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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 116, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/116\hfil Fractional degenerate parabolic equations]
{Stability of entropy solutions for L\'evy mixed hyperbolic-parabolic equations}

\author[K. H. Karlsen, S. Ulusoy\hfil EJDE-2011/116\hfilneg]
{Kenneth H. Karlsen, S\"{u}leyman Ulusoy}  % in alphabetical order

\address{Kenneth Hvistendahl Karlsen \newline
  Centre of Mathematics for Applications \\
  University of Oslo \newline
  P.O. Box 1053, Blindern,  N-0316 Oslo, Norway}
\email{kennethk@math.uio.no \quad http://folk.uio.no/kennethk}


\address{S\"{u}leyman Ulusoy \newline
 Department of Mathematics \\
 Faculty of Education \\
 Zirve University \newline
 Sahinbey, Gaziantep, 27270, Turkey}
\email{suleyman.ulusoy@zirve.edu.tr\quad  http://person.zirve.edu.tr/ulusoy/}

\thanks{Submitted April 25, 2011. Published September 12, 2011.}
\subjclass[2000]{45K05, 35K65, 35L65; 35B65}
\keywords{Degenerate parabolic equation; conservation law; stability;
\hfill\break\indent fractional Laplacian;  
non-local diffusion; entropy solution; 
uniqueness;  continuous dependence} 

\begin{abstract}
 We analyze entropy solutions for a class of L\'evy mixed 
 hyperbolic-parabolic equations containing a non-local (or fractional) 
 diffusion operator originating from a pure jump L\'evy process.
 For these solutions we establish uniqueness ($L^1$ contraction property)
 and continuous dependence results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{definition}[theorem]{Definition}

\newcommand{\abs}[1]{|#1|}
\newcommand{\norm}[1]{\|#1\|}

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

The subject of this paper is uniqueness and stability
results for properly defined entropy solutions of
mixed hyperbolic-parabolic quasilinear equations appended with
a nonlocal (fractional) diffusion operator.
These equations take the form
\begin{equation}\label{eq1}
	\partial_t u + \operatorname{div} f(u) = \operatorname{div} (a(u) \nabla u)
	+ \mathcal{L}[u],
\end{equation}
where $u=u(t,x)$ is the unknown, $(t,x) \in
Q_T := (0,T) \times \mathbb{R}^d$, $d\ge 1$, $T>0$ is a
fixed final time, and $\mathcal{L}$ is a pure jump
L\'evy operator.

Equation \eqref{eq1} is subject to initial data
\begin{equation}\label{in}
	u(0,x)= u_0(x) \in  (L^1 \cap L^{\infty})(\mathbb{R}^d).
\end{equation}
In  \eqref{eq1},
\begin{equation}\label{fregul}
	f= (f_1,\ldots, f_d) \in W^{1,\infty}(\mathbb{R};\mathbb{R}^d)
\end{equation}
is a given vector-valued flux function, $a =(a_{ij})\ge 0$ is a
given symmetric matrix-valued diffusion function of the form
\begin{equation}\label{dif}
	a= \sigma^a (\sigma^a)^{\mathrm{tr}}, \quad
	\sigma^a \in \mathbb{R}^{d \times K},
	\quad 1 \leq K \leq d.
\end{equation}
More precisely, the components of $a$ are
$a_{ij} = \sum_{k=1}^K \sigma_{ik}^a \sigma_{jk}^a$ for $i,j= 1,\ldots,d$.
We assume that the matrix-valued function
$\sigma^a = (\sigma_{ik}^a):\mathbb{R} \to \mathbb{R}^{d \times K}$ satisfies
\begin{equation}\label{sigreg}
	\sigma^a\in W^{1,\infty}(\mathbb{R};\mathbb{R}^{d\times K}).
\end{equation}
Observe that we do not assume the matrix $a(\cdot)$ to be
strictly positive definite, so the operator
$\operatorname{div} (a(u) \nabla u)$ may be strongly degenerate, and
hence the phrase ``mixed hyperbolic-parabolic" is justified.

In terms of its singular integral representation, the nonlocal
operator $\mathcal{L}$ in \eqref{eq1} takes the form
\begin{equation} \label{nlocsplit}
	\mathcal{L}[u](t,x) =  \int_{\mathbb{R}^d \setminus \{0\}}
	\left[ u(t,x+ z) - u(t,x) - z \cdot \nabla u\, \mathbf{1}_{|z|<1}
	\right]\pi(dz),
\end{equation}
where the singular L\'evy measure $\pi(dz)$ is a positive, $\sigma$-finite Borel measure
on $\mathbb{R}^d \setminus \{0\}$ satisfying $\pi(\{0\}) = 0$, $\pi(d(-z))=-\pi(dz)$, and
\begin{equation}\label{intcond1}
	\int_{\mathbb{R}^d \setminus \{0\}}
	\big( \abs{z}^2 \mathbf{1}_{\abs{z}<1}
	+ \abs{z} \mathbf{1}_{\abs{z}\ge 1}\big)
	\pi(dz) < \infty,
\end{equation}
where we note that $z$ can be replaced by a certain
regular jump function $j(z)$ easily throughout the analysis.
A typical example is provided by taking
$$
\pi(z)=\frac{1}{|z|^{d+\alpha}}\,
\mathbf{1}_{|z|<1} \,dz,\quad \alpha \in (0,2).
$$
This example is related to to the fractional
Laplacian $\Delta_\alpha := -(-\Delta)^{\frac{\alpha}{2}}$
on $\mathbb{R}^d$, which can also be defined
in terms of the Fourier transform as
$$
\widehat{\Delta_\alpha v}(\omega)= \abs{\omega}^{\alpha}
\widehat{v}(\omega), \quad \omega\in \mathbb{R}^d.
$$
This definition is employed in \cite{Droniou:2006os}
to prove \eqref{nlocsplit} in this case.

Nonlocal operators like $\Delta_\alpha$ are examples of a
Fourier multiplier operator $\mathcal{P}$ with a symbol $a(\omega)\ge 0$ such that
$\widehat{\mathcal{P}v}(\omega)=a(\omega) \widehat{v}(\omega)$.
The function $e^{-ta(\omega)}$ is positive definite, and thus, by the
L\'evy-Khintchine formula, it can be represented as
$$
a(\omega)=i b\cdot \omega + q(\omega)
+  \int_{\mathbb{R}^d \setminus \{0\}} \left(1-e^{-i z\cdot\omega}
- i z\cdot \omega\, \mathbf{1}_{\abs{z}<1}(z)\right)\pi(dz),
$$
where $b\in \mathbb{R}^d$ represents the drift term, 
$q(\omega)=\sum_{i,j=1}^d q_{ij} \omega_i \omega_j$
is a positive definite quadratic function representing the pure diffusion
part ($q(\omega)=\abs{\omega}^2$ gives raise to the usual Laplacian $-\Delta$),
and the L\'evy measure $\pi(dz)$ accounts for the jump (non-local) part.
In our setting of $\mathcal{L}$, cf.~\eqref{nlocsplit}, we assume
$b\equiv 0$  and $q\equiv 0$, i.e, we are dealing with a pure jump
operator.  For more details about the L\'evy-Khintchine formula 
and L\'evy processes in general, we
refer to \cite{Bertoin:1996ud,Jacob:2001rf,Jacob:2002xp,Jacob:2005ss,Sato:1999qd}.

Integro-partial differential equations, also known as nonlocal, fractional
or L\'evy partial differential equations, appear frequently in many
different areas of research and find many applications in engineering and finance,
including nonlinear acoustics, statistical mechanics, biology, fluid flow,
pricing of financial instruments, and portfolio optimization.
Many authors have recently contributed to advancing the mathematical theory for quasilinear and
fully nonlinear partial differential equations that are supplemented with a
fractional diffusion operator arising as the generator of a L\'evy semigroup, addressing
questions like existence, uniqueness, regularity, formation of
singularities, and asymptotic behavior of solutions.


Another very popular subject recently, where non-local operators
appear, is the so-called quasi-geostrophic equation. This equation can be written
in divergence form and the variational techniques are useful.
Interested readers can consult \cite{Caf-Vas, Ki-Na-Vo, Wu-IUMJ, Wu-EJDE} and
the references therein for further discussion of this subject.

For results with reference to fully nonlinear equations, such as the Hamilton-Jacobi-Bellman equation,
and the (in this context relevant) theory of viscosity solutions, we refer to
\cite{Alibaud:2007dw,Alvarez:1996lq,Arisawa:2006db,Barles:1997xj,Barles:2008lq,Caffarelli:2007hh,Caffarelli:2007qr,Caffarelli:2008bx,Garroni:2002il,Jakobsen:2005jy,Jakobsen:2006aa,Pham:1998zt,Sayah:1991jk,Sayah:1991xy,Silvestre:2006bq,Silvestre:2007qp,Soner:IntegroDif}, see also
\cite{Benth:2001tv,Benth:2002mk,Cont:2004gk} for some concrete applications to finance.

More recently, a number of authors
\cite{Alibaud:2007mi,Alibaud:2007qe,Biler:1998ai,Biler:2001th,Bossy:2002eu,Brandolese:2008sp,Droniou:2006os,Karch:2008dp} have studied questions regarding
existence, uniqueness, regularity, and temporal asymptotics for
quasilinear equations, such as the fractal Burgers equation
\begin{equation}\label{eq:fractional-Burgers}
	\partial_t u + \partial_x (u^2/2)=-(-\partial_{xx}^2)^{\frac{\alpha}{2}}u,
\end{equation}
and more generally multi-dimensional fractional conservation laws
\begin{equation}\label{eq:fCL}
	\partial_t u + \operatorname{div} f(u) =  \Delta_\alpha u,
\end{equation}
where the parameter $\alpha$ is assumed lie in the interval $(0,2)$.
Of course,  the excluded case $\alpha=2$ corresponds to the already fully understood viscous
conservation law  $\partial_t u + \operatorname{div} f(u)=\Delta u$, solutions of which
are always smooth in $t>0$. Regarding the less studied case $\alpha\in [1,2)$, it was recently proved
in \cite{Droniou:2003mz,Kiselev:2008jt} that solutions of the fractional
Burgers equation \eqref{eq:fractional-Burgers} are also smooth in $t>0$.
In the case $\alpha<1$ for the fractional conservation
law \eqref{eq:fCL} the order of the diffusion part
is lower than the first order hyperbolic part, so we do not expect any regularizing
effect to take place. Indeed, for the fractional Burgers
equation \eqref{eq:fractional-Burgers} with $\alpha<1$ it is proved
in \cite{Alibaud:2007qe,Kiselev:2008jt} that solutions can
develop discontinuities in finite time. Consequently, one should employ
a notion of entropy solutions for fractional conservation
laws \eqref{eq:fCL}, i.e., weak solutions satisfying an
additional entropy condition, to ensure the global-in-time well-posedness.
This is well-known for conservation
laws $\partial_t u + \operatorname{div} f(u)=0$, cf.~Kru\u{z}kov \cite{Kruzkov:1970kx}, and
the well-posedness theory of Kru\u{z}kov was recently extended to
fractional conservation laws in \cite{Alibaud:2007mi}.


In recent years the theory of Kru\u{z}kov \cite{Kruzkov:1970kx} has been extended to
quasilinear mixed hyperbolic-parabolic equations of the form
\begin{equation}\label{eq1-local}
	\partial_t u + \operatorname{div} f(u) = \operatorname{div} (a(u) \nabla u),
\end{equation}
where $f$ and $a$ satisfy \eqref{fregul} and \eqref{dif}-\eqref{sigreg}, respectively. Since
the diffusion matrix $a(u)$ is not assumed to be strictly positive definite, \eqref{eq1-local}
is strongly degenerate and will in general posses discontinuous solutions.
In the isotropic case (with $a(\cdot)$ being a scalar function) the
first general uniqueness result is due to Carrillo \cite{Carrillo:1999hq}, who developed
an original extension of Kru\u{z}kov's method of doubling variables to prove
his result, cf.~\cite{Karlsen:2002bh,Karlsen:2003za,Mascia:2002dq,Michel:2003tw}
for some additional applications of his techniques. The anisotropic
case ($a(\cdot)$ being a matrix-valued function) was first treated by Chen
and Perthame \cite{Chen:2003td}, who developed a
kinetic formulation and established the uniqueness result using regularization by convolution.
An alternative proof of the result of Chen and Perthame, adapting
the device of doubling variables, was developed
in \cite{Bendahmane:2004go}, cf.~also \cite{Chen:2006oy,Chen:2005wf,Perthame:2003yq}
some other papers dealing with the anisotropic case.

