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

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 46, pp. 1--31.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE--2003/46\hfil 
Weakly coupled systems of parabolic equations]
{On uniqueness and existence of entropy
 solutions of weakly coupled systems
 of nonlinear degenerate parabolic equations} 

\author[H. Holden,  K. H. Karlsen, \& N. H. Risebro\hfil EJDE--2003/46\hfilneg]
{Helge Holden, Kenneth H. Karlsen, \& Nils H. Risebro}

\address{Helge Holden \newline
    Department of Mathematical Sciences\\
    Norwegian University of Science and Technology\\
    NO--7491 Trondheim, Norway.
  and \newline
  Centre of Mathematics for Applications \\
  Department of Mathematics\\
  University of Oslo\\
  P.O.\ Box 1053, Blindern\\
  N--0316 Oslo, Norway}
\email{holden@math.ntnu.no}
\urladdr{http://www.math.ntnu.no/$\sim$holden}

\address{Kenneth H. Karlsen \newline
  Department of Mathematics\\
  University of Bergen\\
  Johs.\ Brunsgt.\ 12\\
  N--5008 Bergen, Norway.
  and \newline
  Centre of Mathematics for Applications \\
  Department of Mathematics\\
  University of Oslo\\
  P.O.\ Box 1053, Blindern\\
  N--0316 Oslo, Norway}
\email{kennethk@math.uib.no}
\urladdr{www.mi.uib.no/$\sim$kennethk}

\address{Nils H. Risebro \newline
  Centre of Mathematics for Applications \\
  Department of Mathematics\\
  University of Oslo\\
  P.O.\ Box 1053, Blindern\\
  N--0316 Oslo, Norway}
\email{nilshr@math.uio.no}
\urladdr{www.math.uio.no/$\sim$nilshr}


\date{}
\thanks{Submitted October 18, 2002. Published April 22, 2003.}
\thanks{Partially supported by the BeMatA program of the Research
Council of Norway.}
\subjclass[2000]{35K65, 65M12, 35L65}
\keywords{Nonlinear degenerate parabolic
equations, weakly coupled systems, \hfill\break\indent
entropy solution, uniqueness,
existence, finite difference method}


\begin{abstract}
  We prove existence and uniqueness of entropy solutions for
  the Cauchy problem of weakly coupled systems  of nonlinear
  degenerate parabolic equations.  We prove existence of  an
  entropy solution by demonstrating that the  Engquist-Osher
  finite difference scheme is  convergent and that any limit
  function satisfies  the entropy condition. The convergence
  proof is based on  deriving a series of a priori estimates
  and using a general $L^p$ compactness criterion.
  The uniqueness proof is an adaption of Kru\v{z}kov's
  ``doubling of variables'' proof. We also present a numerical
  example motivated by biodegradation in porous media.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\seq}[1]{\left\{#1\right\}}
\newcommand{\norm}[1]{\left\|#1\right\|}
\newcommand{\lipnorm}[1]{\abs{#1}_{\mathrm{Lip}}}
\newcommand{\supnorm}[1]{\Vert#1\Vert_{L^{\infty}(\mathbb{R}^d)}}
\newcommand{\Lennorm}[1]{\Vert#1\Vert_{L^1(\mathbb{R}^d)}}


\section{Introduction}\label{sec:intro}

In this paper we will prove existence and uniqueness of entropy
solutions for  weakly coupled systems of nonlinear (strongly)
degenerate parabolic equations the form
\begin{align}
   \label{CD}
   u^\kappa_t+ \mathop{\rm div} F^\kappa(u^\kappa)& = \Delta A^\kappa(u^{\kappa})
   + g^\kappa(U),\quad
   (x,t)\in \Pi_T,\quad \kappa=1,\dots,K.
\end{align}
Here
$U=(u^1,\dots,u^K)$,
$F^\kappa(u^\kappa)=(F^\kappa_1(u^\kappa),\dots,F^\kappa_d(u^\kappa))$,
$\Pi_T=\mathbb{R}^d\times [0,T]$,
for some $T$ positive.
The system \eqref{CD} can  more compactly be written as
\begin{equation}
   \label{1.1}
   U_t+ \mathop{\rm div} F(U) = \Delta A(U) + G(U),
\end{equation}
when we introduce
\begin{gather*}
   F_i(U)=\left(F_i^1(u^1),\dots,F_i^K(u^K)\right),\quad
   A(U)=\left(A^1(u^1),\dots,A^K(u^K)\right),\\
   G(U)=\left(g^1(U),\dots,g^K(U)\right).
\end{gather*}
We will consider the Cauchy problem for the weakly coupled system
\eqref{CD}; i.e., we require that
\begin{equation}
    \label{CD:data}
    U\big|_{t=0}=U_0\in L^1(\mathbb{R}^d;\mathbb{R}^K)\cap L^{\infty}(\mathbb{R}^d;\mathbb{R}^K).
\end{equation}
We assume that the nonlinear (convection and
diffusion) flux functions satisfy
the general conditions
\begin{equation}    \label{flux_cond}
\begin{gathered}
   F^{\kappa}\in\mathrm{Lip_{loc}}(\mathbb{R};\mathbb{R}^d),
   \quad F^{\kappa}(0)=0,\\
   A^{\kappa}\in\mathrm{Lip_{loc}}(\mathbb{R}),\quad
   A^{\kappa} \text{  is nondecreasing
   with } A^{\kappa}(0)=0,
\end{gathered}
\end{equation}
where $\kappa=1,\dots,K$.
In addition, we assume that
\begin{equation}
   \label{G_cond}
   G\in\mathrm{Lip_{loc}}(\mathbb{R}^K;\mathbb{R}^K), \quad G(0)=0.
\end{equation}

This class of nonlinear partial differential equations includes
several important equations as special cases.  When $g^\kappa$
vanishes identically for all $\kappa$, the equation \eqref{CD}
becomes $K$ scalar partial differential equations.  In particular,
the single conservation law
\[
    u_t+\mathop{\rm div} f(u)= 0,
\]
is a `simple' special case of \eqref{CD}. The regularized
conservation law
$$
 u_t+\mathop{\rm div} f(u)=\Delta u
$$
is another equation within the class analyzed here.
  Included is also the heat equation
\begin{equation}
   \label{heat}
   u_t = \Delta u,
\end{equation}
the porous medium equation
\[
 u_t = \Delta u^m, \quad m\ge 1,
\]
the two-phase reservoir flow equation
\[
 u_t +\Bigl(\frac{u^2}{u^2+ (1-u)^2}\Bigr)_x =
A(u)_{xx}, \quad A(u)=\int^u v(1-v)\, dv,
\]
as well as the nonlinear strongly degenerate convection-diffusion
equation arising the recent theory of sedimentation-consolidation
processes (see \cite{Burger:bok}):
\[
 u_t + \nabla \cdot f(u)= \Delta D(u), \quad D'\ge 0.
\]

Weakly coupled systems arise in relaxation regularizations of conservation
laws, where one studies a linear system of equations of the type
\begin{align*}
    u_{t}-\sqrt{a}u_{x}&=-g(u,v),\\
    v_{t}+\sqrt{a}v_{x}&=g(u,v).
\end{align*}
If $u$ is a scalar, then $a$ is a positive number. Furthermore,
weakly coupled systems also arise in mathematical models of
biodegradation, see Cirpka et al.~\cite{Cirpkaetal:biodegrad}, and
the numerical example in Section~\ref{sec:applic}.

Due to the nonlinearity, the mixed hyperbolic-parabolic problem
\eqref{CD}--\eqref{CD:data} will in general possess shock wave
solutions, a feature that {\it may} reflect the physical phenomenon of
breaking of waves.  This is well-known in the context of conservation
laws.  Consequently, due to this loss of regularity, it is necessary
to work with weak solutions.

A function $u^{\kappa}$ is called a weak solution if
$u^{\kappa}\in L^1\cap L^{\infty}\cap C(0,T;L^1)$,
$\nabla A^{\kappa}(u^{\kappa}) \in L^2$,
$u^{\kappa}$ satisfies \eqref{CD} in the
sense of distributions, and $u^{\kappa}(t)\to u^{\kappa}_0$ in $L^1$
as $t\downarrow 0$.

However, weak solutions are in general not uniquely determined by
their data.  We will here consider weak solutions that satisfy a
so-called Kru\v{z}kov type entropy condition (such solutions are
called entropy solutions):
\begin{equation}
   \label{entropy_cond_Kr}
   \begin{split}
      &|u^{\kappa}-k|_t + \mathop{\rm div}\bigl[\mathop{\rm sign}(u^{\kappa}-k)
      \bigl(F^{\kappa}(u^{\kappa})-F^{\kappa}(k)\bigr)\bigr]
      - \Delta \abs{A^{\kappa}(u^\kappa) - A^{\kappa}(k)} \\
       & \leq \mathop{\rm sign}(u^{\kappa}-k)g^{\kappa}(U)
      \quad\text{in } \mathcal{D}'(\Pi_T) \quad \text{for all $k\in \mathbb{R}$}.
   \end{split}
\end{equation}
For a precise statement of the definition of an entropy solution, see
Section \ref{sect:fund-def}.  For pure hyperbolic equations, the
entropy condition \eqref{entropy_cond} was introduced by Kru\v{z}kov
\cite{Kruzkov} and Vol'pert \cite{Volpert}.  For degenerate parabolic
equations, it must be attributed to Vol'pert and Hudjaev
\cite{VolHud}.  The well-posedness of the entropy solution framework
for weakly coupled system of degenerate parabolic equations is the
content of the following theorem, which is the main contribution of
this paper.


\begin{theorem}   \label{main_thm}
   Assume that the conditions in \eqref{flux_cond}
   and \eqref{G_cond} hold.
   Then there exists a unique entropy
   solution of the Cauchy problem \eqref{CD}, \eqref{CD:data}.
\end{theorem}

We remark that existence and uniqueness of entropy solutions for
weakly coupled system of first-order hyperbolic equations have been
proved by Natalini and Hanouzet \cite{NatHan} and Rohde \cite{Rohde}.

The existence assertion of Theorem \ref{main_thm} follows from the
results in Section \ref{sect:eksistens}. As was done by Evje and
Karlsen \cite{EvjeKarlsen:SJNA} and Karlsen and Risebro in
\cite{KR:Rough_Diff} for scalar equations, existence of an entropy
solution is here proved by establishing convergence of suitable finite
difference approximations.  We mention that for the existence proof
one can replace the difference approximations used in this paper by
proper adoptions of the numerical approximations studied in
\cite{EvjeKarlsen:nm,Eymard_etal_II:2000,BouGuaNat} or the vanishing
viscosity method \cite{VolHud}.  For a partial overview of numerical
methods for entropy solutions of nonlinear degenerate parabolic
equations, we refer to \cite{EspKar}.


We now continue with more details about the convergence proof.  Let
$h>0$ and $\Delta t>0$ denote the spatial and temporal discretization
parameters, respectively.  We then let $u_i^{\kappa,n}$ denote the
finite difference approximation of $u^{\kappa}(i h,n\Delta t)$.  In the
one-dimensional case, the explicit finite difference scheme takes the
form
\begin{equation}
   \label{EO_scheme_intro}
   \frac{u_i^{\kappa,n+1}-u_i^{\kappa,n}}{\Delta t} +
   D_-\Bigl(F^{\kappa,\mathrm{EO}}
   \left(u_i^{\kappa,n},u_{i+1}^{\kappa,n}\right)
   -D_+ A^{\kappa}\left(u_i^{\kappa,n}\right)\Bigr)
   = g^{\kappa}\left(U^n_i\right),
   \quad \kappa=1,\dots,K,
\end{equation}
where $D_-$ and $D_+$ are the usual backward and forward difference
operators, respectively.  In \eqref{EO_scheme_intro},
$F^{\kappa,\mathrm{EO}}$ denotes the Engquist--Osher numerical flux
function \cite{EngquistOsher:81} defined by
$$
F^{\kappa,\mathrm{EO}}(u,v)
= \frac{1}{2}\big(F^{\kappa}(u)+F^{\kappa}(v)\big)
- \int_u^v \big| \frac{d F^{\kappa}}{dr}(r)\big| \,dr.
$$
We refer to Section \ref{sect:eksistens} for precise
statements in the multi-dimensional case.

The convergence proof is based on deriving uniform $L^{\infty}$,
$L^1$, and $BV$ bounds on the approximate solution $u_h$, where
$u_h=u_h(x,t)$ denotes a piecewise constant interpolation of
$\left\{u_i^n\right\}_{i,n}$.  These bounds are readily obtained by
exploiting that the difference operator on the left-hand side of the
equality sign in \eqref{EO_scheme_intro} is $L^1$ contractive, so that
the standard estimates from hyperbolic conservation laws apply.
Equipped with the $BV$ bound, we use the difference scheme itself and
Kru\v{z}kov's interpolation lemma \cite{KruzkovII} to show that
$u_{\Delta t}$ is uniformly $L^1$ continuous in time.  Kolmogorov's
compactness criterion then immediately gives $L^1_{\mathrm{loc}}$
convergence (along a subsequence) of $\left\{u_h\right\}_{h>0}$ to a
function $u\in L^1\cap L^{\infty}\cap C([0,T];L^1)$ such that $u(t)\to
u_0$ in $L^1$ as $t\downarrow 0$.  To ensure that the limit $u$ is the
(unique) entropy solution, we first prove that the difference scheme
satisfies a so-called discrete entropy inequality and hence it
follows, by standard arguments that the entropy condition
\eqref{entropy_cond_Kr} holds true for the limit $u$. Finally, by an
energy type argument we obtain that $A(u_h)$ converges (along a
subsequence) to $A(u)$ in $L^2_{\mathrm{loc}}$ and $\nabla A(u)\in
L^2$.  For other papers dealing with numerical schemes for weakly
coupled systems, see \cite{RohdeII,OhlbergerRohde:WC} (and the
references therein).

The uniqueness assertion of Theorem \ref{main_thm}
follows from the results
in Section \ref{sect:entydighet}. The uniqueness proof
is an adaption of the celebrated ``doubling of variables'' proof
due to Kru\v{z}kov \cite{Kruzkov} for first-order equations
along with an extension to second-order equations
by Carrillo \cite{Carrillo}. To put the second-order
case in a proper perspective, let us illustrate
Kru\v{z}kov's ``doubling'' device on the simple
heat equation \eqref{heat}.

At stake is the integral inequality (from which
uniqueness readily follows\footnote{Formally, one takes a test
  function that is constant in space and
  equals the characteristic function on the time
  interval $[0,t]$. This yields $\norm{u(t)-v(t)}\le\norm{u_0-v_0}$.})
\begin{equation}
   \label{heat_uniq}
   |v-u|_t - \Delta \abs{v - u} \leq 0
   \quad\text{in } \mathcal{D}'(\Pi_T),
\end{equation}
where $v=v(x,t),u=u(x,t)$ are two entropy solutions of the heat
equation \eqref{heat}.


Following Kru\v{z}kov \cite{Kruzkov} closely, we use the entropy
inequalities for $v=v(x,t)$ and $u=u(y,s)$ to derive
\begin{equation}
   \label{heat_tmp1}
   \int \Bigl(|v-u| \left(\partial_t +\partial_s\right)\phi
   +\abs{v - u}\left(\Delta_x + \Delta_y\right) \phi \Bigr)
   \,dt\,dx\,ds\,dy\geq 0,
\end{equation}
where $\phi=\phi(x,t,y,s)$ is a test function on $\Pi_T\times\Pi_T$.
Following the guidelines in \cite{Kruzkov} once more, a
clever choice of test function  is
$$
\phi(x,t,y,s)=\psi\big(\frac{x+y}{2},\frac{t+s}{2}\big)
\omega_{\rho}\big(\frac{x-y}{2},\frac{t-s}{2}\big),
$$
where $\psi$ is again a test function and $\omega_{\rho}$ is an
approximate delta function with smoothing radius $\rho>0$.


With this choice, we have
$$
\left(\partial_t + \partial_s\right)\phi
=\left[ \left(\partial_t + \partial_s\right)
\psi\left(\tfrac{x+y}{2},\tfrac{t+s}{2}\right)\right]
\omega_{\rho}\left(\tfrac{x-y}{2},\tfrac{t-s}{2}\right),
$$
so that the singular (as $\rho\downarrow 0$) term cancels out.
However, with the second-order operator $\Delta_x + \Delta_y$ we run
into problems since there only holds that (see Section
\ref{sect:entydighet})
\begin{equation}
   \label{phi_tmp}
   \left(\Delta_x + 2\nabla_x\cdot \nabla_y  + \Delta_y\right)
   \phi=\left[ \left(\Delta_x + 2\nabla_x\cdot \nabla_y  +
   \Delta_y\right)
   \psi\left(\tfrac{x+y}{2},\tfrac{t+s}{2}\right)\right]
   \omega_{\rho}\left(\tfrac{x-y}{2},\tfrac{t-s}{2}\right).
\end{equation}
With $\psi=\psi\left(\tfrac{x+y}{2},\tfrac{t+s}{2}\right)$ and
$\omega_{\rho}
=\omega_{\rho}\left(\tfrac{x-y}{2},\tfrac{t-s}{2}\right)$,
\eqref{heat_tmp1} then takes the form
\begin{equation}
   \label{heat_tmp2}
   \int \Bigl(|v-u| \left(\partial_t + \partial_s\right)
   \psi
   +\abs{v - u}\left(\Delta_x + 2\nabla_x\cdot\nabla_y +\Delta_y\right)
   \psi \Bigr)
   \omega_{\rho}\,dt\,dx\,ds\,dy\geq \mathrm{RHS},
\end{equation}
where the singular (as $\rho\downarrow 0$) right-hand side is given by
$$
\mathrm{RHS}=2\int \abs{v - u} \nabla_x\cdot\nabla_y \phi\,dt\,dx\,ds\,dy
= -2 \int \nabla_xv\cdot\nabla_y u \mathop{\rm sign}{}'(v - u) \, \phi\,dt\,dx\,ds\,dy.
$$
The question emerges how to get rid of the $\mathrm{RHS}$ term.  At
this stage, one should recall that the ``entropy dissipation'' term
has been thrown away in the course of deriving the entropy inequality
\eqref{entropy_cond_Kr}.  At least formally, the classical derivation
of the entropy condition (see, e.g., Vol'pert and Hudjaev
\cite{VolHud}) would actually produce a right-hand side of
\eqref{heat_tmp1} of the form
\begin{equation}
   \label{heat_tmp3}
   \int \Bigl(\bigl|\nabla_x v\bigr|^2 +
   \bigl|\nabla_y u\bigr|^2\Bigr) \mathop{\rm sign}{}'(v-u)\phi\,dt\,dx\,ds\,dy.
\end{equation}
We see now that if this term is added to $\mathrm{RHS}$, the
result is
$$
\widetilde{\mathrm{RHS}}=
\int \bigl|\nabla_x v -\nabla_y u\bigr|^2\mathop{\rm sign}{}'(v-u)\phi\,dt\,dx\,ds\,dy,
$$
The advantage with this term is that $\widetilde{\mathrm{RHS}}$ has
a definite sign and can therefore be thrown away. The above argument
can be made rigorous by working with a ``smooth'' approximation of
$\mathop{\rm sign}(\cdot)$, see Section \ref{sect:entydighet}. Although the proof
of uniqueness for general second-order equations (and weakly coupled
systems of such) is more technical, the basic ideas are still those
illustrated here on the heat equation.

