\documentstyle[twoside,leqno]{article} %\input amssym.def % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Weak Solutions \hfil EJDE--1996/04}% {EJDE--1996/04\hfil Kazufumi Ito\hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.\ {\bf 1996}(1996), No.\ 4, pp. 1--17. \newline ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113 } \vspace{\bigskipamount} \\ Weak Solutions to the One-dimensional Non-Isentropic Gas Dynamics by the Vanishing Viscosity Method \thanks{ {\em 1991 Mathematics Subject Classifications:} 35L65.\newline\indent {\em Key words and phrases:} Compensated compactness, Conservation laws, Entropy. \newline\indent \copyright 1996 Southwest Texas State University and University of North Texas.\newline\indent Published June 18, 1996.} } \date{} \author{Kazufumi Ito} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \maketitle \begin{abstract} In this paper we consider the non-isentropic equations of gas dynamics with the entropy preserved. Equations are formulated so that the problem is reduced into the $2 \times 2$ system of conservation laws with a forcing term in momentum equation. The method of compensated compactness is then applied to prove the existence of weak solution in the vanishing viscosity method. \end{abstract} \newcommand{\ds}{\displaystyle} \section{Introduction} Consider the one-dimensional gas dynamics equation in the Eulerian coordinate $$\begin{array}{l} \ds \rho_t+(\rho u)_x=0 \\ \ds (\rho u)_t+(\rho u^2+p)_x=0 \\ \ds s_t+us_x=0. \end{array} \leqno (1.1)$$ where $\rho,\;u,\;p$ and $s$ denote the density, velocity, pressure and entropy. Other relevant quantities are the internal energy $e$ and the temperature $T$. We assume that the gas is ideal, so that the equation of state is given by $$p=R\rho T$$ and that it is polytropic, so that $e=c_v\, T$ and $$p=(\gamma-1)e^{s/c_v}\rho^{\gamma} \leqno (1.2)$$ where $\gamma=c_p/c_v >1$ and $R=c_p-c_v$. Define $\phi$ by $$\phi^{1-\gamma}=\gamma(\gamma-1)\,e^{s/c_v}. \leqno (1.3)$$ Then, $\phi$ satisfies $$\phi_t+u\,\phi_x=0.$$ Thus, we consider the Cauchy problem (equivalent to (1.1)) $$\begin{array}{l} \ds \rho_t+m_x=0 \\ \ds m_t+(m^2/\rho+p)_x=0 \\ \ds \phi_t+u\,\phi_x=0\, . \end{array} \leqno (1.4)$$ where $$m=\rho u \quad\mbox{and}\quad p=\frac{1}{\gamma}\,\phi^{1-\gamma} \rho^{\gamma}, \leqno (1.5)$$ with smooth initial data $(\rho_0,m_0)$ in $L^{\infty}(R^2)$ that approaches a constant state $(\bar{\rho},\bar{m})$ at infinity and satisfies $\rho_0(x) \ge \delta_1>0$, and $\phi_0$ in $W^{1,\infty}(R)$ that satisfies $(\phi_0)_x$ converges to 0 at infinity and $$\phi_0(x) \ge \delta_2>0\quad\mbox{and}\quad (\phi_0)_x(x)\ge 0\;\;(\mbox{or}\;\;(\phi_0)_x(x) \le 0). \leqno (1.6)$$ Consider the conservation form of the gas dynamics $$\begin{array}{l} \ds \rho_t+(\rho u)_x=0 \\ \ds (\rho u)_t+(\rho u^2+p)_x=0 \\ \ds [\rho\,(\frac{1}{2}\,u^2+e)]_t+(\rho u\,[\frac{1}{2}\,u^2+e]+pu)_x=0. \end{array} \leqno (1.7)$$ System (1.7) can be written as the hyperbolic system of conservation laws $$y_t+f(y)_x=0$$ where $y=y(t,x)=(\rho,\rho u,\rho\,(\frac{1}{2}\,u^2+e))\in R^3$ and $f$ is a smooth nonlinear mapping from $R^3$ to $R^3$. System (1.4) is equivalent to system (1.7) when solutions are smooth but not necessarily when solutions are weak (e.g.,[Sm, Chapters 16-17]). It is proved in Corollary 3.6 that the viscosity limit of solutions to (1.10) satisfies $\eta_t+q_x \le 0$ in the sense of distributions, i.e., the third equation of (1.7), the conservation of energy $\eta_t+q_x=0$ is replaced by the non-energy production. We also note that the isentropic solution ($\phi=\mbox{const}$) [Di1] is a weak solution of (1.4) but not necessarily of (1.7). In this paper we show the existence of weak solutions to (1.4)-(1.5) using the vanishing viscosity method. The function $(\rho,m,\phi) \in L^{\infty}(\Omega)\times L^{\infty}(\Omega)\times W^{1,\infty}(\Omega)$ with $\Omega=[0,\tau] \times R$ is a weak solution of (1.4)-(1.5) if $\phi$ satisfies the third equation of (1.4) a.e in $\Omega$ and $$\int^{\tau}_0\int^{\infty}_{-\infty} (v\cdot\psi_t+F(\rho,m,\phi)\cdot\psi_x)\,dx\,dt=0 \leqno (1.8)$$ for all $\psi\in C^{\infty}_c(\Omega;R^2)$ where $v=(\rho,m)$ and $$F(\rho,m,\phi)=(m,m^2/\rho+p) \leqno (1.9)$$ We consider the viscous equation of (1.4) with equal diffusion rates $$\begin{array}{l} \ds \rho_t+m_x=\epsilon\,\rho_{xx} \\ \ds m_t+(m^2/\rho+p)_x=\epsilon\,m_{xx} \\ \ds \phi_t+u\,\phi_x=\epsilon\phi_{xx}. \end{array} \leqno (1.10)$$ It will be shown in Theorem 3.5 that the solutions $(\rho^{\epsilon},m^{\epsilon},\phi^{\epsilon})$ to (1.10) converge to a