Stochastic Two-Dimensional Hydrodynamical Systems: Wong-Zakai Approximation and Support Theorem

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 Bénard 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 nonlinear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite dimensional approximation of this process.

Keywords:

• Bénard convection
• Hydrodynamical models
• MHD
• Shell models of turbulence
• Stochastic PDEs
• Support theorem
• Wong–Zakai approximation

Mathematics Subject Classification:

• Primary 60H15
• Secondary 60H30, 76D06, 76M35

Acknowledgments

We would like to thank anonymous referees for pointing out references of related works on the Wong-Zakai approximation of infinite dimensional stochastic evolution equations, and for valuable remarks.