The limit $α\to 0$ of the $α$-Euler equations in the half plane with no-slip boundary conditions and vortex sheet initial data

Abstract

In this article we study the limit when α → 0 of solutions to the α-Euler system in the half-plane, with no-slip boundary conditions, to weak solutions of the 2D incompressible Euler equations with non-negative initial vorticity in the space of bounded Radon measures in H −1 . This result extends the analysis done in [4,13]. It requires a substantially distinct approach, analogous to that used for Delort's Theorem, and a new detailed investigation of the relation between (no-slip) filtered velocity and potential vorticity in the half-plane. H∂ 2 H α (x 1 − y 1 , y 2 )q(y) dy.Above, η 1 and η 2 were introduced in ( 35), (36).