Geodesics on groups of volume preserving maps pdf file See final version in CPAM

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50 pages

English

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MINIMAL GEODESICS ON GROUPS OF VOLUME-PRESERVING MAPS AND GENERALIZED SOLUTIONS OF THE EULER EQUATIONS Yann Brenier Abstra t. The three-dimensional motion of an in ompressible invis id uid is las- si ally des ribed by the Euler equations, but an also be seen, following Arnold [1?, as a geodesi on a group of volume-preserving maps. Lo al ex- isten e and uniqueness of minimal geodesi s have been established by Ebin and Marsden [16?. In the large, for a large lass of data, the existen e of minimal geodesi s may fail, as shown by Shnirelman [26?. For su h 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?. 1. Problems and results. 1.1. Lagrangian des ription of in ompressible uids. Let D be the unit ube [0; 1? d in R d (or the at torus T d = R d =Z d ), let T > 0 be a nite xed time and set Q = [0; T ?D.