1998/07/15 by Steve Shkoller, Shkoller, Steve
Mathematics · #58B20 #58D05 #76E99 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:58B20 #msc:58D05 #msc:76E99
paper · pdf · doi:10.48550/arxiv.math/9807078
AMS-LaTeX, 22 pages, To appear in J. Func. Anal
arxiv created 1998/07/15 · arxiv updated 2009/11/30
Recently, Holm, Marsden, and Ratiu [1998] have derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation V(t) + ∇U(t) V(t) - α2 [∇ U(t)]t ⋅ \triangle U(t) = -grad p(t) where div U=0, and V = (1- α2 \triangle)U. In this model, the momentum V is transported by the velocity U, with the effect that nonlinear interaction between modes corresponding to length scales smaller than α is negligible. We generalize this equation to the setting of an n dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincaré equation associated with the geodesic flow of the H1 right invariant metric on \mathcal Dsμ, the group of volume preserving Hilbert diffeomorphisms of class Hs. We prove that the geodesic spray is continuously differentiable from T\mathcal Dμs(M) into TT\mathcal Dμs(M) so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold [1966]. To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariant H1 metric on \mathcal Dsμ is a bounded trilinear map in the Hs topology, from which it follows that solutions to Jacobi's equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics.