Signed Radon measure-valued solutions of flux saturated scalar conservation laws

TL;DR

This paper establishes existence and uniqueness of signed Radon measure-valued entropy solutions for scalar conservation laws with initial data as superpositions of Dirac masses, extending the understanding of measure-valued solutions.

Contribution

It introduces a new class of measure-valued solutions for scalar conservation laws with signed Radon measures and proves their well-posedness under specific conditions.

Findings

Existence of measure-valued entropy solutions

Uniqueness under additional conditions

Applicable to initial data as superpositions of Dirac masses

Abstract

We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.

Equations

$$F=\{x_1,x_2,\cdots ,x_p\}.$$

$$\left\{\begin{array}[]{ll}u_{t}+[H(u)]_{x}=0&\mbox{in}\ \,(x_{j-1},x_{j})\times(0,T)\\ u=\pm\infty&\mbox{in $\{x_{j-1}\}\times(0,T)$}\\ u=\pm\infty&\mbox{in $\{x_{j}\}\times(0,T)$}\\ u=u_{0r}&\mbox{in $(x_{j-1},x_{j})\times\{0\}$}\end{array}\right.$$

$$\left\{\begin{array}[]{ll}u_{t}+[H(u)]_{x}=0&\mbox{in}\ \,(-\infty,x_{1})\times(0,T)\\ u=\pm\infty&\mbox{in $\{x_{1}\}\times(0,T)$}\\ u=u_{0r}&\mbox{in $(-\infty,x_{1})\times\{0\}$}\,,\end{array}\right.$$

$$\left\{\begin{array}[]{ll}u_{t}+[H(u)]_{x}=0&\mbox{in}\ \,(x_{p},\infty)\times(0,T)\\ u=\pm\infty&\mbox{in $\{x_{p}\}\times(0,T)$}\\ u=u_{0r}&\mbox{in $(x_{p},\infty)\times\{0\}$}\,.\end{array}\right.$$

$$[u{]}_{\pm }:=\max \{\pm u,0\},{\mathrm{sgn}}_{\pm }(u):=\pm {\chi }_{{\mathbb{R}}_{\pm }}(u),\mathrm{sgn}(u):={\mathrm{sgn}}_{-}(u)+{\mathrm{sgn}}_{+}(u).$$

$$\underset{x\to x_0^{\pm }}{\lim }f(x)=l\in \mathbb{R}$$

$$\nu (K)={\int }_{K}fd\mu$$

$$BV(\Omega ):=\left\{z\in L^1(\Omega )|z^{\prime }\in \mathcal{M}(\Omega ),\Vert z^{\prime }{\Vert }_{\mathcal{M}(\Omega )}<\infty \right\},\Vert z{\Vert }_{BV(\Omega )}:=\Vert z{\Vert }_{L^1(\Omega )}+\Vert z^{\prime }{\Vert }_{\mathcal{M}(\Omega )},$$

$${⟨u,\zeta ⟩}_{Q_T}={\int }_0^T{⟨u(\cdot ,t),\zeta (\cdot ,t)⟩}_{\Omega }dt;$$

$$u(E)=\int_{0}^{T}u(\cdot,t)\big{(}E^{t}\big{)}\,dt\,\qquad(E^{t}=\{x\in\Omega\,:\ \,(x,t)\in E\}).$$

$$\left\langle u_{ac},\zeta\right\rangle_{Q_{T}}=\int\!\!\!\int_{Q_{T}}u_{r}\,\zeta\,dxdt\ \$$

$$u_s(\cdot ,t)=[u(\cdot ,t){]}_s,u_{ac}(\cdot ,t)=[u(\cdot ,t){]}_{ac},u_r(\cdot ,t)=[u(\cdot ,t){]}_r,$$

$$u^{\pm }(\cdot ,t)=[u(\cdot ,t){]}^{\pm },|u(\cdot ,t)|=|u|(\cdot ,t)\text{mbox}fora.e.t\in (0,T),$$

$$u_s^{\pm }(\cdot ,t)=[u(\cdot ,t){]}_s^{\pm },|u_s|(\cdot ,t)=\left|[u(\cdot ,t){]}_s\right|\text{mbox}fora.e.t\in (0,T).$$

$$\iint_{Q_{\tau}}\big{[}u_{r}\zeta_{t}+H(u_{r})\,\zeta_{x}\big{]}\,dxdt+\int_{0}^{\tau}\left\langle u_{s}(\cdot,t),\zeta_{t}(\cdot,t)\right\rangle_{\Omega}dt=-\left\langle u_{0},\zeta(\cdot,0)\right\rangle_{\Omega}\,.$$

$${\iint }_{Q_{\tau }}\left\{|u_r-k|{\zeta }_t+\text{sgn}(u_r-k)\left[H(u_r)-H(k)\right]{\zeta }_x\right\}dxdt+$$

$$+{\int }_0^{\tau }{⟨|u_s(\cdot ,t)|,{\zeta }_t(\cdot ,t)⟩}_{\Omega }dt\ge -{\int }_{\Omega }|u_{0r}(x)-k|\zeta (x,0)dx-{⟨|u_{0s}|,\zeta (\cdot ,0)⟩}_{\Omega }$$

$$[u(\cdot ,t_2){]}_s^{\pm }\le [u(\cdot ,t_1){]}_s^{\pm }\le u_{0s}^{\pm }\text{mbox}in\mathcal{M}(\Omega ).$$

$$\{\begin{array}{ll}{u}_s(\cdot ,t)(\{x_j\})\ne 0& \text{if t∈[0,tj),}\\ u_s(\cdot ,t)(\{x_j\})=0& \text{if t∈(tj,T].}\end{array}$$

$${\rm ess}\lim_{x\to x_{j}^{+}}\int_{0}^{t_{j}}\!{\text{\rm sgn}\,}_{\pm}(u_{r}(x,t)-k)\,\big{[}H(u_{r}(x,t))-H(k)\big{]}\,\beta(t)\,dt\leq 0\quad\text{if }\pm c_{j}<0,$$

$${\rm ess}\lim_{x\to x_{j}^{-}}\int_{0}^{t_{j}}\!{\text{\rm sgn}\,}_{\pm}(u_{r}(x,t)-k)\,\big{[}H(u_{r}(x,t))-H(k)\big{]}\,\beta(t)\,dt\geq 0\quad\text{if }\pm c_{j}<0$$

$$\mathrm{ess}\underset{x\to x_j^{\pm }}{\lim }{\int }_0^T{\text{sgn}}_{\pm }(u_r(x,t)-k)[H(u_r(x,t))-H(k)]\beta (t)dt(j=1,\dots ,p),$$

$${\iint }_S(u_r-k){\beta }^{\prime }(t)\rho (x)dxdt+{\int }_0^T{⟨u_s(\cdot ,t),\rho ⟩}_{\mathbb{R}}{\beta }^{\prime }(t)dt+$$

$$+{\iint }_S\left[H(u_r)-H(k)\right]{\rho }^{\prime }(x)\beta (t)dxdt=$$

$$=-\beta (0)\left\{{\int }_{\mathbb{R}}(u_{0r}-k)\rho (x)dx+{⟨u_{0s},\rho ⟩}_{\mathbb{R}}\right\}$$

$${\iint }_S[u_r-k{]}_{\pm }{\beta }^{\prime }(t)\rho (x)dxdt+{\int }_0^T{⟨[u(\cdot ,t){]}_s^{\pm },\rho ⟩}_{\mathbb{R}}{\beta }^{\prime }(t)dt+$$

$$+{\iint }_S{\mathrm{sgn}}_{\pm }(u_r-k)\left[H(u_r)-H(k)\right]{\rho }^{\prime }(x)\beta (t)dxdt\ge$$

$$\ge -\beta (0)\left\{{\int }_{\mathbb{R}}[u_{0r}-k{]}_{\pm }\rho (x)dx+{⟨u_{0s}^{\pm },\rho ⟩}_{\mathbb{R}}\right\},$$

$${\int }_0^T{⟨[u(\cdot ,t){]}_s^{\pm },\rho ⟩}_{\mathbb{R}}{\beta }^{\prime }(t)dt\ge {⟨u_{0s}^{\pm },\rho ⟩}_{\mathbb{R}}.$$

Full text

1. Introduction

We study the Cauchy problem for the scalar conservation law:

[TABLE]

where

[TABLE]

(obviously the condition $H(0)=0$ is not restrictive). The initial condition $u_{0}$ is a signed Radon measure on $\mathbb{R}$. In most of the paper we shall assume that its singular part, $u_{0s}$, is a finite superposition of Dirac masses:

[TABLE]

In that case we denote the support of the singular measure $u_{0s}$ by $F$:

[TABLE]

In [3] we considered the case of nonnegative initial measures $u_{0}$. In the present paper we consider the case of signed measures (see [2, 5, 7, 9] for motivations and related remarks). A specific motivation is the link between measure-valued solutions of $(CL)$ and discontinuous solutions of the Cauchy problem for the Hamilton-Jacobi equation

[TABLE]

where $U_{0}\in BV_{loc}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, $U_{0}^{\prime}\in L^{1}_{loc}(\mathbb{R}\setminus F)$, and $U_{0}(x_{j}^{+})\neq U_{0}(x_{j}^{-})$ if $x_{j}\in F$. If $(A_{1})$ is satisfied, the distributional derivative $U_{0}^{\prime}$ is a Radon measure without singular continuous part, $U_{0}^{\prime}=\sum\limits_{j=1}^{p}\left[U_{0}(x_{j}^{+})-U_{0}(x_{j}^{-})\right]\delta_{x_{j}}+(U_{0}^{\prime})_{ac}$, and problems $(CL)$, $(HJ)$ are formally related by the equality $u=U_{x}$. In a forthcoming paper [4], problem $(HJ)$ will be studied in the context of viscosity solutions.

