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Global stability and asymptotic behavior for the incompressible MHD equations without viscosity or magnetic diffusion

Qunyi Bie, Hui Fang, Yanping Zhou

TL;DR

The paper analyzes the $n$-dimensional incompressible MHD equations on the periodic torus with partial diffusion and a Diophantine background magnetic field. By developing a dimension-independent Fourier-analysis framework and explicit integral representations, it derives precise linear decay rates and then proves nonlinear global stability and decay in all Sobolev norms $H^s$ for $s\in[0,m]$. The authors obtain lower Sobolev regularity requirements than prior work, with $m>4+2r+n/2$ for magnetic diffusion and $m>3+2r+n/2$ for velocity diffusion, and demonstrate decay consistent across all intermediate norms. The results illuminate the stabilizing role of a Diophantine background field in partially dissipative MHD and provide a robust, adaptable framework for similar PDEs in fluid dynamics.

Abstract

Physical experiments and numerical simulations have revealed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper provides a rigorous mathematical justification of this effect for the $n$-dimensional incompressible magnetohydrodynamic equations with partial diffusion on periodic domains. We establish the global stability and derive explicit decay rates for perturbations around an equilibrium magnetic field satisfying the Diophantine condition. Our results yield the \textit{effective decay rates in all intermediate Sobolev norms} and \textit{significantly relax the regularity requirements} on the initial data compared with previous works (\textit{Sci. China Math.} 41:1--10, 2022; \textit{J. Differ. Equ.} 374:267--278, 2023; \textit{Calc. Var. Partial Differ. Equ.} 63:191, 2024). Furthermore, the analytical framework developed here is dimension-independent and can be flexibly adapted to other fluid models with partial dissipation.

Global stability and asymptotic behavior for the incompressible MHD equations without viscosity or magnetic diffusion

TL;DR

The paper analyzes the -dimensional incompressible MHD equations on the periodic torus with partial diffusion and a Diophantine background magnetic field. By developing a dimension-independent Fourier-analysis framework and explicit integral representations, it derives precise linear decay rates and then proves nonlinear global stability and decay in all Sobolev norms for . The authors obtain lower Sobolev regularity requirements than prior work, with for magnetic diffusion and for velocity diffusion, and demonstrate decay consistent across all intermediate norms. The results illuminate the stabilizing role of a Diophantine background field in partially dissipative MHD and provide a robust, adaptable framework for similar PDEs in fluid dynamics.

Abstract

Physical experiments and numerical simulations have revealed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper provides a rigorous mathematical justification of this effect for the -dimensional incompressible magnetohydrodynamic equations with partial diffusion on periodic domains. We establish the global stability and derive explicit decay rates for perturbations around an equilibrium magnetic field satisfying the Diophantine condition. Our results yield the \textit{effective decay rates in all intermediate Sobolev norms} and \textit{significantly relax the regularity requirements} on the initial data compared with previous works (\textit{Sci. China Math.} 41:1--10, 2022; \textit{J. Differ. Equ.} 374:267--278, 2023; \textit{Calc. Var. Partial Differ. Equ.} 63:191, 2024). Furthermore, the analytical framework developed here is dimension-independent and can be flexibly adapted to other fluid models with partial dissipation.

Paper Structure

This paper contains 15 sections, 20 theorems, 247 equations.

Table of Contents

  1. Introduction and main results
  2. Model and related research
  3. Major achievements
  4. On reduced regularity and decay mechanisms
  5. Organization of the paper
  6. Preliminaries
  7. Integral representation and kernel estimates
  8. Linear stability
  9. Proof of Theorem \ref{['thm.2']}
  10. Global-in-time existence
  11. Temporal decay estimates
  12. Proof of Theorem \ref{['thm']}
  13. A priori estimates
  14. Global-in-time existence
  15. Temporal decay estimate

Key Result

Theorem 1.1

Let $n \geq 2$, $r > n-1$, and $\mu=1$, $\nu=0$. Suppose that $\tilde{b}$ satisfies the Diophantine condition, and that $(U_0, B_0) \in H^m(\mathbb{T}^n)$ for some $m \geq 0$, with Let $(U, B)$ be the corresponding solution to the linear system Then, for any $s \in [0, m]$, the following decay estimates hold:

Theorems & Definitions (41)

  • Theorem 1.1: Linear stability I
  • Theorem 1.2: Linear stability II
  • Remark 1.1
  • Theorem 1.3: Nonlinear stability I
  • Theorem 1.4: Nonlinear stability II
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • ...and 31 more