\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 168, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/168\hfil
A bidimensional bi-layer shallow-water model]
{A bidimensional bi-layer shallow-water model}

\author[B. Roamba, J. d. D. Zabsonre \hfil EJDE-2017/168\hfilneg]
{Brahima Roamba, Jean de Dieu Zabsonre}

\address{Brahima Roamba \newline
UFR/ST, IUT, Universit\'e Nazi Boni,
10 BP 1091 Bobo-Dioulasso, Burkina Faso}
\email{braroamba@gmail.com}

\address{Jean de Dieu Zabsonre \newline
IUT, Universit\'e Nazi Boni,
10 BP 1091 Bobo-Dioulasso, Burkina Faso}
\email{jzabsonre@gmail.com}

\dedicatory{Communicated by Vicentiu Radulescu}

\thanks{Submitted January 20, 2017. Published July 5, 2017.}
\subjclass[2010]{35Q30, 76B15}
\keywords{Shallow water; bi-layer; viscous models; energetic consistency;
\hfill\break\indent global weak solution}

\begin{abstract}
 The existence of global weak solutions  in a periodic domain  
 for  a non-linear viscous bi-layer shallow-water model with
 capillarity effects and extra friction terms in a two-dimensional
 space has been proved in \cite{ZaGla}.
 The main contribution of this article is to show the existence of global
 weak solutions without friction term  or capillary effect  following
 the ideas of \cite{alexisv1} for the two dimensional case.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

The shallow-water equations are usually used to model some natural phenomena
such as ocean circulation, coastal areas, rivers, lakes,
avalanches, etc. However  in many situations one layer of shallow-water
cannot be used to model the system. In some cases such as the Strait of Gibraltar
it is necessary to consider two layers  of  shallow-water system to model
the flow. For this purpose many derivations of  bi-layers and multi-layers
shallow-water system have been done
(see  \cite{audusse,NGladys,Peyberne}).

In this  article we study the existence of global weak solutions of  the
bi-layer shallow-water model  derived in \cite{NGladys}.
In \cite{ZaGla}, the authors obtained the existence of  global weak
solutions for a 2D viscous bi-layer shallow-water model derived in \cite{NGladys}.
In their  work they considered  in a periodic domain $\Omega$, a system
composed of  two layers of immiscible fluids with different and
constant densities ($\rho_1$ and $\rho_2$, resp.) and viscosities
($\nu_1$ and $\nu_2$, resp.) and imposed $\nu_1 < \nu_2$.
The system studied in \cite{ZaGla}  reads as follows:
\begin{gather}
 \partial_t h_1 + \operatorname{div}(h_1 u_1)=0;\label{mas1}\\
\begin{aligned}
&\rho_1 \partial_t (h_1 u_1)+\rho_1 \operatorname{div}(h_1
u_1\otimes u_1)-2\nu_1 \operatorname{div}(h_1 D(u_1))+ \rho_1 g h_1 \nabla h_1 \\
&+\rho_2 g h_1 \nabla h_2
 -\Big(1+\frac{c_0 \beta(h_1)h_1}{6\nu_1}\Big)
\operatorname{fric}(u_1,u_2)+c_0 \beta(h_1) u_1\\
&-\alpha_1 h_1 \nabla(\Delta h_1)- \alpha_2 h_1 \nabla(\Delta
h_2)=0;
\end{aligned} \label{mas2} \\
\partial_t h_2 + \operatorname{div}(h_2 u_2)=0;\label{mas3} \\
\begin{aligned}
&\rho_2 \partial_t(h_2 u_2)+\rho_2 \operatorname{div}(h_2
u_2\otimes u_2)-2\nu_2 \operatorname{div}(h_2 D(u_2))
+\rho_2 g h_2\nabla h_2\\
&+\rho_2 g h_2 \nabla h_1
+\operatorname{fric}(u_1,u_2)-\alpha_2 h_2 \nabla (\Delta h)=0.
\end{aligned}\label{mas4}
\end{gather}
where $ \operatorname{fric}(u_1,u_2)=-c_1B(h_1,h_2)(u_1-u_2)$
with
\[
 B(h_1; h_2) =\frac{h_1h_2}{\frac{\nu_1}{\nu_2}h_1+\frac{\nu_2}{\nu_1}h_2}.
\]
  The drag coefficient $B$ is introduced  to  control the friction terms
at the interface.
In their paper the special feature of their  definition of weak solution is based
on the test functions depending on the unknowns used for the momentum
equations namely $h_i \varphi$.
This one has been introduced in \cite{BDcomp} and allowed them
to get the compacticity when height limit  vanishes.
This particular definition of   weak solutions with test functions
$h\varphi$ was firstly introduced  in \cite{Desjardins}.
 With this particular definition of weak solutions the  author in
\cite{jungle} proved the existence of global  weak solutions of quantum
 Navier-Stokes equations in 3D. The main idea
of his paper is to rewrite quantum Navier-Stokes equations as a viscous quantum Euler
system by means of the effective velocity.

In \cite{Lacroix}, the authors proved the existence  of global  weak solutions
 for the compressible quantum
Navier-Stokes equations by the use of a singular cold pressure.
In  \cite{alexisv1} the authors, using the ``BD-entropy" obtained  the existence
of usual global weak solutions for 3D compressible Navier-Stokes equations with
degenerating viscosity. They derived ``Mellet-Vasseur" type inequality
which allows them to get global solutions in time.

For some fluids like electrorheological fluids which can change from a liquid
like state to a solid like viscous state, another approach to study the
existence of weak solution was developed in \cite{MiRa, RaZh}.

Our analysis takes inspiration from the work developed in \cite{alexisv1}.
Our contribution compared with the work performed in \cite{ZaGla} is that
we obtain  the existence of  global usual  weak solutions with test functions
independent  from  the unknowns without friction term and without any
condition on the  two viscosities coefficients.

In   \cite{BDexist,BDdif} the authors obtained  global weak
solutions for a 2D shallow-water system and Korteweg system with diffusion
term of ttype $\nu \operatorname{div}(hD(u))$. 
They proved  that the considered system
is energetically consistent without any restriction on the data.
The key point of this proof is based on an estimate of a new entropy,
called ``mathematical BD entropy", which gives a bound of the term
$\nabla \sqrt{h}$.
We denote that to obtain this result in \cite{BDexist,BDcomp,BDZ}
it was necessary for the  authors  to  add  linear and quadratic terms
of the form $r_0 u$, $r_1h|u|u$ in the momentum equation.
In \cite{melletvas} without additional regularizing terms, the authors
obtained  the existence of  global weak solutions for the barotropic Navier-Stokes
equations. They proved an inequality namely "Mellet-Vasseur"  type inequality
and obtained  a control on $ \int_{\Omega}h_{i}(1+|u_i|^2)\ln(1+|u_i|^2)dx$.

In \cite{Florie} and \cite{Peyberne}, the authors proved the existence of
global weak solutions of a bi-layers shallow-water model without any
friction term but with a diffusion term of the form $\nu\Delta u$.
This analysis used the method developed in \cite{Orenga} and the system
is energetically consistent only for small enough initial data.

  Following \cite{alexisv1}, we add friction terms $r_{i_0}u_i$,
$r_{i_1}h_i|u_i|^2u_i$ and the Bohm potential term
$\kappa_ih_i\nabla(\frac{\Delta\sqrt{h_i}}{\sqrt{h_i}})$  in the momentum equation.
 Following \cite{BDexist} and  \cite{alexisv1} the terms $r_{i_0}u_i$,
$r_{i_1}h_i|u_i|^2u_i$ turn out to be essential to obtain the compactness of
$\sqrt{h_i}u_i$ in $L^2(0,T;L^2(\Omega))$ and $h_iu_i\otimes u_i$ in
$L^1(\Omega)$. The  Bohm potential
$\kappa_ih_i\nabla(\frac{\Delta\sqrt{h_i}}{\sqrt{h_i}})$ allows to deduce an
estimate on $\nabla h_i^{1/4}$ in $L^4([0,T]\times\Omega)$
(see \cite{alexisv1}).
   Another  contribution in this paper is devoted to the convergence when
 the coefficients  $r_{i_0}, r_{i_1}$ and $\kappa_i$   go to $0$ for bi-layers
shallow-water model. We consider in a periodic domain with periodic boundaries
conditions the system
\begin{gather}
\partial_t h_1 + \operatorname{div}(h_1 u_1)=0;\label{mas1b}\\
\begin{aligned}
&\rho_1 \partial_t (h_1 u_1)+\rho_1 \operatorname{div}(h_1
u_1\otimes u_1)-2\nu_1 \operatorname{div}(h_1 D(u_1)) \\
&+ \rho_1 g h_1 \nabla h_1 +\rho_2 g h_1 \nabla h_2 =0;
\end{aligned}\label{mas2b} \\
\partial_t h_2 + \operatorname{div}(h_2 u_2)=0;\label{mas3b} \\
\begin{aligned}
&\rho_2 \partial_t(h_2 u_2)+\rho_2 \operatorname{div}(h_2
u_2\otimes u_2)-2\nu_2 \operatorname{div}(h_2 D(u_2))\\
& +\rho_2 g h_2\nabla h_2+\rho_2 g h_2 \nabla h_1=0
\end{aligned} \label{mas4b}
\end{gather}
with initial conditions:
\begin{equation}\label{Idata}
{h_i}_{|t=0}=h_{i_0}\geq 0,\quad  {h_i u_i}_{|t=0}= m_{i_0},
\end{equation}
for which we assume the following regularities:
\begin{equation}\label{Rdata}
\begin{gathered}
h_{i_0}\in L^2(\Omega), \quad \nabla h_{i_0}\in (L^2(\Omega))^2, \quad
 \nabla \sqrt{h_{i_0}}\in (L^2(\Omega))^2\\
\frac{| m_{i_0}|^2}{h_{i_0}}\in L^1(\Omega), \quad \log_- (h_{i_0})\in
L^1(\Omega).
\end{gathered}
\end{equation}
 for  $i=1,2$.
We denote by $D(u)$ the strain tensor, defined by
$D(u)=\frac{\nabla u+\nabla ^t u}{2}$, and by $A(u)$, the
vorticity tensor such as $A(u)=\frac{\nabla u-\nabla ^tu}{2}$.

