Ergodicity of the 2D Navier-Stokes Equations with Degenerate Multiplicative Noise

TL;DR

This paper proves that the 2D stochastic Navier-Stokes equations with degenerate multiplicative noise exhibit exponential ergodicity and possess the asymptotic strong Feller property, ensuring long-term statistical stability.

Contribution

It establishes the asymptotic strong Feller property and exponential ergodicity for the 2D stochastic Navier-Stokes equations with degenerate multiplicative noise, advancing understanding of their long-term behavior.

Findings

Semigroup P_t is asymptotically strong Feller.

Semigroup P_t is exponentially ergodic.

Results apply to 2D incompressible Navier-Stokes equations on the torus.

Abstract

Consider the two-dimensional, incompressible Navier-Stokes equations on the torus We prove that the semigroup P_t generated by the solutions to stochastic Navier-stokes equations is asymptotically strong Feller. Moreover, we also prove that semigroup P_t is exponentially ergodic in some sense