Asymptotic profiles for solutions to the incompressible Navier-Stokes equations in the critical Fourier-Herz spaces

Ikki Fukuda; Ryosuke Nakasato

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arXiv · 2609.30902

Asymptotic profiles for solutions to the incompressible Navier-Stokes equations in the critical Fourier-Herz spaces

Ikki Fukuda·Ryosuke Nakasato

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Abstract

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.

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Ikki Fukuda, Ryosuke Nakasato. 2026-09-25. Asymptotic profiles for solutions to the incompressible Navier-Stokes equations in the critical Fourier-Herz spaces. https://arxiv.org/abs/2609.30902

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