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

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.

💡 Research Summary

The paper studies the incompressible magnetohydrodynamic (MHD) system on the n‑dimensional torus Tⁿ (n ≥ 2) in two degenerate regimes: the non‑resistive case (magnetic diffusion ν = 0, fluid viscosity μ = 1) and the non‑viscous case (μ = 0, ν = 1). The authors consider perturbations (u,b) of the constant equilibrium (0,eₙ) where eₙ=(0,…,0,1) and assume that the initial data are small in a high Sobolev norm and satisfy one of two symmetry conditions: (i) a mixed even/odd parity with respect to the vertical coordinate xₙ, or (ii) full odd parity for the velocity and even parity for the magnetic field with respect to all spatial directions. These symmetries eliminate the zero‑vertical‑frequency Fourier modes, which is crucial for controlling the loss of dissipation in the degenerate equations.

The main results are two global‑well‑posedness theorems. In the non‑resistive case (Theorem 1.1) the authors require s > n/2 + 3 and ‖u₀‖{H^s}+‖b₀‖{H^s} ≤ ε with ε sufficiently small. They prove the existence of a unique global solution (u,b)∈C(