It is known ([2]) that $(i)$ the singular part $u_{s}$ of a suitably defined entropy solution may persist for some positive time (see [2, Theorem 3.5]) and $(ii)$ entropy solutions are not always uniquely determined by the initial condition $u_{0}$ (see also Remark 3.2). To overcome the latter problem, we introduced in [3] a so-called compatibility condition at those points where $u_{s}(\cdot,t)$ is a Dirac mass, and used it as a uniqueness criterion for nonnegative measure-valued solutions.

The starting point of the present paper is the statement that, for general signed initial measures $u_{0}$, the singular part $u_{s}$ of any local entropy solution $u$ of $(CL)$ (in the sense of Definition 3.2) satisfies a monotonicity result: both the positive and negative part of $u_{s}$, $[u(\cdot,t)]_{s}^{\pm}$, are nonincreasing with respect to $t$ (see Theorem 3.1). For the class of initial measures satisfying $(A_{1})$ this implies that the support of the singular part $u_{s}$ of an entropy solution of problem $(CL)$ is a subset of $F\times[0,T]$ and, in addition, that the sign of $[u(\cdot,t)]_{s}$ is determined by that of $u_{0s}$. Having this in mind it is rather straightforward to adapt the concept of compatibility condition in [3] to signed measure-valued solutions (see Definition 3.3).

The main result of the paper is that if $(A_{0})$ and $(A_{1})$ are satisfied, then $(CL)$ is well-posed in the class of entropy solutions which satisfy the compatibility condition at the $p$ points $x_{j}\in F$.

Existence of a solution is proven by a constructive approach which can be outlined as follows. By $(A_{0})$-$(A_{1})$ there exists a positive time $\tau$ until which all singularities persist (see [2, Theorem 3.5]), thus the real line is the disjoint union of $p+1$ intervals. In each interval we solve the initial-boundary value problem for the conservation law in $(CL)$, the initial data being the restriction of $u_{0r}$ to that interval, with “boundary conditions equal to infinity”. Namely, we consider the singular Dirichlet initial-boundary value problems

[TABLE]

with $j=2,\dots,p$, and

[TABLE]

[TABLE]

The choice between $u=\infty$ and $u=-\infty$ at $x_{j}$ is determined by the sign of $c_{j}$: we choose $\infty$ if $c_{j}>0$ and $-\infty$ if $c_{j}<0$. Existence and uniqueness of an entropy solution to each problem (1.1)-(1.3) is proven in Sections 5-6. In particular, existence follows from an approximation procedure which makes use of BV initial and boundary data, avoiding the $L^{\infty}$-theory of initial-boundary value problems developed in [11] (see Section 6).

The function determined by solutions of (1.1)-(1.3) in $\mathbb{R}\times(0,\tau)$ is, by definition, the regular part of a Radon measure, whose singular part is defined by observing that the variation of mass at each point $x_{j}$ depends on the sweeping effect of the flux across $x_{j}$ (see (7.2), (7.4), (7.6b) and Proposition 5.3). Then it is proven that this measure is the unique entropy solution of $(CL)$ (in the sense of Definition 3.2) which satisfies the compatibility conditions at all $x_{j}\in F$ until the time $t=\tau$. Here we use that the required compatibility condition for the solution of the Cauchy problem $(CL)$ at $x_{j}$ is exactly the entropic formulation of the boundary conditions ”$u=\pm\infty$” for the singular Dirichlet problems (see also Remark 5.3). If $\tau<T$ we iterate the procedure in $\mathbb{R}\times(\tau,T)$ with a smaller number a singularities, thus well-posedness of $(CL)$ follows in a finite number of steps (see Section 7). We observe that the proof of uniqueness of entropy solutions to problem $(CL)$ relies on a general comparison principle between entropy sub and super-solutions of (1.1)–(1.3) (see Definitions 5.2–5.5 and Theorem 5.2 below) which is independent of the above construction procedure. In this sense the comparison results are stronger than those in [3, Theorem 3.2].

The results in the paper can be esaily extended to the case that $u_{0s}$ is a locally finite superposition of Dirac masses (namely, if the number of Dirac masses in every bounded interval is finite).

2. Preliminaries

Let $\chi_{E}$ denote the characteristic function of $E\subseteq\mathbb{R}$. For every $u\in\mathbb{R}$ we set

[TABLE]

For every real function $f$ on $\mathbb{R}$ and $x_{0}\in\mathbb{R}$ we say that

[TABLE]

if there is a null set $E^{*}\subseteq\mathbb{R}$ such that $f(x_{n})\to l$ if $\left\{x_{n}\right\}\subseteq\mathbb{R}\setminus\!(E^{*}\!\cup\!\{x_{0}\})$, $x_{n}\to x_{0}^{\pm}$.

For every open subset $\Omega\subseteq\mathbb{R}$ we denote by $C_{c}(\Omega)$ the space of continuous real functions with compact support in $\Omega$ and by $\mathcal{M}^{+}(\Omega)$ the cone of the nonnegative Radon measures on $\Omega$. According to [6, Section 1.3], we say that $\nu$ is a (signed) Radon measure on $\Omega$ if there exist a (nonnegative) Radon measure $\mu\in\mathcal{M}^{+}(\Omega)$ and a locally $\mu$-summable function $f:\Omega\to[-\infty,\infty]$ such that

[TABLE]

for all compact sets $K\subset\Omega$. The space of (signed) Radon measures on $\Omega$ will be denoted by $\mathcal{M}(\Omega)$.

If $\mu,\nu\in\mathcal{M}(\Omega)$, we say that $\mu\leq\nu$ in $\mathcal{M}(\Omega)$ if $\nu-\mu\in\mathcal{M}^{+}(\Omega)$. We denote by $\left\langle\cdot,\cdot\right\rangle_{\Omega}$ the duality map between $\mathcal{M}(\Omega)$ and $C_{c}(\Omega)$. A sequence $\{\mu_{n}\}$ of Radon measures on $\mathbb{R}$ converges weakly* to a Radon measure $\mu$, $\mu_{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu$, if $\left\langle\mu_{n},\rho\right\rangle_{\mathbb{R}}\to\left\langle\mu,\rho\right\rangle_{\mathbb{R}}$ for all $\rho\in C_{c}(\mathbb{R})$. For any compact $K\in\mathbb{R}$ the space $\mathcal{M}(K)$ is a Banach space with norm $\|\mu\|_{\mathcal{M}(K)}:=|\mu|(K)$, where $|\mu|$ denotes the total variation of $\mu$. A sequence $\{\mu_{n}\}$ converges strongly to $\mu$ in $\mathcal{M}(K)$ if $\|\mu_{n}-\mu\|_{\mathcal{M}(K)}\to 0$ as $n\to\infty$. Similar definitions are used for Radon measures on any subset of $S:=\mathbb{R}\times(0,T)$.

Every $\mu\in\mathcal{M}(\mathbb{R})$ has a unique decomposition $\mu=\mu_{ac}+\mu_{s}$, with $\mu_{ac}\in\mathcal{M}(\mathbb{R})$ absolutely continuous and $\mu_{s}\in\mathcal{M}(\mathbb{R})$ singular with respect to the Lebesgue measure. We denote by $\mu_{r}\in L^{1}_{loc}(\mathbb{R})$ the density of $\mu_{ac}$. Every function $f\in L^{1}_{loc}(\mathbb{R})$ can be identified to an absolutely continuous Radon measure on $\mathbb{R}$; we shall denote this measure by the same symbol $f$ used for the function.

The restriction $\mu\,\lefthalfcup E$ of $\mu\in\mathcal{M}(\mathbb{R})$ to a Borel set $E\subseteq\mathbb{R}$ is defined by $(\mu\,\lefthalfcup E)(A):=\mu(E\cap A)$ for any Borel set $A\subseteq\mathbb{R}$. Similar notations are used for $\mathcal{M}(S)$.

For every open subset $\Omega\subseteq\mathbb{R}$ we denote by $BV(\Omega)$ the Banach space of functions of bounded variation in $\Omega$:

[TABLE]

where $z^{\prime}$ is the first order distributional derivative. The total variation in $\Omega$ of $z$ is $TV(z;\Omega):=\|z^{\prime}\|_{\mathcal{M}(\Omega)}$. We say that $z\in BV_{loc}(\mathbb{R})$ if $z\in BV(\Omega)$ for every open subset $\Omega\subset\subset\mathbb{R}$.

In the remainder of this section $\Omega$ denotes an open subset of $\mathbb{R}$, and $Q_{T}=\Omega\times(0,T)$. By $C([0,T];\mathcal{M}(\Omega))$ we denote the subset of strongly continuous mappings from $[0,T]$ into $\mathcal{M}(\Omega)$ - namely, $u\in C([0,T];\mathcal{M}(\Omega))$ if for all $t_{0}\in[0,T]$ and for every compact $K\in\Omega$ there holds $\|u(\cdot,t)-u(\cdot,t_{0})\|_{\mathcal{M}(K)}\to 0$ as $t\to t_{0}$.

