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

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.