Asymptotic stability of scalar multi-D inviscid shock waves

TL;DR

This paper investigates the detailed structure and asymptotic stability of scalar multi-dimensional shock waves, establishing stability results under certain conditions and providing unconditional results for the multi-D Burgers equation.

Contribution

It characterizes non-planar scalar shock waves in multiple space dimensions and proves their asymptotic stability, with unconditional results for the multi-D Burgers equation.

Findings

Scalar shock waves can be non-planar in multiple dimensions.

Asymptotic stability is proven for non-characteristic shocks.

Unconditional stability results are obtained for the multi-D Burgers equation.

Abstract

In several space dimensions, scalar shock waves between two constant states u $\pm$ are not necessarily planar. We describe them in detail. Then we prove their asymptotic stability, assuming that they are uniformly non-characteristic. Our result is conditional for a general flux, while unconditional for the multi-D Burgers equation.