The main purpose of this paper is to extend the uniqueness and
``continuous dependence on the nonlinearities" results of
\cite{Bendahmane:2004go,Chen:2006oy,Chen:2005wf,Perthame:2003yq} to
fractional degenerate parabolic equations of the form \eqref{eq1}.
We introduce the notion of entropy solutions and state
the main results in Section \ref{sec:entform} .
Sections \ref{prexis} (existence), \ref{pruniq} (uniqueness),
and \ref{prcdep} (continuous dependence on the nonlinearities
and the L\'evy measure) are devoted to the proofs of the main results.



\section{Notion of solution and main results}\label{sec:entform}

For $i =1,\ldots,d$ and $k=1,\ldots,K$, define
$$
\zeta_{ik}^a(u) := \int_0^u \sigma_{ik}^a(\xi)\,d\xi,
\quad \zeta_{ik}^{a,\psi}(u) =
\int_0^u \psi(\xi) \sigma_{ik}^a(\xi)\,d\xi,
\quad  u\in \mathbb{R},
$$
for any $\psi \in C(\mathbb{R})$. Given any convex $C^2$
entropy function $\eta:\mathbb{R} \to \mathbb{R}$, we define
the corresponding entropy fluxes $q=(q_i):\mathbb{R} \to \mathbb{R}^d$
and $r=(r_{ij}):\mathbb{R} \to \mathbb{R}^{d \times d}$ by
$$
q'(u) = \eta'(u) f'(u), \quad r'(u) = \eta'(u) a(u).
$$
We refer to $(\eta, q, r)$ as an entropy-entropy flux triple.
We now introduce the entropy formulation of  \eqref{eq1}-\eqref{in}.

\begin{definition}\label{def:entropy} \rm
An entropy solution of the initial value problem \eqref{eq1}-\eqref{in}
is a measurable function $u : Q_T \to \mathbb{R}$ satisfying the following 
conditions:

\begin{enumerate}
\item $u \in L^{\infty}(Q_T)$, $u \in L^{\infty}(0,T;L^1(\mathbb{R}^d))$,
\begin{equation}\label{l2regu}
\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u)
\in L^2(Q_T), \quad k=1,\ldots,K,
\end{equation}
and
\begin{equation}\label{frintcnd}
	\iint_{Q_T}\int_{\mathbb{R}^d \setminus \{0\}}
	\left(u(t,x+z) - u(t,x) \right)^2 \,
	\pi (dz) \,dx \,dt < +\infty;
\end{equation}

\item For $k= 1,\ldots, K$,
\begin{equation}\label{crule}
	\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^{a,\psi}(u)
	= \psi(u) \sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u),
	\quad \text{a.e.~in $Q_T$ and in $L^2(Q_T)$},
\end{equation}
for any $\psi \in C(\mathbb{R})$;
\label{def:chainrule}

\item For any entropy-entropy flux triple $(\eta, q, r)$,
\begin{equation}\label{entineq}
	\begin{split}
		& \iint_{Q_T} \Bigl ( \eta(u) \partial_t \varphi
		+ \sum_{i=1}^d q_i(u) \partial_{x_i} \varphi
		+ \sum_{i,j=1}^d r_{ij}(u) \partial_{x_ix_j}^2 \varphi \Bigr) \,dx\,dt
		\\ & \quad
		+ \iint_{Q_T} \eta(u) \mathcal{L}[\varphi] \,dx\,dt
		+ \int_{\mathbb{R}^d} \eta(u_0) \varphi(0,x) \,dx
		\geq  n^u + m^u,
	\end{split}
\end{equation}
for all non-negative $\varphi \in C_c^{\infty}([0,T) \times \mathbb{R}^d)$, where
\begin{align*}
	n^u & =  \iint_{Q_T} \eta''(u) \sum_{k=1}^K
	\Bigl(\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u) \Bigr)^2 \varphi(t,x)\,dx \,dt,  \\
	m^u & = \iint_{Q_T}  \int_{\mathbb{R}^d \setminus \{0\}}
	\overline{\eta''}(u;z)
	\left(u(t,x+z)-u(t,x)\right)^2 \varphi(t,x)  \pi(dz)\,dx\,dt,
\end{align*}
and
$$
\overline{\eta''}(u;z)
= \int_0^1 (1-\tau) \eta''((1-\tau)u(t,x)+\tau u(t,x+z)) \,d\tau.
$$
\end{enumerate}
\end{definition}

We remark that the chain rule \eqref{crule}
is automatically fulfilled when $a(\cdot)$ is a scalar or
a diagonal matrix, cf.~Chen and Perthame \cite{Chen:2003td}, and in
this case we can drop \eqref{def:chainrule} from the definition.

Our first result is the expected $L^1$ contraction
property (and thus the uniqueness) of entropy solutions.

\begin{theorem}\label{th:unique}
Suppose $f$ and $a$ satisfy \eqref{fregul}
and \eqref{dif}-\eqref{sigreg}, respectively, and
that the L\'evy measure $\pi(dz)$
satisfies \eqref{intcond1}. Then there exists an
entropy solution of \eqref{eq1}-\eqref{in}.
Let $u,v$ be two entropy solutions of \eqref{eq1} with
initial data $u|_{t=0} =u_0 \in (L^1\cap L^{\infty})(\mathbb{R}^d)$,
$v|_{t=0} = v_0 \in (L^1\cap L^{\infty})(\mathbb{R}^d)$.
For a.e. $t \in (0,T)$,  we have
\begin{equation}\label{eq:contraction}
	\int_{\mathbb{R}^d} \left(u(t,x)-v(t,x)  \right)^+ \,dx
	\leq \int_{\mathbb{R}^d} \left(u_0-v_0 \right)^+ \,dx.
\end{equation}
Consequently, if $u_0 \leq v_0$ a.e.~in $\mathbb{R}^d$ then $u \leq v$ a.e.~in $Q_T$, so
whenever $u_0=v_0$ a.e.~in $\mathbb{R}^d$, then $u=v$ a.e.~in $Q_T$.
\end{theorem}

This theorem generalizes to the ``non-local diffusion" case
the result of Chen and Perthame \cite{Chen:2003td}. The proof follows that
of Bendahmane and Karlsen \cite{Bendahmane:2004go}.

Our second result,  which is a refinement of the previous theorem, reveals
how the entropy solution $u$ depends on  the L\'evy measure $\pi(dz)$, and the
nonlinear fluxes $f,a$ (i.e., it is a ``continuous dependence" estimate).

\begin{theorem}\label{th:contdep}
Suppose $f$ and $a$ satisfy \eqref{fregul} and
\eqref{dif}-\eqref{sigreg}, respectively, and
that the L\'evy measure $\pi(dz)$ satisfies \eqref{intcond1}.
Let $u \in L^{\infty}(0,T;BV(\mathbb{R}^d))$ be the entropy
solution of \eqref{eq1} with $BV$ initial
data $u_0 \in (L^1\cap L^{\infty} \cap BV)(\mathbb{R}^d)$
and with a L\'evy measure of the form $\pi(dz) = m(z)\, dz$
for some measurable function $m:\mathbb{R}^d \setminus \{0\}\to \mathbb{R}_+$.

Replace the data set
$$
(f,a,\pi,u_0), \quad a=\sigma^a(\sigma)^{\mathrm{tr}},
\quad \pi(dz)=m(z)\,dz
$$
by another data set
$$
(\tilde f,\tilde a,\tilde \pi(dz),v_0), \quad
\tilde a=\sigma^{\tilde a}(\sigma^{\tilde a})^{\mathrm{tr}},
\quad \tilde \pi(dz)=\tilde m(z)\,dz,
$$
where $\tilde f,\sigma^{\tilde a}, \tilde\pi, \tilde m$ satisfy the same regularity conditions
as $f, \sigma^a, \pi, m$ and moreover $v_0\in (L^1\cap L^\infty)(\mathbb{R}^d)$.
Denote the corresponding entropy solution by $v$, and assume that $v\in C([0,T];L^1(\mathbb{R}^d))$.
Suppose $u$ and $v$  take values in a closed interval $I \subset \mathbb{R}$.

For any $t \in (0,T)$,
\begin{equation}\label{contdest}
	\begin{split}
&\norm{u(t,\cdot) - v(t,\cdot)}_{L^1(\mathbb{R}^d)}\\
&\leq \norm{u_0-v_0}_{L^1(\mathbb{R}^d)}
 + C_1 t \norm{f-\tilde f}_{W^{1,\infty}(I);\mathbb{R}^d)}\\
&\quad + C_2 \sqrt{t} \norm{\sigma^a-\sigma^{\tilde a}}_{L^{\infty}(I;\mathbb{R}
 ^{d \times K})} 
 + C_3 \sqrt{t}\Big(\int_{|z|<1}\abs{z}^2 \abs{m(z)-\tilde m(z)} \,dz\Big)
 ^{1/2}\\
&\quad + C_4 t\int_{|z|\geq 1}|z| \abs{m(z)-\tilde m(z)} \,dz,
	\end{split}
\end{equation}
where the constants $C_i$, $i=1,\ldots,4$, depend on
the $L^{\infty}(0,T;BV(\mathbb{R}^d))$ norm of $u$.
\end{theorem}

This theorem generalizes results in \cite{Chen:2005wf,Chen:2006oy}
to the ``fractional case". In the hyperbolic
case ($a,\tilde a\equiv 0$), a generalization of this theorem  to
nonlocal operators of the form $\mathcal{L}[A(u)]$, $A:\mathbb{R}\to\mathbb{R}$
Lipschitz and nondecreasing, can be found in \cite{Al-Ci-Ja}.

\section{Proof of Theorem \ref{th:unique} (existence)}\label{prexis}

Although a detailed version of the existence of entropy
solutions to (\ref{eq1}) is presented in \cite{KU2}, to
motivate the entropy condition and to present a brief sketch, let
us consider the following accompanying problem
containing a uniformly parabolic operator depending
on a small parameter $\rho>0$:
\begin{equation}\label{eq1-visc}
	\partial_t u_\rho + \operatorname{div} f(u_\rho) =
	\operatorname{div}( a(u_\rho) \nabla u_\rho)+  \mathcal{L}[u_\rho(t,\cdot)]
	+\rho \Delta u_\rho.
\end{equation}
It is standard to construct a smooth solution $u_\rho$
to \eqref{eq1-visc}, for each fixed $\rho>0$.
Indeed, it can be done using the Galerkin method
and the compactness argument, see Chapter 5 in \cite{Ev} and \cite{Kiselev:2008jt}.

As usual, the game is to pass
to the limit as $\rho \to 0$ and identify the entropy
condition satisfied by the limit function $u$.
We will be brief in establishing the following estimates, since
most of them are similar to the ones in \cite{Chen:2003td} and
we will assume $u_0 \in W^{2,1}\cap H^1 \cap L^{\infty}(\mathbb{R}^d)$,
for general $u_0 \in L^1(\mathbb{R}^d)$ one can follow the
approximation procedure presented in \cite{Chen:2003td}.

