Global Regularity for Non-resistive or Non-viscous MHD System on the Torus

TL;DR

This paper proves the global well-posedness of the incompressible MHD system on a torus in certain non-resistive or non-viscous cases, using novel energy estimates, and establishes stability and decay results, especially in three dimensions.

Contribution

It introduces new energy and commutator estimates in negative Sobolev spaces, relaxes regularity requirements, and proves the first nonlinear stability near the background field in 3D non-viscous MHD.

Findings

Global well-posedness in non-resistive and non-viscous cases.

Relaxed regularity conditions from H^{11} to H^{9/2+} in 3D.

First nonlinear stability result near background field in 3D non-viscous MHD.

Abstract

In this paper, we establish the global well-posedness of the incompressible magnetohydrodynamics (MHD) system on $n-$dimensional $(n\geq 2)$ periodic boxes with either no magnetic diffusivity (non-resistive case) or no fluid viscosity (non-viscous case) under assumption that initial magnetic fields are sufficiently close to the background magnetic field ${\bf e}_n=(0,\cdots,0,1)$. In Eulerian coordinates, we develop novel time-weighted energy estimates and commutator estimates involving Riesz transforms in negative Sobolev spaces to handle two distinct dissipation cases under different initial symmetry assumptions. The analysis becomes much more difficult and delicate in three- or higher-dimensional cases. In particular, for the three-dimensional and non-resistive case, compared with the regularity requirement proposed by Pan, Zhou and Zhu {\it [Arch. Ration. Mech. Anal. 2018]}, our result relaxes it from $H^{11}(\mathbb{T}^3)$ to $H^{\frac{9}{2}+}(\mathbb{T}^3)$. And we further establish precise decay rates and growth bounds for both $u(t)$ and $\partial_n(u(t),b(t))$ in Sobolev norms. For the three-dimensional and non-viscous case, we prove the first nonlinear stability result near the background field $\mathbf{e}_3 = (0,0,1)$. This sharply contrasts with the recent blow-up results on the 3D incompressible Euler equations by Elgindi {\it [Ann. Math. 2021]}, Chen-Hou {\it [Commun. Math. Phys. 2021]} and by Chen-Hou {\it [arXiv:2210.07191]}. Our results show that, under certain symmetry assumptions, magnetic fields near the background field provide enhanced dissipations and suppress potential blow-up mechanisms in non-viscous MHD system.