Hamiltonian ODEs in the Wasserstein space of probability measures

Communications on Pure and Applied Mathematics

Volumn 61, Issue 1, 2008, Pages 18-53

  Ambrosio, Luigi ^a   Gangbo, Wilfred ^b  

^a SCUOLA NORMALE SUPERIORE   (Italy)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 37649010339     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20188     Document Type: Article

References (29)

1. Ambrosio, L. Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158 (2004), no. 2, 227-260.
2. Ambrosio, L; Fusco, N.; Pallara, D. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Clarendon, Oxford University Press, New York, 2000.
3. Ambrosio, L.; Gigli, N.; Savaré, G. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 2005.
4. Benamou, J.-D.; Brenier, Y. Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem. SIAM J. Appl. Math. 58 (1998), no. 5, 1450-1461 (electronic).
5. Benamou, J.-D.; Brenier, Y. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000), no, 3, 375-393.
6. Bouchut, F. Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Ration. Mech. Anal. 157 (2001), no. 1, 75-90.
7. Brenier, Y. Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25 (2000), no. 3-4, 737-754.
8. Brenier,Y.; Loeper, G. A geometric approximation to the Euler equations: the Vlasov-Monge-Ampère system. Geom. Funct. Anal. 14 (2004), no. 6, 1182-1218.
9. Carrillo, J. A.; McCann, R. J.; Villani, C. Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179 (2006), no. 2, 217-263.
10. Cullen, M. J. P.; Feldman, M. Lagrangian solutions of semigeostrophic equations in physical space. SIAM J. Math. Anal. 37 (2006), no. 5, 1371-1395 (electronic).
11. Cullen, M. J. P.; Gangbo, W. A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Ration. Mech. Anal. 156 (2001), no. 3, 241-273.
12. Cullen, M. J. P.; Gangbo, W.; Pisante, G. The semigeostrophic equations discretized in reference and dual variables. Arch. Ration. Mech. Anal., forthcoming.
13. Cullen, M. J. P.; Maroofi, H. The fully compressible semigeostrophic equations from meteorology. Arch. Ration. Mech. Anal. 167 (2003), no. 4, 309-336.
14. Cullen, M. J. P.; Norbury, J.; Purser, R. J. Generalised Lagrangian solutions for atmospheric and oceanic flows. SIAM J. Appl. Math. 51 (1991), no. 1, 20-31.
15. Cullen, M. J. P.; Purser, R. J. An extended Lagrangian theory of semigeostrophic frontogenesis. J. Atmospheric Sci. 41 (1984), no. 9, 1477-1497.
16. Cullen, M. J. P.; Purser, R. J. Properties of the Lagrangian semigeostrophic equations. J. Atmospheric Sci. 46 (1989), no. 17, 2684-2697.
17. Cullen, M. J. P.; Roulstone, J. A geometric model of the nonlinear equilibration of two-dimensional Eady waves. J. Atmospheric Sci. 50 (1993), no. 2, 328-344.
18. DiPerna, R. J.; Lions, P.-L. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), no. 3, 511-547.
19. Gangbo, W; McCann, R. J. The geometry of optimal transportation. Acta Math. 177 113-161, 1996.
20. Gangbo, W.; Pacini, T. Infinite dimensional Hamiltonian systems in terms of the Wasserstein distance. In preparation.
21. Ghoussoub, N.; Moameni, A. On the existence of Hamiltonian paths connecting Lagrangian submanifolds. arXiv:math.AP/0508356, 2005.
22. Jordan, R.; Kinderlehrer, D.; Otto, F. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998), no. 1, 1-17 (electronic).
23. Jost, J. Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 1997.
24. McCann, R. J. A convexity principle for interacting gases. Adv. Math. 128 (1997), no. 1, 153-179.
25. McCann, R. J.; Oberman, A. M. Exact semi-geostrophic flows in an elliptical ocean basin. Nonlinearity 17 (2004), no. 5, 1891-1922.
26. Otto, F. The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26 (2001), no. 1-2, 101-174.
27. Sanz-Serna, J. M.; Vadillo, F. Studies in numerical nonlinear instability. III. Augmented Hamiltonian systems. SIAM J. Appl. Math. 47 (1987), no. 1, 92-108.
28. Villani, C. Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, R.I., 2003.
29. Zheng, Y. X.; Majda, A. Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data. Comm. Pure Appl. Math. 47 (1994), no. 10, 1365-1401.