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

TL;DR

This paper investigates scalar conservation laws with discontinuous flux depending on the gradient, introducing a front tracking algorithm to prove convergence of solutions in the stable case where one flux is always less than the other.

Contribution

It introduces a novel front tracking method for gradient-dependent flux conservation laws and establishes convergence and uniqueness results in the stable case.

Findings

Convergence of piecewise constant approximations to a contractive semigroup.

Equivalence of semigroup trajectories and vanishing viscosity limits in periodic case.

Validation of the method for non-convex flux functions.

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 stable case where $f(u)<g(u)$ for all $u\in {\mathbb R}$, with $f,g$ smooth but possibly not convex. A front tracking algorithm is introduced, proving that piecewise constant approximations converge to the trajectories of a contractive semigroup on $\mathbf{L}^1({\mathbb R})$. In the spatially periodic case, we prove that semigroup trajectories coincide with the unique limits of a suitable class of vanishing viscosity approximations.