Ergodicity for stochastic equation of Navier--Stokes type

Z. Brzeźniak; T. Komorowski; S. Peszat

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arXiv · 2003.08764

Ergodicity for stochastic equation of Navier--Stokes type

Z. Brzeźniak·T. Komorowski·S. Peszat

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Abstract

In the first part of the note we analyze the long time behaviour of a two dimensional stochastic Navier--Stokes equations system on a torus with a degenerate, one dimensional noise. In particular, for some initial data and noises we identify the invariant probability measure for the system and give a sufficient condition under which it is unique and stochastically stable. In the second part of the note, we consider a simple example of a finite-dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic N.S.E., and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant probability measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant probability measure, there exist another one. In particular, the generator of the system is not hypoelliptic.

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BibTeX RIS

Z. Brzeźniak, T. Komorowski, S. Peszat. 2021-08-26. Ergodicity for stochastic equation of Navier--Stokes type. https://arxiv.org/abs/2003.08764

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