Exponential mixing for PDEs in unbounded domains
报告摘要：In the last two decades, there has been a significant progress in the understanding of ergodic properties of white-forced dissipative PDEs. The previous studies mostly focus on equations posed on bounded domains since they rely on different compactness properties and the discreteness of the spectrum of the Laplacian. In this talk, we focus on the damped complex Ginzburg-Landau equation in the whole space driven by a white-in-time noise. Under the assumption that the noise is sufficiently non-degenerate, we establish the uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is based on coupling techniques combined with a generalization of Foiaş-Prodi estimate to the case of the whole space and special space-time weighted estimates which help to handle the behavior of solutions at infinity. This talk is based on a joint work with Meng Zhao (SJTU).