Large time behavior of strong solutions for stochastic Burgers equation with transport noise

We considered 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 were time-asymptotically stable for the deterministic Burgers equation since the pioneer work of A. Ilin and O. Oleinik [29] in 1964. However, the stability of these wave patterns under stochastic perturbation was not known until now. In this paper, we gave 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 was still stable under white noise perturbation and the viscous shock was not stable yet. Moreover, a time-convergence rate toward the rarefaction wave was obtained. To get the desired decay rate, an important inequality (denoted by area inequality) was derived. This inequality played an essential role in the proof, and may have applications in the related problems for both the stochastic and deterministic partial differential equations.