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The Vlasov–Poisson System for Stellar Dynamics in Spaces of Constant Curvature

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Abstract

We obtain a natural extension of the Vlasov–Poisson system for stellar dynamics to spaces of constant Gaussian curvature \({\kappa \ne 0}\): the unit sphere \({\mathbb S^2}\), for \({\kappa > 0}\), and the unit hyperbolic sphere \({\mathbb H^2}\), for \({\kappa < 0}\). These equations can be easily generalized to higher dimensions. When the particles move on a geodesic, the system reduces to a 1-dimensional problem that is more singular than the classical analogue of the Vlasov–Poisson system. In the analysis of this reduced model, we study the well-posedness of the problem and derive Penrose-type conditions for linear stability around homogeneous solutions in the sense of Landau damping.

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References

  1. Andréasson H.: The Einstein–Vlasov system/kinetic theory. Living Rev. Relativ. 14(4), 5–55 (2011)

    MATH  Google Scholar 

  2. Bardos C., Besse N.: The Cauchy problem for the Vlasov–Dirac–Benney equation and related issues in fluid mechanics and semi-classical limits. Kinet. Relat. Models 6(4), 893–917 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bardos C., Degond P.: Global existence for the Vlasov–Poisson equation in three space variables with small initial data. Ann. Inst. H. Poincaré Anal. Non Linéaire 2(2), 101–118 (1985)

    MathSciNet  MATH  Google Scholar 

  4. Batt J.: Gobal symmetric solutions of the initial value problem in stellar dynamics. J. Differ. Equ. 25, 342–364 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bobylev, A., Dukes, P., Illner, R., Victory, Jr. H.: On Vlasov Manev equations I: foundations, properties, and nonglobal existence. J. Stat. Phys. 88(3–4), 885–911 (1997)

  6. Bolyai, W., Bolyai, J.: Geometrische Untersuchungen. Hrsg. P. Stäckel, B.G. Teubner, Leipzig-Berlin (1913)

  7. Cohl, H.S.: Fundamental solution of Laplace’s equation in hyperspherical geometry. SIGMA Symmetry Integrability Geom. Methods Appl. 108 (2011). doi:10.3842/SIGMA.2011.108

  8. Cohl, H.S.: Fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry (2012). arXiv:1201.4406 [math-ph]

  9. Diacu F.: On the singularities of the curved n-body problem. Trans. Am. Math. Soc. 363(4), 2249–2264 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Diacu F.: Polygonal orbits of the curved n-body problem. Trans. Am. Math. Soc. 364(5), 2783–2802 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Diacu F.: Relative Equilibria of the Curved N-Body Problem, vol. 1. Atlantis Press, Amsterdam (2012)

    Book  MATH  Google Scholar 

  12. Diacu F.: Relative equilibria in the 3-dimensional curved n-body problem. Mem. Am. Math. Soc. 228(1071), 1–80 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Diacu F., Kordlou S.: Rotopulsators of the curved n-body problem. J. Differ. Equ. 255, 2709–2750 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Diacu F., Pérez-Chavela E., Santoprete M.: The n-body problem in spaces of constant curvature. Part I: relative equilibria. J. Nonlinear Sci. 22(2), 247–266 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Diacu F., Pérez-Chavela E., Santoprete M.: The n-body problem in spaces of constant curvature. Part II: singularities. J. Nonlinear Sci. 22(2), 267–275 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Faou E., Rousset F.: Landau damping in Sobolev spaces for the Vlasov-HMF model. Arch. Ration. Mech. Anal. 219(2), 887–902 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fowles G.R., Cassiday G.L.: Analytical Mechanics, 7th edn. Thomson Brooks/Cole, Belmont (2005)

    Google Scholar 

  18. Glassey R., Schaeffer J.: Time decay for solutions to the linearized Vlasov equation. Transp. Theory Stat. Phys. 23(4), 411–453 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Grenier E.: Defect measures of the Vlasov–Poisson system in the quasineutral regime. Commun. Partial Differ. Equ. 20, 1189–1215 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Han-Kwan, D., Rousset, F.: Quasineutral limt for Vlasov–Poisson with penrose stable data. arXiv:1508.07600v1

  21. Jabin P.-E., Nouri A.: Analytic solutions to a strongly nonlinear Vlasov equation. C. R. Math. Acad. Sci. Paris 349(910), 541–546 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jin S., Levermore C.D., McLaughlin D.W.: The semiclassical limit of the defocusing NLS hierarchy. Commun. Pure Appl. Math. 52, 613–654 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kurth R.: Das Anfangswertproblem der Stellardynamik. Z. Astrophys. 30, 213–229 (1952)

    ADS  MathSciNet  MATH  Google Scholar 

  24. Liboff R.L.: Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, 3rd edn. Springer-Verlag New York, Inc., New York (2003)

    Google Scholar 

  25. Lions P.L., Perthame B.: Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system. Invent. Math. 105(1), 415–430 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Lobachevsky, N.: The new foundations of geometry with full theory of parallels [in Russian]. In: Collected Works, vol. 2. GITTL, Moscow, p. 159 (1949)

  27. Marsden J.E., Weinstein A.: The Hamiltonian structure of the Maxwell–Vlasov equations. Phys. D 4, 394–406 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mouhot C., Villani C.: On Landau damping. Acta Math. 207(1), 29–201 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Oberhettinger F.: Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin Heidelberg (1990)

    Book  MATH  Google Scholar 

  30. Pfaffelmoser K.: Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differ. Equ. 95(2), 281–303 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Philippe G., Maxime H., Nouri A.: Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinet. Relat. Models 2(4), 707–725 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rendall, A.D.: The Einstein–Vlasov system (2002). arXiv:gr-qc/0208082v1

  33. Villani, C.: Landau Damping. Lecture Notes, CEMRACS (2010)

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Correspondence to Florin Diacu.

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Communicated by C. Mouhot

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Diacu, F., Ibrahim, S., Lind, C. et al. The Vlasov–Poisson System for Stellar Dynamics in Spaces of Constant Curvature. Commun. Math. Phys. 346, 839–875 (2016). https://doi.org/10.1007/s00220-016-2608-9

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