To finish the story, we follow again \cite{Kruzkov} when making the
change of variables $z=(x-y)/2$, $\tau=(t-s)/2$ and $\tilde{x}=(x+y)/2$,
$\tilde{t}=(t+s)/2$, which turns \eqref{heat_tmp2} into the elegant form
\begin{equation}
   \label{heat_tmp4}
   \int \Bigl(|v-u| \psi_{\tilde{t}}
   +\abs{v - u}\Delta_{\tilde{x}}\psi\Bigr)
   \omega_{\rho}(z,\tau)\, d\tilde t\, d\tilde x\, d\tau\,d z\geq 0.
\end{equation}
Sending $\rho\downarrow 0$ in \eqref{heat_tmp4}, we get
\eqref{heat_uniq} since $\psi$ was an arbitrary test function.

As we have seen, at least from point of view of carrying out the
Kru\v{z}kov proof for second-order equations, there seems to be a term
missing in the entropy condition (for the heat equation the form of
this term is hinted in \eqref{heat_tmp3}).  Here a major breakthrough
was found recently by Carrillo \cite{Carrillo}, who exploited the
assumption $\nabla A(u)\in L^2$ to ``test'' the governing equation
against $\mathop{\rm sign}(A(u)-A(c))$, a trick that eventually produced the
``entropy dissipation'' term needed for the Kru\v{z}kov proof to work.
In our context, Carrillo's trick is carried out in the proof of Lemma
\ref{lem:Carrillo} herein. In \cite{Carrillo}, scalar equations with
$f=f(u),A=A(u)$ were studied.  Adopting the ideas in \cite{Carrillo},
uniqueness results for more general scalar equations with $x,t$
dependent coefficients were proved recently by Karlsen and Risebro
\cite{KR:Rough_Unique} and Karlsen and Ohlberger \cite{KO:Unique}.  In
this paper, we follow rather closely the presentation in
\cite{KR:Rough_Unique}. To make the paper self-contained, we have
chosen to give rather detailed proofs, although parts of the proofs
are similar to those in \cite{Carrillo,KR:Rough_Unique}.

It is worthwhile pointing out that different from \cite{Carrillo}, we
work here with all the derivatives on the test functions
(\cite{Carrillo} keeps one derivative on the diffusion function) and
we exploit fully identity \eqref{phi_tmp}. We feel that this slightly
simplifies the uniqueness proof. There is also a similarity here with
the uniqueness proof for \textit{viscosity} solutions of degenerate
second-order equations \cite{Ishii}.


For some other related papers dealing with the Kru\v{z}kov's
``doubling'' device in the context of second-order (scalar) equations
of the type studied herein, see (the list is certainly incomplete)
\cite{BenilanToureIII,Carrillo:94,Otto:L1_contr,ChenDiBen,RouvreGagneux,CockGripen,BurgerEvjeKarlsen:1D_IBVP,BK:Hyp2000,Ohlberger:FVM,Mascia_etal:2000,Eymard_etal_II:2000,Eymard_etal_I:2000,KO:Unique,EvjeKarlsen:ViscError,ChenPerthame}.


Before ending this discussion about uniqueness, we would like to draw
special attention to the paper by Chen and DiBenedetto
\cite{ChenDiBen} (see also Chen and Perthame \cite{ChenPerthame})
cited in the above list, which roughly speaking includes the ``entropy
dissipation'' term into their very definition of an entropy solution.
This is thus another way of circumventing the problem with extending
Kru\v{z}kov's uniqueness proof to second-order equations. However,
from the point of view of establishing existence (i.e., convergence of
approximate solutions), this method is less satisfactory since it is a
more involved process to pass to the limit of approximate solutions in
an entropy inequality that includes the ``entropy dissipation'' term
than in the standard one \eqref{entropy_cond_Kr}.


\section{Definitions and Preliminaries}\label{sect:fund-def}

Recall that a function $\eta\colon\mathbb{R}\to\mathbb{R}$ is called an
\textit{entropy function} if it is convex and $C^2$.  For
$\kappa=1,\dots,K$, a vector-valued function
$q^{\kappa}=(q_1^{\kappa},\dots,q_d^{\kappa})\colon\mathbb{R}\to\mathbb{R}^d$ is
called an \textit{entropy flux} if it satisfies the compatibility
conditions
$$
   \frac{dq^{\kappa}}{du}(u) = \eta'(u) \frac{dF^{\kappa}}{du}(u).
$$
For $\kappa=1,\dots,K$, a function $r^{\kappa}\colon\mathbb{R}\to\mathbb{R}$ is
called a \textit{diffusion entropy flux} if it satisfies the
compatibility conditions
\begin{align*}
   \frac{dr^{\kappa}}{du}(u) = \eta'(u) \frac{dA^{\kappa}}{du}(u).
\end{align*}
For $k\in\mathbb{R}$, the function $\eta(u)=|u-k|$ is called a
\textit{Kru\v{z}kov entropy function}. The associated functions
$$
q^{\kappa}(u)=\mathop{\rm sign}(u-k)
\bigl(F^{\kappa}(u)-F^{\kappa}(k)\bigr),\quad
r^\kappa(u)=\abs{A^\kappa(u)-A^\kappa(k)}
$$
are called the \textit{Kru\v{z}kov entropy fluxes}. Observe that
$r^\kappa(u)=\mathop{\rm sign}(u-k)(A^\kappa(u)-A^\kappa(k))$.  We can now state
the following definition of an entropy solution.


\begin{definition}[Entropy Solution]     \label{def:sol} \rm
    A vector-valued function $U=(u^1,\dots,u^K)\colon\break \Pi_T\to \mathbb{R}^K$
is called an
    \textit{entropy solution} of the Cauchy problem
\eqref{CD},\eqref{CD:data}
    if for all $\kappa=1,\dots,K$:
    \begin{enumerate}
        \item $u^{\kappa}\in L^1(\Pi_T)\cap L^{\infty}(\Pi_T)\cap
          C([0,T];L^1(\mathbb{R}^d))$.
        \item $A^\kappa(u^{\kappa}) \in L^2\left([0,T];H^1(\mathbb{R}^d)\right)$.
        \item For all entropy functions $\eta\colon\mathbb{R}\to\mathbb{R}$ and
        corresponding entropy fluxes $q^{\kappa},r^{\kappa}$,
        \[
            \eta(u^{\kappa})_t + \mathop{\rm div} q^{\kappa}(u^{\kappa}) -
            \Delta r^{\kappa}(u^{\kappa})\le
            \eta'(u^{\kappa})g^{\kappa}(U) \quad \text{in } \mathcal{D}'(\Pi_T);
        \]
        that is, for any non-negative test function
        $\phi(x,t)\in C^{\infty}_0(\Pi_T)$
        \begin{equation}
            \label{entropy_cond}
            \iint_{\Pi_T}\Bigl(\eta(u^{\kappa})\phi_t +
            q^{\kappa}(u^{\kappa})\cdot \nabla \phi +
            r^{\kappa}(u^{\kappa})\Delta \phi\Bigr)\,dt\,dx \ge -
            \iint_{\Pi_T}\eta'(u^{\kappa})g^{\kappa}(U)\phi\,dt\,dx.
        \end{equation}
        \item For any ball $\mathcal{B}_{r}=\bigl\{x\in\mathbb{R}^d\mid \abs{x}\le r\bigr\}$,
        \begin{equation}
            \label{init_data_cond}
            \int_{\mathcal{B}_{r}}|u^{\kappa}(x,t)-u^{\kappa}_0(x)|\,dx \to 0
            \quad \text{essentially as $t\downarrow  0+$}.
        \end{equation}
    \end{enumerate}
\end{definition}
%------------- end of definition

We recall that it is equivalent to require that \eqref{entropy_cond}
holds for the Kru\v{z}kov entropies: for any $k\in\mathbb{R}$ and any
non-negative test function $\phi(x,t)\in C^{\infty}_0(\Pi_T)$,
\begin{equation}   \label{entropy_cond_uniq}
   \begin{aligned}
      &\iint_{\Pi_T}\Bigl(|u^{\kappa}-k|\phi_t
       + \mathop{\rm sign}(u^{\kappa}-k)
\bigl[F^{\kappa}(u^{\kappa})-F^{\kappa}(k)\bigr]\cdot\nabla\phi\\
&\qquad+ \abs{A^{\kappa}(u^{\kappa}) -
A^{\kappa}(k)}\Delta\phi\Bigr)\,dt\,dx \\
&\geq -\iint_{\Pi_T}\mathop{\rm sign}(u^{\kappa}-k) g^{\kappa}(U)\phi\,dt\,dx.
   \end{aligned}
\end{equation}
It is well-known that \eqref{entropy_cond_uniq} in particular
implies that $U$ is a weak solution, that is,
\begin{equation}
   \label{L2_weak1}
   \iint_{\Pi_T}\Bigl(u^{\kappa}\phi_t +
   F^{\kappa}(u^{\kappa})\cdot\nabla\phi
   + A^{\kappa}(u^{\kappa})\Delta\phi\Bigr)\,dt\,dx
   = -\iint_{\Pi_T}g^{\kappa}(U)\phi\,dt\,dx, \
\end{equation}
for $\kappa=1,\dots,K$ and any $\phi \in C^{\infty}_0(\Pi_T)$.  
We shall need the following
five technical lemmas to prove existence of an entropy solution.

\begin{lemma}[Crandall and Tartar \cite{CrandallTartar}]\label{lem:CranTart}
    Let $(\Omega,\mu)$ be a
    measure space and let $D\subset L^1(\Omega)$.
    Assume that if $u$ and $v$ are in $D$, then also
    $u\vee v$ is in $D$. Let $\mathcal{T}$ be a map $D\to D$ such that
    $$
    \int_{\Omega} \mathcal{T}(u)\,d\mu
    = \int_{\Omega} u\,d\mu, \quad u\in D.
    $$
    Then the following statements, valid for all $u$ and $v$ in $D$,
    are equivalent:
    \begin{enumerate}
        \item If $u\le v$, then $\mathcal{T}(u)\le \mathcal{T}(v)$.

        \item $\int_{\Omega} \bigl({(\mathcal{T}(u)-\mathcal{T}(v))}\vee 0\bigr) \,d\mu\le
        \int_{\Omega}\bigl({(u-v)}\vee 0\bigr)\,d\mu$.

        \item $\int_{\Omega}\abs{\mathcal{T}(u)-\mathcal{T}(v)}\,d\mu
        \le \int_{\Omega} \abs{u-v}\,d\mu$.
    \end{enumerate}
\end{lemma}

Let $u\colon\Pi_T \to \mathbb{R}$ be a function such that
$u(\cdot,t)\in L^1(\mathbb{R}^d)$ for all $t\in (0,T)$. By a
modulus of continuity, we mean a nondecreasing continuous function
$\nu\colon[0,\infty)\to [0,\infty)$ such that $\nu(0)=0$. We say that
$u$ has $\nu$ as a spatial modulus of continuity if
\begin{equation}
    \sup_{\abs{y}\le r} \int_{\mathbb{R}^d}
    \abs{u(x+y,t)-u(x,t)} \,dx \le \nu(r;u),
    \label{eq:spatialmodulus}
\end{equation}
(where $\nu$ may depend on $t$). We also say that $u$ has $\omega$ as
a temporal modulus of continuity if there is a modulus of continuity
$\omega(\cdot;u)$ such that for each $\tau\in (0,T)$,
\begin{equation}
    \sup_{0\le s \le \tau}
    \int_{\mathbb{R}^d} \abs{u(x,t+s)-u(x,t)}\, dx \le \omega(\tau;u),
    \quad  t\in (0,T-\tau).
    \label{eq:temporalmodulus}
\end{equation}
For proofs of Lemmas \ref{L1_compact}--\ref{le:Kruzkov} we refer to \cite{FrontBOOK}.
\begin{lemma}[$L^1$ compactness lemma]\label{L1_compact}
  Let $\seq{u_h}_{h>0}$ be a sequence of functions defined on $\Pi_T$
  and assume that we
  have that:
  \begin{enumerate}
  \item There exists a constant $C>0$, independent of $h$, such
    that
    $$
    \|u_h(\cdot,t)\|_{L^1(\mathbb{R}^d)}\leq C,\quad
    \|u_h(\cdot,t)\|_{L^{\infty}(\mathbb{R}^d)} \leq C, \quad t\in (0,T);
    $$

  \item There exists a spatial
    modulus of continuity $\nu$, independent of $h$, such that
    $$
    \|u_h(\cdot+y,t)-u_h(\cdot,t)\|_{L^1(\mathbb{R}^d)}
    \le \nu(|y|;u_h),\quad \in\mathbb{R}^d,\,\, t\in (0,T);
    $$

  \item There exists a temporal
    modulus of continuity $\omega$, independent of $h$,
    $$
    \|u_h(\cdot,t+\tau)-u_h(\cdot,t)\|_{L^1(\mathbb{R}^d)}
    \le \omega(\tau;u_h),
    \quad \text{$\tau\in(0,T)$ and $t\in (0,T-\tau)$}.
    $$
  \end{enumerate}
  Then $\seq{u_h}_{h>0}$ is compact in the strong topology of
  $L^1_{\mathrm{loc}}(\Pi_T)$.
  Moreover, any limit point of $\seq{u_h}_{h>0}$ belongs
  to $L^1(\Pi_T)\cap
  L^{\infty}(\Pi_T)\cap C([0,T];L^1(\mathbb{R}^d))$.
\end{lemma}

\begin{lemma}[$L^2$ compactness lemma]\label{L2_compact}
  Let $\seq{u_h}_{h>0}$ be a sequence of functions defined on $\Pi_T$
  and assume that we have that:
  \begin{enumerate}
  \item There exists a constant $C_1>0$, independent of $h$, such that
    $$
    \|u_h\|_{L^2(\Pi_T)} \leq C_1;
    $$
  \item There exists a constant $C_2>0$, independent of $h$, such that
    $$
    \|u_h(\cdot+y,\cdot)-u_h(\cdot,\cdot)\|_{L^2(\Pi_T)}
    \le C_2 |y|, \quad \text{$y\in \mathbb{R}^d$};
    $$

  \item There exists a constant $C_3>0$, independent of $h$, such that
    $$
    \|u_h(\cdot,\cdot+\tau)
    -u_h(\cdot,\cdot)\|_{L^2(\mathbb{R}^d\times(0,T-\tau))}
    \le C_3 \sqrt{\tau}, \quad
    \text{$\tau\in(0,T)$}.
    $$
  \end{enumerate}
  Then $\seq{u_h}_{h>0}$ is compact in the strong
  topology of $L^2_{\mathrm{loc}}(\Pi_T)$.
  Moreover, any limit point of $\seq{u_h}_{h>0}$ belongs
  to $L^2([0,T];H^1(\mathbb{R}^d))$.
\end{lemma}

\begin{lemma}[Kru\v{z}kov \cite{KruzkovII}]
  \label{le:Kruzkov}
  Let $u(x,t)$ be a bounded measurable function defined on
  $\Pi_T$. For $t\in(0,T)$ assume that $u$ possesses a spatial modulus
  of continuity
  \begin{equation}
    \int_{\mathbb{R}^d} \abs{u\left(x+\varepsilon,t\right)-u(x,t)}\,dx \le
    \nu(\abs{\varepsilon};u),
    \label{eq:Kruzassume1}
  \end{equation}
  where $\nu$ does not depend on $t$.  Suppose that for any
  $\phi\in C^{\infty}_0(\mathbb{R}^d)$ and any $t_1$,$t_2$ $\in (0,T)$,
  \begin{equation}
    \label{eq:Kruzassume2}
    \big| \int_{\mathbb{R}^d} \left(u\left(x,t_2\right)
        -u\left(x,t_1\right)\right)\phi(x)\,dx\big|
   \le \mathrm{Const}_{T}
    \Big(\,\sum_{\abs{\alpha}\le m}c_{\alpha}\norm{D^\alpha
      \phi}_{L^{\infty}\left(\mathbb{R}^d\right)}\,\Big) \abs{t_2 -t_1},
  \end{equation}
  where $\alpha$ denotes a multi-index, and $c_{\alpha}$ are
  constants not depending on $\phi$ or $t$.
  Then for any $t_1,t_2\in (0,T)$ and all $\varepsilon>0$
  \begin{equation}
    \label{eq:time-estimate}
    \int_{\mathbb{R}^d} \abs{u(x,t_2)-u(x,t_1)}\,dx \le
    C\Big(\abs{t_2-t_1}\sum_{\abs{\alpha}\le m}
      \frac{c_{\alpha}}{\varepsilon^{\abs{\alpha}}}
      +\nu(u;\varepsilon)\Big).
  \end{equation}
\end{lemma}

%------------ Section --------------------

\section{Uniqueness of Entropy Solution}
\label{sect:entydighet}

In this section, we prove uniqueness of the entropy solution. Let
$$
M^{\kappa}:=\norm{u^{\kappa}}_{L^{\infty}(\Pi_T)}, \quad
l^{\kappa}=A^{\kappa}(-M^{\kappa}),  \quad
L^{\kappa}=A^{\kappa}(M^{\kappa}),
$$
and define the function
$\big({A^{\kappa}}\big)^{-1}\colon\left[l^{\kappa},L^{\kappa}\right]\to\mathbb{R}$
by
$$
\big({A^{\kappa}}\big)^{-1}(r):=
\min\Bigl\{ \xi\in \left[-M^{\kappa},M^{\kappa}\right]
\mid A^\kappa(\xi)=r \Bigr\}.
$$
Notice that this is a lower semicontinuous function and denote by
$E^{\kappa}$ the set
$$
E^{\kappa} = \bigl\{r\in \left[l^{\kappa},L^{\kappa}\right] :
\text{$\big(A^{\kappa}\big)^{-1}$ is discontinuous at $r$}\bigr\}.
$$
Furthermore, for $\varepsilon>0$,
$$
\mathop{\rm sign}{}_{\varepsilon}(\xi)=
\begin{cases}
   -1, & \xi\le-\varepsilon,\\
   \xi/\varepsilon,& -\varepsilon< \xi< \varepsilon,\\
    1  & \xi\ge \varepsilon.
\end{cases}
$$
To be able to carry out Kru\v{z}kov's uniqueness proof in our
second order context, we need the following version of an important
lemma of Carrillo \cite{Carrillo}.

\begin{lemma}[Entropy Dissipation Term]\label{lem:Carrillo}
  Let $u^{\kappa}$ be the $\kappa$th component of an entropy weak
  solution of \eqref{CD}, \eqref{CD:data}.  Then, for any non-negative
  $\phi\in C^{\infty}_0(\Pi_T)$ and $k\in\mathbb{R}$ such that
  $A^{\kappa}(k)\notin E^{\kappa}$,
  \begin{align}
   \label{eqn:Carrillo}
   \begin{split}
 &\iint_{\Pi_T}\Bigl( |u^{\kappa}-k|\partial_t \phi +
     \mathop{\rm sign}(u^{\kappa}-k)\bigl[F^{\kappa}(u^{\kappa})-
     F^{\kappa}(k)\bigr] \cdot \nabla\phi \\
 &\quad + \abs{A^{\kappa}(u^{\kappa}) - A^{\kappa}(k)}\Delta\phi\Bigr)\,dt\,dx\\
 &= \lim_{\varepsilon\downarrow 0} \iint_{\Pi_T}
     \bigl|\nabla A^{\kappa}(u^{\kappa})\bigr|^2
     \mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(u^{\kappa})
     -A^{\kappa}(k))\phi\,dt\,dx \\
 &\quad - \iint_{\Pi_T} \mathop{\rm sign}(u^{\kappa}-k)g^{\kappa}(U)\phi\,dt\,dx \\
     & = \lim_{\varepsilon\downarrow 0} \frac{1}{\varepsilon}
     \iint_{\abs{A^{\kappa}(u^{\kappa})-A^{\kappa}(k)}< \varepsilon}
     \bigl|\nabla A^{\kappa}(u^{\kappa})\bigr|^2\phi\,dt\,dx -
     \iint_{\Pi_T} \mathop{\rm sign}(u^{\kappa}-k)g^{\kappa}(U)\phi\,dt\,dx.
   \end{split}
\end{align}
\end{lemma}

\begin{proof}
  In what follows, we define $u(t)=u_0$ for $t<0$ and $u(t)=0$ for
  $t>T$.  Throughout this proof, one should keep in mind that
  \begin{equation}
    \label{Grad_abs_Amk}
    \nabla \abs{A^{\kappa}(u^{\kappa}) - A^{\kappa}(k)}
    = \mathop{\rm sign}(u^{\kappa}-k)\nabla A^{\kappa}(u^{\kappa})
    \,\,\text{a.e.~on $\Pi_T$}.
  \end{equation}
  An entropy solution is also a weak solution, and an
  integration by parts in the weak formulation yields
  \begin{align}
    \label{L2_weak}
    \iint_{\Pi_T}&\Bigl(u^{\kappa}\phi_t +
    \bigl[F^{\kappa}(u^{\kappa}) - \nabla
    A^{\kappa}(u^{\kappa})\bigr]\cdot\nabla\phi\Bigr)\,dt\,dx =
    -\iint_{\Pi_T}g^{\kappa}(U)\phi\,dt\,dx,
  \end{align}
  for any $\phi \in C^{\infty}_0(\Pi_T)$.  In view of \eqref{flux_cond},
  \eqref{G_cond}, and Definition \ref{def:sol}, there exists a constant
  such that
  \begin{align*}
    &\Big|\iint_{\Pi_T}\Bigl(\bigl[F^{\kappa}(u^{\kappa}) -
      \nabla A^{\kappa}(u^{\kappa})\bigr]\cdot\nabla\phi +
      g^{\kappa}(U)\phi\Bigr)\,dt\,dx\Big| \\
      & \le
    \mathrm{Const} \bigl(\|u^{\kappa}\|_{L^2(\Pi_T)} +\|\nabla
    A^{\kappa}(u^{\kappa})\|_{L^2(\Pi_T)}\bigr)
    \|\phi\|_{L^2([0,T];H^1(\mathbb{R}^d))}.
  \end{align*}
  This bound implies that \eqref{L2_weak} holds for all
  $\phi \in H^1(\Pi_T)$ with $\phi|_{t=0,T}=0$.

  For $\varepsilon>0$ and $\phi\in C^{\infty}_0(\Pi_T)$, introduce the
  functions
  $$
  \mathcal{A}_{\varepsilon}^{\kappa}(z;k) = \int_k^z
  \mathop{\rm sign}{}_{\varepsilon}(A^{\kappa}(\xi)-A^{\kappa}(k))\,d\xi,\quad
  \psi^{\kappa}_{\varepsilon}\left(u^{\kappa}\right) =
  \mathop{\rm sign}{}_{\varepsilon}(A^{\kappa}(u^{\kappa})-A^{\kappa}(k))\phi.
  $$
  We claim that
  \begin{align}
    \label{Chain_Rule_Claim}
    \iint_{\Pi_T}\Bigl(\mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa};k)\phi_t
    + \bigl[F^{\kappa}(u^{\kappa}) - \nabla
    A^{\kappa}(u^{\kappa})\bigr]\cdot\nabla\psi^{\varepsilon}\Bigr)\,dt\,dx =
    -\iint_{\Pi_T}g^{\kappa}(U)\psi^{\kappa}_{\varepsilon}\,dt\,dx.
  \end{align}
  To show \eqref{Chain_Rule_Claim}, for $\varepsilon>0$, we
  introduce the
  time-regularized test function
  $$
  \psi^{\kappa,\Delta t}_{\varepsilon}(x,t) =\frac{1}{\Delta t}\int_t^{t+\Delta t}
  \psi^{\kappa}_{\varepsilon}(x,s)\,ds.
  $$
  Observe that $\psi^{\kappa}_{\varepsilon}\in L^2([0,T];H^1(\mathbb{R}))$ and
  $\psi^{\kappa,\Delta t}_{\varepsilon}\in H^1(\Pi_T)$, i.e.,
  $\psi^{\kappa,\Delta t}_{\varepsilon}$ is indeed an admissible test function in
  the weak formulation of \eqref{CD}.  Consequently, \eqref{L2_weak}
  reads
  \begin{align*}
&\iint_{\Pi_T}\Bigl(u^{\kappa}\left(\psi^{\kappa,\Delta t}_{\varepsilon}\right)_t
    + \bigl[F^{\kappa}(u^{\kappa}) - \nabla
    A^{\kappa}(u^{\kappa})\bigr]\cdot
    \nabla\psi^{\kappa,\Delta t}_{\varepsilon}\Bigr)\,dt\,dx \\
&= -\iint_{\Pi_T}g^{\kappa}(U)\psi^{\kappa,\Delta t}_{\varepsilon}\,dt\,dx.
  \end{align*}
  Furthermore
  \begin{align*}
    \iint_{\Pi_T}u^{\kappa}\left(\psi^{\kappa,\Delta t}_{\varepsilon}\right)_t\,dt\,dx
    & = \iint_{\Pi_T}u^{\kappa}
    \frac{\psi^\varepsilon(x,t+\Delta t)-\psi^{\kappa}_{\varepsilon}(x,t)}{\Delta t}\,dt\,dx \\& =
    -\iint_{\Pi_T}
    \frac{u^{\kappa}(x,t)-u^{\kappa}(x,t-\Delta t)}{\Delta t}\psi^\kappa_{\varepsilon}\,dt\,dx.
  \end{align*}
  Since $\mathcal{A}_{\varepsilon}^\kappa$ is a convex function, we have
  \begin{equation}
    \label{convex}
    \mathcal{A}_{\varepsilon}^{\kappa}(z_2;k)-\mathcal{A}_{\varepsilon}^{\kappa}(z_1;k)
    \ge \bigl(z_2-z_1\bigr)\mathop{\rm sign}{}_{\varepsilon}(A^\kappa(z_1)-A^\kappa(k)),
    \quad  z_1,z_2\in \mathbb{R}.
  \end{equation}
  In view of \eqref{convex} and the definition of
  $\psi^{\kappa}_{\varepsilon}$, we have, for a.e.~$(x,t)\in \Pi_T$,
  $$
  -\bigl(u^{\kappa}(x,t)-u^{\kappa}(x,t-\Delta t)\bigr)\psi^\kappa_{\varepsilon}
  \le -\bigl(\mathcal{A}_{\varepsilon}^{\kappa}(u^\kappa(x,t);k)
  -\mathcal{A}_{\varepsilon}^{\kappa}(u^\kappa(x,t-\Delta t);k) \bigr)\phi.
  $$
  Using this inequality we get
  \begin{align*}
    \iint_{\Pi_T}u^{\kappa}\left(\psi^{\kappa,\Delta t}_{\varepsilon}\right)_t\,dt
    dx & \le -\iint_{\Pi_T}
    \frac{\mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa}(x,t);k)-
      \mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa}(x,t-\Delta t);k)}{\Delta t}\phi\,dt\,dx
    \\ & = \iint_{\Pi_T}\mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa};k)
    \frac{\phi(x,t+\Delta t)-\phi(x,t)}{\Delta t}\,dt\,dx \\ & \to
    \iint_{\Pi_T}\mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa};k)\phi_t\,dt\,dx
    \quad \text{as $\Delta t\downarrow 0$}.
  \end{align*}
  Keeping in mind that $\psi^{\varepsilon,\Delta t}\to \psi^\varepsilon$ in
  $L^2([0,T];H^1(\mathbb{R}))$ as $\Delta t\downarrow 0$, it hence follows that
  $$
  \iint_{\Pi_T}\Bigl(\mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa};k)\phi_t
  + \bigl[F^{\kappa}(u^{\kappa}) - \nabla
  A^{\kappa}(u^{\kappa})\bigr]\cdot\nabla\psi^{\kappa}_{\varepsilon}\Bigr)\,dt\,dx
  \ge -\iint_{\Pi_T}g^{\kappa}(U)\psi^{\varepsilon}\,dt\,dx
  $$
  which is one half of \eqref{Chain_Rule_Claim}. To prove the opposite
  inequality, one proceeds exactly as before using the
  time-regularized test function $\psi^{\kappa,\Delta t}_{\varepsilon}(x,t)
  =\frac{1}{\Delta t}\int_{t-\Delta t}^t \psi^{\varepsilon}(x,s)\,ds$ and the
  inequality
  $$
  -\bigl(u^{\kappa}(x,t+\Delta t)-u^{\kappa}(x,t)\bigr)\psi^\kappa_{\varepsilon}
  \ge -\bigl(\mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa}(x,t+\Delta t);k)
  -\mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa}(x,t);k) \bigr)\phi.
  $$
  This concludes the proof of our claim \eqref{Chain_Rule_Claim}.

  Let $\phi,k$ be as stated in the lemma.  Then one can easily check
  that, as $\varepsilon\downarrow 0$,
  $$
  \mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa};k) \to
  |u^{\kappa}-k|\quad \text{a.e.~in $\Pi_T$}.
  $$
  Moreover, we have $|\mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa};k)|\le
  \abs{u^\kappa-k}$, so by Lebesgue's dominated convergence theorem
  \[
    \lim_{\varepsilon\downarrow 0} \iint_{\Pi_T}
    \mathcal{A}_{\varepsilon}^{\kappa}(u^{\kappa};k)\partial_t \phi \,dt\,dx =
    \iint_{\Pi_T} |u^{\kappa}-k|\partial_t \phi \,dt\,dx.
  \]
  Furthermore,
  \begin{align*}
    &\lim_{\varepsilon\downarrow 0}\iint_{\Pi_T}
    \Bigl(F^{\kappa}(u^{\kappa})-F^{\kappa}(k) - \nabla
    A^{\kappa}(u^{\kappa})\Bigr)\cdot \nabla \psi^{\kappa}_{\varepsilon}\,dt\,dx
    \\ & = \lim_{\varepsilon\downarrow 0}\iint_{\Pi_T}
    \Bigl(F^{\kappa}(u^{\kappa})-F^{\kappa}(k) - \nabla
    A^{\kappa}(u^{\kappa})\Bigr)\cdot
    \nabla\mathop{\rm sign}{}_{\varepsilon}(A^{\kappa}(u^{\kappa})-A^{\kappa}(k))\phi\,dt\,dx
    \\ &\quad +\lim_{\varepsilon\downarrow 0}\iint_{\Pi_T}
    \mathop{\rm sign}{}_{\varepsilon}(A^{\kappa}(u^{\kappa})-A^{\kappa}(k))
    \Bigl(F^{\kappa}(u^{\kappa})-F^{\kappa}(k) - \nabla
    A^{\kappa}(u^{\kappa})\Bigr)\cdot\nabla\phi\,dt\,dx \\ & =
    \underbrace{\lim_{\varepsilon\downarrow 0}\iint_{\Pi_T}
      \mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(u^{\kappa})-A^{\kappa}(k))
      \Bigl(F^{\kappa}(u^{\kappa})-F^{\kappa}(k)\Bigr) \cdot \nabla
      A^{\kappa}(u^{\kappa})\phi\,dt\,dx}_{I_{1}} \\ & \quad
    -\lim_{\varepsilon\downarrow 0}\iint_{\Pi_T} \bigl|\nabla
    A^{\kappa}(u^{\kappa})\bigr|^2
    \mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(u^{\kappa})-A^{\kappa}(k))\phi\,dt\,dx \\ &
    \quad +\underbrace{\lim_{\varepsilon\downarrow 0}\iint_{\Pi_T}
      \mathop{\rm sign}{}_{\varepsilon}(A^{\kappa}(u^{\kappa})-A^{\kappa}(k))
      \Bigl(F^{\kappa}(u^{\kappa})-F^{\kappa}(k) - \nabla
      A^{\kappa}(u^{\kappa})\Bigr)\cdot \nabla\phi\,dt\,dx}_{I_{2}}.
 \end{align*}
  Regarding $I_1$ we have
  \begin{align*}
    \abs{I_1} &= \Big|\lim_{\varepsilon\downarrow 0}\iint_{\Pi_T}
      \mathop{\rm div}\Big(\int_{A^{\kappa}(k)}^{A^{\kappa}(u^\kappa)}
        \mathop{\rm sign}{}_{\varepsilon}'(r-A^{\kappa}(k)) \\
 &\quad \times \Bigl(F^{\kappa}(\big(A^{\kappa}\big)^{-1}(r))
        -F^{\kappa}\big((A^{\kappa})^{-1}(A^{\kappa}(k))\big)\Bigr)
        \,dr\Big) \phi\,dt\,dx \Big| \\
 & =\Big|\lim_{\varepsilon\downarrow 0}\iint_{\Pi_T}
      \Big(\int_{A^{\kappa}(k)}^{A^{\kappa}(u^\kappa)}
        \mathop{\rm sign}{}_{\varepsilon}'(r-A^{\kappa}(k))\\
 &\quad \times \Bigl(F^{\kappa}(\big(A^{\kappa}\big)^{-1}(r))
        -F^{\kappa}\big((A^{\kappa})^{-1}(A^{\kappa}(k))\big)\Bigr)
        \,dr\Big) \cdot \nabla\phi\,dt\,dx\Big| \\
 & \le C\lim_{\varepsilon\downarrow 0}
    \frac{1}{\varepsilon}\int_{\abs{r-A^{\kappa}(k)}<\varepsilon}
    \Bigl|\big(A^{\kappa}\big)^{-1}(r)
    -\big(A^{\kappa}\big)^{-1}(A^{\kappa}(k))\Bigr|\,dr=0,
  \end{align*}
 since $A^{\kappa}(k)\not\in E^\kappa$.  Here $C$ is
  some constant that
  depends on the Lipschitz constant of $F^{\kappa}$ and $\phi$.
  Furthermore, we have
  \begin{align*}
    I_2 &= \lim_{\varepsilon\downarrow 0}\iint_{\Pi_T}
    \mathop{\rm sign}{}_{\varepsilon}(A^{\kappa}(u^{\kappa})-A^{\kappa}(k))
    \bigl(F^{\kappa}(u^{\kappa})-F^{\kappa}(k) - \nabla
    A^{\kappa}(u^{\kappa})\bigr)\cdot\nabla\phi\,dt\,dx \\&
    =\iint_{\Pi_T} \mathop{\rm sign}(u^{\kappa}-k)
    \bigl(F^{\kappa}(u^{\kappa})-F^{\kappa}(k) - \nabla
    A^{\kappa}(u^{\kappa})\bigr)\cdot \nabla\phi\,dt\,dx.
  \end{align*}
  In addition
  $$
  \lim_{\varepsilon\downarrow 0} \iint_{\Pi_T}
  \mathop{\rm sign}{}_{\varepsilon}(A^{\kappa}(u^{\kappa})-A^{\kappa}(k))g^{\kappa}(U)\phi\,dt\,dx
  =\iint_{\Pi_T} \mathop{\rm sign}(u^{\kappa}-k)g^{\kappa}(U)\phi\,dt\,dx.
  $$
  Consequently, sending $\varepsilon\downarrow 0$ in
  \eqref{Chain_Rule_Claim} and then doing an integration by parts
  (keeping \eqref{Grad_abs_Amk} in mind), we obtain the desired equality
  \eqref{eqn:Carrillo}.
\end{proof}
%--------------- end lemma

\begin{remark} \label{rmk3.2}\rm
   In the proof of Lemma \ref{lem:Carrillo}, we have in effect
   proved the following
   ``weak'' chain rule (see, e.g.,
   \cite{AltLuckhaus,Carrillo,Otto:L1_contr}):
   $$
   - \int_0^T \bigl\langle \partial_t u, \mathop{\rm sign}{}_{\varepsilon}(A(u)-A(k)) \phi
      \bigr\rangle \,dt \\
      =  \iint_{\Pi_T} \Big(\int_k^u
        \mathop{\rm sign}{}_{\varepsilon}(A(\xi)-A(k))\,d\xi\Big)
      \partial_t\phi\,dt\,dx,
   $$
   for every non-negative function $\phi\in C^{\infty}_0$ with
   $\phi|_{t=0}=\phi|_{t=T}=0$.
\end{remark}
We are now ready to prove the following theorem:
%--------- theorem

