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\title[\sc Existence of solution to mixture dynamics equations]{\large Existence and uniqueness of the solution to the initial boundary value problem for one-dimensional isothermal equations of compressible viscous multicomponent media dynamics}
\author[A.E. MAMONTOV, D.A. PROKUDIN, D.A. ZAKORA]{{\bf \large \textsc{A.E. Mamontov}}\protect\orcidicon{0000-0002-0860-2597},
{\bf \large \textsc{D.A. Prokudin}}\protect\orcidicon{0000-0003-3833-0390},
{\bf \large \textsc{\quad\quad\quad\quad D.A. Zakora}}\protect\orcidicon{0000-0002-6084-1114}\\ \\
\textit{Представлено} \textsc{П.П. Петровым}
%\textit{(заполняется редактором)}
}%
\address{Alexander Evgenyevich Mamontov % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Chair of Further Mathematics,
\newline\hphantom{iii} Federal State Institution of Higher Education <<Siberian State
\newline\hphantom{iii} University of Telecommunications and Information Science>>,
\newline\hphantom{iii} 86, st. Kirova,
\newline\hphantom{iii} 630102 Novosibirsk, Russia}
\email{\textcolor{blue}{aem@hydro.nsc.ru}}
\address{Dmitry Alexeyevich Prokudin
\newline\hphantom{iii} Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the
\newline\hphantom{iii} Russian Academy of Sciences,
\newline\hphantom{iii} 15, pr. Lavrent'eva,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}
\email{\textcolor{blue}{prokudin@hydro.nsc.ru}}%
\address{Dmitry Alexandrovich Zakora
\newline\hphantom{iii} V.I.\,Vernadsky Crimean Federal University,
\newline\hphantom{iii} 4, pr. Vernadskogo,
\newline\hphantom{iii} 295007, Simferopol, Russia}
\email{\textcolor{blue}{dmitry.zkr@gmail.com}}

\thanks{\sc Mamontov, A.E., Prokudin, D.A., Zakora, D.A.,
Existence and uniqueness of the solution to the initial boundary value problem for one-di\-men\-sional isothermal equations of compressible viscous multicomponent media dynamics}
\thanks{\copyright \ 2024 Mamontov A.E., Prokudin D.A., Zakora D.A}
\thanks{\rm This work was financially supported by the Ministry of Science and Higher Education of the Russian Federation, project No~075-02-2024-1431}
\thanks{\it Поступила 1 января 2024 г., опубликована 31 декабря 2024 г.}%

\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\bf Abstract:} An initial boundary value problem is considered for one-dimensional equations of the dynamics of compressible  multi\-com\-po\-nent media. A global theorem of existence and  uniqueness of a strong solution is proved without restrictions on the structure of the viscosity matrix except for the standard properties of symmetry and positivity.\medskip

\noindent{\bf Keywords:} compressible viscous medium, multicomponent flows, viscosity matrix, boundary value problem, existence and  unique\-ness theorem.
 \end{quote}
}

\bigskip

\section{Introduction}

Modelling the motion of multi-component media and solving the ma\-the\-ma\-ti\-cal problems arising is of great interest to both physics and ma\-the\-ma\-tics. However it is little-studied, and no unified approach to this field has been developed so far, and no mathematical theory has been constructed concerning the existence, uniqueness and properties of solutions of the initial-boundary value problems that arise in the process of modelling. A detailed survey of this problem area can be obtained from the monographs~\cite{Rajagopal:1995,Nigmatulin:1987} and the papers~\cite{MamontovProkudin.semr:2017,MamontovProkudin.MAA:2013}. In this paper we choose to study one of the numerous versions of modelling the motion of multi-component fluid mixtures, namely, a homogeneous mixture of viscous compressible fluids with multiple velocities. This means that all the components (constituents) of the mixture are present in the same phase at every point of space, but each of them has its own local velocity of motion; the components interact via momentum exchange and viscous friction (and also heat exchange in heatconducting models). The model under consideration is a generalization of the well-known Navier---Sto\-kes system of equations describing the motions of compressible viscous single-component media and includes the continuity and momentum equations, and also the energy equation(s) in heatconducting models. The characteristic feature of these equations, in addition to their nonlinearity, is the presence of higher order derivatives of the velocities of all components in the momentum (and energy) conservation laws, due to the composite structure of the viscous stress tensors~\cite{Rajagopal:1995, Nigmatulin:1987, Kirwan.Fluids:2020, Lee.TI:2013, Malek.NARWA:2008, Atkin.jima:1976, Muller.IMAJAM:1968}. This specificity of multicomponent flows can be described using the concept of viscosity matrix. Unlike the single-component case in which the viscosity is scalar-valued, in the multicomponent case the viscosities form a matrix whose entries describe viscous friction. Diagonal entries describe viscous friction within each component, and non-diagonal entries describe friction between the components. In the case of a diagonal viscosity matrix, the momentum equations are possibly connected via the lower order terms only. In the paper the more complicated case of off-diagonal viscosity matrices is under consideration. The aim of the paper is to analyze the existence and uniqueness of solution to an initial-boundary value problem for isothermal (non-heatconducting) equations of compressible viscous multicomponent media in the case of one-dimensional motions in a bounded domain with non permeable boundaries.

As mentioned above, the multi-velocity model of compressible viscous multicomponent media dynamics under consideration is a generalization of the well-known Navier---Stokes system of equations and hence the ma\-the\-ma\-ti\-cal results for multicomponent media appeared after the progress achieved in the Navier---Stokes theory, for which an invaluable contribution has been made by one-dimensional results~\cite{StraskrabaZlotnik.ZAMP:2003, StraskrabaZlotnik.JMFM:2003, Zlotnik.DE:1995, Zlotnik.MN:1994, Jiang.JDE:1994, Yanagi.MMAS:1993, Hoff.JDE:1992, Vaigant.DSS:1992, Zlotnik.MN:1992, Zlotnik.SMJ:1992, Hoff.ARMA:1991, AKM:1983, Vaigant.DSS:1990, Nagasawa.NHMS:1989, Zlotnik.DANSSR:1988, VaigantPapin.DSS:1987, Nagasawa.JDE:1986, Kawohl.DE:1985, OkadaKawashima.JMKU:1983, Belov.DSS:1982, Kazhikhov.SMJ:1982, KawashimaNishida.JMKU:1981, MatsumuraNishida.PJASA:1979, Shelukhin.DSS:1979, Kazhikhov.DSS:1976, Kazhikhov.DSS:1975}. The unique solvability of the considered one-dimensional equations of compressible viscous mul\-ti\-com\-po\-nent media in the polytropic case is investigated in~\cite{MamontovProkudin.JMFM:2019, MamontovProkudin.SEMR:2018}. Similar issues for related models of multicomponent media are discussed in~\cite{MamontovProkudin.LJM:2024, Prokudin.M:2023, MamontovProkudin.SEMR:2021, Prokudin.SEMR:2021, Prokudin.semr:2017, Prokudin.JFCS:2017, Prokudin.JFCS:2017, Li.ZAMP:2017, BreschHuangLi.CMP:2012, Papin.MN:2010, Zlotnik.MN:2004, Zlotnik.MN:1995, Petrov.DSS:1982, KazhikovPetrov.DSS:1978}. Spectral analysis of some linear models of compressible viscous multicomponent media is carried out in~\cite{a-Zakora-2023-1,a-Zakora-2023-2}.

The structure of the paper is as follows. Section~2 contains the statement of the initial-boundary value problem and the formulation of the main result which is Theorem~1 concerning the existence and uniqueness of the solution. In Section~3, we study the solvability of an approximate initial-boundary value problem which is obtained from the original one via Galerkin method. In Section~4, the solutions to the approximate problem are estimated uni\-form\-ly in the approximation parameter. Basing on these estimates, in Sec\-tion~5 the limit transition is made and the local-in-time existence of a solution to the original initial-boundary value problem is verified. In order to continue the local solution, in Section 6 we prove a priori estimates, in which the constants do not depend on the local existence interval. In Section~7, the uniqueness of the solution to the initial-boundary value problem is proved.

\section{Statement of the initial-boundary value problem and formulation of the existence and uniqueness theorem}

We consider the initial boundary value problem for one-dimensional iso\-ther\-mal equations of the dynamics of compressible viscous multicomponent me\-dia. In the closure $\overline{Q}_{T}$ of a domain ${Q_{T}=(0, T)\times (0, 1)}$ (${T>0}$) the sought values are the densities ${\rho_{i}(t,x)> 0}$ and the velocities $u_{i}(t,x)$ of each com\-po\-nent with the number ${i=1,\ldots,N}$ (${N\in\mathbb{N}}$, ${N \geqslant 2}$), which satisfy the following system of differential equations, initial and boundary conditions:
\begin{equation}\label{mamprok_continuity}
\frac{\partial\rho_{i}}{\partial t}+\frac{\partial(\rho_i v)}{\partial x}=0,\quad i=1,\ldots,N,\end{equation}
\begin{equation}\label{mamprok_momentum}
\rho_{i}\left(\frac{\partial u_{i}}{\partial t}+v\frac{\partial u_{i}}{\partial x}\right)+\alpha_{i}K\frac{\partial \rho}{\partial x}=\sum\limits_{j=1}^N \nu_{ij}\frac{\partial^{2}u_{j}}{\partial x^2},\quad i=1,\ldots,N,
\end{equation}
\begin{equation}\label{mamprok_nachusl}
\rho_{i}|_{t=0}=\rho_{0i}(x), \quad u_{i}|_{t=0}=u_{0i}(x),\quad i=1,\ldots,N,
\end{equation}
\begin{equation}\label{mamprok_boundvelocity}
u_{i}|_{x=0}=u_{i}|_{x=1}=0,\quad i=1,\ldots,N.
\end{equation}
Here $v$~is the average velocity,  ${\displaystyle v=\sum\limits_{j=1}^{N}\alpha_{j}u_{j}}$, ${\alpha_{j}={\rm const}\in(0, 1)}$, ${\displaystyle \sum\limits_{j=1}^{N}\alpha_{j}=1}$, $\rho$ is the total density, ${\displaystyle \rho=\sum\limits_{j=1}^{N}\rho_{j}}$, constant viscosity coefficients $\nu_{ij}$ form the matrix ${\textbf{N}>0}$, the coefficient ${K>0}$, the initial data $\rho_{0i}(x)$, $u_{0i}(x)$, ${i=1,\ldots, N}$ are given.

The given model corresponds to the so-called diffusion approximation appropriate at description of motions in which relative velocities of com\-po\-nents are small in comparison with the general (average) velocity of the flow (mixture).

{\bfseries Definition 1.}
{\it By a strong solution to problem \eqref{mamprok_continuity}--\eqref{mamprok_boundvelocity} we mean $2N$ functions $(\rho_1,\ldots,  \rho_{N}, u_{1},\ldots, u_{N})$
such that for all $i=1,\ldots,N$
\begin{equation}\label{mamprok_oprresh}
\begin{split}
&\displaystyle
\rho_{i}>0,\quad \rho_{i}\in L_{\infty}\big(0, T; W^{1}_{2}(0, 1)\big), \quad  \frac{\partial\rho_{i}}{\partial t}\in L_{\infty}\big(0, T; L_{2}(0, 1)\big),\\
&\displaystyle u_{i}\in L_{\infty}\big(0, T; W^{1}_{2}(0, 1)\big)\bigcap L_{2}\big(0, T; W^{2}_{2}(0, 1)\big),\quad \frac{\partial u_{i}}{\partial t} \in L_{2}(Q_{T}),
\end{split}\end{equation}
equations~\eqref{mamprok_continuity}, \eqref{mamprok_momentum} are satisfied almost everywhere in $Q_{T}$, the initial condi\-tions~\eqref{mamprok_nachusl} are valid for almost all ${x\in (0, 1)}$, and the boundary conditions~\eqref{mamprok_boundvelocity} are accepted for almost all ${ t\in (0, T)}$.
}

{\bfseries Theorem 1.}
{\it
Let the initial data in \eqref{mamprok_nachusl} satisfy the conditions
\begin{equation}\label{mamprok_gladknachusl}
\rho_{0i}>0, \quad \rho_{0i}\in W^{1}_{2}(0, 1),\quad u_{0i}\in \overset{\circ}{W^1_2}(0, 1),\quad i=1,\ldots,N.
\end{equation}
Then there exists a unique strong solution to problem
\eqref{mamprok_continuity}--\eqref{mamprok_boundvelocity} in the sense of Definition~1.
}

{\bfseries Proof} of Theorem~1 is given in Sections 3--7.

\section{Construction of Galerkin approximations}

Let us first prove the solvability of an approximate initial-boundary value problem, obtained from problem~\eqref{mamprok_continuity}--\eqref{mamprok_boundvelocity} by applying
the Galerkin method (in the spatial variable~$x$) to equations \eqref{mamprok_momentum}.

