Well-posedness and dynamics of the stochastic fractional magneto-hydrodynamic equations

The current paper is devoted to the well-posedness and dynamics of the stochastic 2D incompressible fractional Magneto-Hydrodynamic(MHD) equations driven by Gaussian multiplicative noise. The nonlocal fractional diffusion leads to a new difficulty in the convergence since higher order estimates cannot be obtained. The commutator estimates are introduced to overcome these difficulties. Using the stopping time technique and monotonicity arguments, the global existence and uniqueness of the weak solution are obtained in a fixed probability space. Finally, the existence of a random attractor for the random dynamical systems generated by the solution of stochastic MHD equation is presented.