The following estimates can be established for sufficiently regular initial data:
\begin{gather*}
\norm{u_\rho}_{L^\infty(Q_T)}\le C;\quad 
\abs{u_\rho(t,\cdot)}_{BV(\mathbb{R}^d)}\le C;
\\
\norm{u_\rho(t_2,\cdot)-u_\rho(t_1,\cdot)}_{L^1(\mathbb{R}^d)}\to 0,
\quad \text{as $\abs{t_2-t_1}\to 0$, uniformly in $\rho$.}
\end{gather*}
Hence there is a limit $u$ such that, passing if necessary to a 
subsequence as $\rho\to 0$,
\begin{equation}\label{eq:ueps-conv}
	u_\rho\to u\quad \text{a.e.~in $Q_T$ and in $L^p(Q_T)$ for any $p \in [1,\infty)$.}
\end{equation}

Next, we derive an energy estimate. To this end, fix
a convex $C^2$ function $\eta$ and
define $q,r$ by $q'=\eta'f'$, $r'=\eta' a$.
Multiplying \eqref{eq1-visc} by $\eta'$ yields
\begin{equation}\label{eq1-visc-entropy}
	\partial_t \eta(u_\rho) + \operatorname{div} q(u_\rho)
	=\sum_{i,j=1}^d \partial_{ij}^2 r_{ij}(u_\rho) + \mathcal{L}[\eta(u_\rho)]
	+ \rho \Delta \eta(u_\rho) - \nu_\rho
\end{equation}
where $\nu_\rho=\nu_\rho^1+\nu_\rho^2+\nu_\rho^3$ consists of
three parts:
\begin{itemize}
\item[(i)] the entropy dissipation term
$$
\nu_\rho^1:=\rho\Delta \eta(u_\rho)-\rho\eta'(u_\rho)\Delta u_\rho
=\rho \eta''(u_\rho) \abs{\nabla u_\rho}^2;
$$

\item[(ii)] the parabolic dissipation term
$$
\nu_\rho^2:= \sum_{i,j=1}^d \partial_{ij}^2
r_{ij}(u_\rho)- \eta'(u_\rho)\operatorname{div} (a(u_\rho) \nabla u_\rho)
=\eta''(u_\rho) \sum_{k=1}^K
\Big( \sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u_\rho)\Big)^2;
$$

\item[(iii)] the fractional parabolic dissipation term	
$$
\nu_\rho^3 =  \int_{\mathbb{R}^d \setminus \{0\}}\overline{\eta''}(u_\rho;z)
\left(u_\rho(t,x+z)-u_\rho(t,x)\right)^2\pi(dz),
$$
where $\overline{\eta''}(u_\rho;z)= \int_0^1 (1-\tau)
\eta''((1-\tau)u_\rho(t,x) + \tau u_\rho(t,x+z)) \,d\tau$.
\end{itemize}

In deriving \eqref{eq1-visc-entropy}, the ``new" computation
is the one showing that the commutator
$$
\mathcal{L}[\eta(u_\rho)]-\eta'(u_\rho) \mathcal{L}[u_\rho]
$$
equals $\nu_\rho^3$, but this follows easily from
Taylor's formula with integral reminder:
 \begin{equation}\label{eq:Taylor}
\eta(b)-\eta(a) = \eta'(a) (b-a)
  +\Big(\int_0^1 (1-\tau) \eta''((1-\tau)a + \tau b) \,d\tau\Big) (b-a)^2.
\end{equation}

Specifying $\eta(z)=z^2/2$ in \eqref{eq1-visc-entropy} gives
$$
\int_0^T \int_{\mathbb{R}^d}  \sum_{k=1}^K \Big( \sum_{i=1}^d \partial_{x_i}
\zeta_{ik}^a(u_\rho)\Big)^2\,dx \,dt \le C
$$
and
\begin{equation}\label{eq:weakconv-zeta}
	\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u_\rho)
	\rightharpoonup \sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u)
	\quad \text{in $L^2(Q_T)$}.
\end{equation}
From this we easily see, as in \cite{Chen:2003td}, that
\eqref{l2regu} and \eqref{crule} in Definition \ref{def:entropy} hold.

Regarding the non-local operator $\mathcal{L}$, the same choice for $\eta$
reveals that (\ref{frintcnd}) in Definition \ref{def:entropy} holds. Now set
$$
\Pi(dz):= \left( \abs{z}^2 \mathbf{1}_{|z|<1}
+\abs{z}\, \mathbf{1}_{|z|\ge 1}\right)\pi(dz),
$$
and note that $\Pi(dz)$ is a bounded Radon measure.
Introducing the short-hand notation
$$
D_\rho(t,x,z)= \frac{u_\rho(t,x+z)
- u_\rho(t,x)}{\abs{z}\mathbf{1}_{_{|z|<1}}
+ \sqrt{\abs{z}} \mathbf{1}_{_{|z|\ge 1}}}
\quad
d\mu=\Pi(dz)\otimes dx\otimes dt,
$$
Equation \eqref{frintcnd} translates into $D_\rho$ being uniformly bounded
in $L^2((0,T)\times \mathbb{R}^d\times (\mathbb{R}^d \setminus \{0\});d\mu)$. Consequently, we may
assume that there is a limit function $D$ such that
$$
D_\rho \rightharpoonup D \quad \text{in $L^2((0,T)\times \mathbb{R}^d\times (\mathbb{R}^d \setminus \{0\});d\mu)$.}
$$

Let us identify $D$. To this end, fix a smooth function
$\varphi$ in $C^\infty_c(Q_T)$ and observe
\begin{align*}
	&\iint_{Q_T}\int_{\mathbb{R}^d \setminus \{0\}}
	\varphi(t,x)\frac{u_\rho(t,x+z)-u_\rho(t,x)}{\abs{z}\mathbf{1}_{|z|<1}
	+ \sqrt{\abs{z}} \mathbf{1}_{|z|\ge 1}}
	\, \Pi(dz)\, dx\, dt\\ 
	& = \iint_{Q_T}\int_{\mathbb{R}^d \setminus \{0\}}
	\frac{\varphi(t,x+z)-\varphi(t,x)}{\abs{z}\mathbf{1}_{|z|<1}
	+ \sqrt{\abs{z}} \mathbf{1}_{|z|\ge 1}} u_\rho(t,x)
	\, \Pi(dz)\, dx\, dt.
\end{align*}
Now, using that $u_\rho \overset{\rho\to 0}{\longrightarrow} u$ 
a.e.~in $Q_T$, we
conclude that
$$
D_\rho \rightharpoonup \frac{u(t,x+z) - u(t,x)}{\abs{z}\mathbf{1}_{|z|<1}
+ \sqrt{\abs{z}} \mathbf{1}_{|z|\ge 1}}
\quad \text{in $L^2((0,T)\times \mathbb{R}^d\times
(\mathbb{R}^d \setminus \{0\});d\mu)$.}
$$

We are now in a position to pass to the distributional
limit in \eqref{eq1-visc-entropy} to recover the desired
entropy condition satisfied by the limit
$u=\lim_{\rho \to 0} u_\rho$.  Note that to
interpret \eqref{eq1-visc-entropy} in the sense of distributions
we use the formula
\begin{equation}\label{eq:IBP}
	\int_{\mathbb{R}^d} \mathcal{L}[\Phi(x)] \phi(x) \,dx 
  = \int_{\mathbb{R}^d} \Phi(x) \mathcal{L}[\phi(x)] \,dx,
\end{equation}
which holds for all sufficiently regular (say, $C^2$) functions 
$\Phi,\phi:\mathbb{R}^d\to \mathbb{R}$.
This relation is easily obtained by a change of variables
$(t,x,z) \mapsto (t,x+z,-z)$ and an integration by parts in $x$.

We claim that the entropy condition satisfied by the limit
$u=\lim_{\rho \to 0} u_\rho$ takes the following form:
for any convex $C^2$ entropy function $\eta$ and corresponding entropy
fluxes $q,r$ defined by $q'=\eta'f', r'=\eta' a$,
\begin{equation}\label{eq1-limit-entropy}
	\partial_t \eta(u) + \operatorname{div} q(u)
	\le \sum_{i,j} \partial_{x_i x_j} r_{ij}(u) + \mathcal{L}[\eta(u)] 
	- n^{u,\eta}-m^{u,\eta}
\end{equation}
in the sense of distributions, where
$$
n^{u,\eta} = \eta''(u) \sum_{k=1}^K
\Big(\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u)\Big)^2
$$
is the parabolic dissipation measure with respect to $u$ and 	
$$
m^{u,\eta} = \int_{\mathbb{R}^d \setminus \{0\}}\overline{\eta''}(u;z)
\left(u(t,x+z) - u(t,x) \right)^2\pi(dz)
$$
is the fractional parabolic dissipation measure with respect to $u$.

In view of \eqref{eq:ueps-conv}, to verify \eqref{eq1-limit-entropy}
we only need to argue that
$$
\liminf_{\rho \to 0} \iint_{Q_T} \nu_\rho \, dx \, dt \ge
\iint_{Q_T} \left( n^{u,\eta} + m^{u,\eta} \right) \,dx \,dt.
$$
First, $\iint_{Q_T} \nu_\rho^1 \,dx \,dt \ge 0$ for each $\rho>0$. 
Second, thanks
to the weak convergence \eqref{eq:weakconv-zeta} and a standard
weak lower semi-continuity result for quadratic functionals,
\begin{align*}
	& \liminf_{\rho\to 0}
	\int_0^T \int_{\mathbb{R}^d} \eta''(u_\rho)\sum_{k=1}^K
	\Big( \sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u_\rho)  \Big)^2
	\varphi \,dx \,dt \\
 & \ge \int_0^T \int_{\mathbb{R}^d} \eta''(u)\sum_{k=1}^K \Big( \sum_{i=1}^d
	\partial_{x_i} \zeta_{ik}^a(u)  \Big)^2 \varphi \,dx \,dt,
\end{align*}
for all test functions $\varphi\in C^\infty_c$. Similarly,
\begin{align*}
	&\liminf_{\rho\to 0}
	\iint_{Q_T}  \int_{\mathbb{R}^d \setminus \{0\}}
	\overline{\eta''}(u_\rho;z)
	\left(u_\rho(t,x+z)-u_\rho(t,x)\right)^2\varphi\pi(dz)\,dx\,dt
	\\ & 
	\ge \iint_{Q_T} \int_{\mathbb{R}^d \setminus \{0\}}\overline{\eta''}(u;z)
	\left(u(t,x+z)-u(t,x)\right)^2\varphi\pi(dz)\,dx\,dt,
\end{align*}
for all test functions $\varphi\in C^\infty_c$.
Combining, we deduce that \eqref{entineq}
in Definition \ref{def:entropy}  holds. This completes the proof.

\section{Proof of Theorem \ref{th:unique} (uniqueness)}\label{pruniq}

We shall need $C^2$ approximations $\eta_{\varepsilon}^\pm(u)$ of the functions
$$
\eta^\pm(u) :=(u)^\pm=\max\left(\pm(u),0\right),
\quad u\in \mathbb{R}.
$$
We build these by picking nondecreasing $C^1$
approximations $\operatorname{sgn}_{\varepsilon}^\pm(u)$ of
$$
\operatorname{sgn}^+(u):=
\begin{cases}
	0, & \text{if $u \leq 0$},\\
	1, & \text{if $u >0$},
\end{cases}
\quad
\operatorname{sgn}^-(u) :=
\begin{cases}
	-1, & \text{if $u \leq 0$,}\\
	0, & \text{if $u >0$,}
\end{cases}
$$
and defining
$$
\eta_{\varepsilon}^\pm(u)
:= \int_0^u \operatorname{sgn}_{\varepsilon}^\pm(\xi)  \,d\xi,
\quad u\in \mathbb{R}.
$$
For example, we can take
\begin{gather*}
\operatorname{sgn}_{\varepsilon}^+(u)=
\begin{cases}
	0, & \text{if } u<0,\\
	\sin(\pi u/(2\varepsilon)), & \text{if }0\le u \leq \varepsilon,\\
	1, & \text{if }u>\varepsilon.
\end{cases}
\\
\operatorname{sgn}_{\varepsilon}^-(u)=
\begin{cases}
	-1, & \text{if }u<-\varepsilon,\\
	\sin(\pi u/(2\varepsilon)), & \text{if }-\varepsilon\le u \le 0,\\
	0, & \text{if }u>0.
\end{cases}
\end{gather*}

The functions $\eta_{\varepsilon}^\pm$
are $C^2$ and convex. Moreover,
$$
\eta_{\varepsilon}^\pm(u)
\overset{\varepsilon\to 0}{\longrightarrow}
\eta^\pm(u),
\quad u\in \mathbb{R}.
$$
Observe that $\left(\eta_{\varepsilon}^\pm(\cdot-c)\right)_{c \in\mathbb{R}}$
is a family of entropies.
Given these entropies, we introduce the
corresponding entropy fluxes
\begin{gather*}
	q_{\varepsilon}^\pm(u,c) = \int_c^u (\eta_\varepsilon^\pm)'(\xi-c) f'(\xi) d\xi, \quad u,c \in \mathbb{R},\\
	r_{\varepsilon}^\pm(u,c)  = \int_c^u (\eta_{\varepsilon}^\pm)'(\xi-c) a(\xi) \,d\xi,  \quad u,c \in \mathbb{R}.
\end{gather*}
Clearly, as $\varepsilon \to 0$,
\begin{gather*}
	q_{\varepsilon}^\pm(u,c) \to q^\pm(u,c) := \operatorname{sgn}^\pm(u-c) (f(u)-f(c)), \quad u,c\in \mathbb{R},\\
	r_{\varepsilon}^\pm(u,c) \to r^\pm(u,c)  := \operatorname{sgn}^\pm(u-c) (A(u) - A(c)), \quad u,c\in \mathbb{R},
\end{gather*}
where the (matrix-valued) function $A(\cdot)$
is defined by $  A(u) = \int_0^u a(\xi) \,d\xi$.

