Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics
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 $\dot{V}(t) + \nabla_{U(t)} V(t) - \alpha^2 [\nabla U(t)]^t \cdot \triangle U(t) = -\text{grad} p(t)$ where $\text{div} U=0$, and $V = (1- \alpha^2 \trian...
| Main Author: | |
|---|---|
| Format: | Journal article |
| Published: |
1998
|
| _version_ | 1826278955061084160 |
|---|---|
| author | Shkoller, S |
| author_facet | Shkoller, S |
| author_sort | Shkoller, S |
| collection | OXFORD |
| 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 $\dot{V}(t) + \nabla_{U(t)} V(t) - \alpha^2 [\nabla U(t)]^t \cdot \triangle U(t) = -\text{grad} p(t)$ where $\text{div} U=0$, and $V = (1- \alpha^2 \triangle)U$. 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 $\alpha$ is negligible. We generalize this equation to the setting of an $n$ dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincar\'{e} equation associated with the geodesic flow of the $H^1$ right invariant metric on ${\mathcal D}^s_\mu$, 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 ${\mathcal D}^s_\mu$ 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. |
| first_indexed | 2024-03-06T23:51:38Z |
| format | Journal article |
| id | oxford-uuid:72cc1b7a-6981-449a-bdd7-40320d524a28 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T23:51:38Z |
| publishDate | 1998 |
| record_format | dspace |
| spelling | oxford-uuid:72cc1b7a-6981-449a-bdd7-40320d524a282022-03-26T19:52:24ZGeometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamicsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:72cc1b7a-6981-449a-bdd7-40320d524a28Symplectic Elements at Oxford1998Shkoller, SRecently, 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 $\dot{V}(t) + \nabla_{U(t)} V(t) - \alpha^2 [\nabla U(t)]^t \cdot \triangle U(t) = -\text{grad} p(t)$ where $\text{div} U=0$, and $V = (1- \alpha^2 \triangle)U$. 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 $\alpha$ is negligible. We generalize this equation to the setting of an $n$ dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincar\'{e} equation associated with the geodesic flow of the $H^1$ right invariant metric on ${\mathcal D}^s_\mu$, 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 ${\mathcal D}^s_\mu$ 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. |
| spellingShingle | Shkoller, S Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics |
| title | Geometry and curvature of diffeomorphism groups with $H^1$ metric and
mean hydrodynamics |
| title_full | Geometry and curvature of diffeomorphism groups with $H^1$ metric and
mean hydrodynamics |
| title_fullStr | Geometry and curvature of diffeomorphism groups with $H^1$ metric and
mean hydrodynamics |
| title_full_unstemmed | Geometry and curvature of diffeomorphism groups with $H^1$ metric and
mean hydrodynamics |
| title_short | Geometry and curvature of diffeomorphism groups with $H^1$ metric and
mean hydrodynamics |
| title_sort | geometry and curvature of diffeomorphism groups with h 1 metric and mean hydrodynamics |
| work_keys_str_mv | AT shkollers geometryandcurvatureofdiffeomorphismgroupswithh1metricandmeanhydrodynamics |