Figure 1.
Left: the heteroclinic orbit joining the points $(0,0)$ and $(2,2)$ in the Maxwell-Boltzmann case. Right: the mass-density diagram.
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January
2018, 23(1): 45-55.
doi: 10.3934/dcdsb.2018004
Dynamical system modeling fermionic limit
| 1. | Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Ƚódź, Poland |
| 2. | Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland |
The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate value of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero.
Mathematics Subject Classification:
Primary: 35Q85, 70K05, 85A05; Secondary: 34E15, 37N05.
Citation:
Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete and Continuous Dynamical Systems - B,
2018, 23
(1)
: 45-55.
doi: 10.3934/dcdsb.2018004
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References:
| [1] |
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.
doi: 10.4064/cm-67-2-297-308. |
| [2] |
P. Biler and R. Stańczy,
Parabolic-elliptic systems with general density-pressure relations, RIMS Kôkyûroku, 1405 (2004), 31-53.
|
| [3] |
P. Biler, T. Nadzieja and R. Stańczy,
Nonisothermal systems of self-attracting Fermi-Dirac particles, Banach Center Publ., 66 (2004), 61-78.
doi: 10.4064/bc66-0-5. |
| [4] |
D. Bors,
Superlinear elliptic systems with distributed and boundary controls, Control Cybernet., 34 (2005), 987-1004.
|
| [5] |
D. Bors and S. Walczak,
Nonlinear elliptic systems with variable boundary data, Nonlinear Anal., 52 (2003), 1347-1364.
doi: 10.1016/S0362-546X(02)00179-7. |
| [6] |
D. Bors and S. Walczak,
Stability of nonlinear elliptic systems with distributed parameters and variable boundary data, J. Comput. Appl. Math., 164/165 (2004), 117-130.
doi: 10.1016/j.cam.2003.09.014. |
| [7] |
P.-H. Chavanis,
Phase transitions in self-gravitating systems, International Journal of Modern Physics B, 20 (2006), 3113-3198.
doi: 10.1142/S0217979206035400. |
| [8] |
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. |
| [9] |
P.-H. Chavanis, M. Lemou and F. Méhats,
Models of dark matter halos based on statistical mechanics: The classical King model, Phys. Rev. D, 91 (2015), 063531.
doi: 10.1103/PhysRevD.91.063531. |
| [10] |
P.-H. Chavanis, J. Sommeria and R. Robert,
Statistical mechanics of two-dimensional vortices and collisionless stellar systems, Astrophys. J., 471 (1996), p385.
doi: 10.1086/177977. |
| [11] |
J. Dolbeault and R. Stańczy,
Bifurcation diagram and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi-Dirac statistics, Discrete Contin. Dyn. Syst., 35 (2015), 139-154.
doi: 10.3934/dcds.2015.35.139. |
| [12] |
J. Dolbeault and R. Stańczy,
Non-existence and uniqueness results for supercritical semilinear elliptic equations, Ann. Henri Poincaré, 10 (2010), 1311-1333.
doi: 10.1007/s00023-009-0016-9. |
| [13] |
S. Eliezer, A. K. Ghatak and H. Hora,
An Introduction to Equations of State: Theory and Applications, Cambridge University Press, Cambridge, 1986. |
| [14] |
E. Feireisl,
Stability of flows of real monoatomic gases, Comm. Partial Differential Equations, 31 (2006), 325-348.
doi: 10.1080/03605300500358186. |
| [15] |
E. Feireisl and P. Laurençot,
Non-isothermal Smoluchowski-Poisson equations as a singular limit of the Navier-Stokes-Fourier-Poisson system, J. Math. Pures Appl., 88 (2007), 325-349.
doi: 10.1016/j.matpur.2007.07.002. |
| [16] |
E. Feireisl, Mathematics of Complete Fluid Systems available online: http://www.math.cas.cz/fichier/course/filepdf/course_pdf_20121011171111_35.pdf |
| [17] |
E. Feireisl and A. Novotný,
Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser-Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8843-0. |
| [18] |
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. |
| [19] |
F. Golse and L. Saint-Raymond,
The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.
doi: 10.1007/s00222-003-0316-5. |
| [20] |
M. Grendar and R. K. Niven,
Generalized classical, quantum and intermediate statistics and the Pólya urn model, Phys. Lett. A, 373 (2009), 621-626.
doi: 10.1016/j.physleta.2008.12.025. |
| [21] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.
doi: 10.1007/BF00250508. |
| [22] |
A. Krzywicki and T. Nadzieja,
Some results concerning the Poisson-Boltzmann equation, Appl. Math., 21 (1991), 265-272.
|
| [23] |
I. Müller and T. Ruggieri,
Extended Thermodynamics, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4684-0447-0. |
| [24] |
R. Robert,
On the gravitational collapse of stellar systems, Classical Quantum Gravity, 15 (1998), 3827-3840.
doi: 10.1088/0264-9381/15/12/011. |
| [25] |
R. Stańczy,
Steady states for a system describing self-gravitating Fermi-Dirac particles, Differential Integral Equations, 18 (2005), 567-582.
|
| [26] |
R. Stańczy,
The existence of equlibria of many-particle systems, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 623-631.
doi: 10.1017/S0308210508000413. |
| [27] |
R. Stańczy,
On an evolution system describing self-gravitating particles in microcanonical setting, Monatsh. Math., 162 (2011), 197-224.
