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 ℝ3, 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 ργ+m2 for any adiabatic index γ>1. This enables us to control both components of the total pressure, ργ and m2, 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 21/2-dimensional whole space allowing vacuum.
Related
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BibTeX
@misc{li2026global,
title = {{Global Well-Posedness and Large Time Asymptotic Behavior of Strong Solutions to the Compressible Viscous Non-Resistive MHD Equations with Vacuum}},
author = {Jiaxu Li and Boqiang Lü and Bing Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37078},
eprint = {2610.37078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37078},
note = {CC0},
}
RIS
TY - UNPB
TI - Global Well-Posedness and Large Time Asymptotic Behavior of Strong Solutions to the Compressible Viscous Non-Resistive MHD Equations with Vacuum
AU - Jiaxu Li
AU - Boqiang Lü
AU - Bing Yuan
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37078
UR - https://arxiv.org/abs/2610.37078
AB - In this paper, we consider the Cauchy problem to the compressible viscous non-resistive magnetohydrodynamic (MHD) system in 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 21/2-dimensional whole space allowing vacuum.
ER -