Exponential mixing for a class of dissipative PDEs with bounded degenerate noise
Abstract
We study a class of discretetime 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 dualLipschitz 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 nondegenerate 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 NavierStokes system and the nonlinear complex GinzburgLandau equations. The proof of the abstract theorem uses the coupling method, enhanced by the NewtonKantorovichKolmogorov fast convergence.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.03250
 Bibcode:
 2018arXiv180203250K
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Optimization and Control;
 Mathematics  Probability;
 35Q10;
 35Q56;
 35R60;
 37A25;
 37L55;
 60H15;
 76M35
 EPrint:
 64 pages