Stabilization by a background magnetic field: global well-posedness of the full compressible viscous non-resistive MHD system without heat-conductivity
Abstract
We consider the three-dimensional full compressible magnetohydrodynamic(MHD) system on the periodic torus 𝕋3 in the regime where the only dissipative mechanism acting on the system is the viscosity of the fluid: the magnetic field is non-resistive and the flow is non-heat-conducting. We prove that this system admits a unique global smooth solution, together with explicit algebraic decay rates, provided that the perturbation (u0, P0−P̅, H0−n) of the equilibrium state (0,P̅,n) is sufficiently small in a high-order Sobolev space and the background magnetic field n∈ℝ3 satisfies a Diophantine condition. No smallness whatsoever is imposed on the initial density: it is only required to be bounded away from vacuum and from infinity, and may exhibit arbitrarily large variations. The proof uncovers a hidden dissipation mechanism. Although neither the density, nor the pressure, nor the magnetic field is endowed with any diffusion or damping of its own, the coupling of these quantities with the velocity through the background field n, combined with a Poincaré-type inequality of Diophantine origin, generates effective dissipation for both the pressure and the magnetic field perturbations. The large variations of the density are handled by a two-tier energy argument, in which weighted time-decay estimates for the intermediate-order energy compensate exactly for the linear-in-time growth of the highest-order norm of the density.
Related
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BibTeX
@misc{qiao2026stabilization,
title = {{Stabilization by a background magnetic field: global well-posedness of the full compressible viscous non-resistive MHD system without heat-conductivity}},
author = {Liening Qiao and Juntao Sun and Jiahong Wu and Fuyi Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01488},
eprint = {2609.01488},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01488},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stabilization by a background magnetic field: global well-posedness of the full compressible viscous non-resistive MHD system without heat-conductivity
AU - Liening Qiao
AU - Juntao Sun
AU - Jiahong Wu
AU - Fuyi Xu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01488
UR - https://arxiv.org/abs/2609.01488
AB - We consider the three-dimensional full compressible magnetohydrodynamic(MHD) system on the periodic torus 𝕋^3 in the regime where the only dissipative mechanism acting on the system is the viscosity of the fluid: the magnetic field is non-resistive and the flow is non-heat-conducting. We prove that this system admits a unique global smooth solution, together with explicit algebraic decay rates, provided that the perturbation (u_0, P_0−P̅, H_0−n) of the equilibrium state (0,P̅,n) is sufficiently small in a high-order Sobolev space and the background magnetic field n∈ℝ^3 satisfies a Diophantine condition. No smallness whatsoever is imposed on the initial density: it is only required to be bounded away from vacuum and from infinity, and may exhibit arbitrarily large variations. The proof uncovers a hidden dissipation mechanism. Although neither the density, nor the pressure, nor the magnetic field is endowed with any diffusion or damping of its own, the coupling of these quantities with the velocity through the background field n, combined with a Poincaré-type inequality of Diophantine origin, generates effective dissipation for both the pressure and the magnetic field perturbations. The large variations of the density are handled by a two-tier energy argument, in which weighted time-decay estimates for the intermediate-order energy compensate exactly for the linear-in-time growth of the highest-order norm of the density.
ER -