Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces

TL;DR

This paper proves the global existence and decay of solutions to the 3D Navier-Stokes equations within Lei-Lin-Gevrey spaces, under specific initial conditions involving exponential weights, demonstrating long-term decay of solutions.

Contribution

It establishes the existence, uniqueness, and decay properties of solutions to the 3D Navier-Stokes equations in Lei-Lin-Gevrey spaces with exponential weights, extending previous results.

Findings

Existence and uniqueness of global solutions under specified initial conditions.

Solutions decay to zero as time approaches infinity.

Decay rate characterized within Lei-Lin-Gevrey spaces.

Abstract

In this paper, we prove that there exists a unique global solution of $3D$ Navier-Stokes equation if $\exp(a|D|^{1/\sigma})u^0\in{\mathcal{X}}^{-1}(\mathbb R^3)$ and $\|u^0\|_{{\mathcal{X}}^{-1}}<\nu$. Moreover, we will show that $\|\exp(a|D|^{1/\sigma}) u(t)\|_{{\mathcal{X}}^{-1}}$ goes to zero if the time $t$ goes to infinity.