Exponential mixing for a class of dissipative PDEs with bounded degenerate noise. (English) Zbl 1442.35437

Kuksin, Sergei; Nersesyan, Vahagn; Shirikyan, Armen

Geom. Funct. Anal. 30, No. 1, 126-187 (2020).

Summary: 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.

MSC:

35Q56	Ginzburg-Landau equations
35Q30	Navier-Stokes equations
35R60	PDEs with randomness, stochastic partial differential equations
37A25	Ergodicity, mixing, rates of mixing
37L55	Infinite-dimensional random dynamical systems; stochastic equations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
76M35	Stochastic analysis applied to problems in fluid mechanics
93C20	Control/observation systems governed by partial differential equations

Keywords:

Markov process; stationary measure; mixing; Navier-Stokes system; Ginzburg-Landau equations; Newton-Kantorovich-Kolmogorov fast convergence; approximate controllability; Haar series

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