Stability of planar rarefaction waves for scalar viscous conservation law under periodic perturbations

TL;DR

This paper proves that solutions to multi-dimensional viscous conservation laws with periodic initial perturbations tend to planar rarefaction waves over time, providing decay rates and establishing a key inequality in the domain.

Contribution

It demonstrates the asymptotic stability of planar rarefaction waves under periodic perturbations and introduces a Gagliardo-Nirenberg type inequality in the specified domain.

Findings

Solutions tend to planar rarefaction waves over time.

Decay rates of solutions are explicitly obtained.

A Gagliardo-Nirenberg inequality is established in the domain.

Abstract

The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution time-asymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in the domain $ \mathbb R \times \mathbb T^{n-1} (n\geq2) $, where $\mathbb T^{n-1}$ is the $ n-1 $-dimensional torus.