Strong solutions to the initial-boundary-value problem of compressible MHD equations with degenerate viscosities and far field vacuum in 3D exterior domains

TL;DR

This paper proves local existence and uniqueness of strong solutions for 3D compressible MHD equations with degenerate viscosities and far-field vacuum, highlighting the magnetic field's role in controlling singularities.

Contribution

It establishes the first local well-posedness results for the IBVP of compressible MHD with density-dependent viscosities and vacuum in 3D exterior domains, including magnetic field behavior.

Findings

Magnetic field decays faster than density over time.

Strong solutions exist and are unique for large initial data.

Magnetic field influences the handling of singularities from density-dependent viscosities.

Abstract

This paper concerns the initial-boundary-value problem (IBVP) of the compressible Magnetohydrodynamic (MHD) equations in 3D exterior domains with Navier-slip boundary conditions for the velocity and perfect conducting conditions for the magnetic field. For the case that the density approaches far-field vacuum initially and the viscosities are power functions of the density (\rho}{\delta} with 0 < {\delta} < 1), the local existence and uniqueness of strong solutions to the IBVP is established for regular large initial data. In particular, in contrast to the local theory of compressible Navier-Stokes equation Li-L\"u-Yuan [24], we show that the magnetic field maintains the initial quality of decaying faster rate than density throughout the time evolution, which reveals the role of the magnetic field in handling singularities arising from density-dependent viscosities.