Weak solutions for the $\alpha$-Euler equations and convergence to Euler

We consider the limit $\alpha\to0$ for the $\alpha$-Euler equations in a two-dimensional bounded domain with Dirichlet boundary conditions. Assuming that the vorticity is bounded in $L^p$, we prove the existence of a global solution and we show the convergence towards a solution of the incompressible Euler equation with $L^p$ vorticity. The domain can be multiply-connected. We also discuss the case of the second grade fluid when both $\alpha$ and $\nu$ go to 0.