#  $L^{1}$-Contraction Property of Entropy Solutions for Scalar Conservation
Laws with Minimal Regularity Assumptions on the Flux

Paz Hashash

###### Abstract

This paper is concerned with entropy solutions of scalar conservation laws of
the form $\partial_{t}u+\operatorname{div}f=0$ in
$\mathbb{R}^{d}\times(0,\infty)$. The flux $f=f(x,u)$ depends explicitly on
the spatial variable $x$. Using an extension of Kruzkov’s method, we establish
the $L^{1}$-contraction property of entropy solutions under minimal regularity
assumptions on the flux.

###### Contents

  1. 1 Introduction
  2. 2 Remarks about the assumptions on the flux $f$
  3. 3 Lipschitz analysis
  4. 4 Entropy solutions for scalar conservation laws with a Lipschitz flux
  5. 5 $L^{1}$-contraction property

##  1 Introduction

A scalar conservation law is a quasilinear partial differential equation of
the form $\partial_{t}u+\operatorname{div}f=0$ in the half-space
$(x,t)\in\mathbb{R}^{d}\times(0,\infty)$. Here, $u=u(x,t)\in\mathbb{R}$ is the
unknown function, $f=f(x,u)$, and the divergence operator $\operatorname{div}$
is taken with respect to the spatial variable $x$.

As is well known, in general, there is no global (in time) differentiable
solution for the scalar law. Therefore, the concept of weak solutions (or
integral solutions) is introduced. Weak solutions are obtained by multiplying
the scalar law by a test function and using formal111Because $u$ is not
differentiable in general. integration by parts to transfer the derivatives
from $u,f$ to the test function. As was proved in [15], for sufficiently
smooth flux $f$, one can prove the existence of a weak solution by first
proving the existence of a solution for the viscous parabolic equation
$\partial_{t}u_{\epsilon}+\operatorname{div}f(x,u_{\epsilon})=\epsilon\Delta
u_{\epsilon},\epsilon\in(0,\infty),$ and then taking the limit $\epsilon\to
0^{+}$. For a regular flux $f$ and initial data $u_{0}$, the viscous solutions
$u_{\epsilon}$ are unique and regular but, in general, the limiting weak
solution is not regular. Weak solutions for scalar laws are not necessarily
unique in all cases. To establish the uniqueness of weak solutions, the
concept of "entropy solutions" is introduced; these are weak solutions
satisfying suitable "entropy conditions". The $L^{1}$-contraction property is
a key ingredient for proving uniqueness in the class of weak solutions. For a
general introduction to entropy solutions to scalar conservation laws, we
refer to the monographs [8, 10, 13].

The $L^{1}$-contraction property seems to be first established by Vol’pert
[24]. In [15], Kruzkov introduced his method of doubling the variables to
establish the $L^{1}$-contraction property by entropy inequalities. Kruzkov
proved the $L^{1}$-contraction property assuming the flux $f=f(x,k)$ is
continuously differentiable, and the derivatives with respect to the spatial
variables, $\partial_{x_{i}}f(x,k),1\leq i\leq d$, are Lipschitz continuous in
the variable $k$ (see more detailed explanation below).

In this paper, we prove an $L^{1}$-contraction property for scalar
conservation laws of the form $\partial_{t}u+\operatorname{div}f(x,u)=0$,
where the flux $f=f(x,u)$ explicitly depends on the spatial variable $x$ and
has minimal regularity. For the proof, we use a suitable extension of
Kruzkov’s method.

Various cases of fluxes with explicit dependence on the spatial variable have
been treated in the literature. Otto [18] considered the case $f(x,u)=g(u)v$
where $v=v(x)$ is a given velocity field. Seguin and Vovelle [21] established
the $L^{1}$-contraction property in the case where the flux has the form
$f(x,u)=k(x)u(1-u)$ for a discontinuous function $k$. More recently, Bachmann
and Vovelle [3] proved the $L^{1}$-contraction property for a flux of the form
$f(x,u)=g(x,u)+h(u)$ where $g$ is a discontinuous function of the form
$g(x,u)=g_{L}(u)$ for $x<0,g(x,u)=g_{R}(u)$ for $x>0$, where $g_{L}\neq g_{R}$
are Lipschitz functions on an interval, and $h$ is Lipschitz. Many other works
were done in the case where the flux $f=f(x,u)$ is discontinuous, see, e.g.,
[1, 2, 7, 9, 14, 17, 19, 22, 23, 6]. In [5], M. Ben-Artzi and Philippe G.
LeFloch established the $L^{1}$-contraction property on manifolds, assuming
the flux is smooth. In [16], D. Lengeler and T. Müller established the
$L^{1}$-contraction property, assuming the flux and its derivative with
respect to the real variable, $f,\partial_{u}f,f=f(x,u),$ are continuously
differentiable.

In this paper, we establish an $L^{1}$-contraction property under the
assumption that the flux $f=f(x,u)$ satisfies certain regularity hypotheses:

######  Assumption 1.

(Assumptions on the flux $f$) For a function
$f:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d},f=f(x,k),$ we assume:
1a. $f$ is locally Lipschitz on $\mathbb{R}^{d}\times\mathbb{R}$. Furthermore,
we assume that:
1b. $f$ is semi-globally Lipschitz with respect to $k$, which means that, for
every compact set $K\subset\mathbb{R}$, there exists a constant
$C\left(K\right),$ depending on $K$ only, such that

$\left|f\left(x,k_{1}\right)-f\left(x,k_{2}\right)\right|\leq
C\left(K\right)\left|k_{1}-k_{2}\right|,\quad\forall
x\in\mathbb{R}^{d},\quad\forall k_{1},k_{2}\in K.$ (1.1)

2a. There exists a set $\Theta\subset\mathbb{R}^{d}$ such that
$\mathcal{L}^{d}(\Theta)=0$, where $\mathcal{L}^{d}$ is the $d$-dimensional
Lebesgue measure, and for every $k\in\mathbb{R}$, the function
$f(\cdot,k):\mathbb{R}^{d}\to\mathbb{R}^{d}$ is differentiable at every point
$x\in\mathbb{R}^{d}\setminus\Theta$. Moreover, we assume that for every
$x\in\mathbb{R}^{d}\setminus\Theta$, the function $k\mapsto D_{x}f(x,k)$ is
continuous, where $D_{x}f(x,k)$ is the differential of $f(\cdot,k)$ at $x$.
2b. Moreover, at every such point $x$, we assume that for every compact set
$K\subset\mathbb{R}$ we have

$\lim_{y\to x}\left(\sup_{k\in
K}\frac{|f(y,k)-f(x,k)-D_{x}f(x,k)(y-x)|}{|y-x|}\right)=0.$ (1.2)

In this article, our objective is to establish the $L^{1}$-contraction
property while imposing minimal regularity assumptions on the flux function
$f$. We would like to highlight two key distinctions between Kruzkov’s
assumptions on the flux [15] and Assumption 1 above:

  1. 1.

In Kruzkov’s proof, the flux $f=f(x,k)$ is assumed to be continuously
differentiable, while we assume it to be locally Lipschitz. Continuous
differentiability implies (1.2) (see Remark 2.3).

  2. 2.

In Kruzkov’s proof, the derivatives of the flux $f=f(x,k)$ with respect to the
spatial variables $x_{i},1\leq i\leq d,$ are assumed to be Lipschitz
continuous with respect to $k$; while, we only assume that the derivatives of
the flux with respect to the spatial variable exist almost everywhere222The
phrases ”almost everywhere” and ”almost every” always refers to Lebesgue
measure of the relevant dimension. For example, if we write ”for almost every
$x\in\mathbb{R}^{d}$”, we mean almost everywhere with respect to the
$d$-dimensional Lebesgue measure. Similarly, if we write ”for almost every
$(x,t)\in\mathbb{R}^{d}\times(0,\infty)$”, we mean almost everywhere with
respect to the $(d+1)$-dimensional Lebesgue measure, and so on. and they are
continuous functions of the variable $k$. However, the estimate given in (1.1)
is stronger than Kruzkov’s.

The article is organized as follows: Section 2 is devoted to a discussion
about Assumption 1. Section 3 focuses on Lipschitz analysis. In Section 4, we
present the main concepts related to scalar conservation laws and entropy
solutions. In Section 5, we prove the $L^{1}$-contraction property for entropy
solutions.

##  2 Remarks about the assumptions on the flux $f$

In this section we give some remarks about the mentioned above Assumption 1.

######  Remark 2.1.

(Semi-global Lipschitz property of the flux) In this remark, we want to show
that functions $f$ that are globally Lipschitz satisfy conditions 1a and 1b in
Assumption 1, but the opposite is not true in general.

It is easy to check that if $f$ is globally Lipschitz, then it satisfies
conditions 1a and 1b in Assumption 1. Below, we suggest a construction for
non-globally Lipschitz functions that satisfy conditions 1a and 1b in
Assumption 1.

Let $f:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}$ be a function defined
by $f(x,k)=g(x)h(k)$, where $g:\mathbb{R}^{d}\to\mathbb{R}$, and
$h:\mathbb{R}\to\mathbb{R}^{d}$. Note that:

  1. 1.

If $g\in L^{\infty}(\mathbb{R}^{d})$ and $h$ is locally Lipschitz, then $f$
satisfies the semi-global Lipschitz condition (1.1).

Indeed, let $K\subset\mathbb{R}$ be a compact set, and let us denote by $L(K)$
the Lipschitz constant of $h$ on $K$. Then, for every $k_{1},k_{2}\in
K,x\in\mathbb{R}^{d}$, we get

$|f(x,k_{1})-f(x,k_{2})|=|g(x)h(k_{1})-g(x)h(k_{2})|\leq\left|\left|{g}\right|\right|_{L^{\infty}(\mathbb{R}^{d})}L(K)|k_{1}-k_{2}|.$
(2.1)

Therefore, $f$ satisfies the semi-global Lipschitz property (1.1).

  2. 2.

If $g$ is such that $m:=\operatorname{ess-inf}_{x\in\mathbb{R}^{d}}|g(x)|>0$,
and $h$ is differentiable almost everywhere with $h^{\prime}\notin
L^{\infty}(\mathbb{R},\mathbb{R}^{d})$, then $f$ is not globally Lipschitz.

Indeed, let us explain why $\partial_{k}f\notin
L^{\infty}(\mathbb{R}^{d}\times\mathbb{R},\mathbb{R}^{d})$ and, therefore, $f$
is not globally Lipschitz. Take any $M>0$. Let
$E:=\left\\{{k\in\mathbb{R}}\,\middle|\,{|h^{\prime}(k)|>\frac{M}{m}}\right\\}$.
Since $h^{\prime}\notin L^{\infty}(\mathbb{R},\mathbb{R}^{d})$, the one-
dimensional Lebesgue measure of $E$ is positive, $\mathcal{L}^{1}(E)>0$ 333We
denote by $\mathcal{L}^{d}$ the $d$-dimensional Lebesgue measure.. Note that,
up to a set of $(d+1)$-dimensional Lebesgue measure zero, we have
$\mathbb{R}^{d}\times
E\subset\left\\{{(x,k)\in\mathbb{R}^{d}\times\mathbb{R}}\,\middle|\,{|\partial_{k}f(x,k)|>M}\right\\}$.
This is because, for every $x\in\mathbb{R}^{d}$ such that $|g(x)|\geq m$ and
every $k\in E$ at which $h$ is differentiable, we get
$|\partial_{k}f(x,k)|=|g(x)h^{\prime}(k)|\geq m|h^{\prime}(k)|>M$. Since $M$
is arbitrarily large, we conclude that $\partial_{k}f\notin
L^{\infty}(\mathbb{R}^{d}\times\mathbb{R},\mathbb{R}^{d}).$

Now, we can choose $f$ as above with $g,h$ with the properties mentioned in
items 1,2, and in addition to choose $g$ to be locally Lipschitz. Such a
function $f$ satisfies conditions 1a and 1b in Assumption 1 and it is not
globally Lipschitz.

We only use the semi-global Lipschitz property once throughout the article.
See equation (5.56) in the proof of the $L^{1}$-contraction property in
Section 5.

######  Remark 2.2.

(Uniform differentiability almost everywhere) Assume
$f:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d},f=f(x,k),$ is a measurable
function. Let us distinguish between the following two assertions:

  1. 1.

For every $k\in\mathbb{R}$, there exists $\Theta_{k}\subset\mathbb{R}^{d}$
such that $\mathcal{L}^{d}(\Theta_{k})=0$, and for every $k\in\mathbb{R}$, the
function $x\mapsto f(x,k)$ is differentiable at every
$x\in\mathbb{R}^{d}\setminus\Theta_{k}$.

  2. 2.

Uniform differentiability almost everywhere: f is differentiable almost
everywhere in the variable $x$ uniformly with respect to the variable $k$:
there exists $\Theta\subset\mathbb{R}^{d}$ such that
$\mathcal{L}^{d}(\Theta)=0$, and for every $k\in\mathbb{R}$, the function
$x\mapsto f(x,k)$ is differentiable at every
$x\in\mathbb{R}^{d}\setminus\Theta$.

The second assertion implies the first assertion, but not the opposite. In
case $f$ is a Lipschitz function, Rademacher’s Theorem (see Theorem 3.1) tells
us that for every $k\in\mathbb{R}$, the function
$f(\cdot,k):\mathbb{R}^{d}\to\mathbb{R}^{d}$ is differentiable almost
everywhere, and we get the first assertion. However, it does not imply uniform
differentiability almost everywhere. Notice that item 2a in Assumption 1
requires that the flux $f$ has the property of uniform differentiability
almost everywhere. In other words, the family of functions
$\\{f(\cdot,k)\\}_{k\in\mathbb{R}}$ is uniformly differentiable outside a set
of measure zero in $\mathbb{R}^{d}$.

######  Remark 2.3.

(The continuous differentiability of the flux $f$ implies uniform convergence
in the real variable) In this remark, we want to show that if the flux $f$ as
in Assumption 1 is continuously differentiable, then the property (1.2) holds.

It is enough to prove it for scalar functions $f$. Let
$f:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be a continuously
differentiable function. Then, for $x\in\mathbb{R}^{d}$ and a compact set
$K\subset\mathbb{R}$, $D_{x}f$ is uniformly continuous on
$\overline{B}_{1}(x)\times K$, so for arbitrary $\xi\in(0,\infty)$ there
exists $\delta\in(0,1)$ such that, for every
$(x_{1},k_{1}),(x_{2},k_{2})\in\overline{B}_{1}(x)\times K$, such that
$|(x_{1},k_{1})-(x_{2},k_{2})|<\delta$ we have
$|D_{x}f(x_{1},k_{1})-D_{x}f(x_{2},k_{2})|\leq\xi$. Therefore, for every
$(y,k)\in B_{\delta}(x)\times K$, $y\neq x$, we get by the Fundamental Theorem
of Calculus

$\frac{|f(y,k)-f(x,k)-D_{x}f(x,k)(y-x)|}{|y-x|}=\frac{\left|\int_{0}^{1}\frac{d}{dt}f(ty+(1-t)x,k)dt-
D_{x}f(x,k)(y-x)\right|}{|y-x|}\\\
\leq\int_{0}^{1}\big{|}D_{x}f(ty+(1-t)x,k)-D_{x}f(x,k)\big{|}dt\leq\xi.$ (2.2)

Therefore,

$\lim_{y\to x}\left(\sup_{k\in
K}\frac{|f(y,k)-f(x,k)-D_{x}f(x,k)(y-x)|}{|y-x|}\right)=0.$ (2.3)

##  3 Lipschitz analysis

In this section, we state and prove some results about Lipschitz analysis that
will be used throughout the article. We present and prove two technical lemmas
concerning integration by parts and differentiation under the sign of the
integral involving Lipschitz functions. These lemmas are employed to establish
regularity properties for the entropy flux (see Definition 4.1).

