10/6/2005

Anna Mazzucato
Penn State University

Irregular transport and enstrophy dissipation in two-dimensional incompressible flows

We consider the problem of enstrophy dissipation for two-dimensional incompressible flows. Enstrophy is half the space
integral of the vorticity squared and it is a relevant quantity in 2D turbulence. We discuss two notions of enstrophy defects, measuring the rate of dissipation, due respectively to viscosity and to irregular transport by the velocity field. These notions were originally introduced by G. Eyink in order to reconcile the Kraichnan-Batchelor theory of 2D turbulence with properties of weak solutions to 2D Euler equations. Using renormalized solutions in the sense of DiPerna-Lions, we show that, if the initial enstrophy is finite, the total enstrophy is conserved and in the vanishing viscosity limit a well-defined viscous enstrophy defect exists. If the initial vorticity belongs to certain logarithmic refinements of L², then an exact transport equation holds for the corresponding enstrophy density. For rougher data in the Besov space $B⁰_{2,\infty}$, we show that the two notions of enstrophy defect are not equivalent and we produce examples of solutions where true dissipation occurs. This is joint work with Helena and  Milton Lopes, UNICamp, Brazil.