Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping

TL;DR

This paper proves the global well-posedness and decay of solutions for the 3D compressible ideal MHD equations with a background magnetic field and velocity damping, revealing a hidden dissipation mechanism.

Contribution

It establishes the first global existence and stability result for multi-dimensional compressible ideal MHD with a background magnetic field under small perturbations.

Findings

Global smooth solutions exist for small perturbations around equilibrium.

Perturbations decay algebraically over time.

A hidden dissipation mechanism is identified through coupling and Diophantine conditions.

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 $(\bar{\rho},\mathbf{0},\omega)$ in the Sobolev space $H^N(\mathbb{T}^3)$ with $N\geq 6r+4$, 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--Poincar\'{e} 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.