Recall Rademacher’s Theorem [11, 12]:

######  Theorem 3.1.

(Rademacher’s Theorem, [11, 12]) Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be a
locally Lipschitz function. Then, $f$ is differentiable almost everywhere. In
particular, it has partial derivatives that exist almost everywhere and lie in
the space $L^{\infty}_{\text{loc}}(\mathbb{R}^{d})$.

######  Lemma 3.2.

(Differentiation under the sign of the integral for Lipschitz functions) Let
$G:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}$, $G=G(x,\omega)$, be a
locally Lipschitz function, $\xi\in L^{\infty}_{\text{loc}}(\mathbb{R})$, and
let $B\subset\mathbb{R}$ be a bounded Lebesgue measurable set. Then, we have,
for almost every $x\in\mathbb{R}^{d}$,

$D_{x}\int_{B}\xi(\omega)G(x,\omega)d\omega=\int_{B}\xi(\omega)D_{x}G(x,\omega)d\omega.$
(3.1)

###### Proof.

It suffices to prove that for every $1\leq i\leq d,i\in\mathbb{N},$ we get for
almost every $x\in\mathbb{R}^{d}$

$\partial_{x_{i}}\int_{B}\xi(\omega)G(x,\omega)d\omega=\int_{B}\xi(\omega)\partial_{x_{i}}G(x,\omega)d\omega.$
(3.2)

Let $E\subset\mathbb{R}^{d}$ be any bounded Lebesgue measurable set. Since $G$
is locally Lipschitz in $\mathbb{R}^{d}\times\mathbb{R}$ we get by
Rademacher’s Theorem for almost every
$(x,\omega)\in\mathbb{R}^{d}\times\mathbb{R}$

$\lim_{h\to
0}\left(\frac{G(x+he_{i},\omega)-G(x,\omega)}{h}-\partial_{x_{i}}G(x,\omega)\right)=0,$
(3.3)

where $e_{i}:=(0,...,1,...,0)$ is the unit vector with $1$ in the $i$-place.
Let us define a family of functions

$H_{h}(x,\omega):=\xi(\omega)\left(\frac{G(x+he_{i},\omega)-G(x,\omega)}{h}-\partial_{x_{i}}G(x,\omega)\right),\quad
h\in\mathbb{R}\setminus\\{0\\}.$ (3.4)

Let $U\subset\mathbb{R}^{d}$ be an open and bounded set such that
$\overline{E}\subset U$. Since $G$ is locally Lipschitz in
$\mathbb{R}^{d}\times\mathbb{R}$, then $G$ is Lipschitz in $U\times B$. Let us
denote its Lipschitz constant in this set by $L$. Note that, for sufficiently
small $|h|>0$ and almost every $(x,\omega)\in E\times B$, we get

$|H_{h}(x,\omega)|\leq\|\xi\|_{L^{\infty}(B)}\left(L+|\partial_{x_{i}}G(x,\omega)|\right).$
(3.5)

By Dominated Convergence Theorem we get

$\lim_{h\to 0}\int_{E}\int_{B}H_{h}(x,\omega)d\omega
dx=\int_{E}\int_{B}\xi(\omega)\lim_{h\to
0}\left(\frac{G(x+he_{i},\omega)-G(x,\omega)}{h}-\partial_{x_{i}}G(x,\omega)\right)d\omega
dx=0.$ (3.6)

Since $G$ is locally Lipschitz in $\mathbb{R}^{d}\times\mathbb{R}$, then the
function $x\mapsto\int_{B}\xi(\omega)G(x,\omega)d\omega$ is locally Lipschitz
in $\mathbb{R}^{d}$. Therefore, for almost every $x\in\mathbb{R}^{d}$ we
obtain by Rademacher’s Theorem

$\lim_{h\to
0}\left(\frac{\int_{B}\xi(\omega)G(x+he_{i},\omega)d\omega-\int_{B}\xi(\omega)G(x,\omega)d\omega}{h}-\partial_{x_{i}}\int_{B}\xi(\omega)G(x,\omega)d\omega\right)=0.$
(3.7)

Thus, by Dominated Convergence Theorem and equations (3.6), (3.7) we get

$\int_{E}\left(\partial_{x_{i}}\int_{B}\xi(\omega)G(x,\omega)d\omega-\int_{B}\xi(\omega)\partial_{x_{i}}G(x,\omega)d\omega\right)dx\\\
=\int_{E}\left(\lim_{h\to
0}\frac{\int_{B}\xi(\omega)G(x+he_{i},\omega)d\omega-\int_{B}\xi(\omega)G(x,\omega)d\omega}{h}-\int_{B}\xi(\omega)\partial_{x_{i}}G(x,\omega)d\omega\right)dx\\\
=\int_{E}\lim_{h\to
0}\int_{B}\xi(\omega)\left(\frac{G(x+he_{i},\omega)-G(x,\omega)}{h}-\partial_{x_{i}}G(x,\omega)\right)d\omega
dx\\\ =\lim_{h\to
0}\int_{E}\int_{B}\xi(\omega)\left(\frac{G(x+he_{i},\omega)-G(x,\omega)}{h}-\partial_{x_{i}}G(x,\omega)\right)d\omega
dx=0.$ (3.8)

Since $E\subset\mathbb{R}^{d}$ is an arbitrary bounded Lebesgue measurable
set, we get for almost every $x\in\mathbb{R}^{d}$ the formula (3.2).

Let us explain the use of the Dominated Convergence Theorem in (3.8). Since
$G$ is locally Lipschitz in $\mathbb{R}^{d}\times\mathbb{R}$, we obtain, by
Rademacher’s Theorem, that $\partial_{x_{i}}G\in
L^{\infty}_{\text{loc}}(\mathbb{R}^{d}\times\mathbb{R},\mathbb{R}^{d})$, and
so $\partial_{x_{i}}G\in
L^{1}_{\text{loc}}(\mathbb{R}^{d}\times\mathbb{R},\mathbb{R}^{d})$. By
Fubini’s Theorem, we conclude that the function
$x\mapsto\int_{B}\left|\partial_{x_{i}}G(x,\omega)\right|d\omega$ lies in the
space $L^{1}(E)$. Let us denote

$F_{h}(x):=\int_{B}\xi(\omega)\left(\frac{G(x+he_{i},\omega)-G(x,\omega)}{h}-\partial_{x_{i}}G(x,\omega)\right)d\omega.$
(3.9)

Note that, by (3.7), the limit of $F_{h}$ as $h\to 0$ exists almost
everywhere. For an open and bounded set $U\subset\mathbb{R}^{d}$ such that
$\overline{E}\subset U$, let $L$ be the Lipschitz constant of $G$ on the
bounded set $U\times B$. For every $h\in\mathbb{R}\setminus\\{0\\}$ such that
$|h|<\operatorname{dist}(E,\mathbb{R}^{d}\setminus U)$ and almost every $x\in
E$, we have

$|F_{h}(x)|\leq\|\xi\|_{L^{\infty}(B)}\int_{B}\left|\frac{G(x+he_{i},\omega)-G(x,\omega)}{h}\right|d\omega+\|\xi\|_{L^{\infty}(B)}\int_{B}\left|\partial_{x_{i}}G(x,\omega)\right|d\omega\\\
\leq\|\xi\|_{L^{\infty}(B)}\mathcal{L}^{1}(B)L+\|\xi\|_{L^{\infty}(B)}\int_{B}\left|\partial_{x_{i}}G(x,\omega)\right|d\omega.$
(3.10)

Thus, the function in the right hand side of (3.10) lies in $L^{1}(E)$. So we
can use the Dominated Convergence Theorem in the last line of (3.8). ∎

######  Lemma 3.3.

(Integration by parts for Lipschitz functions) Let $g:\mathbb{R}\to\mathbb{R}$
and $h:\mathbb{R}\to\mathbb{R}^{d}$ be locally Lipschitz functions. Then, the
following integration by parts formula holds for every interval
$[k_{0},k]\subset\mathbb{R}$:

$\intop_{k_{0}}^{k}g(\omega)h^{\prime}(\omega)d\omega=-\intop_{k_{0}}^{k}g^{\prime}(\omega)h(\omega)d\omega+g(k)h(k)-g(k_{0})h(k_{0}).$
(3.11)

###### Proof.

Since $g$ and $h$ are locally Lipschitz, their product $gh$ is also locally
Lipschitz. Therefore, $gh$ is absolutely continuous on bounded closed
intervals in $\mathbb{R}$, and it can be differentiated almost everywhere
using the Leibniz rule. Therefore, the Fundamental Theorem of Calculus holds
for the product $gh$, from which we get (3.11). Refer to [20] for a proof of
the Fundamental Theorem of Calculus for absolutely continuous functions. ∎

The following proposition will enable us to define a notion of "entropy flux"
for a given flux $f$. It states that the "entropy flux" $q$ possesses some
regularity properties similar to those of the flux $f$ as specified in
Assumption 1:

######  Proposition 3.4.

(Properties of the entropy flux) Let
$f:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}$ be a function satisfying
$1a$, $2a$ and $2b$ in Assumption 1, $\eta\in C^{2}(\mathbb{R})$ and
$k_{0}\in\mathbb{R}$. Let us set

$q\left(x,k\right):=\intop_{k_{0}}^{k}\eta^{\prime}\left(\omega\right)\partial_{\omega}f\left(x,\omega\right)d\omega.$
(3.12)

Then, for every $(x,k)\in\mathbb{R}^{d}\times\mathbb{R}$, we have the
following integration by parts representation of the function $q$:

$q(x,k)=-\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)f(x,\omega)d\omega+\eta^{\prime}(k)f(x,k)-\eta^{\prime}(k_{0})f(x,k_{0}).$
(3.13)

In particular, we get:

  1. 1.

$q:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}$ is a locally Lipschitz
function (in particular, it is defined everywhere in
$\mathbb{R}^{d}\times\mathbb{R}$).

  2. 2.

For almost every $x\in\mathbb{R}^{d}$ the function $k\mapsto D_{x}q(x,k)$ is
continuous.

  3. 3.

The family of functions $\\{q(\cdot,k)\\}_{k\in\mathbb{R}}$ is uniformly
differentiable almost everywhere as defined in Remark 2.2.

###### Proof.

Let $(x,k)\in\mathbb{R}^{d}\times\mathbb{R}$. Since $f$ is locally Lipschitz,
the function $\omega\mapsto f(x,\omega)$ is locally Lipschitz in $\mathbb{R}$.
Therefore, it is differentiable almost everywhere and
$\partial_{\omega}f(x,\cdot)\in
L^{\infty}_{\text{loc}}(\mathbb{R},\mathbb{R}^{d})$. Consequently, $q$ is
defined everywhere in $\mathbb{R}^{d}\times\mathbb{R}$ with values in
$\mathbb{R}^{d}$. Let us denote $g(\omega):=\eta^{\prime}(\omega)$ and
$h(\omega):=f(x,\omega)$. Note that $g$ and $h$ are locally Lipschitz
functions. Therefore, we get, by integration by parts formula (Lemma 3.3)

$q(x,k)=\intop_{k_{0}}^{k}\eta^{\prime}(\omega)\partial_{\omega}f(x,\omega)d\omega=-\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)f(x,\omega)d\omega+\eta^{\prime}(k)f(x,k)-\eta^{\prime}(k_{0})f(x,k_{0}).$
(3.14)

Since $(x,k)\in\mathbb{R}^{d}\times\mathbb{R}$ was arbitrary, the formula
(3.14) is valid for every $(x,k)\in\mathbb{R}^{d}\times\mathbb{R}$, which
proves (3.13).

  1. 1.

Note that since $f$ is locally Lipschitz in $\mathbb{R}^{d}\times\mathbb{R}$,
the function
$G(x,k):=\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)f(x,\omega)d\omega$ is
also locally Lipschitz. Indeed, let $D\subset\mathbb{R}^{d}$ be any compact
set, and $[a,b]\subset\mathbb{R}$ be a closed and bounded interval. Denote
$S:=D\times[a,b]$. Let $L$ be the Lipschitz constant of $f$ on
$D\times\left[\min\\{k_{0},a\\},\max\\{k_{0},b\\}\right]$. For every
$(x,k),(x^{\prime},k^{\prime})\in S$, we have

$|G(x,k)-G(x^{\prime},k^{\prime})|=\left|\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)f(x,\omega)d\omega-\intop_{k_{0}}^{k^{\prime}}\eta^{\prime\prime}(\omega)f(x^{\prime},\omega)d\omega\right|\\\
=\left|\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)f(x,\omega)d\omega-\left(\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)f(x^{\prime},\omega)d\omega+\intop_{k}^{k^{\prime}}\eta^{\prime\prime}(\omega)f(x^{\prime},\omega)d\omega\right)\right|\\\
\leq\intop_{\min\\{k_{0},k\\}}^{\max\\{k_{0},k\\}}|\eta^{\prime\prime}(\omega)|\left|f(x,\omega)-f(x^{\prime},\omega)\right|d\omega+\intop_{\min\\{k,k^{\prime}\\}}^{\max\\{k,k^{\prime}\\}}|\eta^{\prime\prime}(\omega)||f(x^{\prime},\omega)|d\omega\\\
\leq
L\left(\intop_{\min\\{k_{0},a\\}}^{\max\\{k_{0},b\\}}|\eta^{\prime\prime}(\omega)|d\omega\right)|x-x^{\prime}|+\|\eta^{\prime\prime}\|_{L^{\infty}([a,b])}\|f\|_{L^{\infty}(S)}|k-k^{\prime}|\leq
C|(x,k)-(x^{\prime},k^{\prime})|,$ (3.15)

where $C$ is some constant independent of $x,x^{\prime},k,k^{\prime}$. It
proves that $G$ is locally Lipschitz.

By identity $\eqref{eq:identity for integration by parts}$, we see that $q$ is
a locally Lipschitz function in $\mathbb{R}^{d}\times\mathbb{R}$ as a product
and sum of locally Lipschitz functions in $\mathbb{R}^{d}\times\mathbb{R}$. It
completes the proof of part 1 of the proposition.

  2. 2.

