具有大扰动的三维粘性无阻力MHD系统的全局解和渐近行为

IF 1.3 3区 数学 Q2 MATHEMATICS, APPLIED
Youyi Zhao
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引用次数: 0

摘要

我们重新研究了三维周期域中不可压缩、粘性和非电阻MHD系统的一些大扰动解的整体存在性,其中外加磁场满足Diophantine条件,并且外加磁场强度m比扰动大。Jiang-Jiang证明了速度的最高阶导数随m的增大而增大,非线性系统对线性化问题的收敛速度为\(m^{-1/2}\)。Jiang和S. Jiang, Arch。定量。械甲怪。分析的。生态学报,247(2023),96]。在本文中,我们采用一种不同的方法,利用涡度估计来建立最高阶能量不等式。这种策略防止了随着m增长的项的出现,因此速度的最高阶导数相对于m的增加行为就不会出现。此外,我们使用涡量估计来证明非线性系统在时间或m趋近于无穷大时对线性化问题的收敛速度。值得注意的是,我们的分析表明,与Jiang-Jiang的发现相比,m中的收敛速度更快。最后,我们工作的一个关键贡献是确定了低阶耗散的可积时间衰减。这一发现可以代替低阶能量的时间衰减来关闭最高阶能量不等式,大大放宽了初始扰动的正则性要求。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations

We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by m, is large compared to the perturbations. It was proved by Jiang–Jiang that the highest-order derivatives of the velocity increase with m and the convergence rate of the nonlinear system towards a linearized problem is of \(m^{-1/2}\) in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy inequality. This strategy prevents the appearance of terms that grow with m, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to m does not appear. Additionally, we use the vorticity estimates to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or m approaches infinity. Notably, our analysis reveals that the convergence rate in m is faster compared to the finding of Jiang–Jiang. Finally, a key contribution of our work is identifying an integrable time-decay of the lower-order dissipation. This finding can replace the time-decay of lower-order energy in closing the highest-order energy inequality, significantly relaxing the regularity requirements for the initial perturbations.

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来源期刊
CiteScore
2.00
自引率
15.40%
发文量
97
审稿时长
>12 weeks
期刊介绍: The Journal of Mathematical Fluid Mechanics (JMFM)is a forum for the publication of high-quality peer-reviewed papers on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. As an important part of that, the journal encourages papers dealing with mathematical aspects of computational theory, as well as with applications in science and engineering. The journal also publishes in related areas of mathematics that have a direct bearing on the mathematical theory of fluid mechanics. All papers will be characterized by originality and mathematical rigor. For a paper to be accepted, it is not enough that it contains original results. In fact, results should be highly relevant to the mathematical theory of fluid mechanics, and meet a wide readership.
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