# Asymptotic stability of shock profiles and rarefaction waves under periodic perturbations for 1-d convex scalar viscous conservation laws

## TL;DR

This paper proves the asymptotic stability of shock profiles and rarefaction waves under space-periodic perturbations for 1D convex scalar viscous conservation laws, revealing new phenomena related to the constant shift in shock profiles.

## Contribution

It introduces a novel analysis of shock profile stability under periodic perturbations, highlighting the role of oscillations at infinity and establishing stability results for rarefaction waves.

## Key findings

- Shock profiles approach the background with a constant shift at exponential rates.
- The constant shift is influenced by periodic oscillations at infinity, not just initial mass.
- Stability of rarefaction waves in the $L^inity$ norm is also demonstrated.

## Abstract

This paper studies the asymptotic stability of shock profiles and rarefaction waves under space-periodic perturbations for one-dimensional convex scalar viscous conservation laws. For the shock profile, we show that the solution approaches the background shock profile with a constant shift in the $ L^\infty(\mathbb{R}) $ norm at exponential rates. The new phenomena contrasting to the case of localized perturbations is that the constant shift cannot be determined by the initial excessive mass in general, which indicates that the periodic oscillations at infinities make contributions to this shift. And the vanishing viscosity limit for the shift is also shown. The key elements of the poof consist of the construction of an ansatz which tends to two periodic solutions as $ x \rightarrow \pm\infty, $ respectively, and the anti-derivative variable argument, and an elaborate use of the maximum principle. For the rarefaction wave, we also show the stability in the $ L^\infty(\mathbb{R}) $ norm.