Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence

Details Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence

2004-02-02

arxiv.org/abs/math/0402011v1

Arch. Rat. Mech. Anal. v. 179 no. 3 (2006) 353-387

Mathematics

Analysis of PDEs

49 pages, no figures, submitted for publication

Scientific paper

Enstrophy, half the integral of the square of vorticity, plays a role in 2D turbulence theory analogous to that played by kinetic energy in the Kolmogorov theory of 3D turbulence. It is therefore interesting to obtain a description of the way enstrophy is dissipated at high Reynolds number. In this article we explore the notions of viscous and transport enstrophy defect, which model the spatial structure of the dissipation of enstrophy. These notions were introduced by G. Eyink in an attempt to reconcile the Kraichnan-Batchelor theory of 2D turbulence with current knowledge of the properties of weak solutions of the equations of incompressible and ideal fluid motion. Three natural questions arise from Eyink's theory: (1) Existence of the enstrophy defects (2) Conditions for the equality of transport and viscous enstrophy defects (3) Conditions for the vanishing of the enstrophy defects. In [Nonlinearity, v 14 (2001) 787-802], Eyink proved a number of results related to these questions and formulated a conjecture on how to answer these problems in a physically meaningful context. In the present article we improve and extend some of Eyink's results and present a counterexample to his conjecture.

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