diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmpvs" "b/data_all_eng_slimpj/shuffled/split2/finalzzmpvs" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmpvs" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nWhile most outcomes from the cosmic microwave background (CMB)\nexperiments such as {\\it Wilkinson Microwave Anisotropy Probe (WMAP)}\nand {\\it Planck} satellites agree with the predictions of the\n$\\Lambda$CDM cosmology, some anomalous features in the CMB temperature map persist \\citep{2013ApJS..208...19H, 2014A&A...566A.135G}.\nOne of such anomalies is the CMB Cold Spot centred at $(l,b)=(209^\\circ, -57^\\circ)$ with a temperature decrement $\\Delta T\\sim-100\\mu$K surrounded by a hot ring \\citep{2004ApJ...609...22V, 2005MNRAS.356...29C, 2014A&A...571A..23P, 2016A&A...594A..16P}. \nAlthough a seemingly peculiar 'cold' spot within $\\sim7^\\circ$ from the center is not significant, \na structure of the 'cold' spot surrounded by a hot ring within $\\sim20^\\circ$ is significant at $\\gtrsim3\\sigma$ assuming that the CMB temperature fluctuation is isotropic Gaussian \\citep{2005MNRAS.356...29C, 2010APh....33...69Z, IST10, 2014PhRvD..90j3510N}.\n\nThe anomalous feature can be understood through the effects either on the last scattering surface, or large-scale structures between the last scattering surface and us. \nSome inflation and cosmic texture models could imprint such a feature on the last scattering surface \\citep{2007Sci...318.1612C, 2008MNRAS.390..913C, 2009PhRvL.103g1301B, 2011JCAP...01..019A}.\nOn the other hand, large-scale structures can also generate it. \nThe anomaly can be explained by the linear\/non-linear integrated Sachs-Wolfe (ISW) effect or the Rees-Sciama (RS) effect\n\\citep{1968Natur.217..511R} caused by a supervoid with a radius of $200-300h^{-1}$Mpc in the \nlocal Universe \\citep{2006ApJ...648...23I,2007ApJ...664..650I,Sakai08,Tomita08, IST10} or multiple voids\n\\citep{2016MNRAS.459L..71N}. However, the probability of having such a\nlarge void is low in the $\\Lambda$CDM model and multiple voids cannot explain the feature of a hot ring around the Cold Spot.\n\nMotivated by the theoretical predictions, observational searches for a supervoid in the distribution of galaxies began. \nSpectroscopic surveys and observations with photo-z measurements ruled out a supervoid in the redshift range of $0.350)\\\\\n\\chi &\\hspace{0.5cm}(K=0)\\\\\n\\frac{{\\rm sinh}\\left(\\sqrt{-K}\\chi\\right)}{\\sqrt{-K}} &\\hspace{0.5cm}(K<0).\\\\\n\\end{cases}\n\\end{equation}\n$2 \\pi S_K(\\chi)$ is the circumference of a circle with a radius of $\\chi$ in a space with a constant curvature $K$.\n\nThe lensing shear is related to the convergence as\n\\begin{equation}\n\\gamma\\left(\\bf{\\theta}\\right)=\\frac{1}{\\pi} \\int {\\rm d}\\bm{\\theta'} D\\left(\\bm{\\theta}-\\bm{\\theta'}\\right) \\kappa\\left(\\bm{\\theta}\\right),\n\\end{equation}\nwhere $D\\left(\\bm{\\theta}-\\bm{\\theta'}\\right)$ is \n\\begin{equation}\nD\\left(\\bm{\\theta}-\\bm{\\theta'}\\right) = \\frac{\\theta_2^2-\\theta_1^2-2i\\theta_1\\theta_2}{|\\bm{\\theta}|^4},\n\\end{equation}\nwhere $\\bm{\\theta}=(\\theta_1,\\theta_2)$.\nThe average tangential shear at $\\theta$ is \n\\begin{equation}\n\\langle\\gamma_+\\rangle\\left(\\theta\\right) = \\langle\\kappa\\rangle\\left(<\\theta\\right) - \\bar{\\kappa}\\left(\\theta\\right),\n\\label{eq.tangen}\n\\end{equation}\nwhere $\\langle\\kappa\\rangle\\left(<\\theta\\right)$ and $\\bar{\\kappa}\\left(\\theta\\right)$ are average values of convergence inside a circular aperture and at the edge of the aperture, respectively.\nIn the weak lensing limit ($|\\gamma|\\ll1$ and $|\\kappa|\\ll1$),\ntangential shear can be obtained by measuring galaxy ellipticilities. The galaxy shapes are tangentially (radially) aligned if the tangential\nshear is positive (negative). The tangential shear takes negative values inside a compensating void \\citep{2013MNRAS.432.1021H}. \n\nLensing signals are obtained by taking an ensemble average of background galaxy shapes. \nLight emitted from these galaxies traces matter distributions on the light paths. \nTherefore, uncertainties in shapes of background galaxies, and large-scale structures in front and behind a target have to be considered as measurement errors.\nIn order to estimate the weak lensing detectability of a supervoid taking account of these errors, we define a signal-to-noise ($S\/N$) ratio with these uncertainties as\n\\begin{equation}\n\\left(\\frac{S}{N}\\right)^2=\\sum_{i,j} \\gamma_+(\\theta_i) \\left[\\bm{C}\\right]_{ij}^{-1}\\gamma_+(\\theta_j),\n\\label{eq.sn}\n\\end{equation}\nwhere the indices $i$ and $j$ denote the positions of the bins.\n$\\gamma_+(\\theta_i)$ is a tangential shear at $i$-th bin.\nThe matrix $\\bm{C}$ denotes a covariance matrix and $\\bm{C}^{-1}$ is the inverse matrix.\nIn the estimation of a covariance matrix, we assume that the errors come from the above two components.\nThen it can be written as\n\\begin{equation}\n\\bm{C}=\\bm{C}^{\\rm shape} + \\bm{C}^{\\rm LSS}.\n\\end{equation}\nThe shape noise is estimated by following \\citet{2000MNRAS.313..524V}.\nWe assume the rms amplitude of the intrinsic elliptisity distribution $\\sigma_{\\rm e}=0.4$ \nand the background galaxy density $n_{\\rm g}=10$ arcmin$^{-2}$, which are typical values for weak lensing observations \\citep{2017arXiv170506745M}.\nIn order to estimate the error coming from large-scale structures, we randomly select 200 positions on sky-planes with 20 realisations.\nThen we measure tangential shear values for each selected position and estimate a standard deviation between the profiles.\nWhile we change the number of the profiles for the measurement, the difference in the amplitude of the deviation is $\\lesssim1\\%$ when we use more than 100 random points.\nTherefore, the size of the deviation does not much depend on the number of random points.\n\n\\subsection{void model}\nIn order to model a supervoid, we adopt a compensated Gaussian-like void profile \\citep{2016MNRAS.455.1246F}.\nWe assume that the amplitude of the density contrast is somewhat smaller than 1 and the void radius is sufficiently smaller than the Hubble horizon. \nIn this case, the metric can be treated as a local perturbation in the Friedmann-Lema{\\^i}tre-Robertson-Walker spacetime. \nThen the density profile of the void is described with a parameter $\\alpha$, which characterises the slope of a density profile of a void, as\n\\begin{equation}\n\\delta(a,r)=-\\delta_0g(a)\\left(1-\\frac{2+7\\alpha}{3+3\\alpha}\\frac{r^2}{r_0^2}+\\frac{2\\alpha}{3+3\\alpha}\\frac{r^4}{r_0^4}\\right){\\rm exp}\\left[-\\frac{r^2}{r_0^2}\\right],\n\\label{eq.gaussmodel}\n\\end{equation}\nwhere $\\delta_0$ is a density contrast at the void \ncentre at present and $g(a)$ is a growth factor normalised by the present value. \nFigure~\\ref{fig.denprofile} shows density profiles for the cases of $\\alpha=0, 1$ and 2, respectively.\nThe adopted parameters are $\\delta_0=0.29$, $r_0=198h^{-1}$ Mpc and the redshift of a void $z_{\\rm v}=0.25$. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.9\\columnwidth]{figure\/density_profile.eps}\n\\caption{Density profile of a void model defined with equation~(\\ref{eq.gaussmodel}).\nSolid, dashed and dash-dotted lines indicate profiles with $\\alpha=0,1$ and $2$, respectively.\n$\\delta_0=0.29$, $r_0=198h^{-1}$ Mpc and redshift of a void $z_{\\rm v}=0.25$ are assumed.}\n\\label{fig.denprofile}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{simulation}\nIn order to investigate the properties of supervoids, we used the \npublicly available data of full-sky ray-tracing simulations\\footnote[1]{\\url{http:\/\/cosmo.phys.hirosaki-u.ac.jp\/takahasi\/allsky_raytracing\/}}.\nThe details of the simulations are described in Section~2 in \\citet{2017arXiv170601472T}.\nTo carry out $N$-body simulations, they ran the parallel Tree-Particle Mesh code, Gadget2 \\citep{2005Natur.435..629S} with $2048^3$ particles.\nTo generate lensing maps with different source redshifts, they ran\nsimulations with 14 different box sizes between $450$$h^{-1}$Mpc and $6300$$h^{-1}$Mpc with steps of $450h^{-1}$Mpc. \nThere are 6 simulations with different initial conditions for each box size, which can cover\nthe past light cone of a present observer.\nIn order to generate the initial conditions, \nthe linear matter transfer function was calculated using {\\it CAMB} \\citep{2000ApJ...538..473L}.\n \nRay-tracing simulations were carried out by \\citet{2001MNRAS.327..169H, 2015MNRAS.453.3043S}.\nThe light-ray path and the magnification on lens planes were calculated with the multiple-lens plane algorithm.\nIn order to increase the number of realisations, they chose 18 different positions in each simulation box with the periodic boundary condition.\nThey used the $N$-body simulations to generate projected mass density shells with a width of $150$$h^{-1}$Mpc.\nThe lensing information was calculated for 38 different source redshifts between 0 and 5.3.\nIn order to map lensing signals on a sky-plane, {\\it HEALPIX}\n\\citep{2005ApJ...622..759G} was used to create lensing maps with three\ndifferent resolutions, $N_{\\rm side}=4096$, $8192$ and $16384$.\n\nIn each output of the $N$-body simulations, \nthey created dark matter halo catalogues using a software {\\it ROCKSTAR} \\citep{2013ApJ...762..109B}, which uses the information in the phase space. \nA halo is defined to be a group of more than 50 particles. The adopted\nhalo masses are $M\\geq4\\times10^{10}h^{-1}$M$_\\odot$. An algorithm \nin {\\it HEALPIX} was also used for defining the positions of haloes on the celestial sphere.\n\nSince we focus on the lensing properties of supervoids on large scales, small scale \nstructures residing in each void are not important for our analysis. \nTherefore, we used simulations with an angular resolution parameter $N_{\\rm side}=4096$, which corresponds to the resolution $\\Delta\\theta=\\sqrt{4\\pi\/(12N_{\\rm side}^2)}=0.85$ arcmin. \nThe amplitude of lensing signals depends on the redshifts of a source galaxy and a lens object.\nSince the purpose of this paper is to investigate the detectability of\nweak lensing signals in the on-going and near-future weak lensing surveys, \nwe chose a lens map with a source redshift $z_{\\rm s}=0.508$, which is a\ntypical redshift value for such wide surveys, from the discrete source\nredshift planes in the simulations. We used a virial mass as the\ndefinition of the mass of a halo.\n\n\\subsection{void finding argorithm}\nIn order to find voids with haloes, we ran the publicly available VoidFinder code (see details in \\citealt{2009ApJ...699.1252F, 2013MNRAS.432.1021H}).\nIn the VoidFinder, the space is divided into cells and then \na sphere is enlarged with respect to a cell until the edge of the sphere reaches the third nearest halo.\nThe spheres are sorted by the radii in decreasing order and marginalized\nwith larger spheres with a certain criterion.\n\nIn our analysis, we used haloes with a mass $M_{\\rm vir}\\geq2\\times10^{14}h^{-1}{\\rm M}_\\odot$ in the redshift of $0\\leq z\\leq0.55$.\nThe processes of running the VoidFinder are quite time consuming, especially for the cases in which low mass haloes are used for full-sky simulations.\nSince smoothing convergence maps use only two dimensional information, the computation time can be significantly reduced. \nAs a first step, therefore, we look for underdence regions by smoothing convergence maps.\nThen we ran the VoidFinder in a part of the regions in each simulation. \nWhen running the VoidFinder, we used only haloes within $120\\times120$ degree$^2$ on a sky-plane which is centred on a smoothed convergence map (see Section~\\ref{sec.result}).\nThe adjustable dimensionless parameter and size parameter in the\nVoidFinder are chosen as $\\lambda=2.0$ and $\\xi=20h^{-1}$Mpc, which are\ndefined by equation~(5) in \\citet{2009ApJ...699.1252F}. We note that the number\nof relatively large voids does not much depend on these parameter values (see in \\citealt{2013MNRAS.432.1021H}).\nIn order to select legitimate voids, we exclude voids whose distance\nbetween the void centre and the edges of a selected region is less than the void radius. \n\n\\section{Result}\n\\label{sec.result}\n\n\\begin{figure*}\n\\subfigure{\\includegraphics[width=0.85\\columnwidth,bb=0 0 711 769]{figure\/smoothed_convergence_map_r024_tg=20_peak.eps}}\n\\subfigure{\\includegraphics[width=1.0\\columnwidth,bb=0 0 720 579]{figure\/void_position_on_sky_run024_z=0.25-0.5.eps}}\n\\caption{{\\it Left:} The convergence map smoothed at a scale of $\\theta_{\\rm g}=20$ degree. The source redshift is $z_{\\rm s}=0.508$ and the field of view is $100\\times100$ degree$^2$.\n{\\it Right:} The positions of voids with respect to the negative\n convergence peak in the redshift range of $0.1\\leq z\\leq0.45$.\nThe voids with a radius $R \\geq 90h^{-1}$ Mpc are shown.\nThe horizontal and vertical axes show the coordinates in the simulation. \nThe size of a circle shows the angular radius of a void on the sky-plane. \nThe colours indicate the redshifts estimated by the VoidFinder. \nThe centre of the figure corresponds to the convergence peak in the left panel.}\n\\label{fig.conv_void}\n\\end{figure*}\n\nSince running the VoidFinder for all of the full-sky simulations is time consuming, \nwe used two dimensional convergence maps for finding a most prominent underdense region whose apparent angular size is similar to that of the Cold Spot.\nWe used full-sky map lensing simulations with 6 different initial conditions at 18 different observational points, which correspond to $6\\times18=108$ realizations.\nWe used full-sky convergence maps without shape noise and we smoothed the maps with a Gaussian filter with a smoothing scale of $\\theta_g={\\rm FWHM}\/2\\sqrt{2{\\rm ln}2}=20$ degrees.\nThis is the same value used in \\citet{2016MNRAS.455.1246F}.\nThe left panel in Figure~\\ref{fig.conv_void} shows a smoothed convergence map centred at the largest negative convergence peak.\nThe size of the figure is $100\\times100$ degree$^2$ with a pixel scale of $1.5$ arcmin. \nFinding the position of a maximum of a potential in a large underdense region is not an easy task in real observations.\nThe size of the smoothing scale would shift the peak of the potential and weaken the amplitude of the tangential shear.\nHowever, the position of the largest negative convergence peak is not much affected by\nthe smoothing scale if larger than several tens of degrees.\n\nIn order to measure the properties of voids, we ran \nthe VoidFinder by selecting haloes within $\\sim15,000$ degree$^2$ around\nthe largest negative peak in the smoothed convergence map.\nWe used haloes with a mass of $M\\geq2\\times10^{14}h^{-1}{\\rm M}_\\odot$ in the redshift range of $0\\leq z\\leq0.55$.\nThe average distance between haloes is $\\sim60h^{-1}$Mpc.\nThe right panel in Figure~\\ref{fig.conv_void} shows the positions of\nvoids projected onto the sky-plane. The sizes of the circles and colours\nrepresent the sizes of the voids on the sky-plane and their redshifts, respectively.\nThe centre of the figure corresponds to the convergence peak.\nThe void residing at $(\\Delta\\phi, \\Delta\\theta)\\sim(-2^\\circ, 15^\\circ)$ is the\nlargest one at $z=0.438$ with a radius $r=228~h^{-1}$Mpc.\nAs shown in Figure~\\ref{fig.conv_void}, smaller voids are clustering in front and behind the largest void.\nIn order to estimate the rarity of the underdense region that consist of several tens of voids, \nwe use an 'occupation ratio' defined as a ratio of the total volume occupied by voids to the cosmic volume within a certain angular radius in the redshift range of $0.2 \\leq z \\leq 0.5$. \nTo calculate the average occupation ratios, we selected haloes with a mass of $M\\geq2\\times10^{14}h^{-1}{\\rm M}_\\odot$ from 10 realisations and ran \nthe VoidFinder in a half region of a hemisphere in each realisation.\nWe checked that the number of realisations does not largely affect the significance.\nWe used the Monte Carlo method to calculate the volumes of voids. \nWhen the centre of a void falls within a certain angular distance from \nthe position of the largest negative convergence peak, the volume of the\nvoid is added to the one occupied by voids. \nThe average and the standard deviation of the occupation ratio were\nobtained from values centred at randomly selected 3 points on a\nsky-plane for each realisation, i.e, the total number of random points are $3\\times10=30$ for 10 realisations.\nTable~\\ref{tab.aven} shows the results for 5 different choices of the\nradial distance $r$.\nWe found that the occupation ratio around the largest negative convergence peak shows an excess for $r=25^\\circ - 30^\\circ$. \nAlthough the largest void occupies the most volume at the relevant redshift range, \nthe excess cannot be solely explained by the largest void itself because the angular radius is just $\\sim 15$ degrees.\nVoids in {\\it Void-in-void} mode are preferentially clustered \\citep{2004MNRAS.350..517S,2013MNRAS.434.1435C,2017MNRAS.468.4822L}. \nOur result is consistent with the previous results, and the observed large dip in the smoothed convegence map was created by both the largest and surrounding smaller voids. \n\n\\begin{table}\n\\caption{The void occupation ratios in the vicinity of the largest negative convergence peak in the redshift range of 0.2-0.5.\nThey are estimated from the volumes of voids whose centre is within a certain radius from the peak. \nThe VoidFinder is run in a half region of a hemisphere for 10 realisations.\nThe average and the standard deviation were calculated from 30 samples of 'field' points. \nThe volumes are estimated by the VoidFinder.\nColumn (1): angular radius , Column (2): average and standard\n deviation around 30 'field' points, Column (3):\n ratio around the largest negative convergence peak. }\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\nradius & 30 'field' points & peak\\\\ \\hline\\hline\n$r=25^\\circ$& $0.312\\pm0.024$ & $0.584$\\\\\n$r=28^\\circ$& $0.306\\pm0.021$ & $0.466$\\\\\n$r=30^\\circ$& $0.312\\pm0.019$ & $0.508$\\\\\n$r=33^\\circ$& $0.304\\pm0.013$ & $0.573$\\\\\n$r=35^\\circ$& $0.305\\pm0.017$ & $0.602$\n\\end{tabular}\n\\end{center}\n\\label{tab.aven}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[width=1.\\columnwidth]{figure\/tangential_shear_profile_peak_run024_dbin=1deg_paper_rev2.eps}\n\\caption{The tangential shear profiles for a supervoid.\nThe horizontal axis shows the distance from the largest negative convergence peak. \nThe vertical axis shows the tangential shear. \nGreen points show the profile in the simulation.\nThe grey shaded line shows the average tangential shear profile with respect to 100 randomly selected points within $5$ degrees from the peak. \nThe shaded region indicates the $1\\sigma$ standard deviations. \nThe line with triangles and dotted line (light blue) show the errors from shape noise and large-scale structures, respectively. \nA galaxy number density $n_{\\rm g}=10$ arcmin$^{-2}$, an intrinsic ellipticity dispersion $\\sigma_{\\rm e}=0.4$ and a source redshift $z_{\\rm s}=0.508$ are assumed. \nSolid, dashed and dash-dotted lines show the tangential profiles derived from equation~(\\ref{eq.gaussmodel}) for $\\alpha=0,1$ and 2, respectively. \n$\\delta_0=0.29$, $r_0=198h^{-1}$Mpc and $z_l=0.22$ which were obtained in \\citet{2016MNRAS.455.1246F} are adopted for the model.\nWe assume that the position of the peak correspond to the centre of the void. }\n\\label{fig.shearprofile}\n\\end{figure}\n\nThe analytical tangential shear profile for one single supervoid can be \ncalculated from equation~(\\ref{eq.con}), (\\ref{eq.tangen}) and (\\ref{eq.gaussmodel}). \nThe solid, dashed and dash-dotted lines in Figure~\\ref{fig.shearprofile} show the analytical profiles for different $\\alpha$ values\nby assuming the parameters in \\citet{2016MNRAS.455.1246F}.\nThe line with triangles shows the error from shape noise assuming a galaxy number density $n_{\\rm g}=10$ arcmin$^{-2}$.\nThe dotted line (light blue) shows the errors from large-scale structures.\nAs shown in Figure~\\ref{fig.denprofile}, the shear profiles do not show large differences in the inner region for the three different void profiles.\nHowever, the differences in the steepness of the density profile in the ridge region can generate the differences in the lensing signals in that region.\nThe green points in Figure~\\ref{fig.shearprofile} shows the tangential shear profile in our simulation as a function of distance from the largest negative convergence peak.\nAs noticed, the position of the peak in the smoothed convergence map is not much affected by the smoothing scale if larger than several tens of degrees. \nIn order to estimate the model variability due to ambiguity in the choice of the coordinate centre, however, \nwe measured the tangential shear profile for 100 randomly selected centres within 5 degrees from the peak in the smoothed convergence map. \nThe grey shaded line in figure 3 shows the average tangential shear\nprofile at 100 random points with the $1\\sigma$ standard deviation.\nWe divided the angular distance from 0 to 35 degrees into 35 bins, i.e., the bin size $\\Delta\\theta$ is 1 degree.\nThe source redshift is set to $z_{\\rm s}=0.508$ for the analytical and simulated profiles.\nAlthough we measured the tangential shear profiles with different bin sizes, \nit turned out that the shear profiles do not much depend on the bin size. \nThe estimated signal-to-noise ratio for the tangential shear at angular\ndistances 10 to 35 degrees from the centre is turned out to be $S\/N\\sim 3$. \nThe result indicates that the lensing signals from a locally\nunderdence region can be marginally detected with weak lensing depending on the scale of mass deficiency.\nIn our method, the shear is measured as a function of a distance from the centre of a void.\nIf the shear is measured as a function of a distance from a void boundary, the lensing signal can be enhanced by a factor of two \\citep{2016MNRAS.457.2540C}.\nSince the amplitude of the average tangential shear at the wall of a void is a measure of the average density contrast inside the wall,\nan increase in the amplitude of density contrast at the void centre causes an increase in the amplitude of the average shear, and an increase in the void radius causes an increase in the distance of the negative peak position of the shear measured from the void centre.\n\n\nFigure~\\ref{fig.densdis} shows the density contrast as a function of the\nredshift (see \\citealt{2015MNRAS.450..288S}). \nIn our simulation, haloes with a mass of $M\\geq3\\times 10^{13}h^{-1}$M$_\\odot$ were used to measure\nthe number density at each redshift bin within a certain radius with\nrespect to the largest negative convergence peak. \nThe errors were assumed to be Poissonian. \nIn order to reduce the errors, lower mass haloes were used for this analysis.\nHowever, the results with a threshold of $M\\geq3\\times 10^{13}h^{-1}$M$_\\odot$ and $M\\geq2\\times 10^{14}h^{-1}$M$_\\odot$ are consistent with each other within $2\\sigma$.\nThe average number density at each redshift bin was estimated by using the rest of haloes in the simulation.\nTo investigate the inner density profile of the largest void, we divided the redshift range\n$0\\leq z\\leq0.6$ into 30 bins, i.e., the bin size is $\\Delta z=0.02$.\nAs shown in figure 4, we can see a large dip at $z\\sim 0.36$\nand a small dip at $z \\sim 0.23$ if smoothed within $20^\\circ$ from the\nnegative convergence peak. Since any large underdence regions are\nassociated with multiple voids, such a clustering of multiple voids in\nthe line of sight may be consistent with a recent spectroscopic\nobservation toward the Cold Spot\\citep{2017arXiv170403814M}. \n\n\\begin{figure*}\n\\hspace{2cm}\n\\subfigure{\\includegraphics[width=1.5\\columnwidth]{figure\/redshift_density_run024_convergence_peak_nbin=20_rev2.eps}}\n\\caption{The density contrast towards the largest negative convergence peak as a function of redshift without and with photo-z errors.\nThe horizontal axis shows the redshift and vertical axis shows the density contrast.\nThe circles and squares are the density contrast within a radius of $5$ and $20$ degrees without photo-z errors.\nThe crosses show the density contrast with photo-z errors, respectively.\nThe results for the third, fifth and eighth bins with a radius of 5 degree are out of the plotted range.\nThe density contrast was obtained by using haloes within a certain radius on a sky-plane from the negative convergence peak. \nThe red dotted and green dashed lines show the best fitted curves with the result of a radius of 20 degrees for the underdensities exist at z=0.23 and 0.36, respectively.