Dissipation in Turbulent Solutions of 2-D Euler

We establish local balance equations for smooth functions of the vorticity in the DiPerna-Majda weak solutions of 2D incompressible Euler, analogous to the balance proved by Duchon and Robert for kinetic energy in 3D. The anomalous term or defect distribution therein corresponds to the ``enstrophy cascade'' of 2D turbulence. It is used to define a rather natural notion of ``dissipative Euler solution'' in 2D. However, we show that the DiPerna-Majda solutions with vorticity in $L^p$ for $p>2$ are conservative and have zero defect. Instead, we must seek an alternative approach to dissipative solutions in 2D. If we assume an upper bound on the energy spectrum of 2D incompressible Navier-Stokes solutions by the Kraichnan-Batchelor $k^{-3}$ spectrum, uniformly for high Reynolds number, then we show that the zero viscosity limits of the Navier-Stokes solutions exist, with vorticities in the zero-index Besov space $B^{0,\infty}_2$, and that these give a weak solution of the 2D incompressible Euler equations. We conjecture that for this class of weak solutions enstrophy dissipation may indeed occur, in a sense which is made precise.