Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field

TL;DR

This paper rigorously proves that a background magnetic field stabilizes and damps perturbations in a 3D incompressible MHD system with partial dissipation, demonstrating enhanced dissipation and global stability.

Contribution

It establishes the global stability and decay rates of perturbations near a background magnetic field in a 3D anisotropic MHD system with partial dissipation.

Findings

Global stability of perturbations near background magnetic field in Sobolev space

Explicit decay rates in H^2 norm

Magnetic field induces enhanced dissipation and stabilization

Abstract

Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and damps electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in $\mathbb R^3$. The velocity equation in this system is the 3D Navier-Stokes equation with dissipation only in the $x_1$-direction while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field $(0,1,0)$ is globally stable in the Sobolev setting $H^3(\mathbb R^3)$. In addition, explicit decay rates in $H^2(\mathbb R^3)$ are also obtained. When there is no presence of the magnetic field, the 3D anisotropic Navier-Stokes equation in $\mathbb R^3$ is not well understood and the small data global well-posedness remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid.