American Institute of Mathematical Sciences

2015, 35(1): 139-154. doi: 10.3934/dcds.2015.35.139

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

 1 Ceremade (UMR CNRS no. 7534), Université Paris Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16 2 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Received  September 2013 Revised  May 2014 Published  August 2014

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.
Citation: Jean Dolbeault, Robert Stańczy. Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 139-154. doi: 10.3934/dcds.2015.35.139
References:
 [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. doi: 10.1007/s00605-004-0239-2. [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, (2000), 37. doi: 10.1007/978-1-4613-0017-5_2. [3] 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. [4] P. Biler, P. Laurençot and T. Nadzieja, On an evolution system describing self-gravitating Fermi-Dirac particles,, Adv. Differential Equations, 9 (2004), 563. [5] P. Biler, T. Nadzieja and R. Stańczy, Nonisothermal systems of self-attracting Fermi-Dirac particles,, Nonlocal Elliptic and Parabolic Problems, (2004), 61. doi: 10.4064/bc66-0-5. [6] P. Biler and R. Stańczy, Parabolic-elliptic systems with general density-pressure relations,, Sūrikaisekikenkyūsho Kōkyūroku, 1405 (2004), 31. [7] ________, Mean field models for self-gravitating particles,, Folia Math., 13 (2006), 3. [8] ________, Nonlinear diffusion models for self-gravitating particles,, in Free Boundary Problems, (2007), 107. [9] 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. doi: 10.1007/s006050170032. [10] P.-H. Chavanis, Phase transitions in self-gravitating systems,, International Journal of Modern Physics B, 20 (2006), 3113. doi: 10.1142/S0217979206035400. [11] P.-H. Chavanis, P. Laurençot and M. Lemou, Chapman-Enskog derivation of the generalized Smoluchowski equation,, Phys. A, 341 (2004), 145. doi: 10.1016/j.physa.2004.04.102. [12] P.-H. Chavanis, J. Sommeria and R. Robert, Statistical mechanics of two-dimensional vortices and collisionless stellar systems,, Astrophys. J., 471 (1996). doi: 10.1086/177977. [13] 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. doi: 10.1007/s00205-007-0049-5. [14] J. Dolbeault and R. Stańczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations,, Annales Henri Poincaré, 10 (2009), 1311. doi: 10.1007/s00023-009-0016-9. [15] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. [16] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (): 241. [17] F. Otto, The geometry of dissipative evolution equations: the porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101. doi: 10.1081/PDE-100002243. [18] R. Stańczy, Steady states for a system describing self-gravitating Fermi-Dirac particles,, Differential Integral Equations, 18 (2005), 567. [19] ________, On some parabolic-elliptic system with self-similar pressure term,, in Self-Similar Solutions of Nonlinear PDE, (2006), 205. [20] ________, Reaction-diffusion equations with nonlocal term,, in Equadiff 2007, (2007). [21] ________, Stationary solutions of the generalized Smoluchowski-Poisson equation,, in Parabolic and Navier-Stokes Equations. Part 2, (2008), 493. [22] ________, The existence of equilibria of many-particle systems,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 623. doi: 10.1017/S0308210508000413. [23] ________, On an evolution system describing self-gravitating particles in microcanonical setting,, Monatshefte für Mathematik, 162 (2011), 197. doi: 10.1007/s00605-010-0218-8. [24] G. Wolansky, Critical behaviour of semi-linear elliptic equations with sub-critical exponents,, Nonlinear Analysis, 26 (1996), 971. doi: 10.1016/0362-546X(94)00301-9.

