Ergodicity of 3D Leray-α model with fractional dissipation and degenerate stochastic forcing

Abstract

By using the asymptotic coupling method, the asymptotic log-Harnack inequality is established for the transition semigroup associated to the 3D Leray-[Formula: see text] model with fractional dissipation driven by highly degenerate noise. As applications, we derive the asymptotic strong Feller property and ergodicity for the stochastic 3D Leray-[Formula: see text] model with fractional dissipation, which is the stochastic 3D Navier–Stokes equation regularized through a smoothing kernel of order [Formula: see text] in the nonlinear term and a [Formula: see text]-fractional Laplacian. The main results can be applied to the classical stochastic 3D Leray-[Formula: see text] model ([Formula: see text]), stochastic 3D hyperviscous Navier–Stokes equation ([Formula: see text]) and stochastic 3D critical Leray-[Formula: see text] model ([Formula: see text]).