By (3.14) and differentiation under the sign of the integral (Lemma 3.2), we
get for every $k\in\mathbb{R}$

$D_{x}q(x,k)=-\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)D_{x}f(x,\omega)d\omega+\eta^{\prime}(k)D_{x}f(x,k)-\eta^{\prime}(k_{0})D_{x}f(x,k_{0}),$
(3.16)

for almost every $x\in\mathbb{R}^{d}$. By item 2a of Assumption 1, the
functions
$k\mapsto\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)D_{x}f(x,\omega)d\omega,k\mapsto\eta^{\prime}(k)D_{x}f(x,k)$
are continuous for almost every $x\in\mathbb{R}^{d}$. Therefore, the function
$k\mapsto D_{x}q(x,k)$ is a continuous function for almost every
$x\in\mathbb{R}^{d}.$

  3. 3.

Assume the family $\\{f(\cdot,k)\\}_{k\in\mathbb{R}}$ is uniformly
differentiable at $x_{0}\in\mathbb{R}^{d}$. Let $C\subset\mathbb{R}$ be any
compact set which contains the interval between $k$ and $k_{0}$. Then, for
every $(x,k)\in\mathbb{R}^{d}\times\mathbb{R}$, $x\neq x_{0}$, we get

$\frac{\left|q(x,k)-q(x_{0},k)-\left(\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)D_{x}f(x_{0},\omega)d\omega\right)(x-x_{0})\right|}{|x-x_{0}|}\\\
=\frac{\left|\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)f(x,\omega)d\omega-\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)f(x_{0},\omega)d\omega-\left(\intop_{k_{0}}^{k}\eta^{\prime\prime}(\omega)D_{x}f(x_{0},\omega)d\omega\right)(x-x_{0})\right|}{|x-x_{0}|}\\\
\leq\int_{C}\eta^{\prime\prime}(\omega)\frac{\left|f(x,\omega)-f(x_{0},\omega)-D_{x}f(x_{0},\omega)(x-x_{0})\right|}{|x-x_{0}|}d\omega\\\
\leq\mathcal{L}^{1}(C)\sup_{\omega\in
C}\eta^{\prime\prime}(\omega)\sup_{\omega\in
C}\frac{\left|f(x,\omega)-f(x_{0},\omega)-D_{x}f(x_{0},\omega)(x-x_{0})\right|}{|x-x_{0}|}.$
(3.17)

By assumption $2b$ on the flux $f$, we get that the family
$\\{q(\cdot,k)\\}_{k\in\mathbb{R}}$ is uniformly differentiable at $x_{0}$.

∎

##  4 Entropy solutions for scalar conservation laws with a Lipschitz flux

In this section, we introduce the main concepts of this work: scalar
conservation laws and the notion of entropy solutions for such laws.

For every $k\in\mathbb{R}$, we assume the existence of a function
$f(\cdot,k):\mathbb{R}^{d}\to\mathbb{R}^{d}$, where $f=f(x,k)$. Let
$u_{0}:\mathbb{R}^{d}\to\mathbb{R}$ be a function. Denote $I:=(0,\infty)$.
Consider the Cauchy problem

$\begin{cases}\partial_{t}u\left(x,t\right)+\operatorname{div}f\left(x,u\left(x,t\right)\right)=0&,(x,t)\in\mathbb{R}^{d}\times
I\\\ u\left(x,0\right)=u_{0}\left(x\right)&,x\in\mathbb{R}^{d}\end{cases}.$
(4.1)

Here, $u:\mathbb{R}^{d}\times[0,\infty)\to\mathbb{R}$ is the unknown function.
The function $u_{0}$ represents the initial data. We refer to the equation
$\partial_{t}u(x,t)+\text{div}\,f(x,u(x,t))=0$ as a scalar conservation law,
and we call the function $f$ the flux of the law. The divergence operator
$\operatorname{div}$ is taken with respect to the spatial variable $x$.

By Proposition 3.4 we can define a well-defined notion of an "entropy flux" as
follows:

######  Definition 4.1.

_(Entropy pair)_ Let $f$ be a flux as in Assumption 1. For $C^{2}$ convex
function $\eta:\mathbb{R}\to\mathbb{R}$ and $k_{0}\in\mathbb{R}$ let us define
a function

$q\left(x,k\right):=\intop_{k_{0}}^{k}\eta^{\prime}\left(\omega\right)\partial_{\omega}f\left(x,\omega\right)d\omega,\quad
q:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}.$ (4.2)

We refer to the function $\eta$ as the entropy and the function $q$ as the
entropy flux. The combination $\left(\eta,q\right)$ is termed an entropy pair.

######  Remark 4.2.

(Uniqueness of derivatives of the entropy flux) For a given $\eta$ as in
Definition 4.1, an entropy flux $q$ for which $(\eta,q)$ is an entropy pair is
not unique, in general. Different choices of the number $k_{0}$ result in
different entropy fluxes. More precisely, for $k_{0},k_{1}$, we have

$\intop_{k_{0}}^{k}\eta^{\prime}\left(\omega\right)\partial_{\omega}f\left(x,\omega\right)d\omega=\intop_{k_{0}}^{k_{1}}\eta^{\prime}\left(\omega\right)\partial_{\omega}f\left(x,\omega\right)d\omega+\intop_{k_{1}}^{k}\eta^{\prime}\left(\omega\right)\partial_{\omega}f\left(x,\omega\right)d\omega.$
(4.3)

However, by item 1 of Proposition 3.4 and Rademacher’s Theorem we get for
almost every $(x,k)\in\mathbb{R}^{d}\times\mathbb{R}$

$\partial_{k}\intop_{k_{0}}^{k}\eta^{\prime}\left(\omega\right)\partial_{\omega}f\left(x,\omega\right)d\omega=\partial_{k}\intop_{k_{1}}^{k}\eta^{\prime}\left(\omega\right)\partial_{\omega}f\left(x,\omega\right)d\omega.$
(4.4)

Which means that two different entropy fluxes for a given entropy $\eta$ have
the same derivative with respect to the variable $k$ almost everywhere in
$\mathbb{R}^{d}\times\mathbb{R}$.

We now provide the definition of entropy solutions. Some notation involved in
this definition will be explained in Remark 4.4.

######  Definition 4.3.

(Entropy solution) For $u_{0}\in L^{\infty}\left(\mathbb{R}^{d}\right)$ we say
that a function $u\in
L^{1}_{\text{loc}}\left(I,L^{1}\left(\mathbb{R}^{d}\right)\right)\cap
L^{\infty}\left(\mathbb{R}^{d}\times I\right)$, where $I=(0,\infty)$ is the
time interval, is an entropy solution for $\eqref{eq: The Cauchy problem for
the scalar conservation law}$ if and only if for every entropy pair
$\left(\eta,q\right)$ and test function $0\leq\varphi\in
C_{c}^{0,1}\left(\mathbb{R}^{d}\times\left[0,\infty\right)\right)$444The space
$C_{c}^{0,1}$ consists of Lipschitz functions with compact support in the
variable $(x,t)\in\mathbb{R}^{d}\times[0,\infty)$. it follows that

$\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\left(\partial_{t}\varphi\left(x,t\right)\right)\eta\left(u\left(x,t\right)\right)+\varphi\left(x,t\right)\bigg{(}\operatorname{div}_{x}q\left(x,u\left(x,t\right)\right)-\eta^{\prime}\left(u\left(x,t\right)\right)\operatorname{div}_{x}f\left(x,u\left(x,t\right)\right)\bigg{)}\\\
+\left(\nabla_{x}\varphi\left(x,t\right)\right)\cdot
q\left(x,u\left(x,t\right)\right)\bigg{]}dxdt+\intop_{\mathbb{R}^{d}}\varphi\left(z,0\right)\eta\left(u_{0}\left(z\right)\right)dz\geq
0.$ (4.5)

Omitting variables in the above inequality gives

$\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\partial_{t}\varphi\,\eta(u)+\varphi\bigg{(}\operatorname{div}_{x}q\left(x,u\right)-\eta^{\prime}(u)\operatorname{div}_{x}f\left(x,u\right)\bigg{)}+\nabla_{x}\varphi\cdot
q\left(x,u\right)\bigg{]}dxdt\\\
+\intop_{\mathbb{R}^{d}}\varphi\left(z,0\right)\eta\left(u_{0}\left(z\right)\right)dz\geq
0.$ (4.6)

We call the last inequality an entropy inequality.

######  Remark 4.4.

(Remarks about the definition of entropy solutions)

  1. 1.

We write $u\in
L^{1}_{\text{loc}}\left(I,L^{1}\left(\mathbb{R}^{d}\right)\right)$ if and only
if the function $U:I\to L^{1}(\mathbb{R}^{d}),U(t):=u(\cdot,t),$ is measurable
and for every compact set $K\subset I$, we have
$\int_{K}\|U(t)\|_{L^{1}(\mathbb{R}^{d})}dt<\infty$.

  2. 2.

Note that by Rademacher’s Theorem, the test function $\varphi$ has locally
bounded partial derivatives $\partial_{t}\varphi,\nabla_{x}\varphi$.

  3. 3.

By Definition $\ref{def:entropy pair R},$ the entropy
$\eta:\mathbb{R}\to\mathbb{R}$ is differentiable and we denote its derivative
by $\eta^{\prime}$ and in the entropy inequality $\eqref{eq:entropy
inequality}$ the amount $\eta^{\prime}(u(x,t))$ is the derivative of $\eta$ at
the point $u(x,t).$

  4. 4.

We denote by $\operatorname{div}_{x}f$ the divergence of the flux
$f:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d},f=f(x,k),$ with respect to
the explicit dependence on the variable $x$. In the entropy inequality (4.5),
$\operatorname{div}_{x}f(x,u(x,t))=\operatorname{div}f(x,k)|_{k=u(x,t)}$ and
$\operatorname{div}_{x}q(x,u(x,t))=\operatorname{div}q(x,k)|_{k=u(x,t)}$.

  5. 5.

Notice that according to items 1a,2a of Assumption 1, Proposition 3.4 and
$u\in L^{\infty}(\mathbb{R}^{d}\times I)$, we get that the functions
$\operatorname{div}_{x}f(x,u(x,t))$, $q(x,u(x,t))$,
$\operatorname{div}_{x}q(x,u(x,t))$ are defined for almost every
$(x,t)\in\mathbb{R}^{d}\times I$ and are locally bounded.

  6. 6.

In the entropy inequality (4.5), the term $\nabla_{x}\varphi\cdot
q\left(x,u\right)$ stands for the inner product between the vectors
$\nabla_{x}\varphi(x,t),\,q\left(x,u(x,t)\right)\in\mathbb{R}^{d}$.

Discussion about the entropy inequality
Let us discuss the entropy inequality (4.5). Let us assume the flux $f$ is
$C^{1}$ and the existence of a $C^{1}$ solution $u=u\left(x,t\right)$ for
$\eqref{eq: The Cauchy problem for the scalar conservation law}$, and let
$\left(\eta,q\right)$ be an entropy pair as in Definition 4.1. Note that by
the scalar conversation law we get

$\partial_{t}u=-\operatorname{div}f(x,u)=-\operatorname{div}_{x}f\left(x,u\right)-\partial_{k}f(x,k)|_{k=u(x,t)}\cdot\nabla_{x}u,$
(4.7)

and by the definition of entropy pair (Definition 4.1), we get

$\partial_{k}q(x,k)|_{k=u(x,t)}=\eta^{\prime}\left(u\right)\partial_{k}f(x,k)|_{k=u(x,t)}.$
(4.8)

Therefore, by (4.7) and (4.8) we get

$\partial_{t}\eta\left(u\right)+\operatorname{div}q\left(x,u\right)=\eta^{\prime}\left(u\right)\partial_{t}u+\operatorname{div}_{x}q\left(x,u\right)+\partial_{k}q(x,k)|_{k=u(x,t)}\cdot\nabla_{x}u\\\
=-\eta^{\prime}\left(u\right)\bigg{(}\operatorname{div}_{x}f\left(x,u\right)+\partial_{k}f(x,k)|_{k=u(x,t)}\cdot\nabla_{x}u\bigg{)}+\operatorname{div}_{x}q\left(x,u\right)+\eta^{\prime}\left(u\right)\partial_{k}f(x,k)|_{k=u(x,t)}\cdot\nabla_{x}u\\\
=-\eta^{\prime}\left(u\right)\operatorname{div}_{x}f\left(x,u\right)+\operatorname{div}_{x}q\left(x,u\right).$
(4.9)

Thus,

$\partial_{t}\eta\left(u\right)+\operatorname{div}q\left(x,u\right)+\eta^{\prime}\left(u\right)\operatorname{div}_{x}f\left(x,u\right)-\operatorname{div}_{x}q\left(x,u\right)=0.$
(4.10)

In particular, if a zero viscosity limit solution $u(x,t)$ is not $C^{1}$,
then the equality is replaced by an inequality

$\partial_{t}\eta\left(u\right)+\operatorname{div}q\left(x,u\right)+\eta^{\prime}\left(u\right)\operatorname{div}_{x}f\left(x,u\right)-\operatorname{div}_{x}q\left(x,u\right)\leq
0.$ (4.11)

By multiplying the last inequality by test function $\varphi$ as in the
definition of the entropy solution (Definition 4.3) and using integration by
parts we get the entropy inequality (4.5).

We give now an approximation lemma which tells us that we can take in the
inequality (4.5) an "entropy pair" $(\eta,q)$ such that the entropy $\eta$ is
not $C^{2}$.

######  Lemma 4.5.

(Validity of the entropy inequality for non-regular entropy $\eta$) Let $u$ be
an entropy solution, as defined in Definition 4.3, for the Cauchy problem
(4.1). Fix $k_{0}\in\mathbb{R}$ and define functions
$\eta:\mathbb{R}\to\mathbb{R}$ and
$q:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}$ as follows:

$\eta\left(k\right):=\left|k-k_{0}\right|,$ (4.12)
$q\left(x,k\right):=\operatorname{sign}(k-k_{0})\left(f\left(x,k\right)-f\left(x,k_{0}\right)\right).$
(4.13)

Here, $f$ is the flux given in (4.1) and satisfies items $1a,2a$ of Assumption
1. Then, the pair $\left(\eta,q\right)$ satisfies the entropy inequality (4.5)
together with any $0\leq\varphi\in
C_{c}^{0,1}\left(\mathbb{R}^{d}\times\left[0,\infty\right)\right)$. The pair
$(\eta,q)$ is called Kruzkov’s entropy pair.