The article is organized as follows:
In Section \ref{exist} we give the definition of global weak solutions of
the system \eqref{mas1}-\eqref{mas4} and we state the results of the
existence of weak solutions for the system \eqref{mas1}-\eqref{mas4}.
And moreover we give some Theorems which are  very useful in this current paper.
Section \ref{constr} is devoted to the construction of approximate
 ``Mellet-Vasseur" type inequality for
any weak solutions. In this section we show that  we can control
(uniformly with  respect to $\kappa_i$) this quantity, for any weak
solutions of \eqref{mas11}-\eqref{mas22} with $\kappa_i> 0$.
In Section \ref{recover1}, we study the limits as $\alpha_i$ defined in
\eqref{cisse} approaches $\infty$.
 On the other hand, Section \ref{recover2} is dedicated to the convergence
 of terms when $r_{0_i}, r_{1_i}$ and $\kappa_i$  go to  zero.
In  Section \ref{recover1'} we give the proof of  the ``Mellet-Vasseur"
 type inequality. We denote that  in this section we give also the proof
of  Theorem \ref{theo2} by recovering the limit from
Lemma \ref{lem62}.

\section{Main results}\label{exist}

We  start this section with the definition of weak solutions.

\begin{definition}\label{defit1} \rm
We shall say that $(h_1,h_2,u_1,u_2)$ is a weak solution of
\eqref{mas1}-\eqref{mas4} if \eqref{mas1} and
\eqref{mas3} hold in $(\mathcal{D}'(0,T)\times \Omega)^2$;
\eqref{Idata} holds in
$\mathcal{D}'(\Omega)$; the following assumptions are satisfied:
\begin{equation}
\begin{gathered}
h_i \in L^\infty(0,T;L^2(\Omega));\\
\nabla h_i \in L^2(0,T;(L^2(\Omega))^2) \text{ and } \sqrt{h_i}u_i \in
L^\infty(0,T;(L^2(\Omega))^2);\\
\sqrt{h_i}D(u_i) \in L^2(0,T;(L^2(\Omega))^4);\\
\nabla\sqrt{h_i}\in L^2(0,T;L^2(\Omega)^2);
\end{gathered}
\end{equation}
for any $\varphi\in \mathcal{C}^\infty((0,T)\times \Omega)^2$ with
$\varphi(T,\cdot)=0$, ($\varphi$ with compact support), we have
\begin{equation}\label{defi2}
\begin{aligned}
&-\rho_1 h_{1_0} u_{1_0}\varphi(0,\cdot)-\int_0^T\int_\Omega\rho_1
h_1u_1 {\partial}_t\varphi  -\rho_1 \int_0^T \int_\Omega(h_1 u_1\otimes
 u_1):D(\varphi)\\
&+2\nu_1\int_0^T\int_\Omega h_1 (D(u_1):D(\varphi))+\frac{1}{2}\rho_1 g \int_0^T \int_\Omega
h_1^2\operatorname{div}\varphi+\rho_1 g \int_0^T \int_\Omega
h_1\nabla h_2\varphi\\
&=0
\end{aligned}
\end{equation}
and
\begin{equation}\label{defi1}
\begin{aligned}
&-\rho_2 h_{2_0} u_{2_0}\varphi(0,\cdot)-\int_0^T\int_\Omega\rho_2
h_2u_2 {\partial}_t\varphi -\rho_2 \int_0^T \int_\Omega(h_2 u_2\otimes u_2):D(\varphi) \\
&+2\nu_2\int_0^T\int_\Omega h_2
(D(u_2):D(\varphi))+\frac{1}{2}\rho_2 g \int_0^T \int_\Omega
h_2^2\operatorname{div}\varphi+\rho_2 g
\int_0^T \int_\Omega h_2(\varphi\cdot \nabla h_1)\\
&=0.
\end{aligned}
\end{equation}
\end{definition}

We will prove the following theorem.

\begin{theorem}\label{theorem}
There exists a global weak solution $(h_1,h_2,u_1,u_2)$ of
\eqref{mas1}-\eqref{mas4} satisfying the entropy inequalities
\eqref{en1} and \eqref{en2}.
\end{theorem}

In this section, we give the classical energy estimate and the
``mathematical BD entropy". These two inequalities will allow us to
prove the main theorem.

\begin{lemma}\label{enclas}
Let $(h_1,h_2,u_1,u_2)$ be a solution of the system
\eqref{mas1}-\eqref{mas4}. Then
\begin{equation}\label{en1}
\begin{aligned}
&\frac{1}{2} \rho_1 \frac{d}{d t}\int_\Omega h_1
|u_1|^2+\frac{1}{2} \rho_2\frac{d}{d t}\int_\Omega h_2 |u_2|^2+2\nu_1
\int_\Omega h_1(D(u_1):D(u_1))\\
&+ 2\nu_2 \int_\Omega h_2(D(u_2):D(u_2))+\frac{1}{2}g(\rho_1-\rho_2)\frac{d}{d t}\int_\Omega
|h_1|^2 +\frac{1}{2} \rho_2 g \frac{d}{d t}\int_\Omega |h_1+h_2|^2 \\
& \leq 0.
\end{aligned}
\end{equation}
\end{lemma}

\begin{remark} \label{rmk2.1} \rm
 From the energy estimate \eqref{en1}, we deduce the following:
\begin{gather}\label{hruL2}
 \sqrt{h_1} u_1 \in L^\infty(0,T;(L^2(\Omega))^2); \quad
\sqrt{h_2} u_2 \in L^\infty(0,T;(L^2(\Omega))^2); \\
\label{hL2}
 h_1 \in L^\infty(0,T;L^2(\Omega)); \quad \sqrt{h_1} \,D(u_1) \in L^2(0,T;(L^2(\Omega))^4); \\
 h_2 \in L^\infty(0,T;L^2(\Omega));\quad \sqrt{h_2} \,D(u_2) \in L^2(0,T;(L^2(\Omega))^4).
\end{gather}
However, it is well-known that these estimates are not enough to pass
to the limit and get the stability of the system.
 So we are going to obtain further estimates from the BD entropy that we
state in the following lemma, (see \cite{BDcomp}).
\end{remark}

\begin{lemma}\label{entropy}
If we assume that $(h_1,h_2,u_1,u_2)$ is a smooth solution of
system \eqref{mas1}-\eqref{mas4}, then
\begin{equation}\label{en2}
\begin{aligned}
&\frac{1}{2}\rho_2 \frac{d}{dt} \int_\Omega h_1 |\rho_1 u_1+2\nu_1\nabla\log
h_1|^2+\frac{1}{2}\rho_1 \frac{d}{dt}\int_\Omega h_2 |\rho_2u_2+2\nu_2\nabla\log h_2|^2\\
&+\rho_1\rho_2\Big(\frac{1}{2} g(\rho_1-\rho_2)\frac{d}{dt} \int_\Omega |h_1|^2+\frac{1}{2}\rho_2 g
\frac{d}{dt}\int_\Omega|h_1+h_2|^2\Big)\\
&+2\nu_2\rho_1\rho_2\int_\Omega h_2(A(u_2):A(u_2))+2\nu_1\rho_1\rho_2\int_\Omega
h_1 (A(u_1):A(u_1))\\
&+2\nu_1\rho_1\rho_2 g \int_\Omega
|\nabla h_1|^2+2\nu_2\rho_1\rho_2 g \int_\Omega |\nabla h_2|^2+2\rho_2
g(\rho_2\nu_1+\rho_1\nu_2)\int_\Omega \nabla h_1 \nabla h_2 \\
&\leq 0
\end{aligned}
\end{equation}
\end{lemma}

\begin{remark} \label{rmk2.2} \rm
We would like to point out the boundedness  of the `non usual' terms
appearing above.
\begin{enumerate}
\item In the energy equality \eqref{en2}, it remains to
control the four last terms on left-hand side.
\item The proof of  the previous two lemmas takes inspiration in
\cite{ZaGla}.
 \item The classical  energy  and the
BD entropy allow us to find the estimates:
\begin{equation}
\begin{gathered}
 \nabla \sqrt{h_1} \in L^2(0,T;(L^2(\Omega))^2); \quad
\nabla \sqrt{h_2}\in L^2(0,T;(L^2(\Omega))^2);\\
 \nabla h_1 \in L^2(0,T;(L^2(\Omega))^2); \quad
\nabla h_2 \in L^2(0,T;(L^2(\Omega))^2).
\end{gathered}
\end{equation}
\end{enumerate}
\end{remark}

\section{Construction of the  ``Mellet-Vasseur" type inequality}\label{constr}

Following the idea proposed in \cite{alexisv1} this section is devoted to
 the construction of an approximation of the ``Mellet-Vasseur" type inequality
for any weak solution for the system \eqref{mas11}-\eqref{mas44}, with the initial
 conditions \eqref{Idata}, verifying in additional $h_{i_0}\geq\frac{1}{\alpha_i}$
for $\alpha_i>0$ and $\sqrt{h_{i_0}}u_{i_0}\in L^{\infty}(\Omega)$.

\begin{proposition}\label{propo1}
For any $\kappa\geq 0$ and $\bar{\kappa}\geq 0$, there exists a global weak
 solution to the system
\begin{gather}
\partial_t h_1 + \operatorname{div} (h_1 u_1)=0;\label{mas11}\\
\begin{aligned}
&\rho_1 \partial_t (h_1 u_1)+\rho_1 \operatorname{div}(h_1
u_1\otimes u_1)-2\nu_1 \operatorname{div}(h_1 D(u_1))+ \rho_1 g h_1 \nabla h_1\\
& +\rho_2 g h_1 \nabla h_2
 +r_0u_1+r_1h_1|u_1|^2u_1-\kappa h_1\nabla
\big(\frac{\Delta \sqrt{h_1}}{\sqrt{h_1}}\big) =0;
\end{aligned}\label{mas22} \\
\partial_t h_2 + \operatorname{div}(h_2 u_2)=0;\label{mas33} \\
\begin{aligned}
&\rho_2 \partial_t(h_2 u_2)+\rho_2 \operatorname{div}(h_2
u_2\otimes u_2)-2\nu_2 \operatorname{div}(h_2 D(u_2))\\
&+\rho_2 g h_2\nabla h_2+ \rho_2 g h_2 \nabla h_1
+\bar{r}_0u_2+\bar{r}_1h_2|u_2|^2u_2
 -\bar{\kappa}h_2\nabla(\frac{\Delta \sqrt{h_2}}{\sqrt{h_2}})=0
\end{aligned}\label{mas44}
\end{gather}
with the initial data \eqref{Idata}  satisfying \eqref{Rdata} and
$-r_0\int_{\Omega}\log_{-}h_{i_0} dx<\infty.$ In particular, we have
the  energy inequality
\begin{equation}\label{energy2}
\begin{aligned}
&\frac{d}{dt}E_1(t) +2\nu_1 \int_\Omega h_1(D(u_1):D(u_1))
+ 2\nu_2 \int_\Omega h_2(D(u_2):D(u_2)) +c_0\int_\Omega |u_1|^2 \\
&+r_0\int_\Omega |u_1|^2 +\bar{r}_0\int_\Omega |u_2|^2
+r_1\int_\Omega h_1|u_1|^4+\bar{r}_1\int_\Omega h_2|u_2|^4  = 0,
\end{aligned}
\end{equation}
 where
\begin{align*}
E_1(t)
&=\int_\Omega \Big[\frac{1}{2}\rho_1h_1
|u_1|^2+\frac{1}{2} \rho_2 h_2 |u_2|^2
+\frac{1}{2}g(\rho_1-\rho_2) |h_1|^2
+\frac{1}{2} \rho_2 g|h_1+h_2|^2\\
&\quad +\frac{\kappa}{2}|\nabla\sqrt{h_1}|^2
+\frac{\bar{\kappa}}{2}|\nabla\sqrt{h_2}|^2\Big]
\end{align*}
and the BD-entropy
\begin{equation}\label{Entropy2}
\begin{aligned}
&\frac{d}{dt} E_2(t)+\rho_1\rho_2c_0\int_\Omega|u_1|^2
+2\nu_2\rho_1\rho_2\int_\Omega h_2(A(u_2):A(u_2))\\
&+2\nu_1\rho_1\rho_2\int_\Omega h_1 (A(u_1):A(u_1))
 +2\rho_2 g(\rho_2\nu_1+\rho_1\nu_2)\int_\Omega \nabla h_1 \nabla h_2\\
&+\kappa\int_\Omega h_1|\nabla^2\log h_1|^2
 +\bar{\kappa}\int_\Omega h_1|\nabla^2\log h_2|^2 \\\
&+2\nu_1\rho_1\rho_2 g \int_\Omega |\nabla h_1|^2
 +2\nu_2\rho_1\rho_2 g \int_\Omega |\nabla h_2|^2 = 0
\end{aligned}
\end{equation}
 where
\begin{align*}
E_2(t)
&= \int_\Omega\Big[ \frac{1}{2}\rho_2h_1 |\rho_1 u_1
 +2\nu_1\nabla\log h_1|^2+\frac{1}{2}\rho_1 h_2 |\rho_2v_2
 +2\nu_2\nabla\log h_2|^2 \\
&\quad-\bar{r}_0\log_{-}h_2-r_0\log_{-}h_1
+\frac{\kappa}{2}|\nabla\sqrt{h_1}|^2
 +\frac{\bar{\kappa}}{2}|\nabla\sqrt{h_2}|^2 \\
&\quad +\rho_1\rho_2\Big(\frac{1}{2} g(\rho_1-\rho_2) |h_1|^2+\frac{1}{2}\rho_2 g
|h_1+h_2|^2\Big)\Big].
\end{align*}
\end{proposition}

The proof of the above Proposition takes inspiration in \cite{ZaGla}.
It takes into account the additional terms.

\begin{corollary}\label{corrolaire}
The energy inequalities \eqref{energy2}-\eqref{Entropy2} yield the following
new estimates
\begin{gather}\label{e1}
\|\sqrt{\kappa}\nabla\sqrt{h_1}\|_{L^{\infty}(0,T,L^2(\Omega))}\leq C, \quad
\|\sqrt{\bar{\kappa}}\nabla\sqrt{h_2}\|_{L^{\infty}(0,T,L^2(\Omega))} \leq C, \\
\label{e2}
\|\sqrt{r_0}u_1\|_{L^{2}(0,T,L^2(\Omega))}\leq C,\quad
 \|\sqrt{\bar{r}_0}u_2\|_{L^{2}(0,T,L^2(\Omega))}\leq C, \\
\label{e3}
\|\sqrt[4]{r_1} h_1u_1\|_{L^{4}(0,T,L^4(\Omega))}\leq C,\quad
 \|\sqrt[4]{\bar{r}_1} h_2u_2\|_{L^{4}(0,T,L^4(\Omega))}\leq C,\\
\label{e4}
\|\sqrt{\kappa}\nabla^2\log h_1\|_{L^{2}(0,T,L^2(\Omega))}\leq C,\quad
\|\sqrt{\bar{\kappa}}\nabla^2\log h_2\|_{L^{2}(0,T,L^2(\Omega))}\leq C, \\
\label{e5}
\|\nabla\sqrt{h_1}\|_{L^{\infty}(0,T,L^2(\Omega))}\leq C,\quad
\|\nabla\sqrt{h_2}\|_{L^{\infty}(0,T,L^2(\Omega))}\leq C, \\
\label{e6}
\|\sqrt{h_1}A(u_1)\|_{L^{2}(0,T,L^2(\Omega))}\leq C,\quad
\sqrt{h_1}A(u_1)\|_{L^{2}(0,T,L^2(\Omega))}\leq C.
\end{gather} where $C$ is bounded by the initial data, uniformly on
$r_0, \bar{r}_0, r_1, \bar{r}_1, \kappa,$ and $\bar{{\kappa}}$.
\end{corollary}

\begin{remark} \label{estimate} \rm
(1) The following inequalities hold:
\begin{gather*}
\sqrt{\kappa}\|\sqrt{h_1}\|_{L^2(0,T,H^2(\Omega))}
 +\kappa^{1/4}\|\nabla h_1^{1/4}\|_{L^4(0,T,L^4(\Omega))}\leq C_1,\\
\sqrt{\bar{\kappa}}\|\sqrt{h_2}\|_{L^2(0,T,H^2(\Omega))}
 +\bar{\kappa}^{1/4}\|\nabla h_2^{1/4}\|_{L^4(0,T,L^4(\Omega))}
\leq C_2,
\end{gather*}
 where $C_1$ and $C_2$  depend only on the initial data.
These inequalities are a consequence of the bound on \eqref{e4}

(2) The inequalities \eqref{e5}  and \eqref{hL2} yield
\begin{equation}
\sqrt{h_1}\in L^{\infty}(0,T;L^p(\Omega)),\quad
\sqrt{h_2}\in L^{\infty}(0,T;L^p(\Omega)) \quad\text{for  } p\geq 1.
\end{equation}

(3) The weak formulation reads as follows
\begin{equation}\label{defi2b}
\begin{aligned}
&-\rho_1 h_{1_0} u_{1_0}\varphi(0,\cdot)-\int_0^T\int_\Omega\rho_1
h_1u_1 {\partial}_t\varphi  -\rho_1 \int_0^T \int_\Omega(h_1 u_1\otimes
 u_1):D(\varphi)\\
&+r_0\int_0^T\int_\Omega u_1
 +2\nu_1\int_0^T\int_\Omega h_1
(D(u_1):D(\varphi))+r_1\int_0^T\int_\Omega h_1|u_1|^2u_1\varphi\\
& -\kappa\int_0^T\int_\Omega\Delta\sqrt{h_1}\sqrt{h_1}\operatorname{div}\varphi
-2\kappa\int_0^T\int_\Omega\Delta\sqrt{h_1}\nabla\sqrt{h_1}\varphi\\
&+\frac{1}{2}\rho_1 g \int_0^T \int_\Omega
h_1^2\operatorname{div}\varphi+ \rho_1 g \int_0^T \int_\Omega
h_1\nabla h_2\varphi=0
\end{aligned}
\end{equation}
and
\begin{equation}\label{defi1b}
\begin{aligned}
&-\rho_2 h_{2_0} u_{2_0}\varphi(0,\cdot)-\int_0^T\int_\Omega\rho_2
h_2u_2 {\partial}_t\varphi -\rho_2 \int_0^T \int_\Omega(h_2 u_2\otimes u_2):D(\varphi) \\
&+\bar{r}_0\int_0^T\int_\Omega u_2\varphi
 +2\nu_2\int_0^T\int_\Omega h_2 (D(u_2):D(\varphi))
 +\bar{r}_1\int_0^T\int_\Omega h_2|u_2|^2u_2\varphi \\
&-\bar{\kappa}\int_0^T\int_\Omega\Delta\sqrt{h_2}\sqrt{h_2}\operatorname{div}\varphi
-2\bar{\kappa}\int_0^T\int_\Omega\Delta\sqrt{h_2}\nabla\sqrt{h_2}\varphi \\
& +\frac{1}{2}\rho_2 g \int_0^T \int_\Omega h_2^2\operatorname{div}\varphi
 + \rho_2 g \int_0^T \int_\Omega h_2(\varphi\cdot \nabla h_1)\rm=0.
\end{aligned}
\end{equation}
for any function test  $\varphi$.
\end{remark}

 Next we consider $\varepsilon_1=r$, $\varepsilon_2=1$,
$ r_{1_0}=r_0$, $r_{2_0}=\bar{r}_0$, $r_{1_1}=r_1$, $r_{2_1}=\bar{r}_1$,
$\kappa_1=\kappa$ and $\kappa_2=\bar{\kappa}$.


