Localized minimizers of flat rotating gravitational systems
Dolbeault, Jean ; Fernández, Javier
Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008), p. 1043-1071 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{AIHPC_2008__25_6_1043_0,
     author = {Dolbeault, Jean and Fern\'andez, Javier},
     title = {Localized minimizers of flat rotating gravitational systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {25},
     year = {2008},
     pages = {1043-1071},
     doi = {10.1016/j.anihpc.2007.01.001},
     mrnumber = {2466321},
     zbl = {1157.35112},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2008__25_6_1043_0}
}

Dolbeault, Jean; Fernández, Javier. Localized minimizers of flat rotating gravitational systems. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) pp. 1043-1071. doi : 10.1016/j.anihpc.2007.01.001. http://gdmltest.u-ga.fr/item/AIHPC_2008__25_6_1043_0/

[1] Arnol'D V.I., On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid, Dokl. Akad. Nauk SSSR 162 (1965) 975-978. | MR 180051 | Zbl 0141.43901

[2] Arnol'D V.I., An a priori estimate in the theory of hydrodynamic stability, Izv. Vyssh. Uchebn. Zaved. Mat. 1966 (1966) 3-5. | MR 205552 | Zbl 0158.44904

[3] Bavaud F., Equilibrium properties of the Vlasov functional: the generalized Poisson-Boltzmann-Emden equation, Rev. Modern Phys. 63 (1991) 129-148. | MR 1102194

[4] Ben Abdallah N., Dolbeault J., Relative entropies for the Vlasov-Poisson system in bounded domains, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 867-872. | MR 1771949 | Zbl 0960.35100

[5] Ben Abdallah N., Dolbeault J., Relative entropies for kinetic equations in bounded domains (irreversibility, stationary solutions, uniqueness), Arch. Ration. Mech. Anal. 168 (2003) 253-298. | MR 1994744 | Zbl 1044.76054

[6] Biler P., Laurençot P., Nadzieja T., On an evolution system describing self-gravitating Fermi-Dirac particles, Adv. Differential Equations 9 (2004) 563-586. | MR 2099972 | Zbl 1103.35086

[7] Biler P., Nadzieja T., Stańczy R., Nonisothermal systems of self-attracting Fermi-Dirac particles, in: Nonlocal Elliptic and Parabolic Problems, Banach Center Publ., vol. 66, Polish Acad. Sci., Warsaw, 2004, pp. 61-78. | MR 2143357 | Zbl 1146.35415

[8] P. Biler, R. Stánczy, Parabolic-elliptic systems with general density-pressure relations, Tech. rep., Mathematical Institute of the University of Wrocław, 2004.

[9] Binney J., Tremaine S., Galactic Dynamics, Princeton University Press, Princeton, 1987. | Zbl 1130.85301

[10] Burchard A., Guo Y., Compactness via symmetrization, J. Funct. Anal. 214 (2004) 40-73. | MR 2079885 | Zbl 1065.49006

[11] Catrina F., Wang Z.-Q., On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001) 229-258. | MR 1794994 | Zbl 1072.35506

[12] Chavanis P.-H., Generalized thermodynamics and Fokker-Planck equations: applications to stellar dynamics and two-dimensional turbulence, Phys. Rev. E (2003) 036108.

[13] Chavanis P.-H., Generalized Fokker-Planck equations and effective thermodynamics, Phys. A 340 (2004) 57-65, News and expectations in thermostatistics. | MR 2088323

[14] Chavanis P.-H., Generalized kinetic equations and effective thermodynamics, Banach Center Publ. 66 (2004) 79-102. | MR 2143358 | Zbl pre02122828

[15] Chavanis P.-H., Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski, Physica A 332 (2004) 89. | MR 2048365

[16] Chavanis P.-H., Hamiltonian and Brownian systems with long-range interactions, Physica A 361 (2006) 55-80. | MR 2185919

[17] Chavanis P.-H., Laurencot P., Lemou M., Chapman-Enskog derivation of the generalized Smoluchowski equation, Physica A 341 (2004) 145-164. | MR 2092680

[18] Chavanis P.-H., Ribot M., Rosier C., Sire C., On the analogy between self-gravitating Brownian particles and bacterial populations, in: Banach Center Publ., vol. 66, 2004, pp. 103. | MR 2143359 | Zbl 1055.92005

[19] Chavanis P.-H., Sire C., Anomalous diffusion and collapse of self-gravitating Langevin particles in d dimensions, Phys. Rev. E 69 (2004) 016116.

