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

Quansen Jiu, Yaowei Xie, Zhihong Yan

TL;DR

This work proves global well-posedness for the incompressible MHD system on $n$-dimensional tori in two dissipation regimes: non-resistive ($\nu=0$) and non-viscous ($\mu=0$). The authors develop a robust time-weighted energy framework together with novel commutator estimates for Riesz transforms in negative Sobolev spaces, enabling precise control of the nonlinear couplings between the velocity and magnetic fields. In 3D, they reduce the required initial regularity for the non-resistive case from $H^{11}$ to $H^{\tfrac{9}{2}+}$ and establish explicit decay and growth rates for $u$, $\partial_n u$, and $\partial_n b$, while for the non-viscous case they obtain the first nonlinear stability near $\mathbf{e}_3=(0,0,1)$ in higher dimensions without Diophantine restrictions. The results highlight how proximity to a background magnetic field, under symmetry assumptions, enhances dissipation and suppresses potential blow-up mechanisms, offering a rigorous baseline for stability analyses in non-diffusive MHD on compact domains.

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.

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

TL;DR

This work proves global well-posedness for the incompressible MHD system on -dimensional tori in two dissipation regimes: non-resistive () and non-viscous (). The authors develop a robust time-weighted energy framework together with novel commutator estimates for Riesz transforms in negative Sobolev spaces, enabling precise control of the nonlinear couplings between the velocity and magnetic fields. In 3D, they reduce the required initial regularity for the non-resistive case from to and establish explicit decay and growth rates for , , and , while for the non-viscous case they obtain the first nonlinear stability near in higher dimensions without Diophantine restrictions. The results highlight how proximity to a background magnetic field, under symmetry assumptions, enhances dissipation and suppresses potential blow-up mechanisms, offering a rigorous baseline for stability analyses in non-diffusive MHD on compact domains.

Abstract

In this paper, we establish the global well-posedness of the incompressible magnetohydrodynamics (MHD) system on dimensional 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 . 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 to . And we further establish precise decay rates and growth bounds for both and in Sobolev norms. For the three-dimensional and non-viscous case, we prove the first nonlinear stability result near the background field . 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.
Paper Structure (14 sections, 10 theorems, 171 equations)

This paper contains 14 sections, 10 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Commutator Estimates for Riesz Operators
  4. Standard Energy Estimate
  5. The Non-resistive MHD System
  6. Frequency-space Formulation
  7. Time-weighted Energy Functionals
  8. A Priori Estimates for $\Gamma_{1}(t)-\Gamma_{4}(t)$
  9. Proof of Theorems \ref{['thm1']}
  10. The Non-viscous MHD System
  11. Frequency-space Formulation
  12. Time-weighted Energy Functionals
  13. A Priori Estimates for $\Psi_{1}(t)-\Psi_{4}(t)$
  14. Proof of Theorems \ref{['thm2']}

Key Result

Theorem 1.1

(The non-resistive case)Let $n \geq 2$ and $s > \frac{n}{2} + 3$. Suppose that sym1-perturbation hold and Then there exists $\varepsilon_0 > 0$ such that if the non-resistive MHD system mhd-1 with $\mu=1, \nu=0$ has a unique global solution $(u,b)\in C([0,+\infty);H^s(\mathbb{T}^n))$, satisfying where $0 < \delta < \frac{s-\frac{n}{2}-3}{2}$ and $C>0$ is a constant. Moreover, it holds that for

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 13 more