Stability and inviscid limit of the 3D anisotropic MHD system near a background magnetic field with mixed fractional partial dissipation

TL;DR

This paper proves the global stability of 3D anisotropic MHD equations with mixed fractional dissipation near a magnetic field and investigates the inviscid limit as vertical viscosity vanishes, with convergence in H^1 norm.

Contribution

It establishes the global stability of the 3D anisotropic MHD system with mixed fractional dissipation and analyzes the inviscid limit in the presence of partial magnetic diffusion.

Findings

Global stability of the 3D anisotropic MHD system near a magnetic field.

Convergence of solutions in H^1 norm as vertical viscosity coefficient tends to zero.

Validation of the inviscid limit for the MHD system with mixed fractional dissipation.

Abstract

A main result of this paper establishes the global stability of the three-dimensional MHD equations near a background magnetic field with mixed fractional partial dissipation with $\alpha, \beta\in(\frac{1}{2}, 1]$. Namely, the velocity equations involve dissipation $(\Lambda_1^{2\alpha} + \Lambda_2^{2\alpha}+\sigma \Lambda_3^{2\alpha})u$ with the case $\sigma=1$ and $\sigma=0$. The magnetic equations without partial magnetic diffusion $\Lambda_i^{2\beta} b_i $ but with the diffusion $(-\Delta)^\beta b$, where $\Lambda_i^{s} (s>0)$ with $i=1, 2, 3$ are the directional fractional operators. Then we focus on the vanishing vertical kinematic viscosity coefficient limit of the MHD system with the case $\sigma=1$ to the case $\sigma=0$. The convergent result is obtained in the sense of $H^1$-norm.