On a Lagrangian Formulation of the Incompressible Euler Equation

• 来源期刊: 偏微分方程（英文版）
• 学科: 数学
• 关键词: Euler equation diffeomorphism group
• 年，卷（期）: 2016,（4）
• 页码范围: 320-359
• 页数: 40页
• 分类号: O175.29
• 语种: 英文
• DOI: 10.4208/jpde.v29.n4.5

摘要

In this paper we show that the incompressible Euler equation on the Sobolev space Hs (Rn),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 analytic submanifold of the whole diffeomorphism group.Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.