\nWe put the void model at redshifts of $z=0.25$ and $0.35$ and fitted the data without photo-z errors assuming equation~(\\ref{eq.gaussmodel}), respectively. \n$\\delta_0$, $r_0$ and z are the free parameters in the fitting.\nThe error bars show Poissonian noises. \nThe standard deviation of the photo-z errors is assumed to be $\\sigma_z=0.035(1+z_{\\rm halo})$.}\n\\label{fig.densdis}\n\\end{figure*}\n\n\\begin{table*}\n\\caption{The fitting results with the void model for the density\n profiles at low and high redshifts with a radius of 20 degrees. \nThe fitting was carried out by putting the void model at the redshift of $z=0.25$ and $z=0.35$, respectively. \nThe errors indicate $1\\sigma$.\nThe photo-z errors are randomly added to the redshifts of haloes\n assuming a Gaussian distribution for the errors.\nColumn (1): deviation of photo-z error, Column (2)-(4): fitted values (density contrast, radius and redshift) defined in equation~(\\ref{eq.gaussmodel}).}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n$\\sigma_z\/(1+z_{\\rm halo})$ & $\\delta_0$ & $r_0$ [$h^{-1}$Mpc] & $z_{l}$\\\\ \\hline\\hline\nlow-z underdensity&&&\\\\ \\hline\n0 & $0.1632\\pm0.0044$ & $63\\pm2$ & $0.2315\\pm0.0005$\\\\\n0.035 & $0.0854\\pm0.0014$ & $58\\pm1$ & $0.2290\\pm0.0003$\\\\\n\\hline\\hline\nhigh-z underdensity&&&\\\\\n\\hline\n0 & $0.0895\\pm0.0024$ & $151\\pm4$ & $0.3552\\pm0.0013$\\\\\n0.035 & $0.0674\\pm0.0059$ & $208\\pm2$ & $0.3626\\pm0.0044$\\\\\n\\end{tabular}\n\\end{center}\n\\label{tab.fit}\n\\end{table*}\n\nWe added photo-z errors to the redshifts of haloes \nand investigated the differences of parameters by fitting the density profile.\nTo take into account the photo-z errors, we randomly added \nGaussian errors to the redshifts of haloes. The error value was assumed\nto be $\\sigma_z\/(1+z_{\\rm halo})=0.035$ as in \\citet{2015MNRAS.450..288S}.\nTable~\\ref{tab.fit} shows the fitting results in the cases with and without photo-z errors.\nWe fitted the each underdensity which exist at $z\\sim0.23$ and $z\\sim0.36$ with the void model defined by equation~(\\ref{eq.gaussmodel} ), respectively.\nThe amplitude of density contrast decreases if photo-z errors were added to the data; Structures such as voids, filaments and walls are smoothed out.\nIn \\citet{2015MNRAS.450..288S}, only the density contrast was used to study the void properties. \nThey found that the values of the density contrast measured with smaller radius are higher than that with a larger radius due to the averaging process.\nRecently, \\citet{2017arXiv170403814M} analysed the data with spectroscopic samples within the inner $5$ degrees of the Cold Spot. \nThey found a few voids with a radius of $50-170h^{-1}$Mpc in the redshift range of $0\\leq z\\leq0.5$.\nAlthough the numbers of their samples and the area are small, their results seem to be consistent with our simulation result.\nThe errors in photo-z mitigate small substructures in supervoids which can be seen in spectroscopic observations.\nMoreover, reconstruction of three dimensional mass distributions from weak lensing information is not an easy task \\citep{2001astro.ph.11605T,2002PhRvD..66f3506H,2003MNRAS.344.1307B}.\nA combination of photometric and spectroscopic observations enable to measure precise matter distribution toward supervoids.\n\nAssuming the best-fitted supervoid model in \\citet{2016MNRAS.455.1246F},\nit is effective to observe an annulus region around $\\theta=20$ degrees for detecting the lensing signals \nsince the tangential shear of an underdence region is maximized at $\\theta=20$ degrees on a sky-plane with respect to the centre of the void.\nCompared with the fitting results, in addition, the measured density\ncontrast in \\citet{2016MNRAS.455.1246F} is $1 - 5$ times larger than that in our simulation. \nSince the tangential shear is proportional to the density contrast, and the amplitude of it depends on the redshifts of source galaxies and a supervoid, \nthe lensing signals can be detected at $S\/N\\gtrsim4$ at an angular distance between 10 and 35 degrees in on-going and future\nweak lensing observations.\n\n\n\\section{Conclusions}\n\\label{sec.con}\nWe have studied the expected weak lensing signal from a possible large underdence region towards the Cold Spot.\nIn order to do so, we have used all-sky ray-tracing simulations in a\nconcordant $\\Lambda$CDM model consistent with the WMAP 9yr results. In\norder to find a most prominent underdense region that can contribute to \nthe Cold Spot via the integrated Sachs-Wolfe effect, \nwe smoothed the obtained convergence maps at a scale of $\\sim 20$ degrees.\nThen we ran the VoidFinder in the region for investigating the properties and measured the tangential shear profile around the largest negative peak in the convergence maps. \nIn our simulation, the signal of the negative tangential shear in the locally underdense region expected in a concordant $\\Lambda$CDM model can be detected at $S\/N\\sim3$. We found that a number of voids is\nclustering around the negative convergence peak. \nIf a single supervoid with a radius of $\\sim 200\\,h^{-1}\\,\\textrm{Mpc}$ and a density contrast $\\delta\\sim -0.3$ resides at a redshift $z\\sim 0.2$ as\nobservationally suggested in \\citet{2016MNRAS.455.1246F}, the lensing signal can be detected at $S\/N\\sim4-10$ without resorting to stacking.\n\nWeak lensing measurements can test the existence of a large underdence region toward the Cold Spot\nand the shape of the tangential shear profile can give a direct measurement for the steepness of the matter density in the ridge region of the supervoid.\nBy combining weak lensing measurements, on-going surveys such as {\\it Subaru\/Primary Spectroscopic Survey}\n\\citep{2012SPIE.8446E..0YS, 2014SPIE.9147E..0TS, 2016SPIE.9908E..1MT}\nwould certainly benefit us for investigating a three dimensional matter distribution in the local Universe to the Cold Spot. \n\n \n\\section*{Acknowledgements}\nWe thank an anonymous referee for giving useful comments and improving the manuscript. \nWe thank C. Foster for the use of the VoidFinder.\nWe also thank R. Takahashi, T. Hamana and M. Shirasaki for carrying out the simulations.\nWe would like to thank K. Umetsu, I. Chiu, T. Okumura, Y.Toba for useful comments and discussions.\nYH is supported by ASIAA, Taiwan. \nNumerical computations presented in this paper were in part carried out on the general-purpose PC farm at Center for Computational Astrophysics, CfCA, of National Astronomical Observatory of Japan.\nData analyses were (in part) carried out on common use data analysis computer system at the Astronomy Data Center, ADC, of the National Astronomical Observatory of Japan.\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOptical Character Recogition (OCR) is the problem of reading and extracting information from an image. In order to extract correct information in the image, multiple level detection were made. They range from character-level detection to paragraph level detection. Such set of document structuring problems is called Document Layout Analysis (DLA). The goal of DLA is to decompose an image of document into multiple parts, whose information are then further processed or filtered based on their usefulness. \n\nWith the power of deep learning, DLA can solve the detection problems by using a good object detection model, such as Faster RCNN \\cite{fasterrcnn}, or Mask RCNN \\cite{he2017}. Hence, the problems left is how to obtain a good dataset to train such object detection model. \\cite{Gupta16, long2020unrealtext} proposes ways to generate synthetic scene text image. Such synthetic datasets yield good variety of scene text image with text blending into background. However, we cannot use those datasets directly to train a model for decomposing a document image into level component that are bigger than word-level, since there are differences in distribution between document text and scene text data.\n\nOn the other hand, \\cite{zhong2019publaynet} provided a way to process documented data from the Internet and provided their resulted PublayNet dataset of more than 300000 images and corresponding label. However, PublayNet dataset has the limitation on label-level (only big component such as paragraph, table, list), and limitation on language domain (only in English). \n\nDespite the fact that there are many open-source for both scene-text dataset, at the time of this paper, there is no dataset that covers a full level of label from character up to big page component like paragraph. There is a need for such data providing an end-to-end solution, where all components ranging from smallest like character to biggest like paragraph would be covered thoroughly. In addition, there is also a need for data providing flexibility in languages, especially low-resource language where collecting big dataset is not feasible.\n\nIn this work, we are going to provide a new method to generate synthetic data, with the main focus on multi-level text component, and possible to generate in multiple languages. In short, our main contributions are:\n\\begin{itemize}\n \\item First algorithm to generate synthetic visual dynamic document layout,\n \\item An algorithm that fits in multiple languages document, even low-resource language.\n \\item First big document dataset in Vietnamese with biggest number of images and biggest number of component levels, ranging from character-level to paragraph-level.\n\\end{itemize}\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{0.jpg}\n \\includegraphics[width=0.45\\textwidth]{1.jpg}\n \\caption{Sample images from the dataset. Left: Flexible layout; Right: Fixed columns layout.}\n \\label{fig:fig1}\n\\end{figure*}\n\\section{Related Work}\nThere are two main approaches on gathering big datasets for document layout or any text document dataset in general. The first set of approach is to generate synthetic data that is close to real data that we are going to use. The second set of approach is to crawl real data and either label it manually or automatically.\n\\subsection{Synthetic Text dataset}\nThroughout the years, as deep learning becomes the mainstream in many tasks including optical character recognition, large amount of text images data for training deep model has become more and more important. However, there are many low-resource languages where we cannot crawl much of the data from open source like the Internet. Hence, synthetically generating text dataset comes naturally as an inevitable option for those languages. As far as we know, there are several attempts to generate synthetic data for optical character recognition.\n\n\\cite{Gupta16} provided the first synthetic scene text dataset, with most number of images. The datasets help a lot in pretraining a good model for further fine-tuning on many text detection dataset. In addition, it is well noted that many people used \\cite{Gupta16} open-source code to generate their own custom dataset in multiple languages. It is noted that up to this day, there is no real dataset that can match the number of data instances that SynthText provided as well as the capacity it provided to generate text in multiple languages. \n\nAnother approach by \\cite{long2020unrealtext} combined the power of 3D Unreal Engine in generating persuasive text in the wild data. The data were surreal and can be useful in task like scene-text recognition. It also provides code that people can generate their own custom data and use for many languages.\n\nThere are also frameworks that focuses on generating synthetic text document. However, to the best of our knowledge, there has been no visual synthetic dataset made for document prior to ours.\n\n\\subsection{Document Layout dataset}\n\nThere are several open source dataset on document layout. They range from detecting multiple components of the document at the same level like {\\it paragraph, table, list, etc.} to small text level such as word. For example, \\cite{inbook} provided AR IIIT13k focusing on table detection, but contain several other document components. [author] provided DocBank, claiming the largest dataset that contains both document visual information and textual information. PublayNet [reference] is another dataset that is widely used for pretraining document layout detection due to its utility and usage of real data. \n\nThere are, however, disadvantages behind such data collecting approaches. First, the data is domain limited. For example, DocBank and Publaynet works well only on dataset that are close to medical document or research document when they do zero-shot detection. Second, crawling data is not possible if the targeted language and dataset are either low-resource or private resources. It means that for data whose visual are far from English like Arabic or Thai, those datasets are not very effective.\n\\section{Methods}\n\\label{sec:methods}\nWe aimed at filling the document with 5 forms of document components: \"Title\", \"Paragraph\", \"Table\", \"Image\", \"Formula\". Thus, we divided our algorithm into a 2-step pipeline. First, we generate random math formula and crawl image data for \"Formula\" and \"Image\" class of component. Second, we use one of the two algorithms discussed later to divide a whole page into multiple regions to fill document components into. Finally, we use Pillow library to fill text and fill image component into a background paper. It would result in a large number of document. \nWe provided two options for data generation. The first option is to generate document with layout that are divided into columns. The second option is to make document with more flexibility: we do not have any limitation on either rows or columns. [Figure \\ref{fig:fig2}]\n\n\n\n\\subsection{Fixed columns layout algorithm}\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{Slide1.jpg}\n \\caption{Layout generation algorithm illustration. Up: fixed column; Down: Flexible layout}\n \\label{fig:fig2}\n\\end{figure*}\n\\begin{algorithm}[H]\n\\SetAlgoLined\n\\KwResult{Images and Labels}\nInitialize numColumns;\n\n \\While{generating}{\n {\\it randomize} n in range(1,numColumns);\n \n {\\em divide} a page into n columns;\n \n \\For{each column}{\n {\\it randomize} column break position;\n \n {\\it divide} the column by break points;\n \n fill text component into divided regions;\n }\n}\n \\caption{Fixed layout algorithms}\n \\label{fig:fig3}\n\\end{algorithm}\n\nWith this option, we focus on the range of document whose layouts are only divided into specific number of columns. This was inspired by the style that many research document was presented in. By training the detection algorithm on this generated dataset, we can tell the algorithm to separate paragraph in vertical way even though some of the text columns can be close to each other. In addition, such dataset distribution would have a similar distribution to PublayNet. Thus, we can be use it as an alternative version of PublayNet on low-resource languages.\n\n\\subsection{Flexible layout generation algorithm}\nThe second option is to develop an algorithm that can freely divide a page into multiple sections\/ regions, and fill many information in such regions. Due to the wide variety of document data structure in the wild, this option helps model to explore a wider set of data, thus can represent document model in a better manner. Experiments on this was done and the differences of pre-training models on flexible layout SDL and pre-training models on other limited domain dataset would be shown in the next section.\n\\begin{algorithm}[H]\n\\SetAlgoLined\n\\KwResult{Images and Labels}\n Initialize minimal area of components;\n \n \\While{component area not too small}{\n {\\it randomize} direction to divide;\n \n {\\it divide} component area into 2 parts along the direction;\n}\\ \n\\For{each divided region}{\n fill text component into the region;\n}\n \n \n \\caption{Flexible layout algorithm}\n \\label{fig:fig4}\n\\end{algorithm}\n\n\n\\paragraph{Implementation details}\nFor the Vietnamese synthetic document dataset, we use fontsize ranging from 18 to 31, with the height and width of the image ranging from 1500 to 2500. The text color in the dataset are mainly black with a small variance. The text are filled into 5 different categories: Paragraph, Title, Table, Figure, and Formula. For Paragraph, Title and Table component, text annotations are divided into 4 different levels: component-level, line-level, word-level and character-level.\n\n\\section{Future work}\nThere are several ways that we would like to proceed with our data generation tool. First, multi-level text detection models such as TextFuseNet \\cite{ijcai2020-72} or CRAFT \\cite{baek2019character} can be implemented on our dataset to get a better weight on document model. Second, we can develop a light-weight segmentation model that would take advantages of a multi-level annotations in our dataset. Finally, we would like to have a collaboration on generating dataset in multiple languages other than Vietnamese or English, which would bring benefits to more people who are using low-resource languages.\n\n\n\\bibliographystyle{unsrt} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\subsection{The model} \nKeller and Segel \\cite{Keller_Segel_1970, Keller_Segel_1971} brought in the early 70's the first model governing chemotaxis growth at macroscopic level. It consists of a system of partial differential equations of parabolic type, which is to be solved for the organism and chemoattractant densities as follows. Let $\\Omega\\subset \\mathds{R}^2$ be an open, bounded domain, with $\\boldsymbol{n}$ being its outward-directed unit normal vector to $\\Omega$, and let $[0,T]$ be a time interval. Take $Q=(0,T]\\times \\Omega$ and $\\Sigma=(0,T]\\times\\partial\\Omega$. Then find $n: \\bar Q\\to (0,\\infty)$, the organism density, and $c:\\bar Q \\to [0,\\infty)$, the chemoattractant density, satisfying \n\\begin{equation}\\label{KS}\n\\left\\{\n\\begin{array}{rcll}\n\\partial_t n-\\Delta n&=&-\\nabla\\cdot(n\\nabla c)&\\mbox{ in } Q,\n\\\\\n\\partial_t c -\\Delta c&=&n-c&\\mbox{ in }Q,\n\\end{array}\n\\right.\n\\end{equation}\nsubject to the no-flux boundary conditions\n\\begin{equation}\\label{BC_KS}\n\\nabla n\\cdot \\boldsymbol{n}=0\\quad\\mbox{ and }\\quad \\nabla c\\cdot\\boldsymbol{n}=0\\quad\\mbox{ on }\\quad \\Sigma,\n\\end{equation} \nand the initial conditions \n\\begin{equation}\\label{IC_KS}\nn(0)=n_0\\quad\\mbox{ and }\\quad c(0)=c_0\\quad\\mbox{ in }\\quad \\Omega.\n\\end{equation}\n\n\nThis model exhibits many interesting properties. For instance, solutions to this model can be found, whenever $\\int_\\Omega n_0({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}\\in(0,4\\pi)$, that remain uniformly bounded for all time \\cite{Nagai_Senba_Yoshida_1997}. On the contrary, if $\\int_\\Omega n_0({\\boldsymbol x})$ is larger than such a threshold value, the situation changes drastically; there exists solutions blowing up either in finite or infinite time \\cite{Horstmann_Wang_2001, Senba_Suzuki_2001}\\footnote{Here one needs $\\Omega$ to be simply connected.}. Furthermore, this threshold turns out to be critical in the sense that, for each $\\varepsilon>0$, there exist $n_0$ with $\\int_\\Omega n_0({\\boldsymbol x})\\, {\\rm d}{\\boldsymbol x}>4\\pi +\\varepsilon$ that develop a finite-time collapse into persistent Dirac-type measures.\n\nWhen chemotaxis occurs in a fluid, the original Keller--Segel equations need to be coupled with a Navier--Stokes type of equations. In this context, the fluid flow will be influenced by the self-enhanced chemotactic motion inducing a velocity profile; and more importantly, the converse is further true, a change in the velocity profile will accordingly alter the chemotactic growth. This implies that, if the initial fluid velocity is zero, the evolution of the chemotactic growth will induce a velocity, and this velocity will in turn affect the behaviour of the \nchemotactic growth. \n\nThe Keller--Segel--Navier-Stokes equations for governing the chemotaxis of unicellular organisms under the influence of a fluid flow are written as \n\n\\begin{equation}\\label{KSNS}\n\\left\\{\n\\begin{array}{rclcc}\n\\displaystyle\n\\partial_t n+{\\boldsymbol u}\\cdot\\nabla n-\\Delta n+ \\nabla\\cdot(n\\nabla c)&=&0&\\mbox{ in }& Q ,\n\\\\\n\\displaystyle\n\\partial_t c+{\\boldsymbol u}\\cdot\\nabla c-\\Delta c+ c&=&n&\\mbox{ in }& Q,\n\\\\\n\\partial_t{\\boldsymbol u}+({\\boldsymbol u}\\cdot\\nabla){\\boldsymbol u}-\\Delta{\\boldsymbol u}+\\nabla p-n\\nabla \\Phi&=&\\boldsymbol{0}&\\mbox{ in }& Q,\n\\\\\n\\nabla\\cdot{\\boldsymbol u}&=&0&\\mbox{ in }& Q.\n\\end{array}\n\\right.\n\\end{equation}\nHere $n:\\overline{\\Omega\\times (0,T]}\\to (0,\\infty)$ represents the organism density, $c: \\overline{\\Omega\\times(0,T)}\\to [0,\\infty) $ represents the chemoattractant density, ${\\boldsymbol u}:\\overline{\\Omega\\times(0,T]}\\times\\to\\mathds{R}^2$ is the fluid velocity, and $p:\\overline{\\Omega\\times(0,T]}\\times\\to\\mathds{R}$ is the fluid pressure; furthermore, $\\Phi$ is a gravitational force.\n\nThese equations are supplemented with homogeneous Neumann boundary conditions for the Keller-Segel subsystem and homogeneous Dirichlet boundary conditions for the Navier-Stokes equations, i.e.,\n\\begin{equation}\\label{BC_KSNS}\n\\partial_{\\boldsymbol n} n=0, \\quad\\partial_{\\boldsymbol n} c =0 \\mbox{ and } \\quad {\\boldsymbol u}=0 \\mbox { on } \\quad \\partial \\Omega\\times (0,T),\n\\end{equation}\nand with initial conditions\n\\begin{equation}\\label{IC_KSNS}\nn(0)=n_0,\\quad\nc(0)=c_0\\mbox{ and } {\\boldsymbol u}(0)={\\boldsymbol u}_0\\quad\\mbox{ in }\\quad \\Omega.\n\\end{equation}\n\nAs far as we are concerned, the work of Winkler \\cite{Winkler_2020} is the only mathematical analysis available in the literature for system \\eqref{KSNS}, where a similar mass threshold phenomenon is put forward possibly deciding between the boundedness and unboundedness of solutions. More precisely, he found that, if $\\int_\\Omega n_0({\\boldsymbol x})\\,{\\boldsymbol x}\\in(0,2\\pi)$, problem \\eqref{KSNS}-\\eqref{IC_KSNS} possesses generalised solutions globally in time. Therefore, the theory in studying mathematical properties of solutions to problem \\eqref{KSNS}-\\eqref{IC_KSNS} seems to be in a very primitive stage in comparison with those for problem \\eqref{KS}-\\eqref{IC_KS}. Thus, at this point, two accordingly feasible scenarios can be conjectured. On the one hand, it might occur that the chemotaxis-fluid system inherits the mathematical properties of solutions in a consistent fashion from the fluid-free system thereof, since the influence of the fluid through the transport and gravitational effect is essentially dominated by the cross-diffusion one. On the other hand, the fluid mechanism might interact with the cross-diffusion principle causing that the mass critical takes a certain value between $2\\pi$ and $4\\pi$. This might be due to the creation of extremely large values of density gradients by means of amplifying the nonlinear cross-diffusion process.\n\nDetecting blow-up configurations via the numerical solution to problem \\eqref{KSNS}-\\eqref{IC_KSNS} is extremely difficult because the growth of densities in time is an ubiquitous phenomenom in smooth solutions to problem \\eqref{KSNS}-\\eqref{IC_KSNS} (and \\eqref{KS}-\\eqref{IC_KS}), which must be treated with caution. Therefore, in an evolving smooth solution with a point of actively growing gradients, we must decide whether there is merely a huge growth of the density or actually a finite-time singularity development. Furthermore, it is pointed out in \\cite{GS_RG_2021} that discretising directly without enforcing lower bounds for the chemoattractant and organism densities will lead to unstable numerical solutions.\n\nInspired on \\cite{Badia_Bonilla_GS_2022}, we develop a numerical scheme founded on a finite element method stabilised via adding nonlinear diffusion combined with an Euler time-stepping integrator. The nonlinear diffusion draw on a graph-Laplacian operator together with a shock detector in order to minimise the amount of numerical diffusion introduced in the system. This approach has been turned out efficient in the fluid-free chemotaxis \\cite{Badia_Bonilla_GS_2022}. \n\nThe determination of potential candidate singular solutions from numerical simulation presents a variety of challenging issues inherited from the mathematical analysis that we need to cope with thoroughly. These numerical issues are: \n\\begin{itemize}\n\\item lower bounds: positivity for the chemoattractant density and nonnegativity for the organism density;\n\\item time-independent integrability bounds: particularly mass conservation for the chemoattractant density; and\n\\item time-independent square integrability bounds and time-dependent square integrability for the gradient of the organism density and the fluid velocity. \n\\end{itemize} \n\nProblem \\eqref{KS}--\\eqref{IC_KS} has already studied in the numerical literature with different techniques \\cite{Saito_2012, GS_RG_2021, Strehl_Sokolov_Kuzmin_Turek_2010, Strehl_Sokolov_Kuzmin_Horstmann_Turek_2013, Li_Shu_Yang_2017, Chertock_Epshteyn_Hu_Kurganov_2018, Chertock_Kurganov_2008}, but these have not been applied to contexts of self-enhanced chemotactic motion in the presence of a fluid flow.