Global well-posedness of the inviscid resistive isentropic compressible MHD system

TL;DR

This paper proves the global well-posedness and analyzes the large-time behavior of solutions to the inviscid resistive isentropic compressible MHD system on a three-dimensional torus, highlighting a novel dissipation mechanism due to magnetic field interactions.

Contribution

It establishes the first global well-posedness results for the inviscid resistive isentropic compressible MHD system under small perturbations, revealing a new dissipation mechanism from magnetic interactions.

Findings

Dissipation of density derivatives perpendicular to the magnetic field direction.

Magnetic field interactions generate weak stabilization effects.

Three tiers of dissipative energies are constructed for analysis.

Abstract

Due to the absence of dissipation mechanism to the inviscid compressible systems, it is a challenging problem to prove their global solvability. In this paper, we are concerned with the initial-boundary value problem to the inviscid and resistive isentropic compressible magnetohydrodynamic (MHD) system on three dimensional torus $\mathbb T^3$. Global well-posedness and large time behavior of solutions are established in the first time for the isentropic setting, under the condition that the initial data $(\rho_0, u_0, H_0)$ is a small perturbation around the constant state $(1, 0, w)$, with $w$ satisfying the Diophantine condition. The main observation of this paper is that the spatial derivatives of the density along directions perpendicular to $w$ are dissipated. Such dissipation mechanism is generated from the interaction between the velocity field and the background magnetic field. This verifies the weak stabilizing effects of the magnetic filed on the dynamics in the scenario of inviscid isentropic flows. Due to different dissipation mechanisms for the density, velocity, and magnetic field, three ties of dissipative energies are designed, that is, high order Sobolev norms of the perturbed magnetic field, intermediate order Sobolev norms of the perturbed density, and low order Sobolev norms of the velocity field.