\begin{theorem}[Uniqueness]   \label{thm:uniq}
   Assume that \eqref{flux_cond} and \eqref{G_cond} hold. Let $V,U$
   be  two entropy weak solutions of \eqref{CD}, \eqref{CD:data}
   with initial data $V_0,U_0$, respectively. Then for a.e.~$t\in
   [0,T]$,
   \begin{equation}
      \label{L1_contr}
      \int_{\mathbb{R}^d}\bigl|V(x,t) - U(x,t)\bigr|\,dx
      \le \sqrt{K}\,\exp\bigl(K\|G\|_{\mathrm{Lip}} \; t\bigr)
      \int_{\mathbb{R}^d}\bigl|V_0(x) - U_0(x)\bigr|\,dx.
   \end{equation}
In particular, there exists at most one entropy
weak solution of the Cauchy problem \eqref{CD}, \eqref{CD:data}.
\end{theorem}
\begin{proof}
Let $\phi\in C^{\infty}(\Pi_T\times\Pi_T)$, $\phi\ge 0$,
$\phi=\phi(x,t,y,s)$, and
$$
V=V(x,t)=(v^1(x,t),\dots,v^K(x,t)), \quad
U=U(y,s)=(u^1(y,s),\dots,v^K(y,s)).
$$
Let us introduce the ``hyperbolic'' sets
\begin{gather*}
   \mathcal{E}^{\kappa}_v= \Bigl\{(x,t)\in \Pi_T:
   A^{\kappa}(v^{\kappa}(x,t))\in E^{\kappa}\Bigr\}, \\
\mathcal{E}^{\kappa}_u
   =\Bigl\{(y,s)\in \Pi_T :  A^{\kappa}(u^{\kappa}(y,s))\in E^{\kappa}\Bigr\}.
\end{gather*}
For later use, observe that
$ \mathop{\rm sign}(v^{\kappa}-u^{\kappa})
   =\mathop{\rm sign}(A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa}))$
 a.e.\ in $\bigl[(\Pi_T\setminus
   \mathcal{E}^\kappa_u)\times \Pi_T\bigr] \cup
   \bigl[\Pi_T\times (\Pi_T\setminus \mathcal{E}^\kappa_v)\bigr]$.
Also that  $\nabla_x A^{\kappa}(v^{\kappa})=0$ a.e.~in
$\mathcal{E}^{\kappa}_v$
and $\nabla_y A^{\kappa}(u^{\kappa})=0$ a.e.~in
$\mathcal{E}^{\kappa}_u$.
 From the entropy condition \eqref{entropy_cond_uniq} for
$v^{\kappa}=v^{\kappa}(x,t)$ with $k=u^{\kappa}(y,s)$, we have
\begin{equation}    \label{Entropy_v}
\begin{split}
 &-\iint_{\Pi_T}\Bigl(
      |v^{\kappa}-u^{\kappa}| \phi_t + \mathop{\rm sign}(v^{\kappa}-u^{\kappa})
      \bigl[F^{\kappa}(v^{\kappa})-F^{\kappa}(u^{\kappa})\bigr]\cdot
\nabla_x \phi \\
&\quad+ \abs{A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa})}\Delta_x \phi
\Bigr)\,dt\,dx\\
&\le  \iint_{\Pi_T}
      \mathop{\rm sign}(v^{\kappa}-u^{\kappa}) g^{\kappa}(V)\phi \,dt\,dx.
\end{split}
\end{equation}
Applying  Lemma \ref{lem:Carrillo} with $k$ replaced by $u^{\kappa}$,
we have for all $(y,s)\notin \mathcal{E}^{\kappa}_u$
\begin{equation} \label{Carrillo_v}
\begin{aligned}
&-\iint_{\Pi_T}\Bigl(
 |v^{\kappa}-u^{\kappa}| \phi_t +  \mathop{\rm sign}(v^{\kappa}-u^{\kappa})
 \bigl[F^{\kappa}(v^{\kappa})-F^{\kappa}(u^{\kappa})\bigr]\cdot \nabla_x \phi\\
&\quad+\abs{A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa})}\Delta_x
 \phi\Bigr)\,dt\,dx   \\
&= -\lim_{\varepsilon\downarrow  0}\iint_{\Pi_T}
      \bigl|\nabla_x A^{\kappa}(v^{\kappa})\bigr|^2
      \mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa}))
      \phi\,dt\,dx\\
&\quad+ \iint_{\Pi_T} \mathop{\rm sign}(v^{\kappa}-u^{\kappa}) g^{\kappa}(V)\phi \,dt\,dx.
\end{aligned}
\end{equation}
Integrating over the
additional variables $(y,s)$ in
\eqref{Entropy_v} and \eqref{Carrillo_v} as well
as using Lebesgue's dominated convergence theorem, we find
\begin{align}
      &-\iiiint_{\Pi_T\times\Pi_T}\Bigl(
      |v^{\kappa}-u^{\kappa}| \phi_t + \mathop{\rm sign}(v^{\kappa}-u^{\kappa})
      \bigl[F^{\kappa}(v^{\kappa})-F^{\kappa}(u^{\kappa})\bigr]\cdot
\nabla_x \phi      \notag\\
&\quad  + \abs{A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa})}\Delta_x
\phi\Bigr)      \,dt\,dx\,ds\,dy      \notag\\
& = -\iiiint_{\mathcal{E}^{\kappa}_u\times\Pi_T}
      \Bigl(|v^{\kappa}-u^{\kappa}| \phi_t +
\mathop{\rm sign}(v^{\kappa}-u^{\kappa})
      \bigl[F^{\kappa}(v^{\kappa})-F^{\kappa}(u^{\kappa})\bigr]\cdot
\nabla_x \phi  \notag\\
& \quad + \abs{A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa})}\Delta_x
\phi\Bigr)      \,dt\,dx\,ds\,dy       \notag\\
&\quad -\iiiint_{(\Pi_T\setminus\mathcal{E}^{\kappa}_u)\times\Pi_T}
      \Bigl(|v^{\kappa}-u^{\kappa}| \phi_t +
\mathop{\rm sign}(v^{\kappa}-u^{\kappa})
      \bigl[F^{\kappa}(v^{\kappa})-F^{\kappa}(u^{\kappa})\bigr]\cdot
 \nabla_x \phi      \notag\\
& \quad + \abs{A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa})}\Delta_x
 \phi\Bigr)  \,dt\,dx\,ds\,dy   \label{approx_entropy} \\
&\le  \iint_{\Pi_T\setminus \mathcal{E}^{\kappa}_u}
      \Bigl(-\lim_{\varepsilon\downarrow  0}\iint_{\Pi_T}
      \bigl|\nabla_x A^{\kappa}(v^{\kappa})\bigr|^2
      \mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa}))
      \phi\,dt\,dx\Bigr)\,ds\,dy \notag\\
&\quad+\iiiint_{\Pi_T\times\Pi_T}
      \mathop{\rm sign}(v^{\kappa}-u^{\kappa}) g^{\kappa}(V)\phi \,dt\,dx\,ds\,dy \notag\\
&= -\lim_{\varepsilon\downarrow 0} \iiiint_{(\Pi_T\setminus
\mathcal{E}^{\kappa}_u)\times\Pi_T}
      \bigl|\nabla_x A^{\kappa}(v^{\kappa})\bigr|^2
      \mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa}))
      \phi\,dt\,dx\,ds\,dy  \notag\\
&\quad+\iiiint_{\Pi_T\times\Pi_T}
      \mathop{\rm sign}(v^{\kappa}-u^{\kappa}) g^{\kappa}(V)\phi \,dt\,dx\,ds\,dy
      \notag\\
&= -\lim_{\varepsilon\downarrow  0}\iiiint_{(\Pi_T
      \setminus\mathcal{E}^{\kappa}_u)\times(\Pi_T\setminus
\mathcal{E}^{\kappa}_v)}\bigl|\nabla_xA^{\kappa}(v^{\kappa})\bigr|^2
      \mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(v^{\kappa})
      -A^{\kappa}(u^{\kappa}))\phi\,dt\,dx\,ds\,dy \notag\\
&\quad +\iiiint_{\Pi_T\times\Pi_T}
      \mathop{\rm sign}(v^{\kappa}-u^{\kappa}) g^{\kappa}(V)\phi \,dt\,dx\,ds\,dy. \notag
\end{align}
Similarly, we derive the inequality
\begin{align}
      &-\iiiint_{\Pi_T\times\Pi_T}\Bigl(
      |u^{\kappa}-v^{\kappa}| \phi_s +  \mathop{\rm sign}(u^{\kappa}-v^{\kappa})
      \bigl[F^{\kappa}(u^{\kappa})-F^{\kappa}(v^{\kappa})\bigr]\cdot
\nabla_y \phi  \notag\\
& \quad+ \abs{A^{\kappa}(u^{\kappa})-A^{\kappa}(v^{\kappa})}\Delta_y\phi
      \Bigr)\,dt\,dx\,ds\,dy \label{exact_entropy}\\
&\le  -\lim_{\varepsilon\downarrow 0}\iiiint_{(\Pi_T
        \setminus\mathcal{E}^{\kappa}_u)\times(\Pi_T\setminus
\mathcal{E}^{\kappa}_v)}\bigl|\nabla_yA^{\kappa}(u^{\kappa})\bigr|^2
      \mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(u^{\kappa})
      -A^{\kappa}(v^{\kappa}))\phi\,dt\,dx\,ds\,dy. \notag \\
&\quad + \iiiint_{\Pi_T\times\Pi_T}
\mathop{\rm sign}(u^{\kappa}-v^{\kappa}) g^{\kappa}(U)\phi \,dt\,dx\,ds\,dy. \notag
\end{align}
By adding \eqref{approx_entropy} and \eqref{exact_entropy}, we get
\begin{align}
      &-\iiiint_{\Pi_T\times\Pi_T}\Bigl(
      |v^{\kappa}-u^{\kappa}| \left(\partial_t +
\partial_s\right)\phi
      +\mathop{\rm sign}(v^{\kappa}-u^{\kappa}) \label{total_entropy} \\
&\times \bigl[F^{\kappa}(v^{\kappa})-F^{\kappa}(u^{\kappa})\bigr]
      \cdot\left(\nabla_x  + \nabla_y\right)\phi
      + \abs{A^{\kappa}(v^{\kappa})- A^{\kappa}(u^{\kappa})}
     \left(\Delta_x + \Delta_y\right)\phi\Bigr)\,dt\,dx\,ds\,dy
\notag \\
&\le -\lim_{\varepsilon\downarrow 0}\iiiint_{(\Pi_T
        \setminus\mathcal{E}^{\kappa}_u)\times(\Pi_T\setminus
\mathcal{E}^{\kappa}_v)}\Bigl( \bigl|\nabla_x A^{\kappa}(v^{\kappa})\bigr|^2
       + \bigl|\nabla_yA^{\kappa}(u^{\kappa})\bigr|^2\Bigr)
\mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(v^{\kappa})\notag \\
&-A(u^{\kappa}))\phi\,dt\,dx\,ds\,dy
+ \iiiint_{\Pi_T\times\Pi_T}
      \mathop{\rm sign}(v^{\kappa}-u^{\kappa})\bigl(g^{\kappa}(V) -
      g^{\kappa}(U)\bigr)\phi \,dt\,dx\,ds\,dy. \notag
\end{align}
We now use \eqref{total_entropy} to prove that for
any non-negative test function $\phi(x,t)\in C^{\infty}_0(\Pi_T)$,
\begin{equation} \label{entropy_claim}
\begin{aligned}
&-\iint_{\Pi_T}\Bigl(|v^{\kappa}-u^{\kappa}|\phi_t
      + \mathop{\rm sign}(v^{\kappa}-u^{\kappa})
\bigl[F^{\kappa}(v^{\kappa})-F^{\kappa}(u^{\kappa})\bigr]\cdot\nabla\phi\\
&\quad+ \bigl|A^{\kappa}(v^{\kappa}) - A^{\kappa}(u^{\kappa})\bigr|\Delta
      \phi\Bigr)\,dt\,dx \\
& \leq \iint_{\Pi_T}\mathop{\rm sign}(v^{\kappa}-u^{\kappa})
      \left(g^{\kappa}(V)-g^{\kappa}(U)\right)\phi\,dt\,dx,
\end{aligned}
\end{equation}
where $v^{\kappa}=v^{\kappa}(x,t)$, $u^{\kappa}=u^{\kappa}(x,t)$,
$\kappa=1,\dots,K$.

Following Kru\v{z}kov \cite{Kruzkov}, we introduce a non-negative
function $\delta \in C_{0}^{\infty}$, satisfying
$\delta(\sigma)=\delta(-\sigma)$,
$\delta(\sigma)= 0$ for $|\sigma|\geq 1$, and
$\int_{\mathbb{R}}\delta(\sigma)\,d\sigma=1$. For $\rho>0$ and
$t\in \mathbb{R}$, let $\delta_{\rho}(t)=
\frac{1}{\rho}\delta\bigl(\frac{t}{\rho}\bigr)$.
For $\rho>0$ and $x\in \mathbb{R}^d$, let
$\omega_{\rho}(x)=\frac{1}{\rho}\delta\bigl(\frac{x_1}{\rho}\bigr)
\dots\frac{1}{\rho}\delta\bigl(\frac{x_d}{\rho}\bigr)$.
We take $\phi=\phi(x,t,y,s)\in C^{\infty}_0(\Pi_T\times\Pi_T)$ to be
\begin{equation}
   \label{test}
   \phi(x,t,y,s)=\psi\big(\frac{x+y}{2},\frac{t+s}{2}\big)
   \omega_{\rho}\big(\frac{x-y}{2}\big)
   \delta_{\rho}\big(\frac{t-s}{2}\big),
\end{equation}
where $\psi=\psi(x,t)\in C^{\infty}_0(\Pi_T)$ is
another non-negative test function. Observe that
$$
\left(\partial_t + \partial_s\right)
\delta_{\rho}\big(\frac{t-s}{2}\big)=0,\quad \nabla_{x+y}
\omega_{\rho}\big(\frac{x-y}{2}\big)=0,\quad
\Delta_{xy}\omega_{\rho}\big(\frac{x-y}{2}\big)=0,
$$
where we have introduced the operators
$$
\nabla_{x+y}:=\nabla_x + \nabla_y, \quad \Delta_{xy}:= \Delta_x +
2\nabla_x\cdot\nabla_y + \Delta_y.
$$
After tedious but straightforward
 computations, we find that
\begin{equation} \label{phi_der_prop}
\begin{gathered}
   (\partial_t + \partial_s)\phi(x,t,y,s)
      = \Big[(\partial_t + \partial_s)\psi
      \big(\frac{x+y}{2},\frac{t+s}{2}\big)\Big]
      \omega_{\rho}\big(\frac{x-y}{2}\big)
      \delta_{\rho}\big(\frac{t-s}{2}\big),\\
\nabla_{x+y}\phi(x,t,y,s)
=\Big[\nabla_{x+y}\psi\big(\frac{x+y}{2},\frac{t+s}{2}\big)\Big]
      \omega_{\rho}\big(\frac{x-y}{2}\big)
      \delta_{\rho}\big(\frac{t-s}{2}\big),
      \\
\Delta_{xy}\phi(x,t,y,s)=\Big[\Delta_{xy}
      \psi\big(\frac{x+y}{2},\frac{t+s}{2}\big)\Big]
      \omega_{\rho}\big(\frac{x-y}{2}\big)
      \delta_{\rho}\big(\frac{t-s}{2}\big).
\end{gathered}
\end{equation}
Inserting \eqref{test} into \eqref{total_entropy}
and then using \eqref{phi_der_prop}, we get
\begin{equation}
   \label{total_entropy_nyI}
   \begin{aligned}
      &-\iiiint_{\Pi_T\times\Pi_T}
      \Bigl(\overline{I}^{\kappa}_{\mathrm{time}}(x,t,y,s)
      + \overline{I}^{\kappa}_{\mathrm{conv}}(x,t,y,s)
      + \overline{I}^{\kappa}_{\mathrm{diff}}(x,t,y,s)\Bigr)\\
&\quad\times \omega_{\rho}\big(\frac{x-y}{2}\big)
      \delta_{\rho}\big(\frac{t-s}{2}\big)\,dt\,dx\,ds\,dy \\
&+\lim_{\varepsilon\downarrow  0}\iiiint_{(\Pi_T
        \setminus\mathcal{E}^{\kappa}_u)\times(\Pi_T\setminus
        \mathcal{E}^{\kappa}_v)}\Bigl(
\bigl|\nabla_x A^{\kappa}(v^{\kappa})\bigr|^2
       + \bigl|\nabla_y A^{\kappa}(u^{\kappa})\bigr|^2\Bigr)
      \mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(v^{\kappa})\\
&-A^{\kappa}(u^{\kappa}))\phi\,dt\,dx\,ds\,dy  \\
&+\iiiint_{\Pi_T\times\Pi_T}
      \overline{I}^{\kappa}_{\mathrm{xy}}(x,t,y,s)\,dt\,dx\,ds\,dy\\
&\le \iiiint_{\Pi_T\times\Pi_T}
      \overline{I}^{\kappa}_{\mathrm{sour}}(x,t,y,s)
      \omega_{\rho}\big(\frac{x-y}{2}\big)
      \delta_{\rho}\big(\frac{t-s}{2}\big) \,dt\,dx\,ds\,dy,
\end{aligned}
\end{equation}
where
\begin{gather*}
\overline{I}^{\kappa}_{\mathrm{diff}}(x,t,y,s)=
   \abs{A^{\kappa}(v^{\kappa}(x,t))
      - A^{\kappa}(u^{\kappa}(y,s))}
   \Delta_{xy}\psi\big(\frac{x+y}{2},\frac{t+s}{2}\big),\\
\overline{I}^{\kappa}_{\mathrm{xy}}(x,t,y,s)=
   2\abs{A^{\kappa}(v^{\kappa}(x,t))
      - A^{\kappa}(u^{\kappa}(y,s))}\nabla_x\cdot\nabla_y \phi(t,x,y,s),\\
\overline{I}^{\kappa}_{\mathrm{time}}(x,t,y,s) =
   |v^{\kappa}(x,t)-u^{\kappa}(y,s)|\big(\partial_t +
  \partial_s\big)\psi\big(\frac{x+y}{2},\frac{t+s}{2}\big),\\
\begin{aligned}\overline{I}^{\kappa}_{\mathrm{conv}}(x,t,y,s)&=
   \mathop{\rm sign}(v^{\kappa}(x,t)-u^{\kappa}(y,s))
   \Bigl[F^{\kappa}(v^{\kappa}(x,t))-F^{\kappa}(u^{\kappa}(y,s))\Bigr]\\
 &\quad\cdot  \nabla_{x+y}\psi\big(\frac{x+y}{2},\frac{t+s}{2}\big),
\end{aligned}\\
\begin{aligned}
 \overline{I}^{\kappa}_{\mathrm{sour}}(x,t,y,s)
   &=\mathop{\rm sign}(v^{\kappa}(x,t)-u^{\kappa}(y,s))
  \Bigl[g^{\kappa}(V(x,t)) - g^{\kappa}(U(y,s))\Bigr]\\
  &\quad\times \psi\big(\frac{x+y}{2},\frac{t+s}{2}\big).
\end{aligned}
\end{gather*}
Observe that repeated integration by parts gives
\begin{align*}
   &-\iiiint_{\Pi_T\times\Pi_T}
   \overline{I}^{\kappa}_{\mathrm{xy}}(x,t,y,s) \,dt\,dx\,ds\,dy \\
& =-\lim_{\varepsilon\downarrow  0}
   \iiiint_{\Pi_T\times\Pi_T}
   2\Big(\int_{u^{\kappa}}^{v^{\kappa}}
   \mathop{\rm sign}{}_{\varepsilon}(A^\kappa(\xi)-A^\kappa(v^{\kappa}))\,d\xi\Big)
   \nabla_x\cdot\nabla_y \phi \,dt\,dx\,ds\,dy \\
& =\lim_{\varepsilon\downarrow  0} \iiiint_{\Pi_T\times\Pi_T}
   2\nabla_x A^{\kappa}(v^{\kappa})\cdot
   \nabla_y A^\kappa(u^{\kappa})
\mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(v^{\kappa})-A^\kappa(u^{\kappa}))\phi\,dt\,dx\,ds\,dy\\
& =\lim_{\varepsilon\downarrow  0}\iiiint_{(\Pi_T
        \setminus\mathcal{E}^{\kappa}_u)\times(\Pi_T\setminus
        \mathcal{E}^{\kappa}_v)}
   2\nabla_x A^{\kappa}(v^{\kappa})\cdot
   \nabla_y A^\kappa(u^{\kappa})
\mathop{\rm sign}{}_{\varepsilon}'(A^{\kappa}(v^{\kappa})\\
&\quad -A^\kappa(u^{\kappa}))\phi\,dt\,dx\,ds\,dy.
\end{align*}
Now since
\begin{align*}
&\bigl|\nabla_x A^{\kappa}(v^{\kappa})\bigr|^2
- 2\nabla_x A^{\kappa}(v^{\kappa})\cdot
   \nabla_y A^{\kappa}(u^{\kappa})
+ \bigl|\nabla_yA^{\kappa}(u^{\kappa})\bigr|^2\\
&= \bigl|\nabla_x A^{\kappa}(v^{\kappa})
   -\nabla_yA^{\kappa}(u^{\kappa})\bigr|^2\ge  0,
\end{align*}
it follows from \eqref{total_entropy_nyI} that
\begin{equation}
   \label{total_entropy_nyII}
   \begin{split}
      &-\iiiint_{\Pi_T\times\Pi_T}
      \Bigl(\overline{I}^{\kappa}_{\mathrm{time}}(x,t,y,s)
      + \overline{I}^{\kappa}_{\mathrm{conv}}(x,t,y,s)
      + \overline{I}^{\kappa}_{\mathrm{diff}}(x,t,y,s)\Bigr)\\
&\times \omega_{\rho}\big(\frac{x-y}{2}\big)
      \delta_{\rho}\big(\frac{t-s}{2}\big)\,dt\,dx\,ds\,dy \\
& \le \iiiint_{\Pi_T\times\Pi_T}
      \overline{I}^{\kappa}_{\mathrm{sour}}(x,t,y,s)
      \omega_{\rho}\left(\frac{x-y}{2}\right)
      \delta_{\rho}\big(\frac{t-s}{2}\big) \,dt\,dx\,ds\,dy.
\end{split}
\end{equation}
Let us introduce the change of variables
$$
\tilde{x}=\frac{x+y}{2},\quad \tilde{t}=\frac{t+s}{2},\quad
z=\frac{x-y}{2},\quad \tau=\frac{t-s}{2},
$$
which maps $\Pi_T\times \Pi_T$
into
$$
\Omega=\mathbb{R}^d\times\mathbb{R}^d \times \seq{\left(\tilde{t},\tau\right) :
0\le \tilde{t}+\tau\le T,\; 0\le \tilde{t}-\tau\le T}.
$$
As usual with this change of variables, see, e.g., \cite{Kruzkov},
$$
\left(\partial_t + \partial_s\right)\psi
\big(\frac{x+y}{2},\frac{t+s}{2}\big)=
\psi_{\tilde{t}}(\tilde{x},\tilde{t}),\quad
\nabla_{x+y}\phi(x,t,y,s)=\nabla_{\tilde{x}}\psi(\tilde{x},\tilde{t}).
$$
But in addition it has the wonderful
property of completely diagonalizing the operator $\Delta_{xy}$:
$$
\Delta_{xy}\psi\big(\frac{x+y}{2},\frac{t+s}{2}\big)
= \Delta_{\tilde{x}} \psi(\tilde{x},\tilde{t}).
$$
Keeping in mind that
$x=\tilde{x}+z$, $y=\tilde{x}-z$, $t=\tilde{t}+\tau$,  $s=\tilde{t}-\tau$.
We may now write \eqref{total_entropy_nyII} as
\begin{multline}
   \label{total_entropy_nyIII}-\iiiint_{\Omega}\!
      \Bigl(I^{\kappa}_{\mathrm{time}}(\tilde{x},\tilde{t},z,\tau)
      + I^{\kappa}_{\mathrm{conv}}(\tilde{x},\tilde{t},z,\tau)
      - I^{\kappa}_{\mathrm{diff}}(\tilde{x},\tilde{t},z,\tau)\Bigr)
      \omega_{\rho}(z)\delta_{\rho}(\tau)\,d\tilde{t}\,d\tilde{x}\,d\tau\,dz \\
      \le  \iiiint_{\Omega}\!\!
      I^{\kappa}_{\mathrm{sour}}(\tilde{x},\tilde{t},z,\tau)
      \omega_{\rho}(z)\delta_{\rho}(\tau) \,d\tilde{t}\,d\tilde{x}\,d\tau \,dz,
\end{multline}
where
\begin{gather*}
I^{\kappa}_{\mathrm{diff}}(\tilde{x},\tilde{t},z,\tau)=
   \abs{A^{\kappa}(v^{\kappa}(\tilde{x}+z,\tilde{t}+\tau))
   - A^{\kappa}(u^{\kappa}(\tilde{x}-z,\tilde{t}-\tau))}\Delta_{\tilde{x}}\psi(\tilde{x},\tilde{t}),\\
I^{\kappa}_{\mathrm{time}}(\tilde{x},\tilde{t},z,\tau)
   = |v^{\kappa}(\tilde{x}+z,\tilde{t}+\tau)
   -u^{\kappa}(\tilde{x}-z,\tilde{t}-\tau)|\psi_{\tilde{t}}(\tilde{x},\tilde{t}),   \\
\begin{aligned}
I^{\kappa}_{\mathrm{conv}}(\tilde{x},\tilde{t},z,\tau)
&=\mathop{\rm sign}(v^{\kappa}(\tilde{x}+z,\tilde{t}+\tau)-u^{\kappa}(\tilde{x}-z,\tilde{t}-\tau))   \\
& \quad \times \Bigl[F^{\kappa}(v^{\kappa}(\tilde{x}+z,\tilde{t}+\tau))
   -F^{\kappa}(u^{\kappa}(\tilde{x}-z,\tilde{t}-\tau))\Bigr]
   \cdot \nabla_{\tilde{x}}\psi(\tilde{x},\tilde{t}),
\end{aligned} \\
\begin{aligned}
I^{\kappa}_{\mathrm{sour}}(\tilde{x},\tilde{t},z,\tau)
&=\mathop{\rm sign}(v^{\kappa}(\tilde{x}+z,\tilde{t}+\tau)-u^{\kappa}(\tilde{x}-z,\tilde{t}-\tau))  \\
& \quad \times \Bigl[g^{\kappa}(V(\tilde{x}+z,\tilde{t}+\tau)) -
   g^{\kappa}(U(\tilde{x}-z,\tilde{t}-\tau))\Bigr]\psi(\tilde{x},\tilde{t}).
\end{aligned}
\end{gather*}
After the work of Kru\v{z}kov \cite{Kruzkov}, it
is a routine exercise to use
Lebesgue's differentiation theorem to pass to the limit
in \eqref{total_entropy_nyIII} as $\rho\downarrow 0$
to obtain \eqref{entropy_claim} (with $\psi$ rather than $\phi$).

