Variational models for incompressible Euler equations

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

Variational models for incompressible Euler equations

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

Received November 2007 Revised March 2008 Published November 2008

Abstract
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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 & Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1

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