated'' with these features. The first method demands a complex model,\nwhich involves many more structure parameters, and needs high computing power\nto accomplish the fitting task. \\citet{Lopez2002} is a good example of this\nmethod using 2MASS data. The second method is clearly simpler but needs\njustifications. We observe that (1) the fine structures (e.g., inner bars,\nflares and warps), which have observable contribution on 2MASS star count data,\nare all confined in the Galactic plane region.\n(2) The overdensities or substructures in the outer disk region and the halo are\ndifficult to identify in general.\nTheir contribution is negligible in the 2MASS star count data.\nHere we quote \\citet{Majewski2004} on halo substructure:\n``This substructure is typically subtle and obscured by a substantial\nforeground veil of disk stars, eliciting its presence requires strategies that\noptimize the substructure signal compared to the foreground noise.''\n\nWe prefer the second method in this paper and only use Galactic latitude\n$|b|>30^{\\circ}$ 2MASS $K_s$ band star\ncount data to obtain the structure parameters of a three-component model.\nIn addition, the influence of the near infrared extinction in these\nregions is small. This also allows us to use a simpler extinction model for\ncorrection. We describe our model in section 2 and the analysis method in\nsection 3. Section 4 provides the results and a discussion.\n\n\n\n\\section{The Milky Way Model}\nWe adopt a three-component model for the smooth stellar distribution of the\nMilky Way. It comprises a thin disk, a thick disk and an oblate halo\n\\citep{Bahcall1980, Gilmore1983}. The total stellar density $n(R,Z)$\nat a location $(R,Z)$\nis the sum of the thin disk $D_1$, the think disk $D_2$ and the\nhalo $H$,\n\\begin{equation}\\label{density}\n n(R,Z)=n_0\\left[D_1(R,Z)+D_2(R,Z)+H(R,Z)\\right]\\,,\n\\end{equation}\nwhere $R$ is the galactocentric distance on the Galactic plane, $Z$ is the\ndistance from the the Galactic mid-plane and $n_0$ is the local stellar\ndensity of the thin disk at the solar neighborhood.\n\nThe stellar distribution of the thin disk $D_1$ and the thick disk $D_2$\ndecreases exponentially along $R$ and $Z$ (the so called double exponential\ndisk),\n\\begin{equation}\\label{Di}\n D_i(R,Z)=f_i\\exp\\left[-\\,{(R-R_\\odot)\\over H_{ri}}-\\,{(|Z|-|Z_\\odot|)\\over H_{zi}}\\right]\\,,\n\\end{equation}\nwhere $(R_\\odot,Z_\\odot)$ is the location of the Sun, $H_{ri}$ is the\nscale-length, $H_{zi}$ is the scale-height, and $f_i$ is the density ratio to\nthe thin disk at the solar neighborhood. The subscript $i=1$ stands for the\nthin disk (thus $f_1=1$) and $i=2$ for thick disk. We adopted $R_\\odot=8$ kpc\nin our model \\citep{Reid1993}.\n\nThe halo is a power law decay oblate spheroid flattening in the $Z$ direction,\n\\begin{equation}\\label{SH}\n H(R,Z)=f_h\\left[R^2+(Z\/\\kappa)^2\\over R_\\odot^2+(Z_\\odot\/\\kappa)^2\\right]^{-p\/2}\\,,\n\\end{equation}\nwhere $\\kappa$ is the axis ratio, $p$ is the power index and $f_h$ is the local\nhalo-to-thin disk density ratio.\n\n\\citet{Chang2010} showed that the whole sky $K_s$ band luminosity function can\nbe well approximated by a single power law with a power law index\n$\\gamma=1.85\\pm0.035$, a bright cutoff at $M_b=-7.86\\pm0.60$ and a faint cutoff\nat $M_f=6.88\\pm0.66$. We adopt this luminosity function in the following analysis.