Fokker-Planck equation in bounded domain

Laurent Chupin[1]

  • [1] Université de Lyon INSA de Lyon - Pôle de Mathématiques Institut Camille Jordan - UMR5208 - CNRS 21 av. Jean Capelle 69621 Villeurbanne cedex (France)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 1, page 217-255
  • ISSN: 0373-0956

Abstract

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We study the existence and the uniqueness of a solution  ϕ to the linear Fokker-Planck equation - Δ ϕ + div ( ϕ F ) = f in a bounded domain of  ℝ d when F is a “confinement” vector field. This field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows.

How to cite

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Chupin, Laurent. "Fokker-Planck equation in bounded domain." Annales de l’institut Fourier 60.1 (2010): 217-255. <http://eudml.org/doc/116267>.

@article{Chupin2010,
abstract = {We study the existence and the uniqueness of a solution $\varphi $ to the linear Fokker-Planck equation $-\Delta \varphi + \operatorname\{div\}(\varphi \, \{\mathbf\{F\}\}) = f$ in a bounded domain of $\mathbb\{R\}^d$ when $\mathbf\{F\}$ is a “confinement” vector field. This field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows.},
affiliation = {Université de Lyon INSA de Lyon - Pôle de Mathématiques Institut Camille Jordan - UMR5208 - CNRS 21 av. Jean Capelle 69621 Villeurbanne cedex (France)},
author = {Chupin, Laurent},
journal = {Annales de l’institut Fourier},
keywords = {Fokker-Planck equation; Bounded domain; Stationary solution; Confinement; Fluid mechanics; Polymer flows; bounded domain; stationary solution; confinement; fluid mechanics; polymer flows},
language = {eng},
number = {1},
pages = {217-255},
publisher = {Association des Annales de l’institut Fourier},
title = {Fokker-Planck equation in bounded domain},
url = {http://eudml.org/doc/116267},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Chupin, Laurent
TI - Fokker-Planck equation in bounded domain
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 1
SP - 217
EP - 255
AB - We study the existence and the uniqueness of a solution $\varphi $ to the linear Fokker-Planck equation $-\Delta \varphi + \operatorname{div}(\varphi \, {\mathbf{F}}) = f$ in a bounded domain of $\mathbb{R}^d$ when $\mathbf{F}$ is a “confinement” vector field. This field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows.
LA - eng
KW - Fokker-Planck equation; Bounded domain; Stationary solution; Confinement; Fluid mechanics; Polymer flows; bounded domain; stationary solution; confinement; fluid mechanics; polymer flows
UR - http://eudml.org/doc/116267
ER -

References

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  1. A. Arnold, P. Markowich, G. Toscani, A. Unterreiter, On convex Sobolev Inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Communications in Partial Differential Equations 26 (2001), 43-100 Zbl0982.35113MR1842428
  2. Franck Boyer, Trace theorems and spatial continuity properties for the solutions of the transport equation, Differential Integral Equations 18 (2005), 891-934 Zbl1212.35049MR2150445
  3. Franck Boyer, Pierre Fabrie, Eléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles, 52 (2006), Springer-Verlag, Berlin Zbl1105.76003MR2248409
  4. Herm Jan Brascamp, Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), 366-389 Zbl0334.26009MR450480
  5. L. Chupin, The FENE model for viscoelastic thin film flows: Justification of new models and applications, (2008) 
  6. P. Degond, M. Lemou, M. Picasso, Constitutive relations for viscoelastic fluid models derived from kinetic theory, Dispersive transport equations and multiscale models (Minneapolis, MN, 2000) 136 (2004), 77-89, Springer, New York Zbl1145.76315MR2045232
  7. J. Droniou, Non-coercive linear elliptic problems, Potential Anal. 17 (2002), 181-203 Zbl1161.35362MR1908676
  8. J. Droniou, J.-L. Vazquez, Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var. 34 (2009), 413-434 Zbl1167.35342MR2476418
  9. DF Escande, F. Sattin, When Can the Fokker-Planck Equation Describe Anomalous or Chaotic Transport?, Physical Review Letters 99 (2007) 
  10. I. Ghosh, G.H. McKinley, R.A. Brown, R.C. Armstrong, Deficiencies of FENE dumbbell models in describing the rapid stretching of dilute polymer solutions, Journal of Rheology 45 (2001) 
  11. J. Guíñez, A. D. Rueda, Steady states for a Fokker-Planck equation on S n , Acta Math. Hungar. 94 (2002), 211-221 Zbl1007.58013MR1905726
  12. Juha Heinonen, Tero Kilpeläinen, Olli Martio, Nonlinear potential theory of degenerate elliptic equations, (2006), Dover Publications Inc., Mineola, NY Zbl1115.31001MR2305115
  13. B. Helffer, F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, 1862 (2005), Springer-Verlag, Berlin Zbl1072.35006MR2130405
  14. Frédéric Hérau, Francis Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), 151-218 Zbl1139.82323MR2034753
  15. Benjamin Jourdain, Tony Lelièvre, Claude Le Bris, Existence of solution for a micro-macro model of polymeric fluid: the FENE model, J. Funct. Anal. 209 (2004), 162-193 Zbl1047.76004MR2039220
  16. J. Leray, J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. Éc. Norm. Supér., III. Ser. 51 (1934), 45-78 Zbl0009.07301MR1509338
  17. A. Lozinski, C. Chauvière, A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations: 2D FENE model, J. Comput. Phys. 189 (2003), 607-625 Zbl1060.82525MR1996059
  18. N. Masmoudi, Well posedness for the FENE dumbbell model of polymeric flows, (2007) Zbl1157.35088MR2456183
  19. Guy Métivier, Comportement asymptotique des valeurs propres d’opérateurs elliptiques dégénérés, Journées: Équations aux Dérivées Partielles de Rennes (1975) (1976), 215-249. Astérisque, No. 34-35, Soc. Math. France, Paris Zbl0329.35047MR492952
  20. A. I. Noarov, Generalized solvability of the stationary Fokker-Planck equation, Differ. Uravn. 43 (2007), 813-819, 863 Zbl1149.35020MR2383830
  21. Hans Christian Öttinger, Stochastic processes in polymeric fluids, (1996), Springer-Verlag, Berlin Zbl0995.60098MR1383323
  22. F. Sattin, Fick’s law and Fokker-Planck equation in inhomogeneous environments, Phys. Lett. A 372 (2008), 3941-3945 Zbl1220.82088MR2418394
  23. Edward W. Stredulinsky, Weighted inequalities and degenerate elliptic partial differential equations, 1074 (1984), Springer-Verlag, Berlin Zbl0541.35001MR757718
  24. Hans Triebel, Interpolation theory, function spaces, differential operators, (1995), Johann Ambrosius Barth, Heidelberg Zbl0830.46028MR1328645
  25. Bengt Ove Turesson, Nonlinear potential theory and weighted Sobolev spaces, 1736 (2000), Springer-Verlag, Berlin Zbl0949.31006MR1774162
  26. E. C. Zeeman, Stability of dynamical systems, Nonlinearity 1 (1988), 115-155 Zbl0643.58005MR928950

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