2007/01/31 by Luigi Ambrosio, L. Ambrosio, A. Figalli +1
Mathematics · #Compressibility #Energy (signal processing) #Euler's formula #Eulerian path #Geodesic #Geometric Analysis and Curvature Flows #Lagrangian #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Relaxation (psychology) #Space (punctuation) #math.AP
paper · pdf · doi:10.1007/s00205-008-0189-2
arxiv created 2007/10/21 · openalex publication_date 2008/11/06 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study 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. Moreover, adapting some ideas of Shnirelman, we show that, even for non-deterministic final conditions, generalized flows can be approximated in energy by flows associated to measure-preserving maps.