Ergodicity of the 3D stochastic Navier–Stokes equations driven by mildly degenerate noise

Abstract

We prove that the any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i. e. all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.

Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing

“…Using this idea, the ergodicity of the 2-D stochastic INSE with degenerate noise has been established (see [63]). Some results of ergodicity for the 3-D INSEs driven by mildly degenerate noise relying on the strong asymptotic Feller property have also established (see [93,94] and the references cited therein). Attention is often restricted to the case in which g and h both vanish at the two points a, b ∈ R, and g, h > 0 on (a, b).…”

Uniqueness and blow-up for a stochastic viscous dyadic model

“…In [26, 28] similar results have been obtained with a completely different method, based on the Krylov selection method [35]. Related results can be found in [27, 14,24, 40, 43, 41,44, 42,1].The two methods apply equally well in more general situations (as done for instance in [8]). The purpose of this paper is to analyse, with a view on uniqueness and emergence of blow-up, a much simpler infinite dimensional stochastic equation which anyway retains most of the characteristics of the original problem and which makes the methods of [13] and [28] applicable.…”

Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise

“…Ergodicity results for the two and three dimensional stochastic Navier-Stokes equations (SNSE) on various domains have been established in the literature. The existence and uniqueness of invariant measures for the 2D and 3D SNSE with degenerate and non-degenerate Gaussian noise have been studied by using various methods in [8,9,13,14,22,23,27,34,[39][40][41]. Ergodicity results for the 2D SNSE driven by Lévy noise with non-degenerate Gaussian part is established in [16] and the 2D stochastic magnetohydrodynamic (MHD) equations with Lévy noise is established in [32].…”