{\bfseries Theorem 2.}
{\it
On the assumptions of Theorem~1 for all $m\in{\mathbb N}$ there is a time-interval $(0,  t^{m})\subset (0,T)$, where there exists a solution to the problem
\begin{equation}\label{mamprok_ravlema3}
\frac{\partial\rho_{i}^{m}}{\partial t}+\frac{\partial(\rho_{i}^{m} v^{m})}{\partial x}=0,\quad i=1,\ldots,N,
\end{equation}
\begin{equation}\label{mamprok_ravlema5}
\begin{split}
&\int\limits^{1}_{0}\!\bigg(\rho_{i}^{m}\frac{\partial u_{i}^{m}}{\partial t}\!+\!\rho_{i}^{m}v^{m}\frac{\partial u_{i}^{m}}{\partial x}+\alpha_{i}K\frac{\partial \rho^{m}}{\partial x}
\!-\!\sum\limits_{j=1}^N \nu_{ij}\frac{\partial^{2}u_{j}^{m}}{\partial x^{2}}\bigg)\sin{(\pi kx)}\, dx=0,\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad i=1,\ldots,N, \quad k=1,\ldots,m,
\end{split}
\end{equation}
\begin{equation}\label{mamprok_ravlema4}
\rho_{i}^{m}|_{t=0}=\rho_{0i}(x), \quad i=1,\ldots,N,
\end{equation}
\begin{align}\label{mamprok_ravlema2}
&u_{i}^{m}\!=\!\sum\limits_{s=1}^{m}\xi_{is}^{m}(t)\sin{(\pi sx)},\quad u_{i}^{m}|_{t=0}=\sum\limits_{s=1}^{m}\xi_{0is}^{m}\sin{(\pi sx)}, \quad i=1,\ldots,N,\\
&\xi_{0is}^{m}:=\xi_{is}^{m}(0)=2\int\limits_{0}^{1}u_{0i}(x)\sin{(\pi sx)}\,dx,\quad i=1,\ldots,N,\quad s=1,\ldots,m,\nonumber
\end{align}
where ${\displaystyle v^{m}=\sum\limits_{j=1}^{N}\alpha_{j}u_{j}^{m}}$,  ${\displaystyle\rho^{m}=\sum\limits_{j=1}^{N}\rho_{j}^{m}}$, and we have
\begin{equation}\label{mamprok_1206172}
\begin{split}
\displaystyle
\rho_{i}^{m}>0,&\quad \rho_{i}^{m}\in L_{\infty}\big(0,  t^{m}; W^{1}_{2}(0, 1)\big)\bigcap W^{1}_{\infty}\big(0,  t^{m}; L_{2}(0, 1)\big),\\
&\displaystyle
\xi_{is}^{m}\in C^{1}[0, t^{m}],\quad i=1,\ldots,N,\quad s=1,\ldots,m.
\end{split}
\end{equation}}

{\bfseries Proof.} We fix ${t^m < T}$. We omit the upper index $m$ in the notations of the solutions up to the beginning of Section~5. Consider the set
$$V=\left\{\boldsymbol{\xi}\in(C[0,  t^{m}])^{mN}\big|\;\;\boldsymbol{\xi}(0)=\boldsymbol{\xi}_{0},\;\;\|\boldsymbol{\xi}\|_{(C[0,  t^{m}])^{mN}}\leqslant c\right\},$$
where
$$
\boldsymbol{\xi}=(\boldsymbol{\xi}_{1},\ldots,\boldsymbol{\xi}_{N}), \quad \boldsymbol{\xi}_{i}=({\xi}_{i1},\ldots,{\xi}_{im}), \quad i=1,\ldots,N,
$$
$$
\boldsymbol{\xi}_{0}=(\boldsymbol{\xi}_{01},\ldots,\boldsymbol{\xi}_{0N}), \quad
\boldsymbol{\xi}_{0i}=({\xi}_{0i1},\ldots,{\xi}_{0im}),\quad i=1,\ldots,N,
$$
$$
c^{2}=e\frac{\max\limits_{1\leqslant i\leqslant N}\sup\limits_{[0, 1]}\rho_{0i}}{\min\limits_{1\leqslant i\leqslant N}
\inf\limits_{[0, 1]}\rho_{0i}}\,\|\boldsymbol{\xi}_{0}\|^{2}_{{\mathbb R}^{mN}}+1.
$$
We construct the operator ${A:V\to(C[0,  t^{m}])^{mN}}$, ${{\rm Im}\,A \subset (C^{1}[0,  t^{m}])^{mN}}$,\linebreak ${A(\boldsymbol{\xi})=\boldsymbol{\psi}}$, where $\boldsymbol{\psi}=(\boldsymbol{\psi}_{1},\ldots,\boldsymbol{\psi}_{N})$,  $\boldsymbol{\psi}_{i}=(\psi_{i1},\ldots,\psi_{im})$, $i=1,\ldots,N$, by
the following algorithm. First, we find the functions
$$
\rho_{i}>0,\quad \rho_{i}\in L_{\infty}\big(0,  t^{m}; W^{1}_{2}(0, 1)\big)\bigcap W^{1}_{\infty}\big(0,  t^{m}; L_{2}(0, 1)\big),\quad  i=1,\ldots,N
$$
as solutions to the Cauchy problems \eqref{mamprok_ravlema3}, \eqref{mamprok_ravlema4}, where ${\displaystyle v=\sum\limits_{j=1}^{N}\alpha_{j}u_{j}}$, and $u_{j}$, $j=1,\ldots,N$ are given by \eqref{mamprok_ravlema2} (see \cite{DiPernaLions:1989}). Moreover, the inequalities
\begin{equation}\label{mamprok_ravlema6}
\begin{split}
&\displaystyle \left(\inf\limits_{[0, 1]}\rho_{0i}\right)e^{-\sum\limits_{j=1}^{N}\int\limits_{0}^{t}\sup\limits_{[0, 1]}\Big|\frac{\partial u_{j}}{\partial x}\Big|\,d\tau}
\leqslant\rho_{i}(t, x)\leqslant\left(\sup\limits_{[0, 1]}\rho_{0i}\right)e^{\sum\limits_{j=1}^{N}\int\limits_{0}^{t}\sup\limits_{[0, 1]}\Big|\frac{\partial u_{j}}{\partial x}\Big|\,d\tau},\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad i=1,\ldots,N
\end{split}
\end{equation}
hold, which, in view of the inclusion $\boldsymbol{\xi}\in V$, give the estimates
\begin{equation}\label{mamprok_ravlema7}
\left(\inf\limits_{[0, 1]}\rho_{0i}\right)e^{-\pi m^{2} c N t}
\leqslant\rho_{i}(t, x)
\leqslant\left(\sup\limits_{[0, 1]}\rho_{0i}\right)e^{\pi m^{2} cN t},\quad i=1,\ldots,N.
\end{equation}
Next, we find $\boldsymbol{\psi}$ from the Cauchy problem for the system of $mN$ ordinary first order differential (linear) equations:
\begin{equation}\label{mamprok_ravlema8}
\begin{split}
\int\limits^{1}_{0}\Big(\rho_{i}\frac{\partial U_{i}}{\partial t}+\rho_{i}v\frac{\partial U_{i}}{\partial x}+\alpha_{i}K\frac{\partial \rho}{\partial x}&-\sum\limits_{j=1}^N \nu_{ij}\frac{\partial^{2}U_{j}}{\partial x^{2}}\Big)\sin{(\pi k x)}\, dx=0, \\
&\qquad i=1,\ldots,N,\quad k=1,\ldots,m,
\end{split}
\end{equation}
\begin{equation}
\boldsymbol{\psi}(0)=\boldsymbol{\xi}_{0},\label{mamprok_ravlema9}
\end{equation}
where ${\displaystyle U_{i}=\sum\limits_{s=1}^{m}\psi_{is}(t)\sin{(\pi s x)}}$, ${i=1,\ldots,N}$. The inequality
$$\det M(t)\neq 0,$$
where
\begin{align*}
M(t)&=\left(\begin{array}{cccc}
M_{1}(t) & 0 & \ldots & 0 \\
0 & M_{2}(t) & \ldots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & M_{N}(t) \\
\end{array}
\right),\\
M_{i}(t) &=\left\{\int\limits^{1}_{0}\rho_{i}(t, x)\sin{(\pi k x)}\sin{(\pi s x)}\,dx\right\}_{k, s=1}^{m},\quad i=1,\ldots,N,
\end{align*}
which is valid due to the positivity of $\rho_{i}$, ${i=1,\ldots,N}$, admits to solve system~\eqref{mamprok_ravlema8} with respect to derivatives, which justifies the existence of  $\boldsymbol{\psi}\in (C^{1}[0,  t^{m}])^{mN}$. Thus, for arbitrary ${ t^{m}\in (0,T]}$ we can define the operator ${A: V\to (C^{1}[0,  t^{m}])^{mN}\subset  (C[0,  t^{m}])^{mN}}$, ${\;A(\boldsymbol{\xi})=\boldsymbol{\psi}}$, whose fixed point (if it exists), together with the corresponding functions $\rho_{i}$, $u_{i}$, $i=1,\ldots,N$, is a solution to problem~\eqref{mamprok_ravlema3}--\eqref{mamprok_ravlema2}.

We show that for sufficiently small $t^{m}$ the operator $A$ satisfies the as\-sump\-tions of the Schauder theorem (see \cite[P.\,31]{AKM:1983}), i.\,e.
\begin{enumerate}
\item $V$ is a convex closed bounded set (which is obvious in the case under consideration);
\item $A:\> V\to V$;
\item $A$ is a completely continuous operator.
\end{enumerate}
	
We first show that $A(V)\subset V$. Hereinafter, $C_{i}(\cdot)$, $i\in{\mathbb N}$, denote quantities which accept positive finite values depending on arguments indicated
in the brackets or in comments. Multiplying equations \eqref{mamprok_ravlema8} by $\psi_{ik}(t)$,  summarizing with respect to $i$, $k$ and integrating with respect to $x$, we obtain
due to \eqref{mamprok_ravlema3}, that
\begin{multline*}
\frac{1}{2}\frac{d}{dt}\left(\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}U_{i}^{2}\, dx\right)+
\sum\limits_{i, j=1}^N \nu_{ij}\int\limits^{1}_{0}\left(\frac{\partial U_{i}}{\partial x}\right)\left(\frac{\partial U_{j}}{\partial x}\right) dx=\\
=K\sum\limits_{i=1}^{N}\alpha_{i}\int\limits^{1}_{0}\rho\left(\frac{\partial U_{i}}{\partial x}\right) dx,
\end{multline*}
and taking into account the inequalities (to obtain them, we use \eqref{mamprok_ravlema7} and the fact $\textbf{N}>0$)
\begin{align*}
&\sum\limits_{i, j=1}^N \nu_{ij}\int\limits^{1}_{0}\left(\frac{\partial U_{i}}{\partial x}\right)\left(\frac{\partial U_{j}}{\partial x}\right)dx\geqslant C_{1}(\textbf{N})\sum\limits_{i=1}^N\int\limits^{1}_{0}\left(\frac{\partial U_{i}}{\partial x}\right)^{2}dx,\\
&\quad
K\sum\limits_{i=1}^{N}\alpha_{i}\int\limits^{1}_{0}\rho\left(\frac{\partial U_{i}}{\partial x}\right) dx\leqslant \frac{C_{1}}{2}\sum\limits_{i=1}^N\int\limits^{1}_{0}\left(\frac{\partial U_{i}}{\partial x}\right)^{2}dx+C_{2},
\end{align*}
where
$\displaystyle C_{2}=\frac{K^{2}N^{3}}{2C_{1}}\left(\max\limits_{1\leqslant i\leqslant N}\sup\limits_{[0, 1]}\rho_{0i}\right)^{2}
e^{2\pi m^{2} cNt^{m}}$, we obtain the estimate
$$
\frac{d}{dt}\left(\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}U_{i}^{2}\, dx\right)+C_1\sum\limits_{i=1}^N\int\limits^{1}_{0}\left(\frac{\partial U_{i}}{\partial x}\right)^{2}dx\leqslant 2C_{2},
$$
which, in its turn, leads to
\begin{equation}\label{mamprok.ravlema13}
\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}U_{i}^{2}\, dx\leqslant \sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{0i}U_{0i}^{2}\, dx+2C_{2} t^{m},
\end{equation}
where ${\displaystyle U_{0i}=\sum\limits_{s=1}^{m}\psi_{is}(0)\sin{(\pi s x)}=\sum\limits_{s=1}^{m}\xi_{0is}\sin{(\pi s x)}}$, ${i=1,\ldots,N}$.
Using \eqref{mamprok_ravlema7} for the second time, we obtain from \eqref{mamprok.ravlema13} the inequality
$$
\|\boldsymbol{\psi}\|^{2}_{(C[0,  t^{m}])^{mN}}\leqslant e^{\pi m^{2} c N t^{m}}\,\frac{\max\limits_{1\leqslant i\leqslant N}\sup\limits_{[0, 1]}\rho_{0i}}{\min\limits_{1\leqslant i\leqslant N}\inf\limits_{[0, 1]}\rho_{0i}}\,\|\boldsymbol{\xi}_{0}\|^{2}_{{\mathbb R}^{mN}}+\frac{4C_{2}e^{\pi m^{2} c N t^{m}}}{\min\limits_{1\leqslant i\leqslant N}\inf\limits_{[0, 1]}\rho_{0i}} t^{m}.
$$
Choosing
\begin{equation}\label{mamprok.ravlema15}
t^{m}<\min\left(T, \frac{1}{\pi m^{2}cN}, \frac{\min\limits_{1\leqslant i\leqslant N}\inf\limits_{[0, 1]}\rho_{0i}}{4eC_{3}}\right),
\end{equation}
where
${\displaystyle C_{3}=\frac{K^{2}N^{3}e^{2}}{2C_{1}}\left(\max\limits_{1\leqslant i\leqslant N}
\sup\limits_{[0, 1]}\rho_{0i}\right)^{2}}$,
we obtain that  $C_{2}\leqslant C_{3}$, and arrive at the required estimate
$$
\|\boldsymbol{\psi}\|_{(C[0,  t^{m}])^{mN}}\leqslant c.
$$
Thus, if \eqref{mamprok.ravlema15} holds, the operator $A$ maps the set $V$ to itself.