\nIn normalized form, it is\n\\begin{equation}\\label{LF}\n \\psi(M_{K_s})={2\\log_{\\rm e}10\\ (\\gamma-1)\\over\n 5\\left[10^{2(\\gamma-1)M_{f}\/5}-10^{2(\\gamma-1)M_{b}\/5}\\right]}\\,10^{2(\\gamma-1)M\/5}\\,.\n\\end{equation}\nNote that $\\psi(M)$ includes all luminosity classes.\nIn our analysis the observing magnitude range is $5 \\le K_s \\le 14$ mag.\nThe corresponding distances\nof bright cutoff, faint cutoff and $M_{K_s}=0$ are 3.4 kpc to 216 kpc, 4 pc to\n265 pc and 0.1 kpc to 6.3 kpc, respectively.\n\nAlthough we only use NIR data in the medium and high Galactic latitude regions,\nwe still need to correct possible interstellar extinction. We adopt the new\nCOBE\/IRAS result \\citep{Chen1999} and convert it to the $K_s$ band extinction\nby $A_{K_s}\/E(B-V)=0.367$ \\citep{Schlegel1998}. This extinction model is then\napplied to our simulation data. The extinction values of most of our analyzed\nregions are $A_{K_s}< 0.03$.\n\n\n\n\\section{The Data and Analysis Method}\n\\subsection{The 2MASS Data}\nThe 2MASS Point Source Catalog \\citep[2MASS PSC,][]{Cutri2003} is employed to\ncarry out the $K_s$ band differential star count for the entire Milky Way. We\ndivide the whole sky into 8192 nodes according to level 5 Hierarchical\nTriangular Mesh \\citep[HTM,][]{Kunszt2001}. The level 5 HTM samples the whole\nsky in roughly equal area with an average angular distance about 2 degrees\nbetween any two neighboring nodes. The amount of stars within 1 degree radius\nof each node (i.e., each node covers $\\pi$ square degree) is then retrieved with\na bin size $K_s$=0.5 mag via 2MASS online data service\n\\citep[][catalog]{Cutri2003}. Our selection criterion is: the object must be\ndetected in all $J, H, K_s$ bands and has signal-to-noise ratio $\\ge 5$. Since\nthe limiting magnitude of 2MASS $K_s$ band is 14.3 mag, which has 10\nsignal-to-noise ratio and 99\\% completeness \\citep[see Table~1\nin][]{Skrutskie2006}, and $K_s \\le 5$ mag objects have relatively large\nphotometric error, we only compare 2MASS data with our simulation data from\n$K_s=5$ to $14$ mag.\n\nIn order to minimize the effects coming from the close-to-Galactic-plane fine\nstructures (e.g., flares, warps, arms, budge and bars\\dots etc., which have\nconsiderable contribution to the star count data), and the relatively complex\nextinction correction at the low Galactic latitude region,\nwe avoid low galactic latitudes, and only consider data in Galactic\nlatitude $|b|>30^{\\circ}$. Although several\noverdensities in the halo (such as Sagittarius, Triangulum-Andromeda,\nVirgo\\dots etc.) were identified, they cannot be\npicked up\nfrom the\noverwhelming foreground field stars on star count data without additional\ninformation \\citep[e.g., color, distance, metallicity\\dots\netc.,][]{Majewski2004}. Therefore, their contribution to the 2MASS $K_s$ band\ndifferential star count is negligible and will not affect our result. We also\nexclude the areas around Large and Small Magellanic Clouds for their\nsignificant stellar population.