###### Proof.

The derivative of $\eta$ is given by
$\eta^{\prime}(k)=\operatorname{sign}(k-k_{0})$ for every $k\neq k_{0}$ and we
define $\eta^{\prime}(k_{0}):=0$. Consider a sequence of $C^{2}$ convex
functions $\eta_{n}:\mathbb{R}\to\mathbb{R}$ such that

$\begin{cases}\eta_{n}\to\eta\quad\text{in}\quad
L_{\text{loc}}^{\infty}\left(\mathbb{R}\right),\\\
\eta_{n}^{\prime}(k)\to\operatorname{sign}\left(k-k_{0}\right),\forall
k\in\mathbb{R},\\\ \eta_{n}^{\prime}(k_{0})=0,\quad\forall n\in\mathbb{N},\\\
\exists
0<C<\infty\Longrightarrow\left\|\eta_{n}^{\prime}\right\|_{L^{\infty}\left(\mathbb{R}\right)}\leq
C,\forall n\in\mathbb{N}\end{cases}.$ (4.14)

One can take, for example,
$\eta_{n}\left(k\right):=\sqrt{\left(k-k_{0}\right)^{2}+\frac{1}{n}}$. For
each $\eta_{n}$ define

$q_{n}\left(x,k\right):=\int_{k_{0}}^{k}\eta_{n}^{\prime}\left(w\right)\partial_{w}f\left(x,w\right)dw.$
(4.15)

Hence, $\left(\eta_{n},q_{n}\right)$ is an entropy pair. Since $u$ is an
entropy solution, we obtain for $0\leq\varphi\in
C_{c}^{0,1}\left(\mathbb{R}^{d}\times\left[0,\infty\right)\right)$:

$\intop_{I}\intop_{\mathbb{R}^{d}}\Big{[}\partial_{t}\varphi\,\eta_{n}\left(u\right)+\varphi\,\bigg{(}\operatorname{div}_{x}q_{n}\left(x,u\right)-\eta_{n}^{\prime}\left(u\right)\operatorname{div}_{x}f\left(x,u\right)\bigg{)}+\nabla_{x}\varphi\cdot
q_{n}\left(x,u\right)\Big{]}dxdt\\\
+\intop_{\mathbb{R}^{d}}\varphi\left(z,0\right)\eta_{n}\left(u_{0}\left(z\right)\right)dz\geq
0.$ (4.16)

Recall that $u_{0}$ is the initial data in the Cauchy problem (4.1), and
$I:=(0,\infty)$. We intend to take the limit as $n\to\infty$ in (4.16).

In the following, we use the following notations. We denote by
$\operatorname{Supp}(\varphi)$ the support of $\varphi$; since $\varphi\in
C_{c}^{0,1}(\mathbb{R}^{d}\times[0,\infty))$, then
$\operatorname{Supp}(\varphi)$ is a compact set in
$\mathbb{R}^{d}\times[0,\infty)$. We denote by $\operatorname{Im}u$ the image
of $u$; since $u\in L^{\infty}(\mathbb{R}^{d}\times I)$, then
$\operatorname{Im}u$ is contained in a compact set in $\mathbb{R}$ outside a
set of measure zero. We denote by
$P_{x}:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}$, $P_{x}(x,k):=x$, the
projection onto $\mathbb{R}^{d}$; $P_{x}$ takes compact sets in
$\mathbb{R}^{d}\times\mathbb{R}$ to compact sets in $\mathbb{R}^{d}$.

We prove that:

$\lim_{n\to\infty}\intop_{I}\intop_{\mathbb{R}^{d}}\partial_{t}\varphi\,\eta_{n}(u)dxdt=\intop_{I}\intop_{\mathbb{R}^{d}}\partial_{t}\varphi\,\eta(u)dxdt.$
(4.17)

Note that

$\intop_{I}\intop_{\mathbb{R}^{d}}|\partial_{t}\varphi|\,\left|\eta_{n}(u)-\eta(u)\right|dxdt\leq\mathcal{L}^{d+1}(\operatorname{Supp}(\varphi))\|\partial_{t}\varphi\|_{L^{\infty}(\operatorname{Supp}(\varphi))}\|\eta_{n}-\eta\|_{L^{\infty}(\operatorname{Im}u)}.$
(4.18)

Since $\eta_{n}\to\eta$ in $L_{\text{loc}}^{\infty}(\mathbb{R})$, we obtain
(4.17).

We prove that:

$\lim_{n\to\infty}\intop_{\mathbb{R}^{d}}\varphi(z,0)\eta_{n}(u_{0}(z))dz=\intop_{\mathbb{R}^{d}}\varphi(z,0)\eta(u_{0}(z))dz.$
(4.19)

Note that

$\intop_{\mathbb{R}^{d}}\varphi(z,0)|\eta_{n}(u_{0}(z))-\eta(u_{0}(z))|dz\leq\mathcal{L}^{d}(P_{x}(\operatorname{Supp}(\varphi)))\|\varphi(\cdot,0)\|_{L^{\infty}(P_{x}(\operatorname{Supp}(\varphi)))}\|\eta_{n}-\eta\|_{L^{\infty}(\operatorname{Im}u_{0})}.$
(4.20)

Since $u_{0}\in L^{\infty}(\mathbb{R}^{d})$, we get (4.19).

We prove that:

$\lim_{n\to\infty}\intop_{I}\intop_{\mathbb{R}^{d}}\varphi\,\eta_{n}^{\prime}\left(u\right)\operatorname{div}_{x}f\left(x,u\right)dxdt=\intop_{I}\intop_{\mathbb{R}^{d}}\varphi\,\eta^{\prime}\left(u\right)\operatorname{div}_{x}f\left(x,u\right)dxdt.$
(4.21)

According to Remark 4.4, the function
$(x,t)\mapsto\operatorname{div}_{x}f(x,u(x,t))$ lies in the space
$L^{\infty}_{\text{loc}}(\mathbb{R}^{d}\times I)$. According to the
convergence
$\eta_{n}^{\prime}(k)\to\eta^{\prime}(k):=\operatorname{sign}(k-k_{0})$ for
all $k\in\mathbb{R}$ and the existence of a constant $C$ such that
$\left\|\eta_{n}^{\prime}\right\|_{L^{\infty}(\mathbb{R})}\leq C$ for every
$n\in\mathbb{N}$, we can apply the Dominated Convergence Theorem to get
(4.21).

We now prove that:

$\lim_{n\to\infty}\intop_{I}\intop_{\mathbb{R}^{d}}\nabla_{x}\varphi\cdot
q_{n}\left(x,u\right)dxdt=\intop_{I}\intop_{\mathbb{R}^{d}}\nabla_{x}\varphi\cdot
q\left(x,u\right)dxdt.$ (4.22)

We first prove that, for almost every $(x,t)\in\mathbb{R}^{d}\times I$

$\lim_{n\to\infty}q_{n}\left(x,u(x,t)\right)=q(x,u(x,t)).$ (4.23)

Since $f$ is locally Lipschitz, for every $x\in\mathbb{R}^{d}$, the function
$f_{x}(w):=f(x,w)$ is locally Lipschitz in $\mathbb{R}$. Therefore,
$\partial_{w}f(x,\cdot)=f^{\prime}_{x}\in
L^{\infty}_{\text{loc}}(\mathbb{R})$. By the Dominated Convergence Theorem and
the Fundamental Theorem of Calculus, we have for every $k\in\mathbb{R}$ and
every $x\in\mathbb{R}^{d}$

$\lim_{n\to\infty}q_{n}\left(x,k\right)=\intop_{k_{0}}^{k}\lim_{n\to\infty}\eta_{n}^{\prime}\left(w\right)\partial_{w}f\left(x,w\right)dw=\intop_{k_{0}}^{k}\operatorname{sign}(w-k_{0})\partial_{w}f\left(x,w\right)dw\\\
=\operatorname{sign}(k-k_{0})\intop_{k_{0}}^{k}\partial_{w}f(x,w)dw=\operatorname{sign}(k-k_{0})\left(f(x,k)-f(x,k_{0})\right)=q(x,k).$
(4.24)

It proves (4.23). Now, note that for every $n\in\mathbb{N}$ and
$k\in\mathbb{R}$, we get the estimate

$|q_{n}(x,k)|\leq
C\int_{\min\\{k_{0},k\\}}^{\max\\{k_{0},k\\}}|\partial_{w}f\left(x,w\right)|dw.$
(4.25)

Since $f$ is locally Lipschitz in $\mathbb{R}^{d}\times\mathbb{R}$, we get by
Rademacher’s Theorem that the function

$F(x,t):=\int_{\min\\{k_{0},u(x,t)\\}}^{\max\\{k_{0},u(x,t)\\}}|\partial_{w}f\left(x,w\right)|dw$
(4.26)

lies in the space $L^{1}_{\text{loc}}(\mathbb{R}^{d}\times I)$. Therefore, by
(4.23), (4.25), (4.26) and Dominated Convergence Theorem, we obtain (4.22).

We now prove that:

$\lim_{n\to\infty}\intop_{I}\intop_{\mathbb{R}^{d}}\varphi\,\operatorname{div}_{x}q_{n}(x,u)dxdt=\intop_{I}\intop_{\mathbb{R}^{d}}\varphi\,\operatorname{div}_{x}q(x,u)dxdt.$
(4.27)

First we prove that for almost every $(x,t)\in\mathbb{R}^{d}\times I$ we get

$\lim_{n\to\infty}\operatorname{div}_{x}q_{n}(x,u(x,t))=\operatorname{div}_{x}q(x,u(x,t)).$
(4.28)

For this purpose, let us prove the existence of a set
$\Theta\subset\mathbb{R}^{d}$, $\mathcal{L}^{d}(\Theta)=0$, such that for
every $x\in\mathbb{R}^{d}\setminus\Theta$ and $k\in\mathbb{R}$, we have

$\lim_{n\to\infty}\operatorname{div}q_{n}(x,k)=\operatorname{div}q(x,k).$
(4.29)

From (4.29), we deduce (4.28) because for every $(x,t)\in\mathbb{R}^{d}\times
I$ such that $(x,t)$ lies in the domain of $u$ and $(x,t)\notin\Theta\times
I$, we obtain (4.28) for $x$ and $k=u(x,t)$. Let $\Theta$ be as in item 2a of
Assumption 1. For all $k\in\mathbb{R}$ and for every
$x\in\mathbb{R}^{d}\setminus\Theta$ we obtain

$\operatorname{div}q_{n}(x,k)=\operatorname{div}\left[\intop_{k_{0}}^{k}\eta^{\prime}_{n}(w)\partial_{w}f(x,w)dw\right]\\\
=\operatorname{div}\left[-\intop_{k_{0}}^{k}\eta^{\prime\prime}_{n}(w)f(x,w)dw+\eta_{n}^{\prime}(k)f(x,k)-\eta_{n}^{\prime}(k_{0})f(x,k_{0})\right]\\\
=-\intop_{k_{0}}^{k}\eta^{\prime\prime}_{n}(w)\operatorname{div}f(x,w)dw+\eta_{n}^{\prime}(k)\operatorname{div}f(x,k).$
(4.30)

In the last equality of (4.30) we use $\eta_{n}^{\prime}(k_{0})=0$. The
integration by parts and the differentiation under the sign of the integral
follows from Lemmas 3.2, 3.3. Let us analyse the integral on the right hand
side of (4.30). More precisely, we show that

$\lim_{n\to\infty}\intop_{k_{0}}^{k}\eta^{\prime\prime}_{n}(w)\operatorname{div}f(x,w)dw=\operatorname{sign}(k-k_{0})\operatorname{div}f(x,k_{0}).$
(4.31)

Let $k\in\mathbb{R}$. Let $x\in\mathbb{R}^{d}$ be such that
$w\longmapsto\operatorname{div}f(x,w)$ is continuous (almost every
$x\in\mathbb{R}^{d}$ has this property according to item 2a of Assumption 1).
So it is continuous at $k_{0}$. Let $\epsilon\in(0,\infty).$ There exists
$\delta\in(0,\infty)$ such that
$|\operatorname{div}f(x,w)-\operatorname{div}f(x,k_{0})|<\epsilon$ whenever
$|w-k_{0}|<\delta$. Assume that $k>k_{0}$. It follows that

$\intop_{k_{0}}^{k}\eta^{\prime\prime}_{n}(w)\operatorname{div}f(x,w)dw=\intop_{k_{0}}^{k}\eta^{\prime\prime}_{n}(w)\operatorname{div}f(x,k_{0})dw-\intop_{k_{0}}^{k}\eta^{\prime\prime}_{n}(w)\left(\operatorname{div}f(x,k_{0})-\operatorname{div}f(x,w)\right)dw\\\
=\operatorname{div}f(x,k_{0})\left(\eta^{\prime}_{n}(k)-\eta^{\prime}_{n}(k_{0})\right)-\intop_{k_{0}+\delta}^{k}\eta^{\prime\prime}_{n}(w)\left(\operatorname{div}f(x,k_{0})-\operatorname{div}f(x,w)\right)dw\\\
-\intop_{k_{0}}^{k_{0}+\delta}\eta^{\prime\prime}_{n}(w)\left(\operatorname{div}f(x,k_{0})-\operatorname{div}f(x,w)\right)dw.$
(4.32)

Observe that since $\eta_{n}$ is convex we know that $\eta_{n}^{\prime\prime}$
is a nonnegative function. By the Fundamental Theorem of Calculus we get

$\limsup_{n\to\infty}\intop_{k_{0}+\delta}^{k}\eta^{\prime\prime}_{n}(w)|\operatorname{div}f(x,k_{0})-\operatorname{div}f(x,w)|dw\\\
\leq\sup_{w\in[k_{0},k]}|\operatorname{div}f(x,k_{0})-\operatorname{div}f(x,w)|\lim_{n\to\infty}\left(\eta^{\prime}_{n}(k)-\eta^{\prime}_{n}(k_{0}+\delta)\right)=0.$
(4.33)

In addition,

$\limsup_{n\to\infty}\intop_{k_{0}}^{k_{0}+\delta}\eta^{\prime\prime}_{n}(w)|\operatorname{div}f(x,k_{0})-\operatorname{div}f(x,w)|dw\leq\epsilon\lim_{n\to\infty}\left(\eta^{\prime}_{n}(k_{0}+\delta)-\eta^{\prime}_{n}(k_{0})\right)=\epsilon.$
(4.34)

Therefore, we get (4.31) using (4.32), (4.33) and (4.34).

Taking the limit as $n\to\infty$ in $\eqref{eq:description of partial
derivative of q_n by integration by party}$, and taking into account
$\eqref{eq:limit result involved eta''}$, we obtain (4.29). The case $k<k_{0}$
is similar. It completes the proof of (4.28).

Next, according to $\eqref{eq:description of partial derivative of q_n by
integration by party}$, we get for almost every $(x,t)\in\mathbb{R}^{d}\times
I$

$\displaystyle\operatorname{div}_{x}q_{n}(x,u(x,t))=-\intop_{k_{0}}^{u(x,t)}\eta^{\prime\prime}_{n}(w)\operatorname{div}f(x,w)dw+\eta_{n}^{\prime}(u(x,t))\operatorname{div}_{x}f(x,u(x,t)).$
(4.35)

Notice that, since $u$ is bounded and $f$ is locally Lipschitz, the sequence
of functions $(x,t)\longmapsto\operatorname{div}_{x}q_{n}(x,u(x,t))$ is
uniformly bounded on compact sets in $\mathbb{R}^{d}\times I$, which allows us
to use the Dominated Convergence Theorem to get (4.27).