\n\n\\subsection{Notation} Throughout, we adopt the standard notation for Sobolev spaces. Let $\\mathcal{O}\\subset \\mathds{R}^2$ be an open, bounded domain. For $p\\in[1,\\infty]$, we denote by $L^p(\\Omega)$ the usual Lebesgue space, i.e.,\n$$\nL^p(\\mathcal{O}) = \\{v : \\Omega \\to {\\mathds R}\\, :\\, v \\mbox{ Lebesgue-measurable}, \\int_\\Omega |v({\\boldsymbol x})|^p {\\rm d}{\\boldsymbol x}<\\infty \\}.\n$$\nor\n$$\nL^\\infty(\\mathcal{O}) = \\{v : \\Omega \\to {\\mathds R}\\, :\\, v \\mbox{ Lebesgue-measurable}, {\\rm ess}\\sup_{{\\boldsymbol x}\\in \\Omega} |v({\\boldsymbol x})|<\\infty \\}.\n$$\nThis space is a Banach space endowed with the norm\n$\\|v\\|_{L^p(\\Omega)}=(\\int_{\\Omega}|v({\\boldsymbol x})|^p\\,{\\rm d}{\\boldsymbol x})^{1\/p}$ if $p\\in[1, \\infty)$ or $\\|v\\|_{L^\\infty(\\Omega)}={\\rm ess}\\sup_{{\\boldsymbol x}\\in \\Omega}|v({\\boldsymbol x})|$ if $p=\\infty$. In particular, when $p=2$, $L^2(\\Omega)$ is a Hilbert space. We shall use $(u,v)=\\int_{\\Omega}u({\\boldsymbol x})v({\\boldsymbol x}){\\rm d}{\\boldsymbol x}$ for its inner product\n\nLet $\\alpha = (\\alpha_1, \\alpha_2)\\in \\mathds{N}^2$ be a multi-index with $|\\alpha|=\\alpha_1+\\alpha_2$, and let\n$\\partial^\\alpha$ be the differential operator such that\n$$\\partial^\\alpha=\n\\Big(\\frac{\\partial}{\\partial{x_1}}\\Big)^{\\alpha_1}\\Big(\\frac{\\partial}{\\partial{x_2}}\\Big)^{\\alpha_2}.$$\nFor $m\\ge 0$ and $p\\in[1, \\infty)$, we shall consider $W^{m,p}(\\Omega)$ to be the Sobolev space of all functions whose derivatives up to order $m$ are in $L^p(\\Omega)$, i.e.,\n$$\nW^{m,p}(\\Omega) = \\{v \\in L^p(\\Omega)\\,:\\, \\partial^k v \\in L^2(\\Omega)\\ \\forall ~ |k|\\le m\\},\n$$ associated to the norm\n$$\n\\|f\\|_{W^{m,p}(\\Omega)}=\\left(\\sum_{|\\alpha|\\le m} \\|\\partial^\\alpha f\\|^p_{L^p(\\Omega)}\\right)^{1\/p} \\quad \\hbox{for} \\ 1 \\leq p < \\infty,\n$$\nand\n$$\n\\|f\\|_{W^{m,p}(\\Omega)}=\\max_{|\\alpha|\\le m} \\|\\partial^\\alpha f\\|_{L^\\infty(\\Omega)} \\quad \\hbox{for} \\ p = \\infty.\n$$\nFor $p=2$, we denote $W^{m,2}(\\Omega)=H^m(\\Omega)$.\n\nLet $\\mathcal{D}(\\Omega)$ be the space of infinitely times\ndifferentiable functions with compact support on $\\Omega$. \nThe closure of $\\mathcal {D}(\\Omega)$ in\n$H^{m}(\\Omega)$ is denoted by $H^{m}_0(\\Omega)$. \nWe will also make use of the following space of vector fields:\n$$\n\\boldsymbol{\\mathcal{V}}=\\{{\\boldsymbol v}\\in \\boldsymbol{\\mathcal{D}}(\\Omega): \n\\nabla\\cdot{\\boldsymbol v}=0 \\mbox{ in } \\Omega \\}. \n$$\nThe closure of $\\boldsymbol{\\mathcal{V}}$ in the ${\\boldsymbol L}^2(\\Omega)$ norm is denoted by $\\boldsymbol{H}$ and is characterised \\cite{Temam2001} (for $\\Omega$ being Lipschitz-continuous) by \n$$\n\\boldsymbol{H}= \\{ {\\boldsymbol u} \\in {\\boldsymbol L}^2(\\Omega) : \\nabla\\cdot{\\boldsymbol u} =0 \\mbox{ in } \n\\Omega, {\\boldsymbol u}\\cdot\\boldsymbol{n} = 0 \\hbox{ on }\n\\partial\\Omega \\},\n$$\nwhere ${\\boldsymbol n}$ is the outward normal to $\\Omega$ on $\\partial \\Omega$. \nFinally, we consider \n$$\nL^2_0(\\Omega)= \\{ p \\in L^2(\\Omega) : \\ \\int_\\Omega\np({\\boldsymbol x})\\, d{\\boldsymbol x} =0 \\}.\n$$\n\nDistinction is made between scalar- or vector-valued functions, so spaces of vector-valued functions and their elements are identified with bold font. \n\nFor any sequence $\\{\\eta_h^m \\}_{m=0}^M$, we use the notation $\\delta_t \\eta^{m+1}_h=\\frac{\\eta_h^{m+1}-\\eta^{m}_h}{k}$. We set $B({\\boldsymbol x}_0; r )=\\{ {\\boldsymbol x}\\in\\mathds{R}^2: \\|{\\boldsymbol x}-{\\boldsymbol x}_0\\|_{E}0$, independent of $h$, such that, for all $\\eta\\in L^p(\\Omega)$, \n\\begin{equation\n\\|\\mathcal{I}_h \\eta\\|_{L^p(\\Omega)}\\le C_{\\rm sta} \\| \\eta \\|_{L^p(\\Omega)}\\quad \\mbox{for } 1\\le p\\le\\infty,\n\\end{equation}\nand \n\\begin{equation\n\\mathcal{I}_h \\psi \\ge \\mbox{or} > 0\\quad\\mbox{ if }\\quad \\psi\\ge \\mbox{or} >0. \n\\end{equation}\nIn addition to the above interpolation operators, we consider the Ritz-Darcy projection operator $\\mathcal{RD}_h : \\boldsymbol{H}\\to \\boldsymbol{U}_h $ defined as: given ${\\boldsymbol u}\\in \\boldsymbol{H}$, find $\\mathcal{RD}_h {\\boldsymbol u}$ such that, for all $(\\bar{\\boldsymbol u}_h, \\bar p_h)\\in \\boldsymbol{U}_h\\times P_h$, \n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\n(\\mathcal{RD}_h {\\boldsymbol u}, \\bar {\\boldsymbol u}_h)+(\\nabla p_h, \\bar{\\boldsymbol u}_h )&=&({\\boldsymbol u}, \\bar{\\boldsymbol u}_h ),\n\\\\\n(\\nabla\\cdot\\mathcal{RD}_h {\\boldsymbol u}, \\bar p_h)&=&0\n\\end{array}\n\\right.\n\\end{equation}\nIt is readily to prove that \n$\\|\\mathcal{RD}_h{\\boldsymbol u}\\|_{{\\boldsymbol L}^2(\\Omega)}\\le \\|{\\boldsymbol u}\\|_{{\\boldsymbol L}^2(\\Omega)}$ holds.\n\\subsection{Heuristics} It will be next proceeded without regards to rigour to develop the finite element formulation for approximating system \\eqref{KSNS}--\\eqref{IC_KSNS}. We take as our starting point the standard finite element formulation of system \\eqref{KSNS}--\\eqref{IC_KSNS} combined with a time-stepping integration, which is implicit with respect to the linear terms and semi-implicit with respect to the nonlinear terms, except for the chemotaxis term being implicit. This method it is then read as follows:\n\n\nLet $\\Phi\\in W^{1,\\infty}(\\Omega)$ and assume that $(n_0, c_0,{\\boldsymbol u}_0)\\in L^1(\\Omega)\\times L^2(\\Omega)\\times \\boldsymbol{H}$ with $n_0>0$ and $c_0\\ge0$. Then consider $(n_{0h}, c_{0h}, {\\boldsymbol u}_{0h})\\in N_h\\times C_h\\times \\boldsymbol{H}^1_0(\\Omega)$, where $n_{0h}=\\mathcal{I}_h n_0$, $c_{0h}=\\mathcal{I}_h c_0$, and ${\\boldsymbol u}_{0h}=\\mathcal{RD}_h{\\boldsymbol u}_0$. \n\nLet $\\{t_n \\}_{n=0}^M$ be a sequence of points partitioning $[0,T]$ into subintervals of the same length $k=\\frac{T}{M}$ with $M\\in\\mathds{N}$ and select $n_h^0=n_{0h}$, $c^0_h=c_{0h}$, and ${\\boldsymbol u}^0_h={\\boldsymbol u}_{0h}$. Given $(n^m_h, c^m_h, {\\boldsymbol u}^m_h)\\in N_h\\times C_h\\times \\boldsymbol{U}_h$, find \\linebreak $(n^{m+1}_h, c^{m+1}_h, {\\boldsymbol u}^{m+1}_h, p^{m+1}_h)\\in N_h\\times C_h\\times \\boldsymbol{U}_h\\times Q_h$ such that, for all $(\\bar n_h, \\bar c_h,\\bar{\\boldsymbol u}_h, \\bar p_h)\\in N_h\\times C_h\\times U_h\\times Q_h$, \n\\begin{equation}\\label{eq_aux:n_h}\n(\\delta_t n^{m+1}_h, \\bar n_h)-({\\boldsymbol u}^m_h n^{m+1}_h, \\nabla\\bar n_h)+(\\nabla n^{m+1}_h,\\nabla\\bar n_h)-(n^{m+1}_h\\nabla c^{m+1}_h,\\nabla\\bar n_h)=0,\n\\end{equation}\n\\begin{equation}\\label{eq_aux:c_h}\n\\begin{array}{rcl}\n\\displaystyle\n(\\delta_t c^{m+1}_h, \\bar c_h)+({\\boldsymbol u}^m_h\\cdot\\nabla c^{m+1}_h, \\bar c_h)+\\frac{1}{2} (\\nabla\\cdot{\\boldsymbol u}^{m+1}_h c^{m+1}_h, \\bar c_h)&&\n\\\\[5pt]\n+(\\nabla c^{m+1}_h,\\nabla\\bar c_h)+ (c^{m+1}_h,\\bar c_h)\n-(n^{m+1}_h, \\bar c_h)&=&0,\n\\end{array}\n\\end{equation}\n\\begin{equation}\\label{eq_aux:u_h}\n\\begin{array}{rcl}\n\\displaystyle\n(\\delta_t {\\boldsymbol u}^{m+1}_h, \\bar{\\boldsymbol u}_h)+\n({\\boldsymbol u}^m_h\\cdot\\nabla{\\boldsymbol u}^{m+1}_h,\\bar{\\boldsymbol u}_h)+\\frac{1}{2}(\\nabla\\cdot{\\boldsymbol u}^m_h\\, {\\boldsymbol u}^{m+1}_h, \\bar {\\boldsymbol u}_h)&&\n\\\\\n+(\\nabla {\\boldsymbol u}^{m+1}_h,\\nabla\\bar {\\boldsymbol u}_h)+(\\nabla p^{m+1}_h, \\bar{\\boldsymbol u}_h)-(n^{m+1}_h\\nabla\\Phi, \\bar{\\boldsymbol u}_h)&=&0,\n\\end{array}\n\\end{equation}\nand\n\\begin{equation}\\label{eq_aux:p_h}\n(\\nabla\\cdot{\\boldsymbol u}^{m+1}_h,\\bar p_h)=0. \n\\end{equation}\n\n\nThe obtainment of the underlying properties that lead to deriving a priori estimates for the discrete solutions to \\eqref{eq_aux:n_h}-\\eqref{eq_aux:p_h} is by no means a direct computation. For equation \\eqref{eq_aux:n_h}, we need to control the quantity $\\sum_{m=0}^{M-1} k \\|\\nabla\\log(n^{m+1}_h+1)\\|_{{\\boldsymbol L}^2(\\Omega)}$, which is deduced by testing it against the nonlinear test function $\\frac{1}{n^{m+1}_h}$. Nevertheless, it does not seem evident how to do it directly from \\eqref{eq_aux:n_h}. Positivity and mass conservation are as well required from \\eqref{eq_aux:n_h}. For equation \\eqref{eq_aux:c_h}, a uniform-in-time $L^2(\\Omega)$-bound is sought, but it is apparently connected to $\\sum_{m=0}^{M-1} k \\|\\nabla\\log(n^{m+1}_h+1)\\|_{L^2(\\Omega)}$ for \\eqref{eq_aux:n_h}. Furthermore, the stabilising term $\\frac{1}{2} (\\nabla\\cdot{\\boldsymbol u}^{m+1}_h c^{m+1}_h, \\bar c_h)$ has added to rule out the convective term from \\eqref{eq_aux:c_h} when tested with $c^{m+1}_h$. Nonnegativy and an $L^1(\\Omega)$-bound are additionally needed from \\eqref{eq_aux:c_h}.\nFor equation \\eqref{eq_aux:u_h}, we wish a uniform-in-time ${\\boldsymbol L}^2(\\Omega)$-bound. In doing so, the stabilising term $\\frac{1}{2}(\\nabla\\cdot{\\boldsymbol u}^{n}_h\\, {\\boldsymbol u}^{m+1}_h, \\bar {\\boldsymbol u}_h)$ has been incorporated to deal with the convective term. \n\nIt is interesting to note that $L^1(\\Omega)$ bounds -- in particular, mass conservation -- are straightforwardly derived from \\eqref{eq_aux:n_h} and \\eqref{eq_aux:c_h}; therefore it is desirable to keep them. \n\n\nIn what follows we set forth some modifications for scheme \\eqref{eq_aux:n_h}--\\eqref{eq_aux:p_h} in the forthcoming subsections in order for the above-mentioned properties to hold. \n\n\\subsubsection{Chemotaxis term} The first modification is with regard to the chemotaxis term. It is to be recalled that we seek an estimate for $\\sum_{m=0}^{M-1} k \\|\\nabla\\log(n^{m+1}_h+1)\\|_{{\\boldsymbol L}^2(\\Omega)}$ from \\eqref{eq_aux:n_h}, which stems from testing by $\\frac{1}{n^{m+1}_h}$. We proceed in the spirit of \\cite{Badia_Bonilla_GS_2022} by writing \n$$\n\\begin{array}{rcl}\n( n_h\\nabla c_h,\\nabla \\bar n_h)&=&\\displaystyle\n\\sum_{k,j,i\\in I} n_k c_j \\bar n_i (\\varphi_{{\\boldsymbol a}_k} \\nabla \\varphi_{{\\boldsymbol a}_j}, \\nabla\\varphi_{{\\boldsymbol a}_i}) \n\\\\\n&=&\\displaystyle\n\\sum_{\\tiny\\begin{array}{c}k\\in I \\\\i0$; nevertheless, we cannot ensure such a condition. Consequently, we must allow for negative values of $n_h$. In doing so, the following extension of the logarithmic function to negative values is used. For $0<\\varepsilon<1$, define\n\\begin{equation}\\label{def:g_eps}\ng_\\varepsilon(s)=\\left\\{\n\\begin{array}{ccl}\n\\log s&\\mbox{ if }& s>\\varepsilon,\n\\\\\n\\frac{s}{\\varepsilon}+\\log \\varepsilon-1&\\mbox{ if }& s\\le \\varepsilon. \n\\end{array}\n\\right.\n\\end{equation}\nTherefore it remains \n\\begin{equation}\\label{New_conv_term: extended}\n(n_h {\\boldsymbol u}^m_h,\\nabla \\bar n_h)_*=\\sum_{i0$ such that \n\\begin{equation}\\label{ineq:Moser-Trudinger}\n\\int_{\\Omega} e^{|\\eta({\\boldsymbol x})|}{\\rm d}{\\boldsymbol x}\\le C|\\Omega| e^{\\frac{1}{16\\pi} \\|\\nabla \\eta\\|^2_{{\\boldsymbol L}^2(\\Omega)}}.\n\\end{equation}\n\\end{theorem}\n\nWe next set forth the analogue of a variant of \\eqref{ineq:Moser-Trudinger} given in \\cite[Th. 2.2]{Nagai_Senba_Yoshida_1997} but for polygonal domains. \n\\begin{theorem}\\label{th:Moser-Trundinger}Assume that $\\Omega$ is a bounded polygonal domain in ${\\mathds R}^2$ with $\\alpha_\\Omega$ being the minimum interior angle at the vertices of $\\mathcal{V}_h$. Further suppose that $\\eta\\in H^1(\\Omega)$ with $\\eta>0$. Then there exists $\\beta_\\Omega\\in [1, 2)$, depending upon $\\Omega$, so that, for each $\\lambda>0$, one can find $C_\\lambda>0$ such that \n\\begin{equation}\\label{ineq:Trudinger-Moser_New}\n\\int_\\Omega e^{\\eta({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}\\le C e^{\\displaystyle\\beta^2_\\Omega\\frac{1+\\lambda}{8\\alpha_\\Omega} \\|\\nabla \\eta\\|^2_{{\\boldsymbol L}^2(\\Omega)}+C_\\lambda\\|\\eta\\|^2_{L^1(\\Omega)}}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof} Let $\\{U_j\\}_{j=1}^J$ be a covering of $\\partial\\Omega$, i. e., $\\partial\\Omega\\subset\\bigcup_{j=1}^J U_j$, such that $U_j=B(\\boldsymbol{p}_j; \\rho_j)$, where $\\{\\boldsymbol{p}_j\\}_{j=1}^J\\subset\\partial\\Omega$ satisfying $\\mathcal{V}_h\\subset\\{\\boldsymbol{p}_j\\}_{j=1}^J$ and $\\rho_j>0$. In addition, if $\\boldsymbol{p}_j\\in\\mathcal{V}_h$, then there exist $E, \\tilde E\\in\\mathcal{E}_h $ such that $\\boldsymbol{p}_j=E\\cap\\tilde E$ with $\\rho_j=\\min \\{{\\rm meas}(E), {\\rm meas}(\\tilde E)\\}$; otherwise, if $\\boldsymbol{p}_j\\in {\\rm int }\\, E$ for some $E\\in\\mathcal{E}_h$, then $U_j\\cap \\partial\\Omega\\subset {\\rm int}\\, E$. \n\nIt is well-known \\cite{Adams_Fournier_2003} that there exist a family of functions $\\{\\theta_i\\}_{j=0}^J\\subset C^{\\infty}({\\mathds R}^2)$ such that $0\\le\\theta_j\\le 1$ for all $j=0,\\cdots, J$, with ${\\rm supp}\\, \\theta_j\\subset U_j$ if $j\\not=0$ and ${\\rm supp }\\, \\theta_0\\subset \\Omega$, and $\\sum_{j=0}^J \\theta_j=1$. As a result, one may write\n\\begin{equation}\\label{partition}\n\\begin{array}{rcl}\n\\displaystyle\n\\int_\\Omega e^{\\eta({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}&=&\\displaystyle\n\\int_\\Omega e^{\\sum_{j=0}^J (\\theta_j \\eta)({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}\\le\\sum_{j=0}^J \\int_{U_j\\cap\\Omega} e^{(\\theta_j \\eta)({\\boldsymbol x})}{\\rm d}{\\boldsymbol x},\n\\end{array}\n\\end{equation}\nwhere we denoted $U_0={\\rm supp }\\,\\theta_0$. From inequality \\eqref{ineq:Moser-Trudinger}, we bound\n$$\n\\int_{U_0\\cap \\Omega} e^{(\\theta_0 \\eta)({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}\\le C |U_0\\cap\\Omega| e^{\\frac{1}{16\\pi}\\|\\nabla(\\theta_0 \\eta)\\|^2_{{\\boldsymbol L}^2(U_0\\cap\\Omega)}}.\n$$\nNow Gagliardo--Nirenberg's interpolation and Young's inequality yield\n$$\n\\begin{array}{rcl}\n\\|\\nabla(\\theta_0 \\eta)\\|_{{\\boldsymbol L}^2(U_0\\cap\\Omega)}&\\le&\\|\\nabla \\eta\\|_{{\\boldsymbol L}^2(\\Omega)}+ \\|\\nabla\\theta_0\\|_{{\\boldsymbol L}^\\infty({\\mathds R}^2)} \\|\\eta\\|_{L^2(\\Omega)}\n\\\\\n&\\le&\\|\\nabla \\eta\\|_{{\\boldsymbol L}^2(\\Omega)}+ C_1 \\|\\nabla\\theta_0\\|_{{\\boldsymbol L}^\\infty({\\mathds R}^2)} \\|\\nabla\\eta\\|^{\\frac{1}{2}}_{{\\boldsymbol L}^2(\\Omega)} \\|\\eta\\|_{L^1(\\Omega)}^\\frac{1}{2}+C_2 \\|\\nabla\\theta_0\\|_{{\\boldsymbol L}^\\infty({\\mathds R}^2)} \\|\\eta\\|_{L^1(\\Omega)}\n\\\\\n&\\le&(1+\\tilde\\lambda)\\|\\nabla\\eta\\|_{{\\boldsymbol L}^2(\\Omega)}+ C_{\\tilde\\lambda} \\|\\eta\\|_{L^1(\\Omega)}\n\\end{array}\n$$ \nand therefore\n$$\n\\begin{array}{rcl}\n\\displaystyle\n\\int_{U_0\\cap \\Omega} e^{(\\theta_0\\eta)({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}&\\le& C |U_0\\cap\\Omega| e^{\\frac{1}{16\\pi}((1+ \\tilde\\lambda) \\|\\nabla \\eta\\|_{{\\boldsymbol L}^2(\\Omega)}+ C_{\\tilde\\lambda} \\|\\eta\\|_{L^1(\\Omega)})^2}\n\\\\\n&\\le&\\displaystyle\nC |U_0\\cap\\Omega|e^{\\frac{1}{16\\pi}((1+3\\tilde\\lambda+\\tilde\\lambda^2)\\|\\nabla \\eta\\|^2_{{\\boldsymbol L}^2(\\Omega)}+ C_{\\tilde\\lambda} \\|\\eta\\|^2_{L^1(\\Omega)})}\n\\\\\n&\\le&\\displaystyle\nC|U_0\\cap\\Omega|e^{\\frac{1+\\lambda}{16\\pi}\\|\\nabla\\eta\\|^2_{{\\boldsymbol L}^2(\\Omega)}+ C_{\\lambda} \\|\\eta\\|^2_{L^1(\\Omega)}},\n\\end{array}\n$$\nwhere we chose $\\lambda=3\\tilde\\lambda+\\tilde\\lambda^2$. Let $\\boldsymbol{p}_j\\in\\mathcal{V}_h$ and denote $\\alpha_j$ to be the interior angle at $\\boldsymbol{p}_j$. Without loss of generality, one can assume that $\\boldsymbol{p}_j=\\boldsymbol{0}$ and $U_j\\cap\\Omega=S(\\rho_j, \\alpha_j)$, where\n$$\nS(\\rho_j, \\alpha_j)=\\{{\\boldsymbol x}=r e^{i \\beta}: 0\\le r\\le \\rho_j \\mbox{ and } \\frac{\\pi}{2}-\\frac{\\alpha_j}{2}\\le \\beta\\le \\frac{\\pi}{2}+\\frac{\\alpha_j}{2}\\}.\n$$\nConsider $T_P:{\\mathds R}^2\\to{\\mathds R}^2$ to be the polar coordinate mapping, i.e., $T_P( r,\\beta)=(r\\cos\\beta, r\\sin\\beta)$ and define the invertible mapping $H_j: [0,\\tilde\\rho_j]\\times[0,\\pi]\\to[0,\\rho_j]\\times[ \\frac{\\pi}{2}-\\frac{\\alpha_j}{2}, \\frac{\\pi}{2}+\\frac{\\alpha_j}{2}] $, with $\\tilde\\rho_j>0$, as \n$$H_j(\\tilde r, \\tilde \\beta)= (\\tilde r\\frac{\\rho_j}{\\tilde\\rho_j}, \\frac{\\pi}{2}+\\frac{\\alpha_j}{\\pi} (\\tilde\\beta-\\frac{\\pi}{2})),\n$$\nwhose Jacobian determinant is $ {\\rm det} J_H=\\frac{\\rho_j}{\\tilde\\rho_j}\\frac{\\alpha_j}{\\pi}$. Thus we write \n$$\n\\begin{array}{rcl}\n\\displaystyle\n\\int_{U_j\\cap\\Omega} e^{(\\theta_j\\eta)({\\boldsymbol x})}\\,{\\rm d}{\\boldsymbol x}&=&\\displaystyle\n\\int_{[0,\\rho_j]\\times[ \\frac{\\pi}{2}-\\frac{\\alpha_j}{2}, \\frac{\\pi}{2}+\\frac{\\alpha_j}{2}] } r e^{(\\theta_j \\eta\\circ T_P)(r,\\beta)}\\,dr d\\beta\n\\\\\n&=&\\displaystyle\n\\frac{\\rho_j^2}{\\tilde\\rho_j^2}\\frac{\\alpha_j}{\\pi}\\int_{[0,\\tilde\\rho_j]\\times[0,\\pi]} \\tilde re^{(\\theta_j \\eta\\circ T_P\\circ H_j)(\\tilde r,\\tilde \\beta)}\\,d\\tilde rd\\tilde\\beta\n\\\\\n&=&\\displaystyle\n\\frac{\\rho_j^2}{\\tilde\\rho_j^2}\\frac{\\alpha_j}{\\pi}\\int_{B_+(\\boldsymbol{0},\\tilde\\rho_j)} e^{(\\theta_j \\eta\\circ T_P\\circ H_j\\circ T_P^{-1})({\\boldsymbol x})}\\,{\\rm d}{\\boldsymbol x}.\n\\end{array}\n$$\nWe know that $\\theta_j \\eta\\circ T_P\\circ H_j\\circ T_P^{-1}\\in H^{1}(B_+(\\boldsymbol{0};\\rho_j))$, since $T_P\\circ H_j\\circ T_P^{-1}\\in C_{\\rm Lip}(B_+(\\boldsymbol{0}; \\tilde\\rho_j); S(\\rho_j,\\alpha_j) )$. Indeed, observe that \n\\begin{align*}\nT_P\\circ &H_j\\circ T_P^{-1}(x_1, x_2)\n\\\\\n&=(\\frac{\\rho_j}{\\tilde\\rho_j}\\sqrt{x_1^2+x_2^2} \\cos(\\frac{\\pi}{2}+\\frac{\\alpha_j}{\\pi}(\\arg(x_1,x_2)-\\frac{\\pi}{2})),\\frac{\\rho_j}{\\tilde\\rho_j}\\sqrt{x_1^2+x_2^2} \\sin(\\frac{\\pi}{2}+\\frac{\\alpha_j}{\\pi}(\\arg(x_1,x_2)-\\frac{\\pi}{2}))),\n\\end{align*}\nwhere ${\\rm arg}(\\cdot,\\cdot)\\in [-\\frac{\\pi}{2}, \\frac{3\\pi}{2})$. It is clear that $T_P\\circ H_j\\circ T_P^{-1}\\in C(B_+(\\boldsymbol{0};\\tilde\\rho_j);S(\\rho_j, \\alpha_j))$ and $\\frac{\\partial(T_P\\circ H_j\\circ T_P^{-1})_i}{\\partial x_k}\\in C(B_+(\\boldsymbol{0};\\tilde\\rho_j); S(\\rho_j, \\alpha_j)\\backslash\\{\\boldsymbol{0}\\})$ for $i,k=1,2$. Moreover, there exists $\\beta_j>0$ such that $\\|J_{T_P\\circ H_j\\circ T_P^{-1}}\\|_{L^\\infty(\\Omega)}\\linebreak\\le\\frac{\\rho_j}{\\tilde\\rho_j} \\beta_j$, where $J_{T_P\\circ H_j\\circ T_P^{-1}}$ is the Jacobian matrix. For instance, we have that\n$$\n\\begin{array}{rcl}\n\\displaystyle\n\\frac{\\partial (T_P\\circ H_j\\circ T_P^{-1})_1}{\\partial x_1}(x_1,x_2)&=&\\displaystyle\n\\frac{\\rho_j}{\\tilde\\rho_j \\pi} \\frac{1}{\\sqrt{x_1^2+x_2^2}} \\Big( \\pi x_1 \\sin (\\frac{\\alpha_j}{\\pi} \\arg(x_1,x_2)-\\frac{1}{2})\n\\\\\n&&\\displaystyle\n-\\alpha_j x_2\\cos (\\frac{\\alpha_j}{\\pi} \\arg(x_1,x_2)-\\frac{1}{2}) \\Big);\n\\end{array}\n$$ \nthereby using polar coordinates $\\lim_{r\\to 0}\\Big(\\frac{\\partial (T_P\\circ H_j\\circ T_P^{-1})_1}{\\partial x_1}\\circ T^{-1}_P\\Big)(r,\\beta)$ does not exist, but \n\\begin{align*}\n|\\lim_{r\\to 0}\\Big(\\frac{\\partial (T_P\\circ H_j\\circ T_P^{-1})_1}{\\partial x_1}&\\circ T^{-1}_P\\Big)(r,\\beta)|\n\\\\\n&=|\\frac{\\rho_i}{\\tilde\\rho_j \\pi} \\Big( \\pi \\cos(\\beta) \\sin (\\frac{\\alpha_j}{\\pi} (\\beta-\\frac{\\pi}{2}))\n-\\alpha_j \\sin(\\beta)\\cos (\\frac{\\alpha_j}{\\pi} (\\beta-\\frac{\\pi}{2})) \\Big)|\n\\\\\n&\\le \\frac{\\rho_j}{\\tilde\\rho_j}\\beta^{11}_j,\n\\end{align*}\nwhere $\\beta^{11}_j=\\max_{\\beta\\in[0,\\pi]} \\frac{1}{\\pi}| \\pi \\cos(\\beta) \\sin (\\frac{\\alpha_j}{\\pi}(\\beta-\\frac{\\pi}{2}))\n-\\alpha_j \\sin(\\beta)\\cos (\\frac{\\alpha_j}{\\pi} (\\beta-\\frac{\\pi}{2})|$ and hence take $\\beta_j=max_{i,k=1,2} \\beta^{ik}_j$. \n\nNext define the extension on $B(\\boldsymbol{0};\\tilde\\rho_j)$ by reflection of $\\theta_j n\\circ T_P\\circ H_j\\circ T_P^{-1}$, which is denoted by $(\\theta_j \\eta\\circ T_P\\circ H_j\\circ T_P^{-1})^*$. We thus have that $(\\theta_j \\eta\\circ T_P\\circ H_j\\circ T_P^{-1})^*\\in H^1_0(B(\\boldsymbol{0};\\tilde\\rho_j))$. In view of \\eqref{ineq:Moser-Trudinger}, we find that \n$$\n\\begin{array}{rcl}\n\\displaystyle\n\\int_{U_j\\cap\\Omega} e^{(\\theta_j \\eta)({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}&=&\\displaystyle\n\\frac{\\rho_j^2}{2\\tilde\\rho_j^2}\\frac{\\alpha_j}{\\pi}\\int_{B(\\boldsymbol{0},\\tilde\\rho_j)} e^{(\\theta_j\\eta\\circ T_P\\circ H_j\\circ T_P^{-1})^*({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}\n\\\\\n&\\le&\\displaystyle C \\frac{\\rho_j^2}{2\\tilde\\rho_j^2}\\frac{\\alpha_j}{\\pi} |B(\\boldsymbol{0},\\tilde\\rho_j)|\ne^{\\frac{1}{16\\pi}\\|\\nabla(\\theta_j \\eta\\circ T_P\\circ H_j\\circ T_P^{-1})^*\\|^2_{{\\boldsymbol L}^2(B(\\boldsymbol{0},\\tilde\\rho_j))}}\n\\\\\n&\\le&\\displaystyle C |U_j\\cap \\Omega|\ne^{\\frac{1}{8\\pi}\\|\\nabla(\\theta_j \\eta\\circ T_P\\circ H_j\\circ T_P^{-1})\\|^2_{{\\boldsymbol L}^2(B_+(\\boldsymbol{0},\\tilde\\rho_j))}}\n\\\\\n&\\le&\\displaystyle C |U_j\\cap \\Omega|\ne^{\\beta_j^2\\frac{1}{8\\alpha_j}\\|\\nabla(\\theta_j \\eta)\\|^2_{{\\boldsymbol L}^2(U_j\\cap\\Omega)}}\n\\\\\n&\\le&\\displaystyle C |U_j\\cap \\Omega|\ne^{\\beta_j^2\\frac{1+\\lambda}{8\\alpha_\\Omega}\\|\\nabla \\eta\\|^2_{L^2(\\Omega)}+ C_{\\lambda} \\|\\eta\\|^2_{L^1(\\Omega)}},\n\\end{array}\n$$\nwhere we used the fact that \n$$\n\\begin{array}{rcl}\n\\|\\nabla(\\theta_j \\eta\\circ T_P\\circ H_j\\circ T_P^{-1})\\|_{{\\boldsymbol L}^2(B_+(\\boldsymbol{0},\\tilde\\rho_j))}&\\le&\\displaystyle\n \\|J_{T_P\\circ H_j\\circ T_P^{-1}}\\|_{{\\boldsymbol L}^\\infty(\\Omega)}\\| \\|\\nabla(\\theta_j \\eta)(T_P\\circ H_j\\circ T_P^{-1})\\|_{{\\boldsymbol L}^2(B_+(\\boldsymbol{0};\\tilde\\rho_j))}\n\\\\\n&\\le&\\displaystyle\n\\beta_j \\frac{\\rho_j}{\\tilde\\rho_j} \\|\\nabla(\\theta_j \\eta)(T_P\\circ H_j\\circ T_P^{-1})\\|_{{\\boldsymbol L}^2(B_+(\\boldsymbol{0};\\tilde\\rho_j))}\n\\\\\n&\\le &\\beta_j \\sqrt{\\frac{\\pi}{\\alpha_j}} \\|\\nabla(\\theta_j \\eta)\\|_{{\\boldsymbol L}^2(U_j\\cap\\Omega)}.\n\\end{array}\n$$\nIf $0<\\alpha_j< \\pi $, one can check that $0<\\beta_j<1$; otherwise, if $\\pi\\le\\alpha_j < 2 \\pi$, one has $1\\le\\beta_j<2$.\n\nWhen $\\boldsymbol{p}_j\\not\\in\\mathcal{V}_h$, one does select $\\alpha_j=\\pi$ to define $H_j\\equiv Id$. Thus \n$$\n\\begin{array}{rcl}\n\\displaystyle\n\\int_{U_i\\cap\\Omega} e^{(\\theta_i\\eta)({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}&=&\\displaystyle\n\\frac{1}{2}\\int_{B(\\boldsymbol{p}_j,\\tilde\\rho_j)} e^{(\\theta_j \\eta)^*({\\boldsymbol x})}\\,{\\rm d}{\\boldsymbol x}\n\\\\\n&\\le&\\displaystyle C |U_j\\cap \\Omega|\ne^{\\frac{1}{16\\pi}\\|\\nabla(\\theta_j \\eta)^*\\|^2_{{\\boldsymbol L}^2(B(\\boldsymbol{p}_j,\\tilde\\rho_j))}}\n\\\\\n&\\le&\\displaystyle C |U_j\\cap \\Omega|\ne^{\\frac{1}{8\\pi}\\|\\nabla(\\theta_i \\eta)\\|^2_{{\\boldsymbol L}^2(U_j\\cap\\Omega)}}\n\\\\\n&=&\\displaystyle C |U_j\\cap \\Omega|\ne^{\\frac{1+\\lambda}{8\\alpha_j}\\|\\nabla\\eta\\|^2_{{\\boldsymbol L}^2(\\Omega)}+ C_{\\lambda} \\|\\eta\\|^2_{L^1(\\Omega)}},\n\\end{array}\n$$\n\nwhere we took $\\beta_j=1$.\n\nDefining $\\beta_\\Omega=\\max_{j=1,\\cdots, J} \\beta_j$ and returning to \\eqref{partition}, we obtain, on using the above estimates, that \n$$\n\\begin{array}{rcl}\n\\displaystyle\n\\int_\\Omega e^{\\eta({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}&\\le&\\displaystyle C \n\\sum_{j=0}^J |U_j\\cap \\Omega|\ne^{\\beta^2_\\Omega\\frac{1+\\lambda}{8\\alpha_\\Omega}\\|\\nabla \\eta\\|^2_{{\\boldsymbol L}^2(\\Omega)}+ C_{\\lambda} \\|\\eta\\|^2_{L^1(\\Omega)}}\n\\\\\n&=&\\displaystyle C |\\Omega| e^ {\\beta^2_\\Omega\\frac{1+\\lambda}{8\\alpha_\\Omega} \\|\\nabla \\eta\\|^2_{{\\boldsymbol L}^2(\\Omega)}+ C_{\\lambda} \\|\\eta\\|^2_{L^1(\\Omega)}}.\n\\end{array}\n$$\n\\end{proof}\n\n\\begin{remark} Inequality \\eqref{ineq:Trudinger-Moser_New} may be improved if some geometrical properties of $\\partial\\Omega$ are taken into account. For instance, if $\\Omega=[0,\\ell]\\times[0,\\ell]$ with $\\ell>0$, it takes the form\n\\begin{equation}\\label{ineq:Trudinger-Moser_New_square}\n\\int_\\Omega e^{\\eta({\\boldsymbol x})}{\\rm d}{\\boldsymbol x}\\le C e^{\\displaystyle\\frac{1+\\lambda}{8\\pi} \\|\\nabla \\eta\\|^2_{L^2(\\Omega)}+C_\\lambda\\|\\eta\\|^2_{L^1(\\Omega)}},\n\\end{equation}\nsince $\\alpha_j=\\frac{\\pi}{2}$ and $\\beta_j=\\frac{\\sqrt{2}}{2}$, where $\\partial \\Omega$ may be recovered by $B(\\boldsymbol{p}_j, \\sqrt{2}\\ell)$, with $\\boldsymbol{p}_j\\in\\mathcal{V}_h$.