Let us prove the compactness of the operator $A$. Multiplying \eqref{mamprok_ravlema8} by $\displaystyle \frac{d\psi_{ik}(t)}{dt}$, summarizing with respect to $i$, $k$ and integrating with respect to $x$, we get the relation
\begin{multline}\label{mamprok.ravlema17}
\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial U_{i}}{\partial t}\right)^{2} dx
=\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\left(\vphantom{\sum_{j=1}^{N}}-\rho_{i}v\left(\frac{\partial U_{i}}{\partial x}\right)\left(\frac{\partial U_{i}}{\partial t}\right)+\right.
\\
\left. +\alpha_{i}K\rho\left(\frac{\partial^{2}U_{i}}{\partial t\partial x}\right)-
\sum\limits_{j=1}^N \nu_{ij}\left(\frac{\partial^{2}U_{i}}{\partial t\partial x}\right)\left(\frac{\partial U_{j}}{\partial x}\right)\right) dx.
\end{multline}
Let us estimate the terms on the right-hand side of \eqref{mamprok.ravlema17} by using \eqref{mamprok_ravlema7}, the Cauchy inequality, and the inequalities ${\|\boldsymbol{\xi}\|_{(C[0,  t^{m}])^{mN}}\leqslant c}$, ${\|\boldsymbol{\psi}\|_{(C[0,  t^{m}])^{mN}}\leqslant c}$,
${\displaystyle \left\|\frac{\partial U_{i}}{\partial x}\right\|_{L_{2}(0, 1)}\!\!\!\!\!\leqslant C_{4}(m)\|U_{i}\|_{L_{2}(0, 1)}}$,
${\displaystyle \left\|\frac{\partial^{2}U_{i}}{\partial t\partial x}\right\|_{L_{2}(0, 1)}\!\!\!\!\!\leqslant C_{4}\left\|\frac{\partial U_{i}}{\partial t}\right\|_{L_{2}(0, 1)}}$, ${i=1,\ldots,N}$:
\begin{multline*}
-\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}v\left(\frac{\partial U_{i}}{\partial x}\right)\left(\frac{\partial U_{i}}{\partial t}\right)dx\leqslant\\
\leqslant\frac{1}{6}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial U_{i}}{\partial t}\right)^{2} dx+ C_{5}\left(C_{4},\left\{\sup\limits_{[0, 1]}\rho_{0i}\right\}_{i=1}^{N}, N, c, m,  t^{m}\right),
\end{multline*}
\begin{multline*}
K\sum\limits_{i=1}^{N}\alpha_{i}\int\limits^{1}_{0}\rho\left(\frac{\partial^{2}U_{i}}{\partial t\partial x}\right)dx
\leqslant\frac{1}{6}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial U_{i}}{\partial t}\right)^{2} dx+\\
+C_{6}\left(C_{4}, \left\{\inf\limits_{[0, 1]}\rho_{0i}\right\}_{i=1}^{N},
\left\{\sup\limits_{[0, 1]}\rho_{0i}\right\}_{i=1}^{N}, K, N, c, m,  t^{m}\right),
\end{multline*}
\begin{multline*}
-\sum\limits_{i, j=1}^{N}\nu_{ij}\int\limits^{1}_{0}\left(\frac{\partial^{2}U_{i}}{\partial t\partial x}\right)\left(\frac{\partial U_{j}}{\partial x}\right) dx\leqslant\\
\leqslant\frac{1}{6}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial U_{i}}{\partial t}\right)^{2} dx+ C_{7}\left(C_{4},\left\{\inf\limits_{[0, 1]}\rho_{0i}\right\}_{i=1}^{N}, \textbf{N}, N, c, m,  t^{m}\right).
\end{multline*}
Thus, from \eqref{mamprok.ravlema17} we obtain the inequality
$$\frac{1}{2}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial U_{i}}{\partial t}\right)^{2} dx\leqslant C_{5}+C_{6}+C_{7}.$$
Integrating the last inequality in time and applying \eqref{mamprok_ravlema7}, we obtain the estimate
\begin{equation}\label{mamprok.Ut1}
\sum\limits_{i=1}^{N}\left\|\frac{\partial U_{i}}{\partial t}\right\|_{L_{2}(Q_{t^{m}})}\leqslant C_{8}\left(C_{5}, C_{6}, C_{7}, \left\{\inf\limits_{[0, 1]}\rho_{0i}\right\}_{i=1}^{N}, N, c, m,  t^{m}\right),
\end{equation}
where ${Q_{t^{m}}=(0, t^{m})\times (0,1)}$. Thus, we have obtained the estimate for $\boldsymbol{\psi}$ in $(W^{1}_{2}(0,  t^{m}))^{mN}$. Consequently, $A$ is a compact operator.

We establish the continuity of the operator $A$ from $V$ in $(C[0,  t^{m}])^{mN}$. Let ${\boldsymbol{\xi}^{(1,2)}\in V}$, ${\quad\boldsymbol{\psi}^{(1,2)}=A(\boldsymbol{\xi}^{(1,2)})}$, ${\quad\displaystyle u^{(1,2)}_{i}=\sum\limits_{s=1}^{m}\xi^{(1,2)}_{is}\sin{(\pi s x)}}$,  $\quad\displaystyle U^{(1,2)}_{i}=\sum\limits_{s=1}^{m}\psi^{(1,2)}_{is}\sin{(\pi s x)}$, ${i=1,\ldots,N}$. Let $\rho_{i}^{(1,2)}$, ${i=1,\ldots,N}$ be the solutions to the Cauchy problems \eqref{mamprok_ravlema3}, \eqref{mamprok_ravlema4}, with $v$ instead of ${\displaystyle v^{(1,2)}=\sum\limits_{j=1}^{N}\alpha_{j} u_{j}^{(1,2)}}$ respective\-ly. Denote  ${\rho_{i}=\rho^{(1)}_{i}-\rho^{(2)}_{i}}$, ${u_{i}=u^{(1)}_{i}-u^{(2)}_{i}}$, ${U_{i}=U^{(1)}_{i}-U^{(2)}_{i}}$, ${i=1,\ldots,N}$, ${v=v^{(1)}-v^{(2)}}$, ${\rho=\rho^{(1)}-\rho^{(2)}}$, where  ${\displaystyle \rho^{(1,2)}=\sum\limits_{j=1}^{N}\rho_{j}^{(1,2)}}$.
Differentiating the equations
\begin{equation}\label{mamprok.020820204eq1}
\frac{\partial \rho^{(1,2)}_{i}}{\partial t}+\frac{\partial\left(\rho^{(1,2)}_{i}v^{(1,2)}\right)}{\partial x}=0, \quad i=1,\ldots,N
\end{equation}
(see Remark 1 below) with respect to $x$, multiplying by $\displaystyle \frac{\partial \rho_{i}^{(1,2)}}{\partial x}$, integrating with respect to $x$, $t$, using the initial conditions
\begin{equation}\label{mamprok.020820204eq2}
\rho^{(1,2)}_{i}|_{t=0}=\rho_{0i}, \quad i=1,\ldots,N,
\end{equation}
the inequality (see \eqref{mamprok_ravlema7})
\begin{equation}\label{mamprok.020820204eq3}
\left(\inf\limits_{[0, 1]}\rho_{0i}\right)e^{-\pi m^{2} c N t}
\leqslant\rho_{i}^{(1,2)}
\leqslant\left(\sup\limits_{[0, 1]}\rho_{0i}\right)e^{\pi m^{2} cN t},\quad i=1,\ldots,N
\end{equation}
and Gronwall inequality we get
\begin{equation}\label{mamprok.differ}
\left\|\frac{\partial \rho_{i}^{(1,2)}}{\partial x}\right\|_{L_{2}(0, 1)}\!\!\!\!\!\leqslant C_{9}\left(\left\{\|\rho_{0i}\|_{W^{1}_{2}(0, 1)}\right\}_{i=1}^{N}, N, c, m, t^{m}\right),\quad i=1,\ldots,N.
\end{equation}
Let us note that \eqref{mamprok.020820204eq1}, \eqref{mamprok.020820204eq2} lead to the equalities
\begin{equation}\label{mamprok.ravlema19}
\frac{\partial \rho_{i}}{\partial t}+\frac{\partial\left(\rho_i v^{(1)}\right)}{\partial x}+\frac{\partial\left(\rho_{i}^{(2)} v\right)}{\partial x}=0,\quad \rho_{i}|_{t=0}=0,\quad i=1,\ldots,N.
\end{equation}
Multiplying \eqref{mamprok.ravlema19} by $\rho_{i}$, and integrating with respect to $x$, we arrive at the relations
\begin{multline}\label{mamprok.ravlema21}
\frac{1}{2}\frac{d}{dt}\left(\int\limits^{1}_{0}\rho_{i}^{2}\, dx\right)=
-\!\int\limits^{1}_{0}\left(\frac{1}{2}\rho_{i}^{2}\left(\frac{\partial v^{(1)}}{\partial x}\right)\!+\rho_{i}^{(2)}\rho_{i}\left(\frac{\partial v}{\partial x}\right)\!+\!\left(\frac{\partial\rho_{i}^{(2)}}{\partial x}\right)\rho_{i}v\right)dx\\
\leqslant\frac{1}{2}\left(\sup\limits_{[0, 1]}\left|\frac{\partial v^{(1)}}{\partial x}\right|\int\limits^{1}_{0}\rho_{i}^{2}\, dx+
\sup\limits_{[0, 1]}\rho_{i}^{(2)}\!\int\limits^{1}_{0}\left(\rho_{i}^{2}+\left(\frac{\partial v}{\partial x}\right)^{2}\right)dx+\right.\\
\left.\qquad\qquad\qquad\qquad\qquad\quad+\sup\limits_{[0, 1]}v^{2}\int\limits^{1}_{0}\left(\frac{\partial\rho_{i}^{(2)}}{\partial x}\right)^{2}dx
+\int\limits^{1}_{0}\rho_{i}^{2}\,dx\right)\leqslant \\
\leqslant C_{10}\left(C_{9}, \left\{\sup\limits_{[0, 1]}\rho_{0i}\right\}_{i=1}^{N}, N, c, m, t^{m}\right)\left(\int\limits^{1}_{0}\rho_{i}^{2}\,dx+\sum\limits_{j=1}^{N}\int\limits^{1}_{0}u^{2}_{j}\,dx\right).
\end{multline}
Here we have used obvious relations
\begin{equation}\label{mamprok.1206171}
\begin{split}
\sum\limits_{j=1}^{N}\int\limits_{0}^{1}&u_{j}^{2}\, dx=\frac{1}{2}\sum\limits_{j=1}^{N}\sum\limits_{s=1}^{m}\xi_{js}^{2}(t),\quad
\sup\limits_{[0, 1]}{v^{2}}\leqslant \sum\limits_{i,j=1}^{N}\sum\limits_{s, l=1}^{m}|\xi_{il}(t)\xi_{js}(t)|,\\
& \int\limits_{0}^{1}\left(\frac{\partial v}{\partial x}\right)^{2} dx=\frac{\pi^{2}}{2}\sum\limits_{i,j=1}^{N}\alpha_{i}\alpha_{j}\sum\limits_{s=1}^{m}s^{2}\xi_{js}(t)\xi_{is}(t).
\end{split}
\end{equation}
From \eqref{mamprok.ravlema21}, applying the Gronwall inequality and the initial conditions in~\eqref{mamprok.ravlema19}, we obtain the inequalities
\begin{equation}\label{mamprok.ravlema22}
\int\limits^{1}_{0}\rho_{i}^{2}\, dx\leqslant C_{11}(C_{10},  t^{m})\sum\limits_{j=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}u^{2}_{j}\,dx d\tau,\quad i=1,\ldots,N
\end{equation}
for all ${t\in(0,t_m]}$. Further, from the equations for $U_{i}^{(1,2)}$, ${i=1,\ldots,N}$ (see~\eqref{mamprok_ravlema8}) due to \eqref{mamprok.020820204eq1} we get
\begin{multline}\label{mamprok.ravlema23} \frac{1}{2}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{(1)}U_{i}^{2}\,dx+\sum\limits_{i, j=1}^N \nu_{ij}\int\limits_{0}^{t}\int\limits_{0}^{1}\left(\frac{\partial U_{i}}{\partial x}\right)\left(\frac{\partial U_{j}}{\partial x}\right) dx d\tau=\\
=K\sum\limits_{i=1}^{N}\alpha_{i}\int\limits_{0}^{t}\int\limits_{0}^{1}\rho\left(\frac{\partial U_{i}}{\partial x}\right) dx d\tau-\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}\rho_{i}U_{i}\left(\frac{\partial U_{i}^{(2)}}{\partial \tau }\right)dx d\tau-\\
-\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}\rho_{i}^{(1)}vU_{i}\Bigg(\frac{\partial U_{i}^{(2)}}{\partial x}\Bigg)dx d\tau-
\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}\rho_{i}v^{(2)}U_{i}\Bigg(\frac{\partial U_{i}^{(2)}}{\partial x}\Bigg)dx d\tau
\end{multline}
for all ${t\in(0,t_m]}$. The first term on the left-hand side of \eqref{mamprok.ravlema23} satisfies the estimate
\begin{equation}\label{mamprok.ravlema24}
\frac{1}{2}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{(1)}U_{i}^{2}\,dx\geqslant \frac{e^{-\pi m^{2} c N t^{m}}\min\limits_{1\leqslant i\leqslant N}\inf\limits_{[0, 1]}\rho_{0i}}{2}\;\sum\limits_{i=1}^{N}\int\limits^{1}_{0}U_{i}^{2}\,dx.
\end{equation}
For the second term on the left-hand side of \eqref{mamprok.ravlema23} we have
\begin{equation}\label{mamprok.ravlema25}\sum\limits_{i, j=1}^{N} \int\limits_{0}^{t}\int\limits_{0}^{1}\nu_{ij}\left(\frac{\partial U_{i}}{\partial x}\right)\left(\frac{\partial U_{j}}{\partial x}\right) dx d\tau\geqslant C_{1}\sum\limits_{i=1}^{N}
\int\limits_{0}^{t}\int\limits_{0}^{1}\left(\frac{\partial U_{i}}{\partial x}\right)^{2}dx d\tau.
\end{equation}
For the first term on the right-hand side of \eqref{mamprok.ravlema23} we obtain the relation
\begin{multline}\label{mamprok.ravlema26}
K\sum\limits_{i=1}^{N}\alpha_{i}\int\limits_{0}^{t}\int\limits_{0}^{1}\rho\left(\frac{\partial U_{i}}{\partial x}\right) dx d\tau\leqslant
\frac{C_{1}}{2}\sum\limits_{i=1}^{N} \int\limits_{0}^{t}\int\limits_{0}^{1}\left(\frac{\partial U_{i}}{\partial x}\right)^{2}dx d\tau+ \\
+C_{12}\left(C_{1}, C_{11}, K, N, t^{m}\right)\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}u_{i}^{2}\, dx d\tau.
\end{multline}
For the second term on the right-hand side of \eqref{mamprok.ravlema23} the inequality
\begin{multline*}
-\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}\rho_{i}U_{i}\left(\frac{\partial U_{i}^{(2)}}{\partial \tau }\right)dx d\tau\leqslant\\ \leqslant\frac{\varepsilon}{2}\sum\limits_{i=1}^{N}\sup\limits_{[0,t]\times[0,1]}U_{i}^{2}
\int\limits_{0}^{t}\int\limits_{0}^{1}\left(\frac{\partial U_{i}^{(2)}}{\partial \tau}\right)^{2}dxd\tau
+\frac{C_{11} N t^{m}}{2\varepsilon}\;\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}u_{i}^{2}\, dx d\tau\leqslant\\
\leqslant
\varepsilon mC_{8}^{2}\;\sum\limits_{i=1}^{N}
\sup\limits_{[0,t]}\int\limits_{0}^{1}U_{i}^{2}\, dx+\frac{C_{11} Nt^{m}}{2\varepsilon}\;\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}u_{i}^{2}\, dx d\tau
\end{multline*}
holds, and taking
$$
\varepsilon mC_{8}^{2}=\frac{e^{-\pi m^{2} c N t^{m}}\min\limits_{1\leqslant i\leqslant N}\inf\limits_{[0, 1]}\rho_{0i}}{4N},
$$
we obtain that
\begin{multline}\label{mamprok.ravlema28}
-\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}\rho_{i}U_{i}\left(\frac{\partial U_{i}^{(2)}}{\partial \tau }\right)dx d\tau\leqslant \frac{e^{-\pi m^{2} c N t^{m}}\min\limits_{1\leqslant i\leqslant N}\inf\limits_{[0, 1]}\rho_{0i}}{4N}\sum\limits_{i=1}^{N}
\sup\limits_{[0,t]}\int\limits_{0}^{1}U_{i}^{2}\,dx+\\
+ C_{13}\left(C_{8}, C_{11}, \left\{\inf\limits_{[0, 1]}\rho_{0i}\right\}_{i=1}^{N}, N, c, m, t^{m}\right)\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}u_{i}^{2}\, dx d\tau.
\end{multline}
The third term on the right-hand side of \eqref{mamprok.ravlema23} can be estimated as follows
\begin{multline}\label{mamprok.ravlema29}
-\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}\rho_{i}^{(1)}vU_{i}\Bigg(\frac{\partial U_{i}^{(2)}}{\partial x}\Bigg)dx d\tau\leqslant\\ \leqslant C_{14}\left(\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}u_{i}^{2}\, dx d\tau+\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}U_{i}^{2}\, dx d\tau\right),
\end{multline}
where ${\displaystyle C_{14}=C_{14}\bigg(\Big\{\sup\limits_{[0, 1]}\rho_{0i}\Big\}_{i=1}^{N}, N, c, m,  t^{m}\bigg)}$. Finally, the last term on the right-hand side of \eqref{mamprok.ravlema23} satisfies the relation
\begin{multline}\label{mamprok.ravlema30} -
\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}\rho_{i}v^{(2)}U_{i}\Bigg(\frac{\partial U_{i}^{(2)}}{\partial x}\Bigg)dx d\tau\leqslant\\ \leqslant C_{15}\left(\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}u_{i}^{2}\, dx d\tau+\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}U_{i}^{2}\, dx d\tau\right),
\end{multline}
where ${C_{15}=C_{15}\left(C_{11}, N, c, m, t^{m}\right)}$. Thus, from \eqref{mamprok.ravlema23}, using \eqref{mamprok.ravlema24}--\eqref{mamprok.ravlema30}, the inequality
$$\sum\limits_{i=1}^{N}\int\limits^{1}_{0}U_{i}^{2}\,dx\leqslant C_{16}\left(\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}u_{i}^{2}\, dx d\tau+\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}U_{i}^{2}\, dx d\tau\right)$$
follows, where ${\displaystyle C_{16}=C_{16}\bigg(C_{12},\ldots,C_{15}, \Big\{\inf\limits_{[0, 1]}\rho_{0i}\Big\}_{i=1}^{N}, N, c, m,  t^{m}\bigg)}$,
which, in view of the Gronwall inequality, implies the estimate
$$
\sum\limits_{i=1}^{N}\int\limits^{1}_{0}U_{i}^{2}\,dx\leqslant C_{17}(C_{16},  t^{m})\sum\limits_{i=1}^{N}\int\limits_{0}^{t^{m}}\int\limits_{0}^{1}u_{i}^{2}\, dx dt,
$$
and, consequently,
$$
\|\boldsymbol{\psi}^{(1)}-\boldsymbol{\psi}^{(2)}\|_{(C[0,t^{m}])^{mN}}\leqslant C_{18}(C_{17},  t^{m})\|\boldsymbol{\xi}^{(1)}-\boldsymbol{\xi}^{(2)}\|_{(C[0,  t^{m}])^{mN}}.
$$
The last inequality justifies the continuity of the operator $A$.