Equipped with \eqref{entropy_claim}, we can now
conclude the proof of the
theorem. Pick two (arbitrary but fixed) Lebesgue points
$t_1,t_2\in (0,T)$ of
$\Lennorm{v^{\kappa}(\cdot,t)-u^{\kappa}(\cdot,t)}$,
$\kappa=1,\dots,K$.
For any $\nu\in\bigl(0,\min(t_1,T-t_2)\bigr)$, let
$$
\chi_{\nu}(t)=H_{\nu}(t-t_1) - H_{\nu}(t-t_2), \quad
H_{\nu}(t)=\int_{-\infty}^{t} \delta_{\nu}(\xi)\,d\xi.
$$
Notice that $\chi_{\nu}'(t)=
\delta_{\nu}(t-t_1)-\delta_{\nu}(t-t_2)$.
Pick a function $\psi\in C^{\infty}_0(\mathbb{R}^d)$ such that
\[
   \psi (x)=
   \begin{cases}
      1, & \abs{x}\le 1,\\
      0, & \abs{x}\ge 2,
    \end{cases}
\]
and $0\le \psi\le 1$ when $1<\abs{x}<2$. Let
$\varphi_r(x)=\psi\bigl(\frac{x}{r}\bigr)$, for $r\ge 1$.
We then take the test function $\phi$
in \eqref{entropy_claim} to be of the form
\begin{equation}
     \label{test_funct}
     \phi(x,t,y,s)= \chi_{\nu}(t)\varphi_r(x).
\end{equation}
Since $v^{\kappa},u^{\kappa}\in L^1(\Pi_T)$, we obviously have that
$$
\iint_{\Pi_T}\Bigl(\mathop{\rm sign}(v^{\kappa}-u^{\kappa})
\bigl[F^{\kappa}(v^{\kappa})-F^{\kappa}(u^{\kappa})\bigr]
\cdot\nabla\varphi_r
+ \abs{A^{\kappa}(v^{\kappa}) - A^{\kappa}(u^{\kappa})}
\Delta \varphi_r\Bigr)\,dt\,dx
$$
approaches zero as $r\uparrow \infty$.
Consequently, sending $r\uparrow\infty$ in \eqref{entropy_claim}
yields
\begin{equation}   \label{final_tmpI}
\begin{split}
& - \iint_{\Pi_T}   \abs{v^{\kappa}(x,t)-u^{\kappa}(x,t)}
   \chi_{\nu}'(t)\,dt\,dx\\
&\le \lipnorm{g^{\kappa}}\sum_{\ell=1}^K
   \iint_{\Pi_T}   \abs{v^{\ell}(x,t)-u^{\ell}(x,t)}\chi_{\nu}(t) \,dt\,dx.
\end{split}
\end{equation}
Summing \eqref{final_tmpI} over $\kappa$ we find
\begin{equation}   \label{final_tmpII}
\begin{split}
&  -\sum_{\kappa=1}^K \iint_{\Pi_T}
   \abs{v^{\kappa}(x,t)-u^{\kappa}(x,t)}
   \chi_{\nu}'(t)\,dt\,dx\\
&\le C\sum_{\kappa=1}^K\iint_{\Pi_T}
   \abs{v^{\kappa}(x,t)-u^{\kappa}(x,t)}\chi_{\nu}(t)\,dt\,dx,
\end{split}
\end{equation}
where $C:=K \max_\kappa\big(\lipnorm{g^{\kappa}}\big)$. Sending
$\nu\downarrow 0$ in \eqref{final_tmpII}, we get
\begin{align*}
   &\sum_{\kappa=1}^K \int_{\mathbb{R}^d}
   \abs{v^{\kappa}(x,t_2)-u^{\kappa}(x,t_2)}\,dx\\
& \le \sum_{\kappa=1}^K\int_{\mathbb{R}^d}
   \abs{v^{\kappa}(x,t_1)-u^{\kappa}(x,t_1)}\,dx
   +C\int_{t_1}^{t_2}\Big(\sum_{\kappa=1}^K\int_{\mathbb{R}^d}
   \abs{v^{\kappa}(x,t)-u^{\kappa}(x,t)}\,dx\Big)\,dt.
\end{align*}
An application of Gronwall's inequality now gives
\begin{align*}
&\sum_{\kappa=1}^K \int_{\mathbb{R}^d}
   \abs{v^{\kappa}(x,t_2)-u^{\kappa}(x,t_2)}\,dx\\
& \le \exp\bigl(C (t_2-t_1)\bigr)\sum_{\kappa=1}^K\int_{\mathbb{R}^d}
   \abs{v^{\kappa}(x,t_1)-u^{\kappa}(x,t_1)}\,dx. \\
& \overset{t_1\downarrow 0}{\longrightarrow}
   \exp\bigl(C t_2\bigr)\sum_{\kappa=1}^K\int_{\mathbb{R}^d}
   \abs{v^{\kappa}(x,0)-u^{\kappa}(x,0)}\,dx.
\end{align*}
By using the inequality
$\sum_\kappa \abs{v^\kappa-u^\kappa}\le \sqrt{K}\|V-U\|$
and since $t_2$ is an arbitrary Lebesgue point, the theorem is proved.
\end{proof}

\begin{remark} \label{rmk3.4} \rm
   We note that the proof of Theorem \ref{thm:uniq}
   is slightly different from the corresponding proof
   in \cite{Carrillo}. We here  work with the term
   $$
   \abs{A^{\kappa}(v^{\kappa})-A^{\kappa}(u^{\kappa})}
   \left(\Delta_x \phi + \Delta_y \phi\right),
   $$
   and exploits fully the identity
   \begin{equation}
      \label{rem:tmp}
      \Delta_x \Phi(x-y) + 2\nabla_x\cdot\nabla_y \Phi(x-y)
      + \Delta_y \Phi(x-y)=0,
   \end{equation}
   which holds for any (smooth) function $\Phi\colon\mathbb{R}\to\mathbb{R}$.
   In \cite{Carrillo}, the author works instead with the term
   $$
   \mathop{\rm sign}(v^{\kappa}-u^{\kappa})\Bigl(
   \nabla_x A^{\kappa}(v^{\kappa}) \cdot \nabla_x\phi
   - \nabla_y A^{\kappa}(u^{\kappa})\cdot\nabla_y \phi\Bigr),
   $$
   and exploits eventually the usual ``Kru\v{z}kov identity''
   $$
   \nabla_x \Phi(x-y) + \nabla_y \Phi(x-y) =0.
   $$
   The interested reader is hereby invited to have a look at
   Ishii's paper \cite{Ishii} to see how the identity
   \eqref{rem:tmp} (implicitly) plays a central role in the
   uniqueness proof for \textit{viscosity} solutions
   of degenerate second-order partial differential equations.
\end{remark}

\begin{remark} \label{rmk3.5} \rm
   Following \cite{KR:Rough_Unique} and \cite{KO:Unique}, one can
   prove that Theorem \ref{thm:uniq} holds for more general
   systems of the type
   \[
      u^\kappa_t+ \mathop{\rm div} F^\kappa(x,t,u^\kappa) =
      \Delta_x\bigl(K^{\kappa}(x,t) A^\kappa(u^{\kappa})\bigr)
      + g^\kappa(x,t,U),\quad \kappa=1,\dots,K,
   \]
   where $F^\kappa,K^\kappa,A^\kappa,g^\kappa$ satisfy
   the same assumptions as in \cite{KR:Rough_Unique,KO:Unique}.
   In particular, $K^\kappa$ is a diagonal matrix
   that needs to be bounded away from zero a.e.
\end{remark}


\section{Existence of Entropy Solution}\label{sect:eksistens}

In this section, we prove existence of an entropy
solution by establishing convergence of certain
finite difference approximations.
To this end, we shall assume that
$u_0$ belongs to $L^1(\mathbb{R}^d)\cap BV(\mathbb{R}^d)$
and has compact support, the latter implies that all subsequent sums
over $I$ are finite. Furthermore, we shall
assume that $F$ and $A$ are $C^1$. These additional assumptions
on $u_0,F,A$ will be removed
towards the end of this section (see the
proof of Theorem \ref{thm:exist}).

Let $I=(i_1,\dots,i_d)\in \mathbb{Z}^d$ be a multi-index and
let $e_i\in \mathbb{Z}^d$ the multi-index with with zeros
everywhere except for a $1$ at the $i$th place.
Selecting a mesh size $h>0$, a time step $\Delta t>0$,
and integer $N$ such that $N\Delta t=T$, the value
of our finite difference approximation of $u^{\kappa}$ at the
point $(x_I,t_n)=(hI,n\Delta t)$, with $I\in\mathbb{Z}^d$ and $n=0,\dots,N$,
will be denoted by $u_I^{\kappa,n}$ for $\kappa=1,\dots,K$. Sometimes we
write $U_I^n$ for the vector $\bigl(u_I^{1,n},\dots,u_I^{K,n}\bigr)$.
To simplify the notation, we introduce the (backward
and forward) finite difference operators
$$
D_{i,-} u_I^{\kappa,n} = \frac{1}{h}\big(u_I^{\kappa,n}-u_{I-e_i}^{\kappa,n} \big), \quad D_{i,+} u_I^{\kappa,n} =
\frac{1}{h}\big(u_{I+e_i}^{\kappa,n}-u_I^{\kappa,n}\big), \quad i=1,\dots,d.
$$

As already mentioned in the introduction, we
shall analyze the Engquist--Osher (generalized upwind) scheme.
For a scalar flux function $F_i^{\kappa}(u)$, the associated
Engquist--Osher numerical flux function \cite{EngquistOsher:81}
can be written as
\begin{equation}
    F_i^{\kappa,\mathrm{EO}}(u,v) = \frac{1}{2}\big(F_i^{\kappa}(u)+F_i^{\kappa}(v) \big)
      - \frac12\int_u^v\big| \frac{dF_i^{\kappa}}{dr}(r)\big|\,dr,
    \label{EO_flux}
\end{equation}
which is Lipschitz (actually $C^1$) in both variables with (common)
Lipschitz constant $\lipnorm{F_i^{\kappa}}$.
We may write
\begin{equation}
   \label{EO_decomp}
   F_i^{\kappa,\mathrm{EO}}(u,v) =  F_i^{\kappa,+}(u) +F_i^{\kappa,-}(v),
\end{equation}
where (recall that $F^\kappa_i(0)=0$)
$$
F_i^{\kappa,+}(u)=\int_0^u
\Bigl({\frac{dF^{\kappa}_{i}}{dr}(r)}\vee 0\Bigr)\,dr,\quad
F_i^{\kappa,-}(v)=\int_0^v
\Bigl({\frac{dF^{\kappa}_{i}}{dr}(r)}\wedge 0\Bigr)\,dr.
$$

\begin{remark}
   For a monotone flux function $F_i^{\kappa}$, the
   Engquist-Osher flux reduces to the upwind flux, i.e.,
   $$
   F_i^{\kappa,\mathrm{EO}}(u,v)=F_i^{\kappa}(u)\quad \text{if  $\frac{dF_i^{\kappa}}{dr}\ge 0$}, \quad
   F_i^{\kappa,\mathrm{EO}}(u,v)=F_i^{\kappa}(v)\quad \text{if  $\frac{dF_i^{\kappa}}{dr}< 0$}.
   $$
\end{remark}

The Engquist-Osher finite difference scheme now takes the form
\begin{equation}
   \label{EO_scheme}
   \frac{u_I^{\kappa,n+1}-u_I^{\kappa,n}}{\Delta t} + \sum_{i=1}^d
   D_{i,-} \Bigl(F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)
   -D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)\Bigr)
   = g^{\kappa}\left(U_I^n\right),
\end{equation}
for $\kappa=1,\dots,K$.
Letting $\lambda=\frac{\Delta t}{h}$ and
$\mu=\frac{\Delta t}{h^2}$, we assume hereafter that the
following CFL condition holds:
\begin{equation}
    \label{CFL}
    \mathrm{CFL}^{\kappa}:=\lambda \sum_{i=1}^{d}
    \big\| \frac{dF^{\kappa}_{i}}{du}\big\|_\infty +
    2 \mu d\big\| \frac{d A^{\kappa}}{du}\big\|_\infty \le 1,
    \quad \kappa=1,\dots,K.
\end{equation}
For later use, we note that we may
write the finite difference scheme \eqref{EO_scheme} as
\begin{equation} \label{EO_scheme_new}
\begin{gathered}
\frac{\overline{u}_I^{\kappa,n+1}-u_I^{\kappa,n}}{\Delta t} + \sum_{i=1}^d
D_{i,-} \Bigl(F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)
-D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)\Bigr)=0,\\
u_I^{\kappa,n+1}=\overline{u}_I^{\kappa,n+1} + \Delta t g^{\kappa}\left(U_I^n\right),\quad
 \kappa=1,\dots,K.  \end{gathered}
\end{equation}
Sometimes we will write $\overline{U}_I^n$ for the
vector $\big(\overline{u}_I^{1,n},\dots,\overline{u}_I^{K,n}\big)$.
The approximate solution $U_h=(u^1_h,\dots,u^K_h)$ is
then defined as
\begin{equation}
   \label{eq:uhdef}
   U_h(x,t)=U_I^n,
   \quad\text{for }(x,t)\in \chi_{I} \times [t_{n},t_{n+1}),
\end{equation}
where $\chi_{I}$ denotes the set
\begin{equation}
   \label{def_chi}
   \chi_I=
   \{x\in\mathbb{R}^d : h(i_j-1/2)\le x_j< h(i_j+1/2),\, j=1,\dots,d\},
   \quad I=(i_1,\dots,i_d).
\end{equation}
We initialize the scheme by setting
\begin{equation}
   \label{eq:uodef}
   U_I^0 = \frac{1}{\abs{\chi_{I}}}
   \int_{\chi_{I}} U_0(x)\,dx=h^{-d}\int_{\chi_{I}} U_0(x)\,dx .
\end{equation}
Our first lemma provides  uniform $L^1,L^{\infty},BV$ estimates for
$U_h$.