Definition 2.1**.**

We denote by $L^{\infty}(0,T;\mathcal{M}^{+}(\Omega))$ the set of nonnegative Radon measures $u\in\mathcal{M}^{+}(Q_{T})$ such that for a.e. $t\in(0,T)$ there is a measure $u(\cdot,t)\in\mathcal{M}^{+}(\Omega)$ with the following properties:

$(i)$ if $\zeta\in C([0,T];C_{c}(\Omega))$ the map $t\mapsto\left\langle u(\cdot,t),\zeta(\cdot,t)\right\rangle_{\Omega}$ belongs to $L^{1}(0,T)$ and

[TABLE]

$(ii)$ the map $t\mapsto\|u(\cdot,t)\|_{\mathcal{M}(K)}$ belongs to $L^{\infty}(0,T)$ for every compact $K\subset\Omega$.

Remark 2.1**.**

Definition 2.1 implies that for all $\rho\in C_{c}(\Omega)$ the map $t\mapsto\left\langle u(\cdot,t),\rho\right\rangle_{\Omega}$ is measurable, thus the map $u:(0,T)\to\mathcal{M}^{+}(\Omega)$ is weakly* measurable. For simplicity we prefer the notation $L^{\infty}(0,T;\mathcal{M}^{+}(\Omega))$ to the more correct one $L^{\infty}_{w*}(0,T;\mathcal{M}^{+}(\Omega))$. Moreover, as a consequence of Definition 2.1-$(i)$, it can be seen that for every Borel set $E\subseteq Q_{T}$ the map t\to u(\cdot,t)\big{(}E^{t}\big{)} is Lebesgue measurable and there holds

[TABLE]

If $u\in L^{\infty}(0,T;\mathcal{M}^{+}(\Omega))$, then $u_{ac},\,u_{s}\in L^{\infty}(0,T;\mathcal{M}^{+}(\Omega))$ as well, and $u_{r}\in L^{\infty}(0,T;L^{1}_{\rm loc}(\Omega))$. Moreover, equality (2.1) implies

[TABLE]

Denoting by $[u(\cdot,t)]_{ac},\,[u(\cdot,t)]_{s}\in\mathcal{M}^{+}(\Omega)$ the absolutely continuous and the singular part of the measure $u(\cdot,t)\in\mathcal{M}^{+}(\Omega)$, a routine proof shows that for $a.e.$ $t\in(0,T)$

[TABLE]

where $[u(\cdot,t)]_{r}$ denotes the density of the measure $[u(\cdot,t)]_{ac}$. In view of (2.3), we shall always identify the quantities which appear on either side of equalities (2.3).

We say that a (signed) Radon measure $u\in\mathcal{M}(Q_{T})$ belongs to $L^{\infty}(0,T;\mathcal{M}(\Omega))$ if both $u^{+}$ and $u^{-}$ belong to $L^{\infty}(0,T;\mathcal{M}^{+}(\Omega))$. In particular, this implies that:

$(\alpha)$ the total variation $|u|$ of the measure $u$ belongs to $L^{\infty}(0,T;\mathcal{M}^{+}(\Omega))$;

$(\beta)$ conditions $(i)$ and $(ii)$ of Definition 2.1 hold with $u(\cdot,t):=u^{+}(\cdot,t)-u^{-}(\cdot,t)$ for $a.e.\ t\in(0,T)$.

Moreover, since $u^{+}$ and $u^{-}$ are mutually singular, it follows that for $a.e.$ $t$ the nonnegative measures $u^{+}(\cdot,t)$ and $u^{-}(\cdot,t)$ are mutually singular, whence

[TABLE]

and

[TABLE]

3. Results

For any $\tau\in(0,T]$ and open subset $\Omega\subseteq\mathbb{R}$ set $Q_{\tau}:=\Omega\times(0,\tau]$, $Q_{T}\equiv Q$; set also $S_{\tau}:=\mathbb{R}\times(0,\tau]$, $S_{T}\equiv S$. Solutions of problem $(CL)$ are meant in the following sense.

Definition 3.1**.**

Let $u_{0}$ be a signed Radon measure on $\Omega$ and let $(A_{0})$ be satisfied. A measure $u\in L^{\infty}(0,T;\mathcal{M}(\Omega))$ is a solution of problem $(CL)$ in $Q_{\tau}$ if for all $\zeta\in C^{1}([0,\tau];C^{1}_{c}(\Omega))$, $\zeta(\cdot,\tau)=0$ in $\Omega$ there holds

[TABLE]

Solutions of $(CL)$ in $S$ are simply referred to as “solutions of $(CL)$”.

Definition 3.2**.**

Let $u_{0}$ be a signed Radon measure on $\Omega$ and let $(A_{0})$ be satisfied. A solution of $(CL)$ in $Q_{\tau}$ is called an entropy solution in $Q_{\tau}$ if it satisfies the entropy inequality

[TABLE]

for all $\zeta\in C^{1}([0,\tau];C^{1}_{c}(\Omega))$, $\zeta\geq 0$, $\zeta(\cdot,\tau)=0$ in $\Omega$, and for all $k\in\mathbb{R}$;

If $Q_{\tau}\neq Q_{T}$, an (entropy) solution in $Q_{\tau}$ can be considered as a local (entropy) solution of $(CL)$. For general initial measures, local entropy solutions satisfy the following monotonicity result.

Theorem 3.1**.**

Let $(A_{0})$ be satisfied, let $u_{0}$ be a signed Radon measure on $\Omega$ and let $u$ be an entropy solution $u$ of problem $(CL)$ in $Q_{T}$. Then, for a.e. $0<t_{1}<t_{2}<T$, there holds

[TABLE]

Now we consider the case that $u_{0s}$ is the sum of a finite number of Dirac masses with support $F$.

Remark 3.1**.**

Let $(A_{0})-(A_{1})$ be satisfied and let $u$ be an entropy solution of problem $(CL)$ in $Q_{T}$. Arguing as in the proof of Proposition 3.20 in [2], it follows that $u\in C((0,T];\mathcal{M}(\Omega))$.

Corollary 3.2**.**

Let $(A_{0})-(A_{1})$ be satisfied and let $u$ be an entropy solution $u$ of problem $(CL)$ in $Q_{T}$. Then $u_{s}\in C([0,T];\mathcal{M}(\Omega))$, $[u(\cdot,0)]_{s}=u_{0s}$, (3.3) holds for any $0\leq t_{1}\leq t_{2}\leq T$ and for every $x_{j}\in F\cap\Omega$ there exists $t_{j}\in(0,T]$ such that

[TABLE]

We observe that the proof of Corollary 3.2 provides an explicit lower bound for $t_{j}$.

If $x_{j}\in F$ and $t_{j}\in(0,T]$ as in Corollary 3.2, Theorem 3.1 implies that the support of the singular part of any entropy solution is a subset of $F\times[0,T]$ and that the Delta mass at $x_{j}\in F$ does not change sign in the interval $[0,t_{j})$. Therefore we may formulate a compatibility condition at $x_{j}$ which depends on the sign of $c_{j}$, i.e. on the sign of the initial Delta mass at $x_{j}$:

Definition 3.3**.**

Let $(A_{0})-(A_{1})$ be satisfied. An entropy solution $u$ of $(CL)$ in $Q_{T}$ is said to satisfy the compatibility condition at $x_{j}\in F\cap\Omega$ if

[TABLE]

for all $\beta\in C^{1}_{c}(0,t_{j})$, $\beta\geq 0$ and $k\in\mathbb{R}$, where $t_{j}\in(0,T]$ is defined by Corollary 3.2.

We shall prove below (see Remark 5.4) that, if $(A_{0})$-$(A_{1})$ hold, for every entropy solution $u$ of $(CL)$ the limits

[TABLE]

with $\beta$ and $k$ as above, exist and are finite. Hence Definition 3.3 is well-posed.

The main result of the paper is the well-posedness of problem $(CL)$, if $u_{0}$ satisfies $(A_{1})$, in the class of entropy solutions in $C([0,T];\mathcal{M}(\mathbb{R}))$ which satisfy the compatibility condition in $\text{\rm supp}\,u_{0}$.

Theorem 3.3**.**

Let $(A_{0})$-$(A_{1})$ be satisfied. Then there exists a unique entropy solution of problem $(CL)$ which belongs to $C([0,T];\mathcal{M}(\mathbb{R}))$ and satisfies the compatibility condition at all $x_{j}\in\text{\rm supp}\,u_{0}$.

Remark 3.2**.**

It was already observed in [2] that in general measure-valued entropy solutions are not unique. This is essentially a consequence of the elementary observation that there exists a unique entropy solution for which $[u_{s}(t)]=u_{0s}$ for a.e. $t\in(0,T)$ (it is enough to set $u=u_{0}+\tilde{u}$, where $\tilde{u}$ is the entropy solution with initial data $u_{0r}$). But if $u_{0}$ satisfies $(A_{1})$ and $F\neq\emptyset$, one easily checks that if the function $H$, satisfying $(A_{0})$, is not constant in intervals of the type $(a,\infty)$ and $(-\infty,b)$, then such solution does not satisfy the compatibility condition at $x_{j}\in F$. In particular, it does not coincide with the solution defined by Theorem 3.3.

4. Monotonicity of $u_{s}$.

In this section we prove Theorem 3.1 and Corollary 3.2.

Proof of Theorem 3.1. By (3.1), for every $k\in\mathbb{R}$ we get

[TABLE]

for all $\rho\in C^{1}_{c}(\Omega)$ and $\beta\in C^{1}_{c}([0,T))$. By summing and subtracting the above equality from the entropy inequality (3.2), for every nonnegative $\rho$ and $\beta$ as above we obtain

[TABLE]

Letting $k\to\infty$ with ”+” and $k\to-\infty$ with ”-”, we obtain that

[TABLE]

Let $0<t_{1}<t_{2}\leq T$. By standard approximation arguments we can choose

[TABLE]

Arguing as in the proof of Proposition 3.8$(i)$ in [2], there exists a null set $N\in(0,T)$ which does not depend on the function $\rho$ such that, letting $n\to\infty$,

[TABLE]

Hence the first inequality in (3.3) follows from the arbitrariness of $\rho$.

