Asymptotic expansion in Gevrey spaces for solutions of Navier-Stokes equations. (English) Zbl 1375.35317

Hoang, Luan T.; Martinez, Vincent R.

Asymptotic Anal. 104, No. 3-4, 167-190 (2017).

Summary: 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 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.

MSC:

35Q30	Navier-Stokes equations
76D05	Navier-Stokes equations for incompressible viscous fluids
35C20	Asymptotic expansions of solutions to PDEs
35D30	Weak solutions to PDEs
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

3D Navier-Stokes equations; Leray-Hopf weak solutions; asymptotic expansions; eventual regularity; Gevrey class