Fokker-Planck equation in bounded domain
Annales de l'Institut Fourier, Volume 60 (2010) no. 1, p. 217-255

We study the existence and the uniqueness of a solution ϕ to the linear Fokker-Planck equation -Δϕ+div(ϕF)=f in a bounded domain of  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.

On étudie l’existence et l’unicité de solution ϕ à l’équation de Fokker-Planck linéaire -Δϕ+div(ϕF)=f sur un domaine borné de  d lorsque F est un champ de vecteurs “confinant” comme par exemple l’inverse de la distance au bord. Une illustration des résultats obtenus est donnée dans le cadre de la mécanique des fluides et des écoulements de polymères.

DOI : https://doi.org/10.5802/aif.2521
Classification:  35J25,  35Q35,  35R60,  76A05,  82D60
Keywords: Fokker-Planck equation, Bounded domain, Stationary solution, Confinement, Fluid mechanics, Polymer flows
@article{AIF_2010__60_1_217_0,
     author = {Chupin, Laurent},
     title = {Fokker-Planck equation in bounded domain},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {1},
     year = {2010},
     pages = {217-255},
     doi = {10.5802/aif.2521},
     mrnumber = {2664314},
     zbl = {1200.35305},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_1_217_0}
}
Chupin, Laurent. Fokker-Planck equation in bounded domain. Annales de l'Institut Fourier, Volume 60 (2010) no. 1, pp. 217-255. doi : 10.5802/aif.2521. http://www.numdam.org/item/AIF_2010__60_1_217_0/

[1] Arnold, A.; Markowich, P.; Toscani, G.; Unterreiter, A. On convex Sobolev Inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Communications in Partial Differential Equations, Tome 26 (2001) no. 1, pp. 43-100 | Article | MR 1842428 | Zbl 0982.35113

[2] Boyer, Franck Trace theorems and spatial continuity properties for the solutions of the transport equation, Differential Integral Equations, Tome 18 (2005) no. 8, pp. 891-934 | MR 2150445

[3] Boyer, Franck; Fabrie, Pierre Eléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles, Springer-Verlag, Berlin, Mathématiques & Applications (Berlin) [Mathematics & Applications], Tome 52 (2006) | MR 2248409 | Zbl 1105.76003

[4] Brascamp, Herm Jan; Lieb, Elliott H. 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, Tome 22 (1976) no. 4, pp. 366-389 | Article | MR 450480 | Zbl 0334.26009

[5] Chupin, L. The FENE model for viscoelastic thin film flows: Justification of new models and applications (2008) (Submitted)

[6] Degond, P.; Lemou, M.; Picasso, M. Constitutive relations for viscoelastic fluid models derived from kinetic theory, Dispersive transport equations and multiscale models (Minneapolis, MN, 2000), Springer, New York (IMA Vol. Math. Appl.) Tome 136 (2004), pp. 77-89 | MR 2045232 | Zbl 1145.76315

[7] Droniou, J. Non-coercive linear elliptic problems, Potential Anal., Tome 17 (2002) no. 2, pp. 181-203 | Article | MR 1908676 | Zbl 1161.35362

[8] Droniou, J.; Vazquez, J.-L. Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var., Tome 34 (2009) no. 4, pp. 413-434 | Article | MR 2476418 | Zbl 1167.35342

[9] Escande, Df; Sattin, F. When Can the Fokker-Planck Equation Describe Anomalous or Chaotic Transport?, Physical Review Letters, Tome 99 (2007) no. 18, pp. 185005 | Article

[10] Ghosh, I.; Mckinley, G.H.; Brown, R.A.; Armstrong, R.C. Deficiencies of FENE dumbbell models in describing the rapid stretching of dilute polymer solutions, Journal of Rheology, Tome 45 (2001), pp. 721 | Article

[11] Guíñez, J.; Rueda, A. D. Steady states for a Fokker-Planck equation on S n , Acta Math. Hungar., Tome 94 (2002) no. 3, pp. 211-221 | Article | MR 1905726 | Zbl 1007.58013

[12] Heinonen, Juha; Kilpeläinen, Tero; Martio, Olli Nonlinear potential theory of degenerate elliptic equations, Dover Publications Inc., Mineola, NY (2006) (Unabridged republication of the 1993 original) | MR 2305115 | Zbl 1115.31001

[13] Helffer, B.; Nier, F. Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1862 (2005) | MR 2130405 | Zbl 1072.35006

[14] Hérau, Frédéric; Nier, Francis Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., Tome 171 (2004) no. 2, pp. 151-218 | Article | MR 2034753 | Zbl 1139.82323

[15] Jourdain, Benjamin; Lelièvre, Tony; Le Bris, Claude Existence of solution for a micro-macro model of polymeric fluid: the FENE model, J. Funct. Anal., Tome 209 (2004) no. 1, pp. 162-193 | Article | MR 2039220 | Zbl 1047.76004

[16] Leray, J.; Schauder, J. Topologie et équations fonctionnelles, Ann. Sci. Éc. Norm. Supér., III. Ser., Tome 51 (1934), pp. 45-78 | Numdam | MR 1509338 | Zbl 0009.07301

[17] Lozinski, A.; Chauvière, C. A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations: 2D FENE model, J. Comput. Phys., Tome 189 (2003) no. 2, pp. 607-625 | Article | MR 1996059 | Zbl 1060.82525

[18] Masmoudi, N. Well posedness for the FENE dumbbell model of polymeric flows (2007) (Preprint) | MR 2456183 | Zbl 1157.35088

[19] Métivier, Guy Comportement asymptotique des valeurs propres d’opérateurs elliptiques dégénérés, Journées: Équations aux Dérivées Partielles de Rennes (1975), Soc. Math. France, Paris (1976), p. 215-249. Astérisque, No. 34-35 | Numdam | MR 492952

[20] Noarov, A. I. Generalized solvability of the stationary Fokker-Planck equation, Differ. Uravn., Tome 43 (2007) no. 6, p. 813-819, 863 | MR 2383830 | Zbl 1149.35020

[21] Öttinger, Hans Christian Stochastic processes in polymeric fluids, Springer-Verlag, Berlin (1996) (Tools and examples for developing simulation algorithms) | MR 1383323

[22] Sattin, F. Fick’s law and Fokker-Planck equation in inhomogeneous environments, Phys. Lett. A, Tome 372 (2008) no. 22, pp. 3941-3945 | Article | MR 2418394

[23] Stredulinsky, Edward W. Weighted inequalities and degenerate elliptic partial differential equations, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1074 (1984) | MR 757718 | Zbl 0541.35001

[24] Triebel, Hans Interpolation theory, function spaces, differential operators, Johann Ambrosius Barth, Heidelberg (1995) | MR 1328645 | Zbl 0830.46028

[25] Turesson, Bengt Ove Nonlinear potential theory and weighted Sobolev spaces, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1736 (2000) | MR 1774162 | Zbl 0949.31006

[26] Zeeman, E. C. Stability of dynamical systems, Nonlinearity, Tome 1 (1988) no. 1, pp. 115-155 | Article | MR 928950 | Zbl 0643.58005