The second inequality in (3.3) can be proved in a similar way, replacing $\beta_{n}$ by

[TABLE]

$\square$

Proof of Corollary 3.2. Arguing as in the proof of Theorem 3.1, for every $\rho\in C^{1}_{c}(\Omega)$ and $t\in(0,T]$ from (3.1) we get

[TABLE]

Fix any $x_{j}\in F\cap\Omega$. By standard approximation arguments we can choose in (4.8)

[TABLE]

Then letting $n\to\infty$, and observing that

[TABLE]

we obtain

[TABLE]

Since, by Theorem 3.1, $u_{s}(\cdot,t)=u_{s}(\cdot,t)\lefthalfcup F$ and $F$ contains $p$ points, we obtain that $\|u_{s}(\cdot,t)-u_{0s}\|\leq 2p\|H\|_{\infty}t$. Hence $u_{s}\in C([0,T];\mathcal{M}(\Omega))$ and $[u(\cdot,0)]_{s}=u_{0s}$ (see also Remark 3.1).

If $u_{0s}\lefthalfcup\{x_{j}\}=\pm u_{0s}^{\pm}\lefthalfcup\{x_{j}\}$, it follows from (3.3) that $[u(\cdot,t)]_{s}^{\mp}\lefthalfcup\{x_{j}\}=0$ for any $t\in[0,T]$. Then inequality (4.9) gives

[TABLE]

for any $t\in[0,t_{j})$, with $t_{j}:=\frac{u_{0s}^{\pm}\left(\{x_{j}\}\right)}{2\|H\|_{\infty}}$. Then by the monotonicity of the mappings $t\mapsto u_{s}^{\pm}(\cdot,t)$ (see (3.3)) the conclusion follows. $\square$

5. Problem $(D)$: comparison and uniqueness

As already said, to address $(CL)$ we need results concerning singular Dirichlet initial-boundary value problems for the scalar conservation law:

[TABLE]

where $\Omega=(a,b)$ is a bounded interval, $m_{1}=\pm\infty$, $m_{2}=\pm\infty$, and $u_{0}:\Omega\mapsto\mathbb{R}$. Similar problems will be considered also for half-lines, either $\Omega\equiv(a,\infty)$, or $\Omega\equiv(-\infty,b)$; obviously, the above condition at $\{b\}\times(0,T)$ is omitted when $\Omega\equiv(a,\infty)$, and that at $\{a\}\times(0,T)$ is omitted when $\Omega\equiv(-\infty,b)$.

We shall denote problem $(D)$ by $(D_{S})$ when $m_{1}=\pm\infty$, $m_{2}=\pm\infty$, or by $(D_{R})$ when both $m_{1}$ and $m_{2}$ are finite. When $\Omega=(a,b)$ problem $(D_{S})$ stands for four different initial-boundary value problems, which we denote by $(D_{+}^{+})$, $(D_{-}^{-})$, $(D_{+}^{-})$ and $(D_{-}^{+})$ according to the four choices $m_{1}=m_{2}=\infty$, $m_{1}=m_{2}=-\infty$, $m_{1}=\infty,m_{2}=-\infty$ and $m_{1}=-\infty,m_{2}=\infty$. In the case of half-lines problem $(D_{S})$ consists only of two cases, namely

[TABLE]

if $\Omega=(a,\infty)$, and

[TABLE]

if $\Omega=(-\infty,b)$. We shall write that a statement holds for problem $(D_{S})$, if it collectively holds for all problems $(D_{\pm}^{\pm})$.

The following definition concerns problem $(D_{R})$ (see [13]).

Definition 5.1**.**

Let $\Omega=(a,b)$, $u_{0}\in BV(\Omega)$.

$(i)$ An entropy subsolution of $(D_{R})$ is any $\underline{u}\in BV(Q)$ such that:

$(a)$ for every $k\in\mathbb{R}$ and for all $\zeta\in C^{1}([0,T];C^{1}_{c}(\Omega))$, $\zeta(\cdot,T)=0$ in $\Omega$, $\zeta\geq 0$ in $Q$,

[TABLE]

$(b)$ for a.e. $t\in(0,T)$ there holds

[TABLE]

[TABLE]

$(ii)$ An entropy supersolution of $(D_{R})$ is any $\overline{u}\in BV(Q)$ such that:

$(a^{\prime})$ for every $k\in\mathbb{R}$ and if $\zeta\in C^{1}([0,T];C^{1}_{c}(\Omega))$, $\zeta(\cdot,T)=0$ in $\Omega$, $\zeta\geq 0$ in $Q$,

[TABLE]

$(b^{\prime})$ for a.e. $t\in(0,T)$ there holds

[TABLE]

[TABLE]

$(iii)$ A function $\overline{u}\in BV(Q)\cap C([0,T];L^{1}(\Omega))$ is an entropy solution of $(D_{R})$ if it is both an entropy subsolution and an entropy supersolution.

When $\Omega=(a,\infty)$ entropy sub- and supersolutions of $(D_{R})$ are defined as above, only dropping conditions (5.3) and (5.6); similarly, conditions (5.2) and (5.5) are omitted if $\Omega=(-\infty,b)$. Moreover, in these cases we require that $\underline{u},\overline{u}$ belong to $BV_{loc}(Q)\cap L^{\infty}(Q)$.

Remark 5.1**.**

If $\underline{u},\overline{u}\in BV(Q)$, the traces $\underline{u}(a^{+},t):={\rm ess}\lim_{\xi\to a^{+}}\underline{u}(\xi,t)$, $\underline{u}(b^{-},t):={\rm ess}\lim_{\eta\to b^{-}}\underline{u}(\eta,t)$ exist for a.e. $t\in(0,T)$, and similarly for $\overline{u}$. Hence the above definitions are well-posed. By the same token, conditions (5.2)-(5.3) and (5.5)-(5.6) can be reformulated as follows: for every $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$,

[TABLE]

The following definitions for problem $(D_{S})$ are formulated for a wider class of initial data.

Definition 5.2**.**

Let $\Omega=(a,b)$, $u_{0}\in L^{1}(\Omega)$.

$(i)$ An entropy subsolution of $(D_{+}^{+})$ is any $\underline{u}\in C([0,T];L^{1}(\Omega))$ such that:

$(a)$ for every $k\in\mathbb{R}$ and $\zeta\in C^{1}_{c}(Q)$, $\zeta\geq 0$ in $Q$

[TABLE]

and for any interval $I\subseteq\Omega$

[TABLE]

$(ii)$ An entropy subsolution of $(D_{-}^{-})$ is any $\underline{u}\in C([0,T];L^{1}(\Omega))$ such that $(a)$ holds, and for every $k\in\mathbb{R}$, $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$,

[TABLE]

$(iii)$ An entropy subsolution of $(D_{+}^{-})$ is any $\underline{u}\in C([0,T];L^{1}(\Omega))$ such that $(a)$ holds, and for every $k\in\mathbb{R}$, $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$ inequality (5.10b) holds.

$(iv)$ An entropy subsolution of $(D_{-}^{+})$ is any $\underline{u}\in C([0,T];L^{1}(\Omega))$ such that $(a)$ holds, and for every $k\in\mathbb{R}$, $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$ inequality (5.10a) holds.

Definition 5.3**.**

Let $\Omega=(a,b)$, $u_{0}\in L^{1}(\Omega)$.

$(i)$ An entropy supersolution of $(D_{+}^{+})$ is any $\overline{u}\in C([0,T];L^{1}(\Omega))$ such that:

$(a^{\prime})$ for every $k\in\mathbb{R}$ and $\zeta\in C^{1}_{c}(Q)$, $\zeta\geq 0$ in $Q$

[TABLE]

and for any interval $I\subseteq\Omega$

[TABLE]

$(b^{\prime})$ for every $k\in\mathbb{R}$ and $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$,

[TABLE]

$(ii)$ An entropy supersolution of $(D_{-}^{-})$ is any $\overline{u}\in C([0,T];L^{1}(\Omega))$ such that $(a^{\prime})$ holds.

$(iii)$ An entropy supersolution of $(D_{+}^{-})$ is any $\overline{u}\in C([0,T];L^{1}(\Omega))$ such that $(a^{\prime})$ holds, and for every $k\in\mathbb{R}$, $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$, inequality (5.13a) holds.

$(iv)$ An entropy supersolution of $(D_{-}^{+})$ is any $\overline{u}\in C([0,T];L^{1}(\Omega))$ such that $(a^{\prime})$ holds, and for every $k\in\mathbb{R}$, $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$, inequality (5.13b) holds.

Definition 5.4**.**

A function $u\in C([0,T];L^{1}(\Omega))$ is called an entropy solution of $(D_{S})$ if it is both an entropy subsolution and an entropy supersolution of $(D_{S})$.

Observe that (5.13a)-(5.13b) can be regarded as limiting cases of (5.7c)-(5.7d), since for every $k\in\mathbb{R}$ there holds ${\rm sgn}_{-}(m_{i}-k)\to 0$ as $m_{i}\to\infty$ $(i=1,2)$. Similarly, (5.10a)-(5.10b) can be regarded as limiting cases of (5.7a)-(5.7b) as $m_{i}\to-\infty$.

