Asymptotic expansion in Gevrey spaces for solutions of Navier-Stokes equations

TL;DR

This paper proves that the asymptotic expansion for solutions of the 3D Navier-Stokes equations holds in all Gevrey spaces, extending previous Sobolev space results and simplifying the proof method.

Contribution

It extends the Foias-Saut asymptotic expansion to all Gevrey classes and all Leray-Hopf weak solutions, improving and simplifying prior proofs.

Findings

Asymptotic expansion holds in all Gevrey classes.

Proof method is simplified using Gevrey-norm technique.

Expansion extended to all Leray-Hopf weak solutions.

Abstract

In this paper, we study the asymptotic behavior of solutions to the three-dimensional incompressible Navier-Stokes equations (NSE) with periodic boundary conditions and potential body forces. In particular, we prove that the Foias-Saut asymptotic expansion for the regular solutions of the NSE in fact holds in {\textit{all Gevrey classes}}. This strengthens the previous result obtained in Sobolev spaces by Foias-Saut. By using the Gevrey-norm technique of Foias-Temam, the proof of our improved result simplifies the original argument of Foias-Saut, thereby, increasing its adaptability to other dissipative systems. Moreover, the expansion is extended to all Leray-Hopf weak solutions.