Mixing via controllability for randomly forced nonlinear dissipative PDEs

Sergei Kuksin,Vahagn Nersesyan,Armen Shirikyan

doi:10.5802/jep.130

Sergei Kuksin, Vahagn Nersesyan + Show 1 more

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https://doi.org/10.5802/jep.130

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Abstract

We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.

Highlights

In the last twenty years, there was a substantial progress in the question of description of the long-time behaviour of solutions for PDEs with random forcing
For a large class of PDEs the resulting random flow possesses a unique stationary distribution, which attracts the laws of all the solutions with an exponential rate
It was proved by Hairer and Mattingly [HM06, HM11] that the Navier–Stokes flow is exponentially mixing in the dual-Lipschitz metric, provided that the random perturbation is white in time

Summary

Introduction

In the last twenty years, there was a substantial progress in the question of description of the long-time behaviour of solutions for PDEs with random forcing. To illustrate our general result, let us consider the following example of a randomly forced parabolic PDE which defines (under suitable hypotheses) a random dynamical system in the Sobolev space Hs(Td) with an arbitrary integer s 0:. While in the general setting this question is not likely to have a satisfactory answer, when the noise is sufficiently small, one can prove that, for any stable stationary state, there is a unique stationary measure supported in its neighbourhood In this context, a challenging open problem is the description of the behaviour of stationary measures in the vanishing noise limit, in the spirit of the Freidlin–Wentzell theory developed for finite-dimensional diffusion processes; see [FW84, Chap. We denote by Hs(D) the Sobolev space of order s 0 with the usual norm · s

Main result

Preliminary results

Proof of the main theorem

Application