Remark 5.2**.**

Let us prove that every entropy solution of $(D_{S})$ satisfies the weak formulation

[TABLE]

for every $\zeta\in C^{1}_{c}(Q)$.

To this aim, we fix any sequence $k_{j}\to-\infty$. By (5.8), for all $\zeta\in C^{1}_{c}(Q)$, $\zeta\geq 0$, there holds

[TABLE]

Let us take the limit as $j\to\infty$ in (5.15). Since $u\in L^{1}(Q)$ and $H$ is bounded, we have

[TABLE]

and

[TABLE]

(here we have used that $\left|\iint_{\{u\leq k_{j}\}}k_{j}\zeta_{t}\,dxdt\right|\leq\iint_{\{u\leq k_{j}\}}|u|\,|\zeta_{t}|\,dxdt\to 0$, as $k_{j}\to-\infty$). In view of (5.16)–(5.17), letting $j\to\infty$ in (5.15) gives

[TABLE]

for every $\zeta\in C^{1}_{c}(Q)$, $\zeta\geq 0$. Analogously, letting $k_{j}\to\infty$ in

[TABLE]

(see (5.11)) gives, for every $\zeta$ as above,

[TABLE]

Therefore the conclusion follows combining (5.18) and (5.20).

Remark 5.3**.**

The conditions (5.10a-5.10b) and (5.13a-5.13b) are entropy boundary conditions for singular Dirichlet problems and give a meaning, in a hyperbolic sense, to the boundary conditions ”$u=-\infty$” and ”$u=\infty$”. As already mentioned in the Introduction, they coincide with the compatibility conditions (3.5a) and (3.5b) for entropy solutions of (CL) at points $x_{j}$ where a signed Dirac mass is concentrated.

Remark 5.4**.**

Let $u$ denote either $\underline{u}$ in (5.8), or $\overline{u}$ in (5.11). Choosing $\zeta(x,t)=\alpha(x)\beta(t)$ with $\alpha\in C^{1}_{c}(\Omega)$, $\beta\in C^{1}_{c}(0,T)$, $\alpha,\beta\geq 0$, gives

[TABLE]

for any $k\in\mathbb{R}$ . Since $0\leq[u_{r}-k]_{\pm}\leq[u]_{\pm}+|k|$, from the above inequality we get

[TABLE]

for every $c\in\overline{\Omega}$. Hence the distributional derivative of the function

[TABLE]

is nonpositive. Therefore, the limits

[TABLE]

exist and are finite, thus the above definitions are well-posed.

The same statement can be applied to entropy solutions of $(CL)$, since they satisfy inequalities (5.21) in every domain $Q_{j}=(x_{j},x_{j+1})\times(0,T)$ ($j=1,\dots,p-1$), or $Q^{-}=(-\infty,x_{1})\times(0,T)$, $Q^{+}=(x_{p},\infty)\times(0,T)$ (recall that by Theorem 3.1 and assumption $(A_{1})$ the singular part of an entropy solution of $(CL)$ is not supported in these domains).

Remark 5.5**.**

Conditions (5.10a)-(5.10b) for subsolutions of $(D_{-}^{-})$ can be equivalently rewritten as follows: for all $k\in\mathbb{R}$ and $\beta$ as above and for a.e. $\xi,\eta\in(a,b)$

[TABLE]

Similarly, conditions (5.13a)-(5.13b) for supersolutions of $(D_{+}^{+})$ equivalently read: for all $k\in\mathbb{R}$ and for a.e. $\xi,\eta\in(a,b)$

[TABLE]

When $\Omega=(a,\infty)$ we have the following definition (we omit the formulation for the case $\Omega=(-\infty,b)$).

Definition 5.5**.**

Let $\Omega=(a,\infty)$, $u_{0}\in L^{1}_{loc}(\Omega)$.

$(i)$ An entropy subsolution of $(D_{+})$ is any $\underline{u}\in C([0,T];L^{1}_{loc}(\Omega))$ such that $(a)$ of Definition 5.2 holds.

$(ii)$ An entropy subsolution of $(D_{-})$ is any $\underline{u}\in C([0,T];L^{1}_{loc}(\Omega))$ such that $(a)$ of Definition 5.2 holds, and for every $k\in\mathbb{R}$, $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$ inequality (5.10a) holds.

$(iii)$ An entropy supersolution of $(D_{+})$ is any $\overline{u}\in C([0,T];L^{1}_{loc}(\Omega))$ such that $(a^{\prime})$ of Definition 5.3 holds, and for every $k\in\mathbb{R}$, $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$, inequality (5.13a) holds.

$(iv)$ An entropy supersolution of $(D_{-})$ is any $\overline{u}\in C([0,T];L^{1}_{loc}(\Omega))$ such that $(a^{\prime})$ of Definition 5.3 holds.

$(v)$ A function $u\in C([0,T];L^{1}_{loc}(\Omega))$ is called an entropy solution of $(D_{S})$ if it is both an entropy subsolution and an entropy supersolution of $(D_{S})$.

Comparison and uniqueness results for problem $(D_{R})$ are given by the following theorem (see [13, Theorem 1.1]).

Theorem 5.1**.**

Let $\Omega=(a,b)$. Let $u_{0},v_{0}\in BV(\Omega)$, and $m_{1},m_{2},n_{1},n_{2}\in\mathbb{R}$. Let $\underline{u}$ be an entropy subsolution of $(D_{R})$, and $\overline{v}$ be an entropy supersolution of $(D_{R})$ with $u_{0}$, $m_{1}$ and $m_{2}$ replaced by $v_{0}$, $n_{1}$ and $n_{2}$. Then for a.e. $t\in(0,T)$

[TABLE]

Similar results hold for $\Omega=(a,\infty)$ and $\Omega=(-\infty,b)$ if $u_{0},v_{0}\in BV_{loc}(\Omega)\cap L^{\infty}(\Omega)$. In these cases for a.e. $t\in(0,T)$ there holds

[TABLE]

for every $R>a$ if $\Omega=(a,\infty)$, respectively

[TABLE]

for every $R<b$ if $\Omega=(-\infty,b)$. Therefore, in all cases there exists at most one solution of $(D_{R})$.

As for problem $(D_{S})$, the following holds.

Theorem 5.2**.**

Let $(A_{0})$ hold. Let $\underline{u},\,\overline{u}$ be an entropy sub- and supersolution of $(D_{S})$ with the same boundary conditions. Then $\underline{u}\leq\overline{u}$ a.e. in $Q$. In particular, there exists at most one entropy solution of $(D_{S})$.

Proof.

We only give the proof for $(D_{+}^{-})$, as in the other cases of $(D_{S})$ it is similar. We use the Kružkov doubling method adapted to boundary valued problems (see [3, 10, 11, 12]). Let $\rho_{\epsilon}$ $(\epsilon>0)$ be a symmetric mollifier in $\mathbb{R}$, and set

[TABLE]

with $\sigma_{1}\in C^{1}_{c}(\Omega)$, $\sigma_{2}\in C^{1}_{c}(0,T)$, $\sigma_{1}\geq 0$, $\sigma_{2}\geq 0$. From (5.11) and (5.8) we get

[TABLE]

[TABLE]

Recalling that $[u]_{+}=[-u]_{-}$ and $\text{\rm sgn}\,_{+}(u)=-\text{\rm sgn}\,_{-}(-u)$ ($u\in\mathbb{R}$), we sum the above inequalities integrated over $Q$ :

[TABLE]

Set

[TABLE]

[TABLE]

Observe that the difference between $I_{1}$ and the first term in (5.27) vanishes as $\epsilon_{1}\to 0^{+}$; the same holds for the difference between $I_{2}$ and the second term in (5.27).

Let $a<\xi<\eta<b$ be fixed. By standard approximation arguments we can choose

[TABLE]

where $n\in\mathbb{N}$ and $y\in\Omega$, thus

[TABLE]

With this choice of $\sigma_{1}$, $I_{2}$ reads

[TABLE]

By (5.24a) and (5.23b), from (5.28) we obtain for a.e. $\xi,\eta\in(a,b)$

[TABLE]

As $\epsilon_{1}\to 0^{+}$ we obtain that

[TABLE]

Clearly, there holds

[TABLE]

On the other hand, since $\int_{0}^{T}|\underline{u}(\xi,s)|\,ds\geq 0$ and the map $\xi\to\int_{0}^{T}|\underline{u}(\xi,s)|\,ds$ belongs to $L^{1}(\Omega)$, there holds

[TABLE]

• for, otherwise there would exist $c,\delta>0$ such that $(\xi-a)\!\!\int_{0}^{T}|\underline{u}(\xi,s)|\,ds\geq c$ for a.e. $\xi\in(a,a+\delta)$, thus $\xi\to\int_{0}^{T}|\underline{u}(\xi,s)|\,ds\not\in L^{1}(\Omega)$. Therefore, for every $n\in\mathbb{N}$ there exist $\delta_{n}>0$ and $E_{n}\subseteq(a,a+\delta_{n})$, $|E_{n}|>0$, such that $(\xi-a)\!\int_{0}^{T}|\underline{u}(\xi,s)|\,ds<1/n$ for a.e. $\xi\in E_{n}$. It follows that a sequence $\{\xi_{n}\}\subseteq\Omega$ exists, such that:

$(i)$ $\xi_{n}$ is a Lebesgue point of $\int_{0}^{T}|\underline{u}(\xi,s)|\,ds$,

$(ii)$ $\xi_{n}\to a^{+}$ as $n\to\infty$, and $(\xi_{n}-a)\!\int_{0}^{T}|\underline{u}(\xi_{n},s)|\,ds<\frac{1}{n}$ for all $n\in\mathbb{N}$ .

