  After reformulate the incompressible Euler-$\alpha$ equations in 3D smooth
domain with Drichlet data, we obtain the unique classical solutions to
Euler-$\alpha$ equations exist in uniform time interval independent of
$\alpha$. We also show the solution of the Euler-$\alpha$ converge to the
corresponding solution of Euler equation in $L^2$ in space, uniformly in time.
In the sequel, it follows that the $H^s$ $(s>\frac{n}{2}+1)$ solutions of
Euler-$\alpha$ equations exist in any fixed sub-interval of the maximum
existent interval for the Euler equations provided that initial is regular
enough and $\alpha$ is small sufficiently.
