American Institute of Mathematical Sciences

Advanced
  • Previous Article
    NLS-like equations in bounded domains: Parabolic approximation procedure
  • DCDS-B Home
  • This Issue
  • Next Article
    Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data
January  2018, 23(1): 45-55. doi: 10.3934/dcdsb.2018004

Dynamical system modeling fermionic limit

Dorota Bors 1, and  Robert Stańczy 2,

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

Received  October 2016 Revised  April 2017 Published  January 2018

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.

Keywords: Dynamical system, Fermi-Dirac statistics, multiple solutions.
Mathematics Subject Classification: Primary: 35Q85, 70K05, 85A05; Secondary: 34E15, 37N05.
Citation: Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 45-55. doi: 10.3934/dcdsb.2018004
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.  Google Scholar

[2]

P. Biler and R. Stańczy, Parabolic-elliptic systems with general density-pressure relations, RIMS Kôkyûroku, 1405 (2004), 31-53.   Google Scholar

[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.  Google Scholar

[4]

D. Bors, Superlinear elliptic systems with distributed and boundary controls, Control Cybernet., 34 (2005), 987-1004.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Eliezer, A. K. Ghatak and H. Hora, An Introduction to Equations of State: Theory and Applications, Cambridge University Press, Cambridge, 1986. Google Scholar

[14]

E. Feireisl, Stability of flows of real monoatomic gases, Comm. Partial Differential Equations, 31 (2006), 325-348.  doi: 10.1080/03605300500358186.  Google Scholar

[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.  Google Scholar

[16]

E. Feireisl, Mathematics of Complete Fluid Systems available online: http://www.math.cas.cz/fichier/course/filepdf/course_pdf_20121011171111_35.pdf Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Krzywicki and T. Nadzieja, Some results concerning the Poisson-Boltzmann equation, Appl. Math., 21 (1991), 265-272.   Google Scholar

[23]

I. Müller and T. Ruggieri, Extended Thermodynamics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4684-0447-0.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

P. Biler and R. Stańczy, Parabolic-elliptic systems with general density-pressure relations, RIMS Kôkyûroku, 1405 (2004), 31-53.   Google Scholar

[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.  Google Scholar

[4]

D. Bors, Superlinear elliptic systems with distributed and boundary controls, Control Cybernet., 34 (2005), 987-1004.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Eliezer, A. K. Ghatak and H. Hora, An Introduction to Equations of State: Theory and Applications, Cambridge University Press, Cambridge, 1986. Google Scholar

[14]

E. Feireisl, Stability of flows of real monoatomic gases, Comm. Partial Differential Equations, 31 (2006), 325-348.  doi: 10.1080/03605300500358186.  Google Scholar

[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.  Google Scholar

[16]

E. Feireisl, Mathematics of Complete Fluid Systems available online: http://www.math.cas.cz/fichier/course/filepdf/course_pdf_20121011171111_35.pdf Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Krzywicki and T. Nadzieja, Some results concerning the Poisson-Boltzmann equation, Appl. Math., 21 (1991), 265-272.   Google Scholar

[23]

I. Müller and T. Ruggieri, Extended Thermodynamics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4684-0447-0.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Left: the heteroclinic orbit joining the points $(0,0)$ and $(2,2)$ in the Maxwell-Boltzmann case. Right: the mass-density diagram.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jean Dolbeault, Robert Stańczy. Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 139-154. doi: 10.3934/dcds.2015.35.139
[2]

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
[3]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003
[4]

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
[5]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[6]

Xiangjin Xu. Multiple solutions of super-quadratic second order dynamical systems. Conference Publications, 2003, 2003 (Special) : 926-934. doi: 10.3934/proc.2003.2003.926
[7]

Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems. Journal of Geometric Mechanics, 2014, 6 (2) : 167-236. doi: 10.3934/jgm.2014.6.167
[8]

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
[9]

Mónica Clapp, Jorge Faya. Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3265-3289. doi: 10.3934/dcds.2019135
[10]

Alexandru Kristály, Ildikó-Ilona Mezei. Multiple solutions for a perturbed system on strip-like domains. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 789-796. doi: 10.3934/dcdss.2012.5.789
[11]

Pablo Amster, Mariel Paula Kuna, Gonzalo Robledo. Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1695-1709. doi: 10.3934/cpaa.2019080
[12]

Shi-Liang Wu, Cheng-Hsiung Hsu. Entire solutions with merging fronts to a bistable periodic lattice dynamical system. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2329-2346. doi: 10.3934/dcds.2016.36.2329
[13]

Fang-Di Dong, Wan-Tong Li, Li Zhang. Entire solutions in a two-dimensional nonlocal lattice dynamical system. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2517-2545. doi: 10.3934/cpaa.2018120
[14]

Francesco Paparella, Alessandro Portaluri. Geometry of stationary solutions for a system of vortex filaments: A dynamical approach. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3011-3042. doi: 10.3934/dcds.2013.33.3011
[15]

Xiaoyan Lin, Xianhua Tang. Solutions of nonlinear periodic Dirac equations with periodic potentials. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2051-2061. doi: 10.3934/dcdss.2019132
[16]

Yu Chen, Yanheng Ding, Tian Xu. Potential well and multiplicity of solutions for nonlinear Dirac equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 587-607. doi: 10.3934/cpaa.2020028
[17]

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
[18]

Yancong Xu, Deming Zhu, Xingbo Liu. Bifurcations of multiple homoclinics in general dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 945-963. doi: 10.3934/dcds.2011.30.945
[19]

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
[20]

P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (134)
  • HTML views (135)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences