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

TL;DR

This paper establishes the existence, uniqueness, and asymptotic stability of periodic strong solutions for magnetohydrodynamic equations with inhomogeneous boundary conditions, including special results for the Navier-Stokes case when the magnetic field is zero.

Contribution

It introduces a spectral Galerkin method combined with compactness arguments to analyze periodic solutions and stability for MHD equations with inhomogeneous boundaries, extending to Navier-Stokes equations.

Findings

Existence and uniqueness of periodic strong solutions for MHD equations.

Asymptotic stability of these solutions.

Special case results for Navier-Stokes equations when magnetic field is zero.

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.