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 μ(ρ)=αρδ, 0<δ≤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<δ<1 and one for the linear case δ=1, in a nontrivial invariant class of magnetic fields. The class is described by the frozen-in flux h=b/ρ and its mass derivative k=ρ−1hx. The key identity [∂t+u∂x,ρ−1∂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<δ<1, γ≥δ+1/2, and δ=1, γ>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.
Related
- Global Well-Posedness and Large Time Asymptotic Behavior of Strong Solutions to the Compressible Viscous Non-Resistive MHD Equations with Vacuum
- Finite-Time Vanishing of Vacuum and Large-Time Behavior of the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data
- Global Strong Solutions to the One-Dimensional Planar Compressible Magneto-Micropolar Equations with Vacuum
- Local and Global Existence of Strong Solutions to the One-Dimensional Compressible Navier--Stokes Equations with General Pressure Law
- Global solutions to the two-dimensional compressible MHD equations with density-dependent bulk viscosity
BibTeX
@misc{shang2026global,
title = {{Global Regular Solutions for One-Dimensional Degenerate Compressible Non-Resistive MHD Equations with Large Data and Far-Field Vacuum}},
author = {Zhaoyang Shang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13248},
eprint = {2610.13248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13248},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global Regular Solutions for One-Dimensional Degenerate Compressible Non-Resistive MHD Equations with Large Data and Far-Field Vacuum
AU - Zhaoyang Shang
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13248
UR - https://arxiv.org/abs/2610.13248
AB - We consider the Cauchy problem on the real line for a one-dimensional transverse-field isentropic non-resistive magnetohydrodynamic system with viscosity μ(ρ)=αρ^δ, 0<δ≤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<δ<1 and one for the linear case δ=1, in a nontrivial invariant class of magnetic fields. The class is described by the frozen-in flux h=b/ρ and its mass derivative k=ρ^−1h_x. The key identity [∂_t+u∂_x,ρ^−1∂_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<δ<1, γ≥δ+1/2, and δ=1, γ>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.
ER -