Our first main result is the next theorem which gives the  ``Mellet-Vasseur"
type inequality (see \cite{alexisv1}).

\begin{theorem}\label{fresult}
For any $\delta_i\in (0, 2)$, there exists  $C_i$ depending only on $\delta_i$,
and the weak solutions  $(h_1,h_2, u_1, u_2)$ to \eqref{mas11}-\eqref{mas22}
with $\kappa_i=0$  verify all the properties of Proposition \ref{propo1},
and satisfy the following ``Mellet-Vasseur" type inequality for every $T>0$,
and almost every $T>t$:
\begin{align*}
&\int_{\Omega}h_i(1+|u_i|^2)\ln(1+|u_i|^2) \\
& \leq\int_{\Omega} h_{i_0} (1+|u_{i_0}|^2)\ln(1+|u_{i_0}|^2)
 +C_i \int_{\Omega}\big(\frac{h_{i_0}|u_{i_0}|^2}{2}+\frac{1}{2}|h_i|^2
 +|\nabla \sqrt{h_{i_0}}|^2\big) \\
&+C_i\int_0^T\Big(\int_{\Omega}\big(h_i^{3-\frac{\delta_i}{2}}
 \big)^{\frac{2}{2-\delta_i}}\Big)^{\frac{2-\delta_i}{2}}
 \Big(\int_{\Omega}h_i\big(2+\ln(1+|u_i|^2)\big)^{\frac{\delta_i}{2}}\Big),
\end{align*}
for  $i=1, 2$.
\end{theorem}

\begin{remark} \label{rmk3.3} \rm
(1) The right hand side constant $C_i$ of the above inequality  does not
depend on $r_{i_0}$ and $r_{i_1}$. This theorem will  be crucial  to prove
 the strong convergence of $\sqrt{h_i}u_i$ in the space $L^2(0,T;L^2(\Omega))$
when  $r_{i_0}$ and $r_{i_1}$ converge to 0.

(2) For a global weak solution of \eqref{mas11}-\eqref{mas44}, under hypothesis
of Definition \ref{defit1}, we  need that
 \eqref{mas11}-\eqref{mas44} hold in $D'([0,T]\times\Omega)$ and the following be
 satisfied
\begin{gather*}
h_i\geq 0,\quad h_i\in L^{\infty}(0,T;L^2(\Omega)),\\
h_i(1+|u_i|^2)\ln(1+|u_i|^2)\in L^{\infty}(0,T;L^1(\Omega)),\\
\nabla h_i\in L^{2}(0,T;L^2(\Omega)),\quad
\nabla \sqrt{h_i}\in L^{\infty}(0,T;L^2(\Omega)),\\
\sqrt{h_i}u_i\in L^{\infty}(0,T;L^2(\Omega)),\sqrt{h_i}\nabla u_i
\in L^2(0,T;L^2(\Omega)),
\end{gather*}
\end{remark}

As a sequence of Theorem \ref{fresult}, we have the  same result as
in \cite{alexisv1}:

\begin{theorem}\label{theo2}
Let $(h_{i_0}, m_{i_0})$ satisfy \eqref{Idata} and
$$
\int_{\Omega}h_{i_0}(1+|u_{i_0}|^2)\ln(1+|u_{i_0}|^2)dx< \infty.
$$
Then for $T>0$, there exists a weak solution of \eqref{mas11}-\eqref{mas44}
 on $(0,T)$.
\end{theorem}

 The ``Mellet-Vasseur" type inequality does not  work  for the solutions of
\eqref{mas11}-\eqref{mas44} for  $\kappa_i\geq 0$.
The idea is  to construct as in \cite{alexisv1} an approximation of the
``Mellet-Vasseur" type inequality.
We define four $\mathcal{C}^{\infty}$ non-negative cut-off functions
$\phi_{\alpha_i}$ and $\phi_ {\beta_i}$ as follows.
\begin{equation}\label{cisse}
\phi_{\alpha_i}(h_i)=1\text{ for any } h_i> \frac{1}{\alpha_i}, \quad
\phi_{\alpha_i}(h_i)=0 \text{ for any } h_i< \frac{1}{2\alpha_i},
\end{equation}
where $\alpha_i> 0$ is any real number, and otherwise,
$|\phi'_{\alpha_i}|\leq2\alpha_i$; and
$\phi_{\beta_i}(h_i)\in \mathcal{C}^{\infty}(\mathbb{R})$
is a non-negative function such that
\begin{equation}
\phi_{\beta_i}(h_i)=1\text{ for any } h_i<\beta_i, \quad
\phi_{\beta_i}(h_i)=0 \text{ for any } h_i> 2\beta_i,
\end{equation}
where $\beta_i> 0$ is any real number, and $|\phi'_{\beta_i}|\leq \frac{2}{\beta_i}$.

 We define $ v_i=\phi_i(h_i)u_i$, and
$\phi_i(h_i)=\phi_{\alpha_i}(h_i)\phi_{\beta_i}(h_i)$.
The   lemmas will be very useful to construct the approximation of the
``Mellet-Vasseur" type inequality.

\begin{lemma}\label{41}
For any fixed $\kappa_i> 0$, we have
$$
\|\nabla v_i\rm\|_{L^2(0,T;L^2(\Omega))}\leq C_i,
$$
where the constant $C_i$  depends on  $\kappa_i > 0$, $r_{i_1}$, $\beta_i$
and $\alpha_i$; and
$$
\partial_t h_i\in L^4(0,T;L^{\frac{6}{5}}(\Omega))+L^2(0,T;L^{\frac{3}{2}}(\Omega)).
$$
\end{lemma}

For a proof of the above lemma, see \cite{alexisv1}.
Following  the ideas in \cite{alexisv1}, we introduce a new
$\mathcal{C}^{ \infty}(\mathbb{R}^2)$, non-negative cut-off function
$\varphi_{in}(h_i)$ which is given by
\begin{equation}
\varphi_{in}(x)=\begin{cases}
(1+|x|^2)\ln(1+|x|^2)&\text{if } 0\leq |x|<n,\\
(1+8n^2)\ln(1+4n^2)  &\text{if }|x|\geq 2n
\end{cases}
\end{equation}
where $n > 0$ are large, and
$$
|\varphi'_{in}(x)|+|\varphi"_{in}(x)|\leq \frac{C_i}{n}\quad \text{for any }
|x|\geq n.
$$
The first step of constructing the approximation of the Mellet-Vasseur
type inequality is the following lemma.

\begin{lemma}\label{42}
 For the weak solutions to \eqref{mas11}-\eqref{mas44} constructed in
Proposition \ref{propo1}, and any $\psi_i(t)\in \mathcal{D}(-1,+\infty)$, we have
\begin{equation}\label{eq42}
\begin{aligned}
&\int_0^T\int_{\Omega} \partial_t\psi_i(t)h_i\varphi_{in}( v_i)\,dx\,dt
-\int_0^T\int_{\Omega} \psi_i(t)\varphi'_{in}( v_i) F_i\,dx\,dt \\
&+\int_0^T\int_{\Omega} \psi_i(t) S_i:\nabla(\varphi'_{in}( v_i))
\,dx\,dt \\
&=\int_{\Omega}h_{i_0}\varphi_{in}( v_{i0})\psi_i(0)\,dx\,dt
\end{aligned}
\end{equation}
where
\begin{equation}\label{SF}
\begin{gathered}
 S_i=h_i\phi_i(h_i)(D(u_i)+\kappa_i\frac{\Delta \sqrt{h_i}}{ \sqrt{h_i}})\mathbb{I},
\\
\begin{aligned}
 F_i
&=h_i^2u_i\phi'_i(h_i)\operatorname{div}\rm u_i+gh_i\nabla h_i\phi_i(h_i)
+h_i\nabla \phi_i(h_i)D(u_i)+ g\varepsilon_i h_i\nabla h_j\phi_i(h_i)  \\
&\quad +r_{i_0}u_i\phi_i(h_i) +r_{i_1}h_i|u_i|^2u_i\phi_i(h_i)
 +\kappa_i\sqrt{h_i}\nabla\phi_i(h_i)\Delta\sqrt{h_i} \\
&\quad +2\kappa_i\phi_i(h_i)\nabla\sqrt{h_i}\Delta\sqrt{h_i},
\end{aligned}
\end{gathered}
\end{equation}
where $\mathbb{I}$ is an identity matrix.
\end{lemma}

\begin{proof}
 To obtain the result it suffices to multiply   equations
 \eqref{mas22} and \eqref{mas44} by $\phi_1(h_1)$ and $\phi_2(h_2)$  respectively;
 we have
\begin{align*}
&\partial_t(h_i v_i)-h_iu_i\phi'_i(h_i)\partial_th_i
 +\operatorname{div}(h_iu_i\otimes v_i)-h_iu_i\otimes u_i\nabla\phi_i(h_i)
 +h_i\nabla h_i\phi_i(h_i)\\
&-\operatorname{div}(\phi_i(h_i)h_i\mathbb{D}u_i)
+h_i\nabla\phi_i(h_i)\mathbb{D}u_i+r_{i_0}h_iu_i\phi_i(h_i)
+r_{i_1}h_i|u_i|^2u_i\phi_i(h_i) \\
&+ g\varepsilon_ih_i\nabla h_j\phi_i(h_i)
 -\kappa_i\nabla(\sqrt{h_i}\phi_i(h_i)\Delta\sqrt{h_i}) \\
&+\kappa_i\sqrt{h_i}\nabla\phi_i(h_i)\Delta\sqrt{h_i}
 +2\kappa_i\phi_i(h_i)\nabla\sqrt{h_i}\Delta\sqrt{h_i}=0.
\end{align*}
Here we did a successive integration.
\end{proof}

\begin{remark} \label{rmk3.4} \rm
Both $\nabla\sqrt{h_i}$ and $\partial_t h_i$ are functions, so the above
equalities are justified by regularizing $h_i$ and passing into the limit.
We can rewrite the above equation as follows
\begin{equation}\label{AS}
\partial_t(h_iv_i)\rm+\operatorname{div}(h_iu_i\otimes v_i)
-\operatorname{div}S_i+F_i=0
\end{equation}
where $ S_i$ and $ F_i$ are as in \eqref{SF}, and we used
\begin{align*}
h_iu_i\phi'_i(h_i)\partial_t(h_i)+h_iu_i\otimes u_i\phi'_i(h_i)\nabla h_i
&=h_iu_i\phi'_i(h_i)(\partial_th_i+\nabla h_i\cdot u_i) \\
&=-h_i^2u_i\phi'_i(h_i)\operatorname{div}u_i.
\end{align*}
We should remark that, thanks to Corollary \ref{corrolaire} and Remark
 \ref{estimate},
$$
\| F_i\|_{L^{\frac{4}{3}}(0,T;L^1(\Omega))}\leq C_i,\quad
\| S_i\|_{L^2(0,T;L^2(\Omega))}\leq C_i,
$$
since $\sqrt{h_i}\phi_i(h_i)$  and $h_i\phi_i(h_i)$ are bounded.
Those bounds depend on $\beta_i$ and $\kappa_i$.
\end{remark}

