\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{ {\noindent\small 2006 International Conference in Honor of Jacqueline Fleckinger. \newline {\em Electronic Journal of Differential Equations}, Conference 16, 2007, pp. 15--28. \newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document}\setcounter{page}{15} \title[\hfilneg EJDE/Conf/16 \hfil Nonlinear parabolic-hyperbolic equations] {Nonlinear multidimensional parabolic-hyperbolic equations} \thanks{This work was supported by CTP, the Pyrenean Work Community} \author[G.~Aguilar, L.~L\'{e}vi, M.~Madaune-Tort \hfil EJDE/Conf/16 \hfilneg] {Gloria Aguilar, Laurent L\'{e}vi, Monique Madaune-Tort}% in alphabetical order \address{Gloria Aguilar \newline Departamento de Matem\'{a}tica Aplicada, Universidad de Zaragoza, CPS, Maria de Luna~3, E-50018 Zaragoza, Spain} \email{gaguilar@unizar.es} \address{Laurent L\'{e}vi \newline Laboratoire de Math\'{e}matiques Appliqu\'{e}es, UMR 5142, Universit\'{e} de Pau, IPRA, BP 1155, F-64013 Pau Cedex, France} \email{laurent.levi@univ-pau.fr} \address{Monique Madaune-Tort \newline Laboratoire de Math\'{e}matiques Appliqu\'{e}es, UMR 5142, Universit\'{e} de Pau, IPRA, BP 1155, F-64013 Pau Cedex, France} \email{monique.madaune-tort@univ-pau.fr} \thanks{Published May 15, 2007.} \subjclass[2000]{35F25, 35K65} \keywords{Coupling problem; degenerate parabolic-hyperbolic equation; \hfill\break\indent entropy solution} \dedicatory{Dedicated to Jacqueline Fleckinger on the occasion \\ of an international conference in her honor} \begin{abstract} This paper deals with the coupling of a quasilinear parabolic problem with a first order hyperbolic one in a multidimensional bounded domain $\Omega$. In a region $\Omega_{p}$ a diffusion-advection-reaction type equation is set while in the complementary $\Omega_h\equiv \Omega \backslash \Omega_{p}$, only advection-reaction terms are taken into account. Suitable transmission conditions at the interface $\partial\Omega_{p}\cap \partial\Omega_h$ are required. We find a weak solution characterized by an entropy inequality on the whole domain. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} We are interested in a coupling of a quasilinear parabolic equation with an hyperbolic first-order one in a bounded domain $\Omega $ of $\mathbb{R}^{n}$, $n\geq 1$. The main motivation for considering this problem is the study of infiltration processes in an heterogeneous porous media. For instance, in a stratified subsoil made up of layers with different geological characteristics, the effects of diffusivity may be negligible in some layers. Such a coupled problem occurs also in fluid-dynamical theory for viscous-compressible flows around a rigid profile so that near this profile the viscosity effects have to be taken into account while at a distance they can be neglected. Another example arises in heat transfer studies as mentioned in \cite{gastaldi-quateroni}. We consider the case of two layers, that is sufficient. Then, the geometrical configuration is such that: $\overline{\Omega }=\overline{\Omega _h}\cup \overline{\Omega _{p}}$; $\Omega _h$ and $\Omega _{p}$ are two disjoint bounded domains with Lipschitz boundaries denoted by $\Gamma _{l}=\partial \Omega _{l}$, $l\in \{h,p\}$ and $\Gamma _{hp}=\Gamma _h\cap \Gamma _{p}$. In addition we set $Q=]0,T[\times \Omega $ and for $l$ in $\{h,p\}$, $Q_{l}=]0,T[\times \Omega _{l}$, $\Sigma _{l}=]0,T[\times \Gamma _{l}$. Now, for $q$ in $[0,n+1]$, $\mathcal{H}^{q}$ is the $q$-dimensional Hausdorff measure over $\mathbb{R}^{n+1}$ and for $l$ in $\{h,p\}$, $\nu _{l}$ is the outward normal unit vector defined $\mathcal{H}^{n}$-a.e. on $\Sigma_{l}$. So the interface, denoted by $\Sigma _{hp}=]0,T[\times \Gamma _{hp}$, is such that $\mathcal{H}^{n}(\overline{\Sigma_{hp}}\cap (\overline{\Sigma _{l}\backslash \Sigma _{hp}}))=0$. Now, due to a combination of conservation laws and Darcy's law, the physical model is described as follows: For any positive and finite real $T$, find a measurable and bounded function $u$ on $Q$ such that, \begin{gather} \partial _tu-\sum_{i=1}^{n}\partial _{x_{i}}(f(u)\partial _{x_{i}}{P })+g(t,x,u) =0\quad \text{in }Q_h, \label{equation dans Qh} \\ \partial _tu-\sum_{i=1}^{n}\partial _{x_{i}}(f(u)\partial _{x_{i}}{P })+g(t,x,u) =\Delta \phi (u)\quad \text{in }Q_{p}, \label{equation dans Qp} \\ u = 0\quad \text{on }]0,T[\times \partial \Omega , \label{condition de bord formelles} \\ u(0,.) = u_{0}\quad \text{on }\Omega . \label{condition initiale formelle} \end{gather} Then, suitable conditions on $u$ across the interface $\Sigma _{hp}$ must be added. As for the linear problem studied by F. Gastaldi and \textit{al.} in \cite{gastaldi-quateroni} or for the one dimensional nonlinear problem studied by G. Aguilar and \textit{al.} in \cite{aguilar-lisbona-madaune}, these transmission conditions include the continuity property of the flux through the interface formally written here as \begin{equation} -f(u)\nabla P.\nu _h=(\nabla \phi (u)+f(u)\nabla P).\nu _{p} \quad \text{on } \Sigma _{hp}. \label{conditions de raccord formelles} \end{equation} Let us mention that this problem has already been studied by the authors in \cite{aguilar-levi-madaune} for a nondecreasing flux function $f$ when $ \nabla P.\nu _h\leq 0$ a.e. on $\Gamma _{hp}$. Here, we still consider a nondecreasing flux function $f$, but we give an existence and uniqueness result holding even when $\nabla P.\nu _h\geq 0$ a.e. on $\Gamma _{hp}$. \subsection{Assumptions and notation} \label{assumption on data}The pressure $P$ is a known stationary function belonging to $W^{2,+\infty }(\Omega )$ and such that $\Delta P=0$ which is not restrictive as soon as (\ref{equation dans Qh}) and (\ref{equation dans Qp}) include some reaction terms. In addition, \begin{equation} \nabla P.