Abstract. Rigorous estimates for the total -(kinetic) energy plus pressure -flux in R 3 are obtained from the three dimensional Navier-Stokes equations. The bounds are used to establish a condition -involving Taylor length scale and the size of the domain -sufficient for existence of the inertial range and the energy cascade in decaying turbulence (zero driving force, non-increasing global energy). Several manifestations of the locality of the flux under this condition are obtained. All the scales involved are actual physical scales in R 3 and no regularity or homogeneity/scaling assumptions are made.
Existence of 2D enstrophy cascade in a suitable mathematical setting, and under suitable conditions compatible with 2D turbulence phenomenology, is known both in the Fourier and in the physical scales. The goal of this paper is to show that the same geometric condition preventing the formation of singularities -1 2 -Hölder coherence of the vorticity direction -coupled with a suitable condition on a modified Kraichnan scale, and under a certain modulation assumption on evolution of the vorticity, leads to existence of 3D enstrophy cascade in physical scales of the flow.Date
We construct semi-integral curves which bound the projection of the global attractor of the 2-D Navier-Stokes equations in the plane spanned by enstrophy and palinstrophy. Of particular interest are certain regions of the plane where palinstrophy dominates enstrophy. Previous work shows that if solutions on the global attractor spend a significant amount of time in such a region, then there is a cascade of enstrophy to smaller length scales, one of the main features of 2-D turbulence theory. The semi-integral curves divide the plane into regions having limited ranges for the direction of the flow. This allows us to estimate the average time it would take for an intermittent solution to burst into a region of large palinstrophy. We also derive a sharp, universal upper bound on the average palinstrophy and show that it is achieved only for forces that admit statistical steady states where the nonlinear term is zero.
A mathematical evidence-in a statistically significant sense-of a geometric scenario leading to criticality of the Navier-Stokes problem is presented. C 2012 American Institute of Physics. [http://dx.
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