Observe that $\left(\eta_{\varepsilon}^\pm(\cdot-c),q_{\varepsilon}^\pm(\cdot,c),
r_{\varepsilon}^\pm(\cdot,c)\right)_{c \in \mathbb{R}}$
is a family of entropy-entropy flux triples, so choosing
$\eta = \eta_{\varepsilon}^\pm$ in \eqref{entineq} yields
\begin{equation}\label{in1}
	\begin{split}
& \iint_{Q_T} \Bigl( \eta_{\varepsilon}^\pm(u-c) \partial_t \varphi
 +  \sum_{i=1}^d  q_{\varepsilon,i}^\pm(u,c) \partial_{x_i} \varphi
 + \sum_{i,j=1}^d  r_{\varepsilon,ij}^\pm(u,c) \partial_{x_ix_j}^2 \varphi \Bigr)\,dx \,dt
\\ 
& + \iint_{Q_T}  \eta_{\varepsilon}^\pm(u-c) \mathcal{L}[\varphi]\,dx \,dt
		+ \int_{\mathbb{R}^d} \eta^\pm(u_0-c) \varphi(0,x) \,dx
		\\ 
& \geq  \iint_{Q_T} (\eta_{\varepsilon}^\pm)''(u-c)
 \sum_{k=1}^K \Bigl(\sum_{i=1}^d  \partial_{x_i} \zeta_{ik}^a(u)\Bigr)^2 \varphi \,dx \,dt \\
&\quad + \iint_{Q_T} \int_{\mathbb{R}^d \setminus \{0\}}
		\overline{(\eta_{\varepsilon}^\pm)''}(u-c;z)
		\left(u(t,x+z)-u(t,x)\right)^2 \varphi \pi(dz) \,dx \,dt.
	\end{split}
\end{equation}
Moreover,
\begin{align*}
	\overline{(\eta_\varepsilon^\pm)''}(u-c;z)
& = \int_0^1 (1-\tau) (\eta_\varepsilon^\pm)''\Bigl((1-\tau)u(t,x)+\tau u(t,x+z),c\Bigr) \,d\tau
	\\ 
& =  \int_0^1 (1-\tau) (\operatorname{sgn}_\varepsilon^\pm)'\Bigl((1-\tau)(u(t,x)-c)+\tau (u(t,x+z)-c)\Bigr)\,d\tau.
\end{align*}

To proceed, the following simple observations will be useful:
\begin{itemize}
	\item  $\operatorname{sgn}_{\varepsilon}^-(u-c) =-\operatorname{sgn}_{\varepsilon}^+(c-u)$ and $\eta_{\varepsilon}^-(u-c) = \eta_{\varepsilon}^+(c-u)$;
	\item  $q_{\varepsilon}^-(u,c) = q_{\varepsilon}^+(c,u)$ and $r_\varepsilon^-(u,c) = r_\varepsilon^+(c,u)$;
	\item  $(\eta_{\varepsilon}^-)''(u-c) = (\eta_{\varepsilon}^+)''(c-u)$.
\end{itemize}
Employing these observations, we can rewrite the ``$-$" part of \eqref {in1} as
\begin{equation}\label{in3}
	\begin{split}
& \iint_{Q_T} \Bigl( \eta_{\varepsilon}^+(c-u) \partial_t \varphi
		+  \sum_{i=1}^d  q_{\varepsilon,i}^+(c,u) \partial_{x_i} \varphi
		+ \sum_{i,j=1}^d  r_{\varepsilon,ij}^+(c,u) \partial_{x_ix_j}^2 \varphi \Bigr)\,dx \,dt
\\ 
&  + \iint_{Q_T}  \eta_{\varepsilon}^+(c-u) \mathcal{L}[\varphi]\,dx \,dt
 + \int_{\mathbb{R}^d} \eta_{\varepsilon}^+(c-u_0) \varphi(0,x) \,dx \\ 
& \geq  \iint_{Q_T} (\eta_{\varepsilon}^+)''(c-u)
		\sum_{k=1}^K \Bigl(\sum_{i=1}^d  \partial_{x_i} \zeta_{ik}^a(u)\Bigr)^2 \varphi \,dx \,dt \\
&\quad + \iint_{Q_T} \int_{\mathbb{R}^d \setminus \{0\}}
		\overline{(\eta_{\varepsilon}^+)''}(c-u;z)
		\left(u(t,x+z)-u(t,x)\right)^2 \varphi \pi(dz) \,dx \,dt.
\end{split}
\end{equation}

To establish the $L^1$ contraction property \eqref{eq:contraction} we shall
employ the doubling-of-variables device of Kru\u{z}kov \cite{Kruzkov:1970kx}.
Let $u=u(t,x)$, $v=v(s,y)$ be two entropy solutions as stated
in Theorem \ref{th:unique}. Moreover, let $\varphi=\varphi(t,x,s,y)$ be a
test function in the doubled variables $(t,x,s,y)$.
To simplify the presentation, we introduce the following notation (with $\nabla_{x+y}$ being
short-hand for $\nabla_x+\nabla_y$)
\begin{gather*}
	\mathcal{L}_x[\varphi]  := \int_{\mathbb{R}^d \setminus \{0\}}
	\left[ \varphi(t,x+z,s,y) - \varphi
	- z \cdot \nabla_x \varphi
	\mathbf{1}_{|z|<1} \right]\pi(dz), \\
	\mathcal{L}_y[\varphi]   =  \int_{\mathbb{R}^d \setminus \{0\}}
	\left[ \varphi(t,x,s,y+ z) - \varphi
	-z \cdot \nabla_y \varphi
	\mathbf{1}_{|z|<1} \right]\pi(dz), \\
	\mathcal{L}_{x+y}[\varphi]   =  \int_{\mathbb{R}^d \setminus \{0\}}
	\Bigl[ \varphi(t,x+z,s,y+ z)-\varphi
	-z \cdot \nabla_{x+y}\varphi
	\mathbf{1}_{|z|<1} \Bigr]\pi(dz),
\end{gather*}


In the ``$+$" part of \eqref{in1} written the entropy solution $u(t,x)$ we choose $c=v(s,y)$
and integrate the result over $(s,y)$, obtaining
\begin{equation}\label{ineqt04}
	\begin{split}
& \iiiint  \Bigl( \eta_{\varepsilon}^+(u-v) \partial_t \varphi
		+  \sum_{i=1}^d  q_{\varepsilon,i}^+(u,v) \partial_{x_i} \varphi
		+ \sum_{i,j=1}^d  r_{\varepsilon,ij}^+(u,c) \partial_{x_ix_j}^2 \varphi \Bigr)\,dx \,dt \,dy\,ds
		\\ 
& + \iiiint \eta_{\varepsilon}^+(u-v) \mathcal{L}_x[\varphi]\,dx\,dt\,dy\,ds
		+ \iiint \eta_{\varepsilon}^+(u_0-v) \varphi(0,x,s,y) \,dx\,dy\,ds
		\\ & \geq  \iint_{Q_T} (\eta_{\varepsilon}^+)''(u-v)
		\sum_{k=1}^K \Bigl(\sum_{i=1}^d
		\partial_{x_i} \zeta_{ik}^a(u)\Bigr)^2 \varphi \,dx\,dt\,dy\,ds \\
&\quad+ \iiiint\!\!\int_{\mathbb{R}^d \setminus \{0\}}
		\overline{(\eta_{\varepsilon}^+)''}(u(t,\cdot)-v;z)
		\big(u(t,x+z)-u(t,x)\big)^2 \varphi \pi(dz) \,dx\,dt\,dy\,ds.
	\end{split}
\end{equation}

Similarly, in \eqref{in3} written for the entropy solution $v(s,y)$ we choose $c=u(t,x)$ and
integrate over $(t,x)$, thereby obtaining
\begin{equation}\label{ineek05}
	\begin{split}
& \iiiint \Bigl( \eta_{\varepsilon}^+(u-v) \partial_s \varphi
		+  \sum_{i=1}^d  q_{\varepsilon,i}^+(u,v) \partial_{y_i} \varphi
		+ \sum_{i,j=1}^d  r_{\varepsilon,ij}^+(u,v) \partial_{y_iy_j}^2 \varphi \Bigr)\,dx\,dt\,dy\,ds
		\\ 
&  + \iiiint  \eta_{\varepsilon}^+(u-v) \mathcal{L}_y[\varphi]\,dx\,dt\,dy\,ds
		+ \iiint \eta_{\varepsilon}^+(u-v_0) \varphi(t,x,0,y) \,dx\,dt\,dy
		\\ & \geq  \iiiint (\eta_{\varepsilon}^+)''(u-v)
		\sum_{k=1}^K \Bigl(\sum_{i=1}^d\partial_{y_i} \zeta_{ik}^a(v)\Bigr)^2
		\varphi \,dx \,dt\,dy\,ds \\
&\quad + \iiiint\!\!\int_{\mathbb{R}^d \setminus \{0\}}
		\overline{(\eta_{\varepsilon}^+)''}(u-v(s,\cdot);z)
		\left(v(s,y+z)-v(s,y)\right)^2 \varphi \pi(dz) \,dx\,dt\,dy\,ds.
	\end{split}
\end{equation}

Adding \eqref{ineqt04} and \eqref{ineek05} yields
\begin{equation}\label{in6}
	I_{\mathrm{time}}(\varepsilon)+I_{\mathrm{conv}}(\varepsilon)+I_{\mathrm{diff}}(\varepsilon)
	+I_{\mathrm{fdiff}}(\varepsilon)+	I_{\mathrm{init}}(\varepsilon)
	\ge I_{\mathrm{diss}}(\varepsilon)+I_{\mathrm{fdiss}}(\varepsilon),
\end{equation}
where
\begin{gather*}
I_{\mathrm{time}}(\varepsilon)  = \iiiint \eta_{\varepsilon}^+(u-v) (\partial_t +\partial_s) \varphi  \,dx\,dt\,dy\,ds \\
I_{\mathrm{conv}}(\varepsilon)  = \iiiint \sum_{i=1}^d  q_{\varepsilon,i}^+(u,v)
	(\partial_{x_i} + \partial_{y_i}) \varphi \,dx\,dt\,dy\,ds \\
I_{\mathrm{diff}}(\varepsilon)  = \iiiint \sum_{i,j=1}^d  r_{\varepsilon,ij}^+(u,v)
	(\partial_{x_ix_j}^2+\partial_{y_iy_j}^2) \varphi \,dx\,dt\,dy\,ds \\
I_{\mathrm{fdiff}}(\varepsilon)  = \iiiint  \eta_{\varepsilon}^+(u-v)
	\Bigl(\mathcal{L}_x[\varphi]+\mathcal{L}_y[\varphi]\Bigr)\,dx\,dt\,dy\,ds
\\
\begin{aligned}
I_{\mathrm{init}}(\varepsilon) &
 = \iiint \eta_{\varepsilon}^+(u_0-v)  \varphi(0,x,s,y) \,dx\,dy\,ds \\
&\quad  +\iiint \eta_{\varepsilon}^+(u-v_0) \varphi(t,x,0,y) \,dx\,dt\,dy
\end{aligned}\\
\begin{aligned}
	I_{\mathrm{diss}}(\varepsilon) 
& = \iiiint  (\eta_{\varepsilon}^+)''(u-v)\\ 
& \quad \times \sum_{k=1}^K \Big[ \Bigl(\sum_{i}^d \partial_{x_i} \zeta_{ik}^a(u)\Bigr)^2
	+\Bigl(\sum_{i=1}^d\partial_{y_i} \zeta_{ik}^a(v)\Bigr)^2\Big]
	\varphi \,dx \,dt\,dy\,ds
\end{aligned} \\
\begin{aligned}
	I_{\mathrm{fdiss}}(\varepsilon) 
& =  \iiiint\!\!\int_{\mathbb{R}^d \setminus \{0\}}
	\Bigl [ \overline{(\eta_{\varepsilon}^+)''}(u(t,\cdot)-v;z) 
	\left(u(t,x+z)-u(t,x)\right)^2
	\\ 
& \quad	+ \overline{(\eta_{\varepsilon}^+)''}(u,v(s,\cdot);z) 
\left(v(s,y+z)-v(s,y)\right)^2 \Bigr]
 \varphi \pi(dz) \,dx\,dt\,dy\,ds.
\end{aligned}
\end{gather*}

