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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, toweak solutions of the 2D incompressible Euler equations with non-negativeinitial vorticity in the space of bounded Radon measures in $H^{-1}$. Thisresult extends the analysis done in arXiv:1611.05300 and arXiv:1403.5682. Itrequires a substantially distinct approach, analogous to that used for Delort'sTheorem, and a new detailed investigation of the relation between (no-slip)filtered velocity and potential vorticity in the half-plane.

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