\nu _h\text{ has a constant sign all along }\Gamma _{hp}\text{.} \label{hyp sur P interface} \end{equation} The reaction function $g$ belongs to $W^{1,+\infty }(]0,T[\times \Omega \times \mathbb{R})$ and we set \[ M_g'=\mathop{\rm ess\,sup}_{(t,x,u)\in ]0,T[\times \Omega \times \mathbb{R}} |\partial _{u}g(t,x,u)|\quad\text{and}\quad M_{0}=\mathop{\rm ess\,sup}_{]0,T[\times \Omega }|g_h(t,x,0)|. \] The initial data $u_{0}$ belongs to $L^{\infty }(\Omega )$. Thus we can define the nondecreasing time-depending function \begin{equation} M:t\in [0,T]\to M(t)=\Vert u_{0}\Vert _{L^{\infty }(\Omega )}e^{M_g'\,t}+M_{0}\dfrac{e^{M_g'\,t}-1}{M_g'}. \label{M(t)} \end{equation} To simplify we write $M=M(T)$. Now, we assume local hypotheses on $f$ and $\phi $. \noindent (i) The flux function $f$ is a nondecreasing Lipschitzian function on $[-M,M]$ with constant $M_{f}'$ and such that $f(0)=0$. To express the boundary conditions on the frontier of the hyperbolic area, we introduce the nonnegative function $\mathcal{F}$ defined on $[-M,M]^{3}$ by \begin{equation} \mathcal{F}(a,b,c)=\dfrac{1}{2}\{|f(a)-f(b)|-|f(c)-f(b)|+|f(a)-f(c)|\}. \label{flux} \end{equation} \noindent (ii) $\phi $ is an increasing Lipschitzian function on $[-M,M]$ such that $\phi ^{-1}$ is H\"{o}lder continuous and $\phi (0)=0$. \noindent (iii) $f\circ \phi ^{-1}$ is H\"{o}lder continuous with exponent $ \theta $ in $[1/2,+\infty [$ that is there exists a positive constant $ \mathcal{C}$ such that \begin{equation} \forall (x,y)\in [-M,M]^{2},\quad |(f\circ \phi ^{-1})(x)-(f\circ \phi ^{-1})(y)|\leq \mathcal{C}|x-y|^{\theta }. \label{hyp on Kp circ phi^-1} \end{equation} \begin{remark} \label{ondes}\rm The monotonicity of $f$ and the condition (\ref{hyp sur P interface}) involve that if a.e. on $\Gamma _{hp},\nabla P.\nu _h\leq 0$, then $\Sigma _{hp}$ is included in the set of outward characteristics for the first-order operator in the hyperbolic domain and along the interface the information is leaving the hyperbolic domain. This property has been used in \cite {aguilar-levi-madaune} to split the problem by first considering the behavior of a solution in the hyperbolic area and then in the parabolic one; if a.e. on $\Gamma _{hp},\nabla P.\nu _h\geq 0$, then $\Sigma _{hp}$ is included in the set of inward characteristics for the first-order operator in the hyperbolic domain and along the interface the information is now entering the hyperbolic domain. This property will also be used to first consider the behavior of a solution in the parabolic area and then in the hyperbolic one. \end{remark} At last, for any positive real $\mu $, $\mathop{\rm sgn}{}_{\mu }$ is the Lipschitzian approximation of the function $sgn$ defined by: \begin{equation} \forall x\in [0,+\infty[ ,\quad \mathop{\rm sgn}{}_{\mu }(x)=\min ( \frac{x}{\mu },1), \quad \mathop{\rm sgn}{}_{\mu }(-x)=-\mathop{\rm sgn}{}_{\mu }(x). \label{signe} \end{equation} For the rest of this work, $\sigma $ (resp. $\bar{\sigma}$) denotes the variable on $\Sigma _{l}$ (resp. $\Gamma _{l}$), $l\in \{h,hp,p\}$. This way, for any $t$ of $[0,T]$,$\;\sigma =(t,\bar{\sigma})$. \subsection{Functional spaces} In the sequel, $W(0,T)$ is the Hilbert space \[ W(0,T)\equiv \{v\in L^{2}(0,T;H_{0}^{1}(\Omega ));\partial _tv\in L^{2}(0,T;H^{-1}(\Omega ))\} \] equipped with the norm $\Vert w\Vert _{W(0,T)}=\big( \Vert \partial _tw\Vert _{L^{2}(0,T;H^{-1}(\Omega ))}^{2}+\Vert \nabla w\Vert _{L^{2}(Q)^{n}}^{2}\big) ^{1/2}$ and $V$ is the Hilbert space \[ V=\{v\in H^{1}(\Omega _{p}),v=0\text{ a.e. on }\Gamma _{p}\backslash \Gamma _{hp}\} \] equipped with the norm $\Vert v\Vert _{V}=\Vert \nabla v\Vert _{L^{2}(\Omega _{p})^{n}}$. We denote $\langle .,.\rangle$ the pairing between $V$ and $V'$. At last $BV( \mathcal{O}) $ with $\mathcal{O}=\Omega _h$ or $\mathcal{O}=Q_h$ is the space of summable functions $v$ with bounded total variation on $\mathcal{O}$ where the total variation is given by \[ TV_{\mathcal{O}}(v) =\sup \big\{ \int_{\mathcal{O}}v( x) div\Phi (x) dx,\;\Phi \in (\mathcal{D}( \mathcal{O}) ) ^{p}, \| \Phi \| _{(L^{\infty }(\mathcal{O}) ) ^{p}}\leq 1\big\} \] where $p$ is the dimension of the open set $\mathcal{O}$. Moreover, we denote by $\gamma v$ the trace on $\Gamma _{hp}$ or $\Sigma _{hp}$ of a function $v$ belonging to $BV(\mathcal{O})$. The concept of a weak entropy solution to (\ref{equation dans Qh})-(\ref {conditions de raccord formelles}) is defined in Section 2 through an entropy inequality in the whole domain, the boundary conditions on the outer frontier of the hyperbolic area being expressed by referring to \cite{Otto}. Then, we show some properties of such a solution in the hyperbolic area and in the parabolic one. The proof of the existence result is given in Section 3 and the uniqueness property is established in Section 4. \section{The Entropy Formulation} \subsection{Weak entropy solution} \label{definition} The definition of a weak entropy solution to (\ref {equation dans Qh})-(\ref{conditions de raccord formelles}) has to include an entropy criterion in $Q_h$ where the quasilinear first-order hyperbolic operator is set. Problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}) can be viewed as an evolutional problem for a quasilinear parabolic equation that \textit{strongly degenerates} in a fixed subdomain $Q_h$ of $Q$. As in \cite{aguilar-lisbona-madaune} or \cite {aguilar-levi-madaune}, we propose a weak formulation through a global entropy inequality in the whole $Q$, the latter giving rise to a variational equality in the parabolic domain and to an entropy inequality in the hyperbolic one so as to ensure the uniqueness. \begin{definition} \label{definition generale} \rm A function $u$ is a weak entropy solution to the coupling problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}) if $u\in L^{\infty }(Q)$, $\phi (u)\in L^{2}(0,T;V)$ and for all $\varphi \in \mathcal{D}(Q)$, $\varphi \geq 0$, for all $k\in \mathbb{R}$, \begin{equation} \begin{aligned} &\int_Q|u-k|\partial _t\varphi \,dx\,dt- \int_{Q_{p}}\nabla |\phi (u)-\phi (k)|.