show all references

References:
 [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. doi: 10.1007/s00605-004-0239-2. [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, (2000), 37. doi: 10.1007/978-1-4613-0017-5_2. [3] 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. [4] P. Biler, P. Laurençot and T. Nadzieja, On an evolution system describing self-gravitating Fermi-Dirac particles,, Adv. Differential Equations, 9 (2004), 563. [5] P. Biler, T. Nadzieja and R. Stańczy, Nonisothermal systems of self-attracting Fermi-Dirac particles,, Nonlocal Elliptic and Parabolic Problems, (2004), 61. doi: 10.4064/bc66-0-5. [6] P. Biler and R. Stańczy, Parabolic-elliptic systems with general density-pressure relations,, Sūrikaisekikenkyūsho Kōkyūroku, 1405 (2004), 31. [7] ________, Mean field models for self-gravitating particles,, Folia Math., 13 (2006), 3. [8] ________, Nonlinear diffusion models for self-gravitating particles,, in Free Boundary Problems, (2007), 107. [9] 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. doi: 10.1007/s006050170032. [10] P.-H. Chavanis, Phase transitions in self-gravitating systems,, International Journal of Modern Physics B, 20 (2006), 3113. doi: 10.1142/S0217979206035400. [11] P.-H. Chavanis, P. Laurençot and M. Lemou, Chapman-Enskog derivation of the generalized Smoluchowski equation,, Phys. A, 341 (2004), 145. doi: 10.1016/j.physa.2004.04.102. [12] P.-H. Chavanis, J. Sommeria and R. Robert, Statistical mechanics of two-dimensional vortices and collisionless stellar systems,, Astrophys. J., 471 (1996). doi: 10.1086/177977. [13] 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. doi: 10.1007/s00205-007-0049-5. [14] J. Dolbeault and R. Stańczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations,, Annales Henri Poincaré, 10 (2009), 1311. doi: 10.1007/s00023-009-0016-9. [15] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. [16] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (): 241. [17] F. Otto, The geometry of dissipative evolution equations: the porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101. doi: 10.1081/PDE-100002243. [18] R. Stańczy, Steady states for a system describing self-gravitating Fermi-Dirac particles,, Differential Integral Equations, 18 (2005), 567. [19] ________, On some parabolic-elliptic system with self-similar pressure term,, in Self-Similar Solutions of Nonlinear PDE, (2006), 205. [20] ________, Reaction-diffusion equations with nonlocal term,, in Equadiff 2007, (2007). [21] ________, Stationary solutions of the generalized Smoluchowski-Poisson equation,, in Parabolic and Navier-Stokes Equations. Part 2, (2008), 493. [22] ________, The existence of equilibria of many-particle systems,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 623. doi: 10.1017/S0308210508000413. [23] ________, On an evolution system describing self-gravitating particles in microcanonical setting,, Monatshefte für Mathematik, 162 (2011), 197. doi: 10.1007/s00605-010-0218-8. [24] G. Wolansky, Critical behaviour of semi-linear elliptic equations with sub-critical exponents,, Nonlinear Analysis, 26 (1996), 971. doi: 10.1016/0362-546X(94)00301-9.
 [1] Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic & Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014 [2] Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117 [3] Simone Paleari, Tiziano Penati. Equipartition times in a Fermi-Pasta-Ulam system. Conference Publications, 2005, 2005 (Special) : 710-719. doi: 10.3934/proc.2005.2005.710 [4] Kousuke Kuto, Tohru Tsujikawa. Bifurcation structure of steady-states for bistable equations with nonlocal constraint. Conference Publications, 2013, 2013 (special) : 467-476. doi: 10.3934/proc.2013.2013.467 [5] Antonio Giorgilli, Simone Paleari, Tiziano Penati. Local chaotic behaviour in the Fermi-Pasta-Ulam system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 991-1004. doi: 10.3934/dcdsb.2005.5.991 [6] Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic & Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645 [7] Carmen Cortázar, Manuel Elgueta, Jorge García-Melián, Salomé Martínez. Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1409-1419. doi: 10.3934/dcds.2015.35.1409 [8] Francisco Guillén-González, Mamadou Sy. Iterative method for mass diffusion model with density dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 823-841. doi: 10.3934/dcdsb.2008.10.823 [9] Santiago Cano-Casanova. Decay rate at infinity of the positive solutions of a generalized class of $T$homas-Fermi equations. Conference Publications, 2011, 2011 (Special) : 240-249. doi: 10.3934/proc.2011.2011.240 [10] Jacopo De Simoi. Stability and instability results in a model of Fermi acceleration. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 719-750. doi: 10.3934/dcds.2009.25.719 [11] Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic & Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1 [12] Gisèle Ruiz Goldstein, Jerome A. Goldstein, Naima Naheed. A convexified energy functional for the Fermi-Amaldi correction. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 41-65. doi: 10.3934/dcds.2010.28.41 [13] Andrew Comech, David Stuart. Small amplitude solitary waves in the Dirac-Maxwell system. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1349-1370. doi: 10.3934/cpaa.2018066 [14] Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics. Networks & Heterogeneous Media, 2017, 12 (2) : 217-243. doi: 10.3934/nhm.2017009 [15] Hao Yang, Hang Qiu, Leiting Chen. An optimized direction statistics for detecting and removing random-valued impulse noise. Journal of Industrial & Management Optimization, 2018, 14 (2) : 597-611. doi: 10.3934/jimo.2017062 [16] Shinsuke Koyama, Lubomir Kostal. The effect of interspike interval statistics on the information gain under the rate coding hypothesis. Mathematical Biosciences & Engineering, 2014, 11 (1) : 63-80. doi: 10.3934/mbe.2014.11.63 [17] Kseniia Kravchuk, Alexander Vidybida. Non-Markovian spiking statistics of a neuron with delayed feedback in presence of refractoriness. Mathematical Biosciences & Engineering, 2014, 11 (1) : 81-104. doi: 10.3934/mbe.2014.11.81 [18] M. Pellicer, J. Solà-Morales. Spectral analysis and limit behaviours in a spring-mass system. Communications on Pure & Applied Analysis, 2008, 7 (3) : 563-577. doi: 10.3934/cpaa.2008.7.563 [19] Stefano Scrobogna. Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 5979-6034. doi: 10.3934/dcds.2017259 [20] Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks & Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897

2017 Impact Factor: 1.179