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

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 L2$L<^>2$ spaces.