Global Well-Posedness and Large Time Asymptotic Behavior of Strong Solutions to the Compressible Viscous Non-Resistive MHD Equations with Vacuum

Abstract

In this paper, we consider the Cauchy problem to the compressible viscous non-resistive magnetohydrodynamic (MHD) system in $\mathbb{R}^3$, in which the velocity is planar while the magnetic field is vertical. For regular initial data with small energy but possibly large oscillations, we establish the global existence and uniqueness of strong solution with vacuum. In addition, the large-time behavior of the solution is investigated. A key ingredient is an application of the Bourgain--Brezis divergence estimate to handle the total pressure $ρ^γ+m^2$ for any adiabatic index $γ>1$. This enables us to control both components of the total pressure, $ρ^γ$ and $m^2$, by a single estimate that remains valid in the presence of vacuum. To the best of our knowledge, our result is the first one on global solvability to the compressible, viscous, non-resistive MHD system in the special 2$\frac12$-dimensional whole space allowing vacuum.