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Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics

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  • This paper is devoted to multiplicity results of solutions to nonlocal elliptic equations modeling gravitating systems. By considering the case of Fermi--Dirac statistics as a singular perturbation of Maxwell--Boltzmann statistics, we are able to produce multiplicity results. Our method is based on cumulated mass densities and a logarithmic change of coordinates that allow us to describe the set of all solutions by a non-autonomous perturbation of an autonomous dynamical system. This has interesting consequences in terms of bifurcation diagrams, which are illustrated by some numerical computations. More specifically, we study a model based on the Fermi function as well as a simplified one for which estimates are easier to establish. The main difficulty comes from the fact that the mass enters in the equation as a parameter which makes the whole problem non-local.
    Mathematics Subject Classification: Primary: 35Q85, 70K05, 85A05; Secondary: 34E15, 37N05.


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  • [1]

    A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of many-particle systems: An essay on recent research, Monatshefte für Mathematik, 142 (2004), 35-43.doi: 10.1007/s00605-004-0239-2.


    P. Biler, J. Dolbeault, M. Esteban, T. Nadzieja and P. Markowich, Steady states for Streater's energy-transport models of self-gravitating particles, Transport in Transition Regimes (Minneapolis, MN, 2000), IMA Volumes in Mathematics and Its Applications, 135, Springer, New York, 2004, 37-56.doi: 10.1007/978-1-4613-0017-5_2.


    P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, II, Colloq. Math., 67 (1994), 297-308.


    P. Biler, P. Laurençot and T. Nadzieja, On an evolution system describing self-gravitating Fermi-Dirac particles, Adv. Differential Equations, 9 (2004), 563-586.


    P. Biler, T. Nadzieja and R. Stańczy, Nonisothermal systems of self-attracting Fermi-Dirac particles, Nonlocal Elliptic and Parabolic Problems, Banach Center Publ., 66, Polish Acad. Sci., Warsaw, 2004, 61-78.doi: 10.4064/bc66-0-5.


    P. Biler and R. Stańczy, Parabolic-elliptic systems with general density-pressure relations, Sūrikaisekikenkyūsho Kōkyūroku, 1405 (2004), 31-53.


    ________, Mean field models for self-gravitating particles, Folia Math., 13 (2006), 3-19.


    ________, Nonlinear diffusion models for self-gravitating particles, in Free Boundary Problems, Internat. Ser. Numer. Math., 154, Birkhäuser, Basel, 2007, 107-116.


    J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte für Mathematik, 133 (2001), 1-82.doi: 10.1007/s006050170032.


    P.-H. Chavanis, Phase transitions in self-gravitating systems, International Journal of Modern Physics B, 20 (2006), 3113-3198.doi: 10.1142/S0217979206035400.


    P.-H. Chavanis, P. Laurençot and M. Lemou, Chapman-Enskog derivation of the generalized Smoluchowski equation, Phys. A, 341 (2004), 145-164.doi: 10.1016/j.physa.2004.04.102.


    P.-H. Chavanis, J. Sommeria and R. Robert, Statistical mechanics of two-dimensional vortices and collisionless stellar systems, Astrophys. J., 471 (1996), 385.doi: 10.1086/177977.


    J. Dolbeault, P. Markowich, D. Oelz and C. Schmeiser, Non linear diffusions as limit of kinetic equations with relaxation collision kernels, Arch. Ration. Mech. Anal., 186 (2007), 133-158.doi: 10.1007/s00205-007-0049-5.


    J. Dolbeault and R. Stańczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations, Annales Henri Poincaré, 10 (2009), 1311-1333.doi: 10.1007/s00023-009-0016-9.


    B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.doi: 10.1007/BF01221125.


    D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.


    F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.doi: 10.1081/PDE-100002243.


    R. Stańczy, Steady states for a system describing self-gravitating Fermi-Dirac particles, Differential Integral Equations, 18 (2005), 567-582.


    ________, On some parabolic-elliptic system with self-similar pressure term, in Self-Similar Solutions of Nonlinear PDE, Banach Center Publ., 74, Polish Acad. Sci., Warsaw, 2006, 205-215.


    ________, Reaction-diffusion equations with nonlocal term, in Equadiff 2007, Wien, 2007.


    ________, Stationary solutions of the generalized Smoluchowski-Poisson equation, in Parabolic and Navier-Stokes Equations. Part 2, Banach Center Publ., 81, Polish Acad. Sci. Inst. Math., Warsaw, 2008, 493-500.


    ________, The existence of equilibria of many-particle systems, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 623-631.doi: 10.1017/S0308210508000413.


    ________, On an evolution system describing self-gravitating particles in microcanonical setting, Monatshefte für Mathematik, 162 (2011), 197-224.doi: 10.1007/s00605-010-0218-8.


    G. Wolansky, Critical behaviour of semi-linear elliptic equations with sub-critical exponents, Nonlinear Analysis, 26 (1996), 971-995.doi: 10.1016/0362-546X(94)00301-9.

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