发表机构
Koç University; University of Victoria; University of Sharjah（科奇大学; 维多利亚大学; 沙迦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Navier-Stokes-Maxwell系统在临界空间中小初值全局解的存在唯一性，并借助Lyapunov泛函与Fourier分裂方法建立最优时间衰减率，且结果在光速极限下一致。
AI 中文摘要
我们证明了在任意维数$d\geq 3$下，对于临界Fujita-Kato空间$\dot H^{\frac{d}{2}-1}(\mathbb R^d)$中的小初值，Navier-Stokes-Maxwell（NSM）系统存在全局时间解且解唯一。在额外假设初值属于Besov空间$\dot B^{-\frac{d}{2}}_{2,\infty}(\mathbb R^d)$的条件下，我们还建立了对于任意$s\in\left(-\frac{d}{2},\frac{d}{2}-1\right]$，解在$\dot H^s(\mathbb R^d)$中无穷远处的衰减率$t^{-\frac{s}{2}-\frac{d}{4}}$是最优的。这一结果是通过构造一个在Fourier侧等价于点态能量的Lyapunov泛函，并将其与Fourier分裂方法的改进相结合以建立其最优衰减率而实现的。这里进行的所有分析——从全局存在性理论到大时间行为的研究——都在一个相对于光速$c\in(0,\infty)$一致的框架内进行。特别是，在非相对论极限$c\to\infty$下，这使我们能够恢复相应的极限磁流体动力学系统的相同结果。
英文摘要
We prove the existence and uniqueness of global-in-time solutions to the Navier--Stokes--Maxwell (NSM) system for small initial data in the critical Fujita--Kato space $\dot H^{\frac{d}{2}-1}(\mathbb R^d)$, in any dimension $d\geq 3$. Under the additional assumption that the initial data belong to the Besov space $\dot B^{-\frac{d}{2}}_{2,\infty}(\mathbb R^d)$, we also establish the optimal decay rate $t^{-\frac{s}{2}-\frac{d}{4}}$ at infinity of the solution in $\dot H^s(\mathbb R^d)$, for any $s\in\left(-\frac{d}{2},\frac{d}{2}-1\right]$. This is achieved by constructing a Lyapunov functional that is equivalent to the pointwise energy on the Fourier side, and by combining it with an adaptation of the Fourier splitting method to establish its optimal decay rate. All the analysis carried out here - from the global existence theory to the study of the large-time behavior - is performed within a framework that is uniform with respect to the speed of light $c\in(0,\infty)$. In particular, in the non-relativistic limit $c\to\infty$, this allows us to recover the same results for the corresponding limiting magnetohydrodynamic system.
Comments26 pages
