Localized minimizers of flat rotating gravitational systems

Jean Dolbeault; Javier Fernández

Annales de l'I.H.P. Analyse non linéaire (2008)

  • Volume: 25, Issue: 6, page 1043-1071
  • ISSN: 0294-1449

How to cite

top

Dolbeault, Jean, and Fernández, Javier. "Localized minimizers of flat rotating gravitational systems." Annales de l'I.H.P. Analyse non linéaire 25.6 (2008): 1043-1071. <http://eudml.org/doc/78823>.

@article{Dolbeault2008,
author = {Dolbeault, Jean, Fernández, Javier},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {stellar dynamics; gravitation; mass; rotation; angular velocity; entropy; diffusion limit; drift-diffusion; Hardy-Littlewood-Sobolev inequality; bounded solutions; minimization; solutions with compact support; radial solutions; localized minimizers},
language = {eng},
number = {6},
pages = {1043-1071},
publisher = {Elsevier},
title = {Localized minimizers of flat rotating gravitational systems},
url = {http://eudml.org/doc/78823},
volume = {25},
year = {2008},
}

TY - JOUR
AU - Dolbeault, Jean
AU - Fernández, Javier
TI - Localized minimizers of flat rotating gravitational systems
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2008
PB - Elsevier
VL - 25
IS - 6
SP - 1043
EP - 1071
LA - eng
KW - stellar dynamics; gravitation; mass; rotation; angular velocity; entropy; diffusion limit; drift-diffusion; Hardy-Littlewood-Sobolev inequality; bounded solutions; minimization; solutions with compact support; radial solutions; localized minimizers
UR - http://eudml.org/doc/78823
ER -

