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Asymptotic analysis of the Navier-Stokes equations. (English) Zbl 0584.35007

A remarkable property of any two-dimensional solution of the viscous incompressible Navier-Stokes equations is that, for large times, their asymptotic behavior is totally determined by that of a finite number N of spatial Fourier modes. For both theoretical and practical purposes it is important to get close estimates of N.
In this paper the authors present new rigorous estimates from above of N. The best bound available is nearly proportional to a suitable generalized Grashof number (which depends on the driving force) and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow. The estimates are derived for vanishing boundary data in a bounded domain as well as for space-periodic solutions in an unbounded domain. The first N modes usually determine solutions for large times in the mean square sense. In the last part of the paper conditions are discussed under which the above results hold in the pointwise sense.
Reviewer: F.Rosso

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35Q30 Navier-Stokes equations
49M15 Newton-type methods
35D05 Existence of generalized solutions of PDE (MSC2000)
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