Long time decay for 3D-NSE in Gevrey-Sobolev spaces

TL;DR

This paper proves that solutions to the 3D Navier-Stokes equations in Gevrey-Sobolev spaces decay to zero over time, using Fourier analysis to establish long-term decay behavior.

Contribution

It demonstrates the decay of global solutions in Gevrey-Sobolev spaces, extending understanding of solution behavior in these function spaces.

Findings

Solutions decay to zero as time approaches infinity

Fourier analysis effectively characterizes decay rates

Results apply to global solutions in Sobolev-Gevrey spaces

Abstract

In this paper we prove, if $u$ is a global solution to Navier-Stokes equations in the Sobolev-Gevrey spaces $H^1_{a,\sigma}(\mathbb R^3)$, then $\|u(t)\|_{H^1_{a,\sigma}}$ decays to zero as time goes to infinity. Fourier analysis is used.