Stability and regularity problems on the 2D magnetohydrodynamic equations

Fluid dynamics is a branch of fluid mechanics that examines the behavior of moving fluids such as liquids and gases. The Navier-Stokes equation is a fundamental equation in this field. It describes fluid motion under the influence of shearing forces. This thesis focuses on a closely related system: the magnetohydrodynamic(MHD) equations, which describe electrically conducting fluids such as liquid metals and plasmas. Our primary focus is to study the MHD equations with magnetic diffusion. In chapter two, we discuss the constraints and difficulty of solving the global regularity problem for 2D resistive MHD equations in R² and solve it in the periodic domain(T²) with fractional magnetic dissipation. In chapter three, we discuss the influence of the background magnetic field on the 2D resistive MHD system. In chapter four, we address the global regularity problem for the 2D MHD system with damped wave type in the spatial domain Ω=[0,π]×R.