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]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[2]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297
[3]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
[4]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044
[5]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[6]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[7]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273
[8]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[9]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467
[10]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[11]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[12]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[13]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341
[14]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[15]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382
[16]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468
[17]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[18]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345
[19]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[20]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (91)
  • HTML views (107)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences