A class of nonlocal parabolic problems occurring in statistical mechanics

Piotr Biler; Tadeusz Nadzieja

Colloquium Mathematicae (1993)

  • Volume: 66, Issue: 1, page 131-145
  • ISSN: 0010-1354

Abstract

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We consider parabolic equations with nonlocal coefficients obtained from the Vlasov-Fokker-Planck equations with potentials. This class of equations includes the classical Debye system from electrochemistry as well as an evolution model of self-attracting clusters under friction and fluctuations. The local in time existence of solutions to these equations (with no-flux boundary conditions) and properties of stationary solutions are studied.

How to cite

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Biler, Piotr, and Nadzieja, Tadeusz. "A class of nonlocal parabolic problems occurring in statistical mechanics." Colloquium Mathematicae 66.1 (1993): 131-145. <http://eudml.org/doc/210226>.

@article{Biler1993,
abstract = {We consider parabolic equations with nonlocal coefficients obtained from the Vlasov-Fokker-Planck equations with potentials. This class of equations includes the classical Debye system from electrochemistry as well as an evolution model of self-attracting clusters under friction and fluctuations. The local in time existence of solutions to these equations (with no-flux boundary conditions) and properties of stationary solutions are studied.},
author = {Biler, Piotr, Nadzieja, Tadeusz},
journal = {Colloquium Mathematicae},
keywords = {nonlinear boundary conditions; stationary solutions; existence of solutions; parabolic-elliptic system; nonlocal coefficients; Vlasov-Fokker-Planck equations},
language = {eng},
number = {1},
pages = {131-145},
title = {A class of nonlocal parabolic problems occurring in statistical mechanics},
url = {http://eudml.org/doc/210226},
volume = {66},
year = {1993},
}

TY - JOUR
AU - Biler, Piotr
AU - Nadzieja, Tadeusz
TI - A class of nonlocal parabolic problems occurring in statistical mechanics
JO - Colloquium Mathematicae
PY - 1993
VL - 66
IS - 1
SP - 131
EP - 145
AB - We consider parabolic equations with nonlocal coefficients obtained from the Vlasov-Fokker-Planck equations with potentials. This class of equations includes the classical Debye system from electrochemistry as well as an evolution model of self-attracting clusters under friction and fluctuations. The local in time existence of solutions to these equations (with no-flux boundary conditions) and properties of stationary solutions are studied.
LA - eng
KW - nonlinear boundary conditions; stationary solutions; existence of solutions; parabolic-elliptic system; nonlocal coefficients; Vlasov-Fokker-Planck equations
UR - http://eudml.org/doc/210226
ER -

References

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  1. [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
  2. [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, preprint, 1993, 119 pp. 
  3. [3] P. Biler, Existence and asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux boundary conditions, Nonlinear Anal. 19 (1992), 1121-1136. Zbl0781.35025
  4. [4] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Mathematical Institute, University of Wrocław, Report no 23 (1992), 24 pp. Zbl0814.35054
  5. [5] P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, I, II, to appear. Zbl0832.35015
  6. [6] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer, Berlin, 1990. Zbl0683.35001
  7. [7] H. Gajewski and K. Gröger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. 113 (1986), 12-35. Zbl0642.35038
  8. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983. Zbl0562.35001
  9. [9] L. V. Kantorovich and G. P. Akilov, Functional Analysis, 2nd ed., Pergamon Press, Oxford, 1982. Zbl0484.46003
  10. [10] A. Krzywicki and T. Nadzieja, Some results concerning the Poisson-Boltzmann equation, Zastos. Mat. 21 (1991), 265-272. Zbl0756.35029
  11. [11] A. Krzywicki and T. Nadzieja, A nonstationary problem in the theory of electrolytes, Quart. Appl. Math. 50 (1992), 105-107. Zbl0754.35142
  12. [12] A. Krzywicki and T. Nadzieja, A note on the Poisson-Boltzmann equation, Zastos. Mat. 21 (1993), 591-595. Zbl0780.35033
  13. [13] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1988. 
  14. [14] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1968. 
  15. [15] G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rational Mech. Anal. 119 (1992), 355-391. Zbl0774.76069

Citations in EuDML Documents

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  1. Piotr Biler, Wojbor Woyczyński, Nonlocal quadratic evolution problems
  2. Benoît Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic
  3. Piotr Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles, III
  4. Piotr Biler, Tadeusz Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, I
  5. Tadeusz Nadzieja, A model of a radially symmetric cloud of self-attracting particles
  6. Andrzej Krzywicki, Tadeusz Nadzieja, Nonlocal elliptic problems
  7. Andrzej Raczyński, On a nonlocal elliptic problem
  8. Piotr Biler, Growth and accretion of mass in an astrophysical model

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