Minimal Geodesics on Groups
نویسندگان
چکیده
The three-dimensional motion of an incompressible inviscid uid is classically described by the Euler equations, but can also be seen, following Arnold 1], as a geodesic on a group of volume-preserving maps. Local existence and uniqueness of minimal geodesics have been established by Ebin and Marsden 16]. In the large, for a large class of data, the existence of minimal geodesics may fail, as shown by Shnirelman 26]. For such data, we show that the limits of approximate solutions are solutions of a suitable extension of the Euler equations or, equivalently, as sharp measure-valued solutions to the Euler equations in the sense of DiPerna and Majda 14].
منابع مشابه
Minimal Geodesics and Nilpotent Fundamental Groups
Hedlund 18] constructed Riemannian metrics on n-tori, n 3 for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics 4] are optimal for nilpotent fundamental groups.
متن کاملv 1 2 0 M ar 1 99 6 Minimal Geodesics and Nilpotent Fundamental Groups ∗ Bernd Ammann March 1996
Hedlund [He] constructed Riemannian metrics on n-tori, n ≥ 3 for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert’s existence results of minimal geodesics [Ba2] are optimal for nilpotent fundamental groups.
متن کاملDifferentiability of Minimal Geodesics in Metrics of Low Regularity
In Riemannian metrics that are only Hölder continuous of order α, 0 α 1, let minimal geodesics be the continuous curves realizing the shortest distance between two points. It is shown that for 0 < α 1, minimal geodesics are differentiable with a derivative which is at least Hölder continuous of order α=2 for 0 < α < 1 and which is Hölder continuous of order 1 for α= 1.
متن کامل0 Fe b 20 01 THE RIEMANNIAN GEOMETRY OF ORBIT SPACES . THE METRIC , GEODESICS , AND INTEGRABLE SYSTEMS
We investigate the rudiments of Riemannian geometry on orbit spaces M/G for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space M/G and they can hit strata which are more singular only at the end points. This is phrased as convexity result. The geodesic spray, viewed as a (strata-preserving) vector field on TM/G,...
متن کاملFrom a Closed Piecewise Geodesic to a Constriction on a Closed Triangulated Surface
Constrictions on a surface are defined as simple closed curves whose length is locally minimal. In particular, constrictions are periodic geodesics. We use constrictions in order to segment objects. In [4], we proposed an approach based on progressive surface simplification and local geodesic computation. The drawback of this approach is that constrictions are approximated by closed piecewise g...
متن کامل