Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations
Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations
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DOI：
10.1002/(sici)1097-0312(199904)52:4
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发表时间：
1999-04
影响因子：
3
通讯作者：
Y. Brenier
中科院分区：
文献类型：
--
作者：
Y. Brenier
不可压缩无粘流体的三维运动经典地由欧拉方程描述，但也可以按照Arnold [1]，作为一组体积保持映射上的测地线。Ebin和Marsden [16]建立了极小测地线的局部存在唯一性。在大范围内，对于大类数据，最小测地线的存在可能会失败，如Shnirelman [26]所示。对于这样的数据，我们证明近似解的极限是欧拉方程的适当扩展的解，或者等价地，是DiPerna和Majda意义下的欧拉方程的尖锐测度值解。© 1999 John Wiley & Sons，Inc.
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.