Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping
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arXiv
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2026
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| author | Qiao, Liening Wu, Jiahong Xu, Fuyi Zhai, Xiaoping |
| author_facet | Qiao, Liening Wu, Jiahong Xu, Fuyi Zhai, Xiaoping |
| contents | 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 $\mathbb{T}^3$. The system admits a steady equilibrium consisting of a constant density $\barρ$ and a uniform background magnetic field $ω\in\mathbb{R}^3$. We prove that this equilibrium is nonlinearly stable. More precisely, we show that if the initial data are a sufficiently small perturbation of $(\barρ,\mathbf{0},ω)$ in the Sobolev space $H^N(\mathbb{T}^3)$ with $N\geq 6r+4$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04462 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping Qiao, Liening Wu, Jiahong Xu, Fuyi Zhai, Xiaoping Analysis of PDEs 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 $\mathbb{T}^3$. The system admits a steady equilibrium consisting of a constant density $\barρ$ and a uniform background magnetic field $ω\in\mathbb{R}^3$. We prove that this equilibrium is nonlinearly stable. More precisely, we show that if the initial data are a sufficiently small perturbation of $(\barρ,\mathbf{0},ω)$ in the Sobolev space $H^N(\mathbb{T}^3)$ with $N\geq 6r+4$, 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. |
| title | Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.04462 |