Large Time Behavior of Strong Solutions for Stochastic Burgers Equation with Transport Noise
Time: 2021-09-20
Published By: He Liu
Speaker(s): Houqi Su (AMSS-CAS)
Time: 15:15-17:00 October 13, 2021
Venue: Room 29, Quan Zhai, BICMR
We consider the large time behavior of strong solutions to the stochastic Burgers equation with transport 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 [31] in 1964. However, the stability of these wave patterns under stochastic perturbation is not known until now. In this talk, we give a definite answer to the stability problem of the rarefaction and viscous shock waves for the 1-d stochastic Burgers equation with transport noise. 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.
