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- W2022157864 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 2d Navier-Stokes equations which took the form cG23(1 + logG)13, 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’ 〈κn,r2〉 (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, 〈κ3,12〉. For the higher members, 〈κn,12〉 (n ≥ 4), these estimates rapidly approach cG(1 + logG)12 for large n. When applied to the 3d 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 〈κn,1〉 which takes the form L〈κn,1〉 ≤ cn (LΛK)4, n ≥ 2, where ΛK is the Kolmogorov length, which is bounded from below. Other 3d Navier-Stokes results can be derived from this in a simple fashion." @default.
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- W2022157864 date "1996-05-01" @default.
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- W2022157864 title "A voyage around the Navier-Stokes equations" @default.
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