Taking the limit as $n\to\infty$ in $\eqref{eq:entropy inequlity for convex
pair (eta_n,q_n)}$ and taking into account (4.17), (4.19), (4.21), (4.22) and
(4.27), we see that the pair $(\eta,q)$, as defined in (4.12),(4.13),
satisfies the entropy inequality $\eqref{eq:entropy inequality}.$ ∎

##  5 $L^{1}$-contraction property

In this section, we prove the $L^{1}$-contraction property.

Recall the Lebesgue Differentiation Theorem [11]:

######  Theorem 5.1.

(Lebesgue Differentiation Theorem) Let $h\in
L^{1}_{\text{loc}}(\mathbb{R}^{d})$. Then, almost every point in
$\mathbb{R}^{d}$ is a Lebesgue point of $h$, i.e. a point $x\in\mathbb{R}^{d}$
such that

$\lim_{\epsilon\to
0^{+}}\fintop_{B_{\epsilon}\left(x\right)}\left|h\left(x\right)-h\left(y\right)\right|dy:=\lim_{\epsilon\to
0^{+}}\frac{1}{\mathcal{L}^{d}(B_{\epsilon}\left(x\right))}\intop_{B_{\epsilon}\left(x\right)}\left|h\left(x\right)-h\left(y\right)\right|dy=0.$
(5.1)

Here $B_{\epsilon}(x)$ is a closed ball in $\mathbb{R}^{d}$ centred at $x$
with radius $\epsilon$.

Recall that the $d-$dimensional standard mollifier kernel $\rho_{\epsilon}$ on
$\mathbb{R}^{d}$ is given by

$\rho_{\epsilon}\left(x\right):=\frac{1}{\epsilon^{d}}\rho\left(\frac{x}{\epsilon}\right),\quad\rho\left(x\right)=\begin{cases}C\,\exp\left(\frac{1}{\left|x\right|^{2}-1}\right)&\left|x\right|<1\\\
0&\left|x\right|\geq 1\end{cases},$ (5.2)

where $C>0$ is a constant such that
$\intop_{\mathbb{R}^{d}}\rho\left(x\right)dx=1$.

The following theorem is the main result of this paper. In the treatment of
Cauchy problem $\eqref{eq: The Cauchy problem for the scalar conservation
law}$ the $L^{1}$-contraction property gives us uniqueness in the class of
entropy solutions (see Corollary 5.3).

######  Theorem 5.2.

($L^{1}$-contraction property) Let $u$,$\tilde{u}$ be two entropy solutions of
$\eqref{eq: The Cauchy problem for the scalar conservation law}$ with initial
data $u_{0}$,$\tilde{u}_{0}$, respectively. Then, there exists a set
$\mathcal{N}\subset I=(0,\infty)$ such that $\mathcal{L}^{1}(\mathcal{N})=0$
and

$\intop_{\mathbb{R}^{d}}\left|u\left(x,t\right)-\tilde{u}\left(x,t\right)\right|dx\leq\intop_{\mathbb{R}^{d}}\left|u\left(x,s\right)-\tilde{u}\left(x,s\right)\right|dx,$
(5.3)

for every $s,t\in I\setminus\mathcal{N}$ with $s\leq t$. In other words, the
function
$t\mapsto\left\|u\left(\cdot,t\right)-\tilde{u}\left(\cdot,t\right)\right\|_{L^{1}(\mathbb{R}^{d})}$
is non-increasing on $I$ outside a set of measure zero.

###### Proof.

The proof is divided into $3$ steps.

Step 1
Let us fixed $y\in\mathbb{R}^{d}$, $k_{2}\in\mathbb{R}$ and $s\in I$. We
choose a pair of functions $\left(\eta,q\right)$ defined by

$\eta=\eta\left(k_{1},k_{2}\right)=\left|k_{1}-k_{2}\right|,\quad
q=q\left(x,k_{1},k_{2}\right)=\operatorname{sign}\left(k_{1}-k_{2}\right)\left(f\left(x,k_{1}\right)-f\left(x,k_{2}\right)\right).$
(5.4)

By Lemma 4.5 we know that this pair satisfies the entropy inequality (4.5).
Let us choose a test function of the form

$\varphi=\varphi\left(x,t,y,s\right):=\psi_{1}\left(t\right)\psi_{2}\left(x\right)\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right).$
(5.5)

Here $\omega_{\epsilon}$ stands for the $1$-dimensional standard mollifier
kernel and $\rho_{\epsilon}$ stands for the $d$-dimensional standard mollifier
kernel as in (5.2), $\psi_{1}\in C_{c}^{0,1}\left(I\right)$ (recall that
$I=(0,\infty)$), $\psi_{2}\in C_{c}^{0,1}\left(\mathbb{R}^{d}\right)$. Since
$u$ is an entropy solution, we get by Definition 4.3

$\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\partial_{t}\varphi\,\eta\left(u,k_{2}\right)+\varphi\,\bigg{(}\operatorname{div}_{x}q\left(x,u,k_{2}\right)-\partial_{1}\eta\left(u,k_{2}\right)\operatorname{div}_{x}f\left(x,u\right)\bigg{)}+\nabla_{x}\varphi\cdot
q\left(x,u,k_{2}\right)\bigg{]}dxdt\\\
+\intop_{\mathbb{R}^{d}}\varphi\left(z,0,y,s\right)\eta(u_{0}\left(z\right),k_{2})dz\geq
0.$ (5.6)

Here $u=u\left(x,t\right)$, and $\partial_{1}\eta$ stands for the partial
derivative of $\eta$ with respect to the first variable. We choose
$k_{2}=\tilde{u}\left(y,s\right)$. By integrating the last inequality on
$\mathbb{R}^{d}\times I$ with respect to $dyds$ and using Fubini’s Theorem we
get

$\intop_{I}\intop_{\mathbb{R}^{d}}\Bigg{\\{}\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\partial_{t}\varphi\,\eta\left(u,\tilde{u}\right)+\varphi\bigg{(}\operatorname{div}_{x}q\left(x,u,\tilde{u}\right)-\partial_{1}\eta\left(u,\tilde{u}\right)\operatorname{div}_{x}f\left(x,u\right)\bigg{)}+\nabla_{x}\varphi\cdot
q\left(x,u,\tilde{u}\right)\bigg{]}dyds\\\
+\intop_{\mathbb{R}^{d}}\varphi\left(z,0,x,t\right)\eta\left(u_{0}\left(z\right),\tilde{u}\left(x,t\right)\right)dz\Bigg{\\}}dxdt\geq
0.$ (5.7)

Note that in the boundary term
$\intop_{\mathbb{R}^{d}}\varphi\left(z,0,x,t\right)\eta\left(u_{0}\left(z\right),\tilde{u}\left(x,t\right)\right)dz$
we replaced $y,s$ by $x,t$ in order to write a one integral with respect to
$dxdt$. In a similar manner, for fixed $x\in\mathbb{R}^{d}$,
$k_{1}\in\mathbb{R}$ and $t\in I$ , by the assumption that $\tilde{u}$ is an
entropy solution we have

$\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\partial_{s}\varphi\,\eta\left(k_{1},\tilde{u}\right)+\varphi\,\bigg{(}\operatorname{div}_{y}q\left(y,k_{1},\tilde{u}\right)-\partial_{2}\eta\left(k_{1},\tilde{u}\right)\operatorname{div}_{y}f\left(y,\tilde{u}\right)\bigg{)}+\nabla_{y}\varphi\cdot
q\left(y,k_{1},\tilde{u}\right)\bigg{]}dyds\\\
+\intop_{\mathbb{R}^{d}}\varphi\left(x,t,z,0\right)\eta\left(k_{1},\tilde{u}{}_{0}\left(z\right)\right)dz\geq
0.$ (5.8)

Here $\tilde{u}=\tilde{u}\left(y,s\right)$ and $\partial_{2}\eta$ is the
derivative of $\eta$ with respect to the second variable. We choose
$k_{1}=u\left(x,t\right)$. By integrating the last inequality on
$\mathbb{R}^{d}\times I$ with respect to $dxdt$ we get

$\intop_{I}\intop_{\mathbb{R}^{d}}\Bigg{\\{}\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\partial_{s}\varphi\,\eta\left(u,\tilde{u}\right)+\varphi\,\bigg{(}\operatorname{div}_{y}q\left(y,u,\tilde{u}\right)-\partial_{2}\eta\left(u,\tilde{u}\right)\operatorname{div}_{y}f\left(y,\tilde{u}\right)\bigg{)}+\nabla_{y}\varphi\cdot
q\left(y,u,\tilde{u}\right)\bigg{]}dyds\\\
+\intop_{\mathbb{R}^{d}}\varphi\left(x,t,z,0\right)\eta\left(u\left(x,t\right),\tilde{u}{}_{0}\left(z\right)\right)dz\Bigg{\\}}dxdt\geq
0.$ (5.9)

We add up (5.7) and (5.9), and we get

$\intop_{I}\intop_{\mathbb{R}^{d}}\Bigg{\\{}\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\partial_{t}\varphi\,\eta\left(u,\tilde{u}\right)+\varphi\,\bigg{(}\operatorname{div}_{x}q\left(x,u,\tilde{u}\right)-\partial_{1}\eta\left(u,\tilde{u}\right)\operatorname{div}_{x}f\left(x,u\right)\bigg{)}+\nabla_{x}\varphi\cdot
q\left(x,u,\tilde{u}\right)\bigg{]}dyds\\\
+\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\partial_{s}\varphi\,\left(u,\tilde{u}\right)+\varphi\,\bigg{(}\operatorname{div}_{y}q\left(y,u,\tilde{u}\right)-\partial_{2}\eta\left(u,\tilde{u}\right)\operatorname{div}_{y}f\left(y,\tilde{u}\right)\bigg{)}+\nabla_{y}\varphi\cdot
q\left(y,u,\tilde{u}\right)\bigg{]}dyds\\\
+\intop_{\mathbb{R}^{d}}\varphi\left(z,0,x,t\right)\eta\left(u_{0}\left(z\right),\tilde{u}\left(x,t\right)\right)dz+\intop_{\mathbb{R}^{d}}\varphi(x,t,z,0)\eta\left(u\left(x,t\right),\tilde{u}{}_{0}\left(z\right)\right)dz\Bigg{\\}}dxdt\geq
0.$ (5.10)

In the first two lines of (5.10) we have $\varphi=\varphi(x,t,y,s)$. Note that
we have in (5.10) an integral of the form
$\intop_{I}\intop_{\mathbb{R}^{d}}\left\\{\cdot\right\\}\,dxdt$, and within
$\left\\{\cdot\right\\}$, we observe analogous terms characterized by the
interchange of roles between $x,t$ and $y,s$. For instance, the term
$\intop_{I}\intop_{\mathbb{R}^{d}}\partial_{t}\varphi\,\eta\left(u,\tilde{u}\right)\,dyds$
is analogous to the term
$\intop_{I}\intop_{\mathbb{R}^{d}}\partial_{s}\varphi\,\left(u,\tilde{u}\right)\,dyds$.

We sum up analogous terms from $\left\\{\cdot\right\\}$ in the inequality
(5.10).

By the definition of $\varphi$, we get

$\intop_{I}\intop_{\mathbb{R}^{d}}\partial_{t}\varphi\,\eta\left(u,\tilde{u}\right)dyds+\intop_{I}\intop_{\mathbb{R}^{d}}\partial_{s}\varphi\,\eta\left(u,\tilde{u}\right)dyds\\\
=\intop_{I}\intop_{\mathbb{R}^{d}}\biggl{(}\psi^{\prime}_{1}\left(t\right)\omega_{\epsilon}(t-s)+\psi_{1}\left(t\right)\partial_{t}\omega_{\epsilon}(t-s)+\psi_{1}\left(t\right)\partial_{s}\omega_{\epsilon}(t-s)\biggr{)}\psi{}_{2}\left(x\right)\rho_{\epsilon}\left(x-y\right)\eta\left(u,\tilde{u}\right)dyds\\\
=\psi^{\prime}_{1}\left(t\right)\psi{}_{2}\left(x\right)\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)\eta\left(u,\tilde{u}\right)dyds,$
(5.11)

where in the second equality of (5.11) we have
$\psi_{1}\left(t\right)\partial_{t}\omega_{\epsilon}(t-s)+\psi_{1}\left(t\right)\partial_{s}\omega_{\epsilon}(t-s)=0$.

Next, we sum up analogous terms that involve $\varphi$. Before doing that,
note that

$\operatorname{div}_{x}q\left(x,u,\tilde{u}\right)-\partial_{1}\eta\left(u,\tilde{u}\right)\operatorname{div}_{x}f\left(x,u\right)\\\
=\operatorname{sign}\left(u-\tilde{u}\right)\bigg{(}\operatorname{div}_{x}f\left(x,u\right)-\operatorname{div}_{x}f\left(x,\tilde{u}\right)\bigg{)}-\operatorname{sign}\left(u-\tilde{u}\right)\operatorname{div}_{x}f\left(x,u\right)\\\
=-\operatorname{sign}\left(u-\tilde{u}\right)\operatorname{div}_{x}f\left(x,\tilde{u}\right);$
(5.12)

$\operatorname{div}_{y}q\left(y,u,\tilde{u}\right)-\partial_{2}\eta\left(u,\tilde{u}\right)\operatorname{div}_{y}f\left(y,\tilde{u}\right)\\\
=\operatorname{sign}\left(u-\tilde{u}\right)\bigg{(}\operatorname{div}_{y}f\left(y,u\right)-\operatorname{div}_{y}f\left(y,\tilde{u}\right)\bigg{)}+\operatorname{sign}\left(u-\tilde{u}\right)\operatorname{div}_{y}f\left(y,\tilde{u}\right)\\\
=\operatorname{sign}\left(u-\tilde{u}\right)\operatorname{div}_{y}f\left(y,u\right).$
(5.13)

Therefore, according to (5.12) and (5.13), we get

$\bigg{[}\operatorname{div}_{x}q\left(x,u,\tilde{u}\right)-\partial_{1}\eta\left(u,\tilde{u}\right)\operatorname{div}_{x}f\left(x,u\right)\bigg{]}+\bigg{[}\operatorname{div}_{y}q\left(y,u,\tilde{u}\right)-\partial_{2}\eta\left(u,\tilde{u}\right)\operatorname{div}_{y}f\left(y,\tilde{u}\right)\bigg{]}\\\
=\operatorname{sign}\left(u-\tilde{u}\right)\big{(}\operatorname{div}_{y}f\left(y,u\right)-\operatorname{div}_{x}f\left(x,\tilde{u}\right)\big{)}.$
(5.14)

Therefore,

$\intop_{I}\intop_{\mathbb{R}^{d}}\varphi\,\bigg{(}\operatorname{div}_{x}q\left(x,u,\tilde{u}\right)-\partial_{1}\eta\left(u,\tilde{u}\right)\operatorname{div}_{x}f\left(x,u\right)\bigg{)}dyds\\\
+\intop_{I}\intop_{\mathbb{R}^{d}}\varphi\,\bigg{(}\operatorname{div}_{y}q\left(y,u,\tilde{u}\right)-\partial_{2}\eta\left(u,\tilde{u}\right)\operatorname{div}_{y}f\left(y,\tilde{u}\right)\bigg{)}dyds\\\
=\psi_{1}\left(t\right)\psi_{2}\left(x\right)\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)\operatorname{sign}\left(u-\tilde{u}\right)\bigg{(}\operatorname{div}_{y}f\left(y,u\right)-\operatorname{div}_{x}f\left(x,\tilde{u}\right)\bigg{)}dyds.$
(5.15)