doi: 10.1007/s00605-010-0218-8. |
| [28] |
R. Stańczy,
On stationary and radially symmetric solutions to some drift-diffusion equations with nonlocal term, Appl. Anal., 95 (2016), 97-104.
doi: 10.1080/00036811.2014.998408. |
show all references
References:
| [1] |
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.
doi: 10.4064/cm-67-2-297-308. |
| [2] |
P. Biler and R. Stańczy,
Parabolic-elliptic systems with general density-pressure relations, RIMS Kôkyûroku, 1405 (2004), 31-53.
|
| [3] |
P. Biler, T. Nadzieja and R. Stańczy,
Nonisothermal systems of self-attracting Fermi-Dirac particles, Banach Center Publ., 66 (2004), 61-78.
doi: 10.4064/bc66-0-5. |
| [4] |
D. Bors,
Superlinear elliptic systems with distributed and boundary controls, Control Cybernet., 34 (2005), 987-1004.
|
| [5] |
D. Bors and S. Walczak,
Nonlinear elliptic systems with variable boundary data, Nonlinear Anal., 52 (2003), 1347-1364.
doi: 10.1016/S0362-546X(02)00179-7. |
| [6] |
D. Bors and S. Walczak,
Stability of nonlinear elliptic systems with distributed parameters and variable boundary data, J. Comput. Appl. Math., 164/165 (2004), 117-130.
doi: 10.1016/j.cam.2003.09.014. |
| [7] |
P.-H. Chavanis,
Phase transitions in self-gravitating systems, International Journal of Modern Physics B, 20 (2006), 3113-3198.
doi: 10.1142/S0217979206035400. |
| [8] |
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. |
| [9] |
P.-H. Chavanis, M. Lemou and F. Méhats,
Models of dark matter halos based on statistical mechanics: The classical King model, Phys. Rev. D, 91 (2015), 063531.
doi: 10.1103/PhysRevD.91.063531. |
| [10] |
P.-H. Chavanis, J. Sommeria and R. Robert,
Statistical mechanics of two-dimensional vortices and collisionless stellar systems, Astrophys. J., 471 (1996), p385.
doi: 10.1086/177977. |
| [11] |
J. Dolbeault and R. Stańczy,
Bifurcation diagram and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi-Dirac statistics, Discrete Contin. Dyn. Syst., 35 (2015), 139-154.
doi: 10.3934/dcds.2015.35.139. |
| [12] |
J. Dolbeault and R. Stańczy,
Non-existence and uniqueness results for supercritical semilinear elliptic equations, Ann. Henri Poincaré, 10 (2010), 1311-1333.
doi: 10.1007/s00023-009-0016-9. |
| [13] |
S. Eliezer, A. K. Ghatak and H. Hora,
An Introduction to Equations of State: Theory and Applications, Cambridge University Press, Cambridge, 1986. |
| [14] |
E. Feireisl,
Stability of flows of real monoatomic gases, Comm. Partial Differential Equations, 31 (2006), 325-348.
doi: 10.1080/03605300500358186. |
| [15] |
E. Feireisl and P. Laurençot,
Non-isothermal Smoluchowski-Poisson equations as a singular limit of the Navier-Stokes-Fourier-Poisson system, J. Math. Pures Appl., 88 (2007), 325-349.
doi: 10.1016/j.matpur.2007.07.002. |
| [16] |
E. Feireisl, Mathematics of Complete Fluid Systems available online: http://www.math.cas.cz/fichier/course/filepdf/course_pdf_20121011171111_35.pdf |
| [17] |
E. Feireisl and A. Novotný,
Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser-Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8843-0. |
| [18] |
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. |
| [19] |
F. Golse and L. Saint-Raymond,
The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.
doi: 10.1007/s00222-003-0316-5. |
| [20] |
M. Grendar and R. K. Niven,
Generalized classical, quantum and intermediate statistics and the Pólya urn model, Phys. Lett. A, 373 (2009), 621-626.
doi: 10.1016/j.physleta.2008.12.025. |
| [21] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.
doi: 10.1007/BF00250508. |
| [22] |
A. Krzywicki and T. Nadzieja,
Some results concerning the Poisson-Boltzmann equation, Appl. Math., 21 (1991), 265-272.
|
| [23] |
I. Müller and T. Ruggieri,
Extended Thermodynamics, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4684-0447-0. |
| [24] |
R. Robert,
On the gravitational collapse of stellar systems, Classical Quantum Gravity, 15 (1998), 3827-3840.
doi: 10.1088/0264-9381/15/12/011. |
| [25] |
R. Stańczy,
Steady states for a system describing self-gravitating Fermi-Dirac particles, Differential Integral Equations, 18 (2005), 567-582.
|
| [26] |
R. Stańczy,
The existence of equlibria of many-particle systems, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 623-631.
doi: 10.1017/S0308210508000413. |
| [27] |
R. Stańczy,
On an evolution system describing self-gravitating particles in microcanonical setting, Monatsh. Math., 162 (2011), 197-224.
doi: 10.1007/s00605-010-0218-8. |
| [28] |
R. Stańczy,
On stationary and radially symmetric solutions to some drift-diffusion equations with nonlocal term, Appl. Anal., 95 (2016), 97-104.
doi: 10.1080/00036811.2014.998408. |
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