Existence and stability of planar shocks of viscous scalar conservation laws with space-periodic flux

TL;DR

This paper proves the existence and stability of planar shocks in viscous scalar conservation laws with space-periodic flux in multiple dimensions, extending previous one-dimensional results using PDE techniques.

Contribution

It extends the existence and stability results of shocks from one-dimensional to multi-dimensional scalar conservation laws with space-periodic flux, employing a novel approximation and normalization approach.

Findings

Existence of multi-dimensional shocks established

Shocks are shown to be stable under perturbations

Method relies on elliptic PDE theory rather than ODEs

Abstract

The goal of this paper is to prove the existence and stability of shocks for viscous scalar conservation laws with space periodic flux, in the multi-dimensional case. Such a result had been proved by the first author in one space dimension, but the extension to a multi-dimensional setting makes the existence proof non-trivial. We construct approximate solutions by restricting the size of the domain and then passing to the limit as the size of the domain goes to infinity. One of the key steps is a " normalization " procedure, which ensures that the limit objects obtained by the approximation scheme are indeed shocks. The proofs rely on elliptic PDE theory rather than ODE arguments as in the 1d case. Once the existence of shocks is proved, their stability follows from classical arguments based on the theory of dynamical systems.