locally defined (in time) weak solution $(\rho,m,\phi)$ of (1.4). Our approach is based on the following observation. Suppose $\phi$ is a constant. Then equation (1.4) reduces to the isentropic gas dynamics. For the isentropic equation it is shown in DiPerna [Di1] that (1.4) has a weak solution by the vanishing viscosity method and using the theory of compensated compactness. The key steps in [Di1] are given as follows. First, if $\phi$ is a constant and $\theta=(\gamma-1)/2$ then $$w=G_1(\rho,m,\phi)=\frac{m}{\rho}+\frac{1}{\theta}\,\phi^{-\theta}\rho^{\theta} \quad\mbox{and} \quad -z=G_2(\rho,m,\phi)=-\frac{m}{\rho} +\frac{1}{\theta}\,\phi^{-\theta}\rho^{\theta} \leqno (1.11)$$ are the Riemann invariants so that $\nabla_v G_1$ and $\nabla_v G_2$ are the two left eigenvectors of the $2\times 2$ matrix $$\nabla_v F=\left(\begin{array}{cc} 0 & 1 \\ \ds -\frac{m^2}{\rho^2}+\rho^{2\theta}\phi^{-2\theta} & \ds \frac{2m}{\rho} \end{array}\right).$$ where $\phi$ is assumed to be a positive constant. The method of invariant regions ([CCS],[Sm]) is applied to $G_1,\;G_2$ to obtain that $0\le \rho^{\epsilon}\le\mbox{const}$, $|m^{\epsilon}/\rho^{\epsilon}| \le\mbox{const}$. Then, there exist a subsequence of $v^{\epsilon}=(\rho^{\epsilon}, m^{\epsilon})$, still denoted by $v^{\epsilon}$ and a Young measure $\nu_{t,x}$ such that for each $\Phi\in C(R^2)$ we have $\Phi(v^{\epsilon})$ converges weak star to $\bar{\Phi}$ in $L^{\infty}(\Omega)$ where $$\bar{\Phi}(t,x)=\langle \nu,\Phi\rangle=\int_{\Omega} \Phi(y)\,d\nu_{t,x}(y), \;\mbox{a.e.}\; (t,x)\in \Omega.$$ Using the entropy fields [La] and the div-curl theorem of Murat [Mu] and Tatar [Ta] for bilinear maps in the weak topology, $$\langle \nu ,\eta_1q_2-\eta_2 q_1\rangle=\langle \nu,\eta_1\rangle \langle \nu,q_2 \rangle-\langle \nu,\eta_2\rangle \langle \nu, q_1\rangle \leqno (1.12)$$ for all entropy/entropy flux pairs $(\eta_i,q_i)$ so that $$\nabla_v\eta\nabla_v F=\nabla_v q.$$ Then, using the weak entropy pairs (i.e., $\eta(0,\cdot)=0$) it is shown that $\nu$ reduces to a point mass, i.e., $v_{\epsilon}$ converges to $v$, a.e. in $\Omega$. We will apply the method described above for the non-isentropic equation (1.10). We need to overcome the two major difficulties. First, $G_1,\;G_2$ are no longer the Riemann invariants of the $3 \times 3$ matrix $M$: $$M=\nabla F=\left(\begin{array}{ccc} 0 & 1 & 0 \\ \ds -\frac{m^2}{\rho^2}+\rho^{2\theta}\phi^{-2\theta} & \ds \frac{2m}{\rho} & \ds -\frac{2\theta}{\gamma}\rho^{\gamma}\phi^{-2\theta-1} \\ 0 & 0 &\ds \frac{m}{\rho}\end{array}\right).$$ Next, equation (1.12) should be extended to the $3 \times 3$ system. We resolve these difficulties by the following steps. Note (see Lemma 2.4) that $G_i=G_i(\rho,m,\phi),\;i=1,2$ satisfies $$(G_i)_t+\lambda_i\nabla G_i\cdot y_x+\frac{1}{\gamma}\left\{ \begin{array}{ll} \ds \rho^{2\theta}\phi^{-2\theta-1}\phi_x, & i=1 \\ \ds -\rho^{2\theta}\phi^{-2\theta-1}\phi_x, & i=2 \end{array} \right.= \epsilon\,((G_i)_{xx}-\nabla^2 G_i(y_x,y_x)) \leqno (1.13)$$ where $y=(\rho,m,\phi)$ is a solution to (1.10) and $\lambda_1=u+\rho^{\theta} \phi^{-\theta},\;\lambda_2=u-\rho^{\theta}\phi^{-\theta}$ are the eigenvalues of the $2 \times 2$ matrix $\nabla_v F$. Here, $G_i,\;i=1,2$ are quasi-convex functions of $(\rho,m,\phi)$ (see Lemma 2.5), i.e., $$r\cdot\nabla G_i=0\;\;\mbox{implies}\;\;\nabla^2 G_i(r,r)\ge 0.$$ Note that from the third equation of (1.10) that $\phi^{\epsilon} \ge \delta_2$ and $|\phi^{\epsilon}|_{\infty} \le |\phi_0|_{\infty}$ (see Lemma 2.1) and moreover $\phi^{\epsilon}_x$ satisfies $$(\phi^{\epsilon}_x)_t+(u^{\epsilon}\,\phi^{\epsilon}_x)_x= \epsilon\,(\phi^{\epsilon}_x)_{xx}$$ Observing that if $(\rho,m,\phi)$ is a solution to (1.10) then $\xi=\mbox{log}(\frac{\phi_x}{\rho})$ satisfies $$\xi_t+u\,\xi_x=\epsilon\,(\xi_{xx}+|(\mbox{log}\phi_x)_x|^2- |(\mbox{log}\rho)_x|^2), \leqno (1.14)$$ we show that if $(\phi_0)_x \ge 0$ (resp. $\le 0$) then $\phi^{\epsilon}_x \ge 0$ (resp. $\le 0$) and $|\phi^{\epsilon}_x|\le c\,\rho^{\epsilon}$ in $\Omega$ provided that $|(\phi_0)_x|\le c\,\rho_0$ in $R$ (see Lemma 2.3). It thus follows from (1.13) and the quasi-convexity of $G_i,\;i=1,2$ that $\max_{x} G_2(t,x) \le \max_{x} G_2(0,x)$ (resp. $\max_{x} G_1(t,x) \le \max_{x} G_1(0,x)$) and $$|\rho^{2\theta}\phi^{-2\theta-1}\phi_x|\le c\,\left(\frac{\rho}{\phi}\right) ^{2\theta+1}.