January  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 and Continuous Dynamical Systems, 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), 35-43. 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, 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.

[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), 297-308.

[4]

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

[5]

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.

[6]

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

[7]

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

[8]

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

[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), 1-82. doi: 10.1007/s006050170032.

[10]

P.-H. Chavanis, Phase transitions in self-gravitating systems, International Journal of Modern Physics B, 20 (2006), 3113-3198. 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), 145-164. 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), 385. 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), 133-158. 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), 1311-1333. 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), 209-243. 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), 101-174. doi: 10.1081/PDE-100002243.

[18]

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

[19]

________, 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.

[20]

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

[21]

________, 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.

[22]

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

[23]

________, 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.

[24]

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.

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), 35-43. 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, 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.

[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), 297-308.

[4]

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

[5]

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.

[6]

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

[7]

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

[8]

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

[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), 1-82. doi: 10.1007/s006050170032.

[10]

P.-H. Chavanis, Phase transitions in self-gravitating systems, International Journal of Modern Physics B, 20 (2006), 3113-3198. 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), 145-164. 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), 385. 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), 133-158. 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), 1311-1333. 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), 209-243. 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), 101-174. doi: 10.1081/PDE-100002243.

[18]

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

[19]

________, 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.

[20]

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

[21]

________, 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.

[22]

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

[23]

________, 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.

[24]

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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