\nFurthermore, if $\\Omega$ is convex, one has $\\beta_\\Omega=1$. \n\\end{remark}\n\n\nA further generalisation will play a crucial role in connection with the finite element framework. \n\n\\begin{theorem} Let $\\eta_h\\in X_h$ with $\\eta_h>0$. Then, for $\\lambda>0$, there exists a constant $C_\\lambda>0$, independent of $h$, such that\n\\begin{equation}\\label{functional_ineq_I}\n\\int_\\Omega i_h (e^{\\eta_h({\\boldsymbol x})})\\,{\\rm d}{\\boldsymbol x}\\le C (1+ \\|\\nabla \\eta_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}) e^{\\displaystyle\\beta_\\Omega^2\\frac{1+\\lambda}{8\\alpha_\\Omega} \\|\\nabla \\eta_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}+C_\\lambda\\|\\eta_h\\|^2_{L^1(\\Omega)}}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof} By invoking the inequality \\cite[Co. 2.6]{GS_RG_2021}, for any $m\\in\\mathds{N}$, \n$$\n\\|\\eta_h^m-i_h(\\eta_h^m)\\|_{L^1(\\Omega)}\\le\nC m(m-1)h^2 \\int_\\Omega |\\eta_h({\\boldsymbol x})|^{m-2} |\\nabla \\eta_h({\\boldsymbol x})|^2\\, {\\rm d}{\\boldsymbol x},\n$$\nwe have\n\\begin{equation}\\label{co2.8-lab1}\n\\begin{array}{rcl}\n\\displaystyle\n\\int_\\Omega i_h (e^{\\eta_h({\\boldsymbol x})})\\,{\\rm d}{\\boldsymbol x}&=&\\displaystyle\n\\int_\\Omega (1+\\eta_h({\\boldsymbol x}))\\,{\\rm d}{\\boldsymbol x}+\\sum_{m=2}^\\infty\\frac{1}{m!} \\int_\\Omega i_h(\\eta_h^m({\\boldsymbol x})){\\rm d}{\\boldsymbol x}\n\\\\\n&\\le&\\displaystyle\n\\sum_{n=0}^\\infty\\frac{1}{m!} \\int_\\Omega \\eta^m_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}\n \\\\\n&&\\displaystyle +\\sum_{m=2}^\\infty\\frac{C m(m-1) h^2}{m!} \\int_\\Omega |\\nabla \\eta_h({\\boldsymbol x})|^2 \\eta_h^{m-2}({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}\n\\\\\n&=&\\displaystyle\n\\int_\\Omega (1+ C h^2 |\\nabla \\eta_h({\\boldsymbol x})|^2) e^{\\eta_h({\\boldsymbol x})}\\,{\\rm d}{\\boldsymbol x}\n\\\\\n&\\le&\\displaystyle\n(1+ C\\|\\nabla \\eta_h({\\boldsymbol x})\\|^2_{{\\boldsymbol L}^2(\\Omega)}) \\int_\\Omega e^{\\eta_h({\\boldsymbol x})}\\,{\\rm d}{\\boldsymbol x}.\n\\end{array}\n\\end{equation}\nIn the last line the inverse inequality $ \\|\\nabla \\eta_h\\|_{{\\boldsymbol L}^\\infty(\\Omega)}\\le C h^{-1} \\|\\nabla \\eta_h\\|_{{\\boldsymbol L}^2(\\Omega)}$ was utilised. Inequality \\eqref{functional_ineq_I} follows on applying \\eqref{ineq:Trudinger-Moser_New}. \n\\end{proof}\n\nFor convenience, we use $\\chi_\\Omega$ for a shorthand of $\\frac{\\alpha_\\Omega}{\\beta^2_\\Omega}$, where $\\alpha_\\Omega$ and $\\beta_\\Omega$ were defined in Theorem~\\ref{th:Moser-Trundinger}.\n\\begin{corollary} Let $\\phi_h, \\psi_h\\in X_h$ with $\\phi_h, \\psi_h>0$. Then, for any $\\lambda,\\delta, \\mu>0$, there exists $M=M(\\lambda,\\delta, \\Omega)>0$ such that \n\\begin{equation}\\label{functional_ineq_II}\n\\begin{array}{rcl}\n(\\phi_h, \\psi_h)_h&\\le&\\displaystyle\\frac{1}{\\mu} (\\phi_h, \\log\\frac{\\phi_h}{\\bar \\phi_h} )_h\n\\\\\n&&\\displaystyle\n+ \\mu (\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega})\\|\\phi_h\\|_{L^1(\\Omega)} \\|\\nabla \\psi_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\\n&&\\displaystyle\n +M \\mu \\|\\phi_h\\|_{L^1(\\Omega)} \\|\\psi_h\\|^2_{L^1(\\Omega)}\n\\\\\n&&\\displaystyle\n+\\frac{M}{\\mu} \\|\\phi_h\\|_{L^1(\\Omega)}.\n\\end{array} \n\\end{equation}\n\\end{corollary}\n\\begin{proof} On the one hand, from Jensen's inequality, we obtain\n$$\n\\begin{array}{rcl}\n\\displaystyle\n\\log \\int_\\Omega i_h (e^{\\mu \\psi_h({\\boldsymbol x})})\\,{\\rm d}{\\boldsymbol x}&=&\\displaystyle\n\\log \\sum_{i\\in I} \\frac{e^{\\mu\\psi_i} \\|\\phi_h\\|_{L^1(\\Omega)}}{\\phi_i} \\frac{\\phi_i}{\\|\\phi_h\\|_{L^1(\\Omega)}}\\int_\\Omega \\varphi_{{\\boldsymbol a}_i}({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}\n\\\\\n&\\ge&\\displaystyle\n\\sum_{i\\in I} \\log \\left(\\frac{e^{\\mu\\psi_i} \\|\\phi_h\\|_{L^1(\\Omega)}}{\\phi_i}\\right) \\frac{\\phi_i}{ \\|\\phi_h\\|_{L^1(\\Omega)}}\\int_\\Omega \\varphi_{{\\boldsymbol a}_i}({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}\n\\\\\n&=&\\displaystyle\n\\sum_{i\\in I} \\Big(\\mu\\psi_i+\\log \\|\\phi_h\\|_{L^1(\\Omega)}- \\log \\phi_i\\Big)\\frac{\\phi_i}{\\|\\phi_h\\|_{L^1(\\Omega)}}\\int_\\Omega \\varphi_{{\\boldsymbol a}_i}({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}\n\\\\\n&=&\\displaystyle\n\\frac{\\mu}{ \\|\\phi_h\\|_{L^1(\\Omega)}} (\\phi_h, \\psi_h)_h-\\frac{1}{ \\|\\phi_h\\|_{L^1(\\Omega)}} (\\phi_h, \\log\\frac{\\phi_h}{\\bar\\phi_h} )_h+\\log |\\Omega|.\n\\end{array}\n$$\nOn the other hand, from \\eqref{functional_ineq_I}, we get \n$$\n\\log \\int_\\Omega i_h (e^{\\mu \\psi_h({\\boldsymbol x})})\\,{\\rm d}{\\boldsymbol x}\\le \\log \\frac{C}{\\delta}+ \\log(\\delta(1+ \\mu^2\\|\\nabla \\psi_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}))+ \\frac{(1+\\lambda)\\mu^2}{8\\chi_\\Omega} \\|\\nabla \\psi_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}+C_\\lambda \\mu^2\\|\\psi_h\\|^2_{L^1(\\Omega)},\n$$\nwhence \n$$\n\\log \\int_\\Omega i_h (e^{\\mu \\psi_h({\\boldsymbol x})})\\,{\\rm d}{\\boldsymbol x}\\le \\log \\frac{C}{\\delta}+\\delta+(\\delta +\\frac{1+\\lambda}{8\\chi_\\Omega})\\mu^2 \\|\\nabla \\psi_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}+C_\\lambda \\mu^2\\|\\psi_h\\|^2_{L^1(\\Omega)}.\n$$\nCombining the above inequalities yields \n$$\n\\begin{array}{rcl}\n(\\phi_h, \\psi_h)_h&\\le&\\displaystyle\n\\frac{1}{\\mu} (\\phi_h, \\log\\frac{\\phi_h}{\\bar \\phi_h} )_h\n\\\\\n&&\\displaystyle\n+\\frac{\\|\\phi_h\\|_{L^1(\\Omega)}}{\\mu}(\\log\\frac{C }{\\delta |\\Omega|}+\\delta)\n\\\\\n&&\\displaystyle\n+ (\\delta+\\frac{1+\\lambda}{8\\chi_\\Omega}) \\mu \\|\\phi\\|_{L^1(\\Omega)} \\|\\nabla \\psi_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\\n&&\\displaystyle\n+C_\\lambda \\mu \\|\\phi\\|_{L^1(\\Omega)} \\|\\psi_h\\|^2_{L^1(\\Omega)}.\n\\end{array}\n$$\nSelecting $M=\\max\\{\\log\\frac{C_\\lambda }{\\delta |\\Omega|}+\\delta, C_\\lambda\\}$ completes the proof. \n\\end{proof}\n\\begin{corollary} Let $\\phi_h\\in X_h$ with $\\phi_h>0$. Then, for any $\\lambda>0$, there exists $M=M(\\lambda, \\Omega)>0$ such that\n\\begin{equation}\\label{functional_ineq_III}\n\\begin{array}{rcl}\n(\\phi_h, \\log(\\phi_h+1))_h&\\le&\\displaystyle\n4(\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega}) \\|\\phi_h\\|_{L^1(\\Omega)} \\|\\nabla i_h(\\log(\\phi_h+1))\\|^2_{{\\boldsymbol L}^2(\\Omega)}+4 M \\|\\phi_h\\|_{L^1(\\Omega)}^3\n\\\\\n&&\\displaystyle\n+(M-\\log \\bar\\phi_h) \\|\\phi_h\\|_{L^1(\\Omega)}.\n\\end{array}\n\\end{equation}\n\\end{corollary}\n\\begin{proof} Select $\\psi_h=i_h(\\log(\\phi_h+1))$ and $\\mu=2$ in \\eqref{functional_ineq_II} to get\n$$\n\\begin{array}{rcl}\n(\\phi_h, \\log(\\phi_h+1))_h&\\le&\\displaystyle\n\\frac{1}{2} (\\phi_h, \\log\\frac{\\phi_h}{\\bar \\phi_h} )_h+\\frac{M}{2} \\|\\phi_h\\|_{L^1(\\Omega)}\n\\\\\n&&\\displaystyle\n + 2 (\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega}) \\|\\phi_h\\|_{L^1(\\Omega)} \\|\\nabla i_h(\\log(\\phi_h+1))\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n \\\\\n &&\n \\displaystyle\n +2 M\\|\\phi_h\\|_{L^1(\\Omega)} \\|i_h(\\log(\\phi_h+1))\\|^2_{L^1(\\Omega)}.\n\\end{array} \n$$\nOn noting that \n$$\n\\begin{array}{rcl}\n\\displaystyle\n\\frac{1}{2}(\\phi_h, \\log\\frac{\\phi_h}{\\bar \\phi_h})_h&=&\\displaystyle\n\\frac{1}{2}(\\phi_h, \\log\\phi_h)_h-\\frac{1}{2} \\log \\bar\\phi_h\\|\\phi_h\\|_{L^1(\\Omega)} \n\\\\\n&\\le&\\displaystyle\n\\frac{1}{2}(\\phi_h, \\log(\\phi_h+1))_h-\\frac{1}{2}\\log\\bar\\phi_h \\|\\phi_h\\|_{L^1(\\Omega)}\n\\end{array}\n$$\nand that $\\| i_h(\\log(\\phi_h+1))\\|_{L^1(\\Omega)}\\le \\|\\phi_h\\|_{L^1(\\Omega)}$, we complete the proof.\n\\end{proof}\n\nWe end up this section with a pointwise inequality.\n\\begin{proposition} It follows that, for all $x, y>0$, \n\\begin{equation}\\label{ineq:log}\n(\\log(x+1)-\\log(y+1))^2\\le \\frac{(x-y)^2}{(x+1)(y+1)}\n\\end{equation}\nholds.\n\\end{proposition}\n\\begin{proof} Let $g(z)= (z-1)^2-z \\log^2 z$. We know that $g(z)\\ge 0$ for $z>0$ since $g$ attains its unique global minimum $0$ at $z=1$. Therefore, we deduce that $\\log^2 z\\le \\frac{(z-1)^2}{z}$ for all $z>0$. Taking $z=\\frac{x+1}{y+1}$ completes the proof.\n\\end{proof}\n\n\\section{Main results}\nIn this section the main theoretical results of this work are stated and proved: lower bounds, time-independent and -dependent integrability bounds. These will be successfully attained as an upshot of the new discretisation for the convective and chemotaxis terms in \\eqref{eq:n_h} and of the design of the stabilising terms $B_n$ and $B_c$ in \\eqref{Bn} and $\\eqref{Bc}$, which have been neatly devised. Nor should the role of the functional inequalities \\eqref{functional_ineq_II} and \\eqref{functional_ineq_III} be forgotten, which will be of great importance; its last consequence is deriving a priori bounds under a smallness condition for $\\|n_0\\|_{L^1(\\Omega)}$.\n\n\n\\subsection{Lower bounds} We open our discussion with the proof of the lower bounds for $(n^{m+1}_h, c^{m+1}_h)$, because they are closely connected with the $L^1(\\Omega)$ bounds and later on with the a priori energy bounds. \n\\begin{lemma} It follows that the discrete solution pair $(n^{m+1}_h, c^{m+1}_h)$ of \\eqref{eq:n_h}-\\eqref{eq:c_h} satisfies \n\\begin{equation}\\label{lower_bounds}\nn^{m+1}_h>0\\quad\\mbox{ and }\\quad c^{m+1}_h\\ge 0.\n\\end{equation} \n\\end{lemma}\n\\begin{proof} We establish \\eqref{lower_bounds} by induction on $m$; the case $m=0$ being entirely analogue to the general one is omitted. It will be firstly shown that $n^{m+1}_h$ cannot take non-positive values. Suppose the contrary at a certain node $\\boldsymbol{a}_i\\in\\mathcal{N}_h$, i.e., $n^{m+1}_i:=n^{m+1}_h(\\boldsymbol{a}_i)\\le 0$, which is a local minimum. For the sake of simplicity and with no loss in generality, one may order the indexes such $in^{m+1}_i\\ge n^m_i >0. \n$$ \nThis gives a contradiction since $n^m_i>0$ by the induction hypothesis. \n\nWith little change in argument, one can prove $c^{m+1}_h>0$. Just as before, let ${\\boldsymbol a}_i\\in\\mathcal{N}_h$ be a local minimum of $c^{m+1}_h$ such that $c^{m+1}_i<0$ and set $\\bar c_h=\\varphi_{{\\boldsymbol a}_i}$ in \\eqref{eq:c_h} to find \n$$\n\\begin{array}{c}\n\\displaystyle\n \\sum_{j\\in I(\\Omega_{{\\boldsymbol a}_i})} c^{n+1}_j\\Big[(1+k^{-1})(\\varphi_{{\\boldsymbol a}_j},\\varphi_{{\\boldsymbol a}_i})+(\\nabla\\varphi_{{\\boldsymbol a}_j},\\nabla\\varphi_{{\\boldsymbol a}_i})+({\\boldsymbol u}^m_h\\cdot\\nabla \\varphi_{{\\boldsymbol a}_j},\\varphi_{{\\boldsymbol a}_i}) \n \\\\\n \\displaystyle\n+\\frac{1}{2} (\\nabla\\cdot{\\boldsymbol u}^m_h\\,\\varphi_{{\\boldsymbol a}_j}, \\varphi_{{\\boldsymbol a}_i})+ (B_c(c^{m+1}_h, {\\boldsymbol u}^m_h)\\varphi_{{\\boldsymbol a}_j},\\varphi_{{\\boldsymbol a}_i}) \\Big]= n^{n+1}_i(1,\\varphi_{{\\boldsymbol a}_i}) + k^{-1}\\sum_{j\\in I(\\Omega_{{\\boldsymbol a}_i})} c^{m}_j (\\varphi_{{\\boldsymbol a}_j},\\varphi_{{\\boldsymbol a}_i}).\n\\end{array}\n$$\nOn account of \\eqref{Bc} and $\\alpha_i(c^{m+1}_h)=1$ from Lemma \\ref{lm: alpha_i}, one has \n\\begin{align*}\nk^{-1}(\\varphi_{{\\boldsymbol a}_j}, \\varphi_{{\\boldsymbol a}_i})+({\\boldsymbol u}^m_h\\cdot\\nabla\\varphi_{{\\boldsymbol a}_j}, \\varphi_{{\\boldsymbol a}_i})+\\frac{1}{2}(\\nabla\\cdot {\\boldsymbol u}^m_h \\varphi_{{\\boldsymbol a}_j}, \\varphi_{{\\boldsymbol a}_i}) &\n\\\\\n+(\\nabla\\varphi_{{\\boldsymbol a}_j},\\nabla\\varphi_{{\\boldsymbol a}_i})+ (\\varphi_{{\\boldsymbol a}_j}, \\varphi_{{\\boldsymbol a}_i}) + (B_c(c^{m+1}_h, {\\boldsymbol u}^m_h)\\varphi_{{\\boldsymbol a}_j},\\varphi_{{\\boldsymbol a}_i}) &\\le 0\\quad \\mbox{ for }\\quad i\\not=j.\n\\end{align*}\nThus, \n$$\n\\begin{array}{rcl}\n\\displaystyle\n\\sum_{j\\in I(\\Omega_{{\\boldsymbol a}_i})} c^{n+1}_i\\Big[(1+k^{-1})(\\varphi_{{\\boldsymbol a}_j},\\varphi_{{\\boldsymbol a}_i}) +(\\nabla\\varphi_{{\\boldsymbol a}_j},\\nabla\\varphi_{{\\boldsymbol a}_i}) \n+({\\boldsymbol u}^m_h\\cdot\\nabla\\varphi_{{\\boldsymbol a}_j}, \\varphi_{{\\boldsymbol a}_i})\n\\\\\n\\displaystyle\n+\\frac{1}{2}(\\nabla\\cdot {\\boldsymbol u}^m_h \\varphi_{{\\boldsymbol a}_j}, \\varphi_{{\\boldsymbol a}_i})\n+ (B_c(c^{m+1}_h)\\varphi_{{\\boldsymbol a}_j},\\varphi_{{\\boldsymbol a}_i}) \\Big] & \\ge &\n\\\\\n\\displaystyle\n\\sum_{j\\in I(\\Omega_{{\\boldsymbol a}_i})} (\\varphi_{{\\boldsymbol a}_j},\\varphi_{{\\boldsymbol a}_i}) (k^{-1}c_j^m + n^{m+1}_j) &\\ge& 0.\n\\end{array}\n$$\nObserve that $(\\nabla1,\\nabla \\varphi_{{\\boldsymbol a}_j})=0$, $({\\boldsymbol u}^m_h\\cdot\\nabla 1, \\varphi_{{\\boldsymbol a}_i})+\\frac{1}{2}(\\nabla\\cdot {\\boldsymbol u}^m_h, \\varphi_{{\\boldsymbol a}_i})=0$ and $(B_c(c^{m+1}_h, {\\boldsymbol u}^m_h) 1,\\varphi_{{\\boldsymbol a}_i})=0$ from \\eqref{eq:p_h} and \\eqref{Bc}-\\eqref{def:nu^c_ii}, respectively, which in turn imply that \n$$\n0>c^{m+1}_i (1+k^{-1}) \\ge (n_j^{m+1}+k^{-1}c_j^m ) \\ge 0,\n$$\nwhich is a contradiction; thereby closing the proof.\n\\end{proof}\n\n\\begin{remark} Now that it is known that $n^{m+1}_h$ is positive, we are allowed to rule out the truncating operator $[\\cdot]_+$ in \\eqref{def: gamma_ij_chem} defining \\eqref{New_KS_term}. Furthermore, we can consider $\\gamma^{\\rm c}_{ji}$ defined as in \\eqref{def: gamma_ij_conv not extended} rather than in \\eqref{def: gamma_ij_conv extended}, since $g_\\varepsilon(n^{m+1}_h+1)$ is nothing but $\\log(n^{m+1}_h+1)$ from our choice of $\\varepsilon$.\n\\end{remark}\n\n\n\\subsection{\\texorpdfstring{$L^1(\\Omega)$}{Lg} bounds}\nThe $L^1(\\Omega)$ bounds for $(n^{m+1}_h, c^{m+1}_h)$ are the second step to regarding from \\eqref{eq:n_h} and \\eqref{eq:c_h}. They are somehow naturally inherited from our starting algorithm; namely, from equations \\eqref{eq_aux:n_h} and \\eqref{eq_aux:c_h} and from the conservation structures \\eqref{def:nu^n_ii} and \\eqref{def:nu^c_ii} for the stabilising operators $B_n$ and $B_c$ in \\eqref{Bn} and \\eqref{Bc}, respectively. \n\\begin{lemma}[$L^1(\\Omega)$-bounds] There holds that the discrete solution pair $(n^{m+1}_h, c_h^{n+1})\\in N_h\\times C_h$ computed via \\eqref{eq:n_h} and \\eqref{eq:c_h} fulfils\n\\begin{equation}\\label{L1-Bound-nh}\n\\|n^{m+1}_h\\|_{L^1(\\Omega)}=\\|n_h^0\\|_{L^1(\\Omega)}\n\\end{equation}\nand\n\\begin{equation}\\label{L1-Bound-ch}\n\\|c^{m+1}_h\\|_{L^1(\\Omega)}\\le \\frac{1}{(1+k)^{m+1}} \\|c^0_h\\|_{L^1(\\Omega)}+(1-\\frac{1}{(1+k)^m})\\|n^0_h\\|_{L^1(\\Omega)}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} On choosing $\\bar n_h=1$ in \\eqref{eq:n_h} and on noting that $(B_n(n^{m+1}_h,{\\boldsymbol u}^m_n) n^{m+1}_h, 1)=0$ holds, it follows immediately after a telescoping cancellation that\n\\begin{equation}\\label{lm5.2-lab1}\n\\int_\\Omega n^{m+1}_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}=\\int_\\Omega n^0_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}.\n\\end{equation}\nConsequently we get that \\eqref{L1-Bound-nh} holds by the positivity of $n^{n+1}_h$ and $n^0_h$. Likewise let $\\bar c_h=1$ in \\eqref{eq:c_h} to get, on noting $(B_c(n^{m+1}_h,{\\boldsymbol u}^m_n) c^{m+1}_h, 1)=0$ as well, that \n$$\n\\int_\\Omega c^{m+1}_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}+k \\int_\\Omega c^{m+1}_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}=\\int_\\Omega c^m_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}+k\\int_\\Omega n^{m+1}_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}.\n$$\nA simple calculation shows that\n$$\n\\int_\\Omega c^{m+1}_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}=\\frac{1}{(1+k)^{m+1}} \\int_\\Omega c^0_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}+\\left(\\int_\\Omega n^0_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}\\right) \\sum_{j=1}^{m+1}\\frac{k}{(1+k)^j} ,\n$$\nwhere use was made of \\eqref{lm5.2-lab1}. Thus we infer that \n$$\n\\int_\\Omega c^{m+1}_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}=\\frac{1}{(1+k)^{m+1}} \\int_\\Omega c^0_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}+ (1-\\frac{1}{(1+k)^m})\\int_\\Omega n^0_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x},\n$$\nthereby concluding the proof of \\eqref{L1-Bound-ch} by the non-negativity of $c^{m+1}_h$ and $c^0_h$ and again the positivity of $n^0_h$. \n\\end{proof}\n\n\\subsection{A priori energy bounds}\nWe turn finally to a discussion of our third objective, which is showing a priori bounds for $(n^{m+1}_h, c^{m+1}_h, {\\boldsymbol u}^{m+1}_h)$ as an application of \\eqref{functional_ineq_II} and \\eqref{functional_ineq_III}. We will need on the way to assume that $\\mathcal{T}_h$ is weakly acute, i.e. if every sum of two angles opposite to an interior edge does not exceed $\\pi$. As a result, there holds\n\\begin{equation}\\label{cond:acute}\n(\\nabla\\varphi_{{\\boldsymbol a}_i}, \\nabla\\varphi_{{\\boldsymbol a}_j})\\le 0 \\quad \\mbox{ for all } \\quad i\\not= j\\in I.\n\\end{equation}\n\nLet us define the following energy-like functional: \n$$\n\\mathcal{F}_h(n_h, c_h) =-(\\log(n_h+1), 1)_h+\\|c_h\\|^2_{L^2(\\Omega)}.\n$$\nOur first step is determining the evolution of this functional, which is independent of the fluid part. \n\\begin{lemma}\\label{lm:a_priori_KS} Let $n_h^0\\in N_h$ be such that \n\\begin{equation}\\label{mass_condition: n_0}\n\\int_\\Omega n_h^0({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}<2\\chi_\\Omega.\n\\end{equation} \nThen there exist two constants $F(n^0_h,c^0_h)>0$ and $ D(n^0_h)>0$ such that the discrete solution pair $(n^{m+1}_h, c^{m+1}_h)\\in N_h\\times C_h$ computed via \\eqref{eq:n_h} and \\eqref{eq:c_h} satisfies \n\\begin{equation}\\label{ineq:energy_nh_ch}\n\\begin{array}{rcl}\n\\displaystyle\n\\mathcal{F}(n^m_h, c^m_h)-\\mathcal{F}(n^{m+1}_h, c^{m+1}_h)&&\n\\\\\n\\displaystyle\n+ \\frac{k}{2} \\Big(\\frac{1}{(n^{m+\\theta}_h)^2},(\\delta_t u^{m+1}_h)^2\\Big)_h+ k D(n^0_h) \\|\\nabla i_h(\\log (n^{m+1}_h+1))\\|^2_{{\\boldsymbol L}^2(\\Omega)}&&\n\\\\\n\\displaystyle\n+k D(n^0_h)\\|\\nabla c^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}+\\lambda (n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\bar n^0_h} )_h&\\le& F(n^0_h, c^0_h).\n\\end{array}\n\\end{equation} \n\\end{lemma}\n\\begin{proof} Select $\\bar n_h=-i_h \\frac{1}{n^{m+1}_h+1}$ in \\eqref{eq:n_h} and $\\bar c_h= c^{m+1}_h$ in \\eqref{eq:c_h} to get\n\\begin{equation}\\label{lm4.2-lab1}\n\\begin{array}{rcl}\n\\displaystyle\n-(\\delta_t n^{m+1}_h, i_h \\frac{1}{n^{m+1}_h+1})_h&+&\\displaystyle\n(n^{m+1}_h{\\boldsymbol u}^m_h, \\nabla i_h \\frac{1}{n^{m+1}_h+1})_*\n\\\\\n&-&\\displaystyle\n(\\nabla n^{m+1}_h, \\nabla i_h \\frac{1}{n^{m+1}_h+1})\n\\\\\n&+&\\displaystyle\n(n^{m+1}_h\\nabla c^{m+1}_h, \\nabla i_h \\frac{1}{n^{m+1}_h+1})_*\n\\\\\n&-&\\displaystyle\n(B_n(n^{m+1}_h, {\\boldsymbol u}^m_h) n^{m+1}_h,i_h \\frac{1}{n^{m+1}_h+1})=0\n\\end{array}\n\\end{equation}\nand\n\\begin{equation}\\label{lm4.2-lab2}\n\\frac{1}{2 k}\\|c^{m+1}_h\\|^2_{L^2(\\Omega)}-\\frac{1}{2 k}\\|c^m_h\\|^2_{L^2(\\Omega)}+\\frac{1}{2k}\\|c^{m+1}_h-c^m_h\\|^2_{L^2(\\Omega)}\n+\\|c^{m+1}_h\\|^2_{H^1(\\Omega)}=(n_h^{m+1}, c^{m+1}_h)_h.\n\\end{equation}\n\nTaylor's theorem applied to $\\log(s+1)$ for $n^{m+1}_h$ and evaluated at $n^m_h$ gives\n$$\n\\log(n^m_h+1)=\\log(n^{m+1}_h+1)+\\frac{1}{n^{m+1}_h+1}(n^m_h-n^{m+1}_h)-\\frac{1}{2(n^{m+\\theta}_h)^2}(n^{m}_h-n^{m+1}_h)^2,\n$$\nwhere $\\theta\\in(0,1)$ is such that $n^{n+\\theta}_h=\\theta n^{m+1}_h+(1-\\theta) n^n_h$. Hence, \n$$\n-\\Big(\\delta_t n^{m+1}_h, \\frac{1}{n^m_h+1}\\Big)_h=\\frac{1}{k}\\Big(\\log(n^m_h+1),1\\Big)_h-\\frac{1}{k}\\Big(\\log(n^{m+1}_h+1),1\\Big)_h+ \\frac{k}{2} \\Big(\\frac{1}{(n^{m+\\theta}_h)^2},(\\delta_t n^{m+1}_h)^2\\Big)_h.\n$$\n\nThe convective term is handled, on noting \\eqref{New_conv_term: extended} together with \\eqref{def: gamma_ij_conv not extended}, as: \n$$\n\\begin{array}{rcl}\n\\displaystyle\n-( n^{m+1}_h{\\boldsymbol u}^m_h, \\nabla i_h \\frac{1}{n^{m+1}_h+1})_*&=&\\displaystyle\n\\sum_{i,j\\in I} (\\log(n^{m+1}_j+1)-\\log(n^{m+1}_i+1))\n(\\varphi_{{\\boldsymbol a}_j} {\\boldsymbol u}^m_h, \\nabla\\varphi_{{\\boldsymbol a}_i})\n\\\\\n&=&\n-({\\boldsymbol u}^m_h,\\nabla i_h(\\log(n^{m+1}_h+1))=0,\n\\end{array}\n$$\nwhere we used twice that $({\\boldsymbol u}^m_h, \\nabla \\bar n_h)=0$, for all $\\bar n_h\\in N_h$, from the incompressibility condition \\eqref{eq:p_h}.\n\nThe diffusion term is treated as:\n$$\n-(\\nabla n^{m+1}_h, \\nabla i_h \\frac{1}{n^{m+1}_h+1})=-\\sum_{i0$, for $ i\\not=j$, from \\eqref{def:nu_n}.\n\nPlugging all the above computations yields \n\\begin{equation}\\label{ineq:energy_I}\n\\begin{array}{rcl}\n\\displaystyle\n\\mathcal{F}_h(n^{m+1}_h, c^{m+1}_h)-\\mathcal{F}_h(n^m_h, c^m+_h)&&\n\\\\\n\\displaystyle\n+ \\frac{k^2}{2} \\Big(\\frac{1}{(n^{n+\\theta}_h)^2},(\\delta_t u^{n+1}_h)^2\\Big)_h-k (B_n(n^{m+1}_h, {\\boldsymbol u}^m_h) n^{m+1}_h, \\frac{1}{n^{m+1}_h+1})&&\n\\\\\n\\displaystyle\n-\\frac{k}{2}\\sum_{i0$ such that \n\\begin{equation}\\label{ineq:mu}\n\\frac{\\|n^0_h\\|_{L^1(\\Omega)}}{\\chi_\\Omega}<\\mu<\\frac{4\\chi_\\Omega}{\\|n^0_h\\|_{L^1(\\Omega)}}.\n\\end{equation}\nThus we can select $\\delta, \\lambda >0$ such that \n\\begin{equation}\\label{ineq:mu_eps_I}\n4(\\lambda+\\frac{1}{\\mu}) \\left(\\delta+\\frac{1+\\lambda}{8\\chi_\\Omega}\\right)\\|n^0_h\\|_{L^1(\\Omega)}<\\frac{1}{2}\n\\end{equation}\nand\n\\begin{equation}\\label{ineq:mu_eps_II}\n\\mu\\left(\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega}\\right)\\|n^0_h\\|_{L^1(\\Omega)}<\\frac{1}{2}.\n\\end{equation}\nIn view of \\eqref{functional_ineq_II} and \\eqref{L1-Bound-nh} for the above election of $\\lambda$ and $\\mu$, we have\n$$\n\\begin{array}{rcl}\n(n^{m+1}_h, c^{m+1}_h)_h&\\le&\\displaystyle\n\\frac{1}{\\mu} (n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\|n^0_h\\|_{L^1(\\Omega)}} )_h\n\\\\\n&&\\displaystyle\n + \\mu (\\delta + \\frac{(1+\\lambda)}{8\\chi_\\Omega}) \\|n^0_h\\|_{L^1(\\Omega)} \\|\\nabla c^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\ \n&&\\displaystyle\n+M \\mu\\|n^0_h\\|_{L^1(\\Omega)} \\|c^{m+1}_h\\|^2_{L^1(\\Omega)}\n\\\\\n&&\\displaystyle\n+\\frac{M}{\\mu} \\|n^0_h\\|_{L^1(\\Omega)}.\n\\end{array} \n$$\nWe continue by applying \\eqref{functional_ineq_III} to get \n\\begin{equation}\\label{ineq:rhs}\n\\begin{array}{rcl}\n(n^{m+1}_h, c^{m+1}_h)_h&+&\\displaystyle\\lambda (n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\bar n_{h}^0})_h\n\\\\\n&\\le&\\displaystyle\n(\\lambda+\\frac{1}{\\mu}) (n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\|n^0_h\\|_{L^1(\\Omega)}} )_h\n\\\\\n&&\\displaystyle\n + \\mu (\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega}) \\|n^0_h\\|_{L^1(\\Omega)} \\|\\nabla c^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\ \n&&\\displaystyle\n+M \\mu\\|n^0_h\\|_{L^1(\\Omega)} \\|c^{m+1}_h\\|^2_{L^1(\\Omega)}\n\\\\\n&&\\displaystyle\n+\\frac{M}{\\mu} \\|n^0_h\\|_{L^1(\\Omega)}\n\\\\\n&\\le&\\displaystyle\n(\\lambda+\\frac{1}{\\mu})\\Big\\{4(\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega}) \\|n^0_h\\|_{L^1(\\Omega)} \\|\\nabla i_h(\\log(\\phi_h+1))\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\\n&&\n+4 M \\|n^0_h\\|_{L^1(\\Omega)}^3\n\\\\\n&&\n+(M-\\log \\bar n^0_h) \\|n^0_h\\|_{L^1(\\Omega)}\\Big\\}\n\\\\\n&&\\displaystyle\n-(\\lambda+\\frac{1}{\\mu})\\|n^0_h\\|_{L^1(\\Omega)}\\log \\bar n^0_h\n\\\\\n&&\\displaystyle\n+ \\mu(\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega}) \\|n^0_h\\|_{L^1(\\Omega)} \\|\\nabla c^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\ \n&&\\displaystyle\n+ M \\mu\\|n^0_h\\|_{L^1(\\Omega)} \\|c^{m+1}_h\\|^2_{L^1(\\Omega)}\n\\\\\n&&\\displaystyle\n+\\frac{M}{\\mu} \\|n^0_h\\|_{L^1(\\Omega)}\n\\\\\n&\\le&\\displaystyle\n4(\\lambda+\\frac{1}{\\mu})(\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega}) \\|n^0_h\\|_{L^1(\\Omega)} \\|\\nabla i_h(\\log(n^{m+1}_h+1))\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\\n&&\\displaystyle\n+4 (\\lambda+\\frac{1}{\\mu}) M \\|n^0_h\\|_{L^1(\\Omega)}^3\n\\\\\n&&\\displaystyle\n+ M (\\lambda+\\frac{1}{\\mu}) \\|n^0_h\\|_{L^1(\\Omega)}\n\\\\\n&&\\displaystyle\n + \\mu (\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega}) \\|n^0_h\\|_{L^1(\\Omega)} \\|\\nabla c^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\ \n&&\\displaystyle\n+M \\mu\\|n^0_h\\|_{L^1(\\Omega)} \\|c^{m+1}_h\\|^2_{L^1(\\Omega)}\n\\\\\n&&\\displaystyle\n+\\frac{M}{\\mu} \\|n^0_h\\|_{L^1(\\Omega)}\n\\\\\n&&\\displaystyle\n-2 (\\lambda+\\frac{1}{\\mu}) \\|n^0_h\\|_{L^1(\\Omega)} \\log_- \\bar n^0_h.\n\\end{array} \n\\end{equation}\nOn account of \\eqref{L1-Bound-ch} and \\eqref{ineq:mu}, one has\n$$\n\\begin{array}{rcl}\nM \\mu\\|n^0_h\\|_{L^1(\\Omega)} \\|c^{m+1}_h\\|^2_{L^1(\\Omega)}&\\le& \\displaystyle\nM \\mu \\|n^0_h\\|_{L^1(\\Omega)} \\Big(\\frac{1}{(1+k)^{m+1}} \\|c^0_h\\|_{L^1(\\Omega)}+\\|n^0_h\\|_{L^1(\\Omega)}\\Big)^2\n\\\\\n&\\le& \\displaystyle\nM \\mu \\|n^0_h\\|_{L^1(\\Omega)} \\Big(2 \\|n^0_h\\|^2_{L^1(\\Omega)}+ \\frac{2}{(1+k)^{2(n+1)}} \\|c^0_h\\|^2_{L^1(\\Omega)}\\Big).