$$ By the maximum principle (see Theorem 2.6), there exists a $\tau=\tau_c>0$ with $c \to \tau_c$ monotonically decreasing and $\tau_0=\infty$ such that $\max_{t\in [0,\tau],\,x\in R}\; G_1(t,x)$ is less than a constant independent of $\epsilon>0$. Hence, we obtain $0\le \rho^{\epsilon}\le\mbox{const}$, $|m^{\epsilon}/\rho^{\epsilon}|\le\mbox{const}$ and $|\phi_x| \le\mbox{const}$ in $\Omega$. Second, in contrast to the isentropic case, the system (1.10) is not endowed with a rich family of entropy-entropy flux pairs. Thus, in order to prove that the Young measures $\nu_{t,x}$ of a weakly star convergent subsequence of $(\rho^{\epsilon},m^{\epsilon},\phi^{\epsilon})$ reduce to a point mass, we first note that $\{\phi^{\epsilon}(t,x)\}$ is precompact in $L^2_{loc}(\Omega)$ and thus $\phi^{\epsilon}$ converges $\phi$ a.e. in $\Omega$ (see Lemma 3.4). Also, we note that dividing the first two equations (1.10) by $\phi$, we obtain $$\begin{array}{l} \ds \hat{\rho}_t+\hat{m}_x=\epsilon\,(\hat{\rho}_{xx}+2\frac{\rho_x\phi_x} {\phi^2}) \\ \ds \hat{m}_t+(\hat{m}^2/\hat{\rho}+\frac{1}{\gamma}\,\hat{\rho}^{\gamma})_x +\frac{p\phi_x}{\phi^2}=\epsilon\,(\hat{m}_{xx}+2\,\frac{m_x\phi_x} {\phi^2}). \end{array} \leqno (1.15)$$ where $\ds \hat{\rho}=\frac{\rho}{\phi}$ and $\ds \hat{m}=\frac{m}{\phi}$ (see Lemma 3.2). This implies that $\hat{v}=(\hat{\rho},\hat{m})$ satisfies the (viscous) isentropic gas-dynamics with the forcing term $-p\phi_x/\phi^2$ in the momentum equation. Since $\ds \frac{p^{\epsilon}\phi^{\epsilon}_x}{(\phi^{\epsilon})^2} \in L^{\infty}(\Omega)$ uniformly in $\epsilon>0$, thus $\ds \{ \frac{p^{\epsilon}\phi^{\epsilon}_x} {(\phi^{\epsilon})^2}\}_{\epsilon>0}$ is precompact in $H^{-1,q}_{loc}(\Omega),\; 1 \le q <2$. Hence, the method of compensated compactness in [Di1],[Ch] can be applied to the functions $(\hat{\rho}^{\epsilon},\hat{m}^{\epsilon})$ to show that $\nu_{t,x}$ is a point mass provided that $1<\gamma\le 5/3$. Regarding work on existence of weak solutions for conservation laws, we refer the reader to an excellent treatise by DiPerna [Di3] and references therein. Concerning basic framework on conservation laws, we refer the reader to [La],[Sm] and for the functional analytic framework of compensated compactness we refer [Mu],[Ta1],[Ev] and [Di2]. For scalar conservation laws the vanishing viscosity method is employed (e.g., in [Ol],[Kr] and references in [Sm]) to define the unique entropy solution. Also, the vanishing viscosity method is used to develop the viscosity solution to the Hamilton-Jacobi equation in [CL]. The finite-difference methods (e.g., Lax-Friedrichs and Gudunov schemes) are also used to construct weak solutions to a scalar and $2 \times 2$ system of conservation laws (e.g., see [Di2],[Ch] and [Sm]). In the case where the initial data have small total variation, Glimm [Gl] proved the global existence of BV-solutions for a general class of hyperbolic systems as the strong limit of random choice approximations. However, the problem of existence of solutions to (1.7) with large initial data is still unsolved. In [CD] the vanishing viscosity method is applied to the system (1.7) under a special class of constitutive relations in Lagrangian coordinates. \section{The Viscosity Method} In this section we establish the uniform $L^{\infty}$ bound of $y^{\epsilon}=(\rho^{\epsilon},m^{\epsilon},\phi^{\epsilon})$. \noindent{\bf Lemma 2.1} {\it If $\phi \in C^{1,2}([0,\tau]\times R)$ satisfies $\phi_t+u\,\phi_x=\epsilon\,\phi_{xx}$ then $$\min_x\, \phi_0(x) \le \phi(t,x) \le \max_x\, \phi_0(x).$$} \noindent{\bf Proof:} Using the same arguments as in the proof of Theorem 2.6, we can show that $\max_x\,\phi(t,x) \le \max_x\,\phi_0(x)$ and $\min_x\,\phi(t,x) \ge \min_x\,\phi_0(x)$. $\Box$ \vspace{2mm} It will be shown in Section 3 (see (3.4) and Lemma 3.1) that the normalized mechanical energy $$E(\rho,u,\phi)=\frac{1}{2}\,\rho(u-\bar{u})^2 +\frac{1}{\gamma}\,(\rho^{\gamma}-\gamma\,\bar{\rho}^{\gamma-1} (\gamma-\bar{\gamma})-\bar{\rho}^{\gamma})\phi^{1-\gamma} \leqno (2.1)$$ satisfies $$\int^{\infty}_{-\infty}E(\rho(t,x),u(t,x),\phi(t,x))\,dx \le \int^{\infty}_{-\infty}E(\rho_0(x),u_0(x),\phi_0(x))\,dx. \leqno (2.2)$$ The following lemma shows the lower bound of $\rho_{\epsilon}$. \vspace{2mm} \noindent{\bf Lemma 2.2} {\it If $\rho \in C^{1,2}([0,\tau] \times R)$ satisfies $$\rho_t+(u\,\rho)_x=\epsilon\,\rho_{xx} \leqno (2.3)$$ with $\rho(0,\cdot) \ge 0$ and $u \in C^1(\Omega)$, then $\rho(t,\cdot) \ge 0$. Moreover, if $\rho(0,\cdot) \ge \delta>0$ and $$\int^{\tau}_0\int^{\infty}_{-\infty}\rho\,|u-u_0|^2\,dx\,dt \le \mbox{const}, \leqno (2.4)$$ then $\rho(t,\cdot)\ge \delta(\epsilon,\tau)>0$ on $(0,\tau)$.} \vspace{2mm} \noindent{\bf Proof:} Choose $\psi=\min\,(\rho(t,x),0)$. Then we have $$\int^{\infty}_{-\infty} \frac{1}{2}\,|\psi(t,x)|^2 +\int^t_0\int^{\infty}_{-\infty} (\epsilon\,|\psi_x|^2-\psi u\,\psi_x)\,dx\,ds=0.