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

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 struct...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Kuksin, Sergei, Nersesyan, Vahagn, Shirikyan, Armen
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 28.10.2019
Subjects
Control theory
Decay rate
Decomposition
Discrete time systems
Fluid dynamics
Landau-Ginzburg equations
Markov processes
Nonlinear equations
Nonlinearity
Observability (systems)
Stability
Online AccessGet full text

Cover

More Information
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.
ISSN:2331-8422