Since the operator $A$ satisfies the assumptions of the Schauder theorem listed above, in $V$ there exists a fixed point $\boldsymbol{\xi}$ of the operator $A$ which, together with
the corresponding functions $\rho_{i}$, ${i=1,\ldots,N}$, is a solution to  problem \eqref{mamprok_ravlema3}--\eqref{mamprok_ravlema2}. Theorem 2 is proved.

\section{Uniform estimates of Galerkin approximations}

Let us obtain estimates of solutions to the approximate initial-boundary value problem \eqref{mamprok_ravlema3}--\eqref{mamprok_ravlema2},
which would be uniform with respect to the pa\-ra\-me\-ter $m$ and would allow us to pass to the limit as $m\rightarrow \infty$. Let us denote
\begin{equation}\label{mamprok.lema313}
\alpha(t)=\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits^{1}_{0}\left(\rho_{i}\left(\frac{\partial u_{i}}{\partial t}\right)^{2}+\left(\frac{\partial^{2} u_{i}}{\partial x^{2}}\right)^{2}\right)dxd\tau,\quad \alpha^{\prime}(t)\geqslant 0.
\end{equation}
Equations \eqref{mamprok_ravlema3} imply inequalities $\eqref{mamprok_ravlema6}$ which, in turn, imply the estimates
\begin{equation}\label{mamprok.lema314}
C_{19}^{-1}e^{-C_{19}\alpha(t)}\leqslant\rho_{i}(x,t)\leqslant C_{19}e^{C_{19}\alpha(t)},\quad i=1,\ldots,N,
\end{equation}
where ${C_{19}=C_{19}\bigg(\Big\{\sup\limits_{[0, 1]}\rho_{0i}\Big\}_{i=1}^{N},\Big\{\inf\limits_{[0, 1]}\rho_{0i}\Big\}_{i=1}^{N}, N, T\bigg)}$.
We note that \eqref{mamprok_ravlema3} imply the equalities
\begin{equation}\label{mamprok.lema315}
\rho_{i}\frac{\partial^{2}}{\partial t \partial x}\bigg(\frac{1}{\rho_{i}}\bigg)+\rho_{i}v\frac{\partial^{2}}{\partial x^{2}}\bigg(\frac{1}{\rho_{i}}\bigg)=\frac{\partial^{2}v}{\partial x^{2}},\quad i=1,\ldots,N,\end{equation}
which yield
\begin{align}
&\frac{d}{dt}\left(\int\limits^{1}_{0}\rho_{i}\bigg(\frac{\partial}{\partial x}\bigg(\frac{1}{\rho_{i}}\bigg)\bigg)^{2}dx\right)=2\int\limits^{1}_{0}\bigg(\frac{\partial}{\partial x}\bigg(\frac{1}{\rho_{i}}\bigg)\bigg)\bigg(\frac{\partial^{2}v}{\partial x^{2}}\bigg) dx, \label{mamprok.lema317}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad i=1,\ldots,N.\nonumber
\end{align}
From \eqref{mamprok.lema317}, the relations
\begin{multline}\label{mamprok.lema319}
\int\limits^{1}_{0}\rho_{i}\bigg(\frac{\partial}{\partial x}\bigg(\frac{1}{\rho_{i}}\bigg)\bigg)^{2}dx\leqslant \int\limits^{1}_{0}\rho_{0i}\bigg(\bigg(\frac{1}{\rho_{0i}}\bigg)^{\prime}\bigg)^{2}dx+ \\
+\int\limits_{0}^{t}\int\limits^{1}_{0}\left(\rho_{i}\bigg(\frac{\partial}{\partial x}\bigg(\frac{1}{\rho_{i}}\bigg)\bigg)^{2}+\frac{1}{\rho_{i}}\left(\frac{\partial^{2} v}{\partial x^{2}}\right)^{2}\right) dxd\tau,\quad i=1,\ldots,N
\end{multline}
follow.

{\bfseries Remark 1.} {\it In order to obtain \eqref{mamprok.lema319}, we need $($in \eqref{mamprok.lema315}, \eqref{mamprok.lema317}$)$ an additional smoothness of $\rho_{i}$, $i=1,\ldots,N$ in comparison with \eqref{mamprok_1206172}, however the for\-mu\-la\-tion of  relations \eqref{mamprok.lema319} does not require any additional regularity. This means that \eqref{mamprok.lema319} can be obtained via regularization of $\rho_{0i}$, $i=1,\ldots,N$, derivation of~\eqref{mamprok.lema319} for the solutions of the corresponding problems, and then the limit via the regularization parameter. The derivation of relations~\eqref{mamprok.differ} should be understood in a similar way.}