$$ By the H\"older inequality, we obtain $$\int^{\infty}_{-\infty}|\psi(t,0)|^2 \le \frac{|u|_{\infty}}{2\epsilon} \int^t_0\int^{\infty}_{-\infty} |\psi|^2\,dx\,ds$$ where $|u|_{\infty}=\sup_{(t,x) \in (0,\tau)\times R}\,|u(t,x)|$, and the Gronwall's inequality implies $\psi=0$. Thus, $\rho \ge 0$. Next, we prove $\rho(t,\cdot) \ge \delta=\delta(\epsilon,\tau)>0$ if $\varphi(0,\cdot )\ge\delta>0$ by using the Stampacchia's lemma (e.g., see [FI],[Tr]), i.e., suppose $\chi(c)$ is a nonnegative, non-increasing function on $[c_0,\infty)$, and there exist positive constants $K,\;s$ and $t$ such that $$\chi(\hat{c})\le Kc^s(\hat{c}-c)^{-s}\,\chi(c)^{1+t} \quad\mbox{for all}\;\;\hat{c}>c \ge c_0,$$ then $$\chi(c^*)=0 \quad \mbox{for}\;\;c^*=2c_0\,(1+2^{\frac{1+2t}{t^2}} K^{\frac{1+t}{st}}\,\chi(c_0)^{\frac{1+t}{s}}).$$ First, we establish {\em a priori} bound. We consider the class $K$ [Di1] of strictly convex $C^2$ functions $h$ with following properties: $$h(\bar{\rho})=h^{\prime}(\bar{\rho})=0,\quad h(\rho)=\rho^{-\alpha}\;\;\mbox{on}\;\;(0,\bar{\rho}/2)\; \mbox{for some} \;\;0 < \alpha < 1.$$ Premultiplying the first equation of (1.10) by $h^{\prime}(\rho)$ we obtain $$h(\rho)_t+(h^{\prime}(\rho)\rho u)_x-h^{\prime\prime}(\rho)\rho_x\rho u= \epsilon\,(h(\rho)_{xx}-h^{\prime\prime}(\rho)\rho_x^2)$$ Integration of this over $(0,t) \times R$ yields \begin{eqnarray*} \lefteqn{\int^{\infty}_{-\infty} h(\rho(t,x))-h(\rho_0(x))\,dx +\epsilon\int^t_0\int^{\infty}_{-\infty} h^{\prime\prime} (\rho)\rho_x^2\,dx\,dt} \\ \hfill&=&\int^t_0\int^{\infty}_{-\infty} h^{\prime\prime}(\rho)\rho_{x} \rho(u-\bar{u})\,dx\,dt\,. \end{eqnarray*} Note that $$h^{\prime\prime}(\rho)\rho_{x}\rho(u-\bar{u})\le\frac{\epsilon}{2}\, h^{\prime\prime}(\rho)\rho_x^2+\frac{1}{2\epsilon}\,h^{\prime\prime}(\rho) \rho^2(u-\bar{u})^2.$$ Since there exists some constant $\beta>0$ such that $$\begin{array}{l} \rho^2\,h^{\prime\prime}(\rho)\le \beta\,\rho\quad \mbox{for}\;\; \bar{\rho}/2 \le \rho \le M \\ \rho^2\,h^{\prime\prime}(\rho) \le \beta\,h(\rho)\quad \mbox{for}\;\; 0 < \rho <\bar{\rho}/2 \end{array}$$ it follows that $\rho^2\,h^{\prime\prime}(\rho)(u-\bar{u})^2\le \beta\, (\rho\,(u-\bar{u})^2+h(\rho))$. Hence, \begin{eqnarray*} \lefteqn{\int^{\infty}_{-\infty} h(\rho(t,x))-h(\rho_0(x))\,dx +\frac{\epsilon}{2}\int^t_0\int^{\infty}_{-\infty} h^{\prime\prime}(\rho) \rho_x^2\,dx\,dt} \\ &&\le \frac{\beta}{2\epsilon}\int^t_0\int^{\infty}_{-\infty}\rho(u-\bar{u})^2 +h(\rho)\,dx\,dt\,. \end{eqnarray*} and it follows from (2.4) and Gronwall's inequality that $$\int^{\infty}_{-\infty}h(\rho(t,x))\,dx\le \mbox{const on}\;\; [0,\tau]. \leqno (2.5)$$ Set $\eta=1/\rho$. Then from (2.3) $\eta$ satisfies $$\eta_t+u\,\eta_x-u_x\eta=\epsilon\,(\eta_{xx}-\frac{2|\eta_x|^2}{\eta}).$$ We further introduce $\hat{\eta}=e^{\omega\,t}\eta$ with $\omega>0$ to be determined later. The equation for $\hat{\eta}$ becomes $$\hat{\eta}_t+\omega\,\hat{\eta}+u\,\hat{\eta}_x-u_x\hat{\eta} =\epsilon\,(\hat{\eta}_{xx}-\frac{2|\hat{\eta}_x|^2}{\hat{\eta}}). \leqno (2.6)$$ Define $\xi=\xi_c=\max\,(0,\hat{\eta}-c)$ with $c \ge c_0=\delta^{-1}$. Pre-multiplication of (2.6) by $\xi^3$ and integration over $R$ yields $$\int^{\infty}_{-\infty}(\frac{1}{4}\,(\xi^4)_t+\omega\,(\xi^4+c\,\xi^3)+ \frac{3\epsilon}{4}|(\xi^2)_x|^2) \le -\int^{\infty}_{-\infty} u\,(5\,\xi^3+3c\,\xi^2)\xi_x\,dx. \leqno (2.7)$$ Note that $$-\int^{\infty}_{-\infty} 5u\,\xi^3\xi_x \le -\int^{\infty}_{-\infty}\frac{5}{2}u\xi^2(\xi^2)_x\,dx\le \frac{\epsilon}{4} \int^{\infty}_{-\infty}|(\xi^2)_x|^2\,dx+\frac{25}{4\epsilon}|u|^2_{\infty} \int^{\infty}_{-\infty}|\xi|^4\,dx$$ To estimate the second term on the right hand side of (2.7), we define $$I_c(t)=\{x \in R:\xi(t,x) > 0 \}=\{x \in R:\eta(t,x) > c\,e^{\omega t}\}.$$ Then, using the H\"older inequality, we have $$\begin{array}{l} \ds -\int^{\infty}_{-\infty} 3c\,\xi^2\xi_x\,dx= -\int^{\infty}_{-\infty}\frac{3c}{2}\,u\xi(\xi^2)_x\,dx \\ \ds \quad \le\frac{3c}{2}|u|_{\infty}\left( \int^{\infty}_{-\infty}|(\xi^2)_x|^2\,dx\right)^{1/2} \left(\int^{\infty}_{-\infty}|\xi|^6\,dx\right)^{1/6}|I_c(t)|^{1/3} \\ \ds \quad \le \frac{\epsilon}{4}\int^{\infty}_{-\infty}|(\xi^2)_x|^2\,dx +\frac{9c^2}{4\epsilon}\,|u|_{\infty}^2 \left(\int^{\infty}_{-\infty}|\xi|^6\,dx\right)^{1/3}|I_c(t)|^{2/3}. \end{array}$$ Substituting these estimates into (2.7), choosing $\ds \omega=\frac{\epsilon}{4}+\frac{25}{4\epsilon}\,|u|_{\infty}^2$, and integrating on the interval $[0,t]$, we obtain \setcounter{equation}{7} \begin{eqnarray} \lefteqn{\int^{\infty}_{-\infty} |\xi|^4\,dx+\epsilon\int^t_0 \int^{\infty}_{-\infty} (|\xi^2|^2+|(\xi^2)_x|^2)\,dx\,ds} \nonumber\\ &&\le \frac{9c^2}{\epsilon} \,|u|_{\infty}^2\int^t_0 \left(\int^{\infty}_{-\infty}|\xi|^6\,dx\right)^{1/3}|I_c(s)|^{2/3}\,ds. \end{eqnarray} Since $$|\xi^2|_{\infty} \le \sqrt{2}\left(\int^{\infty}_{-\infty} (|\xi^2|^2+|(\xi^2)_x|^2)\,dx\right)^{1/2}\,,$$ we have $$\left(\int^{\infty}_{-\infty}|\xi|^6\,dx\right)^{1/3}\le 2^{1/6} \left(\int^{\infty}_{-\infty}(|\xi^2|^2+|(\xi^2)_x|^2)\,dx\right)^{1/2}.