Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains

We prove the global well-posedness and regularity of the (isotropic) Lagrangian averaged Navier-Stokes (LANS-alpha) equations on a three-dimensional bounded domain with a smooth boundary with no-slip boundary conditions for initial data in the set (u is an element of H-s boolean AND H-0(1) / Au. = 0 on partial derivative Omega div u = 0), s is an element of [3: 5), where A is the Stokes operator. As with the Navier-Stokes equations, one has parabolic-type regularity; that is, the solutions instantaneously become space-time smooth when the forcing is smooth (or zero). The equations are an ensemble average of the Navier-Stokes equations over initial data in an a-radius phase-space ball, and converge to the Navier-Stokes equations as alpha --> 0, We also show that classical solutions of the LANS-alpha equations converge almost all in H-s for s is an element of (2.5,3), to solutions of the inviscid equations (v = 0). called the Lagrangian averaged Euler (LAE-alpha) equations, even on domains with boundary, for time-intervals governed by the time of existence of solutions of the LAE=alpha equations.