Periodic Solution and Asymptotic Stability for the Magnetohydrodynamic Equations with Inhomogeneous Boundary Condition

Abstract

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.

Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations

“…Proof. (of Proposition 3) Let (u 1 ∞ , h 1 ∞ ) be a slow-flow solution of (9), that is, a weak solution satisfying (11) and (12) , and let (u 2 ∞ , h 2 ∞ ) be another tentative weak solution of (9). By setting…”

“…Further, we assume that (u ∞ , h ∞ ) is a weak slow-flow solution of (9), i.e., (11) and (12) hold. Then there exists a positive constant β 0 > 0 such that, for every…”