Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations

Journal of Nonlinear Science

Volumn 19, Issue 5, 2009, Pages 547-570

  Brenier, Yann ^a  

^a CNRS   (France)

Author keywords

Calculus of variations; Convection; Fluid mechanics; Optimal transport

Indexed keywords

EID: 84867942761     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-009-9044-3     Document Type: Article

References (33)

1. L. Ambrosio 2004 Transport equation and Cauchy problem for BV vector fields Invent. Math. 158 227 260 10.1007/s00222-004-0367-2 2096794 1075.35087 (Pubitemid 39453950)
2. Arnold, V., Khesin, B.: Topological Methods in Hydrodynamics. Applied Mathematical Sciences, vol. 125. Springer, Berlin (1998)
3. S. Angenent S. Haker A. Tannenbaum 2003 Minimizing flows for the Monge-Kantorovich problem SIAM J. Math. Anal. 35 61 97 10.1137/S0036141002410927 2001465 1042.49040 (Pubitemid 38260513)
4. J.-D. Benamou Y. Brenier 1998 Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem SIAM J. Appl. Math. 58 1450 1461 10.1137/S0036139995294111 1627555 0915.35024 (Pubitemid 128689570)
5. Y. Brenier 1991 Polar factorization and monotone rearrangement of vector-valued functions Commun. Pure Appl. Math. 64 375 417 10.1002/cpa. 3160440402 1100809
6. L. Caffarelli 1992 Boundary regularity of maps with convex potentials Commun. Pure Appl. Math. 45 1141 1151 10.1002/cpa.3160450905 1177479 0778.35015
7. D. Chae 2006 Global regularity for the 2D Boussinesq equations with partial viscosity terms Adv. Math. 203 497 513 10.1016/j.aim.2005.05.001 2227730 1100.35084 (Pubitemid 43705740)
8. F. Chalub P. Markowich B. Perthame C. Schmeiser 2004 Kinetic models for chemotaxis and their drift-diffusion limits Mon. Math. 142 123 141 10.1007/s00605-004-0234-7 2065025 1052.92005 (Pubitemid 38808997)
9. M. Cullen J. Norbury J. Purser 1991 Generalised Lagrangian solutions for atmospheric and oceanic flows SIAM J. Appl. Math. 51 20 31 10.1137/0151002 1089128 0713.76026
10. M. Cullen W. Gangbo G. Pisante 2007 The semigeostrophic equations discretized in reference and dual variables Arch. Ration. Mech. Anal. 185 341 363 10.1007/s00205-006-0040-6 2317792 1139.86003
11. R. Di Perna P.-L. Lions 1989 Ordinary differential equations, transport theory and Sobolev spaces Invent. Math. 98 511 547 10.1007/BF01393835 1022305
12. C. Doering F. Otto M. Reznikoff 2006 Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh-Bénard convection J. Fluid Mech. 560 229 241 10.1017/S0022112006000097 2265709 1122.76080 (Pubitemid 44316967)
13. U. Frisch S. Matarrese R. Mohayaee A. Sobolevski 2002 A reconstruction of the initial conditions of the Universe by optimal mass reconstruction Nature 417 260 262 10.1038/417260a (Pubitemid 34534702)
14. S. Haker L. Zhu A. Tannenbaum S. Angenent 2004 Optimal mass transport for registration and warping Int. J. Comput. Vis. 60 3 225 240 10.1023/B:VISI. 0000036836.66311.97
15. Hoskins, B.: The mathematical theory of frontogenesis. In: Annual Review of Fluid Mechanics, vol. 14, pp. 131-151. Palo Alto (1982)
16. T. Hou C. Li 2005 Global well-posedness of the viscous Boussinesq equations Discrete Contin. Dyn. Syst. 12 1 12 2121245 02154350 (Pubitemid 40426348)
17. Incropere, F.P., DeWitt, D.P.: Heat and Mass Transfer. Wiley, New York (1996)
18. W. Jäger S. Luckhaus 1992 On explosions of solutions to a system of partial differential equations modeling chemotaxis Trans. Am. Math. Soc. 329 819 824 10.2307/2153966 0746.35002
19. E. Keller L. Segel 1971 Model for chemotaxis J. Theor. Biol. 30 225 234 10.1016/0022-5193(71)90050-6
20. Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, New York (1996)
21. G. Loeper 2006 A fully nonlinear version of the incompressible Euler equations: the semigeostrophic system SIAM J. Math. Anal. 38 795 823 10.1137/050629070 2262943 1134.35046 (Pubitemid 46796974)
22. R. McCann 2001 Polar factorization of maps on Riemannian manifolds Geom. Funct. Anal. 11 589 608 10.1007/PL00001679 1844080 1011.58009
23. Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean. AMS and CIMS (2000)
24. Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids. Springer, New York (1994)
25. H.K. Moffatt 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. I. Fundamentals J. Fluid Mech. 159 359 378 10.1017/S0022112085003251 819398 0616.76121 (Pubitemid 15607339)
26. Moffatt, H.: Relaxation under topological constraints. In: Topological Aspects of the Dynamics of Fluids and Plasmas. NATO Adv. Sci. Inst. Ser. E, Appl. Sci., vol. 218. Kluwer, Dordrecht (1992)
27. J. Nieto F. Poupaud J. Soler 2001 High-field limit for the Vlasov-Poisson-Fokker-Planck system Arch. Ration. Mech. Anal. 158 29 59 10.1007/s002050100139 1834113 1038.82068
28. T. Nishiyama 2007 Magnetohydrodynamic approaches to measure-valued solutions of the two-dimensional stationary Euler equations Bull. Inst. Math. Acad. Sin. (N.S.) 2 139 154 2325710 05227677
29. Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1979)
30. Schonbek, M.: Decay of solutions to non-oscillating magneto hydrodynamics equations. In: Theory of the Navier-Stokes Equations. Ser. Adv. Math. Appl. Sci., vol. 47, pp. 179-184. World Sci., Singapore (1998)
31. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. AMS, Providence (2003)
32. V.A. Vladimirov H.K. Moffatt K.I. Ilin 1999 On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. IV J. Fluid Mech. 390 127 150 10.1017/S0022112099004991 1708045 1003.76096
33. X. Wang 2004 Infinite Prandtl number limit of Rayleigh-Bénard convection Commun. Pure Appl. Math. 57 1265 1282 10.1002/cpa.3047 1112.76032 (Pubitemid 39234317)