Existence of Vortex Sheets with Reflection Symmetry in Two Space Dimensions

Abstract

The main purpose of this work is to establish the existence of a weak solution to the incompressible 2D Euler equations with initial vorticity consisting of a Radon measure with distinguished sign in H − 1, compactly supported in the closed right half-plane, superimposed on its odd reflection in the left half-plane. We make use of a new a priori estimate to control the interaction between positive and negative vorticity at the symmetry axis. We prove that a weak limit of a sequence of approximations obtained by either regularizing the initial data or by using the vanishing viscosity method is a weak solution of the incompressible 2D Euler equations. We also establish the equivalence at the level of weak solutions between mirror symmetric flows in the full plane and flows in the half-plane. Finally, we extend our existence result to odd L 1 perturbations, without distinguished sign, of our original initial vorticity.