\n\n\n\\subsection{The Analysis Method}\\label{method}\nThe Maximum Likelihood Method (Bienayme et al. 1987) is applied to compare the\n$K_s$ band 2MASS differential star counts and the simulation data to search for\nthe best-fit structure parameters of the three-component Milky Way model. Our\nfitting strategy is as follows:\n\\begin{enumerate}\\label{fitting_steps}\n\\item Take $R_\\odot=8$ kpc;\n\\item choose one $Z_\\odot$ and work out the maximum likelihood value by fitting the 9 parameters\n$(n_0,H_{z1},H_{r1},f_2,H_{z2},H_{r2},f_h,\\kappa,p)$;\n\\item repeat step 2 for other $Z_\\odot$;\n\\item pick the $Z_\\odot$ corresponds to the maximum of the maximum likelihood values in step 3;\n\\item repeat steps 2 to 4 for finer grid size of the 9-parameter fit and a narrower range of\n$Z_\\odot$ around the one found in step 4;\n\\item the uncertainty is estimated by adding Poisson noise on the simulation data to see\nhow the likelihood varies. The difference of the likelihoods of 500\nrealizations of the same model, differed by the Poisson statistics only,\ngives a range of likelihood around the maximum likelihood that defines the\nconfidence level.\n\\end{enumerate}\nTable~\\ref{para_space} lists our searching parameter space and the finest grid\nsize we used. The key parameters in our study are $n_0$, $H_{z1}$, $f_2$ and\n$H_{z2}$ (see Eqs. (1)-(2)). The first two play a primary role on the variation\nof the likelihood value and the latter two play a secondary role. The other\nfive parameters $H_{r1}$, $H_{r2}$, $f_h$, $\\kappa$ and $p$ (see Eqs. (1)-(3))\nare non-key parameters, which play a minor role and do not affect the\nlikelihood value as much as the key parameters.\n\n\\section{The Results and Discussion}\nTable~\\ref{result} lists our best-fit results with the corresponding\nuncertainties. Fig.~\\ref{2Dcon} shows contour plots of likelihood against\ndifferent pairs of parameters. The contour changes dramatically along the key\nparameters $n_0$, $H_{z1}$, $f_2$ and $H_{z2}$, but relatively mild along other\nparameters $H_{r1}$, $H_{r2}$, $f_h$, $p$ and $\\kappa$.\nThis indicates the importance of the key parameters in determining the best-fit result.\n\nDegeneracies exist between some pairs of key parameters, such as $(n_0,\nH_{z1})$, $(n_0, f_2)$, $(H_{z1}, f_2)$\\dots etc. Here degeneracy means that\nthe likelihood value stays almost the same when the pairs of parameters change\ntogether in a particular way. Similar degeneracy between the local\nthick-to-thin disk ratio $f_2$ and the scale-height of the thick disk $H_{z2}$\nhas been reported in \\citet{Chen2001,Siegel2002,Juric2008,Bilir2008}.\nConsequently, it is possible that different combinations of parameters can be\nregarded as `acceptable' fitting. For example, if we choose a higher local\nstellar density $n_0$, then we can pick a smaller $H_{z1}$ such that the\nlikelihood value is very close to the maximum likelihood value and assign it as\nthe `best-match' scale-height of the thin disk. Therefore, a thin light color\ndiagonal strip shows on the $n_0$ against $H_{z1}$ contour plot in\nFig.~\\ref{2Dcon}. Besides, similar trends happen in other pairs of key\nparameters. When one parameter of the pair is higher, we can get a similar\nlikelihood value by lowering the other parameter (see the corresponding\ntwo-parameter contour plots of key parameters in Fig.