Characterization of the law for 3D stochastic hyperviscous fluids

TL;DR

This paper analyzes the 3D hyperviscous Navier-Stokes equations in vorticity form, showing that for sufficiently large correction term c, the equations are equivalent in law to simpler reference equations, improving previous bounds.

Contribution

It establishes that for c > 1/2, the vorticity equations are equivalent in law to simpler equations, refining earlier results requiring c > 3/2.

Findings

Equivalence in law for c > 1/2

Improved bounds over previous work

Application of Girsanov transform to hyperviscous fluids

Abstract

We consider the 3D hyperviscous Navier-Stokes equations in vorticity form, where the dissipative term $-\Delta \vec \xi$ of the Navier-Stokes equations is substituted by $(-\Delta)^{1+c} \vec \xi $. 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 $c>\frac 12$, improving previous results obtained with $c>\frac 32$ in a different setting in [5,14].