In this paper we show that the incompressible Euler equation on the Sobolev space H^S（R^n）, s 〉 n/2＋1, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreove...In this paper we show that the incompressible Euler equation on the Sobolev space H^S（R^n）, s 〉 n/2＋1, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an ana- lytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.展开更多
文摘In this paper we show that the incompressible Euler equation on the Sobolev space H^S（R^n）, s 〉 n/2＋1, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an ana- lytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.
