Global strong solution for the two‐dimensional magnetohydrodynamics equations with shearing‐periodic boundary conditions

Abstract In this paper, we investigate the two‐dimensional (2D), two‐field magnetohydrodynamics (MHD) equations in the presence of a shear flow, assuming positive plasma viscosity and resistivity. We establish the global‐in‐time existence and uniqueness of a strong solution for the 2D two‐field MHD equations under shearing‐periodic boundary conditions, as proposed by Hawley et al. Moreover, we establish the existence and uniqueness of a strong solution for the linear advection‐diffusion equation under shearing‐periodic boundary condition by employing uniformly local spaces.