$$ Thus, from (2.8), \begin{eqnarray*} \lefteqn{\int^{\infty}_{-\infty} |\xi|^4\,dx+\epsilon \int^t_0\int^{\infty}_{-\infty} (|\xi^2|^2+|(\xi^2)_x|^2)\,dx\,ds} \\ &&\le 2^{1/3}\frac{81c^4}{4\epsilon}\,|u|_{\infty}^4\int^t_0|I_c(s)|^{4/3}\,ds \le Kc^4\,\chi(c)^{4/3} \end{eqnarray*} where we define $$\chi(c)=\sup_{t\in [0,\tau]}\;|I_c(t)|\quad\mbox{and} \quad K=2^{1/3}\frac{81\tau}{4\epsilon}.$$ Clearly $\chi(c)$ is nonnegative, non-increasing on $[c_0,\infty)$. Moreover, for $\hat{c}>c$, $$\int^{\infty}_{-\infty}|\xi|^4\,dx \ge \int_{I_{\hat{c}}}|\xi|^4\,dx \ge(\hat{c}-c)^4\,I_{\hat{c}}(t).$$ Hence, $$I_{\hat{c}}(t)\le K\,c^4(\hat{c}-c)^{-4}\chi(c)^{4/3}$$ and by taking the sup over $t$ we obtain $$\chi(\hat{c})\le Kc^4(\hat{c}-c)^{-4}\,\chi(c)^{4/3}.$$ It follows from (2.5) that $\chi(2/\bar{\rho}) < \infty$ and thus from Stampacchia's lemma that $\chi(c^*)=0$ for some $c^* \ge c_0$ and hence $$\eta(t,x) \le c^*e^{\omega t} \quad\mbox{and}\quad \rho(t,x)\ge (c^*)^{-1}\,e^{-\omega t}\,,$$ where $c^*$ depends on $\epsilon$ and $\tau$. $\Box$ \vspace{2mm} The following lemma shows that $|\phi^{\epsilon}_x(t,\cdot)|$ is uniformly bounded by $\rho^{\epsilon}(t,\cdot)$ for every $t \in [0,\tau]$ under assumption (1.7). \vspace{2mm} \noindent{\bf Lemma 2.3} {\it Assume that $\phi_0$ satisfies (1.7) and $|(\phi_0)_x| \le c\,\rho_0$ in $R$. Then $|\phi_x(t,\cdot)| \le c\, \rho(t,x)$ in $R$ for every $t \in [0,\tau]$.} \vspace{2mm} \noindent{\bf Proof:} If the initial condition $(\rho_0,m_0,\phi_0)$ is sufficiently smooth and $(\phi_0)_x \ge \delta_3$ then the solution to (1.10) satisfies $\rho,\;u,\;\phi \in C^3(\Omega)$ and $\phi_x(t,\cdot)>0$. Note that $\phi_x$ satisfies $$(\phi_x)_t+(u\,\phi_x)_x=\epsilon\,(\phi_x)_{xx}. \leqno (2.9)$$ Then, it is not difficult to show that if we define $\xi=\mbox{log}(\frac{\phi_x}{\rho})$ then $\xi$ satisfies $$\xi_t+u\,\xi_x=\epsilon\,(\xi_{xx}+|(\mbox{log}\phi_x)_x|^2- |(\mbox{log}\rho)_x|^2)$$ Suppose $\xi(t,x_0)=\max_x\;\xi(t,x)$. Then $$\xi_x(t,x_0)=(\log\phi_x)_x(t,x_0)-(\log\rho)_x(t,x_0)=0$$ and $\xi_{xx}(t,x_0) \le 0$. Thus, $\partial_t(\max_x\;\xi(t,x)) \le 0$, which implies the lemma. Since the solution to (2.9) continuously depends on the initial data $(\phi_0)_x$ the estimate holds for when $(\phi_0)_x \ge 0$. $\Box$ \vspace{2mm} The following lemmas provide the technical properties of the functions $G_i(t)$, $i=1,2$ defined by (1.11). \vspace{2mm} \noindent{\bf Lemma 2.4} {\it If $y=(\rho,m,\phi) \in C^{1,2}((0,\tau)\times R)^3$ is a solution to (1.10), then (1.13) holds.} \vspace{2mm} \noindent{\bf Proof:} First, note that the $3 \times 3$ matrix $M=\nabla F$ has the eigenvalues $\ds \lambda_1=\frac{m}{\rho}+\rho^{\theta}\phi^{-\theta}, \;\lambda_2=\frac{m}{\rho}-\rho^{\theta}\phi^{-\theta}$ and $\ds\frac{m}{\rho}$ and that $\nabla_v G_i,\;i=1,2$ are the left-eigenvectors of the sub-matrix $\nabla_v F$ corresponding to $\lambda_i$. Thus, $$(G_1)_t+\lambda_1\,(\nabla G_i\cdot y_x+\rho^{\theta}\phi^{-\theta-1} \phi_x) -(\frac{2\theta}{\gamma}\rho^{2\theta-1}\phi^{-2\theta-1} +u\rho^{\theta} \phi^{-\theta-1})\phi_x=\epsilon\nabla G_1\cdot y_{xx}\,.$$ Since $\nabla G_1\cdot y_{xx}=(G_1)_{xx}-\nabla^2G_1(y_x,y_x)$ we obtain (1.13) for $G_1$. The same calculation applies to $G_2$. $\Box$ \vspace{2mm} \noindent{\bf Lemma 2.5} {\it If $\rho>0,\;\phi>0$ then $G_i,\;i=1,2$, are quasi-convex.} \vspace{2mm} \noindent{\bf Proof:} We prove $\ds G_1=\frac{m}{\rho}+\frac{1}{\theta}\, \rho^{\theta}\phi^{-\theta}$ is quai-convex. The same proof applies to $G_2$. Note that $$\nabla G_1=\left(\begin{array}{c} \ds -\frac{m}{\rho^2}+\rho^{\theta-1} \phi^{-\theta} \\ \ds \frac{1}{\rho} \\ \ds -\rho^{\theta}\phi^{-\theta-1} \end{array} \right)$$ $$\nabla^2 G_1=\left(\begin{array}{ccc} \ds-\frac{2m}{\rho^3}+ (\theta-1)\,\rho^{\theta-2}\phi^{-\theta} & \ds -\frac{1}{\rho^2} & \ds -\theta\,\rho^{\theta-1}\phi^{-\theta-1} \\ \ds -\frac{1}{\rho^2} & 0 & 0 \\ \ds -\theta\,\rho^{\theta-1}\phi^{-\theta-1} & 0 & \ds (\theta+1)\,\rho^{\theta}\phi^{-\theta-2}\end{array}\right).