Periodic solution and asymptotic stability for the magnetohydrodynamic equations with inhomogeneous boundary condition

We show, using the spectral Galerkin method together with compactness arguments, the existence and uniqueness of the periodic strong solutions for the magnetohydrodynamic-type equations with inhomogeneous boundary conditions. Furthermore, we study the asymptotic stability for the time periodic solution for this system. In particular, when the magnetic field h(x, t) is zero, we obtain the existence, uniqueness, and asymptotic behavior of the strong solutions to the Navier-Stokes equations with inhomogeneous boundary conditions.