Large deviations of invariant measure for the 3D stochastic hyperdissipative Navier-Stokes equations

TL;DR

This paper investigates the probability of rare events in the long-term behavior of the 3D stochastic hyperdissipative Navier-Stokes equations, establishing large deviation principles for their invariant measures.

Contribution

It proves the unique ergodicity and uniform large deviations of the invariant measure for these equations, which was not previously established.

Findings

Proved unique ergodicity of the invariant measure.

Established uniform large deviations principle.

Derived large deviations bounds for the invariant measure.

Abstract

In this paper, we consider the large deviations of invariant measure for the 3D stochastic hyperdissipative Navier-Stokes equations driven by additive noise. The unique ergodicity of invariant measure as a preliminary result is proved using a deterministic argument by the exponential moment and exponential stability estimates. Then, the uniform large deviations is established by the uniform contraction principle. Finally, using the unique ergodicity and the uniform large deviations results, we prove the large deviations of invariant measure by verifying the Freidlin-Wentzell large deviations upper and lower bounds.