Vanishing Viscosity Solutions for Conservation Laws with Regulated Flux

TL;DR

This paper introduces a new framework for scalar conservation laws with regulated flux functions, proving the existence and uniqueness of solutions via vanishing viscosity methods, and applies it to certain degenerate systems.

Contribution

It develops the concept of regulated functions for fluxes and establishes vanishing viscosity limits for conservation laws with discontinuous flux.

Findings

Existence and uniqueness of solutions for the considered conservation laws.

Extension of vanishing viscosity method to fluxes with regulated discontinuities.

Application to 2x2 triangular systems with hyperbolic degeneracy.

Abstract

In this paper we introduce a concept of "regulated function" $v(t,x)$ of two variables, which reduces to the classical definition when $v$ is independent of $t$. We then consider a scalar conservation law of the form $u_t+F(v(t,x),u)_x=0$, where $F$ is smooth and $v$ is a regulated function, possibly discontinuous w.r.t.both $t$ and $x$. By adding a small viscosity, one obtains a well posed parabolic equation. As the viscous term goes to zero, the uniqueness of the vanishing viscosity limit is proved, relying on comparison estimates for solutions to the corresponding Hamilton--Jacobi equation. As an application, we obtain the existence and uniqueness of solutions for a class of $2\times2$ triangular systems of conservation laws with hyperbolic degeneracy.