Similarly, there holds

[TABLE]

hence there exists $\{\eta_{n}\}\subseteq\Omega$, $\eta_{n}\to b^{-}$ as $n\to\infty$, with properties analogous to $(i)$-$(ii)$ above. Then writing (5.29) with $\xi=\xi_{n}$, $\eta=\eta_{n}$ and letting $n\to\infty$ we obtain that the right-hand side of (5.29) goes to zero.

To sum up, following the above procedure and letting $\epsilon_{2}\to 0^{+}$ from (5.27), we get for any $\sigma_{2}\in C^{1}_{c}(0,T)$, $\sigma_{2}\geq 0$,

[TABLE]

Let $0<t_{1}<t_{2}\leq T$ be fixed. By standard approximation arguments we can choose

[TABLE]

Then from (5.30) we get for all $n$

[TABLE]

whence as $n\to\infty$

[TABLE]

Since $\underline{u},\overline{u}\in C([0,T];L^{1}(\Omega))$, as $t_{1}\to 0^{+}$ by (5.9) and (5.12) there holds

[TABLE]

This proves the result. ∎

For future reference we prove the following generalization of [3, Lemma 4.4].

Proposition 5.3**.**

$(i)$* Let $u$ be an entropy solution either of $(D_{+}^{+})$, or of $(D_{+}^{-})$. Then there exists $f_{a}^{+}\in L^{\infty}(0,T)$ such that*

[TABLE]

for every $\beta\in C^{1}_{c}(0,T)$, and

[TABLE]

$(ii)$* Let $u$ be an entropy solution either of $(D_{-}^{+})$, or of $(D_{-}^{-})$. Then there exists $f_{a}^{-}\in L^{\infty}(0,T)$ such that*

[TABLE]

for every $\beta\in C^{1}_{c}(0,T)$, and

[TABLE]

$(iii)$* Let $u$ be an entropy solution either of $(D_{-}^{+})$, or of $(D_{+}^{+})$. Then there exists $f_{b}^{+}\in L^{\infty}(0,T)$ such that*

[TABLE]

for every $\beta\in C^{1}_{c}(0,T)$, and

[TABLE]

$(iv)$* Let $u$ be an entropy solution either of $(D_{+}^{-})$, or of $(D_{-}^{-})$. Then there exists $f_{b}^{-}\in L^{\infty}(0,T)$ such that*

[TABLE]

for every $\beta\in C^{1}_{c}(0,T)$, and

[TABLE]

Proof.

The existence of the limits in the left-hand side of (5.31), (5.33), (5.35) and (5.37) follows from (5.22a)-(5.22b), since for $a.e.$ $x\in\Omega$ there holds

[TABLE]

(recall that $H(0)=0$). On the other hand, for every sequence $\{x_{n}\}$, $x_{n}\to a^{+}$, the sequence $\{H(u(x_{n},\cdot))\}$ is bounded in $L^{\infty}(0,T)$. Hence there exist a subsequence $\{x_{n_{k}}\}\subseteq\{x_{n}\}$ and a function $f_{a}^{+}\in L^{\infty}(0,T)$ (independent of $\{x_{n_{k}}\}$) such that $H(u(x_{n_{k}},\cdot))\stackrel{{\scriptstyle*}}{{\rightharpoonup}}f_{a}^{+}$ in $L^{\infty}(0,T)$, thus (5.31) follows. Equalities (5.33), (5.35) and (5.37) are similarly proven.

Let us prove (5.32). Clearly, there holds $f_{a}^{+}(t)\leq\sup_{u\in\mathbb{R}}H(u)$ for a.e. $t\in(0,T)$. To prove the first inequality, let us choose in (5.8) $\zeta(x,t)=\rho(x)\beta(t)$ with $\rho\in C^{1}_{c}([a,b))$, $\rho\geq 0$, $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$. By standard arguments we can also choose $\rho=\alpha\sigma_{\epsilon}$ with $\alpha\in C^{1}_{c}([a,b))$, $\alpha\geq 0$, and

[TABLE]

Then for every $k\in\mathbb{R}$ we obtain that

[TABLE]

Letting $\epsilon\to 0^{+}$ and using (5.13a) and (5.31), we get that for every $k\in\mathbb{R}$

[TABLE]

Letting $k\to\infty$ in the above inequality gives

[TABLE]

whence by the arbitrariness of $\beta$ inequality (5.32) follows.

To prove (5.34) we argue as for (5.32), using inequality (5.10a), (5.11) and (5.33) instead of (5.8), (5.13a) and (5.31). Then we get for every $k\in\mathbb{R}$

[TABLE]

As $k\to-\infty$ in the above inequality, by the arbitrariness of $\beta$ we obtain

[TABLE]

thus (5.34) follows. The proof of (5.36) and (5.38) is similar to that of (5.32) and (5.34), using

[TABLE]

instead of (5.39); we leave the details to the reader. ∎

Finally we prove the following result.

Lemma 5.4**.**

Let $u$ be an entropy solution of $(D_{R})$. Then for every $t\in(0,T]$

[TABLE]

Proof.

By (5.1) and (5.4) there holds

[TABLE]

for every $k\in\mathbb{R}$ and $\zeta$ as above. By standard arguments we can choose $\zeta(x,s)=\alpha_{p}(x)\beta_{q}(s)$ with

[TABLE]

[TABLE]

for any fixed $t\in(0,T]$ and $p,q\in\mathbb{N}$ sufficiently large. Then for $k=0$ as $q\to\infty$ we get

[TABLE]

whence as $p\to\infty$ (5.41) follows. ∎

6. Problem $(D)$: existence

Let us recall the following result (see [1, 13]).

Theorem 6.1**.**

Let $\Omega=(a,b)$, and let $u_{0}\in BV(\Omega)$. Then there exists a unique entropy solution $u\in BV(Q)\cap C([0,T];L^{1}(\Omega))$ of problem $(D_{R})$. Moreover,

[TABLE]

The same holds for  $\Omega=(a,\infty)$ and $\Omega=(-\infty,b)$ with $u_{0}\in BV_{loc}(\Omega)\cap L^{\infty}(\Omega)$, ${\rm supp}\,u_{0}$ compact. In these cases there holds $u\in BV_{loc}(Q)\cap L^{\infty}(Q)\cap C([0,T];L^{1}_{loc}(\Omega))$, and inequality (6.1) is replaced by

[TABLE]

if $\Omega=(a,\infty)$, respectively by $\|u\|_{L^{\infty}(Q)}\leq\max\left\{|m_{2}|,\|u_{0}\|_{L^{\infty}(\Omega)}\right\}$ if $\Omega=(-\infty,b)$.

The above uniqueness claim follows from Theorem 5.1. Let us outline the proof of the existence part; we limit ourselves to the case $\Omega=(a,\infty)$, the proof being the same for $\Omega=(-\infty,b)$ and easier for $\Omega=(a,b)$.

Let $f_{1,\epsilon},f_{2,\epsilon}\in C^{\infty}(\mathbb{R})$ ($0<\epsilon<1$) be a partition of unity:

[TABLE]

such that, for $i=1,2$,

[TABLE]

Let $u_{0}\in BV_{loc}(\Omega)\cap L^{\infty}(\Omega)$ have compact support. Set

[TABLE]

where $\{\sigma_{\epsilon}\}$ is a family of standard mollifiers with supp$\,\sigma_{\epsilon}\subseteq[-\sqrt{\epsilon},\sqrt{\epsilon}]$. Then there holds $u_{0\epsilon}\in C^{\infty}(\mathbb{R})$, $u_{0\epsilon}=m_{1}$ in $[a,a+\sqrt{\epsilon}]$, ${\rm supp}\,u_{0\epsilon}$ compact. Moreover,

[TABLE]

[TABLE]

[TABLE]

Let $H\in W^{1,\infty}(\mathbb{R})$, $H(0)=0$. Set

[TABLE]

where the family $\{g_{\epsilon}\}\in C^{\infty}_{c}(\mathbb{R})$ satisfies $g_{\epsilon}=1$ in $(-1/\epsilon,1/\epsilon)$, $0\leq g_{\epsilon}(x)\leq 1$ in $\mathbb{R}$, supp$\,g_{\epsilon}\subseteq(-2/\epsilon,2/\epsilon)$. It is easily seen that

[TABLE]

Let $u_{\epsilon}\in C^{2,1}(\overline{Q})$ be the unique classical solution of the parabolic problem

[TABLE]

with $m_{1}\in\mathbb{R}$, $u_{0\epsilon}$ and $H_{\epsilon}$ as above ($e.g.$, see [8]).

Lemma 6.2**.**

There holds

[TABLE]

and there exists $c>0$ only depending on $m_{1}$, $TV(u_{0};\Omega)$, and $\|H\|_{W^{1,\infty}(\mathbb{R})}$) such that

[TABLE]

[TABLE]

[TABLE]

Proof.