In view of the inequality ``$a^2+b^2\ge 2ab$", we have
$I_{\mathrm{diss}}(\varepsilon)  
\ge \widetilde{I}_{\mathrm{diss}}(\varepsilon)$, with
$$
\widetilde{I}_{\mathrm{diss}}(\varepsilon)
= 2\iiiint  (\eta_{\varepsilon}^+)''(u-v)
\sum_{k=1}^K \sum_{i,j=1}^d
\partial_{x_i} \zeta_{ik}^a(u) \partial_{y_j} \zeta_{jk}^a(v)
\varphi \,dx \,dt\,dy\,ds.
$$
Arguing exactly as in \cite{Bendahmane:2004go}, it follows that
\begin{equation}\label{eq:diff-diss-final}
	\begin{split}
& \lim_{\varepsilon\to 0} \Bigl( I_{\mathrm{diff}}(\varepsilon)
		-\widetilde{I}_{\mathrm{diss}}(\varepsilon)\Bigr)
		\\ 
& \le \iiiint \sum_{i,j=1}^d  r_{ij}^+(u,v)
		(\partial_{x_ix_j}^2+2\partial_{x_iy_j}^2+\partial_{y_iy_j}^2) \varphi \,dx\,dt\,dy\,ds.
	\end{split}
\end{equation}

Fix a small number $\kappa>0$, and let us split $\mathcal{L}$ into two parts
\begin{align*}
	\mathcal{L}[\phi] & =\int_{\abs{z}\le \kappa}\left[ \phi(t,x+ z) - \phi(t,x)
	- z \cdot \nabla \phi \mathbf{1}_{|z|<1} \right]\pi(dz)
	\\ & \quad +\int_{\abs{z}> \kappa}
	\left[ \phi(t,x+ z) - \phi(t,x) - z \cdot \nabla
	\phi \mathbf{1}_{|z|<1} \right]\pi(dz)
	\\ & =: \mathcal{L}_\kappa[\phi]+\mathcal{L}^\kappa[\phi],
	\quad \forall \phi\in C^2,
\end{align*}
and similarly
$$
\mathcal{L}_x = \mathcal{L}_{x,\kappa} + \mathcal{L}_x^\kappa, \quad
\mathcal{L}_y = \mathcal{L}_{y,\kappa} + \mathcal{L}_y^\kappa, \quad
\mathcal{L}_{x+y} = \mathcal{L}_{x+y,\kappa} + \mathcal{L}_{x+y}^\kappa.
$$
The corresponding splitting of
$I_{\mathrm{fdiff}}(\varepsilon)$ is written
$$
I_{\mathrm{fdiff}}(\varepsilon)=
I_{\mathrm{fdiff},\kappa}(\varepsilon)
+I_{\mathrm{fdiff}}^\kappa(\varepsilon).
$$
We also need to introduce the
operator $\widetilde {\mathcal{L}}^\kappa$ defined by writing
\[
	\mathcal{L}^\kappa[\varphi]
	 = \widetilde {\mathcal{L}}^\kappa [\varphi]
	-\Big( \int_{\abs{z}>\kappa} z
	\mathbf{1}_{|z|<1}\,
	\pi(dz)\Big) \cdot \nabla_x \varphi,
\]
with similar definitions for $\widetilde {\mathcal{L}}^\kappa_x$, 
$\widetilde{\mathcal{L}}^\kappa_y$,  and
$\widetilde{\mathcal{L}}^\kappa_{x+y}$. Observe that \eqref{eq:IBP}
continues to hold for all these operators. The function obtained
by replacing $\mathcal{L}^\kappa$ with $\widetilde{\mathcal{L}}^\kappa$ in the
definition of $I_{\mathrm{fdiff}}^\kappa(\varepsilon)$ will be named
$\widetilde I_{\mathrm{fdiff}}^\kappa(\varepsilon)$.

Clearly, in view of \eqref{intcond1},
\begin{equation}\label{eq:kappa-est}
	\abs{I_{\mathrm{fdiff},\kappa}(\varepsilon)}\le
	C \norm{D^2\varphi}_{L^1(Q_T\times Q_T)}
	\int_{\abs{z}\le \kappa} \abs{z}^2\pi(dz)
	\overset{\kappa\to 0}{\longrightarrow} 0,
\end{equation}
for some constant $C$ independent of $\kappa$ and $\varepsilon$.

Let us analyze $\widetilde I_{\mathrm{fdiff}}^\kappa(\varepsilon)$.
By \eqref{eq:IBP},
\begin{align*}
	\widetilde I_{\mathrm{fdiff}}^\kappa(\varepsilon)
	= \iiiint  \Bigl(\widetilde{\mathcal{L}}^\kappa_x\left[\eta_\varepsilon^+(u-v)\right]
	+ \widetilde{\mathcal{L}}^\kappa_y\left[\eta_\varepsilon^+(u-v)\right]\Bigr)
	\varphi \,dt \,dx \,dy\,ds.
\end{align*}
Specifying $a=u(t,x)-v(s,y)$ and $b=u(t,x+z)-v(s,y)$ in \eqref{eq:Taylor} yields
\begin{equation}\label{eq:Taylor-1}
	\begin{split}
		&\eta_{\varepsilon}^+(u(t,x+z)-v(s,y))-\eta_{\varepsilon}^+(u(t,x)-v(s,y))
		\\ & = (\eta_{\varepsilon}^+)'(u(t,x)-v(s,y)) \left(u(t,x+z) - u(t,x)\right)
		\\ & \quad 
		+\overline{(\eta_{\varepsilon}^+)''}(u(t,\cdot)- v;z)\left(u(t,x+z) - u(t,x)\right)^2.
	\end{split}
\end{equation}

Similarly, taking $a=u(t,x)-v(s,y)$, $b=u(t,x)-v(s,y+z)$ in \eqref{eq:Taylor} yields
\begin{equation}\label{eq:Taylor-2}
	\begin{split}
		&\eta_{\varepsilon}^+(u(t,x)-v(s,y+z))-\eta_{\varepsilon}^+(u(t,x)-v(s,y))
		\\ & = -(\eta_{\varepsilon}^+)'(u(t,x)-v(s,y)) \left(v(s,y+z) - v(s,y)\right)
		\\ & \quad 
		+\overline{(\eta_{\varepsilon}^+)''}(u-v(s,\cdot);z) \left(v(s,y+z) - v(s,y)\right)^2.
	\end{split}
\end{equation}

Adding the first term on the right-hand side of \eqref{eq:Taylor-1} to the first
term on the right-hand side of \eqref{eq:Taylor-2} yields
\begin{align*}
&(\eta_{\varepsilon}^+)'(u(t,x)-v(s,y)) \left(u(t,x+z) - u(t,x)\right)\\
&  -(\eta_{\varepsilon}^+)'(u(t,x)-v(s,y)) \left(v(s,y+z) - v(s,y)\right)\\ 
& = (\eta_{\varepsilon}^+)'(u(t,x)-v(s,y))\Bigl[ \left(u(t,x+z) - v(s,y+z)\right)
	-\left(u(t,x) - v(s,y)\right)\Bigr]\\ 
& \le  \eta_\varepsilon^+(u(t,x+z) - v(s,y+z))
	-\eta_\varepsilon^+(u(t,x) - v(s,y)),
\end{align*}
where we have used the convexity of $\eta_\varepsilon$ to derive the 
last inequality.

In view of these findings, we can rewrite
$\widetilde I_{\mathrm{fdiff}}^\kappa(\varepsilon)$ as follows:
\begin{equation}\label{eq:mono-property}
	\begin{split}
		\widetilde I_{\mathrm{fdiff}}^\kappa(\varepsilon)-I_{\mathrm{fdiss}}^\kappa(\varepsilon)
		& \le  \iiiint  \widetilde{\mathcal{L}}^\kappa_{x+y}
		\left[\eta_\varepsilon^+(u(t,\cdot)-v(s,\cdot))\right]\varphi \,dt \,dx \,dy\,ds
		\\ & \overset{\eqref{eq:IBP}}{=}
		\iiiint  \eta_\varepsilon^+(u-v)\widetilde{\mathcal{L}}^\kappa_{x+y}[\varphi] \,dt \,dx \,dy\,ds,
	\end{split}
\end{equation}
where
\begin{align*}
	I_{\mathrm{fdiss}}^\kappa(\varepsilon) 
& =  \iiiint\!\!\int_{\abs{z}>\kappa}
	\Bigl [ \overline{(\eta_{\varepsilon}^+)''}(u(t,\cdot)-v;z) \left(u(t,x+z)-u(t,x)\right)^2
	\\ 
& \quad	+ \overline{(\eta_{\varepsilon}^+)''}(u-v(s,\cdot);z) 
\left(v(s,y+z)-v(s,y)\right)^2 \Bigr]
 \varphi \pi(dz) \,dx\,dt\,dy\,ds.
\end{align*}
Consequently,
\[
	I_{\mathrm{fdiff}}^\kappa(\varepsilon)-I_{\mathrm{fdiss}}^\kappa(\varepsilon)
	\le \iiiint  \eta_\varepsilon^+(u-v)
	\mathcal{L}^\kappa_{x+y}[\varphi] \,dt \,dx \,dy\,ds,
\]
The next step is to first send $\kappa\to 0$ and then $\varepsilon\to 0$. 
Related to this, observe that
$$
\lim_{\kappa\to 0} I_{\mathrm{fdiff}}^\kappa(\varepsilon)
= I_{\mathrm{fdiff}}(\varepsilon), \quad
\lim_{\kappa\to 0} I_{\mathrm{fdiss}}^\kappa(\varepsilon)
= I_{\mathrm{fdiss}}(\varepsilon)
$$
for each fixed $\varepsilon>0$, by the dominated convergence theorem.
Moreover, we clearly have
$\lim_{\kappa\to 0} \mathcal{L}^\kappa_{x+y}[\varphi]
= \mathcal{L}_{x+y}[\varphi]$. In view of this
and \eqref{eq:kappa-est}, we conclude that
\begin{equation}\label{eq:fdiff-fdiss-final}
	I_{\mathrm{fdiff}}(\varepsilon)-I_{\mathrm{fdiss}}(\varepsilon)
	\le \iiiint  \eta_\varepsilon^+(u-v) \mathcal{L}_{x+y}[\varphi] \,dt \,dx \,dy\,ds.
\end{equation}

By \eqref{eq:diff-diss-final} and \eqref{eq:fdiff-fdiss-final}, it
follows from \eqref{in6} and sending $\varepsilon\to 0$ that
\begin{equation}\label{in7}
	\begin{split}
		& \iiiint \Bigl( (u-v)^+ (\partial_t+\partial_s)\varphi
		+ \sum_{i=1}^d  q_{i}^+(u,v) (\partial_{x_i}+\partial_{y_i}) \varphi
		\\ 
& +\sum_{i,j=1}^d  r_{ij}^+(u,v)
		(\partial_{x_ix_j}^2+2\partial_{x_iy_j}^2+\partial_{y_iy_j}^2) \varphi
		+ (u-v)^+ \mathcal{L}_{x+y}[\varphi] \Bigr)\,dx\,dt\,dy\,ds
		\\ 
& + \iiint (u_0-v)^+ \varphi(0,x,s,y) \,dx\,dy\,ds
		+ \iiint (u-v_0)^+ \varphi(t,x,0,y) \,dx\,dt\,dy\\
& \ge 0.
	\end{split}
\end{equation}

