Affiliations: [a] Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX 79409-1042, USA. E-mail: [email protected] | [b] Mathematics Department, Tulane University, 6823 St. Charles Ave, New Orleans, LA 70118, USA. E-mail: [email protected]
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 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.
Keywords: 3D Navier–Stokes equations, Leray–Hopf weak solutions, asymptotic expansions, eventual regularity, Gevrey class