\nabla \varphi \,dx\,dt \\ &-\int_Q|f(u)-f(k)|\nabla P.\nabla \varphi \,dx\,dt- \int_Q\mathop{\rm sgn}(u-k) g(t,x,u)\varphi \,dx\,dt\geq 0, \end{aligned}\label{equation generale sur Q} \end{equation} for all $\zeta \in L^{1}(\Sigma _h\backslash \Sigma _{hp})$, $\zeta \geq 0$, for all $k\in \mathbb{R}$, \begin{gather} \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h\backslash \Sigma _{hp}} \mathcal{F}(u(\sigma +\tau \nu_h),0,k)\nabla P(\bar{\sigma}).\nu _h \zeta d\mathcal{H}^{n}\leq 0, \label{condition de bord dans Omega_h} \\ \mathop{\rm ess\,lim}_{t\to 0^{+}}\int_{\Omega }|u(t,x)-u_{0}(x)|dx=0. \label{condition initiale} \end{gather} \end{definition} \subsection{An entropy inequality in the hyperbolic zone} We derive from (\ref{equation generale sur Q}) and (\ref{condition de bord dans Omega_h}) an entropy inequality in the hyperbolic domain. \begin{proposition} \label{inegalite dans la zone hyperbolique} Let $u$ be a weak entropy solution to the coupling problem \eqref{equation dans Qh}-\eqref{conditions de raccord formelles}. Then for any real $k$ and any $\varphi $ of $\mathcal{D}(]0,T[\times \mathbb{R}^{n})$, $\varphi \geq 0$, \begin{equation} \begin{aligned} &-\int_{Q_h}(|u-k|\partial _t\varphi -|f(u)-f(k)|\nabla P.\nabla \varphi -\mathop{\rm sgn}(u-k) g(t,x,u)\varphi ) \,dx\,dt \\ &\leq \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}|f(u(\sigma +\tau \nu _h))-f(k)|\nabla P(\bar{\sigma}).\nu _h\varphi (\sigma )d\mathcal{H}^{n} \\ &\quad +\int_{\Sigma _h\backslash \Sigma _{hp}}|f(k)|\nabla P(\bar{\sigma }).\nu _h\varphi (\sigma )d\mathcal{H}^{n} \\ &\quad -\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h\backslash \Sigma _{hp}}|f(u(\sigma +\tau \nu _h))|\nabla P(\bar{\sigma }).\nu _h\varphi (\sigma )d\mathcal{H}^{n}. \label{rel uniq zone hyp} \end{aligned} \end{equation} \end{proposition} \begin{proof} From (\ref{equation generale sur Q}) it comes that for $\varphi $ in $\mathcal{D}(Q_h)$, $\varphi \geq 0$, \begin{equation} \int_{Q_h}(|u-k|\partial _t\varphi -|f(u)-f(k)|\nabla P.\nabla \varphi -\mathop{\rm sgn}(u-k) g(t,x,u)\varphi ) \,dx\,dt\geq 0. \label{equation generale sur Q-h} \end{equation} First, by referring to F.Otto's works in \cite{Otto}, we deduce from (\ref {equation generale sur Q-h}) that, for any real $k$ and any $\beta $ in $ L^{1}(\Sigma _h)$, the following limit exists: \begin{equation} \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h}|f(u(\sigma +\tau \nu _h))-f(k)|\nabla P(\bar{\sigma}).\nu _h\beta (\sigma )d\mathcal{H}^{n}. \label{existence sur Sigma-h} \end{equation} Then, it results from (\ref{equation generale sur Q-h}) (see \cite{Otto}) that, for any real $k$ and any $\varphi $ in $\mathcal{D}(]0,T[\times \mathbb{R} ^{n})$, $\varphi \geq 0$, \begin{align*} &-\int_{Q_h}(|u-k|\partial _t\varphi -|f(u)-f(k)|\nabla P.\nabla \varphi -\mathop{\rm sgn}(u-k) g(t,x,u)\varphi ) \,dx\,dt \\ &\leq \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h}|f(u(\sigma +\tau \nu _h))-f(k)|\nabla P(\bar{\sigma}).\nu _h\varphi (\sigma )d\mathcal{H}^{n}. \end{align*} To conclude we share the frontier of $\Omega _h$ into $\Gamma _{hp}$ and $\Gamma _h\backslash \Gamma _{hp}$ and we use boundary condition (\ref {condition de bord dans Omega_h}) on $\Sigma_h\backslash \Sigma _{hp}$. \end{proof} \subsection{A variational equality in the parabolic zone} \label{etude zone parabolique}We give now some information on the regularity for $\partial _tu$ in $Q_{p}$ and we derive from (\ref{equation generale sur Q}) a variational equality satisfied by any weak entropy solution $u$ to the coupling problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}). \begin{proposition} Let $u$ be a weak entropy solution to the coupling problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}). Then $\partial _tu$ belongs to $L^{2}(0,T;V')$. Furthermore, for any $\varphi $ in $ L^{2}(0,T;V)$, \begin{equation} \begin{aligned} &\int_{0}^{T}\langle \partial _tu,\varphi \rangle dt +\int_{Q_{p}}\nabla \phi (u).\nabla \varphi \,dx\,dt +\int_{Q_{p}}f(u)\nabla P.\nabla \varphi\,dx\,dt\\ &+\int_{Q_{p}}g(t,x,u)\varphi \,dx\,dt +\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}f(u(\sigma +\tau \nu _h))\nabla P(\bar{\sigma}).\nu _h\varphi d \mathcal{H}^{n}=0. \end{aligned} \label{relation dans Omega-p} \end{equation} \end{proposition} \begin{remark} \rm This proposition is proved in \cite[Proposition 3.4]{aguilar-levi-madaune} independently of any condition on the hyperbolic characteristics on $\Sigma _{hp}$. \end{remark} \section{The Existence Result} In this section, we will prove the existence of a weak entropy solution. \begin{theorem}\label{ existence gene} The coupling problem \eqref{equation dans Qh}--\eqref{conditions de raccord formelles} has at least a weak entropy solution. \end{theorem} To construct a weak entropy solution to Problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}), we work successively in the hyperbolic domain and in the parabolic one or vice-versa. Indeed, thanks to Remark \ref{ondes}, when a.e. on $\Gamma _{hp}\;\nabla P.\nu _h\leq 0$, we can begin by working in the hyperbolic zone while, when a.e. on $\Gamma _{hp}\;\nabla P.