Using \eqref{mamprok.lema313}, \eqref{mamprok.lema314} and the Gronwall inequality, from \eqref{mamprok.lema319} we obtain the estimates
\begin{equation}\label{mamprok.lema321}
\int\limits^{1}_{0}\bigg(\frac{\partial}{\partial x}\bigg(\frac{1}{\rho_{i}}\bigg)\bigg)^{2}dx+\int\limits^{1}_{0}\left(\frac{\partial\rho_{i}}{\partial x}\right)^{2}dx\leqslant C_{20}e^{C_{20}\alpha(t)},\quad i=1,\ldots,N,
\end{equation}
where ${\displaystyle C_{20}=C_{20}\bigg(C_{19}, \Big\{\|\rho_{0i}\|_{W^{1}_{2}(0, 1)}\Big\}_{i=1}^{N}, \Big\{\inf\limits_{[0, 1]}\rho_{0i}\Big\}_{i=1}^{N}, N, T\bigg)}$.
Further, mul\-ti\-ply\-ing \eqref{mamprok_ravlema5} by $\psi_{ik}^{\prime}+\pi^{2}k^{2}\psi_{ik}$, summarizing with respect to $i$, $k$, and taking into account \eqref{mamprok_ravlema3}, \eqref{mamprok_ravlema2}, we obtain the relation (here, we also use the symmetry of the matrix $\textbf{N}$)
\begin{multline}\label{mamprok.lema323}
\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial u_{i}}{\partial t}\right)^{2}dx+\sum\limits_{i, j=1}^{N}\nu_{ij}\int\limits^{1}_{0}\left(\frac{\partial^{2} u_{i}}{\partial x^{2}}\right)\left(\frac{\partial^{2} u_{j}}{\partial x^{2}}\right)dx+\displaybreak\\ +\frac{1}{2}\frac{d}{dt}\left(\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial u_{i}}{\partial x}\right)^{2}dx\right) +\frac{1}{2}\frac{d}{dt}\left(\sum\limits_{i,j=1}^{N}\nu_{ij}\int\limits^{1}_{0}\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial u_{j}}{\partial x}\right)dx\right)=\\
=-\sum\limits_{i=1}^{N}\int\limits^{1}_{0}
\rho_{i}v\left(\frac{\partial u_{i}}{\partial t}\right)\left(\frac{\partial u_{i}}{\partial x}\right)dx -K\int\limits^{1}_{0}\left(\frac{\partial \rho}{\partial x}\right)\left(\frac{\partial v}{\partial t}\right)dx+\\
+K\int\limits^{1}_{0}\left(\frac{\partial \rho}{\partial x}\right)\left(\frac{\partial^{2} v}{\partial x^{2}}\right)dx-
\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\left(\frac{\partial \rho_{i}}{\partial x}\right)\left(\frac{\partial u_{i}}{\partial t}\right)\left(\frac{\partial u_{i}}{\partial x}\right)dx+\\
+2\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}v\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial^{2} u_{i}}{\partial x^{2}}\right)dx.
\end{multline}
The left-hand side of \eqref{mamprok.lema323} satisfies the estimate (since $\textbf{N}>0$)
\begin{multline}\label{mamprok.lema324}
\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial u_{i}}{\partial t}\right)^{2}dx+\sum\limits_{i, j=1}^{N}\nu_{ij}\int\limits^{1}_{0}\left(\frac{\partial^{2} u_{i}}{\partial x^{2}}\right)\left(\frac{\partial^{2} u_{j}}{\partial x^{2}}\right)dx+\\ +\frac{1}{2}\frac{d}{dt}\left(\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial u_{i}}{\partial x}\right)^{2}dx\right) +\frac{1}{2}\frac{d}{dt}\left(\sum\limits_{i,j=1}^{N}\nu_{ij}\int\limits^{1}_{0}\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial u_{j}}{\partial x}\right)dx\right)\geqslant\\
\geqslant C_{21}(C_{1})\alpha^{\prime}(t)+\beta^{\prime}(t),
\end{multline}
where
$$\beta(t)=\frac{1}{2}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial u_{i}}{\partial x}\right)^{2}dx+
\frac{1}{2}\sum\limits_{i, j=1}^{N}\nu_{ij}\int\limits^{1}_{0}\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial u_{j}}{\partial x}\right)dx.$$
We separately consider each term on the right-hand side of  \eqref{mamprok.lema323}. For the first term on the right-hand side of \eqref{mamprok.lema323} we have
\begin{multline}\label{mamprok.lema325}
-\sum\limits_{i=1}^{N}\int\limits^{1}_{0}
\rho_{i}v\left(\frac{\partial u_{i}}{\partial t}\right)\left(\frac{\partial u_{i}}{\partial x}\right)dx\leqslant\\
\leqslant \sum\limits_{i=1}^{N}\sup\limits_{[0, 1]}|v|\left(\int\limits_{0}^{1}\rho_{i}\left(\frac{\partial u_{i}}{\partial t}\right)^{2}dx\right)^{1/2}\left(\int\limits_{0}^{1}\rho_{i}\left(\frac{\partial u_{i}}{\partial x}\right)^{2}dx\right)^{1/2}\leqslant \\
\leqslant \frac{C_{21}}{10}\alpha^{\prime}(t)+C_{22}\,\beta^{2}(t)e^{C_{22}\alpha(t)},
\end{multline}
where $C_{22}=C_{22}(C_{19}, C_{21}, N)$. For the second and third terms on the right-hand side of \eqref{mamprok.lema323} we obtain
\begin{align}
-K\int\limits^{1}_{0}\left(\frac{\partial \rho}{\partial x}\right)\left(\frac{\partial v}{\partial t}\right)dx&\leqslant \frac{C_{21}}{10}\alpha^{\prime}(t)+C_{23}e^{C_{23}\alpha(t)},\label{mamprok.lema326}\\
K\int\limits^{1}_{0}\left(\frac{\partial \rho}{\partial x}\right)\left(\frac{\partial^{2} v}{\partial x^{2}}\right)dx
&\leqslant \frac{C_{21}}{10}\alpha^{\prime}(t)+C_{24}e^{C_{24}\alpha(t)}\label{mamprok.lema327}
\end{align}
respectively, where $C_{23}=C_{23}(C_{19}, C_{20}, C_{21}, K, N)$, $C_{24}=C_{24}(C_{20}, C_{21}, K, N)$. For the fourth term on the right-hand side of \eqref{mamprok.lema323},
using the interpolation estimate
$$
\sup\limits_{[0, 1]}\left|\frac{\partial u_{i}}{\partial x}\right|\leqslant \sqrt{2}\left(\int\limits^{1}_{0}\left(\frac{\partial u_{i}}{\partial x}\right)^{2}dx\right)^{1/2}\left(\int\limits^{1}_{0}\left(\frac{\partial^{2} u_{i}}{\partial x^{2}}\right)^{2}dx\right)^{1/2},
$$
we deduce
\begin{multline}\label{mamprok.lema329}
-\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\left(\frac{\partial \rho_{i}}{\partial x}\right)\left(\frac{\partial u_{i}}{\partial t}\right)\left(\frac{\partial u_{i}}{\partial x}\right)dx\leqslant\\
\qquad\leqslant\sum\limits_{i=1}^{N}
\sup\limits_{[0, 1]}\left|\frac{\partial u_{i}}{\partial x}\right|\left(\int\limits^{1}_{0}\frac{1}{\rho_{i}}\left(\frac{\partial \rho_{i}}{\partial x}\right)^{2}dx\right)^{1/2}\left(\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial u_{i}}{\partial t}\right)^{2}dx\right)^{1/2}\leqslant\\
\leqslant \frac{C_{21}}{10}\alpha^{\prime}(t)+C_{25}\,{\beta(t)}e^{C_{25}\alpha(t)}, \quad C_{25}=C_{25}(C_{19}, C_{20}, C_{21}, N).
\end{multline}
Finally, for the last term on the right-hand side of \eqref{mamprok.lema323} we obtain the relation
\begin{multline}\label{mamprok.lema328}
2\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}v\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial^{2} u_{i}}{\partial x^{2}}\right)dx\leqslant\\
\leqslant 2\sum\limits_{i=1}^{N}\sup\limits_{[0, 1]}\left|v\right|\sup\limits_{[0, 1]}\sqrt{\rho_{i}}\left(\int\limits^{1}_{0}\rho_{i}\left(\frac{\partial u_{i}}{\partial x}\right)^{2}dx\right)^{1/2}\left(\int\limits^{1}_{0}\left(\frac{\partial^{2} u_{i}}{\partial x^{2}}\right)^{2}dx\right)^{1/2}\leqslant\\
\leqslant \frac{C_{21}}{10}\alpha^{\prime}(t)+C_{26}\,\beta^{2}(t)e^{C_{26}\alpha(t)}, \quad C_{26}=C_{26}(C_{19}, C_{21}, N).
\end{multline}
Thus, from \eqref{mamprok.lema325}--\eqref{mamprok.lema329} it follows that the right-hand side of \eqref{mamprok.lema323}
\begin{multline}\label{mamprok.lema330}
-\sum\limits_{i=1}^{N}\int\limits^{1}_{0}
\rho_{i}v\left(\frac{\partial u_{i}}{\partial t}\right)\left(\frac{\partial u_{i}}{\partial x}\right)dx -K\int\limits^{1}_{0}\left(\frac{\partial \rho}{\partial x}\right)\left(\frac{\partial v}{\partial t}\right)dx+\displaybreak\\
+K\int\limits^{1}_{0}\left(\frac{\partial \rho}{\partial x}\right)\left(\frac{\partial^{2} v}{\partial x^{2}}\right)dx-
\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\left(\frac{\partial \rho_{i}}{\partial x}\right)\left(\frac{\partial u_{i}}{\partial t}\right)\left(\frac{\partial u_{i}}{\partial x}\right)dx+\\
+2\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}v\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial^{2} u_{i}}{\partial x^{2}}\right)dx\leqslant\\ \leqslant \frac{C_{21}}{2}\alpha^{\prime}(t)+C_{27}(1+\beta^{2}(t))e^{C_{27}\alpha(t)}, \quad C_{27}=C_{27}(C_{22},\ldots, C_{26}).
\end{multline}
Combining relations \eqref{mamprok.lema324} and \eqref{mamprok.lema330}, from \eqref{mamprok.lema323} we obtain the inequality
\begin{equation}\label{mamprok.lema331}
\left(\frac{C_{21}}{2}\alpha(t)+\beta(t)\right)^{\prime}\leqslant C_{28}e^{C_{28}\left(C_{21}\frac{\alpha(t)}{2}+\beta(t)\right)},\quad C_{28}=C_{28}(C_{21}, C_{27}).
\end{equation}
We take any $C_{29}>\beta(0)$, for example, $C_{29}=2\beta(0)$. Then for
\begin{equation}\label{mamprok.lema3311}t_0=\min\left(T, \frac{e^{-C_{28}\beta(0)}-e^{-C_{28}C_{29}}}{C_{28}^2}\right)\end{equation}
we derive from \eqref{mamprok.lema331} that the estimate
$$\sup\limits_{0\leqslant t\leqslant t_{0}}(\alpha+\beta)\leqslant \left(1+\frac{2}{C_{21}}\right)C_{30},\quad C_{30}=
\frac{1}{C_{28}}\ln\left(\frac{1}{e^{-C_{28}\beta(0)}-C_{28}^2 t_0}\right)$$
holds, which together with \eqref{mamprok_ravlema3}, \eqref{mamprok.lema314} and \eqref{mamprok.lema321} yields that
\begin{multline}\label{mamprok.lema312}
\sum\limits_{i=1}^{N}\left(\vphantom{\left\|\frac{\partial}{\partial}\right\|_{L_2}}\|\rho_{i}\|_{L_{\infty}\big(0, t_{0}; W^{1}_{2}(0, 1)\big)}+\|u_{i}\|_{L_{\infty}\big(0, t_{0}; W^{1}_{2}(0, 1)\big)}+\|u_{i}\|_{L_{2}\big(0, t_{0}; W^{2}_{2}(0, 1)\big)}+\right.\\ \left.
+\left\|\frac{\partial \rho_{i}}{\partial t}\right\|_{L_{\infty}\big(0, t_{0}; L_{2}(0, 1)\big)}+\left\|\frac{\partial u_{i}}{\partial t}\right\|_{L_{2}(Q_{t_{0}})}+\left\|\frac{1}{\rho_{i}}\right\|_{L_{\infty}(Q_{t_{0}})}\right)\leqslant C_{31},
\end{multline}
where ${Q_{t_{0}}=(0, t_{0})\times(0,1)}$, ${C_{31}=C_{31}\left(C_{19}, C_{20}, C_{21},C_{30}, N\right)}$.

\section{Convergence of Galerkin approximations}

We have constructed the solutions $(\rho_{1}^{m},\ldots,\rho_{N}^{m}, {u}_{1}^{m}, \ldots, u_{N}^{m})$ to  prob\-lems \eqref{mamprok_ravlema3}--\eqref{mamprok_ravlema2} for all
$m\in{\mathbb N}$, and now extend them, if necessary, for $(0,t_0)$, and we can use estimate~\eqref{mamprok.lema312} for them. Based on this estimate, we can extract a subsequence from the mentioned sequence (keeping the same notation, below this procedure is implied as necessary) such that
$$\rho_{i}^{m}\rightarrow\rho_i\;\;\text{weakly* in}\;\;L_{\infty}\big(0,  t_{0}; W^{1}_{2}(0, 1)\big),$$
$$u_{i}^{m}\rightarrow u_{i}\;\;\text{weakly* in}\;\;L_{\infty}\big(0,  t_{0}; W_{2}^{1}(0, 1)\big)\;\;\text{and weakly in}\; L_{2}\big(0,t_{0}; W_{2}^{2}(0, 1)\big)$$
as $m\rightarrow \infty$ for all $i=1, \ldots, N$. The other properties listed in \eqref{mamprok_oprresh} are also satisfied by this sequence in $Q_{t_0}$ uniformly with respect to $m$. Consequently, the limit functions belong to the corresponding classes. We show that\linebreak $(\rho_1,\ldots,  \rho_{N}, u_{1},\ldots, u_{N})$
is a strong solution to  problem~\eqref{mamprok_continuity}--\eqref{mamprok_boundvelocity} on $(0, t_{0})$.

By the Arzela---Ascoli theorem (see~\cite[Theorem\,1.70, P.\,58]{NovotnyStraskraba:2004}) and the uniform estimates for $\rho_{i}^{m}$, $u_{i}^{m}$ in
$L_{\infty}\big(0,t_{0}; W^{1}_{2}(0, 1)\big)$ and for $\displaystyle \frac{\partial \rho_{i}^{m}}{\partial t}$, $\displaystyle \frac{\partial u_{i}^{m}}{\partial t}$ in $L_{2}(Q_{ t_{0}})$ (cf. \eqref{mamprok.lema312}) we have
\begin{align}
&\rho_{i}^{m}\rightarrow\rho_{i}\;\;\text{as}\;\; m\rightarrow\infty\;\;\text{strongly in}\;\;C\big([0,  t_{0}]; L_{2}(0, 1)\big),\quad i=1,\ldots,N,\label{mamprok.paragraf45}\\
&u_{i}^{m}\rightarrow u_{i}\;\;\text{as}\;\;m\rightarrow\infty\;\;\text{strongly in}\;\;C\big([0,  t_{0}];L_{2}(0, 1)\big),\quad i=1,\ldots,N.\label{mamprok.paragraf46}
\end{align}
Since $\displaystyle \frac{\partial u_{i}^{m}}{\partial t}$ are bounded in $L_{2}(Q_{ t_{0}})$,
an uniform estimate for $\displaystyle \frac{\partial^{2} u_{i}^{m}}{\partial t \partial x}$ in\linebreak
$L_{2}(0,  t_{0}; W^{-1}_{2}(0,1))$ is valid, which, together with the estimate for  $\displaystyle \frac{\partial u_{i}^{m}}{\partial x}$ in $L_{2}(0,  t_{0}; W^{1}_{2}(0, 1))$ means via Lions--Aubin’s lemma (see~\cite[Theorem\,1.71, P.\,59]{NovotnyStraskraba:2004}) that
\begin{equation}\label{mamprok.paragraf47}
\frac{\partial u_{i}^{m}}{\partial x}\rightarrow \frac{\partial u_{i}}{\partial x}\;\;\text{as}\;\; m\rightarrow\infty\;\;\text{strongly in}\;\; L_{2}(Q_{ t_{0}}),\quad i=1,\ldots,N.
\end{equation}
Hence \eqref{mamprok.paragraf46} leads to the relations
\begin{equation}\label{mamprok.paragraf48}u_{i}^{m}\rightarrow u_{i}\;\;\text{as}\;\;m\rightarrow\infty\;\;\text{strongly in}\;\;L_{2}\big(0,  t_{0}; C[0,1]\big),\quad i=1,\ldots,N.
\end{equation}
Thus, the limit functions $\rho_{i}$, $u_{i}$, ${i=1,\ldots,N}$, satisfy (almost everywhere in $Q_{ t_{0}}$) the continuity equations~\eqref{mamprok_continuity}, in which
${\displaystyle v=\sum\limits_{j=1}^{N}\alpha_{j}u_{j}}$,
the initial data~\eqref{mamprok_nachusl} for almost all $x\in (0, 1)$
and the boundary conditions~\eqref{mamprok_boundvelocity} for almost all $t\in (0, t_{0})$.

The boundedness of $\displaystyle \frac{\partial u_{i}^{m}}{\partial t}$ in $L_{2}(Q_{ t_{0}})$ implies the weak convergence of $\displaystyle \frac{\partial u_{i}^{m}}{\partial t}$
to $\displaystyle \frac{\partial u_{i}}{\partial t}$ in $L_{2}(Q_{ t_{0}})$, which, together with \eqref{mamprok.paragraf45} and the boundedness of
$\displaystyle \rho_{i}^{m}\frac{\partial u_{i}^{m}}{\partial t}$ in $L_{2}(Q_{ t_{0}})$ implies
$$
\rho_{i}^{m}\frac{\partial u_{i}^{m}}{\partial t}\rightarrow \rho_{i}\frac{\partial u_{i}}{\partial t}\;\;\text{as}\;\; m\rightarrow\infty\;\;\text{weakly in}\;\; L_{2}(Q_{ t_{0}}),\quad i=1,\ldots,N.
$$
Further, from \eqref{mamprok.paragraf45} and \eqref{mamprok.paragraf47} it follows that
$$
\rho_{i}^{m}\frac{\partial u_{i}^{m}}{\partial x}\rightarrow\rho_{i}\frac{\partial u_{i}}{\partial x}\;\;\text{as}\;\;m\rightarrow\infty\;\;\text{strongly in}\;\;L_{2}\big(0,  t_{0};L_{1}(0, 1)\big),\;\; i=1,\ldots,N,$$
and hence \eqref{mamprok.paragraf48} yields that the convergences
\begin{equation}\label{mamprok.paragraf50}
\left(\rho_{i}^{m}\frac{\partial u_{i}^{m}}{\partial x}\right)u_{j}^{m}\rightarrow \left(\rho_{i}\frac{\partial u_{i}}{\partial x}\right)u_{j}\;\;\text{as}\;\; m\rightarrow\infty\;\;\text{strongly in}\;\;L_{1}(Q_{ t_{0}})
\end{equation}
are valid for all $i, j=1,\ldots,N$.

By \eqref{mamprok_ravlema5}, for any functions of the form $(i=1,\ldots,N)$
\begin{align}\label{mamprok.paragraf51}
\varphi_{i}=\sum\limits_{k=1}^{M}\eta_{ik}(t)\sin(\pi k x),\quad \eta_{ik}\in C[0,  t_{0}],\quad k=1,\ldots,M,\quad M\leqslant m,
\end{align}
we have the equalities
\begin{align*}
&\int\limits_{0}^{t_{0}}\int\limits_{0}^{1}
\bigg(\rho_{i}^{m}\frac{\partial u_{i}^{m}}{\partial t}+\rho_{i}^{m}v^{m}\frac{\partial u_{i}^{m}}{\partial x}+\alpha_{i}K\frac{\partial \rho^{m}}{\partial x}-\sum\limits_{j=1}^N \nu_{ij}\frac{\partial^{2} u_{j}^{m}}{\partial x^{2}}\bigg)\varphi_{i}\, dxdt=0,\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad i=1,\ldots,N,
\end{align*}
passing in which to the limit as ${m\rightarrow \infty}$ (by the proved convergences), we find (since the set of functions $\varphi_{i}$, ${i=1,\ldots,N}$ of  form~\eqref{mamprok.paragraf51} is everywhere dense in $L_{2}(Q_{ t_{0}})$) that the momentum equations \eqref{mamprok_momentum} hold for the limit functions $\rho_{i}$, $u_{i}$, $i=1,\ldots,N$ almost everywhere $Q_{ t_{0}}$, with $\displaystyle \rho=\sum\limits_{i=1}^{N}\rho_{i}$.

Thus, we have proved the existence of a solution to the initial-boundary value problem \eqref{mamprok_continuity}--\eqref{mamprok_boundvelocity} in small time.
In order to continue the local solution defined on the interval $(0, t_{0})$ into the entire target interval $(0, T)$, it is necessary to obtain a priori estimates for this local solution which contain constants depending on the input data of the problem and on the value $T$, but not on the parameter $t_{0}$ (see, for example, \cite[P.\,40]{AKM:1983}).

\section{Global a priori estimates}

During the further study of the unique solvability of problem \eqref{mamprok_continuity}--\eqref{mamprok_boundvelocity}, the use of the Lagrangian mass coordinates is convenient.
Let us accept $t$ and ${\displaystyle y(t, x)=\int\limits_{0}^{x}\rho(t, s)\,ds}$ as new independent variables. Then system~\eqref{mamprok_continuity}, \eqref{mamprok_momentum} takes the form
\begin{align}
&\qquad\qquad \frac{\partial\widetilde{\rho}_{i}}{\partial t}+\widetilde{\rho}\widetilde{\rho}_{i}\frac{\partial \widetilde{v}}{\partial y}=0,\quad i=1,\ldots,N, \quad \widetilde{v}=\sum\limits_{j=1}^{N}\alpha_{j}\widetilde{u}_{j},\label{1newcontinuity1lagr}\\
&
\frac{\widetilde{\rho}_{i}}{\widetilde{\rho}}\frac{\partial \widetilde{u}_{i}}{\partial t}+\alpha_{i}K\frac{\partial \widetilde{\rho}}{\partial y}=
\sum\limits_{j=1}^N \nu_{ij}\frac{\partial}{\partial y}\left(\widetilde{\rho}\frac{\partial \widetilde{u}_{j}}{\partial y}\right),\quad i=1,\ldots,N,\quad \widetilde{\rho}=\sum\limits_{j=1}^{N}\widetilde{\rho_{j}}.\label{1newmomentum1lagr}
\end{align}
The domain $Q_{T}$ is transformed into the rectangular ${\Pi_{T}=(0, T)\times(0, d)}$, where ${\displaystyle d=\int\limits_{0}^{1}\rho_{0}\,dx>0}$, ${\displaystyle \rho_{0}=\sum\limits_{j=1}^{N}\rho_{0j}}$, and the initial and boundary conditions  take the form
\begin{align}
&\widetilde{\rho}_{i}|_{t=0}=\widetilde{\rho}_{0i}(y), \quad \widetilde{u}_{i}|_{t=0}=\widetilde{u}_{0i}(y),\quad y\in [0, d],\quad i=1,\ldots,N,\label{1nachusl1lagr}\\
&\qquad\quad
\widetilde{u}_{i}|_{y=0}=\widetilde{u}_{i}|_{y=d}=0,\quad t\in [0, T],\quad i=1,\ldots,N.\label{1boundvelocity1lagr}
\end{align}

Let us construct a priori estimates. First of all, we note that the summation of \eqref{1newcontinuity1lagr} with respect to $i$ gives
\begin{align}\label{1602171}
\frac{\partial\widetilde{\rho}}{\partial t}+\widetilde{\rho}^{2}\frac{\partial \widetilde{v}}{\partial y}=0,
\end{align}
and hence
$$
\frac{\partial}{\partial t}\left(\frac{\widetilde{\rho}_{i}}{\widetilde{\rho}}\right)=0, \quad i=1,\ldots,N.
$$
Hence, due to \eqref{1nachusl1lagr} we get the equalities
\begin{align}\label{1602173}
\frac{\widetilde{\rho}_{i}(t,y)}{\widetilde{\rho}(t,y)}=\frac{\widetilde{\rho}_{0i}(y)}{\widetilde{\rho}_{0}(y)}\quad\text{as}\quad(t,y)\in[0, T]\times[0, d]
\end{align}
for all $i=1,\ldots,N$, where $\displaystyle \widetilde{\rho}_{0}=\sum\limits_{j=1}^{N}\widetilde{\rho}_{0j}$.
In the Eulerian coordinates the ratios ${\rho_{i}/\rho}$, ${i=1,\ldots,N}$, satisfy the transport equations, and we only have the inequalities
\begin{align}\label{2602171}
0<\inf\limits_{[0,1]}\frac{\rho_{0i}}{\rho_{0}}\leqslant\frac{\rho_{i}(t,x)}{\rho(t,x)}\leqslant\sup\limits_{[0,1]}\frac{\rho_{0i}}{\rho_{0}}\leqslant 1
\quad\text{for}\quad(t,x)\in[0, T]\times[0, 1].
\end{align}

Let us multiply  equations~\eqref{mamprok_momentum} by $ u_{i}$, integrate over $x$ and sum with respect to $i$.
In view of~\eqref{mamprok_continuity}, \eqref{mamprok_boundvelocity} and the condition $\textbf{N}>0$, the following relations hold
\begin{align}
&\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\Big(\rho_{i}\frac{\partial u_{i}}{\partial t}+\rho_{i} v\frac{\partial u_{i}}{\partial x}\Big)u_{i}\,dx= \frac{1}{2}\frac{d}{dt}\left(\sum\limits_{i=1}^{N}
\int\limits^{1}_{0}\rho_{i} u_{i}^{2}\, dx\right),\nonumber\\
&
\sum\limits_{i=1}^{N}\alpha_{i}K\int\limits^{1}_{0}u_{i}\frac{\partial\rho}{\partial x}\,dx
=-K\int\limits^{1}_{0}\rho\frac{\partial v}{\partial x}\, dx=
K\frac{d}{dt}\int\limits^{1}_{0}\big(\rho\ln{\rho}-(\ln{d}+1)\rho+d\big)dx,\nonumber\\
&
\sum\limits_{i,j=1}^N \nu_{ij}\int\limits\limits_{0}^{1}\left(\frac{\partial^{2} u_{j}}{\partial x^{2}}\right)u_{i}\,
dx=-\sum\limits_{i,j=1}^N \nu_{ij}\int\limits^{1}_{0}\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial u_{j}}{\partial x}\right)dx\leqslant\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leqslant-C_{1}\sum\limits_{i=1}^N\int\limits\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)^{2}dx,\label{lemma51.4}
\end{align}
and hence we get the inequality
\begin{multline}\label{19082024_eq1}
\frac{d}{dt}\int\limits^{1}_{0}\left(\frac{1}{2}\sum\limits_{i=1}^{N}\rho_{i} u_{i}^{2}+K\left(\rho\ln{\rho}-(\ln{d}+1)\rho+d\right)\right)dx+\\
+C_{1}\sum\limits_{i=1}^N\int\limits\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)^{2}dx\leqslant 0.
\end{multline}
Integrating~\eqref{19082024_eq1} with respect to $t$, and using \eqref{mamprok_nachusl}, we obtain the bound
\begin{multline*}
\int\limits^{1}_{0}\!\!\left(\frac{1}{2}\sum\limits_{i=1}^{N}\rho_{i} u_{i}^{2}+K\big(\rho\ln{\rho}-(\ln{d}+1)\rho+d\big)\!\right)dx+C_{1}\sum\limits_{i=1}^N\int\limits_{0}^{t}\int\limits\limits_{0}^{1}\!\!\left(\frac{\partial u_{i}}{\partial x}\right)^{2}\!dxd\tau\displaybreak\\
\leqslant \int\limits^{1}_{0}\bigg(\frac{1}{2}\sum\limits_{i=1}^{N}\rho_{0i} u_{0i}^{2}+K\big(\rho_{0}\ln{\rho_{0}}-(\ln{d}+1)\rho_{0}+d\big)\bigg)dx,
\end{multline*}
which, due to \eqref{2602171}, implies the estimate
\begin{multline}\label{190820224_lemma1lagr}
\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho u_{i}^{2}dx+\int\limits^{1}_{0}\big(\rho\ln{\rho}-(\ln{d}+1)\rho+d\big)dx+\\
+\sum\limits_{i=1}^N\int\limits_{0}^{T}\int\limits\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)^{2}dxdt
\leqslant C_{32},
\end{multline}
where ${\displaystyle C_{32}\!=\!C_{32}\bigg(\!C_{1}, \bigg\{\inf\limits_{[0, 1]}\frac{\rho_{0i}}{\rho_{0}}\bigg\}_{i=1}^{N}, \bigg\{\sup\limits_{[0, 1]}\rho_{0i}\bigg\}_{i=1}^{N}, \left\{\|u_{0i}\|_{L_{2}(0, 1)}\right\}_{i=1}^{N}, K, N, d\!\bigg)}$.
Let us rewrite \eqref{190820224_lemma1lagr}, using the Lagrangian mass coordinates, in the form
\begin{multline}\label{lemma1lagr}
\sum\limits_{i=1}^{N}\int\limits^{d}_{0}\widetilde{u}_{i}^{2}dy+\int\limits^{d}_{0}\frac{\widetilde{\rho}\ln{\widetilde{\rho}}-(\ln{d}+1)\widetilde{\rho}+d}{\widetilde{\rho}}\,dy+\\
+\sum\limits_{i=1}^N\int\limits_{0}^{T}\int\limits\limits_{0}^{d}\widetilde{\rho}\left(\frac{\partial \widetilde{u}_{i}}{\partial y}\right)^{2}dydt
\leqslant C_{32}.
\end{multline}
Let us note that, from  estimates \eqref{190820224_lemma1lagr}, in view of \eqref{mamprok_boundvelocity}, the inequality
\begin{align}\label{19082024_eq2}
\sum\limits_{i=1}^{N}\int\limits^{T}_{0}\bigg(\sup\limits_{[0, 1]}|u_{i}|\bigg)^{2}dt\leqslant C_{32}
\end{align}
obviously follows.

Let us rewrite  equations \eqref{mamprok_momentum}, using \eqref{mamprok_continuity}, in the form
\begin{align}
&\sum\limits_{j=1}^N \widetilde{\nu}_{ij}\left(\frac{\partial(\rho_{j}u_{j})}{\partial t}+\frac{\partial(\rho_{j}v u_{j})}{\partial x}\right)+K\left(\sum\limits_{j=1}^N \widetilde{\nu}_{ij}\alpha_{j}\right)\frac{\partial \rho}{\partial x}=\frac{\partial^{2}u_{i}}{\partial x^{2}}, \label{new1602171}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad i=1,\ldots,N,\nonumber
\end{align}
where $\widetilde{\nu}_{ij}$ are the entries of the symmetric matrix ${\widetilde{\textbf{N}}=\textbf{N}^{-1}>0}$. We multi\-ply~\eqref{new1602171} by $\alpha_{i}$ and sum with respect to $i$, and we obtain
\begin{align}\label{new1602172}
\frac{\partial V}{\partial t}=\frac{\partial}{\partial x}\left(\frac{\partial v}{\partial x}-\widetilde{K}\rho-vV\right),
\end{align}
where $\displaystyle V=\sum\limits_{i,j=1}^N \widetilde{\nu}_{ij}\alpha_{i}\rho_{j}u_{j}$,
$\displaystyle \widetilde{K}=K\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}\alpha_{j}>0$.
We denote
\begin{equation}\label{lemma52.1}
\gamma(t, x)=\int\limits_{0}^t\left(\frac{\partial v}{\partial x}-\widetilde{K}\rho- v V\right)d\tau+\int\limits_{0}^xV_{0}\,ds,
\end{equation}
where $\displaystyle V_{0}(x)=V(0, x)=\sum\limits_{i,j=1}^N \widetilde{\nu}_{ij}\alpha_{i}\rho_{0j}u_{0j}$.
In view of \eqref{190820224_lemma1lagr}, we have
\begin{align*}
&\qquad \sup\limits_{[0, T]}\int\limits_{0}^{1}\left|\frac{\partial \gamma}{\partial x}\right|dx=\sup\limits_{[0, T]}\int\limits_{0}^{1}\left|V\right|dx\leqslant C_{33}(C_{32}, \textbf{N}, N, d),\\
&\sup\limits_{[0, T]}\left|\int\limits^{1}_{0}\gamma\,dx\right| \leqslant
C_{34}\left(C_{32}, \{\|\rho_{0i}u_{0i}\|_{L_{1}(0, 1)}\}_{i=1}^{N},\widetilde{\textbf{N}}, \widetilde{K}, N, T, d\right),
\end{align*}
and hence, using Poincar\'e’s inequality (see~\cite[Lemma\,1.43, P.\,44]{NovotnyStraskraba:2004}), we get
$$
\sup\limits_{[0, T]}\int\limits^{1}_{0}|\gamma|\, dx \leqslant  C_{35}(C_{33}, C_{34}),
$$
and we arrive at the boundedness of $\gamma$ in $L_{\infty}\big(0,T;W_{1}^{1}(0, 1)\big)$.
Using this and the fact $W_{1}^{1}(0, 1)\hookrightarrow L_{\infty}(0,1)$, we obtain the estimate
$$
\|\gamma\|_{L_{\infty}(Q_{T})}\leqslant C_{36}\left(C_{33}, C_{35}\right).
$$
Let us note that, in view of \eqref{mamprok_continuity}, \eqref{new1602172} and \eqref{lemma52.1}, the following relations hold
$$
\frac{\partial (\rho e^{\gamma})}{\partial t}+v\frac{\partial (\rho e^{\gamma})}{\partial x}=-\widetilde{K}e^{\gamma}\rho^{2}\leqslant 0,
$$
and hence
$$
\rho(t,x) e^{\gamma(t,x)}\leqslant\bigg(\sup\limits_{[0,1]}{\rho_{0}}\bigg)e^{\int\limits_{0}^{1}|V_{0}|\,dx},
$$
so that the estimate
\begin{equation}\label{supernew1602171}
\rho(t,x)\leqslant C_{37}\quad\text{as}\quad(t,x)\in[0, T]\times[0, 1]
\end{equation}
is valid, where ${\displaystyle C_{37}=C_{37}\bigg(C_{36}, \Big\{\sup\limits_{[0,1]}{\rho_{0i}}\Big\}_{i=1}^{N}, \{\|\rho_{0i}u_{0i}\|_{L_{1}(0, 1)}\}_{i=1}^{N}, \widetilde{\textbf{N}}, N\bigg)}$.

Let us use equations~\eqref{mamprok_continuity}, \eqref{mamprok_momentum} in the form~\eqref{1newcontinuity1lagr}, \eqref{1newmomentum1lagr}. We rewrite  equations~\eqref{1newmomentum1lagr} as
\begin{align}\label{newnewmomentum.double}
\sum\limits_{j=1}^{N}\widetilde{\nu}_{ij}\frac{\widetilde{\rho}_{j}}{\widetilde{\rho}}\frac{\partial \widetilde{u}_{j}}{\partial t}+K\left(\sum\limits_{j=1}^{N}\widetilde{\nu}_{ij}\alpha_{j}\right)\frac{\partial \widetilde{\rho}}{\partial y}=
\frac{\partial}{\partial y}\left(\widetilde{\rho}\frac{\partial \widetilde{u}_{i}}{\partial y}\right),\quad i=1,\ldots,N,
\end{align}
and then multiply \eqref{newnewmomentum.double} by $\alpha_{i}$ and sum with respect to $i$. In view of \eqref{1602173}, we obtain
\begin{align}\label{newnewmomentum1newnew}
\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}\frac{\partial \widetilde{u}_{j}}{\partial t}+\widetilde{K}\frac{\partial \widetilde{\rho}}{\partial y}=
\frac{\partial}{\partial y}\left(\widetilde{\rho}\frac{\partial \widetilde{v}}{\partial y}\right).
\end{align}
We extract from \eqref{1602171} that
\begin{align}\label{eq0102173}
\widetilde{\rho}\frac{\partial \widetilde{v}}{\partial y}=-\frac{\partial\ln\widetilde{\rho}}{\partial t}
\end{align}
and substitute this into \eqref{newnewmomentum1newnew}, then we get
$$
\frac{\partial^{2}\ln\widetilde{\rho}}{\partial t \partial y}+\widetilde{K}\frac{\partial\widetilde{\rho}}{\partial y}=-\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}
\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}\frac{\partial\widetilde{u}_{j}}{\partial t}.
$$
We multiply this equality by $\displaystyle \frac{\partial\ln\widetilde{\rho}}{\partial y}$ and integrate with respect to $y$, then we obtain the relation
\begin{multline}\label{eq0102172}
\frac{1}{2}\frac{d}{dt}\left(\int\limits_{0}^{d}\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)^{2} dy\right)+\widetilde{K}\int\limits_{0}^{d}\widetilde{\rho}\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)^{2}dy=\\
=-\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}\int\limits_{0}^{d}\left(\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}\frac{\partial \widetilde{u}_{j}}{\partial t}\right)\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)dy.
\end{multline}
Let us transform the right-hand side of \eqref{eq0102172} via the integration by parts and using \eqref{eq0102173}:
\begin{multline}\label{eq01021777}
-\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}\int\limits_{0}^{d}\left(\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}\frac{\partial \widetilde{u}_{j}}{\partial t}\right)\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)dy=\\
=-\frac{d}{dt}\left(\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}\int\limits_{0}^{d} \left(\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}u_{j}\right)\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)dy\right)+\\
+\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}\int\limits_{0}^{d} \left(\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}\right)^{\prime}\widetilde{\rho}\,\widetilde{u}_{j}\left(\frac{\partial \widetilde{v}}{\partial y}\right)dy+
\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}\int\limits_{0}^{d}\frac{\widetilde{\rho}_{0j}\widetilde{\rho}}{\widetilde{\rho}_{0}}\left(\frac{\partial \widetilde{u}_{j}}{\partial y}\right)\left(\frac{\partial \widetilde{v}}{\partial y}\right)dy.
\end{multline}
Thus, after integration of~\eqref{eq0102172} with respect to $t$, taking into account~\eqref{supernew1602171} and~\eqref{eq01021777}, we get
\begin{multline*}
\int\limits_{0}^{d}\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)^{2} dy+2\widetilde{K}\int\limits_{0}^{t}\int\limits_{0}^{d}\widetilde{\rho}\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)^{2}dyd\tau
\leqslant\int\limits_{0}^{d}\left((\ln{\widetilde{\rho}_{0}})^{\prime}\right)^{2} dy-\displaybreak\\
-2\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}\int\limits_{0}^{d} \left(\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}u_{j}\right)\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)dy
+2\sum\limits_{i, j=1}^{N}\widetilde{\nu}_{ij}\alpha_{i}\int\limits_{0}^{d} \left(\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}\widetilde{u}_{0j}\right)(\ln{\widetilde{\rho}_{0}})^{\prime}dy+\\
+2\sqrt{C_{37}}\sum\limits_{i, j=1}^{N}|\widetilde{\nu}_{ij}|\int\limits_{0}^{t}\left\|\left(\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}\right)^{\prime}\right\|_{L_{2}(0, d)}\|\widetilde{u}_{j}\|_{L_{\infty}(0, d)}\left\|\sqrt{\widetilde{\rho}}\frac{\partial \widetilde{v}}{\partial y}\right\|_{L_{2}(0, d)}d\tau+\\
+2\sum\limits_{i, j=1}^{N}|\widetilde{\nu}_{ij}|\sup\limits_{[0,d]}\frac{\widetilde{\rho}_{0j}}{\widetilde{\rho}_{0}}\int\limits_{0}^{t}\left\|\widetilde{\rho}\left(\frac{\partial \widetilde{u}_{j}}{\partial y}\right) \left(\frac{\partial \widetilde{v}}{\partial y}\right)\right\|_{L_{1}(0, d)}d\tau.
\end{multline*}
Using estimates \eqref{lemma1lagr}, \eqref{19082024_eq2} and \eqref{supernew1602171}, we derive from this the inequality
\begin{align}\label{eq01021710}
\int\limits_{0}^{d}\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)^{2} dy+\int\limits_{0}^{T}\int\limits_{0}^{d}\widetilde{\rho}\left(\frac{\partial\ln\widetilde{\rho}}{\partial y}\right)^{2}dydt\leqslant C_{38},
\end{align}
where ${\displaystyle C_{38}=C_{38}\bigg(C_{32}, C_{37},  \bigg\{\left\|\frac{\widetilde{\rho}_{0i}}{\widetilde{\rho}_{0}}\right\|_{W^{1}_{2}(0,d)}\bigg\}_{i=1}^{N},
\|\left(\ln{\widetilde{\rho}}_{0}\right)^{\prime}\|_{L_{2}(0,d)}, \widetilde{K}, \widetilde{\textbf{N}}, N}$, \linebreak ${\left\{\|\widetilde{u}_{0i}\|_{L_{2}(0,d)}\right\}_{i=1}^{N}\bigg)}$.

From \eqref{1nachusl1lagr}--\eqref{1602171} it follows obviously that for any $t\in[0,T]$ there exists a point $\delta(t)\in[0, d]$ such that
\begin{align}\label{eq0102176}
\widetilde{\rho}(t, \delta(t))=d.
\end{align}
Hence, we can use the representation
$$
\ln\widetilde{\rho}(t, y)=\ln\widetilde{\rho}(t, \delta(t))+\int\limits_{\delta(t)}^{y}\frac{\partial\ln\widetilde{\rho}(t, s)}{\partial s}\, ds,
$$
from which, via H\"older’s inequality, and using \eqref{eq01021710} and \eqref{eq0102176}, we get
$$
|\ln\widetilde{\rho}(t,y)|\leqslant |\ln{d}|+\sqrt{d}\left\|\frac{\partial\ln\widetilde{\rho}}{\partial y}\right\|_{L_{2}(0,d)}\leqslant C_{39}(C_{38},d).
$$
This leads immediately to
\begin{align}\label{eq01021714}
\widetilde{\rho}(t,y)\geqslant C_{40}(C_{39})\quad\text{as}\quad(t,y)\in[0, T]\times[0, d].
\end{align}
From \eqref{1602173}, \eqref{supernew1602171} and \eqref{eq01021714} we obtain that for all $i=1,\ldots,N$
\begin{align}\label{eq01021714111}
C_{41}\leqslant\widetilde{\rho}_{i}(t,y)\leqslant C_{37}\quad\text{as}\quad(t,y)\in[0, T]\times[0, d],
\end{align}
where ${\displaystyle C_{41}=C_{41}\bigg(C_{40},\bigg\{\inf\limits_{[0,d]}\frac{\widetilde{\rho}_{0i}}{\widetilde{\rho}_{0}}\bigg\}_{i=1}^{N}\bigg)}$. Hence, for all $i=1,\ldots,N$ we have
\begin{align}\label{20082024_eq1}
C_{41}\leqslant \rho_{i}(t,x)\leqslant C_{37}\quad\text{as}\quad(t,x)\in[0, T]\times[0, 1].
\end{align}
From \eqref{1602173}, \eqref{supernew1602171} and \eqref{eq01021710} now it follows that
\begin{align}\label{eq01021715}
\int\limits_{0}^{1}\left(\frac{\partial\rho_{i}}{\partial x}\right)^{2} dx\leqslant C_{42},\quad i=1,\ldots,N,
\end{align}
where ${\displaystyle C_{42}=C_{42}\bigg(C_{37}, C_{38}, \bigg\{\sup\limits_{[0,d]}\frac{\widetilde{\rho}_{0i}}{\widetilde{\rho}_{0}}\bigg\}_{i=1}^{N}, \bigg\{\left\|\left(\frac{\widetilde{\rho}_{0}}{\widetilde{\rho}_{0i}}\right)^{\prime}\right\|_{L_{2}(0, d)}\bigg\}_{i=1}^{N}\bigg)}$, and hence
\begin{align}\label{21082024_eq1}
\int\limits_{0}^{1}\left(\frac{\partial\rho}{\partial x}\right)^{2} dx\leqslant C_{43}(C_{42}, N).
\end{align}

We square  equations~\eqref{mamprok_momentum}, divide by $\rho_{i}$ and sum the result with respect to $i$, then we get
\begin{multline}\label{1303171}
\sum\limits_{i=1}^{N}\rho_{i}\left(\frac{\partial u_{i}}{\partial t}\right)^{2}+\sum\limits_{i=1}^{N}\frac{1}{\rho_{i}}
\left(\sum\limits_{j=1}^{N}\nu_{ij}\frac{\partial^{2} u_{j}}{\partial x^{2}}\right)^{2}-2\sum\limits_{i=1}^{N}\left(\frac{\partial u_{i}}{\partial t}\right)\left(\sum\limits_{j=1}^{N}\nu_{ij}\frac{\partial^{2} u_{j}}{\partial x^{2}}\right)=\\
=\sum\limits_{i=1}^{N}\rho_{i}\left(v\frac{\partial u_{i}}{\partial x}+\frac{\alpha_{i}K}{\rho_{i}}\frac{\partial \rho}{\partial x}\right)^{2}.
\end{multline}
Let us introduce the function $\theta(t)$ via the relation:
\begin{multline*}
\theta(t)=\sum\limits_{i, j=1}^{N}\nu_{ij}\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial u_{j}}{\partial x}\right) dx+\\
+\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}\left(\rho_{i}\left(\frac{\partial u_{i}}{\partial \tau}\right)^{2}+\frac{1}{\rho_{i}}\left(\sum\limits_{j=1}^{N}\nu_{ij}\frac{\partial^{2} u_{j}}{\partial x^{2}}\right)^{2}\right) dxd\tau.
\end{multline*}
Then \eqref{1303171} and inequalities \eqref{lemma51.4}, \eqref{20082024_eq1} and \eqref{21082024_eq1} give the estimate
(here the symmetry of the matrix $\textbf{N}$ is used)
\begin{multline*}
\theta^{\prime}(t)\leqslant C_{44}+C_{45}\left(\sum\limits_{j=1}^{N}\|u_{j}\|^{2}_{L_{\infty}(0,1)}\right)\left(\sum\limits_{i, j=1}^{N}\nu_{ij}
\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial u_{j}}{\partial x}\right) dx\right)\leqslant\\
\leqslant C_{44}+C_{45}\left(\sum\limits_{j=1}^{N}\|u_{j}\|^{2}_{L_{\infty}(0,1)}\right)\theta(t),
\end{multline*}
where ${C_{44}=C_{44}(C_{41}, C_{43}, K, N)}$, ${C_{16}=C_{16}(C_{1}, C_{37}, N)}$, from which, via Gronwall’s lemma (see also \eqref{19082024_eq2}), it follows that
\begin{align}\label{21082024_eq2}
\theta(t)\leqslant C_{46}\left(C_{32}, C_{44}, C_{45}, \{\|u_{0i}^{\prime}\|_{L_{2}(0, 1)}\}_{i=1}^{N}, \textbf{N}, N, T\right).
\end{align}
It follows immediately from \eqref{21082024_eq2} that
\begin{multline}\label{22082024new_eq1}
\sum\limits_{i=1}^{N}\Bigg(\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)^{2} dx+\int\limits_{0}^{T}\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial t}\right)^{2} dxdt+\int\limits_{0}^{T}\int\limits_{0}^{1}\left(\frac{\partial^{2} u_{i}}{\partial x^{2}}\right)^{2}dxdt\Bigg)\leqslant\\ \leqslant C_{47}(C_{1}, C_{37}, C_{41}, C_{46}, N).
\end{multline}

Finally, from the continuity equations~\eqref{mamprok_continuity} and the estimates~\eqref{20082024_eq1}, \eqref{eq01021715} and~\eqref{22082024new_eq1} we obtain that
$$
\int\limits_{0}^{1}\left(\frac{\partial\rho_{i}}{\partial t}\right)^{2} dx\leqslant C_{48}(C_{37}, C_{42}, C_{47}, N),\quad i=1,\ldots,N.
$$

Thereby, we have obtained all estimates which are necessary (and sufficient) to continue the local solution to the initial-boundary value problem \eqref{mamprok_continuity}--\eqref{mamprok_boundvelocity} from the interval $(0, t_{0})$ into the entire target interval $(0, T)$. In order to conclude the proof of Theorem 1 we need to justify the uniqueness of the solution to the initial-boundary value problem \eqref{mamprok_continuity}--\eqref{mamprok_boundvelocity}.

\section{Uniqueness of the solution}

We assume that $\big(\rho^{(1)}_{1},\ldots,\rho^{(1)}_{N}, u^{(1)}_{1},\ldots, u^{(1)}_{N}\big)$ and $\big(\rho^{(2)}_{1},\ldots,\rho^{(2)}_{N}, u^{(2)}_{1},\ldots, u^{(2)}_{N}\big)$ are two solutions to the initial-boundary value problem \eqref{mamprok_continuity}--\eqref{mamprok_boundvelocity}, and let
${\displaystyle v^{(1,2)}=\sum\limits_{j=1}^{N}\alpha_{j}u_{j}^{(1,2)}}$, ${\displaystyle \rho^{(1,2)}=\sum\limits_{j=1}^{N}\rho_{j}^{(1,2)}}$. We set ${\rho_{i}=\rho^{(1)}_{i}-\rho^{(2)}_{i}}$,  ${u_{i}=u^{(1)}_{i}-u^{(2)}_{i}}$, ${i=1,\ldots,N}$, ${v=v^{(1)}-v^{(2)}}$, ${\rho=\rho^{(1)}-\rho^{(2)}}$.

From \eqref{mamprok_continuity}, \eqref{mamprok_nachusl} we have (see \eqref{mamprok.ravlema19})
\begin{align}\label{mamprok.ravlema52}
\frac{\partial \rho_{i}}{\partial t}+\partial_{x}\left(\rho_i v^{(1)}\right)+\partial_{x}\left(\rho_{i}^{(2)} v\right)=0,\quad \rho_{i}|_{t=0}=0,\quad i=1,\ldots,N.
\end{align}
Multiplying \eqref{mamprok.ravlema52} by $2\rho_{i}$ and integrating with respect to $x$, we obtain
\begin{align}
&\frac{d}{dt}\left(\int\limits^{1}_{0}\rho_{i}^{2}\, dx\right)=-\int\limits^{1}_{0}\Bigg(\rho_{i}^{2}\left(\frac{\partial v^{(1)}}{\partial x}\right)+2\rho_{i}^{(2)}\rho_{i}\left(\frac{\partial v}{\partial x}\right)+2\rho_{i}v\left(\frac{\partial \rho_{i}^{(2)}}{\partial x}\right)\Bigg) dx,\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad i=1,\ldots,N.\label{mamprok.ravlema53}
\end{align}
The terms on the right-hand side of \eqref{mamprok.ravlema53} can be estimated as follows:
$$
-\int\limits^{1}_{0}\rho_{i}^{2}\left(\frac{\partial v^{(1)}}{\partial x}\right)dx\leqslant \left(\sum\limits_{j=1}^{N}\left\|\frac{\partial u_{j}}{\partial x}^{(1)}\right\|_{L_{\infty}(0, 1)}\right)\left(\int\limits^{1}_{0}\rho_{i}^{2}\, dx\right),\quad i=1,\ldots,N,
$$
\begin{multline*}
-2\int\limits^{1}_{0}\rho_{i}^{(2)}\rho_{i}
\left(\frac{\partial v}{\partial x}\right)dx\leqslant\left\|\rho_{i}^{(2)}\right\|_{L_{\infty}(Q_{T})}^{2}\left(\int\limits^{1}_{0}\rho_{i}^{2}\, dx\right)+\\
+N\sum\limits_{j=1}^{N}
\int\limits^{1}_{0}\left(\frac{\partial u_{j}}{\partial x}\right)^{2}dx,\quad i=1,\ldots,N,
\end{multline*}
\begin{multline*}
-2\int\limits^{1}_{0}\rho_{i}v\left(\frac{\partial \rho_{i}^{(2)}}{\partial x}\right)dx\leqslant
\int\limits^{1}_{0}\rho_{i}^{2}\, dx+\left\|\frac{\partial\rho_{i}^{(2)}}{\partial x}\right\|_{L_{\infty}\big(0, T; L_{2}(0, 1)\big)}^{2}\|v\|^{2}_{L_{\infty}(0, 1)}\leqslant\\
\leqslant \int\limits^{1}_{0}\rho_{i}^{2}\, dx+N\left\|\frac{\partial\rho_{i}^{(2)}}{\partial x}\right\|_{L_{\infty}\big(0, T; L_{2}(0, 1)\big)}^{2}\left(\sum\limits_{j=1}^{N}\int\limits^{1}_{0}\left(\frac{\partial u_{j}}{\partial x}\right)^{2}dx\right),
\quad i=1,\ldots,N.
\end{multline*}
By the inclusions
\begin{align*}
&\rho_{i}^{(2)}\in L_{\infty}(Q_{T}),\quad \frac{\partial\rho_{i}^{(2)}}{\partial x}\in L_{\infty}\big(0, T; L_{2}(0, 1)\big), \quad i=1,\ldots,N,\\
&\qquad\qquad
\frac{\partial u_{i}^{(1)}}{\partial x}\in L_{2}\big(0, T; L_{\infty}(0, 1)\big),\quad i=1,\ldots,N,
\end{align*}
we obtain the estimates
$$
\frac{d}{dt}\left(\int\limits^{1}_{0}\rho_{i}^{2}\, dx\right)\leqslant C_{49}(t)\int\limits^{1}_{0}\rho_{i}^{2}\, dx+C_{50}\sum\limits_{j=1}^{N}\int\limits^{1}_{0}\left(\frac{\partial u_{j}}{\partial x}\right)^{2}dx,\quad i=1,\ldots,N,
$$
where
${\displaystyle C_{49}=C_{49}\bigg(\bigg\{\bigg\|\frac{\partial u_{i}^{(1)}}{\partial x}\bigg\|_{L_{\infty}(0, 1)}\bigg\}_{i=1}^{N}, \left\{\|\rho_{i}^{(2)}\|_{L_{\infty}(Q_{T})}\right\}_{i=1}^{N}\bigg)}$,
${C_{49}\in L_{2}(0,T)}$,
${\displaystyle C_{50}=C_{50}\bigg(\bigg\{\bigg\|\frac{\partial\rho_{i}^{(2)}}{\partial x}\bigg\|_{L_{\infty}\big(0, T; L_{2}(0, 1)\big)}\bigg\}_{i=1}^{N}, N\bigg)}$.
We can apply the Gronwall inequality to get inequalities
\begin{align}\label{mamprok.ravlema54}
\int\limits^{1}_{0}\rho_{i}^{2}\, dx\leqslant C_{51}\sum\limits_{j=1}^{N}\int\limits^{t}_{0}\int\limits^{1}_{0}\left(\frac{\partial u_{j}}{\partial x}\right)^{2}dx d\tau, \quad i=1,\ldots,N,
\end{align}
where $\displaystyle C_{51}=C_{51}\left(C_{50}, \|C_{49}\|_{L_{1}(0, T)}\right)$.

Further, from equations~\eqref{mamprok_momentum} and boundary conditions~\eqref{mamprok_boundvelocity} we obtain (see~\eqref{mamprok.ravlema23})
\begin{multline*}
\frac{1}{2}\frac{d}{dt}\left(\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{(1)}u_{i}^{2}\,dx\right)+\sum\limits_{i, j=1}^N
\nu_{ij}\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial u_{j}}{\partial x}\right) dx=\displaybreak\\
=K\int\limits_{0}^{1}\rho\left(\frac{\partial v}{\partial x}\right) dx
-\sum\limits_{i=1}^{N}\int\limits_{0}^{1}\rho_{i}u_{i}\left(\frac{\partial u_{i}^{(2)}}{\partial t}\right)dx-\\
-\sum\limits_{i=1}^{N}\int\limits_{0}^{1}\rho_{i}^{(1)}vu_{i}\left(\frac{\partial u_{i}^{(2)}}{\partial x}\right)dx-
\sum\limits_{i=1}^{N}\int\limits_{0}^{1}\rho_{i}v^{(2)}u_{i}\left(\frac{\partial u_{i}^{(2)}}{\partial x}\right)dx,
\end{multline*}
which implies
\begin{multline}\label{mamprok.ravlemanew53}
\frac{d}{dt}\left(\frac{1}{2}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{(1)}u_{i}^{2}\,dx+\sum\limits_{i, j=1}^{N}
\nu_{ij}\int\limits_{0}^{t}\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)\left(\frac{\partial u_{j}}{\partial x}\right) dxd\tau\right)=\\
= K\int\limits_{0}^{1}\rho\left(\frac{\partial v}{\partial x}\right) dx
-\sum\limits_{i=1}^{N}\int\limits_{0}^{1}\rho_{i}u_{i}\left(\frac{\partial u_{i}^{(2)}}{\partial t}\right)dx-\\
-\sum\limits_{i=1}^{N}\int\limits_{0}^{1}\rho_{i}^{(1)}vu_{i}\left(\frac{\partial u_{i}^{(2)}}{\partial x}\right)dx-
\sum\limits_{i=1}^{N}\int\limits_{0}^{1}\rho_{i}v^{(2)}u_{i}\left(\frac{\partial u_{i}^{(2)}}{\partial x}\right)dx.
\end{multline}
We estimate the terms on the right-hand side of \eqref{mamprok.ravlemanew53} as follows:
\begin{multline*}
K\int\limits_{0}^{1}\rho\left(\frac{\partial v}{\partial x}\right) dx \leqslant
\frac{C_{1}}{4}\sum\limits_{i=1}^{N}
\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)^{2} dx
+C_{52}\left(C_{1}, K, N \right)\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{2}\,dx,
\end{multline*}
\begin{multline*}
-\sum\limits_{i=1}^{N}\int\limits_{0}^{1}\rho_{i}u_{i}\left(\frac{\partial u_{i}^{(2)}}{\partial t}\right)dx
\leqslant\frac{C_{1}}{4}\sum\limits_{i=1}^{N}
\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)^{2} dx+\\
+C_{53}\Bigg(C_{1}, \Bigg\{\left\|\frac{\partial u_{i}^{(2)}}{\partial t}\right\|_{L_{2}(0, 1)}\Bigg\}_{i=1}^{N}\Bigg)\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{2}\,dx,
\end{multline*}
\begin{multline*}
-\sum\limits_{i=1}^{N}\int\limits_{0}^{1}\rho_{i}^{(1)}vu_{i}\left(\frac{\partial u_{i}^{(2)}}{\partial x}\right)dx\leqslant\\
\leqslant C_{54}
\Bigg(\Bigg\{\left\|\frac{\partial u_{i}^{(2)}}{\partial x}\right\|_{L_{\infty}(0, 1)}\Bigg\}_{i=1}^{N}, N\Bigg)\frac{1}{2}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{(1)}u_{i}^{2}\,dx,
\end{multline*}
\begin{multline*}
-\sum\limits_{i=1}^{N}\int\limits_{0}^{1}\rho_{i}v^{(2)}u_{i}\left(\frac{\partial u_{i}^{(2)}}{\partial x}\right)dx\leqslant C_{55}\left(\left\{\|u_{i}^{(2)}\|_{L_{\infty}(Q_{T})}\right\}_{i=1}^{N}, N\right)\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{2}\,dx+\displaybreak\\
+C_{56}\Bigg(\Bigg\{\left\|\frac{1}{\rho_{i}^{(1)}}\right\|_{L_{\infty}(Q_{T})}\Bigg\}_{i=1}^{N}, \Bigg\{\left\|\frac{\partial u_{i}^{(2)}}{\partial x}\right\|_{L_{\infty}(0, 1)}\Bigg\}_{i=1}^{N} \Bigg)\frac{1}{2}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{(1)}u_{i}^{2}\,dx,
\end{multline*}
where $C_{53}$, ${C_{56}\in L_{1}(0, T)}$, ${C_{54}\in L_{2}(0, T)}$.
Hence, from \eqref{mamprok.ravlemanew53}, using the re\-la\-tion
$$
\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{2}\,dx\leqslant C_{57}(C_{1}, C_{51}, N)\frac{C_{1}}{2}\sum\limits_{i=1}^{N}\int\limits_{0}^{t}\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)^{2} dxd\tau
$$
proved above (see \eqref{mamprok.ravlema54}), we deduce
\begin{multline*}
\frac{1}{2}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{(1)}u_{i}^{2}\,dx+\frac{C_{1}}{2}\sum\limits_{i=1}^{N}
\int\limits_{0}^{t}\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)^{2} dxd\tau\leqslant\\
\leqslant \int\limits_{0}^{t}C_{58}(C_{52},\ldots,C_{57})\Bigg(\frac{1}{2}\sum\limits_{i=1}^{N}\int\limits^{1}_{0}\rho_{i}^{(1)}u_{i}^{2}\,dx+\frac{C_{1}}{2}\sum\limits_{i=1}^{N}
\int\limits_{0}^{\tau}\int\limits_{0}^{1}\left(\frac{\partial u_{i}}{\partial x}\right)^{2} dxds\Bigg)d\tau,
\end{multline*}
where ${C_{58}\in L_{1}(0, T)}$, which yields the identities ${\rho_{i}\equiv 0}$, ${u_{i}\equiv 0}$, ${i=1,\ldots,N}$. Theorem~1 is proved.

\bigskip
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\end{document}
