Global Solutions to an initial boundary problem for the compressible 3-D MHD equations with Navier-slip and perfectly conducting boundary conditions in exterior domains

TL;DR

This paper proves the global existence and decay of smooth solutions for the 3D compressible MHD equations with Navier-slip and perfect conduction boundary conditions in exterior domains, extending to Navier-Stokes solutions.

Contribution

It establishes the first known global existence results for classical solutions of compressible MHD with these boundary conditions in exterior 3D domains.

Findings

Global smooth solutions exist near a constant state.

Explicit decay rates for solutions are provided.

Results imply existence for compressible Navier-Stokes with similar boundary conditions.

Abstract

An initial boundary value problem for compressible Magnetohydrodynamics (MHD) is considered on an exterior domain (with the first Betti number vanishes) in $R^3$ in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier-Stokes equations with Navier-slip boundary conditions on exterior domains in three dimensions, which is not available in literature, to the best of knowledge of the authors'.