CRM: Centro De Giorgi
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Optimal transport: theory and applications

seminar: Variational models for incompressible Euler equations and measures on action-minimizing paths

speaker: Luigi Ambrosio (Scuola Normale Superiore)

abstract: I will describe some work in progress with A.Figalli on the variational models for incompressible Euler equations introduced and studied by Brenier in a series of papers from 1989 to 1999. We show that the purely Lagrangian model introduced in the 1989 paper is equivalent to the mixed Eulerian-Lagrangian one of 1999. As a consequence, many results of the two papers on the existence and regularity of the pressure field $P$ can be combined. By introducing more general first variations for this problem, we also investigate necessary and sufficient minimality conditions at the level of single fluid paths, proving a $BV$ regularity of the velocity of a typical path, and a kind of minimality property with respect to the (nonsmooth) Lagrangian $$ L(t,x,p):={1\over 2
2+P(t,x) $$ This provides one more link between optimal transportation and the theory of action-minimizing measures. In the end of the lecture I will illustrate some of the (many) open problems.

Fri 17 Nov, 9:00 - 9:55, Aula Dini
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