Let us specify the test function $\varphi=\varphi(t,x,s,y)$.
To this end, fix a nonnegative test function
$\phi=\phi(t,x)\in C_c^{\infty}([0,\infty) \times \mathbb{R}^d)$, and
pick two sequences $\{\theta_{\nu}\}_{\nu>0} \subset C_c^{\infty}(0,\nu)$,
$\{\delta_{\mu}\}_{\mu>0} \subset C_c^{\infty}(B(0,\mu))$ of
approximate delta functions, where $B(0,\mu)$ denotes the open ball
centered at the origin with radius $\mu$. Then take
\begin{equation}\label{eq:testfunc}
	\varphi(t,x,s,y) = \theta_{\nu}(s-t)
	\delta_{\mu}(y-x) \phi(t,x).
\end{equation}

Simple calculations reveal that
\begin{gather*}
(\partial_t + \partial_s) \varphi
	 =\theta_{\nu}(s-t) \delta_{\mu}(y-x) \partial_t \phi(t,x),\\ 
(\partial_{x_i} + \partial_{y_i})\varphi
	= \theta_{\nu}(s-t) \delta_{\mu}(y-x) \partial_{x_i} \phi(t,x),	\\ 
(\partial_{x_ix_j}^2 +2 \partial_{x_iy_j}^2 + \partial_{y_iy_j}^2) \varphi
 = \theta_{\nu}(s-t) \delta_{\mu}(y-x) \partial_{x_ix_j}^2 \phi(t,x),\\
\varphi(t,x+z,s,y+z) - \varphi(t,x,s,y)
	= \theta_{\nu}(s-t) \delta_{\mu}(y-x)
	\left( \phi(t,x+z)-\phi(t,x) \right).
\end{gather*}

Note that $\theta_{\nu} = 0 $ on $(-\infty,0]$ and so 
$\varphi(t,x,0,y)\equiv 0$.
By the choice of the test function $\varphi$ and the observations above, we
deduce from \eqref{in7} that
\begin{equation}\label{in8}
	\begin{split}
		& \iiiint (u-v)^+ \theta_{\nu}(s-t) \delta_{\mu}(y-x)
		\partial_t \phi(t,x) \,dx\,dt\,dy\,ds \\
		& +\iiiint
		\sum_{i=1}^d q_i^+(u,v) \theta_{\nu}(s-t)
		\delta_{\mu}(y-x)\partial_{x_i}   \phi(t,x)   \,dx\,dt \,dy\,ds \\
		&+ \iiiint \sum_{i,j=1}^d r_{ij}^+(u,v)
		\theta_{\nu}(s-t) \delta_{\mu}(y-x)\partial_{x_ix_j}^2 \phi(t,x) \,dx\,dt\,dy\,ds
		\\ &  + \iiiint (u-v)^+
		\theta_{\nu}(s-t) \delta_{\mu}(y-x) \mathcal{L}[\phi] \,dx\,dt\,dy\,ds
		+ I_{u_0,v}(\nu,\mu)\geq 0,
	\end{split}
\end{equation}
where
\begin{align*}
	I_{u_0,v}(\nu,\mu) 
& :=\iiint (u_0-v)^+	\theta_{\nu}(s)\delta_{\mu}(y-x) \phi(0,x) \,dx\,dy\,ds
	\\ 
& = -\iiint (u_0-v)^+	\partial_s\left(\tilde{\phi}_{\nu}(s)
	\delta_{\mu}(y-x)\phi(0,x)\right) \,dx\,dy\,ds,
\end{align*}
with
$$
\tilde{\phi}_{\nu}(s) := \int_s^T \theta_{\nu}(\tau)\,d\tau
= \int_{\min(s,\nu)}^{\nu} \theta_{\nu}(\tau)\,d\tau
\overset{\nu\to 0}{\longrightarrow} 1.
$$
Specifying $\varphi=\phi(t,x)\tilde{\phi}_{\nu}(s)\delta_{\mu}(y-x)$ in
the entropy inequality for $v$ and noting
that $\theta_{\nu}(s)$ vanishes for $s>\nu$, we obtain
\begin{equation}\label{in9}
	\begin{split}
		& \iint  (u_0-v)^+\partial_s \varphi(s,x,y) \,dy\,ds \\ 
& \le \iint (u_0 -v_0)^+\theta_{\nu}(s)
		\delta_{\mu}(y-x)\phi(0,x) \,dy\,ds+o(\nu)
		\\ 
&  \overset{\nu\to 0}{\longrightarrow}
		\iint (u_0 -v_0)^+ \delta_{\mu}(y-x) \phi(0,x)\,dy\,ds,
	\end{split}
\end{equation}
where the ``$o(\nu)$" term follows from an integrability argument.

Hence, sending $\nu,\mu \to 0$, we deduce
\begin{equation}\label{in11}
\begin{split}
 \limsup_{\mu \to 0} \limsup_{\nu \to 0} I_{u_0,v}(\nu,\mu) 
&  \leq \limsup_{\mu \to 0} \iint
		(u_0 -v_0)^+  \delta_{\mu}(y-x) \phi(0,x)\,dx\,dy\\ 
& = \int (u_0-v_0)^+\phi(0,x)\,dx,
	\end{split}
\end{equation}
with $u_0=u_0(x)$ and $v_0=v_0(x)$.

Keeping in mind \eqref{in11} when sending $\mu,\nu \to 0$
in \eqref{in8}, we conclude that
\begin{equation}\label{in12}
	\begin{split}
& \iint_{Q_T} \Bigl( (u -v)^+ \partial_t \phi
		+ \sum_{i=1}^d q_i^+(u,v)  \partial_{x_i} \phi\\ 
& + \sum_{i,j=1}^d r_{ij}^+(u,v) \partial_{x_ix_j}^2 \phi
		+(u-v)^+\mathcal{L}[\phi] \Bigr) \,dx\,dt
		+ \int_{\mathbb{R}^d} (u_0-v_0)^+ \phi(0,x) \,dx \\
&\geq 0,
\end{split}
\end{equation}
where all the involved functions depend on $(t,x)$.
It now only takes a standard argument to conclude from
\eqref{in12} that Theorem \ref {th:unique} holds. Indeed, one chooses
a sequence of functions $0 \leq \phi \leq 1$ from $C^\infty_c([0,T)\times \mathbb{R}^d)$
that converges to $\mathbf{1}_{[0,t) \times \mathbb{R}^d}$ for a Lebesgue point $t$ of
$\int_{\mathbb{R}^d} (u -v)^+\,dx$ and then use the integrability of $u,v$ to conclude the proof.


This concludes the proof of Theorem \ref{th:unique}.

\section{Proof of Theorem \ref{th:contdep} (continuous dependence)}\label{prcdep}

We again employ the doubling of variables
device as in the previous section, but with a slightly different
choice of the entropy function. For each $\varepsilon>0$, define
$$
\operatorname{sgn}_{\varepsilon}(\xi) =
\begin{cases}
	-1, & \text{if $\xi<-\varepsilon$}\\
	\sin(\pi \xi/(2\varepsilon)), & \text{if $ |\xi| \leq \varepsilon$}\\
	1, & \text{if $\xi > \varepsilon$},
\end{cases}
$$
which is a $C^1$  approximation of $\operatorname{sgn}(\cdot)$.
This choice gives rise to a $C^2$ approximation
$\eta_\varepsilon(z)=\int_0^z \operatorname{sgn}_{\varepsilon}(\xi)\,d\xi$ of the entropy flux $\abs{z}$.
As before, we introduce the corresponding entropy flux functions
$\eta^{\varepsilon}(u,c)$, $q_i^{\varepsilon}(u,c)$, and $r_{ij}^{\varepsilon}(u,c)$.
We now employ the doubling variables technique using the test function
$$
\varphi(t,x,s,y) =   \theta_{\nu}(s-t) \delta_{\mu}(y-x)  \Theta_{\alpha}(t),
$$
where $\theta_{\nu}$, $\delta_{\mu}$ are symmetric approximate
delta functions with support in $(-\nu,\nu)$ and $B(0,\mu)$, respectively.
Fix a time $\tau$ from $(0,T)$.
For any $\alpha>0$ with $0<\alpha<\min(\tau_0,T-\tau)$, we define
$$
\Theta_{\alpha}(t)
=H_{\alpha}(t) - H_{\alpha}(t-\tau),
\quad
H_{\alpha}(t)=\int_{-\infty}^{t}
\theta_{\alpha}(\sigma)\,d\sigma.
$$
so that $\Theta_{\alpha}'(t)=\theta_{\alpha}(t)-\theta_{\alpha}(t-\tau)$.

Proceeding as in the previous section (cf.~also \cite{Chen:2005wf}) and
sending $\varepsilon \to 0$, we find
$$
-\iiiint
\abs{u-v} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
\Theta_\alpha'(t)\,dx\,dt\,dy\,ds
\le I_{\mathrm{conv}} - I_{\mathrm{diff}}+I_{\mathrm{fdiff}},
$$
where
\begin{gather*}
I_{\mathrm{conv}} := \iiiint \left[ G(u,v) - F(u,v) \right]\cdot
\nabla_x \delta_{\mu}(y-x) \theta_{\nu}(s-t) \Theta_\alpha(t)\,dx\,dt\,dy\,ds,
\\
F(u,v) := \operatorname{sgn}(u-v) \left(f(u) - f(v)\right), \quad
G(u,v) := \operatorname{sgn}(u-v) \left(g(u) - g(v)\right),
\\
I_{\mathrm{diff}} :=\!\!
\iiiint \sum_{i,j=1}^d \Theta_\alpha(t) \theta_{\nu}(s-t)
\partial_{x_ix_j}^2\delta_{\mu}(y-x) \int_v^u \!\!\operatorname{sgn}(\xi -v)
\varepsilon_{ij}^{a-b}(\xi) \,d\xi \,dx\,dt\,dy\,ds,
\\
\varepsilon_{ij}^{a-b}(\xi) := \sum_{k=1}^K \left(\sigma_{ik}^a(\xi) \sigma_{jk}^a(\xi)
- 2 \sigma_{ik}^a(\xi) \sigma_{jk}^b(\xi) + \sigma_{ik}^b(\xi)\sigma_{jk}^b(\xi)\right).
\end{gather*}
and $I_{\mathrm{fdiff}}=
I_{\mathrm{fdiff}_1}+I_{\mathrm{fdiff}_2}$ with
\begin{align*}
I_{\mathrm{fdiff}_{1}} 
&:=\iiiint \int_{|z|<1} \abs{u-v}\theta_{\nu}(s-t)\Theta_{\alpha}(t)\\
&\quad\times\bigl[\delta_{\mu}(y-x-z)-\delta_{\mu}(y-x)
	-\nabla \delta_{\mu}(y-x) \cdot z \bigr]
 (m(z) - \tilde{m}(z))	\,dz\,dx\,dt\,dy\,ds
\end{align*}
and
\begin{align*}
	I_{\mathrm{fdiff}_{2}}&:=
	\iiiint \int_{|z|\geq 1}
	\abs{u-v}\theta_{\nu}(s-t)\Theta_{\alpha}(t)
	\bigl[\delta_{\mu}(y-x-z)-\delta_{\mu}(y-x) \bigr]\\ 
& \quad\times (m(z)-\tilde{m}(z))\,dz\,dx\,dt\,dy\,ds.
\end{align*}
By the triangle inequality
\begin{align*}
& -\iiiint \abs{u(t,x)-v(s,y)} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
	\Theta_\alpha'(t)\,dx\,dt\,dy\,ds\\ 
& \geq -\iiiint \abs{u(t,y)-v(t,y)} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
	\abs{\Theta_\alpha'(t)}\,dx\,dt\,dy\,ds	\\ 
& \quad  - \iiiint \abs{v(t,y)-v(s,y)} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
	\abs{\Theta_\alpha'(t)}\,dx\,dt\,dy\,ds	\\ 
& \quad  - \iiiint \abs{u(t,x)-u(t,y)} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
	\abs{\Theta_\alpha'(t)}\,dx\,dt\,dy\,ds	\\ 
& =: L + R_t + R_x.
\end{align*}

