  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.
