Large time behavior of strong solutions for stochastic Burgers equation

We consider the large time behavior of strong solutions to a kind of stochastic Burgers equation, where the positionxis 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 [20] in 1964. However, the stability of these wave patterns under stochastic perturbation is not known until now. In this paper, we give a defifinite 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.

This is joint work with Feimin Huang and Houqi Su.