\n\\end{array}\n$$\nFinally, we deduce, from \\eqref{ineq:energy_I}, \\eqref{ineq:rhs} and \\eqref{ineq:log}, that \n\\begin{align*}\n\\displaystyle\n\\mathcal{F}(n^m_h, c^{m+1}_h)-\\mathcal{F}(n^{m+1}_h, c^m_h)+&D_n k\\|\\nabla i_h(\\log (n^{m+1}_h+1))\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\\n\\displaystyle\n+k D_c\\|\\nabla c^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n&+\\lambda k(n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\bar n^0_h} )_h\n\\\\\n\\le&\\displaystyle\n4 (\\lambda+\\frac{1}{\\mu}+\\frac{\\mu}{2}) M \\|n^0_h\\|_{L^1(\\Omega)}^3\n+ M (\\lambda+\\frac{2}{\\mu}) \\|n^0_h\\|_{L^1(\\Omega)}\n\\\\\n&+2 M \\mu \\|n^0_h\\|_{L^1(\\Omega)} \\|c^0_h\\|^2_{L^1(\\Omega)}\n-2 (\\lambda+\\frac{1}{\\mu}) \\|n^0_h\\|_{L^1(\\Omega)}\\log_- \\bar n^0_h\n\\\\\n:=&F(n^0_h, c^0_h),\n\\end{align*}\nwhere \n$$\nD_n=\\frac{1}{2}-4(\\frac{1}{\\mu}+\\lambda)(\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega})\\|n^0_h\\|_{L^1(\\Omega)}>0\n$$\nand\n$$\nD_c=\\frac{1}{2}-\\mu(\\delta + \\frac{1+\\lambda}{8\\chi_\\Omega})\\|n^0_h\\|_{L^1(\\Omega)}>0.\n$$\nChoosing $D=\\min\\{D_n, D_c, \\lambda\\}$ concludes the proof. \n\\end{proof}\nIn the context of the Navier--Stokes equations, a control of the $L^2(\\Omega)$-norm for the velocity components is a basic estimate to be dealt with. For the Keller--Segel--Navier--Stokes equations, it will be so as well. As the convective term vanishes together with the pressure term when tested against the solution itself owing to the incompressibility condition, the fundamental term to be treated is the potential one for which some functional inequalities need to be invoked. \n\n\\begin{lemma}\\label{lm:a_priori_NS} It follows that there exists $K>0$ such that the solution ${\\boldsymbol u}^{m+1}_h$ to \\eqref{eq:u_h} fulfils \n\\begin{equation}\\label{ineq:energy_uh}\n\\frac{1}{k}\\|{\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}-\\frac{1}{k}\\|{\\boldsymbol u}^m_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}+\\|\\nabla{\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\\le K \\|n^0_h\\|_{L^1(\\Omega)} (n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\bar n^0_h})_h+K \\|n^0_h\\|^2_{L^1(\\Omega)}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} Substituting $\\bar{\\boldsymbol u}_h={\\boldsymbol u}^{m+1}_h$ into \\eqref{eq:u_h} and $\\bar p_h=p^{m+1}_h$ into \\eqref{eq:p_h} and combining both resulting equations yields \n\\begin{equation}\\label{lm4.5-lab1}\n\\frac{1}{2k}\\|{\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}-\\frac{1}{2k}\\|{\\boldsymbol u}^m_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}+\\|\\nabla{\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}=(n^{m+1}_h\\nabla\\Phi, {\\boldsymbol u}^{m+1}_h)_h.\n\\end{equation}\nNow the right-hand side is estimated as follows. On noting the inequality that results from applying \\eqref{functional_ineq_II} for $\\lambda=1$ and that\n$$\n\\|{\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^1(\\Omega)}\\le |\\Omega| \\|{\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\\le C_{P} |\\Omega| \\|\\nabla {\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)},\n$$\none obtains \n$$\n\\begin{array}{rcl}(n^{m+1}_h\\nabla\\Phi, {\\boldsymbol u}^{m+1}_h)_h&\\le&\\displaystyle\n\\|\\nabla\\Phi\\|_{{\\boldsymbol L}^\\infty(\\Omega)} \\sum_{\\ell=1,2} (n^{m+1}_h,|u_{h\\ell}|)\n\\\\\n&\\le&\\displaystyle\\frac{2}{\\mu} \\|\\nabla\\Phi\\|_{{\\boldsymbol L}^\\infty(\\Omega)}(n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\bar n^0_h})_h\n\\\\\n&&\\displaystyle\n+ 2\\mu (\\delta + \\frac{1}{4\\chi_\\Omega})\\|\\nabla\\Phi\\|_{{\\boldsymbol L}^\\infty(\\Omega)}\\|n^0_h\\|_{L^1(\\Omega)} \\|\\nabla {\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\\n&&\\displaystyle\n +2M \\mu \\|\\nabla\\Phi\\|_{{\\boldsymbol L}^\\infty(\\Omega)} \\|n^0_h\\|_{L^1(\\Omega)} \\|{\\boldsymbol u}^{m+1}_h\\|^2_{L^1(\\Omega)}\n\\\\\n&&\\displaystyle\n+\\frac{2M}{\\mu} \\|\\nabla\\Phi\\|_{{\\boldsymbol L}^\\infty(\\Omega)} \\|n^0_h\\|_{L^1(\\Omega)}\n\\\\\n&\\le&\\displaystyle\\frac{2}{\\mu} \\|\\nabla\\Phi\\|_{{\\boldsymbol L}^\\infty(\\Omega)}(n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\bar n^0_h} )_h\n\\\\\n&&\\displaystyle\n+ 2\\mu \\Big(\\delta + \\frac{1}{4\\chi_\\Omega}+M C_P |\\Omega|\\Big)\\|\\nabla\\Phi\\|_{{\\boldsymbol L}^\\infty(\\Omega)} \\|n^0_h\\|_{L^1(\\Omega)} \\|\\nabla {\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}\n\\\\\n&&\\displaystyle\n+\\frac{2M}{\\mu} \\|\\nabla\\Phi\\|_{{\\boldsymbol L}^\\infty(\\Omega)} \\|n^0_h\\|_{L^1(\\Omega)}.\n\\end{array} \n$$\nRearranging the above inequality, on denoting $\\tilde K=2\\Big(\\delta + \\frac{1}{4\\chi_\\Omega}+M C_P |\\Omega|\\Big)\\|\\nabla\\Phi\\|_{{\\boldsymbol L}^\\infty(\\Omega)} $ and taking $\\mu=\\frac{1}{2 \\|n^0_h\\|_{L^1(\\Omega)} K(n^0_h)}$, there results finally \n\\begin{equation}\\label{lm4.5-lab2}\n(n^{m+1}_h\\nabla\\Phi, {\\boldsymbol u}^{m+1}_h)_h\\le 4 \\tilde K\\|n^0_h\\|_{L^1(\\Omega)} (n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\bar n^0_h})+\\frac{1}{2}\\|\\nabla {\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}+4 \\tilde K M \\|n^0_h\\|^2_{L^1(\\Omega)}.\n\\end{equation}\nThe proof is completed by selecting $K=4 \\tilde K \\max\\{1, M\\}$ and combining \\eqref{lm4.5-lab1} and \\eqref{lm4.5-lab2}.\n\\end{proof}\n\nFinally we end up with a concluding theorem compiling the a priori estimates resulting from Lemmas \\ref{lm:a_priori_KS} and \\ref{lm:a_priori_NS}. \n\\begin{theorem} Let $n_h^0\\in N_h$ be such that \n$$\n\\int_\\Omega n^0_h({\\boldsymbol x})\\,{\\rm d}{\\boldsymbol x}<2\\chi_\\Omega.\n$$\nThen there exist three constants $D=D(n^0_h)>0$, $F=F(n^0_h, c^0_h)>0$ and $K>0$ such that the sequence of discrete solutions $\\{(n^m_h, c^m_h, u^m_h)\\}_{m=0}^M$ satisfies \n\\begin{equation}\\label{ineq:energy_nh_ch_II}\n\\begin{array}{rcl}\nk D(n^0_h) &\\displaystyle\\sum_{m=0}^{M-1} & \\Big(\\|\\nabla i_h(\\log (n^{m+1}_h+1))\\|^2_{{\\boldsymbol L}^2(\\Omega)} +\\|\\nabla c^{m+1}_h\\|_{{\\boldsymbol L}^2(\\Omega)}^2\n\\\\\n&&\\displaystyle+(n^{m+1}_h, \\log\\frac{n^{m+1}_h}{\\bar n^0_h} )_h \\Big)\\le T F(n^0_h, c^0_h) + \\|n^0_h\\|_{L^1(\\Omega)}\n\\end{array}\n\\end{equation}\nand \n\\begin{equation}\\label{ineq:energy_uh_II}\n\\begin{array}{rcl}\n\\displaystyle\n\\|{\\boldsymbol u}^{\\ell}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}+ k \\sum_{m=0}^{\\ell-1}\\|\\nabla{\\boldsymbol u}^{m+1}_h\\|^2_{{\\boldsymbol L}^2(\\Omega)}&\\le&\\displaystyle \\|{\\boldsymbol u}^0_h\\|^2_{{\\boldsymbol L}^2(\\Omega)} + K \\frac{F(n^0_h, c^0_h)}{D(n^0_h)} T \\|n^0_h\\|_{L^1(\\Omega)} \n\\\\\n&&\\displaystyle+ K \\Big(T+\\frac{F(n^0_h, c^0_h)}{D(n^0_h)}\\Big) \\|n^0_h\\|^2_{L^1(\\Omega)},\n\\end{array}\n\\end{equation}\nfor all $\\ell=1, \\cdots, M$.\n\\end{theorem}\n\\begin{proof} First of all, we note that $(\\phi_h, \\log\\frac{\\phi_h}{\\bar\\phi_h})_h\\ge 0$ holds on using that $f(x)=x \\log (x)$ is a convex function and Jensen's inequality. Then summing up \\eqref{ineq:energy_nh_ch} over $m$ follows \\eqref{ineq:energy_nh_ch_II} on noting $0\\le(\\log(n^{M}_h+1)_h, 1)_h\\le \\|n^M_h\\|_{L^1(\\Omega)}=\\|n^0_h\\|_{L^1(\\Omega)}$. Secondly, summing up \\eqref{ineq:energy_uh} over $m$ leads to \\eqref{ineq:energy_uh_II} on using \\eqref{ineq:energy_nh_ch_II}. \n\\end{proof}\n\n\\section{Numerical experiments}\nFollowing the existence theory \\cite{Nagai_Senba_Yoshida_1997} of the Keller--Segel equations \\eqref{KS}-\\eqref{IC_KS}, Morse--Trudinger's inequality \\eqref{ineq:Trudinger-Moser_New} -- which is adapted to two-dimensional domains with polygonal boundary -- is the key tool in establishing the critical value $4\\chi_\\Omega$ for the integrability of $n_0$, i.e. $\\|n_0\\|_{L^1(\\Omega)}< 4\\chi_\\Omega$, as an existence threshold, where $ \\chi_\\Omega =\\frac{4}{\\beta_\\Omega^2}\\alpha_\\Omega$ with $\\beta_\\Omega$ depending upon the geometrical shape of domains. This critical value is apparently reduced to $2\\chi_\\Omega $ with use of \\eqref{functional_ineq_II} and \\eqref{functional_ineq_III}, i.e. $\\|n_0\\|_{L^1(\\Omega)}< 2\\chi_\\Omega$, for the Navier--Stokes--Keller--Segel equations \\eqref{KSNS}-\\eqref{IC_KSNS}. Then in this section we attempt to shed light on the outstanding problem of diagnosing whether, on the one hand, the bound $2\\chi_\\Omega$ for $\\int_\\Omega n_0({\\boldsymbol x})\\, {\\rm d}{\\boldsymbol x}$ found in \\cite{Winkler_2020} is critical for the global-in-time existence of generalised solutions to problem \\eqref{KSNS}-\\eqref{IC_KSNS} or, on the other hand, one can expect that problem \\eqref{KSNS}-\\eqref{IC_KSNS} inherits the same mass critical value $4\\chi_\\Omega$ from problem \\eqref{KS}-\\eqref{IC_KS}. Clearly we shall need a good deal more than mere numerical evidence if we are to make rigorous claims for chemotaxis-fluid interaction, but it might pave the way for analysts to focus their effort on proving the optimal threshold for distinguishing between bounded and unbounded solutions.\n\nWith an eye to demonstrating whether or not $2\\chi_\\Omega$ is optimal, we consider an example quite intensely studied in the numerical context of the Keller--Segel equations and hence it can be served as a benchmark \\cite{Strehl_Sokolov_Kuzmin_Turek_2010, GS_RG_2021, Badia_Bonilla_GS_2022} for comparing chemotaxis with and without fluid interaction. As the domain we take a unit ball, namely $\\Omega = B((0,0.1); 1)$, and as the initial conditions we use \n$$\nn_0(x,y)= \\eta_0 e^{-100(x^2+y^2)}\\quad\\mbox{ and }\\quad c_0(x,y)=0\n$$ \nand\n$$\n{\\boldsymbol u}_0(x,y)=\\boldsymbol{0}.\n$$\nWith regard to the potential $\\Phi$, it is selected as being a gravitational-like one, i.e., \n$$\n\\Phi(x,y)= - \\Phi_0 y.\n$$\n\nThe numerical setting for Algorithm \\eqref{eq:n_h}-\\eqref{eq:p_h} is a triangulation $\\mathcal{T}_h$ of $B((0,0.1); 1)$ as despited in Figure \\ref{meshes} (right) on which we take two first-order finite element spaces $N_h$ and $C_h$ for approximating the chemoattractant and organism densities, respectively, and a Taylor--Hood finite element space pair $(\\boldsymbol{U}_h, P_h)$ for approximating velocity and pressure. When needed, we will make use of a finer triangulation $\\mathcal{T}_h^*$ as shown in Figure \\ref{meshes} (left) to emphasise the growth of possible singularities, since it is known \\cite{Badia_Bonilla_GS_2022} that such a growth is controlled by the $L^1(\\Omega)$-norm and the measure of the macroelement on which the singularity evolves, i.e., the maximum reached by the singularity is proportional to the $L^1(\\Omega)$-norm and inversely proportional to the macroelement measure. Hence the macroelement where the singularity is supported is required to be as small as possible if the size of the $L^1(\\Omega)$-norm is not large to boost its enlargement. It should be further noted that when $\\Omega$ is a ball whose boundary is approximated by a polygonal, we have that $\\alpha_\\Omega= \\pi - \\epsilon$ for certain $\\epsilon >0$ being very small depending on $\\mathcal{T}_h$, since $\\beta_\\Omega=1$ due to the convexity of $\\Omega$. The parameter $\\varepsilon$ is taken to be $10^{-6}$. Different time steps are considered ranging from $10^{-2}$ through $10^{-5}$. The latter is meant to be used for large values of $\\eta_0$ and $\\Phi_0$. As a reference to see how evolves the organism density over time, Figure \\ref{Snapshots_n0h} illustrates the initial distribution of $n_{0h}$. \n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.3\\textwidth]{Figures\/Meshes\/Mesh.png}\n \\includegraphics[width=0.3\\textwidth]{Figures\/Meshes\/Mesh_finer.png}\n \\caption{Triangulation $\\mathcal{T}_h$ of $\\Omega$ (right) and triangulation $\\mathcal{T}_h^*$ refined on the singularity path.}\n \\label{meshes}\n\\end{figure} \n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.4\\textwidth]{Figures\/eta_350\/figures_n0_0.pdf}\n \\caption{Initial distribution of the organism density $n_{0h}$.}\\label{Snapshots_n0h}\n\\end{figure} \n\n\nIn the above framework, the numerical solution (if $\\eta_0=1000$) provided by an algorithm \\cite{Badia_Bonilla_GS_2022} similar to that developed in the paper but for approximating the Keller--Segel equations \\eqref{KS}-\\eqref{IC_KS} is that of the highest density concentration of both the chemoattractant and organisms moving from the origin to the point $(0, -0.9)$, where a Dirac-type singularity takes place. Accordingly, we may expect that as the fluid transports the singularity it may enhance its growth; thereby resulting in a reduction of the existence threshold below $4\\chi_\\Omega$. \n \nIn order to see if a fluid can essentially modify the existence threshold for the fluid-free chemotactic behaviour, we will use the parameters $\\eta_0$ and $\\Phi_0$ to control the size of the $L^1(\\Omega)$-norm of $n_0$ and the size of the $W^{1,\\infty}(\\Omega)$-norm of $\\Phi$, respectively, in a set of numerical experiments. \n\nFinally, as a linearisation of Algorithm \\eqref{eq:n_h}-\\eqref{eq:p_h}, a Picard technique is implemented with a stopping criterium being the relative error between two different iterations for a tolerance of $10^{-3}$. \n\n\\subsection{Case: \\texorpdfstring{$\\eta_0=350$}{Lg} and \\texorpdfstring{$\\Phi_0=10$}{Lg}} To begin with, we take $\\Phi_0=100$ and $\\eta_0=350$, which gives the initial chemoattractant mass $\\|n_0\\|_{L^1(\\Omega)}\\approx11.0027<4\\pi$. This amount is preserved for the $L^1(\\Omega)$-norm of $\\{n_h^m\\}_m$, whereas that of $\\{c_h^m\\}_m$ increases; see Figure \\ref{Graphs_350} (left). Observe in Figure \\ref{Graphs_350} (middle) that maxima for $\\{n_h^m\\}_m$ drop rapidly in the very beginning and then start rising until becoming nearly constantly $100$ and for $\\{c_h^m\\}_m$ grow gradually up to over $14$. In contract to maxima for $\\{n_h^m\\}_m$, minima in Figure \\ref{Graphs_350} (right) move inversely towards approximately $3$ and for $\\{c_h^m\\}_m$ increase reaching slightly over $10$. \n\nSnapshots at times $t=0.02$, $0.04$, $1.0$ and $10.0$ of $\\{n_h^m\\}_m$, $\\{c_h^m\\}_m$ and $\\{{\\boldsymbol u}_h^m\\}_m$ are depicted in Figures \\ref{Snapshots_350_nh}, \\ref{Snapshots_350_ch} and \\ref{Snapshots_350_uh}. It is noticeable that the highest density area of both the chemoattractant and organisms moves from $(0,0.1)$ to $(0, -0.9)$; in particular, that of organisms occupies several macroelements around $(0, -0.9)$. The fluid flow in the beginning generates two vortices as a result of the structure of $n_{0h}$, which vanish as the high density area touches the boundary. \n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_n0_02.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_n0_06.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_n1_0.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_n10_0.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.02$, $t=0.04$, $t=1$, and $10$ of $n_h$ for $\\eta_0=350$}\\label{Snapshots_350_nh}\n \\end{figure}\n \\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_c0_02.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_c0_06.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_c1_0.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_c10_0.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.02$, $t=0.04$, $1,$ and $10$ of $c_h$ for $\\eta_0=350$}\\label{Snapshots_350_ch}\n \\end{figure}\n \\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_u0_02.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_u0_06.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_u1_0.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/figures_u10_0.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.02$, $t=0.04$, $1$, and $10$ of ${\\boldsymbol u}_h$ for $\\eta_0=350$}\\label{Snapshots_350_uh}\n\\end{figure}\n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/conservation.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/maxima.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_350\/minima.pdf}\n \\end{subfigure}\n \\caption{Plots of $L^1(\\Omega)$-norm, maxima, minima and energy for $\\eta=350$}\\label{Graphs_350}\n\\end{figure}\n\\subsection{Case: \\texorpdfstring{$\\eta_0=400$}{Lg} and \\texorpdfstring{$\\Phi_0=10$}{Lg}} It will be assumed now that $\\Phi_0=10$ and $\\eta_0=400$. Accordingly this provides $\\|n_0\\|_{L^1(\\Omega)}\\approx12.5745\\gtrsim4\\pi$, whose value keeps for $\\{n_h^m\\}_m$ as before and $\\{c_h^m\\}_m$ gains mass being $\\max_m\\|c_h^m\\|_{L^1(\\Omega)}\\approx 8.2289$ as shown in Figure \\ref{Graphs_400} (left). In Figure \\ref{Graphs_400} (middle) it is seen that maxima for $\\{n_h^m\\}_m$ and $\\{c_h^m\\}_m$ stabilise at $2115$ and $17.5$, respectively. In the evolution of minima we find a considerably important difference with regard to $\\eta=350$, since minima for $\\{n_h^m\\}_m$ become almost null and those for $\\{c_h^m\\}_m$ exceed the value of $8$; cf. Figure \\ref{Graphs_400} (right). In this case, the number of macroelements supporting the highest density values of $\\{n_h^m\\}_m$ are smaller than that for $\\eta_0=350$ as can see from snapshots at $t=0.02$, $0.04$, $0.5$ and $1.78$ in Figure \\ref{Snapshots_400_nh}. The dynamics of $\\{c_h^m\\}_m$ and $\\{{\\boldsymbol u}_h^m\\}_m$ is displayed in Figures \\ref{Snapshots_400_ch} and \\ref{Snapshots_400_uh}, where it is observed that the chemoattractant density distribution is less uniform than for $\\eta=350$ and the velocity field is more active in the region of $(0, - 0.9)$; as a consequence of a higher concentration for $\\{n_h^m\\}_h$ at such a point. \n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_n0_02.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_n0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_n0_5.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_n1_78.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.02$, $t=0.04$, $0.5$ and $1.78$ of $\\{n_h^m\\}_m$ for $\\eta_0=400$}\\label{Snapshots_400_nh}\n \\end{figure}\n \\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_c0_02.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_c0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_c0_5.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_c1_78.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.02$, $t=0.04$, $0.5$ and $1.78$ of $\\{c_h^m\\}_m$ for $\\eta_0=400$}\\label{Snapshots_400_ch}\n \\end{figure}\n \\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_u0_02.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_u0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_u0_5.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/figures_u1_78.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.02$, $t=0.04$, $0.5$ and $1.78$ of $\\{{\\boldsymbol u}_h^m\\}_m$ for $\\eta_0=400$}\\label{Snapshots_400_uh}\n\\end{figure}\n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/conservation.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/maxima.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_400\/minima.pdf}\n \\end{subfigure}\n \\caption{Evolution of total mass, maxima and minima $\\{n_h^m\\}$ and $\\{c_h^m\\}_m$ for $\\eta=400$}\\label{Graphs_400}.\n\\end{figure}\n\n\n\n\\subsection{Case: \\texorpdfstring{$\\eta_0=450$}{Lg} and \\texorpdfstring{$\\Phi_0=10$}{Lg}} We next continue selecting $\\Phi_0=10$ and $\\eta_0=450$, which leads to $\\|n_0\\|_{L^1(\\Omega)}=14.1463$. For such a value we find a potential blowup formation for $\\{n_h^m\\}_m$ (Figure \\ref{Snapshots_450_nh}), in favorable concordance with the theoretical results. This Dirac-type singularity is spontaneously formed at a time smaller than $t=0.7$ as displayed in Figure \\ref{Graphs_450} (left) for maxima of $\\{n_h^m\\}_m$, where $\\max_m \\|n^m_h\\|_{L^\\infty}(\\Omega)\\approx 6\\cdot 10^4$. As a result of the finite-time singularity development, lower density areas for $\\{n_h^m\\}_m$ (Figure \\ref{Snapshots_450_nh}) are dredged as chemotaxis dominates diffusion until being carried close to $0$ as seen in Figure \\ref{Graphs_450} (middle); on the contrary, Figure \\ref{Graphs_450} (middle) shows minina for $\\{c_h^m\\}_m$, which get smaller values than $\\eta_0=400$. Snapshots of $\\{n_h^m\\}_m$, $\\{c_h^m\\}_m$ and $\\{{\\boldsymbol u}_n^m\\}_m$ at $t=0.04$, $0.1$, $0.4$ and $0.8$ are illustrated in Figures \\ref{Snapshots_450_nh}, \\ref{Snapshots_450_ch} and \\ref{Snapshots_450_uh}, which are evidently different from those of $\\eta_0=400$. \n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_n0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_n0_1.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_n0_4.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_n0_8.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.04$, $0.1$, $0.4$ and $0.8$ of $\\{n_h^m\\}_m$ for $\\eta_0=450$ on $\\mathcal{T}_h^*$.}\\label{Snapshots_450_nh}\n\\end{figure}\n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_c0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_c0_1.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_c0_4.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_c0_8.pdf}\n \\end{subfigure}\n\\caption{Snapshots at $t=0.04$, $0.1$, $0.4$ and $0.8$ of $\\{c_h^m\\}_m$ for $\\eta_0=450$ on $\\mathcal{T}_h^*$.}\\label{Snapshots_450_ch}\n\\end{figure}\n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_u0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_u0_1.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_u0_4.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/figures_u0_8.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.04$, $0.1$, $0.4$ and $0.8$ of $\\{{\\boldsymbol u}_h^m\\}_m$ for $\\eta_0=450$ on $\\mathcal{T}_h^*$.}\\label{Snapshots_450_uh}\n\\end{figure}\n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/conservation.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/maxima.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450\/minima.pdf}\n \\end{subfigure}\n \\caption{Plots of total mass, maxima and minima of $\\{n_h^m\\}_m$ and $\\{c_h^m\\}_m$ for $\\eta_0=450$ on $\\mathcal{T}_h^*$.}\\label{Graphs_450}\n\\end{figure}\n\nTo make the singularity formation far more manifest, we use the partially refined mesh $\\mathcal{T}_h^*$ displayed in Figure \\ref{meshes} (right). The first remarkable aspect in Figure \\ref{Graphs_450_finer} is concerned with maxima of $\\{n_h^m\\}_m$, which reaches higher values; $\\max_{m}\\|n_h^{m}\\|\\approx 4\\cdot 10^{5}$. As for the singularity-formation time, it is smaller being $t\\approx0.5$. The qualitative evolution of $\\{n_h^m\\}_m$, $\\{c_h^m\\}_m$ and $\\{{\\boldsymbol u}_h^m\\}_m$ illustrated in Figures \\ref{Snapshots_450_finer_nh}, \\ref{Snapshots_450_finer_ch} and \\ref{Snapshots_450_finer_nh} does not differ from that computed on $\\mathcal{T}_h$. \n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_n0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_n0_1.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_n0_4.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_n0_7.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.04$, $0.1$, $0.4$ and $0.7$ of $\\{n_h^m\\}_m$ for $\\eta_0=450$}\\label{Snapshots_450_finer_nh}\n\\end{figure}\n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_c0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_c0_1.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_c0_4.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_c0_7.pdf}\n \\end{subfigure}\n\\caption{Snapshots at $t=0.04$, $0.1$, $0.4$ and $0.7$ of $\\{c_h^m\\}_m$ for $\\eta_0=450$.}\\label{Snapshots_450_finer_ch}\n\\end{figure}\n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_u0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_u0_1.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_u0_4.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/figures_u0_7.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.04$, $0.1$, $0.4$ and $0.7$ of $\\{{\\boldsymbol u}_h^m\\}_m$ for $\\eta_0=450$.}\\label{Snapshots_450_finer_uh}\n\\end{figure}\n\n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/conservation.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/maxima.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_finer\/minima.pdf}\n \\end{subfigure}\n \\caption{Plots of total mass, maxima and minima of $\\{n^m_h\\}_m$ and $\\{c_h^m\\}_m$ for $\\eta_0=450$.