Analysis on Groups of Diffeomorphisms of Manifolds with Boundary and the Averaged Motion of a Fluid

Abstract

We establish the existence of three new subgroups of the group of volume-preserving diffeomorphisms of a compact n-dimensional (n ≥ 2) Riemannian manifold which are associated with the Dirichlet, Neumann, and Mixed type boundary conditions that arise in second-order elliptic PDEs. We prove that when endowed with the H^s Hilbert-class topologies for s > (n/2) + 1, these subgroups are C^∞ differential manifolds. We consider these new diffeomorphism groups with an H¹-equivalent right invariant metric, and prove the existence of unique smooth geodesics η(t, ·) of this metric, as well as existence and uniqueness of the Jacobi equations associated to this metric. Geodesics on these subgroups are, in fact, the flows of a time-dependent velocity vector field u(t, x), so that ∂_tη(t, ·) = u(t, η(t, ·)) with η(0, x) = x, and remarkably the vector field u(t, x) solves the so-called Lagrangian averaged Euler (LAE-α) equations on M. These equations, and their viscous counterparts, the Lagrangian averaged Navier-Stokes (LANS-α) equations, model the motion of a fluid at scales larger than an a priori fixed parameter α > 0, while averaging (or filtering-out) the small scale motion, and this is achieved without the use of artificial viscosity. We prove that for divergence-free initial data satisfying u = 0 on ∂M, the LAE-α equations are well-posed, globally when n = 2. We also find the boundary conditions that make the LANS-α equations well-posed, globally when n = 3, and prove that solutions of the LANS-α equations converge when n = 2, 3 for almost all t in some fixed time interval (0, T) in H^s, s ∊ (n/2 + 1, 3) to solutions of the LAE-α equations, thus confirming the scaling arguments of Barenblatt & Chorin.