Weak Solutions of 2-D Euler Equations with Initial Vorticity in L(Log L)

Abstract

In this note we prove global in time existence of weak solutions of 2-0 Euler equations for incompressible fluid flows in the whole plane of R2 when the initial vorticity is in the Zygmund class L(log L). The solution is constructed by a vanishing viscosity limit of the sequence of solutions to the Navier-Stokes equations with the same initial data. This is a limiting case of the corresponding results of DiPerna and Majda in which global existence of weak solutions was obtained for initial vorticity in L1 ∩ Lp, 1 < p < ∞