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A Uniqueness Condition for Conservation Laws with Discontinuous Gradient-Dependent Flux

Alberto Bressan, Wen Shen

Abstract

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. In the stable case where $f(u)<g(u)$ for all $u\in R$, it was proved in [1] that the limits of vanishing viscosity approximations form a contractive semigroup w.r.t. the $L^1$ distance. Further, they coincide with the limits of a suitable family of front tracking approximations. In the present paper we introduce a simple condition that guarantees that every weak, entropy admissible solution of a Cauchy problem coincides with the corresponding semigroup trajectory, and hence is unique.

A Uniqueness Condition for Conservation Laws with Discontinuous Gradient-Dependent Flux

Abstract

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions or , when the gradient of the solution is positive or negative, respectively. In the stable case where for all , it was proved in [1] that the limits of vanishing viscosity approximations form a contractive semigroup w.r.t. the distance. Further, they coincide with the limits of a suitable family of front tracking approximations. In the present paper we introduce a simple condition that guarantees that every weak, entropy admissible solution of a Cauchy problem coincides with the corresponding semigroup trajectory, and hence is unique.
Paper Structure (3 sections, 2 theorems, 109 equations, 10 figures)

This paper contains 3 sections, 2 theorems, 109 equations, 10 figures.

Table of Contents

  1. Introduction
  2. A uniqueness theorem
  3. Proof of Theorem \ref{['THM']}

Key Result

Theorem 2.1

Let $f,g$ satisfy the assumptions (A1). Given an initial data $\bar{u}\in \mathbf{L}^1_{per}$, let $u=u(t,x)$ be a Liu admissible weak solution of the Cauchy problem (1)-(2) with initial data (idata). Assume that Then one has In other words, the solution with the above properties is unique and coincides with the corresponding semigroup trajectory.

Figures (10)

  • Figure 1: Upper left: the initial data at (\ref{['ex2id']}). Upper right: the corresponding solution to Burgers' equation. Lower left: the semigroup solution, obtained as limit of front tracking approximations. Lower right: viscous approximations. For this example we have $u(t,\cdot)\not= S_t \bar{u}$ for all $t>0$.
  • Figure 2: Upper left: the solution to Burgers' equation with initial data (\ref{['ex2id']}). Lower left: the semigroup solution. Upper and lower right: the corresponding functions $\theta(t,\cdot)$.
  • Figure 3: An example where, for the semigroup solution $u(t)= S_t \bar{u}$, the corresponding function $\theta(t,x)$ is not continuous w.r.t. time. For every time $t>0$ the function $\theta(t,\cdot)$ is piecewise affine. If $\tau$ is the time when the two horizontal portions of the graph of $u(t,\cdot)$ join, the function $\theta(\tau,\cdot)$ does not coincide with the limit of $\theta(t,\cdot)$ as $t\to \tau-$.
  • Figure 4: The compact set $K\subseteq [0,T]$ and $\Omega\subset {\mathbb R}^2$, considered at (\ref{['Om']}).
  • Figure 5: Top: a BV solution to the conservation law (\ref{['1']})-(\ref{['2']}), at a given time $t>0$. Bottom: the corresponding function $\theta$. Note that $u(t,\cdot)$ is constant on every open interval $\,\bigl]a_i(t), b_i(t)\bigr[\,$.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Definition 1.1
  • Example 1.1
  • Theorem 2.1
  • Lemma 3.1