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

We show, using the spectral Galerkin method together with compactness arguments, existence and uniqueness of the periodic strong solutions for the magnetohydrodynamics's type equations with inhomogeneous boundary conditions. Also, we study the asymptotic stability for time periodic solution for this system. In particular, when the magnetic field h(x,t) is zero, we obtain existence, uniqueness and asymptotic behavior of the strong solutions to the Navier-Stokes equations with inhomogeneous boundary conditions.