~\\ref{2Dcon}). This\nanti-correlation is not unexpected. The number of stars along the line of sight\nin the model increases when any one of the key parameters increases. Thus for a\ngiven observed number of stars, an increase in one key parameter can be\ncompensated by a decrease in the other. Perhaps this is the reason that our\nbest-fit scale-height of the thin disk, $H_{z1}=360$ pc, is somewhat larger\nthan the reported values, $H_{z1}=285$ pc, in \\citet{Lopez2002} study (they\nalso use star count of 2MASS to obtain Galactic model parameters). Our best-fit\nlocal stellar density from $K_s=$-8 to 6.5 mag is $n_0\\sim 0.030$ star\/pc$^3$,\nwhich is about half of $\\sim 0.056$ star\/pc$^3$ of the corresponding value\ncited in \\citet{Eaton1984}, and a bit lower than $\\sim 0.032$ star\/pc$^3$ of\nthe corresponding value cited in \\citet{Lopez2002}. As a result, our fitting\ntends to choose a larger scale-height of the thin disk. If we force our local\nstellar density to be comparable with that of \\citet{Lopez2002}, then the\ncorresponding `best-fit' scale-height of the thin disk would be $\\sim$320 pc,\nwhich is closer to their result. If we choose even higher local stellar\ndensity, then the `best-fit' scale-height of the thin disk would be made\nbetween 200 to 300 pc, which is similar to the most of recent studies (see\nTable~\\ref{table_ref}). Moreover, we do not apply binarism correction in our\nanalysis, and it has been shown that scale-length and scale-height might be\nunderestimated without binarism correction \\citep{Siegel2002, Juric2008,\nIvezic2008, Yaz2010}.\n\n\nFor our purpose, we deem that in order to lift the degeneracy it is crucial to\nhave a reliable near infrared luminosity function by observation or a near\ninfrared local stellar density. Unfortunately, a systematic study in this\ndirection is yet to come. Some related studies, such as synthetic luminosity\nfunction \\citep[see e.g.,][]{Girardi2005} or luminosity function transformed\nfrom optical observation \\citep[see e.g.,][]{Wainscoat1992} do exist, but some\nuncertainties still need to be settled (e.g., the initial mass function,\nmass-luminosity relation for NIR and color transformation between different\nwavelengths\\dots etc.). Once the `true' local stellar density is known, the\n`true' structure of the Milky Way would be revealed.\n\n\n\nIn order to see how good the agreement between 2MASS data and our best-fit\nmodel, we show an all sky map of the ratios of observed to predicted integrated\nstar count from $K_s=$5 to 14 mag for each node as a function of position on\nthe sky in Fig.~\\ref{comparison}, and the color indicates the values of the\nratios. We do not see obvious deviation in the Galactic latitude\n$|b|>30^{\\circ}$ areas, but only in the Large Magellanic Cloud and the Small\nMagellanic Cloud areas. For comfirmation, we plot integrated star count from\n$K_s=$5 to 14 mag along Galactic longitude and latitude for $|b|>30^{\\circ}$\nareas in Figs.~\\ref{surf_b} \\& \\ref{surf_l}. We see 2MASS data and our best-fit\nmodel agree well and only some small deviations in the nodes at the\nanti-Galactic center $b\\sim30^{\\circ}$ areas. The significant spikes in\nFigs.~\\ref{surf_b} \\& \\ref{surf_l} are due to the populations of the Large\nMagellanic Cloud and the Small Magellanic Cloud. For testing how these small\ndeviations affect the key parameters selection, we exclude these small\ndeviations and re-analyze the data with fixed non-key parameters. It shows only\nthe scale-height of thick disk shifts slightly but still within our error\nestimation. Besides, we do not see any significant differences of reported\noverdensities in sky regions with $|b|>30^{\\circ}$. Thus, we conclude that the\nthree-component