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

Abstract

We prove the global well-posedness and regularity of the (isotropic) Lagrangian averaged Navier-Stokes (LANS-α) equations on a three-dimensional bounded domain with a smooth boundary with no-slip boundary conditions for initial data in the set {u ∈ H^s ∩ H_0^1| Au = 0 on ∂Ω, div u = 0}, s ∈ [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 α-radius phase-space ball, and converge to the Navier-Stokes equations as α → 0. We also show that classical solutions of the LANS-α equations converge almost all in H^s for s ∈ (2:5; 3), to solutions of the inviscid equations (v = 0), called the Lagrangian averaged Euler (LAE-α) equations, even on domains with boundary, for time-intervals governed by the time of existence of solutions of the LAE-α equations.