Large time behavior of strong solutions for stochastic Burgers equation

TL;DR

This paper investigates the long-term stability of solutions to the stochastic Burgers equation, showing that rarefaction waves remain stable under noise while viscous shocks do not, and establishes a convergence rate to the rarefaction wave.

Contribution

It provides the first stability analysis of wave patterns in stochastic Burgers equations, including a new inequality crucial for decay rate estimates.

Findings

Rarefaction wave remains stable under white noise perturbation.

Viscous shock wave stability under stochastic perturbation remains unresolved.

A convergence rate toward the rarefaction wave is established.

Abstract

We consider the large time behavior of strong solutions to a kind of stochastic Burgers equation, where the position x is perturbed by a Brownian noise. It is well known that both the rarefaction wave and viscous shock wave are time-asymptotically stable for deterministic Burgers equation since the pioneer work of A. Ilin and O. Oleinik [21] in 1964. However, the stability of these wave patterns under stochastic perturbation is not known until now. In this paper, we give a definite answer to the stability problem of the rarefaction and viscous shock waves for the 1-d stochastic Burgers equation. That is, the rarefaction wave is still stable under white noise perturbation and the viscous shock is not stable yet. Moreover, a time-convergence rate toward the rarefaction wave is obtained. To get the desired decay rate, an important inequality (denoted by Area Inequality) is derived. This inequality plays essential role in the proof, and may have applications in the related problems for both the stochastic and deterministic PDEs.