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Global stability and asymptotic behavior for incompressible ideal MHD equations with velocity damping term

Hui Fang, Pingping Gui, Yanping Zhou

TL;DR

The paper addresses the global stability and asymptotic behavior of the incompressible MHD system with velocity damping under small perturbations of a Diophantine background magnetic field. It combines a Fourier-analytic representation with sharp kernel estimates for the linearized problem and a nonlinear energy framework to obtain global existence and explicit algebraic decay in Sobolev spaces. The results extend prior work to $\mathbb{T}^n$ domains with lower regularity requirements, revealing a dimension-independent decay mechanism and establishing a versatile approach for partially dissipative fluid models. The analysis underscores the stabilizing role of the background field and delivers decay profiles that depend on initial regularity, offering insights into magnetic relaxation phenomena and potential extensions to related PDEs.

Abstract

In this article, we study the stability and large time behavior for an multi-dimensional incompressible magnetohydrodynamical system with a velocity damping term, for small perturbations near a steady-state of magnetic field fulfilling the Diophantine condition. Our results mathematically characterize the background magnetic field exerts the stabilizing effect, and bridge the gap left by previous work with respect to the asymptotic behavior in time. Our proof approach mainly relies on the Fourier analysis and energy estimates. In addition, we provide a versatile analytical framework applicable to many other partially dissipative fluid models.

Global stability and asymptotic behavior for incompressible ideal MHD equations with velocity damping term

TL;DR

The paper addresses the global stability and asymptotic behavior of the incompressible MHD system with velocity damping under small perturbations of a Diophantine background magnetic field. It combines a Fourier-analytic representation with sharp kernel estimates for the linearized problem and a nonlinear energy framework to obtain global existence and explicit algebraic decay in Sobolev spaces. The results extend prior work to domains with lower regularity requirements, revealing a dimension-independent decay mechanism and establishing a versatile approach for partially dissipative fluid models. The analysis underscores the stabilizing role of the background field and delivers decay profiles that depend on initial regularity, offering insights into magnetic relaxation phenomena and potential extensions to related PDEs.

Abstract

In this article, we study the stability and large time behavior for an multi-dimensional incompressible magnetohydrodynamical system with a velocity damping term, for small perturbations near a steady-state of magnetic field fulfilling the Diophantine condition. Our results mathematically characterize the background magnetic field exerts the stabilizing effect, and bridge the gap left by previous work with respect to the asymptotic behavior in time. Our proof approach mainly relies on the Fourier analysis and energy estimates. In addition, we provide a versatile analytical framework applicable to many other partially dissipative fluid models.
Paper Structure (8 sections, 13 theorems, 194 equations)

This paper contains 8 sections, 13 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kernel estimates
  4. Linear stability
  5. Nonlinear stability
  6. A priori estimates
  7. Global-in-time existence
  8. Temporal decay estimate

Key Result

Theorem 1.1

Let $n \ge 2$ and $r > n - 1$, and suppose that the background magnetic field $\widetilde{\mathbf{b} }$ adheres to the Diophantine condition. Assume that the initial data $(\mathbf{V} _0, \mathbf{H} _0) \in H^m(\mathbb{T}^n)$ for some integer $m \ge 1$ satisfy the mean-free conditions Then the corresponding solution $(\mathbf{V} ,\mathbf{H} )$ to lineartheorem fulfills where $C>0$ denotes a cons

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 17 more