Uniform time of existence of the smooth solution for 3D Euler-α equations with periodic boundary conditions

Abstract

After reformulating the incompressible Euler-[Formula: see text] equations in 3D periodic domain, one obtains that there exists a unique classical solution of Euler-[Formula: see text] equations in uniform time interval independent of [Formula: see text]. It is shown that the solutions of the Euler-[Formula: see text] converge to the corresponding solutions of Euler equation in [Formula: see text] in space, uniformly in time. It also follows that the [Formula: see text] [Formula: see text] solutions of Euler-[Formula: see text] equations exist in any fixed sub-interval of the maximum existing interval for the Euler equations provided that initial velocity is regular enough and [Formula: see text] is sufficiently small.