The uniform time of existence of the smooth solution for 3D Euler-α equations with Dirichlet boundary conditions

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After reformulate the incompressible Euler-α equations in 3D smooth domain with Drichlet data, we obtain the unique classical solutions to Euler-α equations exist in uniform time interval independent of α. We also show the solution of the Euler-α converge to the corresponding solution of Euler equation in L² in space, uniformly in time. In the sequel, it follows that the Hˢ (s>n/2+1) solutions of Euler-α equations exist in any fixed sub-interval of the maximum existent interval for the Euler equations provided that initial is regular enough and α is small sufficiently.