Global Well-Posedness of Full Compressible Magnetohydrodynamic System in 3D Bounded Domains with Large Oscillations and Vacuum

TL;DR

This paper proves the global existence and exponential decay of solutions for the 3D full compressible magnetohydrodynamic system in bounded domains, allowing large oscillations and initial vacuum, with new estimates overcoming boundary condition challenges.

Contribution

It establishes the first global well-posedness results for this system with large oscillations and initial vacuum in bounded domains, including exponential decay rates.

Findings

Global existence of classical and weak solutions is proven.

Density and temperature can initially vanish, with solutions still existing.

Oscillation of density can grow exponentially when initial vacuum appears.

Abstract

The three-dimensional (3D) full compressible magnetohydrodynamic system is studied in a general bounded domain with slip boundary condition for the velocity filed, adiabatic condition for the temperature and perfect conduction for the magnetic field. For the regular initial data with small energy but possibly large oscillations, the global existence of classical and weak solution as well as the exponential decay rate to the initial-boundary-value problem of this system is obtained. In particular, the density and temperature of such a classical solution are both allowed to vanish initially. Moreover, it is also shown that for the classical solutions, the oscillation of the density will grow unboundedly with an exponential rate when the initial vacuum appears (even at a point). Some new observations and useful estimates are developed to overcome the difficulties caused by the slip boundary conditions.