}\\label{Graphs_450_finer}\n\\end{figure}\n\\subsection{Case: \\texorpdfstring{$\\eta_0=450$}{Lg} and \\texorpdfstring{$\\Phi_0=50$}{Lg}}\nNow that it is known that there is a finite-time singularity at least numerically for $\\eta_0=450$. We ask ourselves whether or not the fluid flow may modify this configuration. In doing so, we take $\\Phi_0=50$ to speed up the fluid velocity. Surprisingly as indicated in Figure \\ref{Graphs_450_50} (middle) for maxima of $\\{n_h^m\\}_m$ there seems that the fluid velocity kills the singularity formation, since they do not grow beyond $1200$. Furthermore, minima of $\\{n_h^m\\}_m$ in Figure \\ref{Graphs_450_50} (right) became constant over time, but far away from $0$. For this reason chemotaxis mechanism cannot force the dredging at a single point as a consequence of the growth in velocity induced by $\\Phi_0=50$. In addition, convection introduces diffusion in the system. This phenomenological description is shown in Figures \\ref{Snapshots_450_50_nh}, \\ref{Snapshots_450_50_ch} and \\ref{Snapshots_450_50_uh}. \n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_n0_02.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_n0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_n0_2.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_n1_0.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.02$, $t=0.04$, $0.2$ and $1.0$ of $\\{n_h^m\\}_m$ for $\\eta_0=450$ and $\\Phi_0=50$.}\\label{Snapshots_450_50_nh}\n \\end{figure}\n \\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_c0_02.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_c0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_c0_2.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_c1_0.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.02$, $t=0.04$, $0.2$ and $1.0$ of $\\{c_h^m\\}_m$ for $\\eta_0=450$ and $\\Phi_0=50$.}\\label{Snapshots_450_50_ch}\n \\end{figure}\n \\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_u0_02.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_u0_04.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_u0_2.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/figures_u1_0.pdf}\n \\end{subfigure}\n \\caption{Snapshots at $t=0.02$, $t=0.04$, $0.2$ and $1.0$ of $\\{{\\boldsymbol u}_h^m\\}_m$ for $\\eta_0=450$ and $\\Phi_0=50$.}\\label{Snapshots_450_50_uh}\n\\end{figure}\n\\begin{figure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/conservation.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/maxima.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.23\\textwidth}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{Figures\/eta_450_phi_50\/minima.pdf}\n \\end{subfigure}\n \\caption{Plots of total mass, maxima and minima of $\\{n_h^m\\}_m$ and $\\{c_h^m\\}_m$ for $\\eta=450$ and $\\Phi_0=50$.}\\label{Graphs_450_50}\n\\end{figure}\n\n \\section{Conclusion}\n\nIn this paper a numerical method for approximating solutions to the Keller--Segel--Navier--Stokes system has been constructed. It consists of a finite element method together with a stabilising term, whose design is based on a shock capturing technique so as to preserve lower bounds such as positivity and non-negativity.\n\nIt is known that solutions to the Keller--Segel--Navier--Stokes system are uniformly bounded in time providing that the total mass for the organism density is below $2\\pi$. Such a threshold is smaller than that for the Keller--Segel system, which corresponds to $4\\pi$. Then we have made an attempt at answering the question whether or not the value $2\\pi$ is critical through a set of numerical experiments.\nThe evidence found herein puts into new perspective the threshold value for proving boundedness solutions for the Keller--Segel--Navier--Stokes equations. We have discovered that the value $4\\pi$ instead of $2\\pi$ may be critical; thus inheriting it from the Keller--Segel subsystem. Furthermore, we have observed that the fluid intensification may lead to the depletion of chemotaxis and prevent possible singularity formation. Realising this possibility may be a watershed in the knowledge of the phenomenological interaction of chemotaxis in fluid scenarios. \n\nThe above findings are relied on the fact that numerical solutions computed by the proposed algorithm satisfy lower and $L^1(\\Omega)$ bounds and quasi-energy estimates. This latter property results from a new discretization of the chemotactic and convective terms and a Morse-Trudinger's inequality demonstrated for polynomial domains. All in all, we have found that our numerical solutions are robust and reliable in the numerical simulations.\n\nAn improvement of the numerical method that may be regarded is using stabilising techniques for the convective terms at least in the Keller--Segel subsystem, since when taking $\\Phi_0$ very large numerical solutions do not fulfil lower bounds. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nIn proton-proton (pp) collisions with higher energies at the Large\nHadron Collider (LHC), particle production is dominated by multiple\ninteractions of their constituent partons, with most particles from\nthe hardest proton-proton scattering and the radiation and\nfragmentation of secondary partonic actions. The higher\ncentre-of-mass energy leads to enhanced parton densities which cause\na sizable probability for two or more parton-parton scatterings\nwithin the same pp interaction\n\\cite{labDPSTheory1,Lansberg:2014swa}. At LHC, various measurements\nof the differential distributions in W + jets\n\\cite{labATLASW2Jets,labCMSW2Jets} and $J\/\\psi$\\ + W\n\\cite{labATLASWJPsi} show that the excesses above the expectations\nfrom single parton scattering (SPS) are consistent with double\nparton scattering (DPS). Various measurements in other pp and\n$p\\bar{p}$ collisions at $\\sqrt{s}$ = 63~GeV \\cite{labUA2MultiJets},\n630~GeV \\cite{labAFSMultiJets}, and 1.8~TeV \\cite{labCDFMultiJets}\nare consistent with DPS contributions to multijet final states, as\nwell as to $\\gamma$ + 3-jet events at $\\sqrt{s}$ = 1.8~TeV\n\\cite{labCDFGamma3Jets} and 1.96 ~TeV \\cite{labD0Gamma3Jets}. The\nmeasurements of DPS processes can provide valuable information on\nthe transverse distribution of partons in the proton\n\\cite{labDPSTheoryMultiQCD} and on the multi-parton correlations in\nthe hadronic wave function \\cite{labDPSTheoryHWF}. DPS also\nconstitutes the background for new physics searches at the LHC\n\\cite{labDPSBkg1,labDPSBkg2,labDPSBkg3}. Additional searches for DPS\nhave been proposed via double Drell-Yan, four jets and same-sign WW\nproduction \\cite{labDPSDY,labDPS4Jets,labDPSWW}.\n\n\nIn this paper, the cross section for the production of pairs of\nphotons plus two additional jets produced from double parton\nscattering (DPS) in high-energy proton-proton collisions at the LHC\nare calculated for the first time. $\\gamma\\gamma$ final states have\nplayed a crucial role in the recent discovery of a new boson at the\nLHC \\cite{labCMSHiggs,labATLASHiggs} and are also important in many\nNew Physics searches\n\\cite{labATLASDiphotonMET,labATLASDiphotonED,labCMSDiphotonMET,labCMSDiphotonNP},\nin particular the search for extra spatial dimensions or cascade\ndecays of heavy new particles. In particular, diphotons in\ncombination with jets and missing energy occur in gauge mediated\nSUSY scenarios. $\\gamma\\gamma$ or plus two additional jets offers\nalso an important test of both perturbative and non-perturbative\nquantum chromodynamics (QCD)\n\\cite{labCMSDiphoton7TeV,labJTaoDiphoton7TeV,labATLASDiphoton7TeV}.\n$\\gamma\\gamma$ $+$ 2 jets is also the main irreducible background\nfor other physics analyses with $\\gamma\\gamma$ and jets in the final\nstate at the LHC, such as Higgs produced in vector boson fusion. For\nthe production of $\\gamma\\gamma$ $+$ 2 jets, a sizeable contribution\nfrom DPS with $\\gamma\\gamma$ produce in one scattering while the\nsecond scattering yielding two jets can be expected.\n\nThe structure of this paper is organized as follows. In section 2, a\ngeneric way of the DPS cross section as the product of the SPS cross\nsections and its parameter are briefly introduced. The details of\nthe cross section of $\\gamma\\gamma$ $+$ 2 jets calculation and the\ncross section of different SPS processes estimated from higher order\ntheoretical predictions are described in section 3. The results\nincluding the cross section of $\\gamma\\gamma$ $+$ 2 jets with\n$\\sqrt{s}$ = 13~TeV and 14~TeV at LHC and the expected event rates,\nwith typical selections,a re summarized in section 4. The summary\nand outlook are given in section 5.\n\n\\section{Generic formule of DPS}\n\nFor a composite system ($A+B$) in hadronic collisions, its\nproduction cross section from DPS, $\\sigma_{pp \\to AB}^{DPS}$, can\nbe written model-independently as the product of the cross sections\nof $A$ and $B$ originated from single parton scattering, $\\sigma_{pp\n\\to A}^{SPS}$ and $\\sigma_{pp \\to B}^{SPS}$, normalized by an\neffective cross section $\\sigma_{eff}$ \\cite{labDPSgeneral}\n\n\\begin{equation}\n\\begin{split}\n\\label{eqGeneralDPS} \\sigma_{pp \\to AB}^{DPS} =\n\\frac{m}{2}\\frac{\\sigma_{pp \\to A}^{SPS}\\times \\sigma_{pp \\to\nB}^{SPS}}{\\sigma_{eff}},\n\\end{split}\n\\end{equation}\nwhere $m$ ia a symmetry factor accounting for distinguishable\n($m$=2) and indistinguishable ($m$=1) final-states.\n\nThe effective cross section $\\sigma_{eff}$ is a measure of the\ntransverse distribution of partons inside the colliding hadrons and\ntheir overlap in a collision. It is independent of the process and\nof the phase-space under consideration. A number of measurements of\n$\\sigma_{eff}$ have been performed in pp and $p\\bar{p}$ collisions\nat $\\sqrt{s}$ = 63~GeV \\cite{labUA2MultiJets}, 630~GeV\n\\cite{labAFSMultiJets}, 1.8~TeV\n\\cite{labCDFMultiJets,labCDFGamma3Jets,labCDFGamma3JetsCorr}, 1.96\n~TeV \\cite{labD0Gamma3Jets} and also 7~TeV at LHC\n\\cite{labATLASW2Jets,labCMSW2Jets}. The measured values range from\n5mb at the lowest energy to about 20mb from CMS at 7~TeV.\nFig.~\\ref{fig:effsigma} shows a comparison of the effective cross\nsection $\\sigma_{eff}$ measured by different experiments using\ndifferent processes at various centre-of-mass energies.\n\n\\begin{center}\n\\includegraphics[width=7cm]{DPS_SigEff_compWithAll}\n \\figcaption{ $\\sigma_{eff}$ measured by different experiments using different processes\n\\cite{labUA2MultiJets,labAFSMultiJets,labCDFMultiJets,labCDFGamma3Jets,labD0Gamma3Jets,labATLASW2Jets,labCMSW2Jets,labCDFGamma3JetsCorr}.\nThe \"Corrected CDF\" data point indicate the\n$\\sigma_{eff}$ value corrected for the exclusive event selection\n\\cite{labCDFGamma3JetsCorr}. }\n {\\label{fig:effsigma}}\n\\end{center}\n\nThe measured values of $\\sigma_{eff}$ from TeV experiments at\nTevetron (CDF and D0) and LHC (ATLAS and CMS) are consistent with\neach other within their uncertainties. In the following\ncalculations, a numerical value $\\sigma_{eff}\\approx$ 15 mb was used\nto estimate the production cross section of $\\gamma\\gamma$ $+$ 2\njets from DPS with $\\sqrt{s}$ = 13~TeV and 14~TeV at LHC. A number\n5 mb was assigned as its uncertainty to estimate its effects on the\nfinal results. The uncertainty on $\\sigma_{eff}$ is the dominant\nuncertainty for the calculation of the production cross section of\n$\\gamma\\gamma$ $+$ 2 jets from DPS in the following sections\n\n\n\\section{$\\sigma_{pp\\to\\gamma\\gamma+2jets}^{DPS}$ calculation}\n\nAccording to the descripition in above section, the production cross\nsection of $\\gamma\\gamma$ $+$ 2 jets from DPS in pp collisions can\nbe written as\n\n\\begin{equation}\n\\begin{split}\n\\label{eq2g2jDPS} \\sigma_{pp\\to\\gamma\\gamma+2jets}^{DPS} =\n\\frac{\\sigma_{pp\\to\\gamma\\gamma}^{SPS}\\times\n\\sigma_{pp\\to2jets}^{SPS}}{\\sigma_{eff}} +\n\\frac{1}{2}\\frac{\\sigma_{pp\\to\\gamma+jet}^{SPS}\\times\n\\sigma_{pp\\to\\gamma+jet}^{SPS}}{\\sigma_{eff}}.\n\\end{split}\n\\end{equation}\n\n$\\gamma\\gamma$ production has been calculated at\nnext-to-leading-order (NLO) some time ago \\cite{labDiphotonTJ2000},\nsupplemented also by gluon initiated subprocesses beyond the leading\norder \\cite{labDiphotonZL2002} and soft gluon resummation\n\\cite{labDiphotonRS2000,labDiphotonRS2006}. Recently,\nnext-to-next-to-leading-order (NNLO) corrections to direct diphoton\nproduction also have become available \\cite{labDiphotonNNLO}. The\nmeasurements from LHC\n\\cite{labCMSDiphoton7TeV,labJTaoDiphoton7TeV,labATLASDiphoton7TeV}\nshow that the NNLO can give much better agreement with measured data\nthan the lower order predictions. For the integrated cross section,\nthe predicted values by NNLO are almost exactly the same\n\\cite{labATLASDiphoton7TeV} or consistent within the uncertainties\n\\cite{labCMSDiphoton7TeV,labJTaoDiphoton7TeV} with the measured\nones. So for the production cross sections of $\\gamma\\gamma$\nfinal-state from SPS, $\\sigma_{pp\\to\\gamma\\gamma}^{SPS}$, with\ndifferent $\\sqrt{s}$ will be obtained from the NNLO calculation with\nthe package 2$\\gamma$NNLO.\n\nFor the dijet cross section, the measured data at LHC\n\\cite{labCMSJets20112012,labATLASJets2011} can be well described by\nNLO perturbative QCD (pQCD) calculations from NLOJet++ program\n\\cite{labNLOJet++} corrected to account for non-perturbative and\nelectroweak effects. From \\cite{labATLASJets2011}, the\nnon-perturbative correction is within 3\\% for jet reconstructed with\nthe anti-$k_{t}$ clustering algorithm \\cite{labAntiKTAlgo} and\ndistance parameter or cone size R=0.4. The corrections for the\nelectroweak effect can be negligible if the dijet mass less than\nabout 1~TeV. In this analysis, the NLO calculations of\n$\\sigma_{pp\\to2jets}^{SPS}$ are performed using NLOJet++ (version\n4.1.3) within the framework of the fastNLO package (version 2.3.1)\n\\cite{labfastNLO}.\n\nNLO pQCD prediction from the program JETPHOX (version 1.3.1)\n\\cite{labJetPhox} is used for the calculation of $\\gamma+jet$ cross\nsections from SPS in this paper. This program includes a full NLO\nQCD calculation of both the direct-photon and fragmentation\ncontributions to the cross section. The number of flavours was set\nto five. Compared with the measurements of $\\gamma+jet$ cross\nsections at the LHC, the predictions from JETPHOX multiplied by a\nfactor close to unity for the corrections of hadronisation and\nunderlying-event effects give a good description of the\n$E_T^{\\gamma}$ and $p_T^{jet}$ measured cross sections\n\\cite{labATLASGJet2010,labCMSGJet2010}.\n\nDifferent PDF sets are used for the calculations of these three SPS\nprocesses. MSTW2008NNLO \\cite{labMSTW2008NNLO} is used for\n$\\sigma_{pp\\to\\gamma\\gamma}^{SPS}$ calculations with 2$\\gamma$NNLO.\nCT10NLO \\cite{labCT10NLO} is used for both\n$\\sigma_{pp\\to2jets}^{SPS}$ with NLOJet++ in fastNLO package and\n$\\sigma_{pp\\to\\gamma+jet}^{SPS}$ with JETPHOX.\n\nThe calculations of $\\sigma_{pp\\to\\gamma\\gamma}^{SPS}$ are performed\nwith the factorization and renormalization scales equal to the\ninvariant mass of two photons, $\\mu_{F}$ = $\\mu_{R}$ =\n$m_{\\gamma\\gamma}$. The scale uncertainty and PDF uncertainty are\nalso considered. A simplified and less computationally intensive\nestimate of the renormalization ($\\mu_{R}$) and factorization\n($\\mu_{F}$) scale uncertainties is performed by varying these scales\nsimultaneously by a factor of two up and down around\n$m_{\\gamma\\gamma}$, $\\mu_{F}$ = $\\mu_{R}$ = 2$m_{\\gamma\\gamma}$ and\n$\\mu_{F}$ = $\\mu_{R}$ = 0.5$m_{\\gamma\\gamma}$. 41 eigenvector sets\nof MSTW2008NNLO are used to build the PDF uncertainty envelope.\n\nCalculations of $\\sigma_{pp\\to2jets}^{SPS}$ are derived using\nNLOJet++ within the framework of the fastNLO package at a\nfactorization and renormalization scale equal to the average\ntransverse momentum ($p_{T}^{ave}$) of the two jets ($\\mu_{F}$ =\n$\\mu_{R}$ = $p_{T}^{ave}$). The uncertainty due to the choice of\nfactorization and renormalization scales is estimated as the maximum\ndeviation at the six points ($\\mu_{F}$\/$\\mu$, $\\mu_{R}$\/$\\mu$)=\n(0.5, 0.5), (2, 2), (1, 0.5), (1, 2), (0.5, 1), (2, 1) with $\\mu$ =\n$p_{T}^{ave}$. 52 eigenvector sets of CT10NLO are used to build the\nPDF uncertainty envelope.\n\nFor NLO calculations of $\\sigma_{pp\\to\\gamma+jet}^{SPS}$ using\nJETPHOX, the renormalization, factorization and fragmentation\n($\\mu_f$) scales are chosen to be photon's transverse momentum,\n$\\mu_{F}$ = $\\mu_{R}$ = $\\mu_f$ = $E_{T}^{\\gamma}$. Same as above,\n52 eigenvector sets of CT10NLO are used to build the PDF uncertainty\nenvelope.\n\nAbove calculations were performed with the strong coupling constant\nat two-loop order with $\\alpha_s(m_{Z})$ = 0.118 in CT10NLO and\n0.117 in MSTW2008NNLO. The uncertainty on $\\alpha_s(m_{Z})$ was not\nconsidered in this study. The uncertainty from scales, pdf and\n$\\alpha_s(m_{Z})$ is around 10\\%, 5\\% and 1\\% respectively\n\\cite{labATLASDiphoton7TeV,labCMSDiphoton7TeV,labJTaoDiphoton7TeV,labCMSJets20112012,labATLASJets2011,labATLASGJet2010,labCMSGJet2010}.\nCompared to the larger uncertainty on the $\\sigma_{eff}$ with more\nthan 30\\% used in this study as told in the end of section 2, the\neffect on the final results from the uncertainty on\n$\\alpha_s(m_{Z})$ can be negligible.\n\n\n\n\\section{Results of $\\sigma_{pp\\to\\gamma\\gamma+2jets}^{DPS}$ and expected event rates at LHC}\n\nIn this paper, several sets of typical selections at LHC were tried\nto calculate the production cross section of $\\gamma\\gamma$ $+$ 2\njets from DPS in pp collisions,\n$\\sigma_{pp\\to\\gamma\\gamma+2jets}^{DPS}$. Due to the high level\ntrigger requirements for $\\gamma\\gamma$ events at LHC for the higher\nenergy and higher luminosity collisions, five sets of requirements\non the photons' transverse momentum were considered,\n($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$) $>$ (30, 20)~GeV, (30,\n30)~GeV, (40, 20)~GeV, (40, 30)~GeV and (40, 40)~GeV with\n$\\gamma_{1}$ representing the maximum $E_T$ photon and $\\gamma_{2}$\nthe minimum $E_T$ one of two photons. So for single photon\nrequirement in the $\\gamma+jet$, 3 cases with $E_{T}^{\\gamma}>$ (20,\n30, 40)~GeV were considered. The photon should be also constrained\nin the pseudorapidity region $|\\eta|<$2.5. An isolation requirement\nis applied on the photon to fulfill the isolation requirement from\nexperimental measurements\n\\cite{labATLASDiphoton7TeV,labCMSDiphoton7TeV,labJTaoDiphoton7TeV,labATLASGJet2010,labCMSGJet2010}.\nThe standard isolation, the $E_T$ sum of partons in a cone of size\n$\\Delta R$=0.4 around the photon required to less than 5~GeV, is\napplied in JETPHOX for the calculation of\n$\\sigma_{pp\\to\\gamma+jet}^{SPS}$. For 2$\\gamma$NNLO, the smooth\nFrixione isolation \\cite{labFrixioneISO} on the photons is applied\n\n\\begin{equation}\n\\begin{split}\n\\label{eqFrixISO} E_{T}^{iso}(\\Delta R) < \\epsilon \\Big(\n\\frac{1-\\cos(\\Delta R)}{1-\\cos(\\Delta R_0)} \\Big)^{n}\n\\end{split}\n\\end{equation}\nwhere $E_{T}^{iso}(\\Delta R)$ is the $E_T$ sum of partons in a cone\nof size $\\Delta R$, $\\Delta R_0$ = 0.4, $\\epsilon$ = 5GeV, and $n$ =\n0.1. This criterion is found to have the same efficiency as the\nstandard isolation used for the other generators within a few\npercent \\cite{labCMSDiphoton7TeV,labJTaoDiphoton7TeV}. Additional\nthe angular separation between two photons is required to be at\nleast larger than 0.4 ($\\Delta R_{\\gamma\\gamma}>$0.4) to ensures one\nphoton will not enter the isolation cone of the other photon, which\nis similar to the requirement applied in the data analyses at LHC\nexperiments ATLAS and CMS\n\\cite{labATLASDiphoton7TeV,labCMSDiphoton7TeV,labJTaoDiphoton7TeV}.\n\nIn this study, jet is reconstructed with the anti-$k_{t}$ clustering\nalgorithm and cone size R=0.5. Jets are in the acceptance region\nwith $|\\eta^{jet}|<$4.5. Two tries on jet $p_T^{jet}$ were\nperformed, $p_T^{jet}>$ 20 or 25~GeV. For the dijet events, two jets\nshould be separated by requiring their angular distance $\\Delta\nR_{jj}$ greater than 1.0 to avoid the overlapping of two jet cones.\nFor the $\\gamma+jet$ production, the angular distance between\n$\\gamma$ and jet should be greater than 0.5 ($\\Delta R_{\\gamma j}>$\n0.5) to ensure that the partons belong to the jet will not enter to\nthe isolation cone of $\\gamma$.\n\nFig.~\\ref{fig:XSgg} shows the cross sections of\n$\\sigma_{pp\\to\\gamma\\gamma}^{SPS}$ computed from the 2$\\gamma$NNLO\nat $\\sqrt{s}$ = 13~TeV and 14~TeV with scales and pdf uncertainties\nconsidered, for different sets of $E_T$ requirements on diphotons.\nThe scale uncertainty is around 10\\% and the pdf uncertainty is\nabout 4\\%. The selection sets in x-axis are the 5 sets of\nrequirements on ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$), number\n1 for ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (30, 20)~GeV,\n2 for ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (30, 30)~GeV,\n3 for ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (40, 20)~GeV,\n4 for ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (40, 30)~GeV\nand 5 for ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (40,\n40)~GeV. The detailed values can also be found in Table\n\\ref{tabggXS}. For the central values, the cross section with\n$\\sqrt{s}$ = 14~TeV is about 9\\% higher than that with $\\sqrt{s}$ =\n13~TeV with the same selection requirements, which is within the\nscale and pdf uncertainties.\n\n\\begin{center}\n\\includegraphics[width=7cm]{Comparison_TotXS_gg}\n \\figcaption{ Predicted cross sections of $\\gamma\\gamma$ by 2$\\gamma$ NNLO at $\\sqrt{s}$ = 13~TeV and 14~TeV with\n selection set 1 for ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (30, 20)~GeV, 2 for\n($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (30, 30)~GeV, 3 for\n($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (40, 20)~GeV, 4 for\n($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (40, 30)~GeV and 5\nfor ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (40, 40)~GeV.\nScale and pdf uncertainties are included.\n }\n {\\label{fig:XSgg}}\n\\end{center}\n\n\n\\begin{center}\n\\tabcaption{ \\label{tabggXS} Cross sections in unit of $pb$ of\n$\\gamma\\gamma$ predicted by 2$\\gamma$ NNLO at $\\sqrt{s}$ = 13~TeV\nand 14~TeV. The uncertainties include the scale and pdf\nuncertainties.} \\footnotesize\n\\begin{tabular}{|c|c|c|}\n\\hline\n ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ & $\\sqrt{s}$ = 13~TeV & $\\sqrt{s}$ = 14~TeV \\\\\n\\hline\n(30,20)~GeV & 89.3 $\\pm$ 9.7 & 96.9 $\\pm$ 10.5 \\\\\n(30,30)~GeV & 44.9 $\\pm$ 4.9 & 48.7 $\\pm$ 5.3 \\\\\n(40,20)~GeV & 48.0 $\\pm$ 5.2 & 52.1 $\\pm$ 5.7\\\\\n(40,30)~GeV & 31.2 $\\pm$ 3.4 & 33.7 $\\pm$ 3.7\\\\\n(40,40)~GeV & 18.0 $\\pm$ 2.0 & 19.6 $\\pm$ 2.1\\\\\n\\hline\n\\end{tabular}\n\\vspace{0mm}\n\\end{center}\n\nFig.~\\ref{fig:DiffXSjj} shows the differential cross sections, as a\nfunction of the $p_T$ of leading jet with both jet $p_T^{jet}>$\n20~GeV and $|\\eta|<$ 4.5, of $\\sigma_{pp\\to2jets}^{SPS}$ computed\nfrom NLOJet++ within the framework of the fastNLO package at\n$\\sqrt{s}$ = 13~TeV and 14~TeV with scales and pdf uncertainties\nplotted in the same figure. The bottom two plots show the relative\nuncertainties including the scale uncertainty and scale$\\oplus$pdf\nuncertainty combined in quadrature. The contribution of pdf\nuncertainty is tiny. The integrated cross section are\n117.6$_{-7.0}^{+4.8}$(scale)$_{-1.3}^{+1.0}$(pdf) ($\\mu$b) and\n122.3$_{-5.1}^{+4.6}$(scale) $_{-1.4}^{+1.1}$(pdf) ($\\mu$b) for\n$\\sqrt{s}$ = 13~TeV and 14~TeV with both jet $p_T^{jet}>$ 20~GeV and\n$|\\eta^{jet}|<$ 4.5,\n52.2$_{-2.4}^{+1.6}$(scale)$_{-0.5}^{+0.4}$(pdf) ($\\mu$b) and\n56.2$_{-2.3}^{+1.8}$(scale) $_{-0.6}^{+0.5}$(pdf) ($\\mu$b) for\n$\\sqrt{s}$ = 13~TeV and 14~TeV with both jet $p_T^{jet}>$ 25~GeV and\n$|\\eta^{jet}|<$ 4.5. When $p_{T}$ requirements on both jet increase\n5~GeV from 20~GeV to 25~GeV, the cross section are reduced almost by\na factor of 2.\n\n\\begin{center}\n\\includegraphics[width=7cm]{Comparison_DiffXS_jj}\n \\figcaption{ Differential cross sections of $jet+jet$ computed with NLOJet++ within the framework of the fastNLO package,\n after the requirements on both jets with $p_T^{jet}>$\n20~GeV and $|\\eta^{jet}|<$ 4.5. The solid black circles are the\nresults for $\\sqrt{s}$ = 13~TeV while the red diamond for $\\sqrt{s}$\n= 14~TeV. Bottom two plots show the relative uncertainties including\nthe scale uncertainty and scale$\\oplus$pdf uncertainty with\n$\\sqrt{s}$ = 13~TeV in the middle plot and $\\sqrt{s}$ = 14~TeV in\nthe bottom plot.\n }\n {\\label{fig:DiffXSjj}}\n\\end{center}\n\nCombined the photon $E_T^{\\gamma}$ requirements for the\n$\\gamma\\gamma$ productions and the jet $p_T^{jet}$ requirements for\nthe $jet+jet$ productions, cross section of\n$\\sigma_{pp\\to\\gamma+jet}^{SPS}$ with six sets of selections on the\ntransverse momentums of photon and jet with ($E_T^{\\gamma}$,\n$p_T^{jet}$) $>$ (20, 20)~GeV, (30, 20)~GeV, (40, 20)~GeV, (20,\n25)~GeV, (30, 25)~GeV and (40, 25)~GeV were calculated.\nFig.~\\ref{fig:DiffXSgj} shows the differential cross sections of\n$\\sigma_{pp\\to\\gamma+jet}^{SPS}$ as a function of photon\n$E_T^{\\gamma}$ with ($E_T^{\\gamma}$, $p_T^{jet}$) $>$ (40, 20)~GeV,\n$|\\eta^{\\gamma}|<$ 2.5, $|\\eta^{jet}|<$ 4.5 and separation $\\Delta\nR_{\\gamma j}>$ 0.5, computed from JETPHOX with $\\sqrt{s}$ = 14~TeV.