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

We prove the global well-posedness and regularity of the (isotropic) Lagrangian averaged Navier{Stokes (LANS-¬ ) equations on a three-dimensional bounded domain with a smooth boundary with no-slip boundary conditions for initial data in the set fu 2 Hs \H1 0 j Au = 0 on @« ; div u = 0g, s 2 [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 ¬ -radius phase-space ball, and converge to the Navier{Stokes equations as ¬ ! 0. We also show that classical solutions of the LANS-¬ equations converge almost all in Hs for s 2 (2:5; 3), to solutions of the inviscid equations ( ̧ = 0), called the Lagrangian averaged Euler (LAE¬ ) equations, even on domains with boundary, for time-intervals governed by the time of existence of solutions of the LAE-¬ equations.