Navier--Stokes--Maxwell 系统解的全局适定性与衰减估计

中文摘要 AI 辅助

我们研究了二维和三维空间中 Navier--Stokes--Maxwell 系统解的全局可解性及其大时间渐近行为。该系统通过洛伦兹力和欧姆定律描述了粘性不可压缩流体与电磁场之间的相互作用。该系统表现出抛物-双曲耦合，其中耗散的 Navier--Stokes 系统与双曲型 Maxwell 系统相互作用。Maxwell 系统通过带有 Maxwell 修正的欧姆定律进行阻尼。对于具有适当 Sobolev 正则性的足够小的初始数据，我们建立了强解的全局适定性，并导出了解及其高阶空间导数的最优衰减率。此外，我们证明了电场比速度和磁场分量多出额外的因子 $(1+t)^{-1/2}$，因此衰减得更快。我们分析的关键要素之一是对线性化系统的详细研究，该研究揭示了电磁分量的精细衰减性质。这些线性估计在二维情形的分析中起着至关重要的作用，使我们能够克服由临界衰减估计引起的对数损失。对于非线性系统，我们的方法基于时间加权能量方法，该方法专门设计用于捕捉流体和电磁分量不同的耗散性质。我们引入了一个合适的补偿泛函，以利用电场和磁场之间的耦合，从而恢复磁场缺失的耗散。结合对非线性项的仔细分析，这些估计得到了我们所期望的结果。

英文摘要

We investigate the global solvability and the large-time asymptotic behavior of solutions to the Navier--Stokes--Maxwell system in two and three space dimensions. The system describes the interaction between a viscous incompressible fluid and an electromagnetic field through the Lorenz force and Ohm's law. The system exhibits a parabolic--hyperbolic coupling in which the dissipative Navier--Stokes system interacts with the hyperbolic Maxwell system. The Maxwell system is damped through Ohm's law with the Maxwell correction. For sufficiently small initial data with suitable Sobolev regularity, we establish the global well-posedness of strong solutions and derive optimal decay rates for the solution and its higher-order spatial derivatives. In addition, we show that the electric field decays faster by the extra factor $(1+t)^{-1/2}$ compared to the velocity and magnetic components. One of the key ingredients of our analysis is a detailed study of the linearized system, which reveals refined decay properties of the electromagnetic components. These linear estimates play a crucial role in the analysis of the two-dimensional case, allowing us to overcome the logarithmic loss arising from the borderline decay estimates. For the nonlinear system, our approach is based on a time-weighted energy method, specifically designed to capture the distinct dissipative properties of the fluid and electromagnetic components. A suitable compensating functional is introduced to exploit the coupling between the electric and magnetic fields and thereby recover the missing dissipation of the magnetic field. Combined with careful analysis of the nonlinear terms, these estimates yield our desired result.

发表机构

• Department of Mathematics, College of Sciences, University of Sharjah（沙迦大学理学院数学系）

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