Conservation Laws with Discontinuous Gradient-Dependent Flux: the Unstable Case

TL;DR

This paper investigates scalar conservation laws with discontinuous flux depending on the gradient, focusing on the unstable case where flux functions differ and demonstrating the existence and uniqueness of solutions under specific conditions.

Contribution

It constructs solutions for a class of discontinuous flux conservation laws with convex flux functions and establishes conditions for their uniqueness.

Findings

Solutions can be constructed for piecewise monotone initial data.

The Cauchy problem may have infinitely many solutions for smooth initial data.

Uniqueness is achieved when the number of flux-switching interfaces is minimized.

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. We study here the unstable case where $f(u)>g(u)$ for all $u\in {\mathbb R}$. Assuming that both $f$ and $g$ are strictly convex, solutions to the Riemann problem are constructed. Even for a smooth initial data, examples show that the Cauchy problem can have infinitely many solutions. For an initial data which is piecewise monotone, i.e., increasing or decreasing on a finite number of intervals, a solution can be constructed globally in time. It is proved that such solution is unique under the additional requirement that the number of interfaces, where the flux switches between $f$ and $g$, remains as small as possible.