Existence of solutions and Gevrey class regularity for Leray-alpha equations

In this paper we consider some equations similar to Navier–Stokes equations, the threedimensional Leray-alpha equations with space periodic boundary conditions. We establish the regularity of the equations by using the classical Faedo–Galerkin method. Our argument shows that there exist an unique weak solution and an unique strong solution for all the time for the Lerayalpha equations, furthermore, the strong solutions are analytic in time with values in the Gevrey class of functions (for the space variable). The relations between the Leray-alpha equations and the Navier–Stokes equations are also considered.  2004 Elsevier Inc. All rights reserved.