Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics

Abstract

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 $\dot{V}(t)+{\nabla }_{U(t)}V(t)-{\alpha }^2[\nabla U(t){]}^t\cdot △U(t)=-\text{grad}p(t)$ where $\text{div}U=0$, and $V=(1-{\alpha }^2△)U$. 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 $\alpha$ 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 $H^1$ right invariant metric on ${\mathcal{D}}_{\mu }^s$, the group of volume preserving Hilbert diffeomorphisms of class $H^s$. We prove that the geodesic spray is continuously differentiable from $T{\mathcal{D}}_{\mu }^s(M)$ into $TT{\mathcal{D}}_{\mu }^s(M)$ 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 $H^1$ metric on ${\mathcal{D}}_{\mu }^s$ is a bounded trilinear map in the $H^s$ 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.

arXiv commentAMS-LaTeX, 22 pages, To appear in J. Func. Anal

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