$$ If $r=(X,Y,Z)$ satisfies $r\cdot\nabla G_1$ then $\ds Y=-\frac{m}{\rho}+ \rho^{\theta}\phi^{-\theta}\,X +\rho^{\theta+1}\phi^{-\theta-1}\,Z$. Thus, $$\begin{array}{l} \ds \nabla^2 G_i(r,r)=(\theta+1)\,(\rho^{\theta-2}\phi^{-\theta}\,X^2-2\, \rho^{\theta-1}\phi^{-\theta-1}\,XZ+\rho^{\theta}\phi^{-\theta-2}\,Z^2) \\ \ds\qquad =(\theta+1)\rho^{\theta-2}\phi^{-\theta-2}\,(\phi\,X-\rho\,Z)^2 \ge 0. \quad \Box \end{array}$$ \vspace{2mm} We now state the main result of this section that establishes the uniform $L^{\infty}$-bound of $(\rho^{\epsilon},m^{\epsilon},\phi^{\epsilon}_x)$ in $\epsilon>0$. \vspace{2mm} \noindent{\bf Theorem 2.6} {\it Suppose $\phi_0$ satisfies (1.7) and $|(\phi_0)_x| \le c \rho_0$ in $R$. Then, there exists a $\tau=\tau_c>0$ with $c \to \tau_c$ monotonically decreasing and $\tau_0=\infty$ such that $0 \le \rho^{\epsilon} \le\mbox{const}$, $|\frac{m^{\epsilon}}{\rho^{\epsilon}}| \le\mbox{const}$ and $|\phi_x|\le\mbox{const}$ in $\Omega=[0,\tau] \times R$.} \vspace{2mm} \noindent{\bf Proof:} Suppose that $(\phi_0)_x \le 0$. Then, it follows from Lemmas 2.3-2.6 that $\max_x G_2(t,x) \le \max_x G_2(0,x)=A$. Hence, $\ds \frac{m}{\rho}+A \ge \frac{1}{\theta}\,\rho^{\theta}\phi^{-\theta} \ge 0$. Set $G=A+G_1$. It then follows from Lemma 2.4 that $$G_t+\lambda_1\nabla G\cdot y_x+\frac{1}{\gamma}\,\rho^{2\theta} \phi^{-2\theta-1}\phi_x=\epsilon\,(G_{xx}-\nabla^2 G(y_x,y_x))$$ Let $G^k=G(kh),\;h>0$. Then, $$G^k-G^{k-1}+\lambda_1\nabla G^k\cdot y^k_x+\frac{1}{\gamma}\, (\rho^k)^{2\theta}(\phi^k)^{-2\theta-1}(\phi^k)_x =\epsilon\,(G^k_{xx}-\nabla^2 G^k(y^k_x,y^k_x))+\varepsilon(h)$$ where $\varepsilon(h)/h \to 0$ as $h \to 0^+$. Suppose $G^k(x_0)=\max_x\; G^k(x)$. Then, $G^k_x(x_0)=(\nabla G^k\cdot y^k_x)(x_0)=0$ and $G^k_{xx}(x_0) \le 0$. It follows from Lemmas 2.3 and 2.5 that if $\psi(t)=\max_x\;G(t,x)$ then $$\psi(kh)-\psi((k-1)h)-\varepsilon(h) \le \frac{c}{\gamma}\,(\frac{\rho} {\phi}(x_0))^{2\theta+1} \le \frac{c}{\gamma}\theta^{(2\theta+1)/\theta} \psi(kh)^{(2\theta+1)/\theta}.$$ Taking the limit $h \to 0^+$, we obtain $$\psi(t)-\psi(0) \le \int^t_0 \frac{c}{\gamma}\,\theta^{(2\theta+1)/\theta} \,\psi(\tau)^{(2\theta+1)/\theta}\,d\tau$$ and thus $$\psi(t) \le \left(\frac{\psi(0)^{(\theta+1)/\theta}}{1-\frac{c}{\gamma} (\frac{\theta+1}{\theta})\theta^{(2\theta+1)/\theta} \psi(0)^{(\theta+1)/\theta}\,t}\right)^{\theta/(\theta+1)}. \leqno (2.10)$$ In fact, if $$s(t)=\psi(0)+\int^t_0 \frac{c}{\gamma}\,\theta^{(2\theta+1)/\theta} \,\psi(\tau)^{(2\theta+1)/\theta}\,d\tau$$ then $\psi(t) \le s(t)$ and $\dot{s}\le \frac{c}{\gamma} \, \theta^{(2\theta+1)/\theta} s^{(2\theta+1)/\theta}$, which implies (2.10). Since $$0 \le \frac{1}{\theta}\,\rho^{\theta}\phi^{-\theta} \le \frac{1}{2}\,(G_1+G_2) \quad\mbox{and}\quad -G_2 \le \frac{m}{\rho} \le G_1\,,$$ the lemma follows from (2.9). $\Box$ \section{Compensated Compactness} In this section we show that the sequence $\{(\rho^{\epsilon},m^{\epsilon}, \phi^{\epsilon})\}_{\epsilon>0}$ has a subsequence that converges to a weak solution of (1.10) a.e in $\Omega$ using the method of compensated compactness. First note that the mechanical energy $$\eta=\frac{1}{2}\,\frac{m^2}{\rho}+\frac{1}{\gamma(\gamma-1)}\, \rho^{\gamma}\phi^{-\gamma+1} \leqno (3.1)$$ and the corresponding entropy-flux $$\quad q=\frac{\rho}{2}\,(\frac{m}{\rho})^3+\frac{1}{\gamma-1}\, \frac{m}{\rho}\rho^{\gamma}\phi^{-\gamma+1} \leqno (3.2)$$ form an entropy pair, i.e., $$\nabla\eta\,M=\nabla q \leqno (3.3)$$ In order to treat solutions approaching a nonzero state at infinity, we consider a normalized entropy pair $$\begin{array}{l} \tilde{\eta}=\eta(y)-\eta((\bar{v},\phi))-\nabla_v\eta((\bar{v},\phi)) (v-\bar{v}), \\ \tilde{q}=q(y)-q((\bar{v},\phi))-\nabla_v\eta((\bar{v},\phi))F(y) \end{array}$$ where $v=(\rho,m),\;\bar{v}=(\bar{\rho},\bar{m})$ and $y=(v,\phi)$. Premultiplying (1.10) by $\nabla\tilde{\eta}$, we obtain $$\tilde{\eta}_t+\tilde{q}_x=\epsilon\,(\tilde{\eta}_{xx}-\nabla^2\eta(y_x,y_x)).$$ Integration over $\Omega$ yields an energy estimate $$\int^{\infty}_{-\infty}\tilde{\eta}(t,x)\,dx+\epsilon \int^t_0\int^{\infty}_{-\infty} \nabla^2\eta(y_x,y_x)\,dx\,dt=\int^{\infty}_{-\infty}\tilde{\eta}(0,x)\,dx. \leqno (3.4)$$ The following lemma implies the energy estimate (2.2) where $\tilde{\eta}(y)=E(\rho,u,\phi)$. \vspace{2mm} \noindent{\bf Lemma 3.1} {\it For $\rho>0,\;\phi>0$, $\nabla^2\eta$ is non-negative.} \vspace{2mm} \noindent{\bf Proof:} Note that $$\nabla^2\eta=\left(\begin{array}{ccc} \ds \frac{m^2}{\rho^3}+\rho^{\gamma-2} \phi^{-\gamma+1} & \ds -\frac{m}{\rho^2} & \ds -\rho^{\gamma-1}\phi^{-\gamma} \\ \ds -\frac{m}{\rho^2} & \ds \frac{1}{\rho} & 0 \\ \ds -\rho^{\gamma-1}\phi^{-\gamma} & 0 & \ds \rho^{\gamma}\phi^{-\gamma-1} \end{array}\right).$$ Thus, $$\nabla^2\eta\,(y_x,y_x)=\frac{1}{\rho}\,(\frac{m}{\rho}\,\rho_x-m_x)^2 +\rho^{\gamma-2}\phi^{-\gamma-1}\,(\phi\,\rho_x-\rho\,\phi_x)^2 \ge 0 \leqno (3.5)$$ for $y_x=(\rho_x,m_x,\phi_x)$. $\Box$ \vspace{2mm} The following lemma establishes the viscosity estimate which is essential for the method of compensated compactness. \vspace{2mm} \noindent{\bf Lemma 3.2} {\it Assume that $1< \gamma \le 2$ and $\int^{\infty}_{-\infty} \tilde{\eta}(0,x)\,dx<\infty$. Then, if $(\rho,m,\phi)$ is a solution of (1.10) $$\epsilon\int^{\tau}_0\int^{\infty}_{-\infty}(|\rho_x(t,x)|^2+|m_x(t,x)|^2) \,dx\,dt\le \mbox{const}$$ where $\tau>0$ is defined in Theorem 2.6} \vspace{2mm} \noindent{\bf Proof:} From (2.9) and Lemma 2.2 we have $$\int^{\infty}_{-\infty} |\phi_x(t,x)|\,dx= \int^{\infty}_{-\infty}|\phi_x(0,x)|\,dx\,, \quad t \in [0,\tau].$$ It thus follows from Theorem 2.6 that $$\int^{\tau}_0\int^{\infty}_{-\infty} |\phi_x(t,x)|^2\,dx\le \mbox{const}.$$ Since $0< \rho(t,x),\;\phi(t,x) \le\mbox{const}$ in $\Omega$ it follows from (3.5) that $$\nabla^2\,\eta(y_x(t,x),y_x(t,x))+|\phi_x(t,x)|^2 \ge c_1\,|y_x(t,x)|^2$$ for some $c_1>0$. Hence, the lemma follows from (3.4). $\Box$ \vspace{2mm} We apply the method of compensated compactness for the function $\hat{v}^{\epsilon}$ defined by $$\hat{v}^{\epsilon}=(\hat{\rho}^{\epsilon},\hat{m}^{\epsilon}) =(\frac{\rho^{\epsilon}}{\phi^{\epsilon}},\frac{m^{\epsilon}}{\phi^{\epsilon}})$$ The function $\hat{v}^{\epsilon}$ satisfies the $2 \times 2$ viscous conservation law (1.15) with the forcing term which is in $L^{\infty}(\Omega)$. Based on this observation we have \vspace{2mm} \noindent{\bf Lemma 3.3} {\it Assume that the conditions in Theorem 2.6 are satisfied and that $\int^{\infty}_{-\infty} \tilde{\eta}(0,x)\,dx < \infty$. Then, for $1<\gamma\le 2$, the measure set $$\eta(\hat{v}^{\epsilon})_t+q(\hat{v}^{\epsilon})_x$$ lies in a compact subset of $H^{-1}_{loc}(\Omega)$ for all weak entropy/entropy flux pair $(\eta,q)$ of $\nabla_v F$, where $\ds \hat{v}^{\epsilon}=(\frac{\rho^{\epsilon}}{\phi^{\epsilon}}, \frac{m^{\epsilon}}{\phi^{\epsilon}})$.} \vspace{2mm} \noindent{\bf Proof:} Suppose $(\rho,m,\phi)$ is a solution to (1.10). Then, dividing the first two equations of (1.10) by $\phi$, we obtain (1.15) for $\ds \hat{\rho}=\frac{\rho}{\phi}$ and $\ds \hat{m} =\frac{m}{\phi}$. Let $(\eta,q)$ be a weak entropy/entropy flux pair, i.e., $$\nabla\eta\nabla_v F=\nabla q \quad \mbox{and} \quad \eta(0,\cdot)=0. \leqno (3.6)$$ It can be shown that for $0<\rho \le\mbox{const}$, $|\frac{m}{\rho}|\le\mbox{const}$ $$|\nabla\eta|\le\mbox{const}\quad\mbox{and}\quad |\nabla^2\eta(r,r)|\le\mbox{const}\,\nabla^2\eta^*(r,r) \leqno (3.7)$$ where $$\eta^*=\frac{1}{2}\rho\,(\frac{m}{\rho})^2 +\frac{1}{\gamma(\gamma-1)}\,\rho^{\gamma}$$ is the mechanical energy, $r$ is any vector in $R^2$ and constant is independent of $r$. Premultiplying (1.15) by $\nabla\eta$, we obtain $$\eta(\hat{v})_t+q(\hat{v})_x=\epsilon\,(\eta(\hat{v})_{xx}-\nabla^2\eta (\hat{v}_x,\hat{v}_x))+\nabla\eta(\hat{v})A$$ where $$A=2\epsilon\left(\frac{\rho_x\phi_x}{\phi^2},\frac{m_x\phi_x}{\phi^2}\right)- \left(0,\frac{p\phi_x}{\phi^2}\right)$$ It follows from Theorem 2.6 that $\ds \frac{p^{\epsilon}\phi^{\epsilon}_x}{(\phi^{\epsilon})^2} \in L^{\infty}(\Omega)$ uniformly in $\epsilon>0$. It follows from Lemma 3.2 and Theorem 2.6 that $$\epsilon^{1/2}\left(\frac{\rho^{\epsilon}_x\phi^{\epsilon}_x}{(\phi^{\epsilon}) ^2},\frac{m^{\epsilon}_x\phi^{\epsilon}_x}{(\phi^{\epsilon})^2}\right) \in L^2(\Omega)$$ uniformly in $\epsilon>0$. Thus, $\{\nabla\eta(v^{\epsilon})A^{\epsilon}\} _{\epsilon>0}$ is precompact in $W^{-1,q}_{loc}(\Omega)$, $1 \le q <2$. Since $$\int^{\tau}_0 \int^{\infty}_{-\infty} \epsilon\,|\hat{v}^{\epsilon}_x(t,x)|^2 \,dx\,dt \le \mbox{const}$$ The set $\{\epsilon\nabla\eta\hat{v}^{\epsilon}_x\}_{\epsilon>0}$ is precompact in $L^2(\Omega)$ and so is $\{\epsilon\eta(\hat{v}^{\epsilon})_{xx}\}_{\epsilon>0}$ in $H^{-1}(\Omega)$. Hence, the lemma follows from the fact that if set $S$ is compact in $W^{-1,q}(U)$ and bounded in $W^{-1,r}(U)$ then $S$ is compact in $H^{-1}(U)$ for $1 \le q < 2 < r$ and any bounded and open set $U$ in $R^2$. [Ev] $\Box$ \vspace{2mm} In the next lemma we prove that the sequence $\{\phi^{\epsilon}\}_{\epsilon>0}$ is precompact in $L^2_{loc}(\Omega)$. \vspace{2mm} \noindent{\bf Lemma 3.4} {\it For $\epsilon>0$ and $\tau>0$ defined in Theorem 2.6 $$\int^{\tau}_0\int^{\infty}_{-\infty}(|\phi^{\epsilon}_t|^2 +|\phi^{\epsilon}_x|^2)\,dx\,dt\le \mbox{const}.