# Large time behaviour of solutions to the 3D-NSE in $\mathcal X^{\sigma}$ spaces

## TL;DR

This paper investigates the long-term decay behavior of solutions to the 3D Navier-Stokes equations within specific function spaces, revealing decay rates depending on the regularity index and employing Fourier analysis techniques.

## Contribution

It establishes decay rates for solutions in $ ext{X}^\sigma$ spaces for $\sigma > -3/2$, extending understanding of solution behavior over time in these function spaces.

## Key findings

- Solutions decay at least as fast as $t^{-(\sigma+3/2)/2}$ for $\sigma > -3/2$
- Decay estimates are derived using Fourier analysis and standard PDE techniques
- Results apply to solutions in the space $C( ext{R}^+, L^2 igcap ext{X}^{-1})$

## Abstract

In this paper we study the incompressible Navier-Stokes equations in $L^2(\mathbb R^3)\cap\mathcal X^{-1}(\mathbb R^3)$. In the global existence case, we establish that if the solution $u$ is in the space $C(\mathbb R^+,L^2\cap\mathcal X^{-1})$, then for $\sigma>-3/2$ the decay of $\|u(t)\|_{\mathcal X^\sigma}$ is at least of the order of $t^{-\frac{\sigma+\frac{3}{2}}{2}}$. Fourier analysis and standard techniques are used.