\nu _h\geq 0$, we can begin by working in the parabolic area. \subsection{Waves going from $Q_h$ to $Q_{p}$} In this section we suppose that, a.e. on $\Gamma _{hp},\;\nabla P.\nu _h\leq 0$. The existence of a weak entropy solution to Problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}) is already proved in \cite{aguilar-levi-madaune} by the viscosity method. Here, we give a different proof of this result. First, thanks to \cite{Otto}, there exists one and only one function $w_h$ in $L^{\infty }(Q_h)$ such that for all $\varphi \in \mathcal{D}(Q_h)$, $\varphi \geq 0$, for all $k\in \mathbb{R}$, \begin{equation} \int_{Q_h}(|w_h-k|\partial _t\varphi -|f(w_h)-f(k)|\nabla P.\nabla \varphi -\mathop{\rm sgn}( w_h-k) g(t,x,w_h)\varphi ) \,dx\,dt\geq 0, \label{entropie hyperbolique} \end{equation} for all $\zeta \in L^{1}(\Sigma _h)$, $\zeta \geq 0$, for all $k\in \mathbb{R}$, \begin{gather} \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h} \mathcal{F}(w_h(\sigma +\tau \nu _h),0,k)\nabla P(\bar{\sigma}).\nu _h\zeta d\mathcal{H}^{n}\leq 0, \label{condition de bord hyp} \\ \mathop{\rm ess\,lim}_{t\to 0^{+}}\int_{\Omega _h}|w_h(t,x)-u_{0}(x)|dx=0. \label{condition initiale hyp} \end{gather} Then, thanks to \cite{gagneux-madaune}, there exists one and only one function $w_{p}$ in $L^{\infty }(Q_{p})$ such that $\phi (w_{p})\in L^{2}(0,T;V)$, $\partial _tw_{p}\in L^{2}(0,T;V')$ and for all $\varphi \in L^{2}(0,T;V)$, \begin{gather} \begin{aligned} &\int_{0}^{T}\langle \partial _tw_{p},\varphi \rangle dt +\int_{Q_{p}}\nabla \phi (w_{p}).\nabla \varphi \,dx\,dt +\int_{Q_{p}}f(w_{p})\nabla P.\nabla \varphi \,dx\,dt \\ &+\int_{Q_{p}}g(t,x,w_{p})\varphi \,dx\,dt +\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma_{hp}}f(w_h(\sigma +\tau \nu _h))\nabla P(\bar{\sigma}).\nu _h\varphi d\mathcal{H}^{n}=0, \end{aligned} \label{equation parabolique} \\ \mathop{\rm ess\,lim}_{t\to 0^{+}}\int_{\Omega _{p}}|w_{p}(t,x)-u_{0}(x)|dx=0. \label{condition initiale para} \end{gather} Indeed the mapping $$ \varphi \longmapsto - \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}f(w_h(\sigma +\tau \nu _h))\nabla P(\bar{\sigma}).\nu _h\varphi d\mathcal{H}^{n} $$ belongs to $L^{\infty }(0,T;V')$. Therefore to prove Theorem \ref{ existence gene}, we are going to establish the following lemma. \begin{lemma} \label{lemmehp} Let $u$ be defined by $u=w_h$ in $Q_h$ and $u=w_{p}$ in $Q_{p}$. Then $u$ is a weak entropy solution to the coupling problem \eqref{equation dans Qh}-\eqref{conditions de raccord formelles}. Moreover if $u_{0_{\mid {\Omega _h}}}$ belongs to $BV(\Omega _h)$, then $u_{\mid{Q_h}}$ belongs to $BV(Q_h)$ and \[ \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}|f(u(\sigma +\tau \nu _h))-f(\gamma u(\sigma ))|d\mathcal{H}^{n}=0 \] where $\gamma u(\sigma )$ is the trace on $\Sigma _{hp}$ in the BV-sense of the BV-function $u_{\mid{Q_h}}$. \end{lemma} \begin{proof} First note that $u\in L^{\infty }(Q)$ and $\phi (u)\in L^{2}(0,T;V)$. Let $\varphi $ be in $\mathcal{D}(Q)$, $\varphi \geq 0$ and let $k$ be in $\mathbb{R}$. As in the proof of Proposition \ref{inegalite dans la zone hyperbolique}, we derive from \eqref{entropie hyperbolique} the following inequality \begin{equation} \begin{aligned} &\int_{Q_h}(|w_h-k|\partial _t\varphi -|f(w_h) -f(k)|\nabla P.\nabla \varphi -\mathop{\rm sgn}(w_h-k) g(t,x,w_h)\varphi ) \,dx\,dt \\ &\geq -\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}|f(w_h(\sigma +\tau \nu _h))-f(k)|\nabla P(\bar{\sigma}).\nu _h\varphi (\sigma ) d\mathcal{H}^{n}. \end{aligned} \label{1h} \end{equation} Then, we choose in (\ref{equation parabolique}) the test-function $\varphi \mathop{\rm sgn}{}_{\mu }(\phi (w_{p})-\phi (k))$. It follows: \begin{align} &-\int_{0}^{T} \langle \partial _tw_{p},\mathop{\rm sgn}{}_{\mu }(\phi (w_{p})-\phi (k))\varphi \rangle dt \nonumber \\ &-\int_{Q_{p}}\mathop{\rm sgn}{}_{\mu }(\phi (w_{p})-\phi (k))\;\nabla (\phi (w_{p})-\phi (k)).\nabla \varphi \,dx\,dt\;\;\;\; \nonumber \\ &-\int_{Q_{p}}\mathop{\rm sgn}{}_{\mu }(\phi (w_{p})-\phi (k))(f(w_{p})-f(k))\nabla P.\nabla \varphi \,dx\,dt \nonumber \\ &-\int_{Q_{p}}g(t,x,w_{p})\mathop{\rm sgn}{}_{\mu }(\phi (w_{p})-\phi (k))\varphi \,dx\,dt \nonumber \\ &=\int_{Q_{p}}\mathop{\rm sgn}{}_{\mu }'(\phi (w_{p})-\phi (k))\;\nabla (\phi (w_{p})-\phi (k)).\nabla (\phi (w_{p})-\phi (k))\varphi \,dx\,dt \nonumber \\ &\quad +\int_{Q_{p}}\mathop{\rm sgn}{}_{\mu }'(\phi (w_{p})-\phi (k))(f(w_{p})-f(k))\nabla P.\nabla (\phi (w_{p})-\phi (k))\varphi \,dx\,dt \nonumber \\ &\quad +\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}f(w_h(\sigma +\tau \nu _h))\nabla P.\nu _h\mathop{\rm sgn}{}_{\mu }(\phi (w_{p})-\phi (k))\varphi d\mathcal{H}^{n} \nonumber \\ &\quad +\int_{\Sigma _{hp}}\mathop{\rm sgn}{}_{\mu }(\phi (w_{p})-\phi (k))f(k)\nabla P.\nu _{p}\varphi d\mathcal{H}^{n}. \label{wp} \end{align} Thanks to (\ref{hyp on Kp circ phi^-1}) and to the Cauchy-Scharwz inequality there exists a positive constant $\mathcal{C}$ such that : \begin{align*} &\int_{Q_{p}}\mathop{\rm sgn}{}_{\mu }'(\phi (w_{p})-\phi (k))\;\nabla (\phi (w_{p})-\phi (k)).\nabla (\phi (w_{p})-\phi (k))\varphi \,dx\,dt \\ &+\int_{Q_{p}}\mathop{\rm sgn}{}_{\mu }'(\phi (w_{p})-\phi (k))(f(w_{p})-f(k))\nabla P.