Linear stability analysis of 2D incompressible MHD equations with only magnetic diffusion

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics Letters Pub Date : 2025-05-10 DOI:10.1016/j.aml.2025.109600

Jitao Liu, Huning Zhang

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引用次数: 0

Abstract

Although many physical experiments and numerical simulations show that the magnetic field can stabilize and inhibit electrically conducting fluids, whether 2D incompressible MHD equations with only magnetic diffusion develop finite time singularities or not is one of the most challenging problems and remains open. Therefore, this issue has always attracted a lot of attention of mathematicians. Due to its linearized system plays a crucial role, to deeper understand the aforesaid issue, in this paper, we make the first attempt to study its linear stability when the magnetic field close to the equilibrium state

e_{2} = (0, 1)

in the periodic domain and ultimately proposed the linear stability condition (1.4). To be more precise, we show that the solution of its linearized system will be time-asymptotically stable and converge to the equilibrium state in the algebraic rate via the method of spectral analysis, as long as the integrals in the vertical direction of initial perturbations are zeros.

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仅磁扩散的二维不可压缩MHD方程的线性稳定性分析

虽然许多物理实验和数值模拟表明磁场可以稳定和抑制导电流体，但仅磁扩散的二维不可压缩MHD方程是否会产生有限时间奇点是最具挑战性的问题之一，仍然是一个开放的问题。因此，这个问题一直备受数学家们的关注。由于其线性化系统起着至关重要的作用，为了更深入地理解上述问题，本文首次尝试研究了磁场在周期域中接近平衡态e2=(0,1)时的线性稳定性，并最终提出了线性稳定性条件（1.4）。更精确地说，我们通过谱分析的方法证明，只要初始扰动在垂直方向上的积分为零，其线性化系统的解将是时间渐近稳定的，并以代数速率收敛到平衡状态。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Mathematics Letters 数学-应用数学

CiteScore

7.70

自引率

5.40%

发文量

347

审稿时长

10 days

期刊介绍： The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.