Large Deviations Principle for 2D Navier–Stokes Equations with Space-time Localized Noise

Abstract

We consider a stochastic 2D Navier–Stokes equation in a bounded domain. The random force is assumed to be non-degenerate and periodic in time; its law has a support localized with respect to both time and space. Slightly strengthening the conditions in the pioneering work about exponential ergodicity by Shirikyan [21], we prove that the stochastic system satisfies Donsker–Varadhan-type large deviations principle. Our proof is based on a criterion of [10] in which we need to verify uniform irreducibility and uniform Feller property for the related Feynman–Kac semigroup.

Keywords

Stochastic 2D Navier–Stokes equations / Donsker–Varadhan large deviations principle / Space-time localized noises / 35Q30 / 60H15 / 60J05 / 60F10 / 90B05

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School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature