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

We consider the Cauchy problem on the real line for a one-dimensional transverse-field isentropic non-resistive magnetohydrodynamic system with viscosity $\mu (\rho )=\alpha {\rho }^{\delta }$, $0<\delta \le 1$, 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 $h=b\mathrm{/}\rho$ and its mass derivative $k={\rho }^{-1}{h}_x$. The key identity $[{\mathrm{\partial }}_t+u{\mathrm{\partial }}_x,{\rho }^{-1}{\mathrm{\partial }}_x]=0$ 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 $0<\delta <1$, $\gamma \ge \delta +\frac{1}{2}$, and $\delta =1$, $\gamma >\frac{3}{2}$. 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.