\nThe contributions from the direct photon production and photon from\nfragmentation are shown in the same plot. The scales and pdf\nuncertainties are also plotted in the same figure. The integrated\ncross sections are listed in Table \\ref{tabgjXS} for different sets\nof selections and collision energies. The scale and pdf\nuncertainties are also listed in this table, with about 10\\%\nuncertainty from scales and around 4\\% from pdf.\n\n\n\\begin{center}\n\\includegraphics[width=7cm]{Plot_GJet_E14000_PhotonPT40_JetPT20}\n \\figcaption{ Differential cross sections of $\\gamma+jet$ as a function of the photon\n$E_T^{\\gamma}$, computed with JETPHOX and the selections\n($E_T^{\\gamma}$, $p_T^{jet}$) $>$ (40, 20)~GeV, $|\\eta^{\\gamma}|<$\n2.5, $|\\eta^{jet}|<$ 4.5 and separation $\\Delta R_{\\gamma j}>$ 0.5\nat $\\sqrt{s}$ = 14~TeV. The solid circle are the total contributions\nwhile the blue triangles represent the direct contribution and the\nred triangles are the contributions of photon from fragmentation.\nScales and pdf uncertainties are also shown in this plot.\n }\n {\\label{fig:DiffXSgj}}\n\\end{center}\n\n\\begin{center}\n\\tabcaption{ \\label{tabgjXS} Cross sections in unit of $10^3$ $pb$\nof $\\gamma+jet$ predicted by JETPHOX at $\\sqrt{s}$ = 13~TeV and\n14~TeV. The total uncertainties including scale uncertainty and pdf\nuncertainty are also list in this table.} \\footnotesize\n\\begin{tabular}{|c|c|c|}\n\\hline\n($E_T^{\\gamma}$, $p_T^{jet}$)$>$ & $\\sqrt{s}$ = 13~TeV & $\\sqrt{s}$ = 14~TeV \\\\\n\\hline\n(20,20)~GeV & 90.1$_{-9.6}^{+11.0}$ & 97.2$_{-11.6}^{+13.3}$ \\\\\n(30,20)~GeV & 48.7$_{-4.6}^{+5.2}$ & 52.7$_{-5.0}^{+5.7}$ \\\\\n(40,20)~GeV & 20.2$_{-1.8}^{+2.2}$ & 22.0$_{-2.1}^{+2.5}$ \\\\\n(20,25)~GeV & 85.0$_{-8.7}^{+9.8}$ & 91.7$_{-9.1}^{+10.9}$ \\\\\n(30,25)~GeV & 41.5$_{-3.9}^{+4.4}$ & 45.0$_{-4.4}^{+4.7}$ \\\\\n(40,25)~GeV & 19.7$_{-1.8}^{+2.0}$ & 21.4$_{-1.9}^{+2.4}$ \\\\\n\\hline\n\\end{tabular}\n\\vspace{0mm}\n\\end{center}\n\n\nAccording to Eq.\\ref{eq2g2jDPS} and the above cross sections of the\nSPS processes, the cross sections for the production of pairs of\nphotons plus two additional jets produced from double parton\nscattering (DPS) in high-energy proton-proton collisions at the LHC\nare calculated for the first time. The results are summarized in\nTable \\ref{tab2g2jDPSXS}. Two jets in the same\n$pp\\to\\gamma\\gamma+2jets$ event from DPS have the same $p_{T}$ cut\nthresholds, both $p_{T}^{jet}>$ 20~GeV or 25~GeV simultaneously. The\ncalculated cross section can be around 0.1 pb to $\\approx$1 pb with\nthe selections considered in this paper. The uncertainty on the\ncross section ia around 50\\%, with the dominant contribution from\nthe uncertainty of $\\sigma_{eff}$.\n\n\\begin{center}\n\\tabcaption{ \\label{tab2g2jDPSXS} Cross sections in unit of $pb$ of\n$\\sigma_{pp\\to\\gamma\\gamma+2jets}^{DPS}$ calculated for $\\sqrt{s}$ =\n13~TeV and 14~TeV with the selections described in the paper. The\ntotal uncertainties including scale uncertainty, pdf uncertainty and\nalso the $\\sigma_{eff}$ uncertainty are also list in this table.}\n\\footnotesize\n\\begin{tabular}{|c|c|c|}\n\\hline\n($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$, both $p_T^{jet}$)$>$ & $\\sqrt{s}$ = 13~TeV & $\\sqrt{s}$ = 14~TeV \\\\\n\\hline\n(30,20,20)~GeV & 0.846$_{-0.432}^{+0.423}$ & 0.960$_{-0.479}^{+0.481}$ \\\\\n(30,20,25)~GeV & 0.428$_{-0.215}^{+0.213}$ & 0.500$_{-0.250}^{+0.250}$ \\\\\n(30,30,20)~GeV & 0.431$_{-0.219}^{+0.215}$ & 0.489$_{-0.242}^{+0.243}$ \\\\\n(30,30,25)~GeV & 0.428$_{-0.215}^{+0.213}$ & 0.250$_{-0.124}^{+0.124}$ \\\\\n(40,20,20)~GeV & 0.437$_{-0.222}^{+0.218}$ & 0.496$_{-0.247}^{+0.247}$ \\\\\n(40,20,25)~GeV & 0.223$_{-0.111}^{+0.113}$ & 0.261$_{-0.129}^{+0.130}$ \\\\\n(40,30,20)~GeV & 0.277$_{-0.141}^{+0.137}$ & 0.313$_{-0.155}^{+0.155}$ \\\\\n(40,30,25)~GeV & 0.136$_{-0.067}^{+0.068}$ & 0.159$_{-0.078}^{+0.078}$ \\\\\n(40,40,20)~GeV & 0.154$_{-0.079}^{+0.076}$ & 0.176$_{-0.087}^{+0.086}$ \\\\\n(40,40,25)~GeV & 0.076$_{-0.037}^{+0.038}$ & 0.089$_{-0.043}^{+0.044}$ \\\\\n\\hline\n\\end{tabular}\n\\vspace{0mm}\n\\end{center}\n\nWith an integrated luminosity of 100 $fb^{-1}$ at $\\sqrt{s}$ =\n13~TeV accumulated in the following years, about 85k\n$pp\\to\\gamma\\gamma+2jets$ events from DPS can be obtained with the\nloosest selections, diphoton ($E_{T}^{\\gamma_{1}}$,\n$E_{T}^{\\gamma_{2}}$)$>$ (30, 20)~GeV and both jets $p_{T}^{jet}>$\n20~GeV. These events can be triggered by the diphoton paths proposed\nat the LHC for $\\sqrt{s}$ = 13~TeV. When the integrated luminosity\nincreasing at $\\sqrt{s}$ = 14~TeV, tighter $E_T$ thresholds on\ndiphoton for the trigger will be used. With the tighter selections,\ndiphoton ($E_{T}^{\\gamma_{1}}$, $E_{T}^{\\gamma_{2}}$)$>$ (40,\n30)~GeV and both jets $p_{T}^{jet}>$ 20~GeV, about 940k\n$pp\\to\\gamma\\gamma+2jets$ events from DPS can be obtained with an\nintegrated luminosity of 3000 $fb^{-1}$. Even with the tightest\nselections studied in this paper, diphoton ($E_{T}^{\\gamma_{1}}$,\n$E_{T}^{\\gamma_{2}}$)$>$ (40, 40)~GeV and both jets $p_{T}^{jet}>$\n25~GeV, we can also get about 260k $pp\\to\\gamma\\gamma+2jets$ events\nfrom DPS with 3000 $fb^{-1}$ as designed by LHC.\n\n\\section{Summary and outlook}\n\nIn this paper, the cross sections for the production of pairs of\nphotons plus two additional jets produced from double parton\nscattering in high-energy proton-proton collisions at the LHC with\n$\\sqrt{s}$ = 13~TeV and 14~TeV (LHC Run2) are calculated for the\nfirst time. With the generic formula, the cross sections have been\ncomputed based on the theoretical perturbative QCD predictions for\nthe productions of $\\gamma\\gamma$ at next-to-next-to-leading-order ,\njet $+$ jet and $\\gamma$ $+$ jet at next-to-leading-order, with\ntheir corresponding single-scattering cross sections. From the LHC\nmeasurements with the collision data obtained in years 2011 and 2012\n(LHC Run1), these theoretical predictions for these three SPS\nprocesses can give the best agreements with the measured data. With\nthe typical acceptance and selections used at LHC, the cross\nsections $\\sigma_{pp\\to\\gamma\\gamma+2jets}^{DPS}$ can be estimated\nto be around 0.1 pb to 1 pb with the collision energy $\\sqrt{s}$ =\n13~TeV or 14~TeV. The expected event rates for $\\gamma\\gamma$ $+$\n2jets from DPS, with some sets of selections, are given for\nproton-proton collisions with the collision energy $\\sqrt{s}$ =\n13~TeV and an integrated luminosity of 100 $fb^{-1}$ planned for the\nfollowing two years, and also $\\sqrt{s}$ = 14~TeV with 3000\n$fb^{-1}$ of integrated luminosity as LHC designed. The\nuncertainties on the cross section and the events rates are mainly\ndominated by the $\\sigma_{eff}$ uncertainty. The scale and pdf\nuncertainties for the productions of these three SPS processes are\nalso considered.\n\nWith the incoming LHC Run2 data, there are enough\n$pp\\to\\gamma\\gamma+2jets$ events from DPS for investigations. It\nneeds further studies on the variables, such as the angles between\ntwo photons and two jets, to be chosen for the discrimination of\n$pp\\to\\gamma\\gamma+2jets$ events from DPS and\n$pp\\to\\gamma\\gamma+2jets$ events from SPS when performing the data\nanalysis. Also the contributions from the DPS to the whole\n$pp\\to\\gamma\\gamma+2jets$ event rates on the distributions of some\ntypical variables are need detailed investigations in the LHC Run2\ndata analysis.\n\n\n\\vspace{3mm} \\emph{The authors would like to thank Dr. Hua-Sheng\nShao from CERN for helpful discussions.}\n\n\n\\end{multicols}\n\n\n\n\\vspace{-1mm} \\centerline{\\rule{90mm}{0.1pt}} \\vspace{2mm}\n\n\n\n\\begin{multicols}{2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{INTRODUCTION}\nThe Milky Way galaxy is the best site for understanding the formation and evolution processes of disk galaxies, because we can observe individual stars composing the stellar disk in great detail. Many observations of the Galactic disk stars have shown the various detailed properties of the present stellar disk, such as the spatial structure (e.g., Yoshii 1982; Gilmore \\& Reid 1983; Juri\\' c et al. 2008; Bovy et al. 2012, 2016), metallicity distribution (e.g., Wyse \\& Gilmore 1995; Lee et al. 2011; Hayden et al. 2015), and radial metallicity gradient (e.g., Nordstr\\\"om et al. 2004; Allende Prieto et al. 2006; Cheng et al. 2012; Toyouchi \\& Chiba 2014). However there are still many mysteries in the formation history of the Galactic disk (see Feltzing \\& Chiba 2013; Rix \\& Bovy 2013 for reviews). \n\nOne of the mysteries is the bimodal distribution of the Galactic disk stars on the [$\\alpha$\/Fe]-[Fe\/H] plane: the high and low-$\\alpha$ peaks of stellar density on the chemical abundance plane appear around the [$\\alpha$\/Fe] ratio of $\\sim$ 0.2-0.3 and $\\sim$ 0, respectively (e.g., Bensby et al. 2003; Lee et al. 2011; Adibekyan et al. 2012; Anders et al. 2014; Mikolaitis et al. 2014). Previous observations have shown that the high-$\\alpha$ sequence stars belong to a dynamically hotter and geometrically thicker disk component than the low-$\\alpha$ sequence stars (e.g., Bensby et al. 2014; Recio-Blanco et al. 2014). Therefore some studies suggested that the definition of the thick and thin disk stars should be based on the chemical abundances of the disk stars (e.g., Navarro et al. 2011; Lee et al. 2011), although the existence of actually distinct thick and thin disk components in our Galaxy is still a matter of debate (e.g., Bovy et al. 2012).\n\nHow the bimodal distribution of the disk stars on the [$\\alpha$\/Fe]-[Fe\/H] plane are formed in the course of the Galactic disk formation is not clearly understood. A possible scenario to reproduce such a bimodal distribution is a brief cessation of star formation at the early disk formation phase (e.g., Chiappini et al. 1997, 2001). In this scenario, after the early intense disk formation phase, the star formation activity in the disk temporarily ceases. Such a cessation of star formation leads a rapid decrease of [$\\alpha$\/Fe] ratio of interstellar medium because chemical enrichment is mostly dominated by Type Ia supernovae (SNe), not Type II SNe. As a result, the disk stars born before and after this cessation of star formation have distinctly different [$\\alpha$\/Fe] ratios, respectively, thereby reproducing the bimodality on the [$\\alpha$\/Fe]-[Fe\/H] plane. Recently, Haywood et al. (2016) showed based on the simple closed box chemical evolution model that the significant drop of star formation rate from 10 to 8 Gyr ago is an essential ingredient to reproduce the bimodal distribution of the Galactic disk stars on the [$\\alpha$\/Fe]-[Fe\/H] plane. However, realizing this scenario requires a specific change of star formation efficiency, for which it may be difficult to find an appropriate physical interpretation. Thus, it is worth exploring yet another idea to explain the existence of such a bimodal distribution.\n\nOne of the important processes in disk evolution is radial migration of disk stars, generally driven by bar\/spiral structures in the galactic disk (e.g., Sellwood \\& Binney 2002; Ro{\\v s}kar et al. 2008; Loebman et al. 2011). Hayden et al. (2015) showed that angular momentum redistribution processes in the stellar disk are necessary to interpret the radial dependence of the metallicity distribution of the Galactic disk stars revealed by the observation of SDSS-III\/APOGEE. Sch\\\"onrich \\& Binney (2009a, b) constructed the semi-analytic chemo-dynamical evolution model including such an effect of radial migration, and successfully reproduced the metallicity distribution and age-metallicity relation of the disk stars in the solar neighborhood. Moreover, they suggested that the bimodal distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane is naturally understood by the effect of radial migration of stars driven by bar\/spiral structures in the galactic disk. According to their model calculation, the high-$\\alpha$ sequence consists of the old disk stars, which have radially migrated from the inner disk regions, whereas most of the stars belonging to the low-$\\alpha$ sequence are born around the solar annulus. However, the recent chemo-dynamical evolution model of Minchev et al. (2013), in which the chemical and dynamical evolution were independently calculated by the semi-analytical model and the cosmological numerical simulation, respectively, showed that radial migration alone is insufficient to construct the observed bimodal distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane.\n\nIn addition to the internal process associated with bar\/spiral structures, the external perturbation such as minor merging of a satellite galaxy and subsequent dynamical heating of the galactic disk can also drive radial migration by depositing the orbital energy and angular momentum of the merging satellite into the galactic disk stars (e.g., Quinn et al. 1993; Vel\\'{a}zquez \\& White 1999; Villalobos \\& Helmi 2008). The radial redistribution of stars triggered by a minor merger is a more dramatical and discontinuous event than that induced by bar\/spiral structures. Minor merger events, possibly leading such rapid radial expansions of the stellar disk, are expected to commonly occur at an early disk formation phase (e.g., Ruiz-Lara et al. 2016). Indeed, Minchev et al. (2013) pointed out the importance of minor merger heating events to thicken the stellar disk up to the same level as the observed Galactic oldest disk component. However, the impact of such a discontinuous radial migration process on the chemical evolution of the galactic disk remains to be understood.\n\nIn this study, we reconsider the effect of radial migration of disk stars as an key ingredient to reproduce the bimodal distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane based on the calculation of a semi-analytical chemo-dynamical evolution model. We particularly focus on the influence of the different radial migration histories on the chemical evolution of the disk galaxy, and for this purpose we consider not only continuous radial migration as driven by bar\/spiral structures, but also discontinuous one as driven by minor merger of a satellite. We will show below that radial migration of disk stars is an important process to affect the chemical evolution over the galactic disk and that the incident of discontinuous radial migration process may be essential to reproduce the observed bimodal distribution of the disk stars on the [$\\alpha$\/Fe]-[Fe\/H] plane.\n\nThis paper is organized as follows. In Section 2, we introduce our chemo-dynamical evolution model. In Section 3, we show the stellar distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane obtained from our model calculations and its dependence on the assumed radial migration histories. In Section 4, we discuss the dependence of the results on model parameters and suggest the specific radial migration history for the stellar disk of the Milky Way. Finally, our conclusions are drawn in Section 5.\n\n\n\\section{MODEL}\nTo investigate the influence of radial migration on a chemical enrichment history of a disk galaxy, we adopt a standard chemo-dynamical model, which has been studied in many previous works (e.g., Sch{\\\"o}nrich \\& Binney 2009a; Kubryk et al. 2015). In this model, a galactic disk consists of many co-center rings, and by calculating baryonic mass evolution for each ring at each time step, we obtain the surface density of gas, $\\sgmg$, that of stars, $\\sgms$, and a mass fraction of heavy element $i$, $Z_i$, respectively, at any time, $t$, and at any radius, $R$. These calculations are carried out in a radial range from $R$ = 0 to $R_\\mr{out}$ (= 16 kpc) with a grid of $\\Delta R$ = 1 kpc over $t$ = 0 to $t_\\mr{p}$ (= 12 Gyr) with a grid of $\\Delta t$ = 50 Myr. In this paper, we are particularly interested in the evolution of [$\\alpha$\/Fe], which is defined here as the average for the typical $\\alpha$ elements of O, Mg, Si, Ca, and Ti.\n\nIn Section 2.1-2.4, we introduce each important process in our chemical evolution model. The main equations to calculate baryonic mass evolution on a galactic disk are described in Section 2.5. Finally, in Section 2.6 we introduce how to choose the values of the important free parameters in our model.\n\n\\subsection{Gas Inflow Rate}\nA galactic disk is thought to form from an inflowing gas from outside in the course of galaxy formation. In our model, we assume that the surface density of gas inflow rate, $\\sgmin$, as functions of $t$ and $R$ is given as,\n\\begin{eqnarray}\n\\sgmin \\tr \\ = \\ \\frac{\\mtotin}{2 \\pi \\hrin^2} \\frac{ \\mr{exp} \\left ( -R\/{\\hrin} - t\/{\\tauin} \\right )}{\\tauin \\left[1- \\mr{exp}(-t_\\mr{p}\/\\tauin) \\right]} \\ ,\n\\label{eq:inflow}\n\\end{eqnarray}\nwhere $\\mtotin$ and $\\hrin$ are the total mass and scale length of inflowing gas on the disk plane, respectively. $\\tauin$ is a time scale of gas inflow rate at $R$, described with $\\tauinz$, $\\tauine$ and $\\alpha$ in our model as,\n\\begin{eqnarray}\n\\tauin (R) = \\tauinz + (\\tauine-\\tauinz) \\left ( \\frac{R}{8 \\ \\mr{kpc}} \\right )^{\\alpha} \\ .\n\\label{eq:tauin}\n\\end{eqnarray}\nThis model for gas inflow rate has five parameters ($\\mtotin$, $\\hrin$, $\\tauinz$, $\\tauine$, $\\alpha$).\n\n\\subsection{Star Formation Rate}\nThe surface density of star formation rate, $\\sgmsf$, in our model follows the observationally motivated star formation law by Bigiel et al. (2008), in which $\\sgmsf$ is proportional to the surface density of H$_2$ gas. The star formation rate can be written using the mass ratio of H$_2$ to HI gas, $\\rmol$, as follows,\n\\begin{eqnarray}\n\\sgmsf = 1.6 \\ \\frac{\\rmol}{\\rmol+1} \\ \\left ( \\frac{\\sgmg}{M_\\odot \\mr{pc}^{-2}} \\right ) \\ \\ \\ [M_\\odot \\ \\mr{pc}^{-2} \\ \\mr{Gyr}^{-1}] \\ .\n\\label{eq:sfr}\n\\end{eqnarray}\nTo calculate $\\sgmsf$ from this equation, we additionally make use of the semi-empirical law of $\\rmol$ provided by Blitz \\& Rosolowsky (2006), which depends on $\\sgmg$ and $\\sgms$ as follows,\n\\begin{eqnarray}\n\\rmol = 0.23 \\ \\left [ \\left ( \\frac{\\sgmg}{10 \\ M_\\odot \\mr{pc}^{-2}} \\right ) \\left ( \\frac{\\sgms}{35 \\ M_\\odot \\mr{pc}^{-2}} \\right )^{0.5} \\right ]^{0.92} \\ .\n\\label{eq:rmol}\n\\end{eqnarray}\nThese empirical laws of equations (\\ref{eq:sfr}) and (\\ref{eq:rmol}) have been known to reproduce the present spatial distribution of HI, H$_2$ and star formation in the Milky Way (Blitz \\& Rosolowsky 2006; Kubryk et al. 2015).\n\n\\subsection{Gas Outflow Rate}\nFeedback processes associated with star formation activity, such as radiation pressure from massive stars and supernova explosions, may drive a strong galactic gas outflow, which gives a significant impact on the chemical evolution of a galactic disk. In this model, we assume that the surface density of gas outflow rate, $\\sgmout$, is proportional to $\\sgmsf$ with an assumed outflow-mass loading factor, $\\Lambda$, which is described as a following function,\n\\begin{eqnarray}\n\\Lambda(R) = \\Lambda_0 + (\\Lambda_8-\\Lambda_0) \\left ( \\frac{R}{8 \\ \\mr{kpc}} \\right )^{\\beta} \\ ,\n\\label{eq:outflow}\n\\end{eqnarray}\nwhere $\\Lambda_0$, $\\Lambda_8$ and $\\beta$ are parameters characterizing the radial dependence of $\\Lambda$.\n\nIt is worth noting that this expression of gas outflow tightly linking with star formation approximately includes the effect of gas radial flow along a galactic disk, because the radial gas flow is expected to be driven by gravitational torques related to the formation of spiral\/bar or giant molecular clouds, which are also important drivers of star formation, so that radial gas flow may occur with a time scales similar to that of star formation (Yoshii \\& Sommer-Larsen 1989). Therefore, in our model $\\Lambda$ can be negative, when the amount of gas expelled from the disk region as outflow is less than that of gas supplied into the region by radial gas flow, although $\\Lambda$ calculated in Section 2.6 are found to be positive at all radii.\n\nWe also note that while for simplicity we assume that $\\Lambda$ does not evolve with time, this assumption may not be appropriate. According to previous works on chemical evolution models applied to several observational results of extra-galactic star-forming galaxies, the outflow-mass loading factor increases with increasing redshift (e.g. Yabe et al. 2015; Toyouchi \\& Chiba 2015). Therefore, our time-independent mass loading factor may be regarded as an time-averaged one, although it actually changes with time.\n\n\n\\subsection{Radial migration}\nTo take into account the effect of radial migration of disk stars in our model, we make use of the method adopted in Sellwood \\& Binney (2002), which can reproduce the basic properties of radial migration obtained in N-body simulations well. This method provides the probability, in which a star born at radius $\\rf$ and time $\\tf$ is found later in radius $R$ and time $t$, $P(t, \\ R, \\ \\tf, \\ \\rf)$, and this is expressed in terms of the following Gaussian function \n\\begin{eqnarray}\nP(t, \\ R, \\ \\tf, \\ \\rf) = \\frac{1}{\\sqrt{2 \\pi \\srm^2}} \\ \\mr{exp} \\left [ \\ -\\frac{(R-\\rf)^2}{2 \\srm^2} \\ \\right ] \\ ,\n\\label{eq:p_rm}\n\\end{eqnarray}\nwhere $\\srm$ corresponds to the diffusion length of stars by radial migration, which is generally a function of $t$, $\\tf$, and $\\rf$. Thus, in this method the radial migration history can be characterized by the time dependence of $\\srm$. \n\nIn our study, to investigate how the difference in radial migration histories affects the chemical evolution of a galactic disk, we consider three different models for $\\srm$. The first model does not include the effect of radial migration, hereafter no radial migration (NRM) model, where $\\srm$ is always zero. Second is the continuous radial migration (CRM) model, where $\\srm$ monotonically increases with increasing $t-\\tf$, which corresponds to the stellar age, as follows,\n\\begin{eqnarray}\n\\srm(t, \\ \\tf, \\ \\rf) = (-0.0667 \\rf + 2.75) \\left ( \\frac{t-\\tf}{5 \\ \\mr{Gyr}} \\right )^{0.5} \\ \\ [ \\mr{kpc} ] \\ .\n\\label{eq:crm}\n\\end{eqnarray}\nThe evolution of $\\srm$ described by this equation is demonstrated in the upper panel of Figure \\ref{fig:srm}, where we show the cases of ($\\tf$\/Gyr, $\\rf$\/kpc) = (1, 5), (1, 10), (3, 5) and (3, 10). Equation (\\ref{eq:crm}) is based on the result of Kubryk et al. (2013), who analyzed the radial migration history in a bar-dominated disk galaxy in the high-resolution N-body+smoothed particle hydrodynamics simulation. The properties of the radial migration in equation (\\ref{eq:crm}) has been known to be roughly similar to that in the chemo-dynamical model of Sch\\\"onrich \\& Binney (2009a), who adopt a different scheme to represent radial migration (Kubryk et al. 2015). Therefore, we study the CRM model as an appropriate test case to examine the influence of continuous radial migration driven by internal mechanisms, such as interaction between bar\/spiral and stars, on the chemical evolution of a galactic disk.\n\nOur third model is the discontinuous radial migration (DRM) model, where $\\srm$ evolves discontinuously at $t$ = $\\trm$ as follows, \n\\begin{eqnarray}\n\\srm(t, \\ \\tf, \\ \\rf) = \\begin{cases}\n\\scalebox{1.0}{$\\displaystyle \\ \\ \\ \\ 1 \\ \\ [\\mr{kpc}]$} & (t \\leq \\trm) \\\\[11pt]\n\\scalebox{1.0}{$\\displaystyle (\\srmz-1) \\times \\mr{min} \\left [ 1, \\ \\left ( \\frac{t-\\tf}{0.5 \\ \\mr{Gyr}} \\right ) \\right ]+1 \\ \\ [\\mr{kpc}]$} & (t > \\trm \\ \\& \\ \\tf \\leq \\trm) \\\\[11pt]\n\\scalebox{1.0}{$\\displaystyle (-0.0667 \\rf + 2.75) \\left ( \\frac{t-\\tf}{5 \\ \\mr{Gyr}} \\right )^{0.5} \\ \\ [\\mr{kpc}]$} & (t > \\trm \\ \\& \\ \\tf > \\trm) \\ .