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 ${\mathbb{T}}^3$. The system admits a steady equilibrium consisting of a constant density $\bar{\rho }$ and a uniform background magnetic field $\omega \in {\mathbb{R}}^3$. We prove that this equilibrium is nonlinearly stable. More precisely, we show that if the initial data are a sufficiently small perturbation of $(\bar{\rho },0,\omega )$ in the Sobolev space $H^N({\mathbb{T}}^3)$ with $N\ge 6r+4$, and if $\omega$ 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 $\omega$, 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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