发表机构
Faculty of Engineering, Shinshu University（信州大学工学部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究不可压缩 Navier-Stokes 方程在临界 Fourier-Herz 空间中的大时间渐近行为，发现二维情形下出现不同于经典结果的新渐近轮廓。
AI 中文摘要
我们考虑全空间 $\mathbb{R}^d$（$d\ge 2$）中不可压缩 Navier-Stokes 方程初值问题解的大时间渐近行为。当初始速度属于 $L^1(\mathbb{R}^d)$ 时，众所周知，由于无散度相容条件，其空间积分为零。考虑到这一性质，Carpio（1996）和 Fujigaki-Miyakawa（2001）推导了强解的高阶渐近公式。在本文中，我们研究初始速度属于 Fourier-Herz 空间 $\widehat{L}^1(\mathbb{R}^d)$ 时解的大时间行为，该空间提供了比 $L^1(\mathbb{R}^d)$ 更广泛的框架，并在低频区域施加了额外条件。特别地，在二维情形下，我们表明会出现不同于 Carpio 和 Fujigaki-Miyakawa 所获得的渐近轮廓。
英文摘要
We consider the large time asymptotic behavior of solutions to the initial value problem for the incompressible Navier-Stokes equations in the whole space $\mathbb{R}^d$ ($d\ge 2$). When the initial velocity belongs to $L^1(\mathbb{R}^d)$, it is well-known that its spatial integral vanishes as a consequence of the divergence-free compatibility condition. Taking this property into account, Carpio (1996) and Fujigaki-Miyakawa (2001) derived higher-order asymptotic formulas for the strong solutions. In this paper, we investigate the large time behavior of solutions when the initial velocity belongs to the Fourier-Herz space $\widehat{L}^1(\mathbb{R}^d)$, which provides a broader framework than $L^1(\mathbb{R}^d)$ and imposes an additional condition in the low-frequency region. In particular, in the two-dimensional case, we show that an asymptotic profile different from those obtained by Carpio and Fujigaki-Miyakawa arises.
Comments20 pages