Inequality (6.8) follows by the maximum principle and (6.4). Arguing as in the proof of [13, Proposition 3.1] (see also [1]) and using (6.5) gives (6.9)-(6.10). As for (6.11), integrating the first equation of $(D_{\epsilon})$ over $(a,x)$ gives

[TABLE]

whence

[TABLE]

Integrating the above inequality over $(a,a+1)$ and using (6.9)-(6.10) we get

[TABLE]

for some $\tilde{c}>0$ independent of $\epsilon$. Then by (6.9)-(6.10) and (6.12)-(6.13) the estimate in (6.11) follows. ∎

Proof of Theorem 6.1. By estimates (6.8)-(6.10) the family $\{u_{\epsilon}\}$ is bounded in $L^{\infty}(Q)$, and there exists $M>0$ (only depending on $m_{1}$, $TV(u_{0};\Omega)$, $\|H\|_{W^{1,\infty}(\mathbb{R})}$) such that

[TABLE]

Then by embedding theorems there exist a sequence $\{u_{\epsilon_{n}}\}\subseteq\{u_{\epsilon}\}$ and a function $u\in BV(Q)\cap C([0,T];L^{1}(\Omega))$ such that

[TABLE]

Arguing as in [1] shows that $u$ is an entropy solution of problem $(D_{R})$, In fact, let $E,F_{\epsilon}:\mathbb{R}\to\mathbb{R}$, $E\in C^{2}(\mathbb{R})$, $F_{\epsilon}\in C^{1}(\mathbb{R})$ and $F^{\prime}_{\epsilon}=E^{\prime}H^{\prime}_{\epsilon}$ . Multiplying the first equation in $(D_{\epsilon})$ by $E^{\prime}(u_{\epsilon})\,\zeta$ gives for any $\zeta\in C^{1}([0,T];C^{1}_{c}(\overline{\Omega}))$, $\zeta(\cdot,T)=0$ in $\Omega$,

[TABLE]

By standard regularization arguments we can choose in (6) $E(u_{\epsilon})=[u_{\epsilon}-k]_{\pm}$, thus obtaining for all $k\in\mathbb{R}$ and $\zeta$ as above, $\zeta\geq 0$,

[TABLE]

If $\zeta(\cdot,t)\in C^{1}_{c}(\Omega)$ for all $t\in(0,T)$, from (6.16) we obtain

[TABLE]

On the other hand, choosing in (6.16) $\zeta(x,t)=\chi_{[a,\,\xi+1/n)}(x)\beta(t)$ $(\xi\in\Omega,n\in\mathbb{N})$ with $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$, and letting $n\to\infty$ plainly gives for every $k\in\mathbb{R}$

[TABLE]

Multiplying the first equation of $(D_{\epsilon})$ by $\zeta(x,t)=\chi_{[a,\,\xi+1/n)}(x)\beta(t)$ and letting $n\to\infty$, one easily sees that

[TABLE]

From (6.18)-(6.19) we get

[TABLE]

By (6.7), (6.9) and (6.14) we can take the limit as $\epsilon_{n}\to 0^{+}$ in (6.17) and (6.20) (written with $\epsilon=\epsilon_{n}$). It follows that the function $u$ in (6.14) satisfies the following inequalities:

• for every $k\in\mathbb{R}$ and for all $\zeta\in C^{1}([0,T];C^{1}_{c}(\Omega))$, $\zeta(\cdot,T)=0$ in $\Omega$, $\zeta\geq 0$ in $Q$,

[TABLE]

• for every $k\in\mathbb{R}$ and $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$ and for a.e. $\xi\in\Omega$,

[TABLE]

Letting $\xi\to a^{+}$ in the latter inequality and using Remark 5.1 we conclude that $u$ is an entropy solution of $(D_{R})$. Hence the result follows. $\square$

Remark 6.1**.**

In the proof of Theorem 6.1 when $\Omega=(a,b)$ one uses the family of solutions of the problem

[TABLE]

with $m_{1},m_{2}\in\mathbb{R}$, $H_{\epsilon}$ as above and $u_{0\epsilon}$ defined by a suitable partition of unity; we leave the details to the reader.

Concerning $(D_{S})$ the following holds.

Theorem 6.3**.**

Let $(A_{0})$ hold. When $\Omega=(a,b)$ for any $u_{0}\in L^{1}(\Omega)$ there exists an entropy solution of $(D_{S})$. The same holds for any $u_{0}\in L^{1}_{loc}(\Omega)$ if $\Omega=(a,\infty)$, or $\Omega=(-\infty,b)$.

Proof.

Let $\Omega=(a,b)$. Let us prove the result for $(D^{-}_{+})$, the proof being the same for $(D^{+}_{+})$, $(D^{-}_{-})$ and $(D^{+}_{-})$. Let $u_{0}\in BV(\Omega)$. By Theorem 6.1, for all $n,p\in\mathbb{N}$ there exists an entropy solution $u_{n,-p}\in BV(Q)\cap C([0,T];L^{1}(\Omega))$ of problem $(D_{R})$ with $m_{1}=n$, $m_{2}=-p$. In particular, there holds:

$(a)$ by (5.8)-(5.9) and (5.11)-(5.12), for every $k\in\mathbb{R}$ and for all $\zeta\in C^{1}([0,T];C^{1}_{c}(\Omega))$, $\zeta(\cdot,T)=0$ in $\Omega$, $\zeta\geq 0$ in $Q$,

[TABLE]

$(b)$ by (5.24a), for every $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$, for $a.e.$ $\xi\in\Omega$ and for all $k<n$ and $p\in\mathbb{R}$,

[TABLE]

$(d)$ by (5.41), for every $t\in(0,T)$

[TABLE]

Moreover, by inequality (5.25), for all $n,p\in\mathbb{N}$ there holds a.e. in $Q$

[TABLE]

Let $p\in\mathbb{R}$ be fixed. By (6.23) and (6.24a) there exists $u_{\infty,-p}\in L^{\infty}(0,T;L^{1}(\Omega))$ such that

[TABLE]

Then letting $n\to\infty$ in (6.21) gives

[TABLE]

whereas from (6.22a)-(6.22b) we get, for every $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$, for $a.e.$ $\xi,\eta\in\Omega$ and for all $k,p\in\mathbb{R}$:

[TABLE]

Moreover, from (6.23) and (6.24b) we obtain

[TABLE]

[TABLE]

By (6.28)-(6.29) there exists $u_{\infty,-\infty}\in L^{\infty}(0,T;L^{1}(\Omega))$ such that

[TABLE]

Then letting $p\to\infty$ in (6.26) shows that $u_{\infty,-\infty}$ satisfies (5.8) and (5.11). In addition, letting $p\to\infty$ in (6.27a)-(6.27b) proves that $u_{\infty,-\infty}$ satisfies (5.10b) and (5.13a) for every $k\in\mathbb{R}$ (see Remark 5.5). By Remark 3.1 and arguing as in the proof of [2, Proposition 3.20], it can be checked that $u_{\infty,-\infty}\in C([0,T];L^{1}(\Omega))$ and, by construction, $u_{\infty,-\infty}(\cdot,0)=u_{0}$ in $\mathcal{M}(\Omega)$. Therefore (5.9) and (5.12) follow as well, and $u_{\infty,-\infty}$ is an entropy solution of $(D_{+}^{-})$.

It remains to remove the assumption $u_{0}\in BV(\Omega)$. To this purpose, let $v_{0}\in BV(\Omega)$, and let $v_{\infty,-\infty}$ be the entropy solution of $(D_{+}^{-})$ with initial data $v_{0}$ constructed by the above procedure (and with the same boundary conditions considered in the construction of $u_{\infty,-\infty}$). Then by (5.25) there holds

[TABLE]

Let $u_{0}\in L^{1}(\Omega)$, let $\{u_{0j}\}\subseteq BV(\Omega)$ be any sequence such that $u_{0j}\to u_{0}$ in $L^{1}(\Omega)$. Let $\{u_{j}\}\equiv\{(u_{\infty,-\infty})_{j}\}$ be the sequence of entropy solutions to problem $(D_{+}^{-})$ constructed as above, with initial data $u_{0j}$. Then for all $j\in\mathbb{N}$:

$(a)$ for every $k\in\mathbb{R}$ and for all $\zeta\in C^{1}([0,T];C^{1}_{c}(\Omega))$, $\zeta(\cdot,T)=0$ in $\Omega$, $\zeta\geq 0$ in $Q$,

[TABLE]

$(b)$ for all $\beta\in C^{1}_{c}(0,T)$, $\beta\geq 0$, for all $k\in\mathbb{R}$ and for $a.e.$ $\xi,\eta\in\Omega$:

[TABLE]

By (6.31) there holds

[TABLE]

thus there exists $u\in L^{1}(Q)$ such that $u_{j}\to u$ in $L^{1}(Q)$ as $j\to\infty$. As before, there holds $u\in C([0,T];L^{1}(\Omega))$. Then letting $j\to\infty$ in (6.32) and (6.33a)-(6.33b) the result for $(D_{+}^{-})$ follows. The other cases of $(D_{S})$ can be dealt with similarly, hence the conclusion follows if $\Omega=(a,b)$.

