Anna Mazzucato
Department of Mathematics
The Pennsylvania State University
Abstract:
We consider the problem of enstrophy dissipation for two-dimensional turbulence at high Reynolds numbers. We discuss two notions of enstrophy defects associated to corresponding balance equations for regularizations of the Euler equations via smoothing of the initial data and by solving the Navier-Stokes equations with the same initial data. This notion of enstrophy defect was originally introduced by G. Eyink. While the enstrophy is exactly conserved if the initial enstrophy is finite and an exact transport equation holds for the corresponding density in the vanishing viscosity limit (under some additional mild conditions), we show that for rougher data the enstrophy defect depends upon the regularization, and we produce examples of dissipative solutions to 2D Euler, that is, weak solutions for which the enstrophy defect is a positive measure. This is joint work with Helena and Milton Lopes.