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January  2009, 11(1): 1-10. doi: 10.3934/dcdsb.2009.11.1

Variational models for incompressible Euler equations

Luigi Ambrosio 1,

1. 

Scuola Normale Superiore, Piazza Cavalieri 7, 56123 Pisa, Italy

Received  November 2007 Revised  March 2008 Published  November 2008

In this paper we illustrate some recent work [1], [2] on Brenier's variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field.
Keywords: Euler equations, Optimal transportation, Measure-preserving maps.
Mathematics Subject Classification: Primary: 49J45; Secondary: 49K99, 76B99.
Citation: Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1
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