Exponential mixing for a class of dissipative PDEs with bounded degenerate noise

TL;DR

This paper proves exponential mixing for certain dissipative PDEs with bounded, decomposable noise, including 2D Navier-Stokes and Ginzburg-Landau equations, using coupling and Newton-Kantorovich methods.

Contribution

It establishes exponential mixing for nonlinear dissipative PDEs under bounded, decomposable noise, extending previous results to a broader class of stochastic PDEs.

Findings

Unique stationary measure exists and is exponentially mixing.

Applicable to 2D Navier-Stokes and Ginzburg-Landau equations.

Includes noise modeled by random Haar series with decaying coefficients.

Abstract

We study a class of discrete-time random dynamical systems with compact phase space. Assuming that the deterministic counterpart of the system in question possesses a dissipation property, its linearisation is approximately controllable, and the driving noise is bounded and has a decomposable structure, we prove that the corresponding family of Markov processes has a unique stationary measure, which is exponentially mixing in the dual-Lipschitz metric. The abstract result is applicable to nonlinear dissipative PDEs perturbed by a bounded random force which affects only a few Fourier modes. We assume that the nonlinear PDE in question is well posed, its nonlinearity is non-degenerate in the sense of the control theory, and the random force is a regular and bounded function of time which satisfies some decomposability and observability hypotheses. This class of forces includes random Haar series, where the coefficients for high Haar modes decay sufficiently fast. In particular, the result applies to the 2D Navier-Stokes system and the nonlinear complex Ginzburg-Landau equations. The proof of the abstract theorem uses the coupling method, enhanced by the Newton-Kantorovich-Kolmogorov fast convergence.