Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations
✍ Scribed by Yann Brenier
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 454 KB
- Volume
- 52
- Category
- Article
- ISSN
- 0010-3640
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✦ Synopsis
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].