\nabla (\phi (w_{p})-\phi (k))\varphi \,dx\,dt \\ &\geq -\mathcal{C}\int_{Q_{p}}|\phi (w_{p})-\phi (k)|^{2\theta }\mathop{\rm sgn}{}_{\mu }'(\phi (w_{p})-\phi (k))\varphi \,dx\,dt, \end{align*} and the term in the right-hand side goes to $0$ with $\mu $ as $\theta \geq 1/2$ thanks to the Lebesgue's bounded convergence theorem. In the first term of (\ref{wp}), we use an integration by parts formula based on a convexity inequality (see e.g. \cite{gagneux-madaune}, the Mignot-Bamberger Lemma) to obtain \[ -\int_{0}^{T}\langle \partial _tw_{p},\mathop{\rm sgn}{}_{\mu }(\phi (w_{p}) -\phi (k))\varphi \rangle dt =\int_{Q_{p}}\Big(\int_{k}^{w_{p}}\mathop{\rm sgn}{}_{\mu }(\phi (r) -\phi (k))dr\Big) \partial _t\varphi \,dx\,dt. \] Therefore, we are able to pass to the limit in (\ref{wp}) when $\mu $ approaches $0^{+}$ in all the integrals over $Q_{p}$. For the one on $\Sigma _{hp}$, we argue from \eqref{entropie hyperbolique} and \cite{Otto} that \eqref{existence sur Sigma-h} is valid for $w_h$. Therefore there exists $\theta $ in $L^{\infty }(\Sigma _{hp})$ such that for any $\beta $ in $L^{1}(\Sigma _{hp})$, \begin{equation} \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}f(w_h(\sigma +\tau \nu _h))\nabla P(\bar{\sigma}).\nu _h\beta (\sigma )d\mathcal{H}^{n}=\int_{\Sigma _{hp}}\theta (\sigma )\beta (\sigma )d\mathcal{H}^{n}. \label{pbordh} \end{equation} Therefore, we can use that \[ \lim_{\mu \to 0^{+}}\int_{\Sigma _{hp}}\theta (\sigma )\mathop{\rm sgn}{}_{\mu }(\phi (w_{p})-\phi (k))\varphi d\mathcal{H} ^{n}=\int_{\Sigma _{hp}}\theta (\sigma )\mathop{\rm sgn}(\phi (w_{p})-\phi (k))\varphi d\mathcal{H}^{n}. \] After all, we obtain \begin{equation} \begin{aligned} &\int_{Q_{p}}(|w_{p}-k|\partial _t\varphi -|f(w_{p})-f(k)| \nabla P.\nabla \varphi -\mathop{\rm sgn}(w_{p}-k) g(t,x,w_{p})\varphi ) \,dx\,dt \\ &-\int_{Q_{p}}\nabla |\phi (w_p)-\phi (k)|.\nabla \varphi \,dx\,dt \\ &\geq \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}(f(w_h(\sigma +\tau \nu _h))-f(k)) \mathop{\rm sgn}(w_{p}(\sigma )-k) \nabla P.\nu _h\varphi d\mathcal{H}^{n} \end{aligned} \label{1p} \end{equation} where in (\ref{1p}), $w_{p}(\sigma )$ is defined as $\phi ^{-1}(\phi (w_{p}(\sigma )))$ and belongs to $L^{\infty }(\Sigma _{hp})$. By adding the inequalities (\ref{1h}) and (\ref{1p}), we obtain \begin{align*} &\int_Q|u-k|\partial _t\varphi \,dx\,dt- \int_{Q_{p}}\nabla |\phi (u)-\phi (k)|.\nabla \varphi \,dx\,dt \\ &-\int_Q|f(u)-f(k)|\nabla P.\nabla \varphi \,dx\,dt- \int_Q\mathop{\rm sgn}(u-k) g(t,x,u)\varphi \,dx\,dt\\ &\geq \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}(f(w_h(\sigma +\tau \nu _h))-f(k)) \mathop{\rm sgn}(w_{p}(\sigma ) -k) \nabla P(\bar{\sigma}). \nu _h\varphi (\sigma ) d\mathcal{H}^{n} \\ &\quad -\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}|f(w_h(\sigma +\tau \nu _h))-f(k)|\nabla P(\bar{\sigma}).\nu _h\varphi (\sigma ) d\mathcal{H}^{n}. \end{align*} Now by using the condition $\nabla P.\nu _h\leq 0$ a.e. on $\Gamma _{hp}$, we derive that $u$ satisfies Inequality (\ref{equation generale sur Q}). At last, thanks to (\ref{condition de bord hyp}), (\ref{condition initiale hyp}) and (\ref{condition initiale para}), we can conclude that $u$ is a weak entropy solution to the coupling problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}). Now, if $u_{0 \mid {\Omega _h}}$ belongs to $BV(\Omega _h)$, it results from \cite{Bardos-leroux-nedelec} and \cite{Otto} that $w_{h \mid {Q_h}}$ belongs to $BV(Q_h)$. Therefore $u_{\mid {Q_h}}$ belongs to $BV(Q_h)$ and thanks to the properties of the trace operator from $BV(Q_h)$ into $L^{1}(\Sigma _h)$ \[ \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h}|f(u(\sigma +\tau \nu _h)) -f(\gamma u(\sigma ))|d\mathcal{H}^{n}=0 \] where $\gamma u(\sigma )$ is the trace on $\Sigma _h$ in the BV-sense of the BV-function $u_{\mid {Q_h}}$. \end{proof} \subsection{Waves going from $Q_{p}$ to $Q_h$}\label{sub} In this section we suppose that a.e. on $\Gamma _{hp},\;\nabla P.\nu_h\geq 0$. First, thanks to \cite{gagneux-madaune}, there exists one and only one function $w_{p}$ in $L^{\infty }(Q_{p})$ such that $\phi (w_{p})\in L^{2}(0,T;V)$, $\partial _tw_{p}\in L^{2}(0,T;V')$ and for all $\varphi \in L^{2}(0,T;V)$, \begin{gather} \begin{aligned} &\int_{0}^{T}\langle \partial _tw_{p},\varphi \rangle dt +\int_{Q_{p}}\nabla \phi (w_{p}).\nabla \varphi \,dx\,dt +\int_{Q_{p}}f(w_{p})\nabla P.\nabla \varphi \,dx\,dt\\ &+\int_{Q_{p}}g(t,x,w_{p})\varphi \,dx\,dt+\int_{\Sigma _{hp}}f(w_{p}(\sigma ))\nabla P(\bar{\sigma}).\nu _h\varphi d\mathcal{H} ^{n}=0, \end{aligned} \label{equation parabolique2} \\ \mathop{\rm ess\,lim}_{t\to 0^{+}}\int_{\Omega _{p}}|w_{p}(t,x)-u_{0}(x)|dx=0. \label{condition initiale para2} \end{gather} In (\ref{equation parabolique2}), $w_{p}(\sigma )$ is defined as $\phi^{-1}(\phi (w_{p}(\sigma )))$ and belongs to $L^{\infty }(\Sigma _{hp})$. Then, thanks to \cite{Otto}, there exists one and only one function $w_h$ in $L^{\infty }(Q_h)$ such that for all $\varphi \in \mathcal{D}(Q_h)$, $\varphi \geq 0$, for all $k\in \mathbb{R}$, \begin{equation} \int_{Q_h}(|w_h-k|\partial _t\varphi -|f(w_h)-f(k)|\nabla P.\nabla \varphi -\mathop{\rm sgn}( w_h-k) g(t,x,w_h)\varphi ) \,dx\,dt\geq 0; \label{entropie hyperbolique2} \end{equation} for all $\zeta \in L^{1}(\Sigma _h)$, $\zeta \geq 0$, for all $k\in \mathbb{R}$, \begin{gather} \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h\backslash \Sigma _{hp}} \mathcal{F}(w_h(\sigma +\tau \nu _h),0,k)\nabla P(\bar{\sigma}).\nu _h\zeta d\mathcal{H}^{n} \leq 0, \label{condition de bord hyp2} \\ \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}} \mathcal{F}(w_h(\sigma +\tau \nu _h),w_{p}(\sigma),k)\nabla P(\bar{ \sigma}).