Next, we add up analogous terms which contain
$\nabla_{x}\varphi,\nabla_{y}\varphi$. We get

$\intop_{I}\intop_{\mathbb{R}^{d}}\nabla_{x}\varphi\cdot
q\left(x,u,\tilde{u}\right)dyds+\intop_{I}\intop_{\mathbb{R}^{d}}\nabla_{y}\varphi\cdot
q\left(y,u,\tilde{u}\right)dyds\\\
=\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{(}\psi_{1}\left(t\right)\nabla\psi_{2}\left(x\right)\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)+\psi_{1}\left(t\right)\psi_{2}\left(x\right)\omega_{\epsilon}(t-s)\nabla_{x}\rho_{\epsilon}\left(x-y\right)\bigg{)}\cdot
q\left(x,u,\tilde{u}\right)dyds\\\
+\intop_{I}\intop_{\mathbb{R}^{d}}\psi_{1}\left(t\right)\psi_{2}\left(x\right)\omega_{\epsilon}(t-s)\nabla_{y}\rho_{\epsilon}\left(x-y\right)\cdot
q\left(y,u,\tilde{u}\right)dyds\\\
=\psi_{1}\left(t\right)\psi_{2}\left(x\right)\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\bigg{(}\nabla_{x}\rho_{\epsilon}\left(x-y\right)\cdot
q\left(x,u,\tilde{u}\right)+\nabla_{y}\rho_{\epsilon}\left(x-y\right)\cdot
q\left(y,u,\tilde{u}\right)\bigg{)}dyds\\\
+\psi_{1}\left(t\right)\nabla\psi_{2}\left(x\right)\cdot\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)q\left(x,u,\tilde{u}\right)dyds\\\
=\psi_{1}\left(t\right)\psi_{2}\left(x\right)\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\nabla_{y}\rho_{\epsilon}(x-y)\cdot\bigg{(}q(y,u,\tilde{u})-q(x,u,\tilde{u})\bigg{)}dyds\\\
+\psi_{1}\left(t\right)\nabla\psi_{2}\left(x\right)\cdot\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)q\left(x,u,\tilde{u}\right)dyds,$
(5.16)

where in the last equation of (5.16) we use
$\nabla_{x}\rho_{\epsilon}\left(x-y\right)=-\nabla_{y}\rho_{\epsilon}\left(x-y\right)$.

Note that the boundary terms in the third line of (5.10) vanish. Indeed, since
$\psi_{1}\in C^{0,1}_{c}(I)$, we have $\psi_{1}(0)=0$. According to the
equation (5.5) (definition of the test function $\varphi$), we get
$\varphi\left(z,0,x,t\right)=0$, hence

$\intop_{\mathbb{R}^{d}}\varphi\left(z,0,x,t\right)\eta\left(u_{0}\left(z\right),\tilde{u}\left(x,t\right)\right)dz=0.$
(5.17)

In addition, for every small enough $\epsilon\in(0,\infty)$, we have
$\operatorname{Supp}(\psi_{1})\cap\operatorname{Supp}(\omega_{\epsilon})=\emptyset$.
Therefore, for every $t\in I$ and every small enough $\epsilon\in(0,\infty)$,
we get $\varphi(x,t,z,0)=0$. Thus,

$\intop_{\mathbb{R}^{d}}\varphi(x,t,z,0)\eta\left(u\left(x,t\right),\tilde{u}{}_{0}\left(z\right)\right)dz=0.$
(5.18)

We put (5.11), (5.15), (5.16),(5.17) and (5.18) into (5.10) to get for every
small enough $\epsilon\in(0,\infty)$

$\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\psi^{\prime}_{1}\left(t\right)\psi_{2}\left(x\right)I_{1}^{\epsilon}(x,t)+\psi_{1}\left(t\right)\nabla\psi_{2}\left(x\right)\cdot
I_{2}^{\epsilon}(x,t)+\psi_{1}\left(t\right)\psi_{2}\left(x\right)\bigg{(}I_{3}^{\epsilon}(x,t)+I_{4}^{\epsilon}(x,t)\bigg{)}\bigg{]}dxdt\geq
0.$ (5.19)

Here

$I_{1}^{\epsilon}\left(x,t\right):=\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}(x-y)\eta\left(u(x,t),\tilde{u}(y,s)\right)dyds,$
(5.20)
$I_{2}^{\epsilon}\left(x,t\right):=\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)q\left(x,u(x,t),\tilde{u}(y,s)\right)dyds,$
(5.21)

$I_{3}^{\epsilon}\left(x,t\right):=\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)\operatorname{sign}\left(u(x,t)-\tilde{u}(y,s)\right)\bigg{(}\operatorname{div}_{y}f\left(y,u(x,t)\right)-\operatorname{div}_{x}f\left(x,\tilde{u}(y,s)\right)\bigg{)}dyds,$
(5.22)

$I_{4}^{\epsilon}\left(x,t\right):=\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\nabla_{y}\rho_{\epsilon}(x-y)\cdot\biggl{(}q(y,u(x,t),\tilde{u}(y,s))-q(x,u(x,t),\tilde{u}(y,s))\biggr{)}dyds.$
(5.23)

Step 2
We intend to take the limit as $\epsilon\to 0^{+}$ in (5.19) using the
Dominated Convergence Theorem to interchange the limit and the integral. To
apply the Dominated Convergence Theorem, we need to prove the following four
assertions, which tell us that the families of functions $I^{\epsilon}_{j}$,
$j\in\\{1,2,3,4\\}$, indexed by $\epsilon$ with variable
$(x,t)\in\mathbb{R}^{d}\times I$, converge almost everywhere and bounded by
locally integrable functions:

  1. 1.

For almost every $(x,t)\in\mathbb{R}^{d}\times I$ we get

$\begin{cases}\lim_{\epsilon\to
0^{+}}I_{1}^{\epsilon}\left(x,t\right)=\eta(u(x,t),\tilde{u}(x,t))\\\
\sup_{\epsilon\in(0,\infty)}|I_{1}^{\epsilon}\left(x,t\right)|\leq\|\tilde{u}\|_{L^{\infty}(\mathbb{R}^{d}\times
I)}+|u(x,t)|\end{cases},$ (5.24)

and the function on the right hand side of the inequality in (5.24) lies in
the space $L^{1}_{\text{loc}}(\mathbb{R}^{d}\times I)$.

  2. 2.

For almost every $(x,t)\in\mathbb{R}^{d}\times I$ we get

$\begin{cases}\lim_{\epsilon\to
0^{+}}I_{2}^{\epsilon}\left(x,t\right)=q(x,u(x,t),\tilde{u}(x,t))\\\
\sup_{\epsilon\in(0,\infty)}|I_{2}^{\epsilon}\left(x,t\right)|\leq\|f(x,\cdot)\|_{L^{\infty}(\operatorname{Im}\tilde{u})}+|f(x,u(x,t))|\end{cases},$
(5.25)

and the function on the right hand side of the inequality in (5.25) lies in
the space $L^{1}_{\text{loc}}(\mathbb{R}^{d}\times I)$.

  3. 3.

Let $\operatorname{Supp}(\psi_{2})$ be the compact support of $\psi_{2}$, and
let $U_{0}$ be any open and bounded set which contains it. Let us denote
$D:=\operatorname{dist}(\operatorname{Supp}(\psi_{2}),\partial U_{0})>0$. For
almost every $(x,t)\in\operatorname{Supp}(\psi_{2})\times I$ we have

$\begin{cases}\lim_{\epsilon\to
0^{+}}I_{3}^{\epsilon}\left(x,t\right)=\operatorname{div}_{x}q\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\\\
\sup_{\epsilon\in(0,D)}|I_{3}^{\epsilon}\left(x,t\right)|\leq\|\operatorname{div}_{y}f(\cdot,u(x,t))\|_{L^{\infty}(U_{0})}+\|\operatorname{div}_{x}f(x,\cdot)\|_{L^{\infty}(\operatorname{Im}\tilde{u})}\end{cases},$
(5.26)

and the function on the right hand side of the inequality in (5.26) lies in
the space $L^{1}_{\text{loc}}(\mathbb{R}^{d}\times I)$.

  4. 4.

Moreover, let $U_{0}$ and $D$ as above, and let $A\subset\mathbb{R}$ be any
bounded set such that
$\mathcal{L}^{1}\big{(}A\Delta\left[\operatorname{Im}u\cup\operatorname{Im}\tilde{u}\right]\big{)}=0$.
Let us denote by $L$ the Lipschitz constant of $f$ on the bounded set
$U_{0}\times A$. Then, for almost every
$(x,t)\in\operatorname{Supp}(\psi_{2})\times I$ we get

$\begin{cases}\lim_{\epsilon\to
0^{+}}I_{4}^{\epsilon}\left(x,t\right)=-\operatorname{div}_{x}q\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\\\
\sup_{\epsilon\in(0,D)}|I_{4}^{\epsilon}\left(x,t\right)|\leq
2\mathcal{L}^{d}(B_{1}(0))\|\nabla\rho\|_{L^{\infty}(\mathbb{R}^{d})}L\end{cases}.$
(5.27)

Notice that By (5.19), (5.24), (5.25), (5.26), and (5.27), we get by Dominated
Convergence Theorem

$\intop_{I}\intop_{\mathbb{R}^{d}}\bigg{[}\psi^{\prime}_{1}(t)\psi_{2}\left(x\right)\eta\left(u\left(x,t\right),\tilde{u}\left(x,t\right)\right)+\psi_{1}\left(t\right)\nabla\psi_{2}\left(x\right)\cdot
q\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\bigg{]}dxdt\geq
0.$ (5.28)

The proofs of (5.24), (5.25) and (5.26) rely mainly upon Lebesgue
Differentiation Theorem (Theorem 5.1). The proof of (5.27) is a bit more
complicated than the others, and it is established using Lebesgue
Differentiation Theorem along with an approximating argument.

We prove (5.24). Note that for $t\in(0,\infty)$, we get for every sufficiently
small $\epsilon\in(0,\infty)$ that $\intop_{I}\omega_{\epsilon}(t-s)ds=1$.
Hence, by the definition of $\eta$ and the triangle inequality we get

$\Bigl{|}I_{1}^{\epsilon}\left(x,t\right)-\eta\left(u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\Bigr{|}\\\
=\left|\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}(x-y)\eta\left(u(x,t),\tilde{u}(y,s)\right)dyds-\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}(x-y)\eta\left(u(x,t),\tilde{u}(x,t)\right)dyds\right|\\\
\leq\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)\big{|}\eta\left(u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-\eta\left(u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\big{|}dyds\\\
\leq\|\omega\|_{L^{\infty}(\mathbb{R})}\|\rho\|_{L^{\infty}(\mathbb{R}^{d})}\frac{1}{\epsilon}\intop_{t-\epsilon}^{t+\epsilon}\frac{1}{\epsilon^{d}}\intop_{B_{\epsilon}\left(x\right)}\left|\tilde{u}\left(x,t\right)-\tilde{u}\left(y,s\right)\right|dyds.$
(5.29)

Since $\tilde{u}\in L^{\infty}\left(\mathbb{R}^{d}\times I\right)$, then
$\tilde{u}\in L_{\text{loc}}^{1}\left(\mathbb{R}^{d}\times I\right)$ and,
according to Lebesgue Differentiation Theorem, we get for almost every
$(x,t)\in\mathbb{R}^{d}\times I$ the limit in (5.24). The inequality in (5.24)
follows from the definition of $I^{\epsilon}_{1}$ and properties of
$\omega_{\epsilon},\rho_{\epsilon}$: for every $\epsilon\in(0,\infty)$ and for
almost every $(x,t)\in\mathbb{R}^{d}\times I$ we have

$|I_{1}^{\epsilon}\left(x,t\right)|\leq\operatorname{ess-
sup}-sup_{(y,s)\in\mathbb{R}^{d}\times
I}\eta(u(x,t),\tilde{u}(y,s))\leq\|\tilde{u}\|_{L^{\infty}(\mathbb{R}^{d}\times
I)}+|u(x,t)|.$ (5.30)

Since $u,\tilde{u}\in L^{\infty}(\mathbb{R}^{d}\times I)$, the function
$(x,t)\mapsto\|\tilde{u}\|_{L^{\infty}(\mathbb{R}^{d}\times I)}+|u(x,t)|$ lies
in the space $L^{1}_{\text{loc}}(\mathbb{R}^{d}\times I)$. It proves (5.24).

We prove (5.25). We get for almost every $(x,t)\in\mathbb{R}^{d}\times I$ and
every small enough $\epsilon\in(0,\infty)$

$\big{|}I_{2}^{\epsilon}\left(x,t\right)-q\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\big{|}\\\
=\left|\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)q\left(x,u(x,t),\tilde{u}(y,s)\right)dyds-\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)q\left(x,u(x,t),\tilde{u}(x,t)\right)dyds\right|\\\
\leq\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}(x-y)\big{|}q\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-q\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\big{|}dyds\\\
\leq\|\omega\|_{L^{\infty}(\mathbb{R})}\|\rho\|_{L^{\infty}(\mathbb{R}^{d})}\frac{1}{\epsilon}\intop_{t-\epsilon}^{t+\epsilon}\frac{1}{\epsilon^{d}}\intop_{B_{\epsilon}\left(x\right)}\left|q\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-q\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\right|dyds.$
(5.31)

Since $\tilde{u}\in L^{\infty}\left(\mathbb{R}^{d}\times I\right)$ and
$f(x,\cdot)\in L^{\infty}_{\text{loc}}(\mathbb{R})$ for every
$x\in\mathbb{R}^{d}$, we know that the function

$\left(y,s\right)\longmapsto
q\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)=\operatorname{sign}\left(u\left(x,t\right)-\tilde{u}\left(y,s\right)\right)\big{(}f\left(x,u\left(x,t\right)\right)-f\left(x,\tilde{u}\left(y,s\right)\right)\big{)}$
(5.32)

lies in $L_{\text{loc}}^{1}\left(\mathbb{R}^{d}\times I\right)$. Therefore, by
Lebesgue Differentiation Theorem, almost every
$\left(x,t\right)\in\mathbb{R}^{d}\times I$ is a Lebesgue point of this
function, so we get the limit in (5.25). The inequality in (5.25) follows from
the definition of $I^{\epsilon}_{2}$: for every $\epsilon\in(0,\infty)$ and
almost every $(x,t)\in\mathbb{R}^{d}\times I$ we get

$|I_{2}^{\epsilon}\left(x,t\right)|\leq\operatorname{ess-
sup}-sup_{(y,s)\in\mathbb{R}^{d}\times
I}|q(x,u(x,t),\tilde{u}(y,s))|\leq\operatorname{ess-
sup}-sup_{(y,s)\in\mathbb{R}^{d}\times I}|f(x,u(x,t))-f(x,\tilde{u}(y,s))|\\\
\leq|f(x,u(x,t))|+\|f(x,\cdot)\|_{L^{\infty}(\operatorname{Im}\tilde{u})}.$
(5.33)

Since $u,\tilde{u}\in L^{\infty}(\mathbb{R}^{d}\times I)$ and $f\in
L^{\infty}_{\text{loc}}(\mathbb{R}^{d}\times\mathbb{R},\mathbb{R}^{d})$
(because $f$ is continuous), the function
$(x,t)\mapsto|f(x,u(x,t))|+\|f(x,\cdot)\|_{L^{\infty}(\operatorname{Im}\tilde{u})}$
lies in the space $L^{1}_{\text{loc}}(\mathbb{R}^{d}\times I)$. It proves
(5.25).

