A voyage around the Navier-Stokes equations

Abstract

In 1988, Constantin, Foias and Temam found a sharp (to within logarithms) upper bound on the Lyapunov dimension of the attractor of the periodic 2 d Navier-Stokes equations which took the form cG 2 3 (1 + logG) 1 3 , where G is the Grashof number. This result is derived by a method independent of that used to estimate the attractor dimension. It involves defining a set of time averaged squared ‘wavenumbers’ 〈 κ n, r 2〉 ( r < n and n ≥ 1) that can be related to a definition of the number of degrees of freedom of the system and which arise directly out of the Navier-Stokes equations themselves. The Constantin-Foias-Temam result is related to the estimate for the first nontrivial member of this set, 〈 κ 3,1 2〉. For the higher members, 〈 κ n,1 2〉 ( n ≥ 4), these estimates rapidly approach cG(1 + logG) 1 2 for large n. When applied to the 3 d Navier-Stokes equations this method also shows that for weak solutions there is an a priori upper bound on the time averaged infinite set of wavenumbers 〈 κ n,1 〉 which takes the form L〈κ n,1〉 ≤ c n ( L Λ K ) 4 , n ≥ 2, where Λ K is the Kolmogorov length, which is bounded from below. Other 3 d Navier-Stokes results can be derived from this in a simple fashion.