[20] Collet J.F., Extensive Lyapounov functionals for moment-preserving evolution equations, C. R. Math. Acad. Sci. Paris, Ser. I 334 (2002) 429-434. | MR 1892947 | Zbl 1090.82026

[21] Cortázar C., Elgueta M., Felmer P., Existence of signed solutions for a semilinear elliptic boundary value problem, Differential Integral Equations 7 (1994) 293-299. | MR 1250952 | Zbl 0819.35052

[22] Cortázar C., Elgueta M., Felmer P., On a semilinear elliptic problem in R N with a non-Lipschitzian nonlinearity, Adv. Differential Equations 1 (1996) 199-218. | MR 1364001 | Zbl 0845.35031

[23] Dolbeault J., Monokinetic charged particle beams: qualitative behavior of the solutions of the Cauchy problem and 2d time-periodic solutions of the Vlasov-Poisson system, Comm. Partial Differential Equations 25 (2000) 1567-1647. | MR 1778774 | Zbl 0971.35075

[24] J. Dolbeault, P. Felmer, J. Mayorga-Zambrano, Compactness properties for trace-class operators and applications to quantum mechanics, Monatshefte für Mathematik (2008), in press. | MR 2434925 | Zbl 1151.81014

[25] Dolbeault J., Fernández J., Sánchez O., Stability for the gravitational Vlasov-Poisson system in dimension two, Comm. Partial Differential Equations 31 (2006) 1425-1449. | MR 2273960 | Zbl 1107.35009

[26] Dolbeault J., Markowich P., Oelz D., Schmeiser C., Nonlinear diffusions as limit of kinetic equations with relaxation collision kernels, Arch. Rational Mech. Anal. 186 (2007) 133-158. | MR 2338354 | Zbl 1148.76047

[27] Dolbeault J., Sánchez O., Soler J., Asymptotic behaviour for the Vlasov-Poisson system in the stellar-dynamics case, Arch. Ration. Mech. Anal. 171 (2004) 301-327. | MR 2038342 | Zbl 1057.70009

[28] Felli V., Schneider M., Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type, J. Differential Equations 191 (2003) 121-142. | MR 1973285 | Zbl 1088.35023

[29] Fiřt R., Stability of disk-like galaxies. II: The Kuzmin disk, Analysis (Munich) 27 (2007) 405-424. | MR 2373664 | Zbl 1144.35340

[30] Fiřt R., Rein G., Stability of disk-like galaxies. I: Stability via reduction, Analysis (Munich) 26 (2006) 507-525. | MR 2329590 | Zbl 1135.35084

[31] Floer A., Weinstein A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986) 397-408. | MR 867665 | Zbl 0613.35076

[32] Gidas B., Ni W.M., Nirenberg L., Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68 (1979) 209-243. | MR 544879 | Zbl 0425.35020

[33] Gidas B., Ni W.M., Nirenberg L., Symmetry of positive solutions of nonlinear elliptic equations in R n , in: Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York, 1981, pp. 369-402. | MR 634248 | Zbl 0469.35052

[34] Goudon T., Poupaud F., Approximation by homogenization and diffusion of kinetic equations, Comm. Partial Differential Equations 26 (2001) 537-569. | MR 1842041 | Zbl 0988.35023

[35] Guo Y., Rein G., Existence and stability of Camm type steady states in galactic dynamics, Indiana Univ. Math. J. 48 (1999) 1237-1255. | MR 1757074 | Zbl 0945.35003

[36] Guo Y., Rein G., Stable steady states in stellar dynamics, Arch. Ration. Mech. Anal. 147 (1999) 225-243. | MR 1709211 | Zbl 0935.70011

[37] Guo Y., Rein G., Isotropic steady states in galactic dynamics, Commun. Math. Phys. 219 (2001) 607-629. | MR 1838751 | Zbl 0974.35093

[38] Guo Y., Rein G., Stable models of elliptical galaxies, Mon. Not. R. Astronom. (2003).