\nu _h \zeta d\mathcal{H}^{n} \leq 0, \label{condition de bord hyp2hp} \\ \mathop{\rm ess\,lim}_{t\to 0^{+}}\int_{\Omega _h}|w_h(t,x)-u_{0}(x)|dx=0. \label{condition initiale hyp2} \end{gather} Therefore to prove Theorem \ref{ existence gene}, we establish the following lemma. \begin{lemma} \label{lemmeph} Let $u$ be defined by $u=w_h$ in $Q_h$ and $u=w_{p}$ in $Q_{p}$. Then $u$ is a weak entropy solution to the coupling problem \eqref{equation dans Qh}-\eqref{conditions de raccord formelles}. Moreover \[ \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}|f(u(\sigma +\tau \nu _h))-f(u(\sigma ))|d\mathcal{H}^{n}=0 \] where $u(\sigma )$ is defined as $\phi ^{-1}(\phi (u(\sigma )))$. \end{lemma} \begin{proof} First $u\in L^{\infty }(Q)$ and $\phi (u)\in L^{2}(0,T;V)$. Now let $\varphi $ be in $\mathcal{D}(Q)$, $\varphi \geq 0$ and let $k$ be in $\mathbb{R}$. Following the proof of (\ref{1p}) in Lemma \ref{lemmehp}, we deduce from (\ref{equation parabolique2}) that \begin{equation} \begin{aligned} &\int_{Q_{p}}(|w_{p}-k|\partial _t\varphi -|f(w_{p})-f(k)|\nabla P.\nabla \varphi -\mathop{\rm sgn}(w_{p}-k) g(t,x,w_{p})\varphi ) \,dx\,dt \\ &-\int_{Q_{p}}\nabla |\phi (w_p)-\phi (k)|.\nabla \varphi \,dx\,dt \\ &\geq \int_{\Sigma _{hp}}|f(w_{p}(\sigma ))-f(k)|\nabla P(\bar{\sigma }).\nu _h\varphi (\sigma ) d\mathcal{H}^{n}. \end{aligned} \label{1pp2} \end{equation} Moreover, Inequality (\ref{1h}) is still satisfied by $w_h$. By adding the inequalities (\ref{1h}) and (\ref{1pp2}) we obtain \begin{align*} &\int_Q|u-k|\partial _t\varphi \,dx\,dt- \int_{Q_{p}}\nabla |\phi (u)-\phi (k)|.\nabla \varphi \,dx\,dt \\ &-\int_Q|f(u)-f(k)|\nabla P.\nabla \varphi \,dx\,dt- \int_Q\mathop{\rm sgn}( u-k) g(t,x,u)\varphi \,dx\,dt \\ &\geq \int_{\Sigma _{hp}}|f(w_{p}(\sigma ))-f(k)|\nabla P(\bar{\sigma }).\nu _h\varphi (\sigma ) d\mathcal{H}^{n} \\ &\quad -\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}|f(w_h(\sigma +\tau \nu _h))-f(k)|\nabla P(\bar{\sigma}).\nu _h\varphi (\sigma ) d\mathcal{H}^{n}. \end{align*} Then, thanks to (\ref{condition de bord hyp2hp}) and to the condition $\nabla P.\nu _h\geq 0$ a.e. on $\Gamma _{hp}$, we obtain that $u$ satisfies Inequality (\ref{equation generale sur Q}). Now, thanks to (\ref{condition de bord hyp2}), (\ref{condition initiale hyp2}) and (\ref{condition initiale para2}), we conclude that $u$ is a weak entropy solution to the coupling problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}). At last, it results from \cite{Malek} that as $\Sigma _{hp}$ is included in the set of inward characteristics for the first order operator, the solution $w_h$ of Problem (\ref{entropie hyperbolique2})-(\ref{condition initiale hyp2}) satisfies \[ \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}|f(w_h(\sigma +\tau \nu _h))-f(w_{p}(\sigma ))|d\mathcal{H}^{n}=0. \] \end{proof} \section{The Uniqueness Property} We have seen in Lemma \ref{lemmehp} or Lemma \ref{lemmeph} that Problem (\ref {equation dans Qh})-(\ref{conditions de raccord formelles}) has at least a weak entropy solution $u$ for which there exists $\theta \in L^{1}(\Sigma _{hp})$, $|\theta |\leq M$ and \begin{equation}\label{traceforte} \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}|f(u(\sigma +\tau \nu _h))-f(\theta (\sigma ))|\nabla P.\nu _hd\mathcal{H}^{n}=0. \end{equation} Indeed, when $\nabla P.\nu _h\leq 0$ a.e. on $\Gamma _{hp}$, as soon as $u_{0 \mid {\Omega _h}}$ belongs to $BV(\Omega _h)$ then (\ref{traceforte}) is satisfied with $\theta =\gamma u$ where $\gamma u$ is the trace on $\Sigma _{hp}$ in the BV-sense of the BV-function $u_{\mid {Q_h}}$. When $\nabla P.\nu _h\geq 0$ a.e. on $\Gamma _{hp}$, (\ref {traceforte}) is satisfied with $\theta =u$ where $u$ is defined as $\phi ^{-1}(\phi (u))$ and $\phi (u)$ is the trace on $\Sigma _{hp}$ of $\phi (u)_{\mid{{Q_{p}}}}$. In this section, we prove the uniqueness property in the class of weak entropy solutions satisfying (\ref{traceforte}). Indeed, we have justified that Problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}) admits such a solution (under the additional hypothesis $u_{0 \mid{\Omega _h}}$ belongs to $BV(\Omega _h)$ when a.e. on $\Gamma _{hp}\nabla P.\nu _h\leq 0$). \subsection{Preliminaries} To use the method of doubling variables, we introduce a sequence of mollifiers $(W_{\delta })_{\delta >0}$ on $\mathbb{R}^{n+1}$ defined by \[ \forall \delta >0,\,\forall r=(t,x)\in \mathbb{R}^{n+1},\,W_{\delta }(r)=\varpi _{\delta }(t)\prod_{i=1}^{n}\varpi _{\delta }(x_{i}), \] where $(\varpi _{\delta })_{\delta >0}$ is a standard sequence of mollifiers on $\mathbb{R}$. We will use classical results on the Lebesgue set of a summable function on $Q$ and a similar property on $\Sigma $ proved in \cite{peyroutet-madaune}: \begin{lemma} \label{lemme d'unicite} Let $v$ and $w$ be in $L^{\infty }(Q_h)$ such that (\ref{equation generale sur Q-h}) and (\ref{traceforte}) hold. Then for any continuous function $\varphi $ on $\overline{Q_h}$, \begin{align*} &\lim_{\delta \to 0^{+}}\int_{Q_h}\int_{\Sigma _h\backslash \Sigma _{hp}}|f(v(r))|\nabla P(\bar{\sigma}).\nu _h\varphi (\dfrac{\tilde{\sigma} +r}{2})\mathcal{W}_{\delta }(\tilde{\sigma}-r)d\mathcal{H}_{\tilde{\sigma} }^{n}dr \\ &=\dfrac{1}{2}\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h\backslash \Sigma _{hp}}|f(v(\sigma +\tau \nu _h))|\nabla P(\bar{\sigma})\nu _h\varphi (\sigma )d\mathcal{H}^{n}, \end{align*} \begin{align*} &\lim_{\delta \to 0^{+}}\int_{Q_h}\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h\backslash \Sigma _{hp}}|f(v(\sigma +\tau \nu _h))|\nabla P(\bar{\sigma}).\nu _h\varphi ( \dfrac{\sigma +\tilde{r}}{2})\mathcal{W}_{\delta }(\sigma -\tilde{r})d \mathcal{H}_{\sigma }^{n}d\tilde{r} \\ &=\dfrac{1}{2}\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h\backslash \Sigma _{hp}}|f(v(\sigma +\tau \nu _h))|\nabla P(\bar{\sigma}).\nu _h\varphi (\sigma )d\mathcal{H}^{n}, \end{align*} and \begin{align*} &\lim_{\delta \to 0^{+}}\int_{Q_h}\int_{\Sigma _{hp}}|f(\theta _{v}(\sigma ))-f(w(\tilde{r}))|\nabla P(\bar{\sigma}).\nu _h\varphi (\dfrac{\sigma +\tilde{r}}{2})W_{\delta }(\sigma -\tilde{r})d\mathcal{H}_{\sigma }^{n}d \tilde{r}\\ &=\dfrac{1}{2}\int_{\Sigma _{hp}}|f(\theta _{v}(\sigma ))-f(\theta _{w}(\sigma ))|\nabla P(\bar{\sigma})\nu _h\varphi (\sigma )d\mathcal{H} ^{n} \end{align*} where $\theta _{v}$ (resp. $\theta _{w}$) is defined by (\ref{traceforte}) for $v$ (resp. $w$). \end{lemma} \subsection{The uniqueness theorem} \begin{lemma}\label{lemme comparaison} Let $u_1$, $u_2$ be two weak solutions to \eqref{equation dans Qh}-\eqref{conditions de raccord formelles} for initial data respectively $u_{0,1}$, $u_{0,2}$ and such that \eqref{traceforte} holds with $f(\theta_i)\nabla P.\nu_h=f(u_i)\nabla P.\nu_h$, for $i=1,2$, when $\nabla P.\nu_h \geq 0$ a.e. on $\Gamma_{hp}$. Then, for a.e. $t$ of $[0,T]$, \[ \int_{\Omega }|u_1(t,.)-u_2(t,.)|dx\leq e^{M_g'\,t} \int_{\Omega }|u_{0,1}-u_{0,2}|dx. \] \end{lemma} \begin{theorem} Let $u_{0}$ be in $L^{\infty }(\Omega ) $. The coupling problem \eqref{equation dans Qh}-\eqref{conditions de raccord formelles} admits at most one weak entropy solution $u$ such that \eqref{traceforte} holds with $f(\theta)\nabla P .\nu_h=f(u)\nabla P .\nu_h$ when $\nabla P .\nu_h \geq 0$ a.e. on $\Gamma_{hp}$. Moreover, for initial data $u_{0,1}$ and $u_{0,2}$ in $L^{\infty }(\Omega ) $ the corresponding weak entropy solutions $u_1$ and $u_2$ to \eqref{equation dans Qh}-\eqref{conditions de raccord formelles} are such that for a.e. $t$ of $[0,T]$, \[ \int_{\Omega }|u_1(t,.)-u_2(t,.)|dx\leq e^{M_g'\,t} \int_{\Omega }|u_{0,1}-u_{0,2}|dx. \] \end{theorem} \begin{proof}[Proof of Lemma \ref{lemme comparaison}] (i) We first compare the two solutions $u_1$ and $u_2$ in the parabolic zone. The lack of regularity of the time partial derivative of any weak entropy solution to (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}) requires a doubling of the time variable. Therefore, let $\chi $ be a nonnegative element of $\mathcal{D}(0,T)$. We consider $\delta $ a positive real small enough for $\alpha _{\delta }:(\tilde{t},t)\longmapsto \alpha _{\delta }(\tilde{t},t) =\chi ((t+\tilde{t})/2)\varpi _{\delta }((t-\tilde{t})/2) $ to belong to $\mathcal{D}(]0,T[\times ]0,T[)$. Then, for $\mu >0$, in (\ref{relation dans Omega-p}) for $u_1$ written in variables $(t,x)$ we consider $\varphi (t,x)=\mathop{\rm sgn}{}_{\mu }(\phi (u_1)(t,x)-\phi (u_2)(\tilde{t},x))\alpha _{\delta }(\tilde{t},t)$ and in (\ref{relation dans Omega-p}) written in variables $(\tilde{t},x)$ for $u_2$, we consider $\varphi (\tilde{t},x)=-\mathop{\rm sgn}{}_{\mu }(\phi (u_1)(t,x)-\phi (u_2)(\tilde{t},x))\alpha _{\delta }(\tilde{t},t)$. To simplify the writing, we add a ''tilde'' superscript to any function in the $ \tilde{t}$ variable. Moreover, thanks to (\ref{traceforte}) we observe that in (\ref{relation dans Omega-p}), for $i=1,2$, \[ \mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _{hp}}f(u_{i}(\sigma +\tau \nu _h))\nabla P(\bar{\sigma}).\nu _h\varphi d \mathcal{H}^{n}=\int_{\Sigma _{hp}}f(\theta _{i}(\sigma ))\nabla P.\nu _h\varphi d\mathcal{H}^{n}. \] Then, by adding up, it comes: \begin{equation} \begin{aligned} &\int_{0}^{T}\int_{0}^{T} \langle \partial _tu_1 -\partial _{\tilde{t}}\tilde{u}_2,\mathop{\rm sgn}{}_{\mu }(\phi (u_1)-\phi (\tilde{u}_2))\rangle \alpha_{\delta }dtd\tilde{t} \\ &+\int_{]0,T[\times Q_{p}}\nabla (\phi (u_1)-\phi (\tilde{u} _2)).\nabla \mathop{\rm sgn}{}_{\mu }(\phi (u_1)-\phi (\tilde{u}_2))\;\alpha _{\delta }\;\,dx\,dtd\tilde{t} \\ &+\int_{]0,T[\times Q_{p}}(f(u_1)-f(\tilde{u}_2))\nabla P.\nabla \mathop{\rm sgn}{}_{\mu }(\phi (u_1)-\phi (\tilde{u}_2))\;\alpha _{\delta }\;\,dx\,dtd \tilde{t} \\ &+\int_{]0,T[\times Q_{p}}(g(t,x,u_1)-g(\tilde{t},x,\tilde{u} _2))\mathop{\rm sgn}{}_{\mu }(\phi (u_1)-\phi (\tilde{u}_2))\;\alpha _{\delta }\;\,dx\,dtd \tilde{t} \\ &=-\int_{0}^{T}\int_{\Sigma _{hp}}f(\theta _1(\sigma ))\nabla P.\nu _h\mathop{\rm sgn}{}_{\mu }(\phi (u_1)-\phi (u_2(\widetilde{\sigma } )))\alpha _{\delta }d\mathcal{H}_{\sigma }^{n}d\tilde{t} \\ &\quad +\int_{0}^{T}\int_{\Sigma _{hp}}f(\theta _2(\widetilde{ \sigma }))\nabla P.\nu _h\mathop{\rm sgn}{}_{\mu }(\phi (u_1)-\phi (u_2(\widetilde{ \sigma })))\alpha _{\delta }d\mathcal{H}_{\tilde{\sigma}}^{n}dt. \end{aligned} \label{equation pour unicite dans Qp} \end{equation} In the left-hand side, we use the calculus of the proof of Lemma \ref{lemmehp}. So, we are able to pass to the limit in (\ref{equation pour unicite dans Qp}) when $\mu $ approaches $0^{+}$. Therefore, \begin{align*} &-\int_{]0,T[\times Q_{p}}|u_1-\tilde{u}_2|(\partial _t\alpha _{\delta }+\partial _{\tilde{t}}\alpha _{\delta })\,dx\,dtd\tilde{t} \\ &\leq \int_{]0,T[\times Q_{p}}|g(t,x,\tilde{u}_2)-g(\tilde{t},x, \tilde{u}_2)|\alpha _{\delta }\,dx\,dtd\tilde{t} \\ &\quad -\int_{0}^{T}\int_{\Sigma _{hp}}(f(\theta _1(\sigma ))-f(\theta _2(\widetilde{\sigma }))) \nabla P.\nu _h\mathop{\rm sgn}{}_{\mu }(\phi (u_1)-\phi (u_2(\widetilde{\sigma })))\alpha _{\delta }d\mathcal{H }_{\sigma }^{n}d\tilde{t}. \end{align*} Now, we come back to the definition of $\alpha _{\delta }$ to express the sum $\partial _t\alpha _{\delta }+\partial_{\tilde{t}}\alpha _{\delta }$. Then we are able to take the limit with respect to $\delta $ through the notion of the Lebesgue's set of a summable function on $]0,T[$. Therefore, as $g$ is Lipschitzian, for any $\chi $ in $\mathcal{D}(0,T)$, $\chi \geq 0$, \begin{equation} \begin{aligned} &-\int_{Q_{p}}|u_1-u_2|\chi '(t)\,dx\,dt\\ &\leq M_g'\int_{Q_{p}}|u_1-u_2|\chi(t)\,dx\,dt \\ &\quad-\int_{\Sigma _{hp}}(f(\theta _1(\sigma ))-f(\theta _2(\sigma ))) \nabla P.