Well-posedness and exponential mixing for stochastic magneto-hydrodynamic equations with fractional dissipations

Abstract

Consider d-dimensional magneto-hydrodynamic (MHD) equations with fractional dissipations driven by multiplicative noise. First, we prove the existence of martingale solutions for stochastic fractional MHD equations in the case of d = 2, 3 and α ⋀ β > 0, where α, β are the parameters of the fractional dissipations in the equation. Second, for d = 2, 3 and $$\alpha \wedge \beta \geqslant {1 \over 2} + {d \over 4}$$ , we show the pathwise uniqueness of solutions and then obtain the existence and uniqueness of strong solutions using the Yamada-Watanabe theorem. Furthermore, we establish the exponential mixing property for stochastic MHD equations with degenerate multiplicative noise when d = 2, 3 and $$\alpha \wedge \beta \geqslant {1 \over 2} + {d \over 4}$$ .