Stochastic generalized magnetohydrodynamics equations with not regular multiplicative noise: Well-posedness and invariant measure

Abstract

In this paper we study a stochastic magnetohydrodynamics (MHD) system with fractional diffusion and resistivity ( − Δ ) α , α > 0 , in R d , d = 2 , 3 . Our main goal is to identify the conditions on α under which we can prove the existence of a martingale solution, the pathwise uniqueness of solution and the existence of invariant measure when the noises are multiplicative and take values in functional space bigger than the space of square integrable functions. Roughly speaking, we prove that if α ≥ 1 , θ ∈ ( 0 , α ) and the driving noises take values in H − θ , then the stochastic system has at least a weak martingale solution. We also establish the pathwise uniqueness of solution whenever α ≥ d 2 . Finally, under the latter condition and under the addition of linear damping to the equations we are able to establish the existence of an invariant measure.