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Ladder theorems for the \(2D\) and \(3D\) Navier-Stokes equations on a finite periodic domain. (English) Zbl 0736.35084

We use time dependent quantities \(H_ N={\sum\sum}_{i\{n\}=N}\int(D^ n u_ i)^ 2 dx\), to obtain ladder inequalities for both the 2D and 3D Navier-Stokes equations on periodic boundary conditions after the manner of the authors, P. Constantin and M. Gisselfalt [Physica D 44, 421-444 (1990; Zbl 0702.76061)] who found a ladder structure for the complex Ginzburg-Landau equation. The two alternative ladder inequalities are: \[ (1/2)\dot H_ N\leq-{\nu\over 2d}{H^ 2_ N\over H_{N- 1}}+{1\over \nu}c_{N,1}H_ N\|\underline u\|^ 2_ \infty+H_ N^{1/2}\sum_ i\| D^ Nf_ i\|_ 2 \]
\[ (1/2)\dot H_ N\leq- {\nu\over d}{H^ 2_ N\over H_{N-1}}+c_{N,2}H_ N\| Du\|_ \infty+H_ N^{1/2}\sum_ i\| D^ Nf_ i\|_ 2,\qquad N\geq1. \] In 2D it is possible to find long time upper bounds for all the \(H_ N\) making no assumptions, thereby proving the existence of an attractor made up from \(C^ \infty\) functions. In 3D one has to make some a priori assumption to obtain upper bounds for all the \(H_ N\), such as taking \(\|\underline u\|_ \infty\) or \(H_ 1\) bounded above. Length scales \(\ell\) for the smallest feature in the flow appear naturally from this ladder structure. For decaying homogeneous turbulence we find that \(\ell\geq c[\nu/\sup_ t\| Du\|_ \infty]^{1/2}\) which agrees with the minimum scale of Henshaw, Kreiss and Reyna although our decay rate is algebraic and not exponential. Under the assumption that \(\sup_ t\| Du\|_ \infty\approx\sup_ tH_ 1^{1/2}\), the lower bound on \(\ell\) can be shown to be equivalent to the Kolmogorov scale. We also illustrate these ideas by considering thermal convection on periodic boundary conditions as an example. We provide arguments which show that under the same assumption which gave the Kolmogorov scale for homogeneous turbulence, the minimum length scale has a lower bound \(\ell_ N\geq c[(N_ uR_ a)^{1/2}+R_ a]^{-1/2}\) where \(N_ u\) is the Nusselt number. Hence, for most realistic Nusselt Rayleigh number relationships, the minimum scale is \(R_ a^{-1/2}\).

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Citations:

Zbl 0702.76061
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