Nonlinear stability of the composite wave of planar rarefaction waves and planar contact waves for viscous conservation laws with non-convex flux under multi-dimensional periodic perturbations

TL;DR

This paper proves the nonlinear stability of a composite wave made of rarefaction and contact waves in multi-dimensional viscous conservation laws with non-convex flux, considering periodic perturbations that oscillate infinitely.

Contribution

It is the first to establish the stability of such composite waves in multiple dimensions with periodic perturbations in viscous conservation laws.

Findings

Zero modes decay at a satisfactory rate

Non-zero modes decay exponentially

First stability result in several dimensions for this problem

Abstract

In this paper, we study the nonlinear stability of the composite wave consisting of planar rarefaction and planar contact waves for viscous conservation laws with degenerate flux under multi-dimensional periodic perturbations. To the level of our knowledge, it is the first stability result of the composite wave for conservation laws in several dimensions. Moreover, the perturbations studied in the present paper are periodic, which keep constantly oscillating at infinity. Suitable ansatz is constructed to overcome the difficulty caused by this kind of perturbation and delicate estimates are done on zero and non-zero modes of perturbations. We obtain satisfactory decay rates for zero modes and exponential decay rates for non-zero modes.