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

Abstract

In this paper, we study the asymptotic behavior of solutions to the threedimensional 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 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 Leray-Hopf weak solutions and the exponential decay rate for each of its remainders is calculated explicitly.