Stochastic 2D hydrodynamical systems: Wong-Zakai approximation and Support theorem

TL;DR

This paper studies the behavior of solutions to a broad class of 2D hydrodynamical stochastic models, establishing support theorems through Wong-Zakai approximations and convergence results.

Contribution

It introduces a general Wong-Zakai type convergence theorem for nonlinear stochastic PDEs with multiplicative noise, applicable to various 2D hydrodynamical models.

Findings

Support of solution distributions characterized

Wong-Zakai approximation convergence proved

Applicable to Navier-Stokes, MHD, and magnetic Bénard models

Abstract

We deal with a class of abstract nonlinear stochastic models with multiplicative noise, which covers many 2D hydrodynamical models including the 2D Navier-Stokes equations, 2D MHD models and 2D magnetic B\'enard problems as well as some shell models of turbulence. Our main result describes the support of the distribution of solutions. Both inclusions are proved by means of a general Wong-Zakai type result of convergence in probability for non linear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite dimensional approximation of this process.