Keeping in mind that $v\in C(L^1)$ and $u\in L^\infty(BV)$, it
is standard to show that
$$
\lim_{\nu \to 0} R_t = 0, \quad
\limsup_{\alpha\to 0}\abs{R_x} \leq C\mu
$$
and moreover, since also $u(t)\to u_0,v(t)\to v_0$ as $t\to 0$,
$$
\lim_{\alpha\to 0} L
=  \norm{u(\tau,\cdot) - v(\tau,\cdot)}_{L^1(\mathbb{R}^d)}
-\norm{u_0-v_0}_{L^1(\mathbb{R}^d)}.
$$
Following \cite{Chen:2005wf}, using $u\in L^\infty(BV)$ we conclude that
$$
\lim_{\alpha \to 0} \lim_{\nu \to 0} \abs{I_{\mathrm{conv}}}
\leq C \tau \norm{f-g}_{\mathrm{Lip}(I)},
$$
and, exploiting also that $\int \abs{\partial_{x_i} \delta_\mu}\le C/\mu$,
$$
\lim_{\alpha \to 0} \lim_{\nu \to 0} \abs{I_{\mathrm{diff}}}
\leq \frac{C}{\mu}
\tau \norm{(\sigma^a - \sigma^b)
(\sigma^a -\sigma^b)^{\mathrm{tr}}}_{L^{\infty}(I;\mathbb{R}^{d\times d})}.
$$

It remains to estimate $\abs{I_{\mathrm{fdiff}}}$.
First, we nconsider $I_{\mathrm{fdiff}_{1}}$.
Using the Taylor and Fubini theorems we obtain
\begin{align*}
	\abs{I_{\mathrm{fdiff}_{1}}}
	&=\iiint \int_{|z|<1} \int_0^1  (1-\tau)
	\theta_{\mu}(s-t) \Theta_{\alpha}(t) (\tilde{m}(z)-m(z))
	\\ & \quad \times
	\Big( \int_{\mathbb{R}^d} \abs{u(t,x)-v(s,y)} D^2
	\delta_{\mu}(y-x-\tau z)\, z \cdot z
	\,dx \Big)\,d\tau \,dz \,dy \,ds \,dt.
\end{align*}
Thanks to $|u(t,\cdot)-v(s,y)| \in BV(\mathbb{R}^d)$, an integration
 by parts yields
\begin{equation}\label{i22bdd}
	\begin{split}
&I_{\mathrm{fdiff}_{1}} \\
&=\iiint \int_{|z|<1} \int_0^1  (1-\tau)
		\theta_{\mu}(s-t) \Theta_{\alpha}(t) (\tilde{m}(z)-m(z)) \\
&\quad \times \Big( \int_{\mathbb{R}^d} \nabla \delta_{\mu}(y-x-\tau z)\cdot z\,
		D_x\left(\abs{u(t,x)-v(s,y)}\right)\cdot z \,dx \Big)
		\,d\tau \,dz\,dy\,ds\,dt,
	\end{split}
\end{equation}
where the inner integral is taken with respect to the bounded Borel
measure $D\left(|u(t,\cdot)-v(s,y)|\right)\cdot z$.
Since $|D(u(t,\cdot) - v(s,y))| \leq |D(u(t,\cdot))|$, the term inside the
parentheses in \eqref{i22bdd}, is upper bounded by
\[
	|z|^2 \int_{\mathbb{R}^d} \int_{\mathbb{R}^d}
	|\nabla \delta_{\mu}(y-x-\tau z)|\,|dD(u(t,\cdot))(x)|\,dy
	\le |z|^2\abs{u(t,\cdot)}_{BV(\mathbb{R}^d)} \norm{\nabla \delta_{\mu}}_{L^1(\mathbb{R}^d)},
\]
where we have used that $|D u(t,\cdot)|$ is finite and the Fubini's theorem to
first integrate with respect to $y$. Hence,
$$
\lim_{\alpha \to 0} \lim_{\nu \to 0}
\abs{I_{\mathrm{fdiff}_{1}}}
\leq \frac{C}{\mu}
\tau \int_{|z|<1} |z|^2
\abs{m(z)-\tilde{m}(z)}\,dz,
$$
where $C>0$ is a finite constant.

Similarly, relying again on the $L^\infty(BV)$ regularity of $u$, it
is not difficult to deduce via an integration by parts the estimate
$$
\lim_{\nu \to 0} \lim_{\alpha \to 0}
\abs{I_{\mathrm{fdiff}_{2}}}
\leq C\tau\int_{|z|\geq 1}
|z|\abs{m(z)-\tilde{m}(z)}\,dz.
$$

Finally, we collect the bounds we have obtained so far and then
optimize over $\mu$ to obtain the desired
continuous dependence estimate \eqref{contdest}.


\section{More general equations}

To motivate what follows, we recall that (formally)
the L\'evy part of the entropy condition \eqref{entineq} comes from
multiplying the nonlocal operator $\mathcal{L}[u]$ by $\eta'(u)$ and computing the commutator
$\mathcal{L}[\eta(u)]-\eta'(u) \mathcal{L}[u]$.
As an alternative, we can replace the term
$$
\iint_{Q_T} \eta(u) \mathcal{L}[\varphi] \,dx\,dt-m^u
$$
occurring in \eqref{entineq} by
\begin{equation}\label{eq:newsplit}
	\begin{split}
&\iint_{Q_T} \eta(u) \mathcal{L}_\kappa[\varphi] \,dx\,dt
 + \iint_{Q_T} \eta'(u) \widetilde{\mathcal{L}}^\kappa[u]\varphi \,dx\,dt\\ 
 & +u(t,x)\Big( \int_{\abs{z}>\kappa} z\mathbf{1}_{|z|<1}
		\pi(dz)\Big) \cdot \nabla \varphi,
		\quad \kappa\in (0,1),
	\end{split}
\end{equation}
cf.~the proofs of Theorems \ref{th:unique} and \ref{th:contdep}
for the relevant notation.

This formulation of the nonlocal term is directly related to the
formulation used in \cite{Alibaud:2007mi}
for fractional conservation laws.  The proof of
Theorem \ref{th:unique} works equally well with
this formulation of the L\'evy part of the entropy condition.
In fact, (the L\'evy part of)  the proof relies
on two main properties, which both are available
with \eqref{eq:newsplit}:

First, as $\kappa\to 0$, cf.~\eqref{eq:kappa-est},
\begin{equation}\label{eq:essential-I}
	\abs{\iint_{\mathbb{R}^d\times\mathbb{R}^d} \abs{u(x)-u(y)} \Bigl(\mathcal{L}_{x,\kappa}[\phi]
	+\mathcal{L}_{y,\kappa}[\phi]\Bigr) \,dx\,dy}=o(\kappa),
\end{equation}
and second the monotonicity
property, cf.~\eqref{eq:mono-property},
\begin{equation}\label{eq:essential-II}
	\begin{split}
& \iint_{\mathbb{R}^d\times\mathbb{R}^d} \operatorname{sgn}(u(x)-v(y))
		\Bigl(\widetilde{\mathcal{L}}^\kappa[u](x)
		-\widetilde{\mathcal{L}}^\kappa[v](y)\Bigr)\phi(x,y)\, dx\, dy	\\ 
& \le  \iint_{\mathbb{R}^d\times\mathbb{R}^d} \abs{u(x)-v(y))}
		\widetilde{\mathcal{L}}^\kappa_{x+y}[\phi](x,y)\phi(x,y)\, dx\, dy,
	\end{split}
\end{equation}
for $u,v\in L^\infty(\mathbb{R}^d)$ and $0\le \phi\in C^\infty_c(\mathbb{R}^d\times\mathbb{R}^d)$.

Now let $\beta:\mathbb{R}\to\mathbb{R}$ be a nondecreasing Lipschitz
continuous function, and consider the equation
\begin{equation}\label{eq1-sigma}
	\partial_t u + \operatorname{div} f(u) = \operatorname{div} (a(u) \nabla u)
	+ \mathcal{L}[\beta(u)],
\end{equation}
where
\begin{equation} \label{eq:levy-sigma}
	\mathcal{L}[\beta(u)]=  \int_{\mathbb{R}^d \setminus \{0\}}
	\left[ \beta(u(t,x+ z)) - \beta(u(t,x))
	- z \cdot \nabla \beta(u)\, \mathbf{1}_{|z|<1}
	\right]\pi(dz).
\end{equation}

Recently the authors of \cite{Al-Ci-Ja}\footnote{We are grateful to an anonymous referee for
drawing our attention to this paper.} analyzed the equation
$$
\partial_t u + \operatorname{div} f(u) = \mathcal{L}[\beta(u)],
$$
which is a special case of \eqref{eq1-sigma} (set $a\equiv 0$).
Actually, the work \cite{Al-Ci-Ja} allowed for slightly more general 
Levy measures than we do in our framework, but we will not be concerned 
with this  refinement here.
The work \cite{Al-Ci-Ja} provides a series of results
regarding stability and continuous dependence estimates.

Inspired by \cite{Al-Ci-Ja}, our aim here is to outline a uniqueness 
(stability) proof for the more general equation \eqref{eq1-sigma}.
Combining this proof with arguments from  \cite{Al-Ci-Ja}, it is moreover
possible to generalize the continuous dependence estimates
in \cite{Al-Ci-Ja} to \eqref{eq1-sigma} (we leave
the details to the interested reader).

Let $\eta\in C^2(\mathbb{R})$ be convex and $u\in C^2(\mathbb{R})$. Then
$$
\eta'(u(x)) \mathcal{L}[\beta(u)]
=  \eta'(u(x))\mathcal{L}_\kappa[\beta(u)]
+ \eta'(u(x))\mathcal{L}^\kappa[\beta(u)], \quad \kappa \in (0,1).
$$
Define $q_{\beta},S_\beta:\mathbb{R}\to\mathbb{R}$ by
$$
q_{\beta}'=\eta'\beta', \quad S_\beta'=\eta'\circ \beta^{-1},
$$
where $\beta^{-1}$ denotes, say, the left-continuous
inverse of the nondecreasing
function $\beta$, see also \cite[Lemma 2.2]{Andreianov:2009kx}.
One can check that $S_\beta(\beta(u))=q_\beta(u)$.

By the convexity of $S_\beta$,
\begin{align*}
	&\eta'(u(t,x))\Bigl(\beta(u(t,x+ z)) - \beta(u(t,x))
	-z \cdot \nabla \beta(u(t,x)) \mathbf{1}_{|z|<1}\Bigr)
	\\ & 
	= S_\beta'(\beta(u(t,x))\Bigl(\beta(u(t,x+ z)) - \beta(u(t,x))
	-z \cdot \nabla \beta(u(t,x)) \mathbf{1}_{|z|<1}\Bigr)
	\\ & 
	\le S_\beta(\beta(u(t,x+ z))) - S_\beta(\beta(u(t,x)))
	-z \cdot \nabla S_\beta(\beta(u(t,x))) \mathbf{1}_{|z|<1}
	\\ & 
	= q_\beta(u(t,x+ z)) - q_\beta(u(t,x))
	-z \cdot \nabla q_\beta(u(t,x)) \mathbf{1}_{|z|<1}.
\end{align*}
Therefore,
$$
\eta'(u(t,x))\mathcal{L}_\kappa[\beta(u)] \le \mathcal{L}_\kappa[q_\beta(u)],
$$
and so for any non-negative $\phi \in C_c^{\infty}(\mathbb{R}^d)$,
\begin{equation*}%\label{eq:contraction-sigma}
	\int_{\mathbb{R}^d}  \eta'(u(x))\mathcal{L}_\kappa[\beta(u)] \phi(x) \, dx
	\le \int_{\mathbb{R}^d} q_\beta(u(x)) \mathcal{L}_\kappa[\phi]  \, dx.
\end{equation*}

