Global Regular Solutions for One-Dimensional Degenerate Compressible Non-Resistive MHD Equations with Large Data and Far-Field Vacuum

Abstract

We consider the Cauchy problem on the real line for a one-dimensional transverse-field isentropic non-resistive magnetohydrodynamic system with viscosity $\mu(\rho)=\alpha\rho^\delta$, $0<\delta\leq1$, and vacuum at spatial infinity. The viscosity therefore vanishes precisely where the density vanishes. We establish two global large-data theorems, one for the sublinear range $0<\delta<1$ and one for the linear case $\delta=1$, in a nontrivial invariant class of magnetic fields. The class is described by the frozen-in flux $h=b/\rho$ and its mass derivative $k=\rho^{-1}h_x$. The key identity $[\partial_t+u\partial_x,\rho^{-1}\partial_x]=0$ transports both $h$ and $k$, removes all negative powers of the density from the Lorentz force, and propagates the magnetic regularity. A modified effective-velocity entropy contains an additional coercive magnetic term, while its only error is controlled by $k$. Magnetic pressure also strengthens the density upper bound. These estimates close the large-data continuation argument in the ranges $0<\delta<1$, $\gamma\geq\delta+\tfrac12$, and $\delta=1$, $\gamma>\tfrac32$. As a result, we extend the work of Cao, Li, and Zhu (SIAM J. Math. Anal., 2022) from the Navier-Stokes equations to the MHD equations.