[39] Guo Y., Rein G., A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Commun. Math. Phys. 271 (2007) 489-509. | MR 2287914 | Zbl 1130.85002

[40] Hadžić M., Rein G., Global existence and nonlinear stability for the relativistic Vlasov-Poisson system in the gravitational case, Indiana Univ. Math. J. 56 (2007) 2453-2488. | MR 2360616 | Zbl 1133.35011

[41] Lemou M., Méhats F., Raphael P., Orbital stability and singularity formation for Vlasov-Poisson systems, C. R. Math. Acad. Sci. Paris, Ser. I 341 (2005) 269-274. | MR 2164685 | Zbl 1073.70012

[42] Li Y.Y., On uniformly rotating stars, Arch. Ration. Mech. Anal. 115 (1991) 367-393. | MR 1120853 | Zbl 0850.76784

[43] Lieb E.H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983) 349-374. | MR 717827 | Zbl 0527.42011

[44] Lieb E.H., Loss M., Analysis, Graduate Studies in Mathematics, vol. 14, second ed., American Mathematical Society, Providence, RI, 2001. | MR 1817225 | Zbl 0966.26002

[45] Luo T., Smoller J., Rotating fluids with self-gravitation in bounded domains, Arch. Ration. Mech. Anal. 173 (2004) 345-377. | MR 2091509 | Zbl 1060.76125

[46] Markowich P., Rein G., Wolansky G., Existence and nonlinear stability of stationary states of the Schrödinger-Poisson system, J. Statist. Phys. 106 (2002) 1221-1239. | MR 1889607 | Zbl 1001.82107

[47] Mccann R.J., Stable rotating binary stars and fluid in a tube, Houston J. Math. 32 (2006) 603-631, (electronic). | MR 2219334 | Zbl 1096.85006

[48] Padmanabhan T., Statistical mechanics of gravitating systems, Phys. Rep. 188 (1990) 285-362. | MR 1052381

[49] Rein G., Non-linear stability for the Vlasov-Poisson system-the energy-Casimir method, Math. Methods Appl. Sci. 17 (1994) 1129-1140. | MR 1303559 | Zbl 0814.76094

[50] Rein G., Flat steady states in stellar dynamics - existence and stability, Commun. Math. Phys. 205 (1999) 229-247. | MR 1706880 | Zbl 0937.85003

[51] Rein G., Reduction and a concentration-compactness principle for energy-Casimir functionals, SIAM J. Math. Anal. 33 (2001) 896-912, (electronic). | MR 1884728 | Zbl 1019.35003

[52] Rein G., Non-linear stability of gaseous stars, Arch. Ration. Mech. Anal. 168 (2003) 115-130. | MR 1991989 | Zbl 1044.76026

[53] Rein G., Nonlinear stability of Newtonian galaxies and stars from a mathematical perspective, Ann. New York Acad. Sci. 1045 (2005) 103-119.

[54] Sánchez O., Soler J., Orbital stability for polytropic galaxies, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 781-802. | Numdam | MR 2271693 | Zbl 1110.35011

[55] Schaeffer J., Steady states in galactic dynamics, Arch. Ration. Mech. Anal. 172 (2004) 1-19. | MR 2048565 | Zbl 1061.85001

[56] Smets D., Willem M., Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations 18 (2003) 57-75. | MR 2001882 | Zbl pre01993491

[57] Toscani G., Remarks on entropy and equilibrium states, Appl. Math. Lett. 12 (1999) 19-25. | MR 1750055 | Zbl 0940.35168

[58] Tsallis C., Possible generalization of Boltzmann-Gibbs statistics, J. Statist. Phys. 52 (1988) 479-487. | MR 968597 | Zbl 1082.82501

[59] Vladimirov V.A., Ilin K.I., On Arnol'd's variational principles in fluid mechanics, in: The Arnoldfest, Toronto, ON, 1997, Fields Inst. Commun., vol. 24, Amer. Math. Soc., Providence, RI, 1999, pp. 471-495. | Zbl 0981.76071

[60] Wolansky G., On nonlinear stability of polytropic galaxies, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 15-48. | Numdam | MR 1668556 | Zbl 0927.70019

[61] Wolansky G., Ghil M., An extension of Arnol'd's second stability theorem for the Euler equations, Physica D 94 (1996) 161-167. | MR 1398637 | Zbl 0890.58087