\n\\end{cases}\n\\label{eq:drm}\n\\end{eqnarray}\nIn this model, we assume that while at $t < \\trm$, the disk stars have experienced only moderate radial migration with radial diffusion length of 1 kpc, which is roughly consistent with the scale length of the Milky Way-like galaxies at the early disk formation phase. At $t = \\trm$ the rapid radial migration is assumed to occur with the radial diffusion length, $\\srmz$, in which the time scale of the diffusion is 0.5 Gyr, corresponding to the several dynamical times of the Milky Way disk with the circular velocity of $\\sim$ 200 km s$^{-1}$. This evolution of $\\srm$ for $\\tf < \\trm$ in the DRM model is demonstrated in the lower panel of Figure \\ref{fig:srm}, where we show the case of $\\tf$ = 1 Gyr, $\\trm$ = 2 Gyr and $\\srmz$ = 3 kpc. Subsequently the disk stars, which were born at $t > \\trm$, have followed the same radial migration history as the CRM model. In this DRM model, $\\trm$ and $\\srmz$ are key parameters characterizing the properties of the discontinuous radial migration event. As our fiducial DRM model, we choose $\\srmz$ = 3 kpc, which is roughly consistent with the scale length of the present Milky Way disk, and $\\trm$ = 2 Gyr, corresponding to the look back time of 10 Gyr, which is in roughly agreement with the specific time when the age-metallicity and age-[$\\alpha$\/Fe] relation of the Milky Way disk stars in the solar neighborhood dramatically evolve (Haywood et al. 2013). How the choice of the values of $\\srmz$ and $\\trm$ affects our results will be discussed in Section 4.1. \n\nWe note here that such rapid and discontinuous radial migration in the DRM model is supposed to be triggered by an external perturbation, such as minor merger heating, rather than the internal processes as considered in the CRM model. Thus, by comparing these three radial migration models, we explore the impact on the chemical evolution of the galactic disk by both internal and external dynamical heating events.\n\n\n\\subsection{Basic Equations}\nTo obtain the time evolution of $\\sgmg$, $\\sgms$ and $Z_i$, we solve the following equations, including the effects of star formation, gas inflow, gas outflow and radial migration of stars,\n \\begin{eqnarray}\n\\frac{\\partial \\sgmg}{\\partial t} = - (1 - \\ret) \\sgmsf + \\sgmin - \\sgmout \\ ,\n\\label{eq:gas}\n\\end{eqnarray}\n \\begin{eqnarray}\n \\sgms \\tr = (1 - \\ret) \\int^{R_\\mr{out}}_{0} \\int^{t}_{0} \\frac{\\rf}{R} \\ \\sgmsf (\\tf, \\ \\rf) P(t, \\ R, \\ \\tf, \\ \\rf) \\mr{d}\\tf \\mr{d}\\rf \\ ,\n\\label{eq:star}\n\\end{eqnarray}\n \\begin{eqnarray}\n\\frac{\\partial (Z_i \\sgmg)}{\\partial t} = (Y_{\\mr{II},i} + Y_{\\mr{Ia},i}) \\sgmsf - Z_i (1 - \\ret) \\sgmsf + Z_{\\mr{in},i} \\sgmin - Z_{\\mr{out},i} \\sgmout \\;.\n\\label{eq:metal}\n\\end{eqnarray}\nIn equation (\\ref{eq:gas}) the first term on the right hand side represents the net gas consumption by star formation, where $\\ret$ is the mass fraction returned back to interstellar medium (ISM) via stellar mass loss, and the description of this term is based on an instantaneous recycling approximation. In this paper we set $\\ret$ = 0.45 corresponding to the Chabrier initial mass function (Leitner \\& Kravtsov 2011).\n\nEquation (\\ref{eq:star}) implies that the time evolution of $\\sgms$ at any radius depends on the past star formation and radial migration history over the galactic disk. $R_\\mr{out}$ in equation (\\ref{eq:star}) is the outer limit of $R$ in our calculation, where we set $R_\\mr{out}$ = 16 kpc, i.e., much larger than the disk size of the Milky Way to avoid its effect on the calculation.\n\nIn equation (\\ref{eq:metal}) the first term on the right hand side describes the supply of heavy element $i$ newly synthesized in stars, where $Y_{\\mr{II},i}$ and $Y_{\\mr{Ia},i}$ are the nucleosynthetic yields from Type II and Ia SNe (hereafter SN II and SN Ia), respectively. We assume that $Y_{\\mr{II},i}$ is constant and adopt the SN II yield from Fran\\c{c}ois et al. (2004). On the other hand, $Y_{\\mr{Ia},i}$ changes with time and radius, reflecting the past star formation and radial migration history as follows, \n\\begin{eqnarray}\nY_{\\mr{Ia},i} = \\frac{(1 - \\ret) \\ m_{\\mr{Ia},i} \\ \\fia}{\\sgmsf \\tr} \\int^{R_\\mr{out}}_{0} \\int^{t-\\Delta t_\\mr{Ia}}_{0} \\frac{\\rf}{R} \\ \\frac{\\sgmsf (\\tf, \\ \\rf) P(t, \\ R, \\ \\tf, \\ \\rf)}{t-\\tf} \\mr{d}\\tf \\mr{d}\\rf \\ ,\n\\label{eq:yia}\n\\end{eqnarray}\nwhere $\\fia$ is a free parameter controlling the SN Ia rate in the galactic disk, and $\\Delta t_\\mr{Ia}$ is a minimum delayed time of SN Ia. Here we set $\\Delta t_\\mr{Ia}$ = 0.5 Gyr suggested by Homma et al. (2015), which reproduced, with this value of $\\Delta t_\\mr{Ia}$, star formation histories and chemical evolutions of the Galactic dwarf galaxies self-consistently. Another SN Ia parameter, $m_{\\mr{Ia},i}$, in equation (\\ref{eq:yia}) is the released mass of heavy element $i$ per a Type Ia supernova, for which we adopt the SN Ia yield of W7 model in Iwamoto et al. (1999). \n\nThe second term on the right hand side of equation (\\ref{eq:metal}) is the mass of heavy elements finally locked up in stars. The third and fourth terms on the right hand side of equation (\\ref{eq:metal}) denote the mass injection and ejection of heavy elements associated with inflow and outflow, respectively, where $Z_{\\mr{in},i}$ and $Z_{\\mr{out},i}$ are mass fractions of heavy element $i$ in inflowing and outflowing gas, respectively. In this model, we set inflowing gas abundance of [Fe\/H] = -1.5, which is the typical metallicity of the Galactic halo stars (e.g., Carollo et al. 2010), and [$\\alpha$\/Fe] = 0.4 for each $\\alpha$ element, roughly corresponding to the average abundance ratio of $Y_\\mr{II,\\alpha}$ to $Y_\\mr{II, Fe}$. Additionally we assume that the metallicity of outflowing gas corresponds to that of the ISM, implying $Z_{\\mr{out},i} = Z_i$.\n\n \n\\subsection{Determination of Model Parameters}\nThis chemical evolution model contains nine free parameters, which are summarized in Table 1. In this study, we adopt the MCMC method (Metropolis et al. 1953; Hastings 1970) to obtain the best set of these nine free parameters, which reproduce the observed radial profiles of gas, star, [O\/H] and [Fe\/H] in the Milky Way disk. In this model, following the results of several previous works, we adopt the Galactic stellar disk with the scale length of 2.3 kpc and the stellar density at $R$ = 8 kpc of 35 $\\mr{M_\\odot} \\ \\mr{pc}^{-2}$ (Flynn et al. 2006). For the total gas density profile, consisting of HI and H$_2$, in the Galactic disk, we adopt the work of Kubryk et al. (2015) shown in their Figure (A.2), and the radial profile of [O\/H] and [Fe\/H] in the disk is taken from the observation of Cepheids in Luck \\& Lambert (2011). \n\nIn our MCMC procedure, for each radial migration model, we carry out 5 MCMC chains starting from different parameter sets. Each chain consists of 100,000 iterations, and by compiling the later 50,000 iterations in all chains, we get the posterior probability distribution for each parameter. Figure \\ref{fig:mc_nrm}, \\ref{fig:mc_crm} and \\ref{fig:mc_drm} show the posterior probability distributions of the nine parameters for the NRM, CRM, and DRM models, respectively. We find from these figures that for all radial migration models, our MCMC chains for all parameters converge successfully. The results of this estimation of the nine parameters for each radial migration model are summarized in Table 2. \n\nFigure \\ref{fig:rp_nrm}, \\ref{fig:rp_crm} and \\ref{fig:rp_drm} show the time evolution of the radial profiles of gas, star, [O\/H] and [Fe\/H] obtained from the calculation based on the set of best-fit parameters for the NRM, CRM and DRM models, respectively. The green, blue, yellow and red lines in each panel in these figures represent the results at $t$ = 2, 4, 8, and 12 Gyr, respectively, and the black line denotes the observed profiles. From comparison between the red and black lines in these figures, we find that all of our three models can give the moderately good reproduction for the present chemo-structural properties of the Galactic disk. Thus our chemical evolution models are appropriate in comparison of the Galactic stellar disk. In the next section, we present the detailed properties of chemical evolutions provided by these best models with three different assumptions for radial migration.\n\n\n\\section{RESULTS}\nIn this section, by investigating the results of the three radial migration models, we study the difference in the stellar distribution on the [$\\alpha$\/Fe]-[Fe\/H] diagram, arising from the different radial migration histories. \n\nIn the top, middle, and bottom panels of Figure \\ref{fig:plot}, we plot the [$\\alpha$\/Fe] vs. [Fe\/H] for gas in each radial ring at each time step for the NRM, CRM, and DRM models, respectively. The color of each plot in the left and right panels shows the look back time and the radius, respectively. These panels show a general trend that at a fixed [Fe\/H] disk gas at an earlier epoch has a higher [$\\alpha$\/Fe] ratio (left panel) and that at a fixed [$\\alpha$\/Fe] inner disk gas has a higher [Fe\/H] (right panel) for all the radial migration models. There are also some remarkable differences between the three models. Comparing the CRM with NRM models, the former model shows a more rapid decrease of [$\\alpha$\/Fe] with time at a large $R$ than the latter model. This difference seen at a larger $R$ is due to the increase of the number ratio of SNe Ia to SNe II, $N_\\mr{Ia}\/N_\\mr{II}$, caused by radial migration in which intermediate and old disk stars, which eventually explode as SNe Ia, migrate from inner to outer disk regions. However, since this migration event of stars proceeds gradually in this CRM model, the increase of $N_\\mr{Ia}\/N_\\mr{II}$ remains insignificant that the time dependence of [$\\alpha$\/Fe]-[Fe\/H] diagram at each radius does not change so dramatically. \n\nOn the other hand, the DRM model shows a remarkable property compared with other models. In this model, [$\\alpha$\/Fe] at $R \\gtrsim 6$ kpc decreases rapidly at the look back time of $\\sim$ 10 Gyr, corresponding to the timing of the discontinuous radial migration event, so that the density of plotted dots on the [$\\alpha$\/Fe]-[Fe\/H] plane is made sparse near [$\\alpha$\/Fe] $\\sim$ 0.2-0.3 and [Fe\/H] $\\sim$ $-$0.5. In contrast, in the inner disk regions of $R \\lesssim 3$ kpc the declines of [$\\alpha$\/Fe] temporarily suspend immediately after the discontinuous radial migration event, and consequently the plotted dots on the plane are crowded around [$\\alpha$\/Fe] $\\sim$ 0.2-0.3 and [Fe\/H] $\\sim$ 0. These changes in the inner and outer disk regions are explained by the rapid decrease and increase of $N_\\mr{Ia}\/N_\\mr{II}$, respectively, arising from the rapid transfer of disk stars from inner to outer disk regions, which occurs much rapidly in the DRM model. Thus the event of fast radial redistribution of disk stars as supposed in the DRM model provides a remarkable feature in the [$\\alpha$\/Fe]-[Fe\/H] diagram for disk gas.\n\nTo specify how these differences in the chemical evolution of disk gas shown in Figure \\ref{fig:plot} appear in the observed disk stars, we show the present stellar distributions on the [$\\alpha$\/Fe]-[Fe\/H] plane with color maps and contours in Figure \\ref{fig:dist}. The results of the NRM, CRM, and DRM models are shown in the top, middle, and bottom panels, respectively. The left, middle, and right panels for each model represent the stellar distribution observed at the inner ($R$ = 5-7 kpc), solar neighborhood ($R$ = 7-9 kpc), and the outer disk region ($R$ = 9-11 kpc), respectively. For comparison with observations, we plot the black circle and square at ([$\\alpha$\/Fe], [Fe\/H]) = (0.2, $-$0.2) and (0.05, 0.05) in each panel, which delineate the observed [$\\alpha$\/Fe] and [Fe\/H] values of the high and low-$\\alpha$ peaks for the nearby disk stars, respectively (see Figure 3 in Hayden et al. 2015).\n\nAs shown in many previous works, the effect of radial migration of stars makes their distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane diffuse at all radial ranges. In the CRM model, we find that the distribution of stars in this abundance-ratio diagram is unimodal and that the metallicity of their density peak remains higher at inner $R$, suggesting the negative radial metallicity gradient as also seen in the NRM model. In contrast, it is remarkable that at all radii the DRM model produces bimodal stellar density distributions on the [$\\alpha$\/Fe]-[Fe\/H] plane, where the [$\\alpha$\/Fe] ratios of the high and low-$\\alpha$ density peaks are $\\sim$ 0.2-0.3 and $\\sim$ $-$0.05, respectively. The stars located at high-[$\\alpha$\/Fe] density peaks are originally born in inner disk regions and have moved outward. In addition, as a result of the discontinuous migration event, the number density of stars near [$\\alpha$\/Fe] $\\sim$ 0.2-0.3 and [Fe\/H] $\\sim$ $-$0.5, is made sparse related to the same effect seen in the bottom panels of Figure \\ref{fig:dist}, so that the bimodal distributions as observed in the Milky Way disk stars are successfully reproduced in the DRM model. It is also worth remarking that while the metallicity of the high-[$\\alpha$\/Fe] peak is generally independent of radius, that of the low-[$\\alpha$\/Fe] peak decreases with increasing radius, in good agreement with the recent observational results for the Galactic stellar disk (Hayden et al. 2015; Kordopatis et al. 2015). \n\nThe comparisons with the observed high- and low-[$\\alpha$\/Fe] peaks denoted with the black circle and square in the figure show that in our model the metallicity of the high-[$\\alpha$\/Fe] peak and the [$\\alpha$\/Fe] ratio of the low-[$\\alpha$\/Fe] peak are somewhat higher and lower than the observed values, respectively. This small difference from the observational data is due to our simple constraints adopted in the MCMC procedure, namely based on only the present radial profiles of gas, stars and metallicity of the Milky Way disk, without using the distributions of [$\\alpha$\/Fe] and [Fe\/H] as constraints. More refined modelings and extensive searches of model parameters are needed to reproduce the details of the observed abundance distributions, which is however beyond the scope of this work, aiming to demonstrate the impact of the radial migration history on the [$\\alpha$\/Fe] vs. [Fe\/H] diagram.\n\nThus our model experiments suggest that the discontinuous radial migration history can be a candidate solution in explaining the observed bimodal distribution of the Milky Way disk stars on the [$\\alpha$\/Fe]-[Fe\/H] plane. In the next section, we discuss the possibility of such a discontinuous radial migration of stars as represented in the DRM model in more detail and examine its significance in the chemical evolution history of the Galactic stellar disk.\n\n\n\\section{DISCUSSION}\n\\subsection{Dependence of the model results on $\\trm$ and $\\srmz$}\nA discontinuous radial migration event is characterized by the value of $\\trm$ and $\\srmz$, corresponding to the timing and the magnitude of the radial migration. We here examine how different choices of these value can affect the stellar distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane. \n\nFigure \\ref{fig:trm} shows the stellar density distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane for the two DRM models with different values of $\\trm$ from the fiducial DRM model of $\\trm$ = 2 Gyr. The top and bottom panels are the results of the model with $\\trm$ = 1 and 4 Gyr, respectively, for which we set $\\srmz$ = 3 kpc for both models. \n\nWe find from the top panels that the model with a smaller value of $\\trm$ does not produce a clear bimodal distribution. This may be explained by the too early timing of the discontinuous radial migration event, in which the amount of disk stars that are transferred to the outer disk region is not enough to cause a rapid increase of $N_\\mr{Ia}\/N_\\mr{II}$: the mass density of the stellar disk was still small at such an early epoch. On the other hand, although the model with $\\trm$ = 4 Gyr seems to reproduce a bimodal distribution, the two peaks are mostly merged because the [$\\alpha$\/Fe] ratio in the inner disk region is made already low before the radial migration event occurs. Thus the formation of such a bimodal distribution is sensitive to the timing of the discontinuous radial migration event: to reproduce the observed bimodality in our Galactic stellar disk, $\\trm$ is confined to be around 2-3 Gyr.\n\nNext, we investigate the dependence of the model results on $\\srmz$. Figure \\ref{fig:srmz} shows the results for the DRM model when we adopt two different values of $\\srmz$. The top and bottom panels correspond to the cases of $\\srmz$ = 2 and 4 kpc, respectively, where we set $\\trm$ = 2 Gyr for both models. It follows that while the high-$\\alpha$ peak around [$\\alpha$\/Fe] $\\sim$ 0.2-0.3 disappears in the model with $\\srmz$ = 2 kpc, it remains in the model with $\\srmz$ = 4 kpc but more clearly than in the fiducial DRM model with $\\srmz$ = 3 kpc. Such a dependence on $\\srmz$ is simply explained in terms of the increase of the fraction of the inner disk stars with increasing $\\srmz$, which can reach the outer disk region by radial migration. Thus, the value of $\\srmz$, namely the magnitude of the radial migration event, is also an important ingredient for the formation of the bimodal distribution as well as $\\trm$: to produce a clear bimodal distribution we need $\\srmz \\geq$ 3 kpc, where its actual value can be derived from the number ratio between stars belonging to the high and low-$\\alpha$ sequences.\n\nBased on the above experiments, as one of possible mechanisms to produce the observed bimodal distribution of the disk stars on the [$\\alpha$\/Fe] plane, we propose the past event of a rapid and discontinuous radial migration with a diffusion scale of $\\geq$ 3 kpc which may have occurred during 2-3 Gyr after the onset of the disk formation. In the next section we discuss the possibility of this scenario in the context of galaxy evolution.\n\n\\subsection{What would cause discontinuous radial migration?}\nHere, we consider the possibility that the discontinuous radial migration event is triggered by a minor merger of a relatively massive satellite galaxy onto the Milky Way stellar disk. Following the above experiments, this last minor merger event, which can affect the dynamical structure of the disk, may have occurred at the look back time of 9-10 Gyr. This scenario may not be unrealistic in the cosmological context: according to Ruiz-Lara et al. (2016), in disk galaxies like the Milky Way, last minor mergers with satellite galaxies being more massive than 1\/10 virial mass of host galaxies can take place at a look back time of $\\sim$ 9 Gyr on average. \n\nOn the other hand, another condition of $\\srmz \\geq$ 3 kpc seems rather specific when we consider the possibility of vastly various properties of minor mergers. Therefore it is worth assessing the possibility of such a discontinuous radial migration event in terms of the evolution of a scale length of stellar disk rather than $\\srmz$. In Figure \\ref{fig:hr} the black and red solid lines show the stellar density profiles for the fiducial DRM model at $t$ = 2 and 3 Gyr, corresponding to immediately before and after the discontinuous radial migration event, respectively. The dashed lines represent the results of the exponential fitting to these stellar density profiles in the radial range of $R \\leq$ 10 kpc, and the scale lengths obtained from these fittings are also shown in the upper-right rectangle in Figure \\ref{fig:hr}. This figure implies that the discontinuous radial migration event with $\\srmz$ = 3 kpc increases the scale length of the stellar disk from 1.42 kpc to 2.09 kpc. Such a change of the scale length is reasonably achievable in the minor merger of a satellite and associated disk heating. Indeed, the N-body simulations in Villalobos \\& Helmi (2008) show that the scale length of a disk galaxy, in which the stellar mass is similar to that of our model galaxy at the time of the rapid radial migration event, is grown by minor merger heating from 1.65 kpc to 2.1 kpc. Thus the discontinuous radial migration assumed in our DRM model may be the case in a minor merger event between massive a satellites and the Milky Way stellar disk.\n\nWe note here that such a minor merger event may also trigger not only discontinuous radial migration but also discontinuous disk thickening. Bovy et al. (2012) investigated the spatial distribution of the Galactic mono-abundance disk populations and suggested that the vertical structure of the Galactic stellar disk has evolved continuously rather than discontinuously. This implies that a supposed merger event does not increase the disk thickness so dramatically, although this may be unlikely in light of the associated dynamical heating of a stellar disk. To resolve this paradoxical situation, one possible solution is to rely on a special minor merger event, in which the orbit of an accreting satellite is almost parallel to the stellar disk plane of the host galaxy. Villalobos \\& Helmi (2008) show that a minor merger event of a massive satellite with a parallel orbit to the initial stellar disk can thicken the stellar disk with an amount only to the scale height of $\\sim$ 0.6 kpc, which is much thinner than the thickest population of the present Galactic stellar disk. Thus, if such a minor merger event without causing significant disk thickening occurred in the Galactic past, then the issue mentioned above may be resolved.\n\n\n\\section{Summary \\& Conclusion}\nIn this paper, we have calculated the chemo-dynamical model to investigate the influence of radial migration histories on the chemical evolution of a disk galaxy, in particular focusing on stellar distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane. For this purpose, we have examined the three models, where we consider the models with both continuous and discontinuous radial migration events, in comparison with the no radial migration model. \n\nWe found that radial migration can speed up the evolution of [$\\alpha$\/Fe] in outer disk regions by increasing the rate of SNe Ia to SNe II because of the net transfer of intermediate and old disk stars, including progenitors of SNe Ia, from inner to outer disk regions. Moreover, in the model of the rapid and discontinuous radial migration, such an effect is stronger than the case of the continuous radial migration, thereby the properties of the [$\\alpha$\/Fe]-[Fe\/H] diagram in the inner and outer disk regions are made distinctly different. Thus chemical evolution histories of disk galaxies can be significantly affected by their radial migration histories.\n\nWe also found that the discontinuous radial migration can reproduce the bimodal distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane as observed in the Galactic stellar disk. Our model calculation predicts that the high-[$\\alpha$\/Fe] sequence consists of disk stars which were originally born in inner disk regions and have been moved to the solar neighborhood by the rapid and discontinuous radial migration event. On the other hand, the low-[$\\alpha$\/Fe] sequence stars are originated from the outer disk region, where the rapid decrease of [$\\alpha$\/Fe] caused by the rapid increase of the rate of SN Ia to SN II is achieved. A remarkable point of the bimodal distribution reproduced by our discontinuous radial migration model is that while the metallicity of the high-[$\\alpha$\/Fe] sequence is roughly independent of radius, that of the low-[$\\alpha$\/Fe] sequence increases with increasing radius, reflecting the negative radial metallicity gradient, in good agreement with the observational property of the Galactic stellar disk. Therefore, we suggest in this paper that the discontinuous radial migration scenario is a key in understanding the observed bimodal distribution of the Milky Way disk stars on the [$\\alpha$\/Fe]-[Fe\/H] plane.\n\nTo further understand this formation scenario of the bimodal distribution, we have investigated the dependence of the bimodality on the model parameters for radial migration. It is found that the reproduction of a bimodal distribution on the [$\\alpha$\/Fe]-[Fe\/H] plane is very sensitive to the timing and magnitude of the discontinuous radial migration event. According to our model calculations, in order to reproduce such a bimodal distribution on the [$\\alpha$\/Fe] plane observed in the Galactic stellar disk, a rapid and discontinuous radial migration event with the diffusion length of stars of $\\geq$ 3 kpc is required to occur during 2-3 Gyr after the onset of the disk formation. Thus the bimodality in the [$\\alpha$\/Fe]-[Fe\/H] distribution of disk stars is a key in elucidating the detailed radial migration history. Further studies based on detailed numerical simulations for minor merging events in the Galactic past may be necessary to obtain their effects on the mechanism of radial migration of stars and the subsequent role in the chemo-dynamical evolution of a disk galaxy like the Milky Way.\n\n\n\n\\acknowledgments\nWe are grateful to Kenji Bekki for his invaluable comments on this work. This work is supported in part by JSPS Grant-in-Aid for Scientific Research (No. 27-2450 for DT) and MEXT Grant-in-Aid for Scientific Research on Innovative Areas (No. 15H05889 for MC).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}