We prove now (5.26). For almost every $(x,t)\in\mathbb{R}^{d}\times I$ and
sufficiently small $\epsilon\in(0,\infty)$ we have

$\left|I_{3}^{\epsilon}\left(x,t\right)-\operatorname{div}_{x}q(x,u(x,t),\tilde{u}(x,t))\right|\\\
=\Bigg{|}\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)\operatorname{sign}\left(u(x,t)-\tilde{u}(y,s)\right)\bigg{(}\operatorname{div}_{y}f\left(y,u(x,t)\right)-\operatorname{div}_{x}f\left(x,\tilde{u}(y,s)\right)\bigg{)}dyds\\\
-\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)\operatorname{div}_{x}q(x,u(x,t),\tilde{u}(x,t))dyds\Bigg{|}\\\
\leq\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)\bigg{|}\operatorname{sign}\left(u(x,t)-\tilde{u}(y,s)\right)\bigg{(}\operatorname{div}_{y}f\left(y,u(x,t)\right)-\operatorname{div}_{x}f\left(x,\tilde{u}(y,s)\right)\bigg{)}\\\
-\operatorname{div}_{x}q(x,u(x,t),\tilde{u}(x,t))\bigg{|}dyds\\\
\leq\|\omega\|_{L^{\infty}(\mathbb{R})}\|\rho\|_{L^{\infty}(\mathbb{R}^{d})}\frac{1}{\epsilon}\intop_{t-\epsilon}^{t+\epsilon}\frac{1}{\epsilon^{d}}\intop_{B_{\epsilon}\left(x\right)}\bigg{|}\operatorname{sign}\left(u(x,t)-\tilde{u}(y,s)\right)\bigg{(}\operatorname{div}_{y}f\left(y,u(x,t)\right)-\operatorname{div}_{x}f\left(x,\tilde{u}(y,s)\right)\bigg{)}\\\
-\operatorname{div}_{x}q(x,u(x,t),\tilde{u}(x,t))\bigg{|}dyds.$ (5.34)

Since $\tilde{u}\in L^{\infty}(\mathbb{R}^{d}\times I)$ and $f$ is locally
Lipschitz, we get for almost every $(x,t)\in\mathbb{R}^{d}\times I$ that the
function

$(y,s)\longmapsto\operatorname{sign}(u(x,t)-\tilde{u}(y,s))\bigg{(}\operatorname{div}_{y}f(y,u(x,t))-\operatorname{div}_{x}f(x,\tilde{u}(y,s))\bigg{)}$
(5.35)

lies in the space $L^{1}_{\text{loc}}(\mathbb{R}^{d}\times I)$. Indeed, note
that for $k:=u(x,t)$, the function $y\mapsto f(y,k)$ is locally Lipschitz in
$\mathbb{R}^{d}$. Therefore, by Rademacher’s Theorem, we get that
$\operatorname{div}_{y}f(\cdot,k)\in L^{\infty}_{\text{loc}}(\mathbb{R}^{d})$,
so the function $(y,s)\mapsto\operatorname{div}_{y}f(y,u(x,t))$ lies in
$L^{1}_{\text{loc}}(\mathbb{R}^{d}\times I)$. By item 2a in Assumption 1, for
almost every $x\in\mathbb{R}^{d}$, we get that the function
$k\mapsto\operatorname{div}_{x}f(x,k)$ is continuous on $\mathbb{R}$, so it is
locally bounded on $\mathbb{R}$. Since $\tilde{u}\in
L^{\infty}(\mathbb{R}^{d}\times I)$, we have that the function
$(y,s)\mapsto\operatorname{div}_{x}f(x,\tilde{u}(y,s))$ is bounded in
$\mathbb{R}^{d}\times I$. Therefore, we get that the function in (5.35) lies
in $L^{1}_{\text{loc}}(\mathbb{R}^{d}\times I)$. Thus, almost every
$(x,t)\in\mathbb{R}^{d}\times I$ is a Lebesgue point of this function
according to Lebesgue Differentiation Theorem. Therefore, for almost every
$(x,t)\in\mathbb{R}^{d}\times I$, we obtain by (5.34), the limit in (5.26).
The inequality in (5.26) follows from the definition of $I_{3}^{\epsilon}$:
for almost every $(x,t)\in\operatorname{Supp}(\psi_{2})\times I$ and every
$\epsilon\in(0,D)$ we have $B_{\epsilon}(x)\subset U_{0}$ and

$|I_{3}^{\epsilon}\left(x,t\right)|\leq\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\rho_{\epsilon}\left(x-y\right)\bigg{|}\operatorname{div}_{y}f\left(y,u(x,t)\right)-\operatorname{div}_{x}f\left(x,\tilde{u}(y,s)\right)\bigg{|}dyds\\\
\leq\operatorname{ess-sup}-sup_{(y,s)\in U_{0}\times
I}\big{|}\operatorname{div}_{y}f\left(y,u(x,t)\right)-\operatorname{div}_{x}f\left(x,\tilde{u}(y,s)\right)\big{|}\\\
\leq\|\operatorname{div}_{y}f(\cdot,u(x,t))\|_{L^{\infty}(U_{0})}+\|\operatorname{div}_{x}f(x,\cdot)\|_{L^{\infty}(\operatorname{Im}\tilde{u})}.$
(5.36)

It completes the proof of (5.26).

Now we prove (5.27). First note that by the definition of the inner product we
get

$I_{4}^{\epsilon}\left(x,t\right)=-\sum_{i=1}^{d}\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\,\bigg{(}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-q_{i}\left(y,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)\bigg{)}dyds,$
(5.37)

where we denote $q=(q_{1},...,q_{d})$. Therefore, for proving the limit in
(5.27), according to (5.37), it is enough to show that for every natural
number $1\leq i\leq d$ and almost every $(x,t)\in\mathbb{R}^{d}\times I$

$\lim_{\epsilon\to
0^{+}}\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\,\bigg{(}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-q_{i}\left(y,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)\bigg{)}dyds\\\
=(q_{i})_{x_{i}}\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)=e_{i}\cdot\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right),$
(5.38)

where $e_{i}$ is the standard unit vector with $1$ in the $i$-th coordinate
and zero in the other coordinates. Here
$(q_{i})_{x_{i}}\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)=\partial_{x_{i}}q_{i}\left(x,k_{1},k_{2}\right)|_{k_{1}=u(x,t),k_{2}=\tilde{u}(x,t)}$.

Let us fix $1\leq i\leq d,i\in\mathbb{N}$. We now prove that, for almost every
$(x,t)\in\mathbb{R}^{d}\times I$ the following formula holds:

$q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-q_{i}\left(y,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)=\nabla_{x}q_{i}(x,u(x,t),\tilde{u}(y,s))\cdot\left(x-y\right)+\Theta(x,t,y,s)$
(5.39)

for almost every $(y,s)\in\mathbb{R}^{d}\times I$. Here $\Theta(x,t,y,s)$ is a
function with the following property: for every $\xi\in(0,\infty)$ there
exists $\epsilon\in(0,\infty)$ such that for almost every $(y,s)\in
B_{\epsilon}(x)\times I$ we have $|\Theta(x,t,y,s)|\leq\xi|x-y|$.

Let us denote $f=(f_{1},...,f_{d})$. Let $(x_{0},t_{0})\in\mathbb{R}^{d}\times
I$ be any point in the domain of $u$ such that the derivative of the $i$-th
function coordinate of the flux, $f_{i}$, at the point $x_{0}$,
$\nabla_{x}f_{i}(x_{0},k)$, is a continuous function in $k\in\mathbb{R}$; by
item 2a of Assumption 1 almost every $x\in\mathbb{R}^{d}$ has this property.
Let $(y_{0},s_{0})\in\mathbb{R}^{d}\times I$ be a point in the domain of
$\tilde{u}$. Let us denote $k_{1}:=u\left(x_{0},t_{0}\right)$ and
$k_{2}:=\tilde{u}\left(y_{0},s_{0}\right)$. Let us define

$g(y):=q_{i}\left(y,k_{1},k_{2}\right)=\operatorname{sign}\left(k_{1}-k_{2}\right)\left(f_{i}\left(y,k_{1}\right)-f_{i}\left(y,k_{2}\right)\right).$
(5.40)

By the choice of $x_{0}$, the function $g$ is differentiable at $x_{0}$ and

$g(x_{0})-g(y)=\nabla g(x_{0})\cdot(x_{0}-y)+o(x_{0}-y),\quad\forall
y\in\mathbb{R}^{d}.$ (5.41)

Therefore, we get (5.41) for $y=y_{0}$. It proves (5.39). Note that

$\frac{|\Theta(x,t,y,s)|}{|x-y|}=\frac{\left|q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-q_{i}\left(y,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-\nabla_{x}q_{i}(x,u(x,t),\tilde{u}(y,s))\cdot\left(x-y\right)\right|}{|x-y|}\\\
\leq\frac{\left|f_{i}(x,u(x,t))-f_{i}(y,u(x,t))-\nabla_{x}f_{i}(x,u(x,t))\cdot(x-y)\right|}{|x-y|}\\\
+\frac{\left|f_{i}(x,\tilde{u}(y,s))-f_{i}(y,\tilde{u}(y,s))-\nabla_{x}f_{i}(x,\tilde{u}(y,s))\cdot(x-y)\right|}{|x-y|}\\\
\leq 2\sup_{k\in
K}\frac{\left|f_{i}(x,k)-f_{i}(y,k)-\nabla_{x}f_{i}(x,k)\cdot(x-y)\right|}{|x-y|},$
(5.42)

where $K\subset\mathbb{R}$ is any compact set for which
$\mathcal{L}^{1}\left(K\Delta\left[\operatorname{Im}u\cup\operatorname{Im}\tilde{u}\right]\right)=0$.
By item 2b of Assumption 1 we have for arbitrary positive number $\xi$ a
number $\epsilon\in(0,\infty)$ such that $|\Theta(x,t,y,s)|\leq\xi|x-y|$ for
almost every $(y,s)\in B_{\epsilon}(x)\times I$.

We get by (5.39)

$\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\,\bigg{(}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-q_{i}\left(y,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)\bigg{)}dyds\\\
=\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\left(x-y\right)\cdot\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)dyds+o_{\epsilon}\left(1\right),$
(5.43)

where

$o_{\epsilon}\left(1\right):=\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\Theta(x,t,y,s)dyds\quad\text{and}\quad\lim_{\epsilon\to
0^{+}}o_{\epsilon}\left(1\right)=0.$ (5.44)

In order to prove the limit in (5.44), note that for $y\in
B_{\epsilon}\left(x\right)$, we get

$\left|\partial_{y_{i}}\rho_{\epsilon}(x-y)\right||x-y|=\left|\frac{1}{\epsilon^{d}}\nabla\rho\left(\frac{x-y}{\epsilon}\right)\cdot\left(-\frac{1}{\epsilon}e_{i}\right)\right||x-y|\leq\left|\frac{1}{\epsilon^{d}}\nabla\rho\left(\frac{x-y}{\epsilon}\right)\right|\leq\frac{1}{\epsilon^{d}}\|\nabla\rho\|_{L^{\infty}(\mathbb{R}^{d})},$
(5.45)

and for arbitrarily small $\xi\in(0,\infty)$, there exists
$\epsilon\in(0,\infty)$ such that for almost every $(y,s)\in
B_{\epsilon}(x)\times I$ we have $|\Theta(x,t,y,s)|\leq\xi|x-y|$. Therefore,
$|o_{\epsilon}\left(1\right)|\leq\|\nabla\rho\|_{L^{\infty}(\mathbb{R}^{d})}\mathcal{L}^{d}(B_{1}(0))\xi$.

In addition, note that by Fubini’s Theorem and integration by parts we have
for every sufficiently small $\epsilon\in(0,\infty)$

$\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\left(x-y\right)dyds=e_{i}.$
(5.46)

More precisely, note that on the left-hand side of (5.46), we have a vector
due to the expression $x-y$. Examine the $j$-th coordinate of this vector,
meaning that
$\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\left(x_{j}-y_{j}\right)dyds$.
Utilize Fubini’s Theorem and integration by parts to transfer the derivative
$\partial_{y_{i}}$ to $x_{j}-y_{j}$. In the case where $j\neq i$, we obtain
zero; in the case where $j=i$, we obtain $1$.