$$ Thus, the family $\{\phi^{\epsilon}(t,x)\}_{\epsilon>0}$ is compact in $L^2(U)$ for any bounded rectangle $U=(0,\tau) \times (-L,L)$.} \vspace{2mm} \noindent{\bf Proof:} Premultiplying (1.10) by $\phi_{xx}$ and integrating in $(0,\tau) \times R$, we obtain \begin{eqnarray*} \lefteqn{ \frac{1}{2}\int^{\infty}_{-\infty}|\phi_x(\tau,x)|^2\,dx+\frac{\epsilon}{2} \int^{\tau}_0\int^{\infty}_{-\infty}|\phi_{xx}|^2\,dx\,dt} \\ &&\le\frac{1}{2}\int^{\infty}_{-\infty}|\phi_x(0,x)|^2\,dx +\frac{1}{2\epsilon}\,|u|^2_{\infty}\int^{\tau}_0 \int^{\infty}_{-\infty}|\phi_x|^2\,dx\,dt. \end{eqnarray*} where $|u|_{\infty}=\sup_{(t,x)\in (0,\tau) \times R}\,|u(t,x)|$. Thus, $$\int^{\tau}_0\int^{\infty}_{-\infty}|\epsilon\,\phi_{xx}|^2\,dx\,dt \le |u|^2_{\infty} \int^t_0\int^{\infty}_{-\infty}|\phi_x|^2\,dx\,dt +\epsilon\int^{\infty}_{-\infty}|\phi_x(0,x)|^2\,dx$$ and $$\int^{\tau}_0\int^{\infty}_{-\infty}|\phi_t|^2\,dx\,dt \le 4|u|^2_{\infty} \int^{\tau}_0\int^{\infty}_{-\infty}|\phi_x|^2\,dx\,dt+ 2\epsilon\int^{\infty}_{-\infty}|\phi_x(0,x)|^2\,dx$$ which proves the lemma. \vspace{2mm} Now, we state the main result of the paper. \vspace{2mm} \noindent{\bf Theorem 3.5} {\it Assume that the conditions in Theorem 2.6 are satisfied and $\int \tilde{\eta}(0,x)\,dx < \infty$. Then, for $1<\gamma\le 5/3$, there exists a subsequence of $(\rho^{\epsilon}, m^{\epsilon},\phi^{\epsilon})$ such that $$(\rho^{\epsilon}(t,x),m^{\epsilon}(t,x),\phi^{\epsilon}(t,x)) \to (\rho(t,x), m(t,x),\phi(t,x))\quad\mbox{a.e. in}\;\; \Omega= [0,\tau] \times R. \leqno (3.8)$$ where the triple $(\rho,m,\phi) \in L^{\infty}_{+}(\Omega) \times L^{\infty}(\Omega) \times W^{1,\infty}(\Omega)$ is a weak solution to (1.4).} \vspace{2mm} \noindent{\bf Proof:} It follows from Lemma 3.3 that there exists a subsequence of $(\hat{\rho}^{\epsilon},\hat{m}^{\epsilon})$ such that $$(\hat{\rho}^{\epsilon}(t,x),\hat{m}^{\epsilon}(t,x)) \to (\hat{\rho}(t,x), \hat{m}(t,x)) \quad \mbox{a.e. in}\;\; \Omega.$$ by applying the results of [Di1] and [Ch]. It follows from Lemma 3.4 that using a standard diagonal process, there is a subsequence of $\phi^{\epsilon}(t,x)$ that converges a.e. in $\Omega$, weakly in $H^1(\Omega)$ and weakly-star in $W^{1,\infty}(\Omega)$ to $\phi$. Define $\rho(t,x)=\hat{\rho}(t,x)\phi(t,x),\; m(t,x)=\hat{m}(t,x)\phi(t,x)$ a.e. $(t,x) \in \Omega$. Then, the statement (3.8) holds. It follows from the first two equations of (1.10) that $$\int^{\tau}_0\int^{\infty}_{-\infty} \left((\rho^{\epsilon},m^{\epsilon}) \cdot(\psi_t-\epsilon\,\psi_{xx})+F(\rho^{\epsilon},m^{\epsilon}, \phi^{\epsilon})\cdot\psi_x\right)\,dx\,dt=0$$ for all $\psi \in C^{\infty}_c(\Omega;R^2)$. It thus follows from (3.8) and the dominated convergence theorem that (1.8) is satisfied. It follows from the third equation of (1.10) that $$\int^{\tau}_0\int^{\infty}_{-\infty}\left( (\phi^{\epsilon}_t+u^{\epsilon} \phi^{\epsilon}_x) \,\xi+\epsilon\,\phi_x\xi_x\right)\,dx\,dt=0$$ for all $\xi \in C^{\infty}_c(\Omega;R)$. Since $u^{\epsilon} \to u$ in $L^2(U)$ for any bounded rectangle $U= [0,\tau] \times [-L,L]$ and $\phi^{\epsilon} \to \phi$ weakly in $H^1(\Omega)$ it follows that $$\int^{\tau}_0\int^{\infty}_{-\infty} (\phi_t+u\phi_x)\,\xi\,dx\,dt=0$$ for all $\xi \in C^{\infty}_c(\Omega;R)$. Hence $\phi$ satisfies (1.4) a.e. in $\Omega$. $\Box$ \vspace{2mm} \noindent{\bf Corollary 3.6} {\it Suppose the entropy pair $(\eta,q)$ is defined by (3.1)-(3.2). Then $$\int^{\tau}_0\int^{\infty}_{-\infty}\left(\eta\,\xi_t+q\,\xi_x\right)\,dx\,dt \ge 0 \leqno (3.9)$$ for all $\xi \in C^{\infty}_c(\Omega;R)$ satisfying $\xi \ge 0$. That is, the third equation of (1.1) is replaced by the inequality $\eta_t+q_x \le 0$ in the sense of distributions.} \vspace{2mm} \noindent{\bf Proof:} It follows from (3.3) that $$\int^{\tau}_0\int^{\infty}_{-\infty} \left(\eta^{\epsilon}\, (\xi_t-\epsilon\,\xi_{xx}) +q^{\epsilon}\,\xi_x\right)\,dx\,dt=\epsilon\int^{\tau}_0 \int^{\infty}_{-\infty} \nabla^2\eta(y^{\epsilon},y^{\epsilon})\,\xi \,dx\,dt$$ for all $\psi \in C^{\infty}_c(\Omega;R^2)$ satisfying $\xi \ge 0$. It follows from Lemma 3.1 that the right hand side of this equality is nonnegative. Thus, by taking the limit as $\epsilon \to 0^+$ we obtain (3.9) $\Box$ \vspace{5mm} \noindent{\bf References} \vspace{2mm} \begin{enumerate} \item [{[CCS]}] K. Chueh, C. Conley and J. Smoller, Positive invariant regions for systems of nonlinear diffusions equations, Ind. U. Math. J., 26 (1979), 373-393. \item [{[CD]}] G. Q. Chen and C. M. 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