References

top
  1. [1] Arnol'd V.I., On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid, Dokl. Akad. Nauk SSSR162 (1965) 975-978. Zbl0141.43901MR180051
  2. [2] Arnol'd V.I., An a priori estimate in the theory of hydrodynamic stability, Izv. Vyssh. Uchebn. Zaved. Mat.1966 (1966) 3-5. Zbl0158.44904MR205552
  3. [3] Bavaud F., Equilibrium properties of the Vlasov functional: the generalized Poisson–Boltzmann–Emden equation, Rev. Modern Phys.63 (1991) 129-148. MR1102194
  4. [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. Zbl0960.35100MR1771949
  5. [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. Zbl1044.76054MR1994744
  6. [6] Biler P., Laurençot P., Nadzieja T., On an evolution system describing self-gravitating Fermi–Dirac particles, Adv. Differential Equations9 (2004) 563-586. Zbl1103.35086MR2099972
  7. [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. Zbl1146.35415MR2143357
  8. [8] P. Biler, R. Stánczy, Parabolic-elliptic systems with general density–pressure relations, Tech. rep., Mathematical Institute of the University of Wrocław, 2004. 
  9. [9] Binney J., Tremaine S., Galactic Dynamics, Princeton University Press, Princeton, 1987. Zbl1130.85301
  10. [10] Burchard A., Guo Y., Compactness via symmetrization, J. Funct. Anal.214 (2004) 40-73. Zbl1065.49006MR2079885
  11. [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. Zbl1072.35506MR1794994
  12. [12] Chavanis P.-H., Generalized thermodynamics and Fokker–Planck equations: applications to stellar dynamics and two-dimensional turbulence, Phys. Rev. E (2003) 036108. 
  13. [13] Chavanis P.-H., Generalized Fokker–Planck equations and effective thermodynamics, Phys. A340 (2004) 57-65, News and expectations in thermostatistics. MR2088323
  14. [14] Chavanis P.-H., Generalized kinetic equations and effective thermodynamics, Banach Center Publ.66 (2004) 79-102. Zbl1235.82060MR2143358
  15. [15] Chavanis P.-H., Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski, Physica A332 (2004) 89. MR2048365
  16. [16] Chavanis P.-H., Hamiltonian and Brownian systems with long-range interactions, Physica A361 (2006) 55-80. MR2185919
  17. [17] Chavanis P.-H., Laurencot P., Lemou M., Chapman–Enskog derivation of the generalized Smoluchowski equation, Physica A341 (2004) 145-164. MR2092680
  18. [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. Zbl1055.92005MR2143359
  19. [19] Chavanis P.-H., Sire C., Anomalous diffusion and collapse of self-gravitating Langevin particles in d dimensions, Phys. Rev. E69 (2004) 016116. 
  20. [20] Collet J.F., Extensive Lyapounov functionals for moment-preserving evolution equations, C. R. Math. Acad. Sci. Paris, Ser. I334 (2002) 429-434. Zbl1090.82026MR1892947
  21. [21] Cortázar C., Elgueta M., Felmer P., Existence of signed solutions for a semilinear elliptic boundary value problem, Differential Integral Equations7 (1994) 293-299. Zbl0819.35052MR1250952
  22. [22] Cortázar C., Elgueta M., Felmer P., On a semilinear elliptic problem in R N with a non-Lipschitzian nonlinearity, Adv. Differential Equations1 (1996) 199-218. Zbl0845.35031MR1364001
  23. [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 Equations25 (2000) 1567-1647. Zbl0971.35075MR1778774
  24. [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. Zbl1151.81014MR2434925
  25. [25] Dolbeault J., Fernández J., Sánchez O., Stability for the gravitational Vlasov–Poisson system in dimension two, Comm. Partial Differential Equations31 (2006) 1425-1449. Zbl1107.35009MR2273960
  26. [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. Zbl1148.76047MR2338354
  27. [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. Zbl1057.70009MR2038342
  28. [28] Felli V., Schneider M., Perturbation results of critical elliptic equations of Caffarelli–Kohn–Nirenberg type, J. Differential Equations191 (2003) 121-142. Zbl1088.35023MR1973285
  29. [29] Fiřt R., Stability of disk-like galaxies. II: The Kuzmin disk, Analysis (Munich)27 (2007) 405-424. Zbl1144.35340MR2373664
  30. [30] Fiřt R., Rein G., Stability of disk-like galaxies. I: Stability via reduction, Analysis (Munich)26 (2006) 507-525. Zbl1135.35084MR2329590
  31. [31] Floer A., Weinstein A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal.69 (1986) 397-408. Zbl0613.35076MR867665
  32. [32] Gidas B., Ni W.M., Nirenberg L., Symmetry and related properties via the maximum principle, Commun. Math. Phys.68 (1979) 209-243. Zbl0425.35020MR544879
  33. [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. Zbl0469.35052MR634248
  34. [34] Goudon T., Poupaud F., Approximation by homogenization and diffusion of kinetic equations, Comm. Partial Differential Equations26 (2001) 537-569. Zbl0988.35023MR1842041
  35. [35] Guo Y., Rein G., Existence and stability of Camm type steady states in galactic dynamics, Indiana Univ. Math. J.48 (1999) 1237-1255. Zbl0945.35003MR1757074
  36. [36] Guo Y., Rein G., Stable steady states in stellar dynamics, Arch. Ration. Mech. Anal.147 (1999) 225-243. Zbl0935.70011MR1709211
  37. [37] Guo Y., Rein G., Isotropic steady states in galactic dynamics, Commun. Math. Phys.219 (2001) 607-629. Zbl0974.35093MR1838751
  38. [38] Guo Y., Rein G., Stable models of elliptical galaxies, Mon. Not. R. Astronom. (2003). 
  39. [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. Zbl1130.85002MR2287914
  40. [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. Zbl1133.35011MR2360616
  41. [41] Lemou M., Méhats F., Raphael P., Orbital stability and singularity formation for Vlasov–Poisson systems, C. R. Math. Acad. Sci. Paris, Ser. I341 (2005) 269-274. Zbl1073.70012MR2164685
  42. [42] Li Y.Y., On uniformly rotating stars, Arch. Ration. Mech. Anal.115 (1991) 367-393. Zbl0850.76784MR1120853
  43. [43] Lieb E.H., Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2)118 (1983) 349-374. Zbl0527.42011MR717827
  44. [44] Lieb E.H., Loss M., Analysis, Graduate Studies in Mathematics, vol. 14, second ed., American Mathematical Society, Providence, RI, 2001. Zbl0966.26002MR1817225
  45. [45] Luo T., Smoller J., Rotating fluids with self-gravitation in bounded domains, Arch. Ration. Mech. Anal.173 (2004) 345-377. Zbl1060.76125MR2091509
  46. [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. Zbl1001.82107MR1889607
  47. [47] McCann R.J., Stable rotating binary stars and fluid in a tube, Houston J. Math.32 (2006) 603-631, (electronic). Zbl1096.85006MR2219334
  48. [48] Padmanabhan T., Statistical mechanics of gravitating systems, Phys. Rep.188 (1990) 285-362. Zbl1211.82001MR1052381
  49. [49] Rein G., Non-linear stability for the Vlasov–Poisson system—the energy-Casimir method, Math. Methods Appl. Sci.17 (1994) 1129-1140. Zbl0814.76094MR1303559
  50. [50] Rein G., Flat steady states in stellar dynamics – existence and stability, Commun. Math. Phys.205 (1999) 229-247. Zbl0937.85003MR1706880
  51. [51] Rein G., Reduction and a concentration-compactness principle for energy-Casimir functionals, SIAM J. Math. Anal.33 (2001) 896-912, (electronic). Zbl1019.35003MR1884728
  52. [52] Rein G., Non-linear stability of gaseous stars, Arch. Ration. Mech. Anal.168 (2003) 115-130. Zbl1044.76026MR1991989
  53. [53] Rein G., Nonlinear stability of Newtonian galaxies and stars from a mathematical perspective, Ann. New York Acad. Sci.1045 (2005) 103-119. 
  54. [54] Sánchez O., Soler J., Orbital stability for polytropic galaxies, Ann. Inst. H. Poincaré Anal. Non Linéaire23 (2006) 781-802. Zbl1110.35011MR2271693
  55. [55] Schaeffer J., Steady states in galactic dynamics, Arch. Ration. Mech. Anal.172 (2004) 1-19. Zbl1061.85001MR2048565
  56. [56] Smets D., Willem M., Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations18 (2003) 57-75. Zbl1274.35026MR2001882
  57. [57] Toscani G., Remarks on entropy and equilibrium states, Appl. Math. Lett.12 (1999) 19-25. Zbl0940.35168MR1750055
  58. [58] Tsallis C., Possible generalization of Boltzmann–Gibbs statistics, J. Statist. Phys.52 (1988) 479-487. Zbl1082.82501MR968597
  59. [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. Zbl0981.76071
  60. [60] Wolansky G., On nonlinear stability of polytropic galaxies, Ann. Inst. H. Poincaré Anal. Non Linéaire16 (1999) 15-48. Zbl0927.70019MR1668556
  61. [61] Wolansky G., Ghil M., An extension of Arnol'd's second stability theorem for the Euler equations, Physica D94 (1996) 161-167. Zbl0890.58087MR1398637

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.