Hence, using (5.45) and (5.46), we obtain

$\bigg{|}\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\left(x-y\right)\cdot\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)dyds-
e_{i}\cdot\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\bigg{|}\\\
=\bigg{|}\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\left(x-y\right)\cdot\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)dyds\\\
-\left(\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\partial_{y_{i}}\rho_{\epsilon}\left(x-y\right)\left(x-y\right)dyds\right)\cdot\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\bigg{|}\\\
\leq\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)\left|\partial_{y_{i}}\rho_{\epsilon}(x-y)\right||x-y|\bigg{|}\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\bigg{|}dyds\\\
\leq\|\omega\|_{L^{\infty}(\mathbb{R})}\|\nabla\rho\|_{L^{\infty}(\mathbb{R}^{d})}\frac{1}{\epsilon}\intop_{t-\epsilon}^{t+\epsilon}\frac{1}{\epsilon^{d}}\intop_{B_{\epsilon}\left(x\right)}\bigg{|}\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)-\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(x,t\right)\right)\bigg{|}dyds.$
(5.47)

By item 2a of Assumption 1 on the flux $f$ and $\tilde{u}\in
L^{\infty}(\mathbb{R}^{d}\times I)$, we get that

$(y,s)\longmapsto\nabla_{x}q_{i}\left(x,u\left(x,t\right),\tilde{u}\left(y,s\right)\right)$
(5.48)

lies in the space $L_{\text{loc}}^{1}(\mathbb{R}^{d}\times I)$. Therefore, by
Lebesgue Differentiation Theorem, (5.43) and (5.47), we obtain (5.38), from
which we get the limit in $\eqref{eq:convergence of I4}$. For the inequality
in $\eqref{eq:convergence of I4}$, note that, for $k_{1}:=u(x,t)$ and
$k_{2}:=\tilde{u}\left(y,s\right)$ we get

$q(y,k_{1},k_{2})-q(x,k_{1},k_{2})=\operatorname{sign}(k_{1}-k_{2})(f(y,k_{1})-f(y,k_{2}))-\operatorname{sign}(k_{1}-k_{2})(f(x,k_{1})-f(x,k_{2}))\\\
=\operatorname{sign}(k_{1}-k_{2})\bigg{(}f(y,k_{1})-f(x,k_{1})+f(x,k_{2})-f(y,k_{2})\bigg{)}.$
(5.49)

Let $U_{0},A,D$ and $L$ as in the formulation above (5.27). Notice that for
almost every $(x,t)\in\operatorname{Supp}(\psi_{2})\times I$ and for every
$\epsilon\in(0,D)$, we get by (5.49)

$|I_{4}^{\epsilon}\left(x,t\right)|\leq\intop_{I}\intop_{\mathbb{R}^{d}}\omega_{\epsilon}(t-s)|\nabla_{y}\rho_{\epsilon}(x-y)|\Bigl{|}q(y,u(x,t),\tilde{u}(y,s))-q(x,u(x,t),\tilde{u}(y,s))\Bigr{|}dyds\\\
\leq\intop_{I}\intop_{B_{\epsilon}(x)}\omega_{\epsilon}(t-s)|\nabla_{y}\rho_{\epsilon}(x-y)|\Bigl{|}f(x,u(x,t))-f(y,u(x,t))\Bigr{|}dyds\\\
+\intop_{I}\intop_{B_{\epsilon}(x)}\omega_{\epsilon}(t-s)|\nabla_{y}\rho_{\epsilon}(x-y)|\Bigl{|}f(x,\tilde{u}(y,s))-f(y,\tilde{u}(y,s))\Bigr{|}dyds\\\
\leq
2L\intop_{I}\intop_{B_{\epsilon}(x)}\omega_{\epsilon}(t-s)|\nabla_{y}\rho_{\epsilon}(x-y)||x-y|dyds\leq
2L\mathcal{L}^{d}(B_{1}(0))\|\nabla\rho\|_{L^{\infty}(\mathbb{R}^{d})}.$
(5.50)

It completes the proof of (5.27).

Step 3
Next, we fix $0<s<t$ and $\delta\in(0,\infty)$ such that $\delta<t-s$. Let
$\psi_{1}:I\to\mathbb{R}$ be a “table function”:

$\psi_{1}(\tau):=\begin{cases}0\quad&\tau\in(0,s]\\\
\frac{1}{\delta}\tau-\frac{1}{\delta}s\quad&\tau\in(s,s+\delta]\\\
1\quad&\tau\in(s+\delta,t]\\\
-\frac{1}{\delta}\tau+\frac{1}{\delta}(t+\delta)\quad&\tau\in(t,t+\delta]\\\
0\quad&\tau\in(t+\delta,\infty)\end{cases}.$

Note that $\psi_{1}\in C^{0,1}_{c}(I)$. For every $r>0$ we choose a smooth
function $\psi_{2}^{r}:\mathbb{R}^{d}\to\mathbb{R}$ such that
$\psi_{2}^{r}\left(x\right)=1$ in $\left\\{x:\,\left|x\right|\leq r\right\\}$
, $\psi_{2}^{r}\left(x\right)=0$ in $\left\\{x:\,\left|x\right|\geq
r+1\right\\}$ and there exists a constant $c$ such that for every
$x\in\mathbb{R}^{d}$ and $r>0$ we have
$\max\left\\{\left|\psi_{2}^{r}\left(x\right)\right|,\left|\nabla\psi_{2}^{r}\left(x\right)\right|\right\\}\leq
c$. We put $\psi_{1}$ and $\psi_{2}^{r}$ into (5.28) to get

$\frac{1}{\delta}\intop_{s}^{s+\delta}\intop_{\mathbb{R}^{d}}\psi_{2}^{r}\left(x\right)\eta\left(u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)dxd\tau\geq\frac{1}{\delta}\intop_{t}^{t+\delta}\intop_{\mathbb{R}^{d}}\psi_{2}^{r}\left(x\right)\eta\left(u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)dxd\tau\\\
-\intop_{s}^{t+\delta}\intop_{\left\\{z\in\mathbb{R}^{d}:r\leq\left|z\right|\leq
r+1\right\\}}\psi_{1}\left(\tau\right)\nabla\psi{}_{2}^{r}\left(x\right)\cdot
q\left(x,u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)dxd\tau.$
(5.51)

For almost every $(x,\tau)\in\mathbb{R}^{d}\times I$ we have

$\lim_{r\to\infty}\psi_{2}^{r}\left(x\right)\eta\left(u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)=\eta\left(u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right),$
(5.52)

and

$\left|\psi_{2}^{r}\left(x\right)\eta\left(u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)\right|\leq
c\left|\eta\left(u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)\right|.$
(5.53)

By the assumption that $u,\tilde{u}$ are entropy solutions, we have
$u,\tilde{u}\in
L^{1}_{\text{loc}}\left(I,L^{1}\left(\mathbb{R}^{d}\right)\right)$, so the
function on the right hand side in (5.53) lies in
$L^{1}\left(\mathbb{R}^{d}\times K\right)$ for every compact set $K\subset I$.
Dominated Convergence Theorem tells us we can change the order of limit when
$r\rightarrow\infty$ and integral in the first line of (5.51). For the term in
the second line of (5.51), note that for almost all
$(x,\tau)\in\mathbb{R}^{d}\times I$

$\lim_{r\to\infty}\chi_{\left\\{z\in\mathbb{R}^{d}:r\leq\left|z\right|\leq
r+1\right\\}}\left(x\right)\psi_{1}\left(\tau\right)\nabla\psi{}_{2}^{r}\left(x\right)\cdot
q\left(x,u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)=0.$ (5.54)

Here $\chi_{\left\\{z\in\mathbb{R}^{d}:r\leq\left|z\right|\leq
r+1\right\\}}\left(x\right)$ is the characteristic function of the set
$\left\\{z\in\mathbb{R}^{d}:r\leq\left|z\right|\leq r+1\right\\}$ at the point
$x$. In addition,

$\left|\chi_{\left\\{z\in\mathbb{R}^{d}:r\leq\left|z\right|\leq
r+1\right\\}}\left(x\right)\psi_{1}\left(\tau\right)\nabla\psi{}_{2}^{r}\left(x\right)\cdot
q\left(x,u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)\right|\leq
c\left|q\left(x,u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)\right|.$
(5.55)

Notice that

$\intop_{s}^{t+\delta}\intop_{\mathbb{R}^{d}}\left|q\left(x,u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)\right|dxd\tau\leq\intop_{s}^{t+\delta}\intop_{\mathbb{R}^{d}}\left|f\left(x,u\left(x,\tau\right)\right)-f\left(x,\tilde{u}\left(x,\tau\right)\right)\right|dxd\tau\\\
\leq
C\left(K\right)\intop_{s}^{t+\delta}\intop_{\mathbb{R}^{d}}\left|u\left(x,\tau\right)-\tilde{u}\left(x,\tau\right)\right|dxd\tau.$
(5.56)

The second inequality comes from item 1b of Assumption 1 on the flux $f$ for
some compact set $K\subset\mathbb{R}$ which contains the images of
$u$,$\tilde{u}$ up to a set of measure zero in $\mathbb{R}$. The only use of
item 1b in Assumption 1 is to insure the finiteness of the integral

$\intop_{s}^{t+\delta}\intop_{\mathbb{R}^{d}}\left|f\left(x,u\left(x,\tau\right)\right)-f\left(x,\tilde{u}\left(x,\tau\right)\right)\right|dxd\tau.$
(5.57)

The integral on the right hand side of (5.56) is finite because
$u,\tilde{u}\in
L^{1}_{\text{loc}}\left(I,L^{1}\left(\mathbb{R}^{d}\right)\right)$. We take
$r\to\infty$ in (5.51) to get

$\frac{1}{\delta}\intop_{t}^{t+\delta}\intop_{\mathbb{R}^{d}}\eta\left(u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)dxd\tau\leq\frac{1}{\delta}\intop_{s}^{s+\delta}\intop_{\mathbb{R}^{d}}\eta\left(u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)dxd\tau.$
(5.58)

The function
$\tau\longmapsto\intop_{\mathbb{R}^{d}}\eta\left(u\left(x,\tau\right),\tilde{u}\left(x,\tau\right)\right)dx$
lies in $L^{1}_{\text{loc}}(I)$, so almost every point is a Lebesgue point of
this function. Let us denote by $\mathcal{N}$ the set of points which are not
Lebesgue points of this function. We take the limit $\delta\to 0^{+}$ in
(5.58) and we get for every $0<s<t$, $s,t\in I\setminus\mathcal{N}$, that

$\displaystyle\intop_{\mathbb{R}^{d}}\eta\left(u\left(x,t\right),\tilde{u}\left(x,t\right)\right)dx\leq\intop_{\mathbb{R}^{d}}\eta\left(u\left(x,s\right),\tilde{u}\left(x,s\right)\right)dx.$
(5.59)

It completes the proof. ∎

######  Corollary 5.3.

(Uniqueness of entropy solutions) Assume that $u$ is an entropy solution of
the Cauchy problem (4.1). Additionally, assume that $u$ is continuous in the
time variable from the right at $0$ in the $L^{1}$-sense, meaning there exists
a set $\Theta\subset I$ with $\mathcal{L}^{1}(\Theta)=0$ such that

$\lim_{\underset{t\in I\setminus\Theta}{t\to
0^{+}}}\int_{\mathbb{R}^{d}}|u(x,t)-u_{0}(x)|dx=0.$ (5.60)

Then, $u$ is unique in $L^{1}(\mathbb{R}^{d}\times I)$, which means that for
every entropy solution $\tilde{u}$ satisfying Cauchy problem (4.1) and
condition (5.60), we obtain

$\|u-\tilde{u}\|_{L^{1}(\mathbb{R}^{d}\times I)}=0.$ (5.61)

###### Proof.

Assume that $u$ and $\tilde{u}$ are two entropy solutions to the Cauchy
problem (4.1) that satisfy the condition (5.60). According to the
$L^{1}$-contraction property (Theorem 5.2), there exists a set $\Theta\subset
I$ with $\mathcal{L}^{1}(\Theta)=0$ such that for every $s,t\in
I\setminus\Theta$ with $s<t$, we have
$\int_{\mathbb{R}^{d}}\left|u\left(x,t\right)-\tilde{u}\left(x,t\right)\right|dx\leq\int_{\mathbb{R}^{d}}\left|u\left(x,s\right)-\tilde{u}\left(x,s\right)\right|dx.$
Taking the limit as $s\to 0^{+}$, $s\in I\setminus\Theta$, and using the
condition (5.60), we obtain
$\int_{\mathbb{R}^{d}}\left|u\left(x,t\right)-\tilde{u}\left(x,t\right)\right|dx\leq\int_{\mathbb{R}^{d}}\left|u_{0}(x)-u_{0}(x)\right|dx=0$.
Therefore, we get (5.61). ∎

Acknowledgement
This paper is based on a M.Sc. thesis of the author at the Institute of
Mathematics, Hebrew University of Jerusalem. It is a pleasure to thank my
advisor, Professor Matania Ben-Artzi, for his continuous guidance and support.

## References

  * [1] B. Andreianov, K.H.Karlsen and N.H. Risebro, "A Theory of $L^{1}$-dissipative solvers for scalar conservation laws with discontinuous flux", Arch. Rat. Mech. Anal. 201 (2011), 27–86.
  * [2] B. Andreianov and D. Mitrović, "Entropy conditions for scalar conservation laws with discontinuous flux revisited", Annales IHP-Anal. Nonlin. 32 (2015),1307–1335.
  * [3] F. Bachmann and J. Vovelle, "Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients", Comm. PDE 31 (2006),371–395.
  * [4] M.Bank and M.Ben-Arzi. "Scalar conservation laws on a half-line:a parabolic approach", Journal of Hyperbolic differential Equations 7.01 (2010): 165-189.
  * [5] M. Ben-Artzi and Philippe G. LeFloch. "Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds." Annales de l’Institut Henri Poincaré C 24.6 (2007): 989-1008.
  * [6] M. Buliček, P. Gwiazda and A. Świerczewska-Gwiazda, "Multi-dimensional scalar conservation laws with fluxes discontinuous in the unknown and the spacial variable", Math. Models Meth. Appl. Sci. 23 (2013), 407–439.
  * [7] G. Crasta, V. De Cicco and G. De Philippis, "Kinetic formulation and uniqueness for scalar conservation laws with discontinuous flux", Comm. PDE 40 (2015),694–726.
  * [8] C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics, 4-th Ed.", Grundlehren der Mathematischen Wissenschaften, vol. 325, Springer, 2016.
  * [9] S. Diehl, "A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients", J. Hyperbolic Diff. Eqs. 06 (2009), 127–159.
  * [10] L.C.Evans, "Partial Differential Equations", American Mathematical Society, 1998.
  * [11] L.C.Evans and R.F.Gariepy, "Measure theory and fine properties of functions" ,CRC press,2015.
  * [12] H. Federer, "Geometric measure theory", Springer Verlag, Berlin, 1969.
  * [13] E. Godlewski and P.A. Raviart, "Hyperbolic Systems of Conservation Laws", Ellipses, 1991.
  * [14] J. Jimenez, "Mathematical analysis of a scalar multidimensional conservation law with discontinuous flux", J. Evolution Eqs. 11 (2011), 553–576.
  * [15] S. Kruzkov, "First-order quasilinear equations with several space variables", Math.USSR Sb. 10 (1970), 217–243.
  * [16] D. Lengeler and T. Müller. "Scalar conservation laws on constant and time-dependent Riemannian manifolds." Journal of Differential Equations 254.4 (2013): 1705-1727.
  * [17] D. Mitrović, "New entropy conditions for scalar conservation laws with discontinuous flux", Disc. Cont. Dynamical Sys. 30 (2011), 1191–1210.
  * [18] F. Otto, "A regularizing effect of nonlinear transport equations", Quart. Appl. Math. 56 (1998), 355–375.
  * [19] E. Y. Panov, "On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux", J. Hyperbolic Diff. Eqs. 06 (2009), 525–548.
  * [20] Rudin, Walter. "Real and complex analysis. 1987." Cited on 156 (1987): 16.
  * [21] N. Seguin and J. Vovelle, "Analysis and approximation of scalar conservation law with a flux function with discontinuous coefficients", Math. Models and Methods Appl. Sci. 13 (2003),221–257.
  * [22] C. Shen and M. Sun, "The bifurcation phenomenon for scalar conservation laws with discontinuous flux functions", Acta Applic. Mathematicae 121, 69–80.
  * [23] C. Shen and M. Sun, "Wave interactions and stability of the Riemann solutions for a scalar conservation law with a discontinuous flux function", Zeits. Angew. Mathematik Physik 64 (2013), 1025–1056.
  * [24] A.I. Vol’pert, "The spaces BV and quasilinear equations", Math. USSR Sb. 2 (1967), 225- 267

Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653,
Beer Sheva 8410501, Israel
_E-mail:_<EMAIL_ADDRESS>