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.
A.Biswas, Gevrey regularity for a class of dissipative equations with applications to decay, J. Differential Equations253 (2012), 2739–2764. doi:10.1016/j.jde.2012.08.003.
2.
A.Biswas and C.Foias, On the maximal space analyticity radius for the 3D Navier–Stokes equations and energy cascades, Ann. Mat. Pura Appl. (4)193(3) (2014), 739–777. doi:10.1007/s10231-012-0300-z.
3.
A.Biswas, V.R.Martinez and P.Silva, On Gevrey regularity of the supercritical SQG equation in critical Besov spaces, J. Funct. Analysis269(10) (2015), 3083–3119. doi:10.1016/j.jfa.2015.08.010.
4.
P.Constantin and C.Foias, Navier–Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
5.
C.R.Doering and E.S.Titi, Exponential decay rate of the power spectrum for solutions of the Naiver–Stokes equations, Phys. Fluids7(6) (1995), 1384–1390. doi:10.1063/1.868526.
6.
R.H.Dyer and D.E.Edmunds, Lower bounds for solutions of the Navier–Stokes equations, Proc. London Math. Soc.3(18) (1968), 169–178. doi:10.1112/plms/s3-18.1.169.
7.
C.Foias, L.Hoang and B.Nicolaenko, On the helicity in 3D-periodic Navier–Stokes equations. I. The non-statistical case, Proc. Lond. Math. Soc. (3)94(1) (2007), 53–90. doi:10.1112/plms/pdl003.
8.
C.Foias, L.Hoang and B.Nicolaenko, On the helicity in 3D-periodic Navier–Stokes equations. II. The statistical case, Comm. Math. Phys.290(2) (2009), 679–717. doi:10.1007/s00220-009-0827-z.
9.
C.Foias, L.Hoang, E.Olson and M.Ziane, On the solutions to the normal form of the Navier–Stokes equations, Indiana Univ. Math. J.55(2) (2006), 631–686. doi:10.1512/iumj.2006.55.2830.
10.
C.Foias, L.Hoang, E.Olson and M.Ziane, The normal form of the Navier–Stokes equations in suitable normed spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire26(5) (2009), 1635–1673. doi:10.1016/j.anihpc.2008.09.003.
11.
C.Foias, L.Hoang and J.-C.Saut, Asymptotic integration of Navier–Stokes equations with potential forces. II. An explicit Poincaré–Dulac normal form, J. Funct. Anal.260(10) (2011), 3007–3035. doi:10.1016/j.jfa.2011.02.005.
12.
C.Foias, O.Manley, R.Rosa and R.Temam, Navier–Stokes Equations and Turbulence, Encyclopedia of Mathematics and Its Applications, Vol. 83, Cambridge University Press, Cambridge, 2001.
13.
C.Foias and J.-C.Saut, Asymptotic behavior, as , of solutions of Navier–Stokes equations and nonlinear spectral manifolds, Indiana Univ. Math. J.33(3) (1984), 459–477. doi:10.1512/iumj.1984.33.33025.
14.
C.Foias and J.-C.Saut, Linearization and normal form of the Navier–Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire4(1) (1987), 1–47. doi:10.1016/S0294-1449(16)30372-9.
15.
C.Foias and J.-C.Saut, Asymptotic integration of Navier–Stokes equations with potential forces. I, Indiana Univ. Math. J.40(1) (1991), 305–320. doi:10.1512/iumj.1991.40.40015.
16.
C.Foias and R.Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal.87(2) (1989), 359–369. doi:10.1016/0022-1236(89)90015-3.
17.
U.Frisch, Turbulence, Cambridge University Press, Cambridge, 1995, The legacy of A. N. Kolmogorov.
18.
I.Kukavica and E.Reis, Asymptotic expansion for solutions of the Navier–Stokes equations with potential forces, J. Differential Equations250(1) (2011), 607–622. doi:10.1016/j.jde.2010.08.016.
19.
O.A.Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York, 1969, 2nd English edn, revised and enlarged. Translated from the Russian by R.A. Silverman and J. Chu, Mathematics and Its Applications, Vol. 2.
20.
D.Levermore and M.Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations133 (1997), 321–339. doi:10.1006/jdeq.1996.3200.
21.
G.Minea, Investigation of the Foias–Saut normalization in the finite-dimensional case, J. Dynam. Differential Equations10(1) (1998), 189–207. doi:10.1023/A:1022696614020.
22.
M.Oliver and E.S.Titi, Gevrey regularity for the attractor of a partially dissipative model of Bénard convection in a porous medium, J. Differential Equations163(2) (2000), 292–311. doi:10.1006/jdeq.1999.3744.
23.
M.Oliver and E.S.Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier–Stokes equations in , J. Funct. Anal.172(1) (2000), 1–18. doi:10.1006/jfan.1999.3550.
24.
M.Oliver and E.S.Titi, On the domain of analyticity of solutions of second order analytic nonlinear differential equations, J. Differential Equations174(1) (2001), 55–74. doi:10.1006/jdeq.2000.3927.
25.
R.Temam, Navier–Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, Theory and numerical analysis, Reprint of the 1984 edition.