Summarizing, for any $\kappa\in (0,1)$,
\begin{align*}
	& \int_{\mathbb{R}^d}  \eta'(u(x))\mathcal{L}[\beta(u)] \phi(x) \,dx
	\\ & 
	\le \int_{\mathbb{R}^d} q_\beta(u(x)) \mathcal{L}_\kappa[\phi]  \,dx
	+ \int_{\mathbb{R}^d}  \eta'(u(x))\mathcal{L}^\kappa[\beta(u)] \phi(x) \,dx
	\\ & 
	\le \int_{\mathbb{R}^d} q_\beta(u(x)) \mathcal{L}_\kappa[\phi]  \, dx
	+ \int_{\mathbb{R}^d}  \eta'(u(x))\widetilde{\mathcal{L}}^\kappa[\beta(u)] \phi(x) \, dx
	+\int_{\mathbb{R}^d} q_{\beta,\kappa}(u) \cdot \nabla \phi\,dx,
\end{align*}
where $q_{\beta,\kappa}:\mathbb{R}\to\mathbb{R}$ is defined by
$$
q_{\beta,\kappa}'(u)=q_\beta'(u) \cdot
\Big( \int_{\abs{z}>\kappa} z\mathbf{1}_{|z|<1}\pi(dz)\Big)
\overset{z\mapsto -z}{=}q_\beta'(u) \cdot
\Big( -\int_{\abs{z}>\kappa} z\mathbf{1}_{|z|<1}\pi(dz)\Big).
$$

The above formal calculations motivate the following definition.

\begin{definition}\label{def:entropy-sigma} \rm
An entropy solution of the initial value problem \eqref{eq1-sigma}-\eqref{in}
is a measurable function $u : Q_T \to \mathbb{R}$ satisfying the following 
conditions:
\begin{enumerate}

\item $u \in L^{\infty}(Q_T)$, $u \in L^{\infty}(0,T;L^1(\mathbb{R}^d))$,
\begin{equation*}%\label{l2regu-sigma}
	\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u)
	\in L^2(Q_T), \quad k=1,\ldots,K;
\end{equation*}

\item For $k= 1,\ldots, K$,
\begin{equation*}%\label{crule}
	\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^{a,\psi}(u)
	= \psi(u) \sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u),
	\quad \text{a.e.~in $Q_T$ and in $L^2(Q_T)$},
\end{equation*}
for any $\psi \in C(\mathbb{R})$;
%\label{def:chainrule}

\item For any entropy-entropy flux triple $(\eta, q, r)$,
\begin{equation*}%\label{entineq}
	\begin{split}
		& \iint_{Q_T} \Bigl ( \eta(u) \partial_t \varphi
		+ \Bigl(q(u)+q_{\beta,\kappa}(u)\Bigr) \cdot \nabla \varphi
		+ \sum_{i,j=1}^d r_{ij}(u) \partial_{x_ix_j}^2 \varphi \Bigr) \,dx\,dt
		\\ & 
		+\iint_{Q_T} q_{\beta}(u) \mathcal{L}_\kappa[\varphi]  \,dx\,dt
		+\iint_{Q_T}  \eta'(u)\widetilde{\mathcal{L}}^\kappa[\beta(u)] \varphi \,dx\,dt
		\\ &
		+ \int_{\mathbb{R}^d} \eta(u_0) \varphi(0,x) \,dx
		\geq \iint_{Q_T} \eta''(u) \sum_{k=1}^K
		\Bigl(\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u) \Bigr)^2
		\varphi(t,x)\,dx \,dt,
	\end{split}
\end{equation*}
for all $0\le \varphi \in C_c^{\infty}([0,T) \times \mathbb{R}^d)$
and any $\kappa\in (0,1)$.
\end{enumerate}
\end{definition}

Equipped with this definition, we can repeat many of the steps
in the proof of Theorem \ref{th:unique}. Indeed, in the
present context, inequality \eqref{in6} reads
\begin{equation}\label{in6-sigma}
	I_{\mathrm{time}}(\varepsilon)+I_{\mathrm{conv}}(\varepsilon)+I_{\mathrm{diff}}(\varepsilon)
	+I_{\mathrm{fdiff}}(\varepsilon)+	I_{\mathrm{init}}(\varepsilon)
	\ge I_{\mathrm{diss}}(\varepsilon),
\end{equation}
where $I_{\mathrm{time}}, I_{\mathrm{diff}}, I_{\mathrm{diss}}$
are exactly as before, whereas
\begin{gather*}
I_{\mathrm{conv}}(\varepsilon) = \iiiint
	\Bigl(q_{\varepsilon}(u,v)+q_{\beta,\kappa}(u,v)\Bigr)\cdot
	\nabla_{x +y} \varphi \,dx\,dt\,dy\,ds,
	\\ 
\begin{aligned}
I_{\mathrm{fdiff}}(\varepsilon) 
&= I(\kappa)+\iiiint \eta_{\varepsilon}'(u(t,x)-v(s,y))	\\ 
& \quad	\times \Bigl( \widetilde{\mathcal{L}}^\kappa[\beta(u)](t,x)
	-\widetilde{\mathcal{L}}^\kappa[\beta(v)](s,y)\Bigr)\varphi\,dx\,dt\,dy\,ds,
\end{aligned}	\\  
I(\kappa)  = \iiiint  q_{\beta,\varepsilon}(u-v)
	\Bigl(\mathcal{L}_{x,\kappa}[\varphi]
	+\mathcal{L}_{y,\kappa}[\varphi]\Bigr)\,dx\,dt\,dy\,ds,
\end{gather*}
with $q_{\beta,\varepsilon}$ defined by
$q_{\beta,\varepsilon}'=\eta_\varepsilon'\beta'$; $\eta_\varepsilon$ defined
in the proof of Theorem \ref{th:contdep}; and the test
function $\varphi\in C^\infty_c(Q_T\times Q_T)$
defined in \eqref{eq:testfunc}.

Observe that
\begin{equation}\label{eq:Ikappatozero}
	\abs{I(\kappa)} = o(\kappa) \quad \text{(independently of $\varepsilon$)},
\end{equation}
and
\begin{align*}
	&\eta_{\varepsilon}'(u(t,x)-v(s,y))
	\Bigl( \widetilde{\mathcal{L}}^\kappa[\beta(u)](t,x)
	-\widetilde{\mathcal{L}}^\kappa[\beta(v)](s,y)\Bigr)
	\\ & 
	= \int_{\abs{z}>\kappa}
	\Bigl( \eta_{\varepsilon}'(u(t,x)-v(s,y))
	\left(\beta(u(t,x+z))-\beta(v(s,y+z))\right)
	\\ & \quad
	- \eta_{\varepsilon}'(u(t,x)-v(s,y))
	\left(\beta(u(t,x))-\beta(v(s,y))\right)\Bigr)\pi(dz)
	\\ & 
	\le \int_{\abs{z}>\kappa}
	\Bigl(\abs{\beta(u(t,x+z))-\beta(v(s,y+z))}
	\\ & \quad
	-\eta_{\varepsilon}'(u(t,x)-v(s,y))
	\left(\beta(u(t,x))-\beta(v(s,y))\right)\Bigr)\pi(dz)
	\\ & 
	\overset{\varepsilon\to 0}{\to} \int_{\abs{z}>\kappa}
	\Bigl(\abs{\beta(u(t,x+z))-\beta(v(s,y+z))}
	\\ & \quad
	-\abs{\beta(u(t,x))-\beta(v(s,y))}\Bigr)\pi(dz)
	\quad \text{in $L^1(Q_T\times Q_T)$}
	\\ & 
	= \widetilde{\mathcal{L}}_{x+y}^\kappa[\abs{\beta(u)-\beta(v)}],
\end{align*}
where we have used $\abs{\eta_{\varepsilon}'(\cdot)}\le 1$ and 
$\eta_\varepsilon(\cdot)\to \operatorname{sgn}(\cdot)$
as $\varepsilon\to 0$. Consequently,
\begin{equation}\label{eq:Ifdiff-sigma}
	\limsup_{\varepsilon\to 0}I_{\mathrm{fdiff}}(\varepsilon) \le
	\iiiint \abs{\beta(u)-\beta(v)} \widetilde{\mathcal{L}}_{x+y}^\kappa[\varphi]\,dx\,dt\,dy\,ds
	+o(\kappa).
\end{equation}

In view of \eqref{eq:Ikappatozero} and \eqref{eq:Ifdiff-sigma}, we can
proceed as in the proof of Theorem \ref{th:unique}
and eventually send $\varepsilon\to 0$ (keeping $\kappa$ fixed)
in \eqref{in6-sigma}, resulting in the inequality
\begin{equation}\label{in7-sigma}
	\begin{split}
& \iiiint \Bigl( \abs{u-v} (\partial_t+\partial_s)\varphi
		+q(u,v)\cdot \nabla_{x+y} \varphi\\
& +\abs{\beta(u)-\beta(v)}
		\Big(-\int_{\abs{z}>\kappa} z\mathbf{1}_{|z|<1}\pi(dz)\Big)
		\cdot \nabla_{x+y} \varphi\\
& +\sum_{i,j=1}^d  r_{ij}(u,v)
	(\partial_{x_ix_j}^2+2\partial_{x_iy_j}^2+\partial_{y_iy_j}^2) \varphi\\
& + \abs{\beta(u)-\beta(v)}
		\widetilde{\mathcal{L}}_{x+y}^\kappa[\varphi]\Bigr)\,dx\,dt\,dy\,ds
		\\ 
& + \iiint \abs{u_0-v} \varphi(0,x,s,y) \,dx\,dy\,ds
		\\ 
& + \iiint \abs{u-v_0} \varphi(t,x,0,y) \,dx\,dt\,dy \ge o(\kappa),	
\end{split}
\end{equation}
where the integrands on the second and fourth lines added together becomes
$$
\abs{\beta(u)-\beta(v)} \mathcal{L}_{x+y}^\kappa[\varphi].
$$
Sending $\kappa\to 0$ in \eqref{in7-sigma}
we arrive at, compare with \eqref{in7},
\begin{equation*}%\label{in7-sigma}
	\begin{split}
& \iiiint \Bigl( \abs{u-v} (\partial_t+\partial_s)\varphi
		+ \sum_{i=1}^d  q_{i}(u,v) (\partial_{x_i}+\partial_{y_i}) \varphi
		\\ 
& +\sum_{i,j=1}^d  r_{ij}(u,v)
		(\partial_{x_ix_j}^2+2\partial_{x_iy_j}^2+\partial_{y_iy_j}^2) \varphi
		+ \abs{\beta(u)-\beta(v)} \mathcal{L}_{x+y}[\varphi] \Bigr)\,dx\,dt\,dy\,ds
		\\ 
& + \iiint \abs{u_0-v} \varphi(0,x,s,y) \,dx\,dy\,ds
		+ \iiint \abs{u-v_0} \varphi(t,x,0,y) \,dx\,dt\,dy \ge 0.
\end{split}
\end{equation*}

We are now in a position to conclude the $L^1$ stability
and uniqueness of entropy solutions
as in the proof of Theorem \ref{th:unique}.
An existence proof can be given along the lines
outlined in Section \ref{prexis}. We summarize with

\begin{theorem}\label{th:unique-sigma}
Suppose $f$ and $a$ satisfy \eqref{fregul}
and \eqref{dif}--\eqref{sigreg}, respectively, and
that the L\'evy measure $\pi(dz)$
satisfies \eqref{intcond1}. Moreover, suppose $\beta:\mathbb{R}\to\mathbb{R}$ is
a nondecreasing Lipschitz continuous function.
Then there exists an entropy solution of \eqref{eq1-sigma}-\eqref{in}.
Let $u,v$ be two entropy solutions of \eqref{eq1-sigma} with
initial data $u|_{t=0} =u_0 \in (L^1\cap L^{\infty})(\mathbb{R}^d)$,
$v|_{t=0} = v_0 \in (L^1\cap L^{\infty})(\mathbb{R}^d)$.
For a.e. $t \in (0,T)$,  we have
\begin{equation*}%\label{eq:contraction}
	\int_{\mathbb{R}^d} \abs{u(t,x)-v(t,x)} \,dx
	\leq \int_{\mathbb{R}^d} \abs{u_0-v_0} \,dx.
\end{equation*}
\end{theorem}

\subsection*{Acknowledgments}
This work was supported by the Research Council of Norway through
an Outstanding Young Investigators Award of K. H. Karlsen.
This article was written as
part of the the international research program on
Nonlinear Partial Differential Equations at the Centre for
Advanced Study at the Norwegian Academy of Science
and Letters in Oslo during the academic year 2008--09.


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\end{document}