Global well-posedness and large time asymptotic behavior of strong solutions to the 2-D compressible magnetohydrodynamic equations with vacuum

TL;DR

This paper proves the global existence, uniqueness, and decay rates of strong solutions to 2-D and 3-D compressible magnetohydrodynamic equations with vacuum, highlighting their long-term behavior under certain initial conditions.

Contribution

It establishes the global well-posedness and decay properties of strong solutions for the 2-D and 3-D compressible MHD equations with vacuum, extending previous results to possibly large oscillations.

Findings

Global existence and uniqueness of strong solutions in 2-D with small initial energy.

Decay rates for pressure, velocity gradient, and magnetic field over time.

Similar decay results obtained for 3-D case.

Abstract

The authors study the Cauchy problem of the magnetohydrodynamic equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For two spatial dimensions, we establish the global existence and uniqueness of strong solutions (which may be of possibly large oscillations) provided the smooth initial data are of small total energy, and obtain some a priori decay with rates (in large time) for the pressure, the spatial gradient of both the velocity field and the magnetic field. Moreover, for three spatial dimensions case, some similar decay rates are also obtained.