We first introduced a test function $\psi_i(t) \in \mathcal{D}(0,+\infty)$.
Essentially this function vanishes for $t$ close $t = 0$. We will later
extend the result for $\psi_i(t)\in D(-1,+\infty)$. We define a new
function $\Phi_i = \overline{\psi_i(t)\phi'_{in}( v_i)}$, where
$\overline{f_i(t, x) }= f_i\ast\eta_{ik}(t, x)$, $k$ is a small enough number. Note
that, since $\phi_i(t)$ is compactly supported in $(0,\infty)$.
$\Phi_i$ is well defined on $(0,\infty)$ for $k$ small
enough. We use it to test \eqref{AS} to have
$$
\int_0^T \int_{\Omega}\overline{\psi_i(t)\varphi'_{in}(\bar{v\rm}_i)}
[\partial_t ( h_i  v_i)+\operatorname{div}(h_iu_i\otimes v_i)
-\operatorname{div}S_i+ F_i]\,dx\,dt=0
$$
which in turn gives us
\begin{equation}\label{AZ}
\int_0^T \int_{\Omega}\psi_i(t)\varphi'_{in}
(\bar{ v}_i)\overline{[\partial_t ( h_i  v_i)
+\operatorname{div}(h_iu_i\otimes v_i)-\operatorname{div}S_i+ F_i]} \,dx\,dt=0
\end{equation}
The first term in \eqref{AZ} can be calculated as follows:
\begin{align*}
&\int_{\Omega}\psi_i(t)\varphi'_{in}(\bar{ v}_i)
 \overline{\partial_t(h_i v_i)} \,dx\,dt \\
&=\int_{\Omega}\psi_i(t)\varphi'_{in}(\bar{ v}_i)\partial_t(h_i\bar{ v}_i)\,dx\,dt
+\int_{\Omega}\psi_i(t)\varphi'_{in}(\bar{ v}_i)
[\overline{\partial_t(h_i v_i)}-\partial_t(h_i\bar{ v}_i)] \,dx\,dt \\
&=\int_{\Omega}\psi_i(t)\varphi'_{in}(\bar{ v}_i)(\partial_t(h_i)
 \bar{ v}_i+h_i\partial_t (\bar{ v}_i))\,dx\,dt+R_1 \\
&=\int_{\Omega}\psi_i(t)\partial_t h_i\varphi'_{in}(\bar{ v}_i)\bar{ v}_i \,dx\,dt
+\int_0^T\int_{\Omega}\psi_i(t)h_i\varphi_{in}(\partial_t\bar{ v}_i) \,dx\,dt
 + R_1
\end{align*}
where
$$
R_1=\int_0^T\int_{\Omega} \psi_i(t)\varphi'_{in}(\bar{ v}_i)
[\overline{\partial_t(h_i\bar{ v}_i}-\partial_t(h_i\bar{ v}_i)]\,dx\,dt.
$$
Thanks to  equation  \eqref{mas11}, we can rewrite the second term
in \eqref{AZ} as follows
\begin{equation}\label{ZA}
\begin{aligned}
&\int_0^T\int_{\Omega} \psi_i(t)\varphi'_{in}(\bar{ v}_i)
 \overline{\operatorname{div}(h_iu_i\otimes v_i)}\,dx\,dt \\
&=\int_0^T\int_{\Omega} \psi_i(t)\partial_th_i\varphi_{in}(\bar{ v}_i)\,dx\,dt
 -\int_0^T\int_{\Omega} \psi_i(t)\partial_th_i\varphi'_{in}(\bar{ v}_i)
 \bar{ v}_i\,dx\,dt+R_2,
\end{aligned}
\end{equation}
and
$$
R_2=\int_0^T\int_{\Omega} \psi_i(t)\varphi'_{in}(\bar{ v}_i)
[\operatorname{div}(h_iu_i\otimes\bar{ v}_i)
-\overline{\operatorname{div}(h_iu_i\otimes v_i}]\,dx\,dt.
$$
By \eqref{AZ}-\eqref{ZA}, we have
\begin{align*}
&\int_0^T\int_{\Omega} \psi_i(t)\partial_t(h_i\varphi_{in}(\bar{ v}_i))\,dx\,dt
+R_1+R_2
-\int_0^T\int_{\Omega}\psi_i(t)\varphi'_i(\bar{ v}_i)
\overline{\operatorname{div}S_i}\,dx\,dt\\
&+\int_0^T\int_{\Omega} \psi_i(t)\varphi'_{in}(\bar{ v}_i)\bar{ F}_i\,dx\,dt=0.
\end{align*}
Notice that $\bar{ v}_i$ converges to $ v$ almost everywhere and
$$
h_i\varphi_{in}(\bar{ v}_i)\partial_t\psi_i\to
 h_i\varphi_{in}( v_i)\partial_t\psi_i\quad \text{in }L^1((0,T)\times\Omega).
$$
So, up to a subsequence, we have
\begin{equation}\label{e48}
\int_0^T\int_{\Omega}h_i\varphi_{in}(\bar{ v}_i)\partial_t\psi_i \,dx\,dt
\to  \int_0^T\int_{\Omega} h_i\varphi_{in}( v_i)\partial_t\psi_i \,dx\,dt\quad
 \text{as } k\to  0.
\end{equation}
Since $\varphi'_{in}(\bar{ v}_i)$ converges to $\varphi'_{in}( v_i)$
almost everywhere, and is uniformly bounded in $L^{\infty}((0,T)\times\Omega)$,
we have
\begin{equation}
\int_0^T\int_{\Omega} \psi_i(t)\varphi'_{in}(\bar{ v}_i)\bar{F}_i \,dx\,dt
\to  \int_0^T\int_{\Omega} \psi_i(t)\varphi'_{in}( v_i)F_i\quad \text{as }
k\to  0.
\end{equation}
Noticing that $ \nabla  v_i\in L^2(0,T;L^2(\Omega))$,
we have
$$
\overline{\nabla  v_i}\to  \nabla  v_i\quad \text{strongly in } L^2(0,T;L^2(\Omega)).
$$
Since $\bar{S_i}$ converges to $ S_i$ strongly in $L^2(0,T;L^2(\Omega))$,
 and $\varphi_i"(\bar{ v}_i)$ converges to $\varphi_i"( v_i)$  almost
everywhere and uniformly bounded in $L^{\infty}((0,T)\times\Omega))$, we obtain
\begin{equation}
\int_0^T\int_{\Omega}\psi_i(t)\varphi'_{in}(\bar{ v}_i)
\overline{\operatorname{div}S_i}\,dx\,dt
=-\int_0^T\int_{\Omega}\psi_i(t)\bar{ S_i}:\nabla(\varphi'_{in}(\bar{ v}_i))\,dx\,dt,
\end{equation}
which converges to
\begin{equation}
-\int_0^T\int_{\Omega}\psi_i(t) S_i:\nabla(\varphi'_{in}( v_i))\,dx\,dt
\end{equation}
To handle $R_1$ and $R_2$, we use the following lemma due to Lions \cite{Lions}.

\begin{lemma}\label{43}
Let $f\in W^{1, p}(\mathbb{R}^N)$, $g\in L^q(\mathbb{R}^N)$ with
$1\leq p, q\leq \infty$, and $\frac{1}{P}+\frac{1}{q}\leq 1$. Then we have
$$
\|\operatorname{div}(fg)\ast w_{\varepsilon}
-\operatorname{div}(f(g\ast w_{\varepsilon}))\|_{L^r(\mathbb{R}^N)}
\leq C\|f\|_{W^{1, p}(\mathbb{R}^N)}\|g\|_{L^q(\mathbb{R}^N)}
$$
 for some $C\geq 0$ independent of $\varepsilon$, $f$ and $g$, $r$ is
determined by $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. In addition,
 $$
 \operatorname{div}(fg)\ast w_{\varepsilon}-\operatorname{div}
(f(g\ast w_{\varepsilon}))\to  0 \quad\text{in } L^r(\mathbb{R}^N)
 $$
as $\varepsilon\to  0$ if $r< \infty$.
\end{lemma}

This lemma includes the following statement.

\begin{lemma}\label{44}
 Let $\partial_t f\in L^P(0,T)$, $g\in L^q(0,T)$ with $1\leq p, q\leq \infty$, and
$\frac{1}{P}+\frac{1}{q}\leq 1$. Then we have
 $$
\|\partial_t(fg)\ast w_{\varepsilon}-\partial_t(f(g\ast w_{\varepsilon}))\|_{L^r(0,T)}
\leq C\|f\|_{L^p(0,T)}\|g\|_{L^q(0,T)}
$$
for some $C\geq 0$ independent of $\varepsilon$, $f$ and $g$, $r$ is determined
by $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. In addition,
 $$
 \partial_t(fg)\ast w_{\varepsilon}-\partial_t(f(g\ast w_{\varepsilon}))\to  0
\quad\text{in } L^r(0,T)
 $$
as $\varepsilon\to  0$ if $r< \infty$.
\end{lemma}

With Lemmas \ref{43} and \ref{44} in hand, we are ready to handle the terms
$R_1$ and $R_2$.
For $\kappa_i > 0$, by Lemma \ref{41} and Poincare inequality, we have
$ v_i\in L^2(0,T;L^6(\Omega))$. We also have, by Lemma \ref{41},
$$
\partial_t h_i\in L^4(0,T;L^{\frac{6}{5}}(\Omega))+L^2(0,T;L^{\frac{3}{2}}(\Omega)).
$$
Thus, applying Lemma \ref{44},
\begin{equation}\label{e53}
\begin{aligned}
|R_1|&\leq \int_0^T\int_{\Omega}\Big|\psi_i(t)\varphi'(\bar{ v}_i)
[\overline{\partial_t(h_i v_i})-\partial_t(h_i\bar{ v}_i)]\Big|\,dx\,dt \\
&\leq C(\psi_i)\int_0^T\int_{\Omega}\Big|\varphi'(\bar{ v}_i)
 [\overline{\partial_t(h_i v_i})-\partial_t(h_i\bar{ v}_i)]\Big|\,dx\,dt\to  0
 \quad\text{as }k\to  0.
\end{aligned}
\end{equation}
By a similar reasoning and using Lemma \ref{43}, We deduce that
$R_2 \to  0$ as $k\to  0$.
By \eqref{e48}-\eqref{e53}, we have
\begin{equation}\label{varf}
\begin{aligned}
&\int_0^T\int_{\Omega}\partial_t\psi_i(t)h_i\varphi_{in}( v_i)\,dx\,dt
-\int_0^T\int_{\Omega}\psi_i(t)\varphi'_{in}( v_i) F\,dx\,dt \\
&+\int_0^T\int_{\Omega}\psi_i(t) S_i:\nabla(\varphi'_{in}( v_i))\,dx\,dt=0,
\end{aligned}
\end{equation}
for any test function $\psi_i\in \mathcal{D}(0,\infty)$.

