Characterization of the law for 3D stochastic hyperviscous fluids

Abstract

We consider the 3D hyperviscous Navier-Stokes equations in vorticity form, where the dissipative term −∆ ξ of the Navier-Stokes equations is substituted by (−∆) 1+c ξ. We investigate how big the correction term c has to be in order to prove, by means of Girsanov transform, that the vorticity equations are equivalent (in law) to easier reference equations obtained by neglecting the stretching term. This holds as soon as c > 1 2 , improving previous results obtained with c > 3 2 in a different setting in [5,14].MSC2010: 76M35, 60H15, 35Q30.

Stochastic 3D Leray-$α$ Model with Fractional Dissipation

“…Moreover, our global well-posedness results are applicable for the case θ 2 = 1, θ 1 = 1 4 and the case θ 1 = 0, θ 2 ≥ 5 4 , which are corresponding to the the stochastic critical Leray-α model (see [2] for deterministic case) and hyperviscous Navier-Stokes equations. Hence our results cover and generalize some corresponding results in [2,19,22,24,27,44]. And we believe that the methods presented in this paper are also useful for tackling other types of SPDEs with fractional Laplacian.…”

“…In the case of θ 1 = 0, the stochastic fractional (or hyperviscous) Navier-Stokes equations have been intensively studied (see e.g. [19,24,44,46,50] and references within). Chueshov and Millet in [16] proved the well-posedness and large deviation principle of stochastic 3D Leray-α model in the case of θ 1 = θ 2 = 1 (see also [22]).…”

Stochastic Navier-Stokes Equations and Related Models

“…How could one prove uniqueness in law by means of probabilistic arguments? Girsanov theorem is the easiest method but it cannot work for Navier-Stokes equations, as Ferrario has shown in [33]. In general it seems that the Girsanov approach has limitations that are too strong.…”

Large deviations for stochastic 3D Leray-<inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> model with fractional dissipation

“…When θ 1 = 0, the stochastic fractional (or hyperviscous) Navier-Stokes equations have been intensively studied (see e.g. [20,26,38,48,54] and references within). The authors in [19] proved the well-posedness and large deviation principle of stochastic 3D Leray-α model in the case of θ 1 = θ 2 = 1 (see also [22]).…”