\begin{lemma}\label{lem:basic}
   There exists a constant $C$, independent of $h$, such that
   $t\in (0,T)$
   $$
   \Lennorm{U_h(\cdot,t)}\le C, \quad
   \supnorm{U_h(\cdot,t)}\le C, \quad
   |{U_h(\cdot,t)}|_{BV(\mathbb{R}^d)}  \le C.
   $$
\end{lemma}

\begin{proof}
First note that we can write $\overline{u}_I^{\kappa,n+1}=\mathcal{S}^{\kappa}(u^{\kappa,n};I)$, where
$\mathcal{S}^{\kappa}\colon L^1(\mathbb{Z}^d)\to L^1(\mathbb{Z}^d)$ maps the sequence $u^{\kappa,n}=\{u_I^{\kappa,n}\}_I$
according to the formula
$$
\mathcal{S}^{\kappa}(u^{\kappa,n};I)=u_I^{\kappa,n}  - \Delta t \sum_{i=1}^d
D_{i,-} \Bigl(F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)
-D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)\Bigr).
$$
An easy exercise will
reveal that the CFL condition \eqref{CFL} implies
that $\mathcal{S}^{\kappa}(\cdot)$ is a monotone function of all
its arguments.
Since the difference approximation has compact support, we
get $\sum_I \mathcal{S}^{\kappa}(u^{\kappa,n};I)= \sum_I u_I^{\kappa,n}$.
Since $\mathcal{S}^{\kappa}$ is monotone and obviously
commutes with spatial translations, it follows
from Lemma \ref{lem:CranTart} that
\begin{equation}
    \big\|\overline{u}^{\kappa,n+1}\big\|_{L^1(\mathbb{Z}^d)}\le
    \big\|u^{\kappa,n}\big\|_{L^1(\mathbb{Z}^d)}.\label{extra_lign}
\end{equation}
For a grid function $u=\{u_I\}_I$, we recall that
the $L^p$ norms are defined as
$$
\left\|u\right\|_{L^p(\mathbb{Z}^d)}^p = \sum_I\abs{u_I}^p,\quad
p<\infty, \quad \left\|u\right\|_{L^{\infty}(\mathbb{Z}^d)} =
\sup_I\abs{u_I}.
$$
Furthermore, using a standard argument, the inequality
\eqref{extra_lign} is also valid with $L^1(\mathbb{Z}^d)$ replaced by
$L^\infty(\mathbb{Z}^d)$. For completeness we repeat
the argument here in the case $d=1$.  Rewriting \eqref{EO_scheme_new}
and using \eqref{EO_decomp} we find
\begin{equation}
\begin{aligned}
    \overline{u}_I^{\kappa,n+1} &=u_I^{\kappa,n}
    -\lambda\big(F^{\kappa,+}(u_I^{\kappa,n})-F^{\kappa,+}(u_{I-1}^{\kappa,n})+F^{\kappa,-}(u_{I+1}^{\kappa,n})
    -F^{\kappa,-}(u_I^{\kappa,n})\big) \\
    &\quad+\mu\big(A^\kappa(u_{I+1}^{\kappa,n})+A^\kappa(u_{I-1}^{\kappa,n})
    -2A^\kappa(u_I^{\kappa,n})\big) \\
&=\big(1-\mu dA^\kappa_I-\mu dA^\kappa_{I+1}
    -\lambda dF^{\kappa,+}_I+\lambda dF^{\kappa,-}_I \big)u_I^{\kappa,n} \\
&\quad+\big(\mu dA^\kappa_I+\lambda  dF^{\kappa,+}_I\big)u_{I-1}^{\kappa,n}
    +\big(\mu dA^\kappa_{I+1}-\lambda dF^{\kappa,-}_I\big)u_{I+1}^{\kappa,n}.
\end{aligned}    \label{inf_begr}
\end{equation}
Here the quantities $dA^\kappa_I$, $dA^\kappa_{I+1}$,
$dF^{\kappa,\pm}_I$  denote
derivatives of $A^\kappa$ and $F^{\kappa,\pm}$, respectively,
evaluated at points between $u_I^{\kappa,n}$ and $u^{\kappa,n}_{I\pm 1}$
using the mean value theorem.  Applying the CFL condition
\eqref{CFL} we see that
\begin{align*}
    \big|\overline{u}_I^{\kappa,n+1}\big|&\le
    \big(1-\mu dA^\kappa_I-\mu dA^\kappa_{I+1}
    -\lambda dF^{\kappa,+}_I+\lambda dF^{\kappa,-}_I\big)\abs{u_I^{\kappa,n}} \\
    &\quad +\big(\mu dA^\kappa_I+\lambda dF^{\kappa,+}_I\big)
    \abs{u_{I-1}^{\kappa,n}}
    +\big(\mu dA^\kappa_{I+1}-\lambda dF^{\kappa,-}_I\big)\abs{u_{I+1}^{\kappa,n}} \\
    &\le \norm{u^{\kappa,n}}_{L^\infty(\mathbb{Z}^d)}.
\end{align*}
Thus we have shown
\begin{equation}
    \|\overline{u}^{\kappa,n+1}\|_{L^\infty(\mathbb{Z}^d)}\le
    \|u^{\kappa,n}\|_{L^\infty(\mathbb{Z}^d)}.\label{extra_lign_infty}
\end{equation}
Using \eqref{EO_scheme_new}, \eqref{extra_lign} and summing
over $\kappa=1,\dots,K$, we get
$$
\sum_{\kappa=1}^K\left\|u^{\kappa,n+1}\right\|_{L^p(\mathbb{Z}^d)}\le \Bigl(1 +K
\max_{\kappa=1,\dots,K} \bigl(\lipnorm{g^{\kappa}}\bigr)\Delta t\Bigr)
\sum_{\kappa=1}^K \left\|u^{\kappa,n}\right\|_{L^p(\mathbb{Z}^d)}, \quad
p=1,\infty,
$$
from which it follows that
\begin{align*}
   \sum_{\kappa=1}^K\left\|u^{\kappa,n}\right\|_{L^p(\mathbb{Z}^d)}
   &\le \Bigl(1 + K\max_{\kappa=1,\dots,K}
   \bigl(\lipnorm{g^{\kappa}}\bigr)\Delta t\Bigr)^n
   \sum_{\kappa=1}^K \left\|u^{\kappa,0}\right\|_{L^p(\mathbb{Z}^d)},
   \\ & \le \exp\Bigl(\hat{C} t_n\Bigr)
   \sum_{\kappa=1}^K \left\|u^{\kappa,0}\right\|_{L^p(\mathbb{Z}^d)},
   \quad  n=0,\dots,N,\quad p=1,\infty,
\end{align*}
for some constant $\hat{C}$ independent $h$.

Similarly, an application of Lemma \ref{lem:CranTart} gives
$$
\sum_I\big|\overline{u}_I^{\kappa,n+1}-\overline{u}_{I-e_i}^{\kappa,n+1}\big|\le
\sum_I\big|u_I^{\kappa,n}-u_{I-e_i}^{\kappa,n}\big|.
$$
Hence, from \eqref{EO_scheme_new} and after
summing over $\kappa=1,\dots,K$, we get
$$
\sum_{\kappa=1}^K\sum_I\left|u_I^{\kappa,n}-u_{I-e_i}^{\kappa,n}\right|\le
\exp\bigl(\hat{C} t_n\bigr)\sum_{\kappa=1}^K
\sum_I\big|u_I^{\kappa,0}-u_{I-e_i}^{\kappa,0}\big|, \quad  n=0,\dots,N.
$$
This concludes the proof of the lemma.
\end{proof}

The next lemma shows that $U_h(\cdot,t)$ is
$L^1$ H\"older continuous in time.

\begin{lemma}\label{lem:L1cont}
   There exists a constant $C$, independent of $h$, such that
   $$
   \Lennorm{U_h(\cdot,t+\tau)-U_h(\cdot,t)}
   \le C\sqrt{\tau}, \quad \text{$\tau\in (0,T)$
     and  $t\in (0,T-\tau)$}.
   $$
\end{lemma}

\begin{proof}
Let $\phi=\phi(x)$ be a $C^{\infty}_0(\mathbb{R}^d)$ function
and set $\phi_I=\phi(x_I)$. From \eqref{EO_scheme} we get
\begin{align*}
&\Big|h^d \sum_I \phi_I\big( u_I^{\kappa,n+1}-u_I^{\kappa,n}\big)\Big|\\
& \le  \Big( \underbrace{h^d \sum_I\sum_{i=1}^d
   \big| \phi_I D_{i,-} F_i^{\kappa,\mathrm{EO}}(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n})\big|}_{B_1}\\
&\quad+ \underbrace{ h^d \sum_I \sum_{i=1}^d
   \left|D_{i,+} \phi_I D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)\right|}_{B_2}
   + \underbrace{h^d \sum_I\left|\phi_I
       g^{\kappa}\left(U_I^n\right)\right|}_{B_3}\Big)\Delta t.
\end{align*}
Equipped with Lemma \ref{lem:basic}, we get the following
estimates:
\begin{gather*}
   \abs{B_1} \le
   2\max_{i,u}\big|\frac{dF_i^{\kappa}}{du}(u)\big| \supnorm{\phi} h^d
\sum_{I} \sum_{i=1}^d \left|D_{i,-}u_I^{\kappa,n}\right| \le C_1\supnorm{\phi}, \\
 \abs{B_2} \le \max_{i,u}\big|\frac{dA^{\kappa}}{du}(u)\big|
  \max_{i}\supnorm{\phi_{x_i}}
   h^d \sum_{I} \sum_{i=1}^d \left|D_{i,+}u_I^{\kappa,n}\right| \le C_2
 \max_{i}\supnorm{\phi_{x_i}},   \\
 \abs{B_3} \le  \supnorm{\phi}
   \sum_{\kappa=1}^K \bigl(\lipnorm{g^{\kappa}}
   h^d \sum_{I} \left|u_I^{\kappa,n}\right|\bigr)
   \le C_3\supnorm{\phi},
\end{gather*}
for some constant $C_1,C_2,C_3$ that are independent of $h$.
 From these estimates it now follows that
\begin{equation}
   \label{WeakTime:tmpI}
   \big|h^d \sum_I\big( u_I^{\kappa,n+1}-u_I^{\kappa,n}\big)\phi_I\big|
   \le C_4\Bigl(\supnorm{\phi} +
   \max_{i}\supnorm{\phi_{x_i}}\Bigr) \Delta t,
\end{equation}
for some constant $C_4$ that is independent of $h$.
Regarding $u^{\kappa}_h$, we have
\begin{equation} \label{WeakTime:tmpII}
\begin{split}
&\Big|\int_{\mathbb{R}^d}\bigl(u^{\kappa}_h(x,t_{n+1})-u^{\kappa}_h(x,t_n)\bigr)\phi(x)\,dx\Big|\\
& \le \Bigl|h^d \sum_I\big( u_I^{\kappa,n+1}-u_I^{\kappa,n}\big)\phi_I\Big|
      + \underbrace{\sum_I \big|u_I^{\kappa,n+1}-u_I^{\kappa,n} \big|
        \int_{\chi_I} \abs{\phi(x)-\phi_I}\,dx}_{I_4},
   \end{split}
\end{equation}
where $\chi_I$ is defined in \eqref{def_chi}.
Let us estimate the additional error term $I_4$.
Using the finite difference
scheme \eqref{EO_scheme} and Lemma \ref{lem:basic}, we
can do this as we did for $B_1,B_2,B_3$:
\begin{align*}
   \abs{I_4} &\le \max_i\supnorm{\phi_{x_i}}
    h^{d+1}\sum_I \big| u_I^{\kappa,n+1}-u_I^{\kappa,n}\big|   \\
& \le  \max_i\supnorm{\phi_{x_i}}\Big(h^{d+1}\sum_I\sum_{i=1}^d
   \Big|D_{i,-} F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)\Big|\\
&\;+ h^d \sum_I \sum_{i=1}^d
   \abs{D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)}
+ h^d \sum_I \sum_{i=1}^d
   \abs{D_{i,+} A^{\kappa}\left(u_{I-e_i}^{\kappa,n}\right)}
   + h^{d+1} \sum_I\abs{g^{\kappa}}\Big)\Delta t \\
&\le   C_5\max_i\supnorm{\phi_{x_i}}\Delta t,
\end{align*}
for some constant $C_5$ that is independent of $h$.

 From this estimate as well as \eqref{WeakTime:tmpII}
and \eqref{WeakTime:tmpI}, we get
\begin{equation}
   \label{WeakTime:tmpIII}
\begin{split}
&\Big|\int_{\mathbb{R}^d}\Bigl(
      u^{\kappa}_h(x,t_{n+1})-u^{\kappa}_h(x,t_n)\Bigr)\phi(x)\,dx\Big|\\
& \le C_6\Bigl(\supnorm{\phi} +
      \max_{i}\supnorm{\phi_{x_i}}\Bigr) \Delta t,\quad
      C_6:=C_4 + C_5.
\end{split}
\end{equation}
Let $\tau,t$ be as in the lemma. Using \eqref{WeakTime:tmpIII}, it is
not difficult to show that
\begin{equation}
   \label{Weak_Time}
   \Big|\int_{\mathbb{R}^d}\Bigl( u^{\kappa}_h(t+\tau)-u^{\kappa}_h(t)\Bigr)\phi(x)\Big|
   \le C_7\Bigl(\supnorm{\phi} +
   \max_{i}\supnorm{\phi_{x_i}}\Bigr) \tau,
\end{equation}
for some constant $C_7$ that is independent of $h$.
In view of \eqref{Weak_Time}, the lemma now follows from
an application of Lemma \ref{le:Kruzkov}.
\end{proof}

The next lemma provides us with
a uniform $L^2$ space translation estimate
for $A^\kappa(U_h)$.

\begin{lemma}\label{lem:A_space}
   There exists a constant $C$, independent of $h$, such that
   \begin{equation}
      \label{Adiff_L2}
      \left\|A^\kappa\left(U_h(\cdot+y,\cdot)\right)
        - A^\kappa\left(U_h(\cdot,\cdot)\right)\right\|_{L^2(\Pi_T)}
      \le C\sqrt{\abs{y}(\abs{y}+h)}, \quad \text{$y\in \mathbb{R}^d$}.
   \end{equation}
\end{lemma}

\begin{proof}
We shall derive a discrete energy estimate.
Multiplying \eqref{EO_scheme}
by $ \Delta t\, h^d\, u_I^{\kappa,n}$, summing over $n,I$, and then
doing summation by parts in $I$, we find that
\begin{equation}
   \label{WeakBV:I}
   \begin{split}
      &h^d \sum_{n,I} u_I^{\kappa,n} \left(u_I^{\kappa,n+1}-u_I^{\kappa,n}\right)
      + \Delta t\, h^d \sum_{n,I} \sum_{i=1}^d
      u_I^{\kappa,n} D_{i,-} F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)\\
& \quad + \Delta t\, h^d \sum_{n,I} \sum_{i=1}^d D_{i,+} u_I^{\kappa,n} D_{i,+}
A^{\kappa}\left(u_I^{\kappa,n}\right)
      - \Delta t\,h^d\sum_{n,I} u_I^{\kappa,n} g^{\kappa}\left(U_I^n\right) =0.
   \end{split}
\end{equation}
Observe that we can write
$$
u_I^{\kappa,n}\big(u_I^{\kappa,n+1}-u_I^{\kappa,n}\big) =
\frac{1}{2}\Big(\big(u_I^{\kappa,n+1}\big)^2 - \big(u_I^{\kappa,n}\big)^2
-\big(u_I^{\kappa,n+1}-u_I^{\kappa,n}\big)^2 \Big),
$$
Assuming (without loss
of generality) $\max_u dA^{\kappa}(u)/du>0$ and
since $dA^{\kappa}/du\ge 0$, we also have
$$
\frac{1}{\max_u\frac{dA^{\kappa}}{du}(u)}\Bigl(D_{i,+}
A^{\kappa}\left(u_I^{\kappa,n}\right)\Bigr)^2 \le
D_{i,+} u_I^{\kappa,n} D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right) .
$$
 From these observations, we get from \eqref{WeakBV:I} that
\begin{equation} \label{WeakBV:II}
\begin{split}
&\frac{\Delta t\, h^d}{\max_u\frac{dA^{\kappa}}{du}(u)} \sum_{n,I}
      \sum_{i=1}^d
      \big(D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)\big)^2\\
& \le - \frac{h^d}{2}\sum_{n,I}\Big(
      \big(u_I^{\kappa,n+1}\big)^2 - \big(u_I^{\kappa,n}\big)^2\Big)
      + \frac{h^d}{2}\sum_{n,I}\big(u_I^{\kappa,n+1}-u_I^{\kappa,n}\big)^2 \\
& \quad - \Delta t\, h^d \sum_{n,I} \sum_{i=1}^d
      u_I^{\kappa,n} D_{i,-} F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)
      + \Delta t\,h^d\sum_{n,I} u_I^{\kappa,n} g^{\kappa}\left(U_I^n\right) \\
& = \frac{h^d}{2}\sum_{I}\Big(\big(u_I^{\kappa,0}\big)^2 -
 \big(u_I^{\kappa,N}\big)^2\Big)+ \frac{h^d}{2}\sum_{n,I}\big(u_I^{\kappa,n+1}-u_I^{\kappa,n}\big)^2\\
& \quad - \Delta t\, h^d \sum_{n,I} \sum_{i=1}^d
      u_I^{\kappa,n} D_{i,-} F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)
      + \Delta t\,h^d\sum_{n,I} u_I^{\kappa,n} g^{\kappa}\left(U_I^n\right) \\
& \le  C_1 +
      \frac{h^d}{2}\sum_{n,I}\left(u_I^{\kappa,n+1}-u_I^{\kappa,n}\right)^2
      + 2\max_{n,I}\left|u_I^{\kappa,n}\right|\\
&\quad\times \max_{i,u}\big|\frac{dF_i^{\kappa}}{du}(u)\big|\Delta t\,h^d
      \sum_{n,I} \sum_{i=1}^d \left|D_{i,-}u_I^{\kappa,n}\right|
 + \sum_{\kappa=1}^K \Big(\lipnorm{g^{\kappa}}
       \Delta t\,h^d \sum_{n,I}\left|u_I^{\kappa,n}\right|\Big) \\
& \le \frac{h^d}{2}\sum_{n,I}\left(u_I^{\kappa,n+1}-u_I^{\kappa,n}\right)^2 + C_2,
   \end{split}
\end{equation}
for constants $C_1,C_2$ that are independent of $h$. To derive
the last two inequalities, we used that the
finite difference solution is uniformly bounded
in the $L^1$, $L^{\infty}$, and $BV$ norms.

