Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics
| dc.creator | Shkoller, Steve | |
| dc.date | 1998-07-15 | |
| dc.date.accessioned | 2026-07-07T05:25:24Z | |
| dc.date.available | 2026-07-07T05:25:24Z | |
| dc.description | 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 $\dot{V}(t) + \nabla_{U(t)} V(t) - α^2 [\nabla U(t)]^t \cdot \triangle U(t) = -\text{grad} p(t)$ where $\text{div} U=0$, and $V = (1- α^2 \triangle)U$. 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 $H^1$ right invariant metric on ${\mathcal D}^s_μ$, the group of volume preserving Hilbert diffeomorphisms of class $H^s$. We prove that the geodesic spray is continuously differentiable from $T{\mathcal D}_μ^s(M)$ into $TT{\mathcal D}_μ^s(M)$ 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 ${\mathcal D}^s_μ$ 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. | |
| dc.description | AMS-LaTeX, 22 pages, To appear in J. Func. Anal | |
| dc.identifier | https://arxiv.org/abs/math/9807078 | |
| dc.identifier | http://arxiv.org/abs/math/9807078 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/77165 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Differential Geometry | |
| dc.subject | 58B20,58D05;76E99 | |
| dc.title | Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics | |
| dc.type | text |