We need to consider the test function $\psi_i\in \mathcal{D}(-1,\infty)$.
For this, we need the continuity of $h_i(t)$ and $(\sqrt{h_i}u_i)(t)$
in the strong topology at $t = 0$.
In fact, thanks to Proposition \ref{propo1}, we have
$$
\partial_t \sqrt{h_i}\in  L^2(0,T;L^2(\Omega)),\quad
 \sqrt{h_i}\in L^2(0,T;H^2(\Omega)).
$$
This gives us
$$
\sqrt{h_i}\in C([0,T];L^2(\Omega))\quad \text{and}\quad
 \nabla\sqrt{h_i}\in C([0,T];L^2(\Omega))
$$
thanks to \cite[Theorem 3 p. 287]{Evans}. Similarly, we have
 \begin{equation}\label{contin1}
 h_i\in C([0,T];L^2(\Omega))
 \end{equation}
 due to
$$
 \|\nabla h_i\|_{L^2(0,T;L^2(\Omega))}
\leq \|\nabla\sqrt{h_i}\|_{L^4(0,T;L^4(\Omega))}\|\sqrt{h_i}\|_{L^4(0,T;L^4(\Omega))}.
 $$
 We have
 $\sqrt{h_i}\in  L^{\infty}(0,T;L^p(\Omega))$  for any $p\geq 1$,
 and hence
 \begin{equation}\label{hilp}
 \sqrt{h_i}\in  C([0,T];L^p(\Omega)) \text{ for any } p\geq 1.
 \end{equation}
An analogous  reasoning as in \cite{alexisv1} gives us
\begin{equation}\label{contin2}
\sqrt{h_i}u_i \in C([0,T];L^2(\Omega)).
\end{equation}
Indeed: we have
\begin{align*}
&\operatorname{ess}\limsup_{t\to  0} \int_{\Omega}\rho_1|\sqrt{h_1}u_1
 -\sqrt{h_{1_0}}u_{1_0}|^2 dx
+\operatorname{ess}\limsup_{t\to  0} \int_{\Omega}\rho_2|\sqrt{h_2}u_2
 -\sqrt{h_{2_0}}u_{2_0}|^2 dx \\
&\leq  \varepsilon (t,x)-\varepsilon( 0,x)
 +2\operatorname{ess}\limsup_{t\to  0}
  \int_{\Omega}\rho_1\sqrt{h_{1_0}}u_{1_0}(\sqrt{h_{1_0}}u_{1_0}
 -\sqrt{h_{1}}u_{1})dx \\
&\quad +\operatorname{ess}\limsup_{t\to  0} \int_{\Omega}\rho_2
\sqrt{h_{2_0}}u_{2_0}(\sqrt{h_{2_0}}u_{2_0}-\sqrt{h_{2}}u_{2})dx\\
&\quad +\operatorname{ess}\limsup_{t\to  0}
\int_{\Omega}g(\rho_1-\rho_2)|h_1-h_{1_0}|^2dx \\
&\quad +2 \operatorname{ess}\limsup_{t\to  0}  \int_{\Omega}g(\rho_1-\rho_2)
h_1(h_{1_0}-h_1)dx \\
&\quad +\operatorname{ess}\limsup_{t\to  0}
\int_{\Omega}g\rho_2 |h_1+h_2-h_{1_0}-h_{2_0}|^2dx \\
&\quad + 2 \operatorname{ess}\limsup_{t\to  0}
\int_{\Omega}g\rho_2(h_1+h_2)(h_1+h_2-h_{1_0}-h_{2_0})dx \\
&\quad - \frac{3\kappa_1}{2} \operatorname{ess}\limsup_{t\to  0}
\int_{\Omega}|\nabla\sqrt{h_1}-\nabla\sqrt{h_{1_0}}|^2dx \\
&\quad -  \frac{3\kappa_2}{2} \operatorname{ess\,lim}_{t\to  0}
\sup\int_{\Omega}|\nabla\sqrt{h_2}-\nabla\sqrt{h_{2_0}}|^2dx \\
&\quad +3\kappa_1 \operatorname{ess}\limsup_{t\to  0}
\int_{\Omega}\nabla\sqrt{h_{1_0}}(\nabla\sqrt{h_{1}}-\nabla\sqrt{h_{1_0}})dx \\
&\quad +3\kappa_2 \operatorname{ess}\limsup_{t\to  0} \int_{\Omega}\nabla\sqrt{h_{2_0}}
(\nabla\sqrt{h_{2}}-\nabla\sqrt{h_{2_0}})dx,
\end{align*}
where
\begin{align*}
 \varepsilon(t,x)&=\frac{1}{2}\int_{\Omega}\rho_1h_1|u_1|^2
 +\rho_2h_2|u_2|^2+g(\rho_1-\rho_2)|h_1|^2
 +\rho_2|h_1+h_2|^2 \\
&\quad +2\kappa_1|\nabla\sqrt{h_1}|^2  +2\kappa_2|\nabla\sqrt{h_2}|^2.
\end{align*}
We have
\begin{equation}
3\kappa_i \operatorname{ess}\limsup_{t\to  0}
 \int_{\Omega}\nabla\sqrt{h_{i_0}}(\nabla\sqrt{h_{i}}
-\nabla\sqrt{h_{i0}})dx=0 \quad \text{for } i=1,2.
\end{equation}
So, using \eqref{energy2}, \eqref{hilp}  and the convexity of
$h_i\longmapsto h_i^2$, we have
\begin{align*}
&\operatorname{ess}\limsup_{t\to 0}
 \int_{\Omega}\rho_1|\sqrt{h_1}u_1
-\sqrt{h_{1_0}}u_{1_0}|^2 dx
+\operatorname{ess}\limsup_{t\to 0} \int_{\Omega}\rho_2|\sqrt{h_2}u_2
-\sqrt{h_{2_0}}u_{2_0}|^2 dx \\
&\leq 2\operatorname{ess}\limsup_{t\to 0} \int_{\Omega}\rho_1
 \sqrt{h_{1_0}}u_{1_0}(\sqrt{h_{1_0}}u_{1_0}-\sqrt{h_{1}}u_{1})dx\\
&\quad +2\operatorname{ess}\limsup_{t\to 0} \int_{\Omega}\rho_2
 \sqrt{h_{2_0}}u_{2_0}(\sqrt{h_{2_0}}u_{2_0}-\sqrt{h_{2}}u_{2})dx
\end{align*}
Following the line of \cite{alexisv1} the right terms tend to $0$
and we deduce that
$$
\operatorname{ess}\limsup_{t\to 0} \int_{\Omega}|\sqrt{h_{i_0}}u_{i_0}
-\sqrt{h_i}u_i|^2=0  \quad \text{for } i=1,2,
$$
which gives us
$\sqrt{h_i}u_i\in C([0,T];L^2(\Omega))$.
By  \eqref{contin1} and \eqref{contin2}, we obtain
$$
\lim_{\tau \to  0} \frac{1}{\tau}\int_0^T\int_{\Omega}h_i\varphi_{in}( v_i)\,dx\,dt
=\int_{\Omega}h_{i_0}\varphi_{in}( v_{i_0})dx
$$
Considering \eqref{varf} for the test function,
$$
\psi_{\tau i}(t)=\psi_i(t) \text{ for } t\geq \tau,\quad
\psi_{\tau i}(t)=\psi_i(\tau)\frac{t}{\tau} \text{ for } t\leq \tau,
$$
we obtain
\begin{align*}
&\int_{\tau}^T\int_{\Omega}\partial_t\psi_{i}h_i\varphi_{in}( v_i)\,dx\,dt
 -\int_{0}^T\int_{\Omega}\psi_{\tau i}(t)\varphi'_{in}( v_i) F_i\,dx\,dt \\
&+\int_{0}^T\int_{\Omega}\psi_{\tau i}(t) S_i:\nabla (\varphi'_{in}( v_i))\,dx\,dt \\\
&=\frac{\psi_i(\tau)}{\tau}\int_0^\tau\int_{\Omega}h_i\varphi_{in}( v_i)\,dx\,dt.
\end{align*}
Passing into the limit as $\tau\to  0$, this gives us
\begin{equation}\label{passlim}
\begin{aligned}
&\int_{\tau}^T\int_{\Omega}\partial_t\psi_{i}h_i\varphi_{in}( v_i)\,dx\,dt
 -\int_{0}^T\int_{\Omega}\psi_{i}(t)\varphi'_{in}( v_i) F_i\,dx\,dt \\
&+\int_{0}^T\int_{\Omega}\psi_{i}(t) S_i:\nabla (\varphi'_{in}( v_i))\,dx\,dt\\
&=\int_0^\tau\int_{\Omega}h_{i_0}\psi_i(0)\varphi_{in}( v_{i_0})\,dx\,dt
\end{aligned}
\end{equation}

\section{Recover the limits as $\alpha_i\to  \infty$}\label{recover1}

In this section, we want to recover the limits in \eqref{eq42} as
$\alpha_i \to  \infty$. Here, we should remark
that $(h_1, h_2, u_1, u_2)$ is any fixed weak solution to
\eqref{mas11}-\eqref{mas44} satisfying Proposition \ref{propo1} with $\kappa_i > 0$.
 For any fixed weak solution $(h_1, h_2, u_1, u_2)$, we have
$$
\phi_{\alpha_i}(h_i)\to  1 \quad \text{almost everywhere for } (t,x),
$$
and it is uniformly bounded in $L^{\infty}(\Omega)$; we also have
$$
r_{0_i}\phi_{\beta_i}(h_i)u_i\in L^2(0,T;L^2(\Omega)),
$$
and thus
$$
 v_{\alpha_i}=\phi_{\alpha_i}\phi_{\beta_i}u_i\to  \phi_{\beta_i}u_i \quad
\text{almost everywhere for } (t,x)
$$
as $\alpha_i\to  \infty$. By the Dominated Convergence Theorem, we have
$$
 v_{\alpha_i}\to  \phi_{\beta_i}u_i \text{ in } L^2(0,T;L^2(\Omega)),
$$
as $\alpha_i\to  \infty$, and hence, we have
$$
\varphi_{in}( v_{\alpha_i})\to  \varphi_{in}(\phi_{\beta_i}u_i)\quad
 \text{in } L^p((0,T)\times \Omega)
$$
for any $1\leq p\leq \infty$.
For any fixed $h_i$, we have
$$
\phi'_{\alpha_i}(h_i)\to  0 \quad\text{almost everywhere for } (t,x)
$$
as $\alpha_i\to  \infty$.  for any fixed $h_i$.
Since   $|\phi'_{\alpha_i}(h_i)|\leq 2\alpha_i$ as
$\frac{1}{2\alpha_i}\leq h_i\leq \frac{1}{\alpha_i}$ and otherwise,
$\phi'_{\alpha_i}(h_i)=0$, we have
$$
|h_i\phi'_{\alpha_i}(h_i)|\leq 1 \quad \text{for all } h_i.
$$
We  find that
$$
\int_0^T\int_{\Omega}\psi'_i(t)(h_i\varphi_{in}( v_{\alpha_i}))\,dx\,dt\to
 \int_0^T\int_{\Omega}\psi'_i(t)(h_i\varphi_{in}(\phi_{\beta_i}(h_i)u_i))\,dx\,dt
$$
and
$$
\int_{\Omega}h_{i_0}\varphi_{in}( v_{\alpha_i0})\to
 \int_{\Omega}h_{i_0}\varphi_{in}(\phi_{\beta_i}(h_{i_0})u_{i_0})
$$
as $\alpha_i\to  \infty$.

To pass into the limits in \eqref{passlim} as $\alpha_i\to \infty$,
 we rely on the following lemma.

\begin{lemma}
If $\|a_{\alpha_i}\|_{L^{\infty}(0,T;\Omega)}\leq C$,
$a_{\alpha_i}\to  a$ as $\alpha_i\to  \infty$  a.e. for
$(t,x)$ in $L^p((0,T)\times\Omega)$  for any $1\leq p\leq \infty$,
$f\in L^1((0,T)\times\Omega)$, then we have
$$
\int_0^T\int_{\Omega}\phi_{\alpha_i}(h_i)a_{\alpha_i}f\,dx\,dt
\to  \int_0^T\int_{\Omega} af\,dx\,dt \quad \text{as } \alpha_i\to  \infty,
$$
and
$$
\int_0^T\int_{\Omega}|h_i\phi'_{\alpha_i}(h_i)a_{\alpha_i}f|\,dx\,dt\to  0  \quad
\text{as } \alpha_i\to  \infty.
$$
\end{lemma}

For a proof of the above lemma see \cite{alexisv1}.
Now we prove that
\begin{equation}\label{exp1}
\int_0^T\int_{\Omega}\psi_i(t) S_{\alpha_i}:\nabla(\varphi_{in}
( v_{\alpha_i}))\,dx\,dt
\to  \int_0^T\int_{\Omega}\psi_i(t) S:\nabla(\varphi'_{in}
(\phi_{\beta_i}(h_i)u_i))\,dx\,dt
\end{equation}
as $\alpha_i \to  \infty $, where
 $ S_i=\psi_{\beta_i}(h_i)h_i(D(u_i)
+\kappa_i\frac{\Delta\sqrt{h_i}}{\sqrt{h_i}}\mathbb{I})$ and
 \begin{equation}\label{exp2}
 \int_0^T\int_{\Omega}\psi_i(t)\varphi'_{in}( v_{\alpha_i}) F_{\alpha_i}\,dx\,dt
\to  \int_0^T\int_{\Omega}\psi_i(t)\varphi'_{in}(\phi_{\beta_i}(h_i)u) F_{i}\,dx\,dt
 \end{equation}
where
\begin{align*}
 F_i&=h_i^2u_i\phi'_{\beta_i}(h_i)\operatorname{div}u_i+gh_i\nabla h_i\phi_{\beta_i}(h_i)
 +h_i\nabla \phi_{\beta_i}(h_i)D(u_i)\\
&\quad + g\varepsilon_i h_i\nabla h_j\phi_{\beta_i}(h_i)
 +r_{i0}u_i\phi_{\beta_i}(h_i)
 +r_{i1}h_i|u_i|^2u_i\phi_{\beta_i}(h_i) \\
&\quad +\kappa_i\sqrt{h_i}\nabla\phi_{\beta_i}(h_i)\Delta\sqrt{h_i}
 +2\kappa_i\phi_{\beta_i}(h_i)\nabla\sqrt{h_i}\Delta\sqrt{h_i},
\end{align*}
 For the proof of \eqref{exp1} the  reasoning is similarly as in \cite{alexisv1}.
Concerning  \eqref{exp2} we just notice that
$ h_i\nabla h_j\rm \in L^1((0,T)\times\Omega)$.
 Letting $\alpha_i\to  0$ in  \eqref{passlim}, we have
\begin{align*}
&\int_{0}^T\int_{\Omega}\psi'_{i}(h_i\varphi_{in}(\phi_{\beta_i}(h_i)u_i))\,dx\,dt
-\int_{0}^T\int_{\Omega}\psi_{i}(t)\varphi'_{in}(\phi_{\beta_i}(h_i)u_i) F_i\,dx\,dt\\
&+\int_{0}^T\int_{\Omega}\psi_{i}(t) S_i:\nabla (\varphi'_{in}
 (\phi_{\beta_i}(h_i)u_i))\,dx\,dt \\
&=\int_{\Omega}\psi_i(0)
 h_{i_0}\varphi_{in}(\phi_{\beta_i}(h_{i_0})u_{i_0})\,dx\,dt
\end{align*}
which in turn gives us the following lemma.

\begin{lemma} \label{lem4.2}
For any weak solutions to \eqref{mas11}-\eqref{mas44} satisfying
Proposition \ref{propo1}, we have
\begin{equation}\label{passlim1}
\begin{aligned}
&\int_{0}^T\int_{\Omega}\psi'_{i}(t)(h_i\varphi_{in}(\phi_{\beta_i}(h_i)u_i))\,dx\,dt
-\int_{0}^T\int_{\Omega}\psi_{i}(t)\varphi'_{in}(\phi_{\beta_i}(h_i)u_i) F_i\,dx\,dt \\
&+\int_{0}^T\int_{\Omega}\psi_{i}(t) S_i:\nabla (\varphi'_{in}
 (\phi_{\beta_i}(h_i)u_i))\,dx\,dt  \\
&=\int_{\Omega}\psi_i(0)h_{i_0}\varphi'_{in}(\phi_{\beta_i}(h_{i_0})u_{i_0})\,dx\,dt
\end{aligned}
\end{equation}
 where: $ S_i=\psi_{\beta_i}(h_i)h_i(D(u_i)
+\kappa_i\frac{\Delta\sqrt{h_i}}{\sqrt{h_i}}\mathbb{I})$
 and
\begin{align*}
 F_i&=h_i^2u_i\phi'_{\beta_i}(h_i)\operatorname{div}u_i+gh_i\nabla h_i\phi_{\beta_i}(h_i)
 +h_i\nabla \phi_{\beta_i}(h_i)D(u_i)\\
&\quad + g\varepsilon_i h_i\nabla h_j\phi_{\beta_i}(h_i)
 +r_{i_0}u_i\phi_{\beta_i}(h_i)
 +r_{i1}h_i|u_i|^2u_i\phi_{\beta_i}(h_i) \\
&\quad +\kappa_i\sqrt{h_i}\nabla\phi_{\beta_i}(h_i)\Delta\sqrt{h_i}
 +2\kappa_i\phi_{\beta_i}(h_i)\nabla\sqrt{h_i}\Delta\sqrt{h_i},
\end{align*}
 where $\mathbb{I}$ is an identity matrix.
\end{lemma}

 \section{Recover the limits as $\kappa_i$,  $r_{i_0}$ and
$r_{i_1}$ approach $0$}\label{recover2}


 The objective of this section is twofold. Firstly, to recover the limits
in \eqref{passlim1} as $\kappa_i\to  0$ and $\beta_i\to  \infty$. Secondly,
to apply Theorem \ref{theo2}  to prove Theorem \ref{theorem} by letting
as in \cite{alexisv1} $r_{i_0}\to  0$ and $r_{i_1}\to  0$.
We assume that $\beta_i=\kappa_i^{-3/4}$, thus $\beta_i\to  \infty$
when $\kappa_i\to  0$. First, we state the following lemmas.

\begin{lemma}\label{lemme61}
Let $\kappa_i\to  0$ and $\beta_i\to  \infty$, we have
\begin{gather*}
h_{i\kappa_i}\to  h_i \quad \text{strongly in } L^2(0,T;L^2(\Omega)),\\
\nabla h_{i\kappa_i}\rightharpoonup \nabla h_i \quad \text{weakly in }
  L^2(0,T;H^{-1}(\Omega)),\\
h_{i\kappa_i}\varphi_{in}(\phi_{\beta_i}(h_{i\kappa_i})u_{i\kappa_i})
\to  h_i\varphi_{in}(u_i)\quad \text{strongly in } L^1((0,T)\times\Omega),\\
h_{i\kappa_i}\varphi'_{in}(\phi_{\beta}(h_{i\kappa_i})u_{i\kappa_i})
\to  h_i\varphi'_{in}(u_i)\quad \text{strongly in } L^2(0,T;L^2(\Omega)).
\end{gather*}
\end{lemma}

\begin{lemma}\label{lem62}
Let $\beta_i=\kappa_i^{-3/4}$, and $\kappa_i\to  0$, we have
\begin{equation}\label{lemma62}
\begin{aligned}
&\int_0^T\int_{\Omega}|\psi'_i(t)|h_i\varphi_{in}(u_i)\,dx\,dt \\
&\leq \frac{C}{n}+\Big| \int_0^T\int_{\Omega}\psi_i(t)\nabla h_i^2\varphi'_{in}(u_i)
 \,dx\,dt\Big|
 +\Big| \int_0^T\int_{\Omega}\psi_i(t)h_i\nabla h_j\varphi'_{in}(u_i)\,dx\,dt\Big|\\
&\quad +C\int_0^T\int_{\Omega}(\frac{1}{2}h_{i_0}|u_{i_0}|^2
 +\frac{1}{2}gh_{i_0}^2+|\nabla \sqrt{h_{i_0}}|^2)dx
 +\psi_i(0)\int_{\Omega}h_{i0}\varphi_{in}(u_{i_0})dx.
\end{aligned}
\end{equation}
\end{lemma}

For a proof of the above lemma see \cite{alexisv1}.
With above two lemmas in hand, we are ready to recover the limits in
\eqref{passlim1} as $\kappa_i\to 0$ and $\beta_i\to  \infty$.

