Asymptotic behavior of solutions of the compressible MHD equations on the half space

Abstract

In this paper, we study the global well-posedness and large-time asymptotic behavior of strong solutions to the $n$-dimensional ($n\ge 2$) compressible viscous resistive magnetohydrodynamics (MHD) equations in the upper half-space $\mathbb{R}_+^n$. We impose the standard no-slip boundary condition on the velocity field and two types of physically relevant boundary conditions on the magnetic field: the insulating boundary condition and the perfectly conducting boundary condition. Under the assumption that the initial data are small perturbations of the constant equilibrium state $(\barρ, \mathbf{0}, \mathbf{0})$, we first establish the global existence and uniqueness of strong solutions in Sobolev spaces by deriving uniform energy estimates and using a bootstrapping argument. Then, by combining the sharp decay estimates of solutions to the corresponding linearized system with Duhamel's principle, we obtain the decay rates of solutions to the nonlinear problem. Specifically, we prove that the solutions decay at the rate $t^{-\frac{n}{2}\left(1-\frac{1}{p}\right)}$ in $L^p$ norm for $2\le p\le\infty$, and we also establish the decay rates of their first-order derivatives. Furthermore, we show that the global solutions converge to the linearized solutions with an improved decay rate as time goes to infinity. Our results extend the classical results of Kagei and Kobayashi \cite{Kagei-Kobayashi2002, Kagei-Kobayashi2005} for the compressible Navier--Stokes equations on the half space to the compressible MHD system with both insulating and perfectly conducting boundary conditions.