\nu_h\mathop{\rm sgn}(\phi (u_1)-\phi (u_2)) \chi (t)d\mathcal{H}^{n}. \end{aligned}\label{Up} \end{equation} (ii) Now, we work in the hyperbolic domain. We use a doubling method for all the variables . Let $\psi $ be such that $\psi \equiv \chi \zeta $ where $\chi $ is a function in $\mathcal{D}(0,T)$, $\chi \geq 0$, as in Part (i) and $\zeta $ is in $\mathcal{D}(\mathbb{R}^{n})$ such that: $\zeta \geq 0$, $\zeta \equiv 1$ on $Q_h$. We consider $\delta $ a positive real small enough in order that the mapping $(\tilde{t},t)\longmapsto \chi ((t+\tilde{t})/2)w_{\delta }((t-\tilde{t})/2) $ belongs to $\mathcal{D}(]0,T[\times ]0,T[)$. Then, for any positive $\delta $, we define the function $\Psi _{\delta }$ in $]0,T[\times \mathbb{R}^{n}\times ]0,T[\times \mathbb{R}^{n}$ by $\Psi _{\delta}(r,\tilde{r})=\chi ((t+\tilde{t})/2)\zeta ((x+\tilde{x}) /2)W_{\delta }(r-\tilde{r})$. Due to Proposition \ref{inegalite dans la zone hyperbolique}, Inequality (\ref{rel uniq zone hyp}) holds for $u_1$ and $u_2$. We choose in (\ref{rel uniq zone hyp}) written for $u_1$ in variables $(t,x)$, \[ k=\tilde{u}_2\equiv u_2(\tilde{t},\tilde{x})\quad\text{and}\quad \varphi (t,x)=\Psi _{\delta }(t,x,\tilde{t},\tilde{x}) \] and in (\ref{rel uniq zone hyp}) written for $u_2$ in variables $(\tilde{t},\tilde{x})$, \[ k=u_1(t,x)\quad \text{and}\quad \varphi (\tilde{t},\tilde{x})=\Psi _{\delta }(t,x, \tilde{t},\tilde{x}). \] By integrating over $Q_h$ and adding up, it comes by using \eqref{traceforte}: \begin{equation} \begin{aligned} &-\int_{Q_h\times Q_h}\!(|u_1-\tilde{u}_2|(\partial _t\Psi _{\delta }+\partial _{\tilde{t}}\Psi _{\delta })-|f(u_1)-f(\tilde{ u}_2)|(\nabla P.\nabla _x\Psi _{\delta }+\nabla \tilde{P}.\nabla _{ \tilde{x}}\Psi _{\delta })\,dr\,d\tilde{r} \\ &+\int_{Q_h\times Q_h}\mathop{\rm sgn}(u_1- \tilde{u}_2)(g(t,x,u_1)-g(\tilde{t},x,\tilde{u}_2))\Psi _{\delta } \,dr\,d\tilde{r} \\ &\leq \int_{Q_h}\int_{\Sigma _h\backslash \Sigma _{hp}}|f( \tilde{u}_2)|\nabla _xP.\nu _h\Psi _{\delta }(\sigma ,\tilde{r})d \mathcal{H}_{\sigma }^{n}d\tilde{r} \\ &\quad +\int_{Q_h}\int_{\Sigma _h\backslash \Sigma _{hp}}|f(u_1)|\nabla _{\tilde{x}}\tilde{P}.\nu _h\Psi _{\delta }(r, \tilde{\sigma})d\mathcal{H}_{\tilde{\sigma}}^{n}dr \\ &\quad -\int_{Q_h}\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h\backslash \Sigma _{hp}}|f(u_1(\sigma +\tau \nu _h))|\nabla _xP.\nu _h\Psi _{\delta }(\sigma ,\tilde{r})d\mathcal{H }_{\sigma }^{n}d\tilde{r} \\ &\quad -\int_{Q_h}\mathop{\rm ess\,lim}_{\tau \to 0^{-}}\int_{\Sigma _h\backslash \Sigma _{hp}}|f(u_2(\tilde{\sigma} +\tau \nu _h))|\nabla _{\tilde{x}}\tilde{P}.\nu _h\Psi _{\delta }(r, \tilde{\sigma})d\mathcal{H}_{\tilde{\sigma}}^{n}dr \\ &\quad +\int_{Q_h}\int_{\Sigma _{hp}}|f(\theta _1(\sigma ))-f( \tilde{u}_2)|\nabla _xP.\nu _h\Psi _{\delta }(\sigma ,\tilde{r})d \mathcal{H}_{\sigma }^{n}d\tilde{r} \\ &\quad +\int_{Q_h}\int_{\Sigma _{hp}}|f(\theta _2(\tilde{\sigma} ))-f(u_1)|\nabla _{\tilde{x}}\tilde{P}.\nu _h\Psi _{\delta }(r,\tilde{ \sigma})d\mathcal{H}_{\tilde{\sigma}}^{n}dr. \end{aligned}\label{unicite dans hyp zone} \end{equation} Then through a classical reasoning we pass to the limit with $\delta $ on the left-hand side of (\ref{unicite dans hyp zone}). On the right-hand side, we refer to Lemma \ref{lemme d'unicite}. It comes: \begin{align*} -\int_{Q_h}|u_1-u_2|\chi '(t)\,dx\,dt &\leq -\int_{Q_h}\mathop{\rm sgn}(u_1-u_2)(g(t,x,u_1)-g(t,x,u_2))\chi(t)\,dx\,dt \\ &\quad +\int_{\Sigma _{hp}}|f(\theta_1(\sigma ))-f(\theta _2(\sigma ))|\nabla _xP.\nu _h\chi (t)d\mathcal{H}^{n}. \end{align*} The Lipschitz condition for $g$ provides: for any $\chi $ of $\mathcal{D} (0,T)$, $\chi \geq 0$, \begin{equation} \begin{aligned} -\int_{Q_h}|u_1-u_2|\chi '(t)\,dx\,dt &\leq \int_{\Sigma _{hp}}|f(\theta_1(\sigma))-f(\theta _2(\sigma ))|\nabla _xP.\nu _h\chi (t)d\mathcal{H}^{n} \\ &\quad + M_g'\int_{Q_h}|u_1-u_2|\chi (t)\,dx\,dt. \end{aligned}\label{uh} \end{equation} By adding inequalities (\ref{Up}) and (\ref{uh}), we obtain \begin{align*} &-\int_Q|u_1-u_2|\chi '(t)\,dx\,dt\\ &\leq M_g'\int_Q|u_1-u_2|\chi (t)\,dx\,dt +\int_{\Sigma _{hp}}|f(\theta _1(\sigma ))-f(\theta _2(\sigma ))|\nabla _xP.\nu _h\chi (t)d\mathcal{H}^{n} \\ &\quad -\int_{\Sigma _{hp}}(f(\theta _1(\sigma )) -f(\theta _2(\sigma ))) \nabla P.\nu_h\mathop{\rm sgn} (\phi (u_1)-\phi (u_2))\chi (t)d\mathcal{H}^{n}. \end{align*} Therefore, when a.e. on $\Gamma _{hp},\;\nabla P.\nu _h\leq 0$, we have \[ |f(\theta _1(\sigma ))-f(\theta _2(\sigma ))|\nabla P.\nu _h \leq (f(\theta _1(\sigma ))-f(\theta_2(\sigma ))) \mathop{\rm sgn} (\phi (u_1)-\phi (u_2))\nabla P.\nu _h. \] Now, when a.e. on $\Gamma _{hp},\;\nabla P.\nu _h\geq 0$, a.e. on $\Sigma _{hp}$, $$ f(\theta _{i}(\sigma))\nabla P.\nu _h=f(u_{i}(\sigma )) \nabla P.\nu _h,\;i=1,2. $$ As a consequence, a.e. on $\Sigma _{hp}$, \[ (f(\theta _1(\sigma ))-f(\theta _2(\sigma ))) \nabla P.\nu _h \mathop{\rm sgn}(\phi (u_1)-\phi (u_2)) =|f(\theta _1(\sigma ))-f(\theta _2(\sigma ))|\nabla P.\nu _h. \] At last in both cases, we have for any $\chi $ of $\mathcal{D}(0,T)$, $\chi \geq 0$, \[ -\int_Q|u_1-u_2|\chi '(t)\,dx\,dt\leq M_g'\int_Q|u_1-u_2| \chi (t)\,dx\,dt. \] When $\chi $ is the element of a sequence approximating $\mathbb{I}_{[0,t]}$, $t$ being given outside a set of measure zero, the desired inequality of Lemma \ref{lemme comparaison} is obtained thanks to the initial condition (\ref{condition initiale}) for $u_1$ and $u_2$ and to the Gronwall Lemma. \end{proof} \subsection*{Comments} In this paper we have looked for solutions to the coupling problem (\ref{equation dans Qh})-(\ref{conditions de raccord formelles}). 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