Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping
Abstract
We study the Cauchy problem for the three-dimensional isentropic compressible ideal (inviscid and non-resistive) magnetohydrodynamic equations with velocity damping on the periodic torus 𝕋3. The system admits a steady equilibrium consisting of a constant density ρ̅ and a uniform background magnetic field ω∈ℝ3. We prove that this equilibrium is nonlinearly stable. More precisely, we show that if the initial data are a sufficiently small perturbation of (ρ̅,0,ω) in the Sobolev space HN(𝕋3) with N≥ 6r+4, and if ω satisfies a Diophantine condition, then the system admits a unique global smooth solution. Moreover, the perturbations decay algebraically in time. To the best of our knowledge, this is the first global well-posedness result for the multi-dimensional isentropic compressible ideal MHD system. The proof reveals a hidden dissipation mechanism: although neither the density equation nor the magnetic field equation contains explicit diffusion or damping, the coupling between the velocity and the magnetic field through the background field ω, combined with a Diophantine--Poincaré inequality, generates effective dissipation for both the density perturbation and the magnetic field perturbation, which together with the velocity damping yields global regularity and time decay.
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BibTeX
@misc{qiao2026stabilization,
title = {{Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping}},
author = {Liening Qiao and Jiahong Wu and Fuyi Xu and Xiaoping Zhai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.04462},
eprint = {2605.04462},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.04462},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping
AU - Liening Qiao
AU - Jiahong Wu
AU - Fuyi Xu
AU - Xiaoping Zhai
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2605.04462
UR - https://arxiv.org/abs/2605.04462
AB - We study the Cauchy problem for the three-dimensional isentropic compressible ideal (inviscid and non-resistive) magnetohydrodynamic equations with velocity damping on the periodic torus 𝕋^3. The system admits a steady equilibrium consisting of a constant density ρ̅ and a uniform background magnetic field ω∈ℝ^3. We prove that this equilibrium is nonlinearly stable. More precisely, we show that if the initial data are a sufficiently small perturbation of (ρ̅,0,ω) in the Sobolev space H^N(𝕋^3) with N≥ 6r+4, and if ω satisfies a Diophantine condition, then the system admits a unique global smooth solution. Moreover, the perturbations decay algebraically in time. To the best of our knowledge, this is the first global well-posedness result for the multi-dimensional isentropic compressible ideal MHD system. The proof reveals a hidden dissipation mechanism: although neither the density equation nor the magnetic field equation contains explicit diffusion or damping, the coupling between the velocity and the magnetic field through the background field ω, combined with a Diophantine--Poincaré inequality, generates effective dissipation for both the density perturbation and the magnetic field perturbation, which together with the velocity damping yields global regularity and time decay.
ER -