Dissipation in turbulent solutions of 2D Eulerequations

Abstract

We establish local balance equations for smoothfunctions of the vorticity in the DiPerna-Majda weaksolutions of two-dimensional (2D) incompressible Eulerequations, analogous to the balance proved by Duchon andRobert for kinetic energy in three dimensions. Theanomalous term or defect distribution thereincorresponds to the `enstrophy cascade' of 2D turbulence.It is used to define a rather natural notion of a`dissipative Euler solution' in 2D. However, we showthat the DiPerna-Majda solutions with vorticity inLp for p>2 are conservative and have zero defect.Instead, we must seek an alternative approach todissipative solutions in 2D. If we assume an upper boundon the energy spectrum of 2D incompressibleNavier-Stokes solutions by the Kraichnan-Batchelork-3 spectrum, uniformly for high Reynolds number,then we show that the zero viscosity limits of theNavier-Stokes solutions exist, with vorticities in thezero-index Besov space B0,∞2, and that thesegive a weak solution of the 2D incompressible Eulerequations. We conjecture that for this class of weaksolutions enstrophy dissipation may indeed occur, in asense which is made precise.