The geometry and analysis of the averaged Euler equations and a new diffeomorphism group

This paper develops the geometric analysis of geodesic flow of a new right, invariant metric [., .](1) on two subgroups of the volume preserving diffe omorphism group of a smooth n-dimensional compact subset Omega of R-n with C-infinity boundary partial derivative Omega. The geodesic equations are gi ven by the system of PDEs (v) over dot (t) + del (u(t))v(t) - epsilon\del u(t)\(t) . Delta u(t) = -gr ad p(t) in Omega, v = (1 - epsilon Delta)u, div u = 0, u(0) = u(0), which are the averaged Euler (or Euler-alpha) equations when epsilon = alph a(2) is a length scale, and are the equations of an inviscid non-newtonian second grade fluid when epsilon = <(alpha)over bar>(1), a material paramete r. The boundary conditions associated with the geodesic flow on the two gro ups we study are given by either u = 0 on partial derivative Omega or u . n = 0 and (del(n)u)(tan) + S-n(u) = 0 on partial derivative Omega, where n is the outward pointing unit normal on partial derivative Omega, an d where S-n is the second fundamental form of partial derivative Omega. We prove that for initial data u(0) iu H-s, s > (n/2) + 1, the above system of PDE are well-posed, by establishing existence, uniqueness, and smoothness of the geodesic spray of the metric [. , .](1), together with smooth depend ence on initial data. We are then able t,a prove that the limit of zero vis cosity for the corresponding viscous equations is a regular limit.