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

In this article we study the limit when $alpha to 0$ of solutions to the $alpha$-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 arXiv:1611.05300 and arXiv:1403.5682. It requires a substantially distinct approach, analogous to that used for Delorts Theorem, and a new detailed investigation of the relation between (no-slip) filtered velocity and potential vorticity in the half-plane.