model can describe the Milky Way structure sufficiently well\nfor high Galactic latitude regions and the single power law luminosity function\nof \\citet{Chang2010} is a good approximation as well. Since our main purpose is\nto search for the global Galactic model parameters, we do not try to explore\nthe best-fit result for each node individually as what \\citet{Cabrera2007,\nBilir2008, Yaz2010} have done in their studies to seek the variations of disk\nmodel parameters with Galactic longitude. Instead, we treat the differences of\n2MASS data to our best-fit model as deviations from a global smooth\ndistribution. We believe that similar variations in disk model parameters will\nbe obtained if we take similar analysis procedure (i.e., searching best-fit\nparameters for each node), but this is beyond the scope of this work.\nBecause we do not consider flares, warps and other overdensities in our\nmodel, there are some discrepancies between 2MASS data and the model in the low\nGalactic latitude regions (see Fig.~\\ref{comparison}). These fine structures\nmake the star distribution more fluffy in the vertical direction toward the\nedge of the Milky Way. Hence the discrepancy between 2MASS data and the model\nincreases vertically towards the anti-Galactic center region. In addition, the\nabsence of Galactic bulge in our model contributes to the large discrepancy in\nGalactic center areas. The difference in the low Galactic latitude region needs\nmore delicate analysis to rectify \\citep[see, e.g.,][]{Lopez2002, Momany2006}.\n\n\\subsection{Summary}\nIn summary, we set forth to study the global smooth structure of the Milky Way\nby a three-component stellar distribution model which comprises two double\nexponential disks (one thin and one thick) and an oblate halo. The $K_s$ band\n2MASS star count is used to determine the structure parameters. To avoid the\ncomplication introduced by the fine structures and complex extinction\ncorrection close to Galactic plane, we use only Galactic latitude\n$|b|>30^{\\circ}$ data. There are 10 parameters in the model, but only four of\nthem play the dominant role in the fitting process. They are the local stellar\ndensity of the thin disk $n_0$, the local density ratio of thick-to-thin disk\n$f_2$, and the scale-height of the thin and thick disks $H_{z1}$ and $H_{z2}$.\nThe best-fit result is listed in Table~\\ref{result}. In short the scale-height\nof the thin and thick disks are $360\\pm 10$ pc and $1020\\pm 30$ pc,\nrespectively; the scale-length of the thin disk is $3.7\\pm 1.0$ kpc and that of\nthick disk is $5.0\\pm 1.0$ kpc (the uncertainty in scale-length is large\nbecause it is not very sensitive to high latitude data.) The local stellar\ndensity ratio of thick-to-thin disk and halo-to-thin disk are $7\\pm 1$\\% and\n$0.20\\pm 0.10$\\%, respectively. The local stellar density of the thin disk is\n$0.030\\pm 0.002$ stars\/pc$^3$.\n\nAn all sky comparison of the 2MASS data to our best-fit model is shown in\nFig.~\\ref{comparison}. A good agreement in the Galactic latitude\n$|b|>30^{\\circ}$ areas is expected from our fitting procedure. In low Galactic\nlatitude regions, fine structures (such as flares, warps\\dots etc.) increase\nthe effective scale-height towards the edge of the Milky way. This is reflected\nin the fan-like increase in discrepancy towards the anti-Galactic center\nregions.