American Institute of Mathematical Sciences

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

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056
[2]

Gershon Wolansky. Limit theorems for optimal mass transportation and applications to networks. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 365-374. doi: 10.3934/dcds.2011.30.365
[3]

Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353
[4]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[5]

S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343
[6]

H. E. Lomelí, J. D. Meiss. Generating forms for exact volume-preserving maps. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 361-377. doi: 10.3934/dcdss.2009.2.361
[7]

Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651
[8]

Simion Filip. Tropical dynamics of area-preserving maps. Journal of Modern Dynamics, 2019, 14: 179-226. doi: 10.3934/jmd.2019007
[9]

A. Daducci, A. Marigonda, G. Orlandi, R. Posenato. Neuronal Fiber--tracking via optimal mass transportation. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2157-2177. doi: 10.3934/cpaa.2012.11.2157
[10]

Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61
[11]

Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017
[12]

Hans Koch. On hyperbolicity in the renormalization of near-critical area-preserving maps. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7029-7056. doi: 10.3934/dcds.2016106
[13]

Horst R. Thieme. Eigenvectors of homogeneous order-bounded order-preserving maps. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1073-1097. doi: 10.3934/dcdsb.2017053
[14]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[15]

Casimir Emako, Farah Kanbar, Christian Klingenberg, Min Tang. A criterion for asymptotic preserving schemes of kinetic equations to be uniformly stationary preserving. Kinetic & Related Models, 2021, 14 (5) : 847-866. doi: 10.3934/krm.2021026
[16]

Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101
[17]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[18]

Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673
[19]

Xiangfeng Yang. Stability in measure for uncertain heat equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6533-6540. doi: 10.3934/dcdsb.2019152
[20]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy