Stochastic magneto-hydrodynamic equations (MHD): Invariant measures in 2D Poincaré domains

Abstract

We consider stochastic magneto-hydrodynamic equations with a multiplicative noise in 2D and 3D possibly unbounded domains. The noise terms depend both on the velocity and the magnetic field and its spatial derivatives. We prove the existence of a martingale solution and its continuous dependence on the initial state and deterministic external forces. The analysis is based on the stochastic compactness method and the Jakubowski generalization of the Skorokhod Theorem for nonmetric spaces. Moreover, in the case of 2D domains we prove the existence and uniqueness of a strong solution. Using the Maslowski-Seidler result we prove the existence of an invariant measure for the semigroup corresponding to the strong solutions of the MHD equations in 2-dimensional Poincaré domains. The results concerning the continuous dependence play an important role in proving sequentially weak Feller property of this semigroup. The aim of the paper is to extend the results obtained in [4] and [5] for the Navier-Stokes equations to the MHD equations.