 From \eqref{EO_scheme} and the inequality $\left(\sum_{i=1}^r
a_i\right)^2\le
c_r\sum_{i=1}^r \left(a_i\right)^2$ for any
integer $r\ge 1$, we find that
\begin{equation}
  \label{WeakBV:III}
  \begin{split}
     \frac{1}{2}\big(u_I^{\kappa,n+1}-u_I^{\kappa,n}\big)^2
      &\le C_d\underbrace{\Delta t^2\sum_{i=1}^d\Bigl(D_{i,+}
      F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)\Bigr)^2}_{B_1}
      \\ & \quad + C_d\underbrace{\Delta t^2\sum_{i=1}^d
      \Bigl(D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)\Bigr)^2}_{B_2}
      + C_d\underbrace{\Delta t^2\Bigl(g^{\kappa}\left(U_I^n\right)\Bigr)^2}_{B_3},
  \end{split}
\end{equation}
for some constant $C_d$ that is independent of $h$ but
it depends on $d$ (the number of spatial dimensions).
In view of (the hyperbolic part of) \eqref{CFL}
and Lemma \ref{lem:basic}, we have that
\begin{align*}
   \Big|\frac{h^d}{2}\sum_{n,I} B_1 \Big|
   &\le \lambda\max_{i,u}\big|\frac{dF_i^{\kappa}}{du}(u)\big|
   \max_{n,I}\left|u_I^{\kappa,n}\right| \Delta t\,h^d \sum_{n,I}
   \sum_{i=1}^d\big| D_{i,-}
   F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)\big|   \\
&\le \max_{n,I}\left|u_I^{\kappa,n}\right|
   \max_{i,u}\big|\frac{dF_i^{\kappa}}{du}(u)\big| \Delta t\,h^d \sum_{n,I}
   \sum_{i=1}^d\left|D_{i,-} u_I^{\kappa,n}\right| \le C_3,
\end{align*}
for some constant $C_3$ that is independent of $h$.
Similarly, in view of (the parabolic
part of) \eqref{CFL} and the $L^1,L^{\infty}$ bounds in
Lemma \ref{lem:basic}, we have that
\begin{align*}
   \Big|\frac{h^d}{2}\sum_{n,I} B_2 \Big|
   &\le \frac{\Delta t}{h}\max_{u} \frac{dA^{\kappa}}{du}(u)
   \max_{n,I}\left|u_I^{\kappa,n}\right|
   \Delta t\,h^d \sum_{n,I}
   \sum_{i=1}^d\left|D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)\right|
   \\ & \le 2\mu d \big(\max_u \frac{dA^{\kappa}}{du}(u)\big)^2
   \max_{n,I}\left|u_I^{\kappa,n}\right|
   \Delta t\,h^d \sum_{n,I} \left|u_I^{\kappa,n}\right|\le C_4.
\end{align*}
Finally, we have $\big|\frac{h^d}{2}\sum_{n,I} B_3 \big|\le C_5$.

Summing up, from \eqref{WeakBV:III} and the uniform
bounds just obtained for $B_1,B_2,B_3$, we have
$$
\Big|\frac{h^d}{2}\sum_{n,I}\big(u_I^{\kappa,n+1}-u_I^{\kappa,n}\big)^2 \Big|
\le C_6,
$$
for some constant $C_6$ that is independent of $h$.
Inserting this estimate into \eqref{WeakBV:I}, we finally get
\begin{equation}
   \label{WeakBV:Final}
   \Delta t\, h^d \sum_{n,I} \sum_{i=1}^d
   \bigl(D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)\bigr)^2
   \le C_7,
\end{equation}
for some constant $C_7$ that is independent of $h$.

Let us now derive \eqref{Adiff_L2} from \eqref{WeakBV:Final}.  To
facilitate this we introduce some notation inspired by
\cite{Eymardetal}. Let  $\bar{\chi}_I$ denote the
the set
$$
\bar{\chi}_I=\seq{ x\in\mathbb{R}^d : h\left(i_j-1/2\right) \le
x \le h\left(i_j+1/2\right), \; j=1,\ldots,d\,},\quad
I=\left(i_1,\ldots,i_d\right),
$$
and for $x$ and $y$ in $\mathbb{R}^d$ let $\sigma(x,y)$ denote the line from
$x$ to $x+y$.
Then for $x$ and $y$ in $\mathbb{R}^d$ we define
\begin{equation}
  \label{eq:eymardchar}
  \chi_{I+e_i/2}(x,y) =
  \begin{cases}
    1 &\text{if $\sigma(x,y)\cap
    {\bar{\chi}_I}\cap \left(\bar{\chi}_{I+e_i}\right)
    \ne \emptyset$},\\
    0 &\text{otherwise.}
  \end{cases}
\end{equation}
Using this notation, we have
$$
\abs{A^{\kappa}\left(u^{\kappa,n}(x+y)\right)-A^{\kappa}\left(u^{\kappa,n}(x)\right)}\le
\sum_{I} \sum_{i=1}^d \chi_{I+e_i/2}(x,y) \abs{D_{i,+}
A^{\kappa}\left(u_I^{\kappa,n}\right)},
$$
which by the Cauchy-Schwarz inequality implies
\begin{equation}  \label{eq:eymard2}
\begin{aligned}
&\left(A^{\kappa}\left(u^{\kappa,n}(x+y)\right)-A^{\kappa}\left(u^{\kappa,n}(x)\right)\right)^2\\
& \le \sum_{I}\sum_{i=1}^d \chi_{I+e_i/2}(x,y)h^2
  \sum_{I}\sum_{i=1}^d \chi_{I+e_i/2}(x,y) \abs{D_{i,+}
  A^{\kappa}\left(u_I^{\kappa,n}\right)}^2.
\end{aligned}
\end{equation}
If we let $n_i(y)$ denote the number of edges crossed by $y$, then we
find that
$$
n_i(y) \le \mathrm{floor}\big(\frac{\abs{y_i}}{h}\big) + 1,
$$
where $\mathrm{floor}(a)$ denotes the integer part of $a$, and $y_i$
the $i$th component of $y$. Thus
$$
\sum_{I}\sum_{i=1}^d \chi_{I+e_i/2}(x,y)h \le \sum_{i=1}^d n_i(y) h \le
h\sum_{i=1}^d \Big(\big(\frac{\abs{y_i}}{h}\big) + 1\Big)
\le \sqrt{d}\abs{y}+dh.
$$
Furthermore, we have the relation
$$
\int_{\mathbb{R}^d} \chi_{I+e^i/2}(x,y)\,dx = h^{d-1} \abs{y_i}.
$$
Hence, integrating \eqref{eq:eymard2} over $x$, and then over $t$
(which amounts to summing over $n$), we find that
\begin{align*}
&\iint_{\Pi_T}
  \left(A^{\kappa}\left(U_h(x+y,t)\right)-A^{\kappa}\left(U_h(x,t)\right)\right)^2
  \,dt\,dx   \\
& \le  (\abs{y}+h) h \Delta t
  \sum_{n,I}\sum_{i=1}^d \int_{\mathbb{R}^d} \chi_{I+e_i/2}(x,y)\,dx \abs{D_{i,+}
    A^{\kappa}\left(u_I^{\kappa,n}\right)}^2.\\
& \le d^{3/2} (\abs{y}+h) \abs{y}  \Delta t h^d
  \sum_{n,I}\sum_{i=1}^d \abs{D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)}^2\\
& \le C_8 (\abs{y}+h)\abs{y},
  %\label{eq:eymard3}
\end{align*}
by \eqref{WeakBV:Final}. This concludes the proof of
\eqref{Adiff_L2}.
\end{proof}

\begin{remark} \label{rmk4.5} \rm
   It is possible to derive Lemma \ref{lem:A_space} without
   using $BV$ regularity of the approximate
   solution, see \cite{KR:Rough_Diff} and also
   \cite{AAI,Champieretal:93,Eymardetal,Eymardetal:diffusion,Kroner}.
\end{remark}

The next lemma provides us with
a uniform $L^2$ time translation estimate
for $A(U_h)$.

\begin{lemma}\label{lem:A_time}
   There exists a constant $C$, independent of $h$, such that
   \begin{equation}
      \label{A_L2_time}
      \left\|A^\kappa\left(U_h(\cdot,\cdot+\tau)\right)
      -A^\kappa\left(U_h(\cdot,\cdot)\right)\right\|_{L^2(\mathbb{R}^d\times(0,T-\tau))}
      \le C \sqrt{\tau}, \quad  \tau\in(0,T).
  \end{equation}
\end{lemma}

\begin{proof}
We will use the space estimate \eqref{WeakBV:Final} and the
finite difference scheme \eqref{EO_scheme} to show
that $A^{\kappa}\left(u^{\kappa}_h\right)$ is also $L^2$ continuous
in time. For $t\in [t_{n},t_{n+1})$, $t+\tau\in
[t_{n+m_\alpha},t_{n+m_\alpha+1})$ for some $m_\alpha$, $\alpha=1,2$, and
$m_{2}=m_{1}+1$. Furthermore $m_{2}\Delta t\le \tau+\Delta t$. Using this
notation we have that
\begin{equation}\label{eq:obser1}
\begin{aligned}
&\iint_{\Pi_{T-\tau}}\Bigl(A^{\kappa}\left(u^{\kappa}_h(x,t+\tau)\right)
- A^{\kappa}\left(u^{\kappa}_h(x,t)\right)\Bigr)^2 \,dt\,dx \\
&=  \sum_{\alpha=1}^2 c_{\alpha} \Delta t h^d\sum_{n,I}
        \Bigl(A^{\kappa}\left(u^{\kappa,m_{\alpha}}_I\right) -
        A^{\kappa}\left(u^{\kappa,n}_I\right) \Bigr)^2,
\end{aligned}
\end{equation}
for some weights $c_1,c_2\in [0,1]$
satisfying $c_{1}+c_{2}=1$. Now
\begin{align*}
&\bigl(A^{\kappa}\left(u^{\kappa,m_{\alpha}}_I\right) -
    A^{\kappa}\left(u^{\kappa,n}_I\right) \bigr)^2 \\
&\le
    \max_{u} \frac{dA^{\kappa}}{du}(u) \Bigl(
    A^{\kappa}\left(u^{\kappa,n+m_{\alpha}}_I\right) -
    A^{\kappa}\left(u^{\kappa,n}_I\right)\Bigr)\Bigl(u^{\kappa,m_{\alpha}}_I -
    u^{\kappa,n}_I\Bigr) \\
&\le C_{1}\Bigl(
    A^{\kappa}\left(u^{\kappa,n+m_{\alpha}}_I\right) -
    A^{\kappa}\left(u^{\kappa,n}_I\right)\Bigr) \sum_{m=n}^{n+m_{\alpha}-1}
    \Bigl(u^{\kappa,m+1}_I -
    u^{\kappa,m}_I\Bigr)\\
&=C_{1}\Big[-\underbrace{\Bigl(
    A^{\kappa}\left(u^{\kappa,n+m_{\alpha}}_I\right) -
    A^{\kappa}\left(u^{\kappa,n}_I\right)\Bigr)
    \Delta t \sum_{m=n}^{n+m_{\alpha}-1}
    \sum_{i=1}^{d} D_{i,-}
    F_i^{\kappa,\mathrm{EO}}\left(u^{\kappa,m}_{I},u^{\kappa,m}_{I+e_{i}}\right)
    }_{B_{1,\alpha}(n,I)}\\
&\quad +
    \underbrace{\Bigl(
    A^{\kappa}\left(u^{\kappa,n+m_{\alpha}}_I\right) -
    A^{\kappa}\left(u^{\kappa,n}_I\right)\Bigr) \Delta t \sum_{m=n}^{n+m_{\alpha}-1}
    \sum_{i=1}^{d} D_{i,-}D_{i,+}
    A^{\kappa}\left(u^{\kappa,m}_{I}\right)}_{B_{2,\alpha}(n,I)}\\
&\quad +
    \underbrace{\Bigl(
    A^{\kappa}\left(u^{\kappa,n+m_{\alpha}}_I\right) -
    A^{\kappa}\left(u^{\kappa,n}_I\right)\Bigr) \Delta t \sum_{m=n}^{n+m_{\alpha}-1}
    g^{\kappa}\left(U^m_{I}\right)}_{B_{3,\alpha}(n,I)}\Big].
\end{align*}
Using Lemma \ref{lem:basic} (as  before), we get
the uniform bound
\begin{equation}
    \Delta t h^d \sum_{n,I} \abs{B_{1,\alpha}(n,I)} +
    \abs{B_{3,\alpha}(n,I)} \le
    C_{2} \tau,
    \label{eq:obser2}
\end{equation}
where $C_2$ is independent of $h$ and we have used that
$(m_{2}-1)\Delta t\le \tau$.
Regarding $B_{2,\alpha}$, we
use summation by parts to obtain
\begin{align*}
   \sum_{I}B_{2,\alpha}(n,I)
&= - \Delta t \sum_{m=n}^{n+m_{\alpha}-1} \sum_{I}
   \sum_{i=1}^{d} D_{i,+} A^{\kappa}\left(u_I^{\kappa,n+m_{\alpha}}\right)
   D_{i,+} A^{\kappa}\left(u_{I}^{\kappa,m}\right)   \\
& \quad +\Delta t \sum_{m=n}^{n+m_{\alpha}-1} \sum_{I}
   \sum_{i=1}^{d} D_{i,+} A^{\kappa}\left(u_I^{\kappa,n}\right)
   D_{i,+} A^{\kappa}\left(u_{I}^{\kappa,m}\right)\\
&\le \frac{\Delta t}{2} \left(m_{\alpha}-1\right) \sum_{I}
   \Big(\sum_{i=1}^{d} \left(D_{i,+}
   A^{\kappa}\left(u_{I}^{\kappa,n+m_{\alpha}}\right)\right)^2 +
   \left(D_{i,+} A^{\kappa}\left(u_{I}^{\kappa,n}\right)\right)^2 \Big)\\
&\quad + \Delta t \sum_{m=n}^{n+m_{\alpha}-1}
   \sum_{I}\sum_{i=1}^{d} \left(D_{i,+}
   A^{\kappa}\left(u_{I}^{\kappa,m}\right)\right)^2,
\end{align*}
where we have used the identity $ab\le \frac12 \left(a^2 + b^2\right)$
for all $a,b\in \mathbb{R}$. Now that we have used the scheme to get rid of
all time differences we use  \eqref{WeakBV:Final} to conclude that
\begin{equation}
    \Delta t h^d \sum_{n,I} \abs{B_{2,\alpha}(n,I)} \le C_{3}\tau,
    \label{eq:obser3}
\end{equation}
for some constant $C_{3}$ independent of $\Delta t$. Now
\eqref{eq:obser1}, \eqref{eq:obser2} and \eqref{eq:obser3} closes the
proof of \eqref{A_L2_time}.
\end{proof}

\begin{remark} \label{rmk4.7}\rm
   Observe that if we went directly via Lemmas \ref{lem:basic}
   and \ref{lem:L1cont} (interpolating between
   $L^1$ and $L^{\infty}$), then we would have obtained
   the (not optimal) estimate
   $$
   \left\|A\left(U_h(\cdot,t+\tau)\right)
   -A\left(U_h(\cdot,t)\right)\right\|_{L^2(\mathbb{R}^d)}
   \le C \tau^{1/4},\quad  t\in (0,T),
   $$
   for some constant that is independent of $h$.
\end{remark}

We next show that the finite difference scheme satisfies
a discrete entropy condition.
Let $\eta\colon\mathbb{R}\to\mathbb{R}$ be an entropy function. In this case
the associated Engquist--Osher (numerical) entropy flux
$q^{\kappa,\mathrm{EO}}(u,v)=\big(q_1^{\kappa,\mathrm{EO}},\dots,q_d^{\kappa,\mathrm{EO}}\big)$
is defined by (see, e.g., Kr\"oner \cite[p.~184]{Kroner})
\begin{equation}
   \label{EO_entropyflux}
   q_i^{\kappa,\mathrm{EO}}(u,v)= \int_0^u \eta'(\xi)\Bigl(
{\frac{dF_i^{\kappa}}{du}(\xi)}\vee 0\Bigr)\,d\xi
   + \int_0^v \eta'(\xi)\Bigl({\frac{dF_i^{\kappa}}{du}(\xi)}\wedge 0\Bigr)\,d\xi,
   \; i=1,\dots,d.
\end{equation}
The next lemma provides us with a cell entropy inequality for the
Engquist--Osher scheme \eqref{EO_scheme}.

\begin{lemma}\label{lem:cell_entr}
   For any entropy function
   $\eta\colon\mathbb{R}\to\mathbb{R}$ and corresponding entropy fluxes $q^{\kappa,\mathrm{EO}},r^{\kappa}$,
\begin{equation}     \label{eq:cell_entr}
 \begin{split}
&\frac{\eta(u_I^{\kappa,n+1}) - \eta\left(u_I^{\kappa,n}\right)}{\Delta t}
 + \sum_{i=1}^d D_{i,-} q_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)
 - \sum_{i=1}^d D_{i,-}D_{i,+} r^{\kappa}\left(u_I^{\kappa,n}\right)\\
&\le \eta'\left(u_I^{\kappa,n+1}\right)g^{\kappa}\left(U_I^n\right),
\quad \kappa=1,\dots,K.
\end{split}
\end{equation}
\end{lemma}

\begin{proof}
Assume for the moment that the following
inequality holds:
\begin{equation}
   \label{cell_entr_bar}
   \frac{\eta(\overline{u}_I^{\kappa,n+1})
   - \eta\left(u_I^{\kappa,n}\right)}{\Delta t}
   + \sum_{i=1}^d D_{i,-} q_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)
   - \sum_{i=1}^d D_{i,-}D_{i,+} r^{\kappa}\left(u_I^{\kappa,n}\right) \le 0.
\end{equation}
Then using \eqref{EO_scheme_new} and convexity of the
entropy function $\eta$, it follows that
\begin{equation}
   \label{convex_bar}
   \eta\big(\overline{u}_I^{\kappa,n+1}\big)\ge \eta\big(u_I^{\kappa,n+1}\big)
   - \eta'\big(u_I^{\kappa,n+1}\big) \Delta t g^{\kappa}\big(U_I^n\big).
\end{equation}
Combining \eqref{cell_entr_bar} and \eqref{convex_bar}, we
get the desired cell entropy inequality \eqref{eq:cell_entr}.

