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Minimal geodesics on groups of volume‐preserving maps and generalized solutions of the Euler equations
Author(s) -
Brenier Yann
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199904)52:4<411::aid-cpa1>3.0.co;2-3
Subject(s) - inviscid flow , mathematics , euler equations , geodesic , uniqueness , euler's formula , solving the geodesic equations , semi implicit euler method , mathematical analysis , compressibility , backward euler method , classical mechanics , physics , thermodynamics
The three‐dimensional motion of an incompressible inviscid fluid 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, are sharp measure‐valued solutions to the Euler equations in the sense of DiPerna and Majda [14]. © 1999 John Wiley & Sons, Inc.