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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.
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