Large time behavior of strong solutions for stochastic Burgers equation II

主讲人：黄飞敏 中科院数学与系统科学院研究员

时间：2022年10月31日9：00

地点：腾讯会议 754 696 999

举办单位：数理学院

主讲人介绍：黄飞敏，中国科学院数学与系统科学研究院华罗庚首席研究员，主要研究非线性偏微分方程，曾获2013年国家自然科学奖二等奖，国家杰出青年基金，美国工业与应用数学学会杰出论文奖。

内容介绍：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 \cite{Olinik64} 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.