\n\nDegeneracy (i.e., different combinations of parameters give similar likelihood\nvalues) is found in pairs of key parameters\n(see Fig.~\\ref{2Dcon}). Thus different combinations of parameters may fit the\ndata almost as good as the best-fit one, and these are all legitimate\n`acceptable' fitting in view of the uncertainty. Therefore, accompanying our\nlower local stellar density $0.030$ stars\/pc$^3$ (from $K_s=-$8 to 6.5 mag) is\na higher thin disk scale-height $360$ pc. In the context of NIR star count, the\nNIR luminosity function or the NIR local stellar density is imperative to\ndetermine the scale-height and other Milky Way structure parameters. We hope\nthat systematic study on the luminosity function and the local stellar density\nin near infrared will be available in the near future.\n\n\n\\acknowledgements\nWe acknowledge the use of the Two Micron All Sky Survey Point\nSource Catalog (2MASS PSC).\nWe would like to thank the anonymous referee, whose advice greatly improves the\npaper.\nCMK is grateful to S. Kwok and K.S. Cheng for their\nhospitality during his stay at the Department of Physics, Faculty of Science,\nUniversity of Hong Kong. This work is supported in part by the National Science\nCouncil of Taiwan under the grants NSC-98-2923-M-008-001-MY3 and\nNSC-99-2112-M-008-015-MY3.\n\n\n\\begin{deluxetable}{lllllllll}\n\\tabletypesize{\\tiny} \\setlength{\\tabcolsep}{0.03in} \\tablecaption{Previous\nGalactic models. The parentheses are the corrected values for binarism. The\nasterisk denotes the power-law index replacing Re. References indicating with\nthe original result table mean Galactic longitude or limiting magnitude\ndependent Galactic model parameters. \\label{table_ref}} \\tablewidth{0pt}\n\\tablehead{ \\colhead{$H_{z1}$ (pc)} & \\colhead{$H_{r1}$ (kpc)} & \\colhead{$f_2$\n(\\%)} & \\colhead{$H_{z2}$ (kpc)} & \\colhead{$H_{r2}$ (kpc)} &\n\\colhead{$f_h$(\\%)} & \\colhead{Re(S) (kpc)} & \\colhead{$\\kappa$} &\n\\colhead{Reference}} \\startdata\n310 - 325 & - & 0.0125-0.025 & 1.92 - 2.39 & - & - & - & - & \\citet{Yoshii1982} \\\\\n300 & - & 0.02 & 1.45 & - & - & - & - & \\citet{Gilmore1983} \\\\\n325 & - & 0.02 & 1.3 & - & 0.002 & 3 & 0.85 & \\citet{Gilmore1984} \\\\\n280 & - & 0.0028 & 1.9 & - & 0.0012 & - & - & \\citet{Tritton1984} \\\\\n125-475 & - & 0.016 & 1.18 - 2.21 & - & 0.0013 & 3.1* & 0.8 & \\citet{Robin1986} \\\\\n300 & - & 0.02 & 1 & - & 0.001 & - & 0.85 & \\citet{delRio1987} \\\\\n285 & - & 0.015 & 1.3 - 1.5 & - & 0.002 & 2.36 & Flat & \\citet{Fenkart1987} \\\\\n325 & - & 0.0224 & 0.95 & - & 0.001 & 2.9 & 0.9 & \\citet{Yoshii1987} \\\\\n249 & - & 0.041 & 1 & - & 0.002 & 3 & 0.85 & \\citet{Kuijken1989} \\\\\n350 & 3.8 & 0.019 & 0.9 & 3.8 & 0.0011 & 2.7 & 0.84 & \\citet{Yamagata1992} \\\\\n290 & - & - & 0.86 & - & - & 4 & - & \\citet{vonHippel1993} \\\\\n325 & - & 0.020-0.025 & 1.6-1.4 & - & 0.0015 & 2.67 & 0.8 & \\citet{ReidMajewski1993} \\\\\n325 & 3.2 & 0.019 & 0.98 & 4.3 & 0.0024 & 3.3 & 0.48 & \\citet{Larsen1996} \\\\\n250-270 & 2.5 & 0.056 & 0.76 & 2.8 & 0.0015 & 2.44 - 2.75* & 0.60 - 0.85 & \\citet{Robin1996, Robin2000} \\\\\n260 & 2.3 & 0.074 & 0.76 & 3 & - & - & - & \\citet{Ojha1996} \\\\\n290 & 4 & 0.059 & 0.91 & 3 & 0.0005 & 2.69 & 0.84 & \\citet{Buser1998, Buser1999} \\\\\n240 & - & 0.061 & 0.79 & - & - & - & - & \\citet{Ojha1999} \\\\\n280\/267 & - & 0.02 & 1.26\/1.29 & - & - & 2.99* & 0.63 & \\citet{Phleps2000} \\\\\n330 & 2.25 & 0.065 - 0.13 & 0.58 - 0.75 & 3.5 & 0.0013 & - & 0.55 & \\citet{Chen2001} \\\\\n- & 2.8 & 3.5 & 0.86 & 3.7 & - & - & - & \\citet{Ojha2001} \\\\\n280(350) & 2 - 2.5 & 0.06 - 0.10 & 0.7 - 1.0 (0.9 - 1.2) & 3 - 4 & 0.0015 & - & 0.50 - 0.70 & \\citet{Siegel2002} \\\\\n285 & 1.97 & - & - & - & - & - & - & \\citet[][]{Lopez2002} \\\\\n - & 3.5 & 0.02-0.03 & 0.9 & 4.7 & 0.002-0.003 & 4.3 & 0.5-0.6 & \\citet{Larsen2003} \\\\\n320 & - & 0.07 & 0.64 & - & 0.00125 & - & 0.6 & \\citet{Du2003} \\\\\n265-495 & - & 0.052-0.098 & 0.805-0.970 & - & 0.0002-0.0015 & - & 0.6-0.8 & \\citet[][Table 16]{Karaali2004} \\\\\n268 & 2.1 & 0.11 & 1.06 & 3.04 & - & - & - & \\citet{Cabrera2005} \\\\\n300 & - & 0.04-0.10 & 0.9 & - & - & 3\/2.5* & 1\/0.6 & \\citet{Phleps2005} \\\\\n220 & 1.9 & - & - & - & - & - & - & \\citet{Bilir2006a} \\\\\n160-360 & - & 0.033-0.076 & 0.84-0.87 & - & 0.0004-0.0006 & - & 0.06-0.08 & \\citet[][Table 15]{Bilir2006b} \\\\\n301\/259 & - & 0.087\/0.055 & 0.58\/0.93 & - & 0.001 & - & 0.74 & \\citet[][Table 5]{Bilir2006c} \\\\\n220-320 & - & 0.01-0.07 & 0.6-1.1 & - & 0.00125 & - & $>$0.4 & \\citet{Du2006} \\\\\n206\/198 & - & 0.16\/0.10 & 0.49\/0.58 & - & - & - & 0.45 & \\citet{Ak2007} \\\\\n140-269 & - & 0.062-0.145 & 0.80-1.16 & - & - & - & - & \\citet[][Table 1]{Cabrera2007} \\\\\n220-360 & 1.65-2.52 & 0.027-0.099 & 0.62-1.03 & 2.3-4.0 & 0.0001-0.0022 & - & 0.25-0.85 & \\citet[][Table 3]{Karaali2007} \\\\\n167-200 & - & 0.055-0.151 & 0.55-0.72 & - & 0.0007-0.0019 & - & 0.53-0.76 & \\citet[][Table 1]{Bilir2008} \\\\\n245(300) & 2.15(2.6) & 0.13(0.12) & 0.743(0.900) & 3.261(3.600) & 0.0051 & 2.77* & 0.64 & \\citet{Juric2008} \\\\\n325-369 & 1.00-1.68 & 0.0640-0.0659 & 0.860-0.952 & 2.65-5.49 & 0.0033-0.0039 & - & 0.0489-0.0654 & \\citet[][Table 1]{Yaz2010} \\\\\n360 & 3.7 & 0.07 & 1.02 & 5 & 0.002 & 2.6* & 0.55 & This Work \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\begin{deluxetable}{lcc}\n\\tabletypesize{\\scriptsize} \\tablecaption{The searching parameter space.\n\\label{para_space}} \\tablewidth{0pt} \\tablehead{ \\colhead{} & \\colhead{Range} &\n\\colhead{Grid size}} \\startdata\nThin Disk & \\\\\n\\quad $H_{r1}$ & 1.0-6.0 kpc & 100 pc \\\\\n\\quad $H_{z1}$ & 200-450 pc & 10 pc \\\\\n\\quad $n_0$ & 0.02-0.04 stars\/pc$^3$ & 0.002 stars\/pc$^3$ \\\\\n\\quad $Z_\\odot$ & 11-35 pc & 2.5 pc \\\\\nThick Disk & \\\\\n\\quad $H_{r2}$ & 3.0-7.0 kpc & 200 pc \\\\\n\\quad $H_{z2}$ & 400-1200 pc & 20 pc \\\\\n\\quad $f_2$ & 0-20\\% & 1\\% \\\\\nSpheroid & \\\\\n\\quad $\\kappa$ & 0.1-1.0 & 0.05 \\\\\n\\quad $p$ & 2.3-3.3 & 0.1 \\\\\n\\quad $f_h$ & 0-0.45\\% & 0.05\\% \\\\\n\\enddata\n\\end{deluxetable}\n\n\\begin{deluxetable}{lcc}\n\\tabletypesize{\\scriptsize} \\tablecaption{The best-fit Milky Way model.\n\\label{result}} \\tablewidth{0pt} \\tablehead{ \\colhead{} & \\colhead{Value} &\n\\colhead{Uncertainty}} \\startdata\nThin Disk & \\\\\n\\quad $H_{r1}$ & 3.7 kpc & 1.0 kpc \\\\\n\\quad $H_{z1}$ & 360 pc & 10 pc \\\\\n\\quad $n_0$ & 0.030 stars\/pc$^3$ & 0.002 stars\/pc$^3$ \\\\\n\\quad $Z_\\odot$ & 25 pc & 5 pc \\\\\nThick Disk & \\\\\n\\quad $H_{r2}$ & 5.0 kpc & 1.0 kpc \\\\\n\\quad $H_{z2}$ & 1020 pc & 30 pc \\\\\n\\quad $f_2$ & 7\\% & 1 \\% \\\\\nSpheroid & \\\\\n\\quad $\\kappa$ & 0.55 & 0.15 \\\\\n\\quad $p$ & 2.6 & 0.6 \\\\\n\\quad $f_h$ & 0.20\\% & 0.10\\% \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}