Ergodicity of the 2-D Navier-Stokes equation under random perturbations

Abstract

A 2-dimensional Navier-Stokes equation perturbed by a sufficiently distributed white noise is considered. Existence of invariant measures is known from previous works. The aim is to prove uniqueness of the invariant measures, strong law of large numbers, and convergence to equilibrium.

Ergodicity for the Navier‐Stokes equation with degenerate random forcing: Finite‐dimensional approximation

“…The number of determining modes goes to infinity as ν → 0. Previous results on uniqueness of invariant measures [7,8] require that all modes be forced. The second question is whether the condition in Theorem 1.1 is sharp.…”

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

“…In fact, the polynomial mixing of the 2D Navier-Stokes equations driven by compound Poisson processes was treated in [38], but their approach does not apply to our situation as we consider stochastic evolution equation driven by Lévy noise of infinite activity. By adapting the tools for the ergodicity of SPDEs driven by Wiener noise developed in [15,22] and [24], the authors of [19] proved the ergodicity of the 2D Navier-Stokes equations driven by Lévy noise with a non-degenerate Wiener noise. Recently, H. Bessaih and the last two authors proved in [7] the ergodicity of stochastic shell models with tempered stable process.…”

“…The proof of existence and uniqueness follows the same lines as in [24,Appendix] or [8, Proof of Theorem 4.5]. The estimate (32) can be proved using the same idea as in [8, Proof of (5.5)] or as in [24, Proof of (21)].…”

“…For c ∈ L 4 (0, T ; H ) and x ∈ H, let y c (·, x) be the mild solution of (31) with Z replaced by c. As in [24], we also set…”

“…Their studies have generated many important results. We refer, for instance, to [5,20,[22][23][24]27,28,[38][39][40] and references therein for the results obtained and the advances that have been made so far.…”