Study On Stochastic Well-posedness Of A Class Of Fluid Equations

In this paper,we study the well-posedness of 2D magnetic Bénard problem and2 D Magnetohydrodynamics(MHD)equations with horizontal dissipation and horizontal magnetic diffusion.For the 2D magnetic Bénard problem,under the certain condition,the well-posedness and regularity of the 2D magnetic Bénard problem with zero thermal conductivity,local dissipation and mixed partial viscosity are studied.In the process of research,the equations are often disturbed by some uncertain factors.In this paper,we consider the well-posedness of solutions under random perturbations.By using uniform estimations,weak convergence method and monotonicity method,the existence and uniqueness of weak solutions for 2D stochastic magnetic Bénard problem are proved.On this basis,we further study the regularity of the solution of 2D magnetic Bénard problem.For the 2D MHD equations,firstly,we study the local existence of solutions for the 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion under small initial values.Secondly,by using uniform estimations,convergence method,curl equation transformation,It? formula and Burkholder-DavisGundy inequality,the existence of solutions for 2D stochastic MHD equations with horizontal dissipation and horizontal magnetic diffusion is proved.