The above arguments easily extend to the case of half-lines. For instance, let $\Omega=(a,\infty)$, $m_{1}=\infty$ and $u_{0}\in BV(\Omega)\cap L^{\infty}(\Omega)$. Then by Theorem 6.1 and inequality (5.26) there exists a sequence $\{u_{n}\}$ of entropy solutions of $(D_{R})$, such that for every $t\in(0,T)$ $\|u_{n}(\cdot,t)\|_{L^{1}(\Omega)}\leq\|u_{0}\|_{L^{1}(\Omega)}+2\,\|H\|_{\infty}t$, and $u_{n}\leq u_{n+1}$ a.e. in $Q$ for all $n\in\mathbb{N}$ . Then letting $n\to\infty$ we obtain an entropy solution $u_{\infty}$ of $(D_{+})$ in this case. Moreover, if $v_{0}\in BV(\Omega)\cap L^{\infty}(\Omega)$ and $v_{\infty}$ is the corresponding entropy solution of $(D_{+})$ with initial data $v_{0}$ constructed as before, by (5.26) there holds

[TABLE]

Now let $u_{0}\in L^{1}_{loc}(\Omega)$, and let $\{u_{0j}\}\subseteq BV(\Omega)\cap L^{\infty}(\Omega)$, ${\rm supp}\,u_{0j}$ compact, $u_{0j}\to u_{0}$ in $L^{1}_{loc}(\Omega)$. Let $\{u_{j}\}\equiv\{(u_{\infty})_{j}\}$ be the sequence of entropy solutions of $(D_{+})$ constructed as above, with initial data $u_{0j}$. By (6.34) $\{u_{j}\}$ is a Cauchy sequence in $L^{\infty}(0,T;L^{1}(K))$ for every compact subset $K\subset\Omega$. Then by a diagonal argument the conclusion easily follows. ∎

7. Well-posedness of problem $(CL)$

In this section we prove Theorem 3.3.

We first prove the existence claim. Rewrite $(A_{1})$ as follows:

[TABLE]

For every $j=1,\dots,p$ such that $c_{j}>0$ we set

[TABLE]

with $f_{x_{j}^{+}}^{+}$ satisfying (5.31) (written with $x_{j}^{+}$ instead of $a$) and $f_{x_{j}^{-}}^{+}$ satisfying (5.35) (written with $x_{j}^{-}$ instead of $b$); observe that by (5.32) and (5.36) there holds

[TABLE]

Similarly, for every $j=1,\dots,p$ such that $c_{j}<0$ we set

[TABLE]

with $f_{x_{j}^{+}}^{-}$ satisfying (5.33) (written with $x_{j}^{+}$ instead of $a$) and $f_{x_{j}^{-}}^{-}$ satisfying (5.37) (written with $x_{j}^{-}$ instead of $b$); observe that by (5.34) and (5.38) there holds

[TABLE]

Let $\bar{t}_{j}:=\sup\{t\in[0,T]\,|\,C_{j}^{\pm}(t)>0\}>0$ $(j=1,\dots,p)$. Then $\bar{t}_{j}>0$ since $C_{j}^{\pm}(0)=\pm c_{j}>0$. By (7.3) and (7.5) $C_{j}^{\pm}$ is nonincreasing in $(0,T)$, whence $C_{j}^{\pm}>0$ in $[0,\bar{t}_{j})$ and, if $\bar{t}_{j}<T$, there holds $C_{j}^{\pm}=0$ in $[\bar{t}_{j},T]$. Let $\tau_{1}:=\min\,\{\bar{t}_{1},\dots,\bar{t}_{p}\}$, and define $u\in C([0,\tau_{1}];\ \mathcal{M}(\mathbb{R}))$ as follows:

[TABLE]

By Definitions 3.2 and 5.2-5.4 $u$ is an entropy solution of $(CL)$ in $Q_{j,\tau_{1}}$ for $j=1,\dots,p+1$. Hence $u$ is an entropy solution of $(CL)$ in $S_{\tau_{1}}$, if we prove (3.1)-(3.2) with $\Omega=\mathbb{R}$, $\tau=\tau_{1}$ for all $\zeta\in C^{1}([0,\tau_{1}];C^{1}_{c}(\mathbb{R}))$, $\zeta\geq 0$, $\zeta(\cdot,\tau_{1})=0$ in $\mathbb{R}$, such that

[TABLE]

We only give the proof when $\zeta(x,t)=\alpha(x)\beta(t)$ with $\alpha\in C^{1}_{c}(\mathbb{R})$, $\alpha\geq 0$, $\alpha(x_{j})>0$ for a unique $j\in\{1,\dots,p\}$, and $\beta\in C^{1}([0,\tau_{1}])$, $\beta\geq 0$, $\beta(\tau_{1})=0$ (the general case can be dealt with similarly). We also assume $c_{j}>0$, since the proof is similar for $c_{j}<0$. Let us first prove (3.1) in this case, namely

[TABLE]

for all $\zeta$ as above. From (7.2) we obtain

[TABLE]

On the other hand, since $u$ is a solution of $(CL)$ in $Q_{j+1,\tau_{1}}$, by (3.1) there holds

[TABLE]

for all $k\in\mathbb{R}$ and $\xi\in C^{1}([0,\tau_{1}];C^{1}_{c}(I_{j+1}))$, $\xi(\cdot,\tau_{1})=0$ in $I_{j+1}$. Let $\eta_{q}$ be defined by

[TABLE]

and let $\zeta\in C^{1}([0,\tau_{1}];C^{1}_{c}([x_{j},x_{j+1})))$, $\zeta(\cdot,\tau_{1})=0$ in $I_{j+1}$ (here $x_{j+1}=\infty$ if $j=p$). By standard arguments we can choose $\xi=\zeta\eta_{q}$ in the above equality. Then we get

[TABLE]

Letting $q\to\infty$ in the above equality plainly gives (see (5.31) and (7.6a)):

[TABLE]

Since $u$ is an entropy solution of $(CL)$ in $Q_{j+1,\tau_{1}}$, arguing as before we obtain

[TABLE]

for all $\zeta$ as above, $\zeta\geq 0$. Choosing $\zeta(x,t)=\alpha(x)\beta(t)$ with $\alpha\in C^{1}_{c}([x_{j},x_{j+1}))$, $\beta\in C^{1}([0,\tau_{1}])$, $\alpha\geq 0,\,\beta\geq 0$ and $\beta(\tau_{1})=0$, by (5.13a) from the above inequality we obtain

[TABLE]

since $\text{\rm sgn}\,(u)=1+2\,\text{\rm sgn}\,_{-}(u)$.

Replacing $Q_{j+1,\tau_{1}}$ by $Q_{j,\tau_{1}}$ we obtain, similarly to (7.9)-(7.10),

[TABLE]

[TABLE]

Summing (7.9) and (7.11) gives

[TABLE]

Then equality (7) follows from (7.8) and (7.13).

Next we prove (3.2) for all $\zeta$ as above, namely

[TABLE]

Since $u$ is an entropy solution of $(CL)$ in $Q_{j,\tau_{1}}$ and $Q_{j+1,\tau_{1}}$, from (5.31), (5.35), (7.10) and (7.12) it follows that

[TABLE]

Since $|u_{0s}|\lefthalfcup(x_{j-1},x_{j+1})=u_{0s}\lefthalfcup(x_{j-1},x_{j+1})$ (recall that $c_{j}>0$ by assumption) and by (7.6b) $|u_{s}(\cdot,t)|\lefthalfcup(x_{j-1},x_{j+1})=u_{s}(\cdot,t)\lefthalfcup(x_{j-1},x_{j+1})$ for all $t\in[0,T]$, the above inequality together with (7.8) implies (7). Therefore, the measure $u$ defined by (7.6) is an entropy solution of $(CL)$ in $S_{\tau_{1}}$.

If $\tau_{1}<T$, either $u_{s}(\cdot,\tau_{1})=0$, or $u_{s}(\cdot,\tau_{1})\neq 0$. If $u_{s}(\cdot,\tau_{1})=0$, there holds $C_{j}^{\pm}(\tau_{1})=0$ for all $j=1,\dots,p$, thus $u_{s}(\cdot,t)=0$ for all $t\in[\tau_{1},T]$. Then, by the standard theory of scalar conservation laws, we can continue the solution of $(CL)$ in $(\tau_{1},T]$ with initial data $u_{r}(\cdot,\tau_{1})$. On the other hand, if $u_{s}(\cdot,\tau_{1})\neq 0$, then $C_{j}^{\pm}(\tau_{1})\neq 0$ for some $j=1,\dots,p$ and, arguing as before, we can continue the solution of $(CL)$ in $(\tau_{1},\tau_{2}]$, with initial data $u(\cdot,\tau_{1})$, for some $\tau_{2}\in(\tau_{1},T]$. Iterating the procedure $q$ times with $2\leq q\leq p$, we obtain that either $\tau_{q}=T$, or $u_{s}(\cdot,\tau_{q})=0$.

Let us now address uniqueness. Let $u,v\in C([0,T];\mathcal{M}(\mathbb{R}))$ be entropy solutions of $(CL)$, and let $\tau:=\min\,\{t_{u},t_{v}\}$, where

[TABLE]

Arguing as at the end of the existence proof, it is enough to show that $u=v$ in $\mathcal{M}(S_{\tau})$. We claim that this follows, if we prove that

[TABLE]

In fact, equalities (3.1) and (7.15) imply that, for all $\zeta\in C^{1}([0,\tau];C^{1}_{c}(\mathbb{R}))$, $\zeta(\cdot,\tau)=0$ in $\mathbb{R}$,

[TABLE]

Hence $\langle u_{s}(\cdot,t)-v_{s}(\cdot,t),\alpha\rangle_{\mathbb{R}}=0$ for a.e. $t\in(0,\tau)$, for all $\alpha\in C^{1}_{c}(\mathbb{R})$. Therefore $u_{s}=v_{s}$ in $L^{\infty}(0,\tau;\mathcal{M}(\mathbb{R}))$, thus (7.15) implies $u=v$ in $\mathcal{M}(S_{\tau})$.

It remains to prove (7.15), which is equivalent to showing that $u_{r}=v_{r}$ a.e. in $Q_{j,\tau}$ for all $j=1,\dots,p+1$. However, this follows from the uniqueness results provided by Theorem 5.2.. Then the result follows. $\square$

Acknowledgement. M.B. acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.