It remains to prove \eqref{cell_entr_bar}.
The proof is based on a monotonicity property ensured by the CFL
condition. We refer to Kr\"oner \cite{Kroner}
for a similar proof in the context
of hyperbolic conservation laws.
For $\kappa=1,\dots,K$,
define the function $H^{\kappa}\colon\mathbb{R}^{2d}\to \mathbb{R}$ by
\begin{align*}
   &H^{\kappa}\left(u_{I-e_1}^{\kappa,n},u_{I+e_1}^{\kappa,n}, \dots,
   u_{I-e_d}^{\kappa,n},u_{I+e_d}^{\kappa,n}\right)   \\
&  =\eta\left(\overline{u}_I^{\kappa,n+1}\right)- \eta\left(u_I^{\kappa,n}\right)
   + \lambda\sum_{i=1}^d \Bigl(q_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)
      - q_i^{\kappa,\mathrm{EO}}\left(u_{I-e_i}^{\kappa,n},u_I^{\kappa,n}\right)\Bigr)\\
& \quad + \mu \sum_{i=1}^d\Bigl(r^{\kappa}\left(u_{I-e_i}^{\kappa,n}\right)
      - 2 r^{\kappa}\left(u_I^{\kappa,n}\right) + r^{\kappa}\left(u_{I+e_i}^{\kappa,n}\right)\Bigr),
\end{align*}
where
\begin{align*}
\overline{u}_I^{\kappa,n+1}&=u_I^{\kappa,n} - \lambda\sum_{i=1}^d
      \Bigl(F_i^{\kappa,\mathrm{EO}}\left(u_I^{\kappa,n},u_{I+e_i}^{\kappa,n}\right)
      - F_i^{\kappa,\mathrm{EO}}\left(u_{I-e_i}^{\kappa,n},u_I^{\kappa,n}\right)\Bigr)\\
&\quad + \mu \sum_{i=1}^d\Bigl(A^{\kappa}\left(u_{I-e_i}^{\kappa,n}\right)
      - 2 A^{\kappa}\left(u_I^{\kappa,n}\right) + A^{\kappa}\left(u_{I+e_i}^{\kappa,n}\right)\Bigr).
\end{align*}
Observe that
$H^{\kappa}(u_I^{\kappa,n}, \dots,u_I^{\kappa,n})=0$.
Furthermore, using a first-order Taylor expansion along
with the CFL condition \eqref{CFL} and
convexity of $\eta$, it is not hard to check that
\begin{gather*}
   \partial_{\ell}H^{\kappa}\left(u_I^{\kappa,n},
\dots,\xi_{\ell},\dots,u_I^{\kappa,n}\right)
   >0, \quad \xi_{\ell}<u_I^{\kappa,n}, \quad \ell=1,\dots,2d. \\
\partial_{\ell}H^{\kappa}\left(u_I^{\kappa,n},
\dots,\xi_{\ell},\dots,u_I^{\kappa,n}\right)
   <0, \quad \xi_{\ell}>u_I^{\kappa,n}, \quad \ell=1,\dots,2d.
\end{gather*}
 From this we conclude that $H$ is a non-positive function and
hence \eqref{cell_entr_bar} follows.
\end{proof}

We now have the necessary tools to
prove our main result of this section.

\begin{theorem}[Existence]
   \label{thm:exist}
   Assume that \eqref{flux_cond} and \eqref{G_cond} hold.
   There exists an entropy solution of the Cauchy problem
   \eqref{CD}, \eqref{CD:data}. Furthermore, the
   entropy solution can be constructed as the limit
   of a sequence of finite difference approximations.
\end{theorem}

\begin{proof}
Let us first treat the case where
$u_0$ belongs to $L^1(\mathbb{R}^d)\cap L^{\infty}(\mathbb{R}^d)\cap BV(\mathbb{R}^d)$ and
has
compact support. Furthermore, we assume that $F,A$ are $C^1$.
In view of the $h$ uniform estimates
in Lemmas \ref{lem:basic} and
\ref{lem:L1cont},  Lemma \ref{L1_compact} tells us that
the sequence $\seq{u^{\kappa}_h}_{h>0}$ is compact in
$L^1_{\mathrm{loc}}(\Pi_T)$. Moreover, any limit point of this sequence
satisfies (1) and (4) in Definition \ref{def:sol}.
Using Lemma \ref{lem:cell_entr} and
standard arguments analogous to the ones
used to prove the classical Lax--Wendroff theorem, we eventually
conclude that any limit point of $\seq{u^{\kappa}_h}_{h>0}$ satisfies
the entropy condition \eqref{entropy_cond}.
In view of Lemma \ref{Adiff_L2}, Lemma \ref{lem:A_time},
and since $A^{\kappa}(u^{\kappa}_h)$ obviously belongs to $L^2(\Pi_T)$, Lemma
\ref{L2_compact}
tells us that the sequence $\seq{A^{\kappa}\left(u^{\kappa}_h\right)}_{h>0}$ is
compact in
$L^2_{\mathrm{loc}}(\Pi_T)$. Moreover, any limit point of this sequence
satisfies (3) in Definition \eqref{def:sol}.

To treat the general case where
$u_0$ only belongs to $L^1(\mathbb{R}^d)\cap L^{\infty}(\mathbb{R}^d)$, we
use the $L^1$ stability result in
Theorem \ref{thm:uniq} along with an
approximation procedure. This argument
is classical and it is thus omitted,  see instead
Crandall and Majda \cite{CranMaj:Monoton}, for example.
Similarly, the case that $F,A$ are merely Lipschitz continuous
can be treated by approximating $F,A$ with $C^1$ functions
$F_\ell,A_\ell$
and noting that all previous estimates are robust with respect to
sending $\ell\uparrow \infty$.
\end{proof}

\section{A numerical example}\label{sec:applic}

As an illustration of the ideas set forth in this paper, we consider a
simplified model of biodegradation of a contaminant in a
porous medium. Assume that a contaminant (e.g., oil) is injected into
a porous medium containing water with dissolved oxygen. The contaminant
reacts with oxygen to some third component, which we assume does not
influence the model. We also assume that the oxygen is passively
advected along with the flow, and that it dissolves equally well in
the water and the contaminant. To be precise, we study the following
model
\begin{equation}
    \begin{gathered}
        u_{t}+\mathbf{v}\cdot \nabla \left(f(u)\right) = \varepsilon \Delta u + g(u,c),\\
        c_{t}+\mathbf{v}\cdot \nabla c    =\varepsilon \Delta c + g(u,c).
    \end{gathered}
    \label{eq:nummodel}
\end{equation}
Here, $u$ denotes the concentration of the contaminant, and $c$ the
concentration of the oxygen. The velocity field $\mathbf{v}$ is given
by
$$
\mathbf{v}(x,y)=-\frac{\mathbf{r}_{1}}{\abs{\mathbf{r}_{1}}} +
\frac{\mathbf{r}_{2}}{\abs{\mathbf{r}_{2}}}
$$
with $\mathbf{r}_{1}=(x-0.1,y-0.5)$, $\mathbf{r}_{2}=(x-1,y-0.5)$.
The flux function is
$$
f(u)=\frac{u^2}{u^2+(1-u)^2},
$$
and the source term $g$ models the reaction by Monod kinetics via
\begin{equation}
    g(u,c)=K\frac{uc}{(0.2+u)(0.2+c)}, \quad K=3.5.
    \label{eq:numsource}
\end{equation}
Finally, we set $\varepsilon=0.25$. We consider this model in the rectangle
$(x,y)\in [0,1]\times[0,0.5]$. To compute numerical approximations we
use a straightforward modification of the Engquist--Osher scheme
\eqref{EO_scheme}, using Neumann boundary conditions.  We remark that
this model is strongly inspired by a similar model in
\cite{OhlbergerRohde:WC}.
In Figure~\ref{fig:velofig} we show the velocity field $\mathbf{v}$
and the setup for our computations.
The ``inlet'' is at the point $(0.1,0.25)$ and is modeled by setting
$$
u(x,y,0)=\begin{cases}
    1 & (x,y)\in D, \\
    0 & \text{otherwise},
\end{cases}
$$
where
$$
D=\seq{ (x,y) : (x-0.1)^2+(y-0.25)^2\le 0.025\;}.
$$
Furthermore, we also set $u(x,y,t)=1$ for $(x,y)\in D$. The initial
``oxygen'' saturation is everywhere $1$, i.e., $c(x,y,0)=1$.
We used $\Delta x=\Delta y=1/100$ for our simulation.

\begin{figure}[tbp]
\begin{center}
  \includegraphics[width=0.5\linewidth]{fig1.eps} %velos
\end{center}
    \caption{The velocity field and the geometric configuration.}
    \label{fig:velofig}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
 \includegraphics[width=0.7\linewidth]{fig2.eps} %upl-u
\end{center}
    \caption{Numerical solution of the uncoupled equations, the
    $u$-component.}
    \protect\label{fig:uncoupled}
\end{figure}

In Figure~\ref{fig:uncoupled} we show the saturation $u$ at $t=0.4$
if $K=0$ in \eqref{eq:numsource}, i.e., we have a scalar conservation
law. Compare this with Figure~\ref{fig:u-coupled} where we show the
approximate solution of \eqref{eq:nummodel} at $t=0.4$. In
Figure~\ref{fig:c-coupled} we show the corresponding $c$ variable.
It is not difficult to see the effect of the coupling of the equations.

\begin{figure}[tbp]
\begin{center}
 \includegraphics[width=0.7\linewidth]{fig3.eps} % cpl-u
\end{center}
    \caption{Numerical solution of \eqref{eq:nummodel},
    $\Delta x=\Delta y=0.01$.}
    \protect\label{fig:u-coupled}
\end{figure}
\begin{figure}[tbp]
\begin{center}
 \includegraphics[width=0.7\linewidth]{fig4.eps} % cpl-c
\end{center}
    \caption{Numerical solution of \eqref{eq:nummodel}, the
    $c$-component.}
    \protect\label{fig:c-coupled}
\end{figure}


\begin{thebibliography}{00}

\bibitem{AAI}
M.~Afif and B.~Amaziane.
\newblock Convergence of finite volume schemes for a degenerate
  convection-diffusion equation arising in two-phase flow in porous
media.
\newblock  {\em Comput. Methods. Appl. Engrg.}, 191:5265--5286, 2002.

\bibitem{AltLuckhaus}
H.~W. Alt and S.~Luckhaus.
\newblock Quasilinear elliptic-parabolic differential equations.
\newblock {\em Math. Z.}, 183(3):311--341, 1983.

\bibitem{BenilanToureIII}
P.~B{\'e}nilan and H.~Tour{\'e}.
\newblock Sur l'\'equation g\'en\'erale $u\sb
t=a(\cdot,u,\phi(\cdot,u)\sb
  x)\sb x+v$ dans ${L}\sp 1$. {I}{I}. {L}e probl\`eme d'\'evolution.
\newblock {\em Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire},
12(6):727--761,
  1995.

\bibitem{BouGuaNat}
F.~Bouchut, F.~R. Guarguaglini, and R.~Natalini.
\newblock Diffusive {BGK} approximations for nonlinear
multidimensional
  parabolic equations.
\newblock {\em Indiana Univ. Math. J.}, 49:723--749, 2000.

\bibitem{BurgerEvjeKarlsen:1D_IBVP}
R.~B\"urger, S.~Evje, and K.~H. Karlsen.
\newblock On strongly degenerate convection-diffusion problems
modeling
  sedimentation-consolidation processes.
\newblock {\em J. Math. Anal. Appl.}, 247(2):517--556, 2000.

\bibitem{BK:Hyp2000}
R.~B\"urger and K.~H. Karlsen.
\newblock A strongly degenerate convection-diffusion problem modeling
  centrifugation of flocculated suspensions.
\newblock In {\em Int. Series of Numerical Mathematics}, Vol.
140 (2001), 207-216, Birkhäuser Verlag.

\bibitem{Burger:bok}
M.~C. Bustos, F.~Concha, R.~B\"{u}rger, and E.~M. Tory.
\newblock {\em Sedimentation and Thickening: Phenomenological
Foundation and
  Mathematical Theory}.
\newblock Kluwer Academic Publishers, Dordrecht, The Netherlands,
1999.

\bibitem{Carrillo:94}
J.~Carrillo.
\newblock On the uniqueness of the solution of the evolution dam
problem.
\newblock {\em Nonlinear Anal.}, 22(5):573--607, 1994.

\bibitem{Carrillo}
J.~Carrillo.
\newblock Entropy solutions for nonlinear degenerate problems.
\newblock {\em Arch. Rational Mech. Anal.}, 147(4):269--361, 1999.

\bibitem{Champieretal:93}
S.~Champier, T.~Gallou{\"e}t, and R.~Herbin.
\newblock Convergence of an upstream finite volume scheme for a
nonlinear
  hyperbolic equation on a triangular mesh.
\newblock {\em Numer. Math.}, 66(2):139--157, 1993.

\bibitem{ChenDiBen}
G.-Q. Chen and E.~DiBenedetto.
\newblock Stability of entropy solutions to the {C}auchy problem for
a class of
  hyperbolic-parabolic equations.
\newblock{\em SIAM J. Math. Anal.}, 33(4): 751--762, 2001.

\bibitem{ChenPerthame}
G.-Q. Chen and B.~Perthame.
\newblock Well-posedness for anisotropic
degenerate parabolic-hyperbolic equations.
\newblock \newblock {\em Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire},
to appear.


\bibitem{Cirpkaetal:biodegrad}
O.~Cirpka, E.~Frind, and R.~Helmig.
\newblock Numerical simulation of biodegradation controlled by
transverse mixing. \newblock{\em J.~Contaminant Hydrology},
40:159--182, 1999.

\bibitem{CockGripen}
B.~Cockburn and G.~Gripenberg.
\newblock Continuous dependence on the nonlinearities of solutions of
  degenerate parabolic equations.
\newblock {\em J. Differential Equations}, 151(2):231--251, 1999.

\bibitem{CranMaj:Monoton}
M.~G. Crandall and A.~Majda.
\newblock Monotone difference approximations for scalar conservation
laws.
\newblock {\em Math. Comp.}, 34(149):1--21, 1980.

\bibitem{CrandallTartar}
M.~G. Crandall and L.~Tartar.
\newblock Some relations between nonexpansive and order preserving
mappings.
\newblock {\em Proc. Amer. Math. Soc.}, 78(3):385--390, 1980.

\bibitem{EngquistOsher:81}
B.~Engquist and S.~Osher.
\newblock One-sided difference approximations for nonlinear
conservation laws.
\newblock {\em Math. Comp.}, 36(154):321--351, 1981.

\bibitem{EspKar}
M.~S. Espedal and K.~H. Karlsen.
\newblock Numerical solution of reservoir flow models based on large
time step operator splitting algorithms.
\newblock In {\em Filtration in Porous Media and Industrial
Applications (Cetraro, Italy, 1998)}, volume 1734 of {\em Lecture Notes in
Mathematics}, pages 9--77. Springer, Berlin, 2000.

\bibitem{EvjeKarlsen:nm}
S.~Evje and K.~H. Karlsen.
\newblock Viscous splitting approximation of mixed
hyperbolic-parabolic convection-diffusion equations.
\newblock {\em Numer. Math.}, 83(1):107--137, 1999.

\bibitem{EvjeKarlsen:SJNA}
S.~Evje and K.~H. Karlsen.
\newblock Monotone difference approximations of {$BV$} solutions to
degenerate convection-diffusion equations.
\newblock {\em SIAM J. Numer. Anal.}, 37(6):1838--1860,
2000.

\bibitem{EvjeKarlsen:ViscError}
S.~Evje and K.~H. Karlsen.
\newblock An error estimate for viscous approximate solutions of
degenerate parabolic equations.
\newblock {\em J. Nonlinear Math. Phys.}, 9:262--281, 2002.

\bibitem{Eymardetal}
R.~Eymard, T.~Gallou{\"e}t, M.~Ghilani, and R.~Herbin.
\newblock Error estimates for the approximate solutions of a nonlinear
  hyperbolic equation given by finite volume schemes.
\newblock {\em IMA J. Numer. Anal.}, 18(4):563--594, 1998.

\bibitem{Eymard_etal_I:2000}
R.~Eymard, T.~Gallou{\"e}t, and R.~Herbin.
\newblock Error estimate for approximate solutions of a nonlinear
  convection-diffusion problem.
\newblock {\em Advances in Differential Equations}, 7(4):419--440, 2002.

\bibitem{Eymard_etal_II:2000}
R.~Eymard, T.~Gallou{\"e}t, R.~Herbin, and A.~Michel.
\newblock Convergence of a finite volume scheme for nonlinear
degenerate  parabolic equations.
\newblock {\em Numer. Math.}, 92(1):41--82, 2002.

\bibitem{Eymardetal:diffusion}
R.~Eymard, T.~Gallou{\"e}t, D.~Hilhorst, and Y.~Na{\"\i}t~Slimane.
\newblock Finite volumes and nonlinear diffusion equations.
\newblock {\em RAIRO Mod\'el. Math. Anal. Num\'er.},
32(6):747--761, 1998.

\bibitem{FrontBOOK}
H.~Holden and N.~H. Risebro.
\newblock {\em Front Tracking for Hyperbolic Conservation Laws.}
\newblock Springer, New York, 2002.

\bibitem{Ishii}
H.~Ishii.
\newblock On uniqueness and existence of viscosity solutions of fully
nonlinear second-order elliptic {P}{D}{E}s.
\newblock {\em Comm. Pure Appl. Math.}, 42(1):15--45, 1989.

\bibitem{KO:Unique}
K.~H. Karlsen and M.~Ohlberger.
\newblock A note on the uniqueness of entropy solutions of nonlinear
degenerate parabolic equations.
\newblock {\em J. Math. Anal. Appl.}, 275(1):439-458, 2002.

\bibitem{KR:Rough_Diff}
K.~H. Karlsen and N.~H. Risebro.
\newblock Convergence of finite difference schemes for viscous and
inviscid conservation laws with rough coefficients.
\newblock {\em M2AN Math. Model. Numer. Anal.}, 35(2):239--269, 2001.

\bibitem{KR:Rough_Unique}
K.~H. Karlsen and N.~H. Risebro.
\newblock On the uniqueness and stability of entropy solutions of
nonlinear degenerate parabolic equations with rough coefficients.
\newblock {\em Discrete Contin. Dynam.
Systems - Series A}, 9(5):1081--1104, 2003.

\bibitem{Kroner}
D.~Kr{\"o}ner.
\newblock {\em Numerical Schemes for Conservation Laws}.
\newblock John Wiley \& Sons Ltd., Chichester, 1997.

\bibitem{KruzkovII}
S.~N. Kru{\v{z}}kov.
\newblock Results concerning the nature of the continuity
of solutions of parabolic
  equations, and some of their applications.
\newblock {\em Mat. Notes}, 6:517--523, 1969.

\bibitem{Kruzkov}
S.~N. Kru{\v{z}}kov.
\newblock First order quasi-linear equations
in several independent variables.
\newblock {\em Math. USSR Sbornik}, 10(2):217--243, 1970.

\bibitem{Mascia_etal:2000}
C.~Mascia, A.~Porretta, and A.~Terracina.
\newblock Nonhomogeneous Dirichlet problems
for degenerate parabolic-hyperbolic equations.
\newblock {\em Arch. Ration. Mech. Anal.}, 163:87--124, 2002.

\bibitem{NatHan}
R.~Natalini and B.~Hanouzet.
\newblock Weakly coupled systems of quasilinear hyperbolic equations.
\newblock {\em Differential Integral Equations}, 9(6):1279--1292, 1996.

\bibitem{Ohlberger:FVM}
M.~Ohlberger.
\newblock A posteriori error estimates for vertex centered finite
volume approximations of convection-diffusion-reaction equations.
\newblock {\em M2AN Math. Model. Numer. Anal.}, 35(2):355-387, 2001.

\bibitem{OhlbergerRohde:WC} M.~Ohlberger and C.~Rohde.
\newblock{Adaptive finite volume approximations for weakly
coupled convection dominated parabolic systems.}
\newblock{\em IMA J. Numer. Anal.}
\newblock 22(2):253--280, 2002.

\bibitem{Otto:L1_contr}
F.~Otto.
\newblock ${L}\sp 1$-contraction and uniqueness for quasilinear
  elliptic-parabolic equations.
\newblock {\em J. Differential Equations}, 131(1):20--38, 1996.

\bibitem{Rohde}
C.~Rohde.
\newblock Entropy solutions for weakly coupled hyperbolic systems in
several space dimensions.
\newblock {\em Z. Angew. Math. Phys.}, 49(3):470--499, 1998.

\bibitem{RohdeII}
C.~Rohde.
\newblock Upwind finite volume schemes for weakly coupled hyperbolic
systems of conservation laws in 2D.
\newblock {\em Numer. Math.}, 81(1):85--123, 1998.

\bibitem{RouvreGagneux}
{\'E}.~Rouvre and G.~Gagneux.
\newblock Solution forte entropique de lois scalaires
  hyperboliques-paraboliques d\'eg\'en\'er\'ees.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I Math.},
329(7):599--602, 1999.

\bibitem{Volpert}
A.~I. Vol'pert.
\newblock The spaces {BV} and quasi-linear equations.
\newblock {\em Math. USSR Sbornik}, 2(2):225--267, 1967.

\bibitem{VolHud}
A.~I. Vol'pert and S.~I. Hudjaev.
\newblock Cauchy's problem for degenerate second
order quasilinear parabolic equations.
\newblock {\em Math. USSR Sbornik}, 7(3):365--387, 1969.

\end{thebibliography}


\end{document}









)