Let $r_i = r_{i_0} = r_{i_1}$, we use $(h_1^{r_1}, h_2^{r_2}, u_1^{r_1}, u_2^{r_2})$
to denote the weak solutions to \eqref{mas11}-\eqref{mas44}
verifying Proposition \ref{propo1} with $\kappa_i= 0$. Here, we remark
 that the initial data should satisfy the following conditions, more precisely,
\begin{gather*}
h_{i0}^{r_i}\to  h_{i_0}\quad \text{strongly in } L^2(\Omega),\\
\sqrt{h_{i_0}^{r_i}}u_{i0}^{r_i}\to  \sqrt{h_{i_0}}u_{i_0} \quad \text{strongly in }
 L^2(\Omega)
\end{gather*}
as $r_i\to  0$ and
\begin{equation}\label{eq64}
\begin{gathered}
h_{i_0} \text{ is bounded in } L^1(\Omega)\cap L^2(\Omega),\quad
 h_{i_0} \geq 0 \text{ a.e. in } \Omega,\\
h_{i0}|u_{i_0}|^2=\frac{ m_{i_0}^2}{h_{i_0}} \text{ is bounded in } L^1(\Omega),\\
\nabla\sqrt{h_{i_0}}\text{ is bounded in } L^2(\Omega),\\
 \frac{1}{2}\int_{\Omega}h_{i_0}(1+|u_{i_0}|^2)\ln(1+|u_{i_0}|^2)dx\leq C< \infty
\end{gathered}
\end{equation}
 By \eqref{energy2}-\eqref{Entropy2} one obtains the following estimates:
 \begin{equation}\label{eq65}
 \begin{gathered}
\|\sqrt{h_{i}^{r_i}}u_i^{r_i} \|_{L^{\infty}(0,T;L^2(\Omega))}\leq C,\quad
\|h_i^{r_i}\|_{L^{\infty}(0,T;L^1\cap L^2(\Omega))}\leq C,\\
\|\nabla h_i^{r_i}\|_{L^{2}(0,T;L^2(\Omega))} \leq C,\quad
\|\nabla \sqrt{h_{i}^{r_i}}\\|_{L^{\infty}(0,T;L^2(\Omega))}\leq C,\\
\|\sqrt{h_{i}^{r_i}}\nabla u_i^{r_i} \|_{L^{2}(0,T;L^2(\Omega))}\leq C;
\end{gathered}
 \end{equation}
and by Theorem \ref{fresult}, we have
 \begin{equation}\label{eq66}
  \sup_{t\in[0,T]}\int_{\Omega}h_{i}^{r_i}|u_i^{r_i}|^2\ln(1+|u_i^{r_i}|^2)dx
\leq C.
 \end{equation}
In line with the ideas developed in \cite{alexisv1}, we have
\begin{equation}
 \int_0^T\int_{\Omega}r_i|u_i^{r_i}|^2\,dx\,dt\leq C,\quad
 \int_0^T\int_{\Omega}r_i|h_i^{r_i}u_i^{r_i}|^4\,dx\,dt\leq C,
\end{equation}
where the constant $C$ only depends on the initial data and we can pass
into the limits as $r_i\to  0$.
 In particular,
\begin{gather}
\sqrt{h_i^{r_i}}\to  \sqrt{h_i} \quad\text{almost everywhere and strongly in }
  L^2_{\rm loc}((0,T)\times\Omega),\\
h_i^{r_i}\to  h_i \quad \text{in } C^0 (0,T; L^{\frac{3}{2}}_{\rm loc}(\Omega)), \\
 {h_i^{r_i}}^2\to  h_i^2 \quad \text{strongly in } L_{\rm loc}^1((0,T)\times\Omega)),\\
\sqrt{h_i^{r_i}}u_i^{r_i}\to  \sqrt{h_i}u_i \quad
 \text{strongly in }L_{\rm loc}^2((0,T)\times\Omega)), \\
h_i^{r_i}u_i^{r_i}\to  h_iu_i \quad \text{strongly in }
 L^2(0,T;L_{\rm loc}^p(\Omega))\quad \text{for } p\in[1,\frac{3}{2}) .
\end{gather}
and the convergence of the diffusion terms
\begin{equation}
\begin{gathered}
h_i^{r_i}\nabla u_i^{r_i}\to  h_i\nabla u_i \quad\text{in } \mathcal{D}',\\
h_i^{r_i} (\nabla)^t u_i^{r_i}\to  h_i (\nabla)^t u_i \quad \text{in } \mathcal{D}'
\end{gathered}
\end{equation}
For the  proof of the convergence of the terms   $r_{i}u_i^{r_i}$ and
$r_ih_i^{r_i}|u_i^{r_i}|^2 u_i^{r_i}$  to zero  when  $r_i \to  0$
we refer the reader to  \cite{alexisv1}.

 \section{Recover the limits as $n\to  \infty$}\label{recover1'}

 We want in this section to recover the ``Mellet-Vasseur" type inequality
by  letting $n\to  \infty$. In particular, we prove Theorem \ref{fresult}
 by recovering the limit from Lemma \ref{lem62}. In
this section, $(h_1, h_2, u_1, u_2)$ are the fixed weak solutions.
Following the ideas proposed in  \cite{alexisv1} we only have to control the term
\[
 \Big| \int_0^T\int_{\Omega}\psi_i(t)h_i\nabla h_j\varphi'_{in}(u_i)\,dx\,dt\Big|
\quad  (i\neq j)
\]
 in the right term of \eqref{lemma62} for the other terms the proof
is the same as in \cite{alexisv1}.
We have
\begin{align*}
&\Big| \int_0^T\int_{\Omega}\psi_i(t)h_i\nabla h_j\varphi'_{in}(u_i)\,dx\,dt\Big| \\
&\leq \Big| \int_0^T\int_{\Omega}\psi_i(t)h_i\nabla h_j\varphi'_{in}(u_i)
1_{|u_i|\geq n} \,dx\,dt\Big| \\
&\quad +\Big| \int_0^T\int_{\Omega}\psi_i(t)h_i\nabla h_j\varphi'_{in}(u_i)
1_{|u_i|\leq n} \,dx\,dt\Big|,\\
\end{align*}
where $1_A$ is the indicator function that yields on $A$ and zero outside $A$.
So we have
\begin{gather*}
\Big| \int_0^T\int_{\Omega}\psi_i(t)h_i\nabla h_j\varphi'_{in}(u_i)
1_{|u_i|\geq n} \,dx\,dt\Big| 
\leq \frac{ C}{n}\|h_i\|_{L^2(0,T;L^2(\Omega))}\|\nabla h_j
\|_{L^2(0,T;L^2(\Omega))} \,, \\
\begin{aligned}
&\Big| \int_0^T\int_{\Omega}\psi_i(t)h_i\nabla h_j\varphi'_{in}(u_i)
 1_{|u_i|\leq n} \,dx\,dt\Big| \\
&\leq \Big| \int_0^T\int_{\Omega}\psi_i(t)h_j\nabla h_i\varphi'_{in}(u_i)
 1_{|u_i|\leq n} \,dx\,dt\Big| \\
&\quad +C\int_0^T\int_{\Omega}\psi_i(t)h_ih_j\frac{2u_{il}u_{ik}}{1+|u_i|^2}
 \partial_lu_{ik} 1_{|u_i|\leq n}\,dx\,dt\Big| \\
&\quad +C \Big| \int_0^T\int_{\Omega}\psi_i(t)h_ih_j
 (1+\ln(1+|u_i|^2)\operatorname{div}(u_i)1_{|u_i|\leq n}\,dx\,dt\Big| \\
&\leq  \frac{C}{n}\|h_j\|_{L^2(0,T;L^2(\Omega))}\|\nabla h_i
 \|_{L^2(0,T;L^2(\Omega))} \\
&\quad +C\|\sqrt{h_i}\mathbb{D}u_i\|_{L^2(0,T;(L^2(\Omega))^4)}
 \|h_i\|_{L^2(0,T;L^2(\Omega))}\|h_j\|_{L^4(0,T;L^4(\Omega))} \\
&\quad +C \Big| \int_0^T\int_{\Omega}\psi_i(t)h_ih_j
 (1+\ln(1+|u_i|^2)\operatorname{div}(u_i)1_{|u_i|\leq n}\,dx\,dt\Big|,
\end{aligned} 
\end{gather*}
and
\begin{align*}
&\Big| \int_0^T\int_{\Omega}\psi_i(t)h_ih_j(1+\ln(1+|u_i|^2)\operatorname{div}(u_i)
1_{|u_i|\leq n}\,dx\,dt\Big|\\
&\leq C \Big| \int_0^T\int_{\Omega}(1+\ln(1+|u_i|^2)h_i|\mathbb{D}u_i|^2
 1_{|u_i|\leq n}\,dx\,dt\Big| \\
&\quad +C\Big| \int_0^T\int_{\Omega} h_ih_j^2(1+\ln(1+|u_i|^2)
 1_{|u_i|\leq n}\,dx\,dt\Big|,
\end{align*} 
where
\begin{align*}
& \Big| \int_0^T\int_{\Omega} h_ih_j^2(1+\ln(1+|u_i|^2)1_{|u_i|\leq n}
\,dx\,dt\Big| \\
&\leq C\|\sqrt{h_i}u_i\|_{L^2(0,T;L^2(\Omega))}
 \|h_i\|_{L^2(0,T;L^2(\Omega))}\|h_j\|_{L^8(0,T;L^8(\Omega))}
\end{align*}


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\end{document}