A fully nonlinear version of the incompressible Euler equations: The semigeostrophic system

SIAM Journal on Mathematical Analysis

Volumn 38, Issue 3, 2006, Pages 795-823

  Loeper, G ^a  

^a UNIVERSITÉ LYON 1   (France)

Author keywords

Monge Amp re equations; Optimal transportation; Semigeostrophic equations

Indexed keywords

BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; METEOROLOGY; NONLINEAR EQUATIONS; OPTIMIZATION;

INCOMPRESSIBLE EULER EQUATIONS; OPTIMAL TRANSPORTATION; SEMIGEOSTROPHIC SYSTEM;

EULER EQUATIONS;

EID: 34249066806     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/050629070     Document Type: Article

References (26)

1. J.-D. BENAMOU AND Y. BRENIER, Weak existence for the semigeostrophic equations formulated as a coupled Monge- Ampère/ transport problem, SIAM J. Appl. Math., 58 (1998), pp. 1450-1461.
2. J.-D. BENAMOU AND Y. BRENIER, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375-393.
3. Y. BRENIER, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), pp. 375-417.
4. Y. BRENIER, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), pp. 737-754.
5. Y. BRENIER AND G. LOEPER, A geometric approximation to the Euler equations: The Vlasov-Monge-Ampère system, Geom. Funct. Anal., 14 (2004), pp. 1182-1218.
6. 2,p estimates for solutions of the Monge-Ampère equation, Ann. of Math. 2, 131 (1990), pp. 135-150.
7. L. A. CAFFARELLI, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math. (2), 131 (1990), pp. 129-134.
8. L. A. CAFFARELLI, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), pp. 99-104.
9. D. CORDERO-ERAUSQUIN, Sur le transport de mesures périodiques, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), pp. 199-202.
10. M. CULLEN AND W. GANGBO, A variational approach for the 2-dimensional semi-geostrophic shallow water equations, Arch. Ration. Mech. Anal., 156 (2001), pp. 241-273.
11. M. J. P. CULLEN AND H. MAROOFI, The fully compressible semi-geostrophic system from meteorology, Arch. Ration. Mech. Anal., 167 (2003), pp. 309-336.
12. M. J. P. CULLEN AND R. J. PURSER, Properties of the Lagrangian semi-geostrophic equations, J. Atmospheric Sci., 46 (1989), pp. 2684-2697.
13. C. M. DAFERMOS, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss. 325, Springer-Verlag, Berlin, 2000.
14. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer-Verlag, Berlin, 1983.
15. J. KOVATS, Dini-Campanato spaces and applications to nonlinear elliptic equations, Electron. J. Differential Equations, 37 (1999).
16. J.-L. LIONS, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.
17. G. LOEPER, Quasi-neutral limit of the Euler-Poisson and Euler-Monge-Ampère systems, Comm. Partial Differential Equations, 30 (2005), pp. 1141-1167.
18. G. LOEPER, Uniqueness of solutions to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., in press.
19. G. LOEPER, On the regularity of the polar factorization for time dependent maps, Calc. Var. Partial Differential Equations, 22 (2005), pp. 343-374.
20. M. C. LOPES FILHO AND H. J. NUSSENZVEIG LOPES, Existence of a weak solution for the semigeostrophic equation with integrable initial data, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), pp. 329-339.
21. A. J. MAJDA AND A. L. BERTOZZI, Vorticity and Incompressible Flow, Cambridge Texts Appl. Math. 27, Cambridge University Press, Cambridge, UK, 2002.
22. R. J. MCCANN, A convexity principle for interacting gases, Adv. Math., 128 (1997), pp. 153-179.
23. R. J. MCCANN, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., 11 (2001), pp. 589-608.
24. C. VILLANI, Topics in Optimal Transportation, Grad. Stud. Math. 58, AMS, Providence, RI, 2003.
25. X. J. WANG, Remarks on the regularity of Monge-Ampère equations, in Proceedings of the International Conference on Nonlinear PDE (Hangzhou, 1992), F. H. Lin and G. C. Dong, eds., Academic Press, Beijing, 1992, pp. 257-263.
26. V. YOUDOVICH, Non-stationary flows of an ideal incompressible fluid, Zh. Vych. Mat., 3 (1963), pp. 1032-1066.