Convergence of an entropic semi-discretization for nonlinear Fokker-Planck equations in Rd

Publicacions Matematiques

Volumn 52, Issue 2, 2008, Pages 413-433

  Carrillo, J A ^a   Gualdani, M P ^b   Jungel A ^c  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^c VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

Degenerate parabolic equation; Drift diffusion equation; Existence of weak solutions; Fokker planck equation; Implicit Euler scheme; Nonnegativity; Relative entropy; Uniqueness of solutions

Indexed keywords

EID: 52649181558     PISSN: 02141493     EISSN: None     Source Type: Journal    
DOI: 10.5565/PUBLMAT_52208_08     Document Type: Article

References (25)

1. L. AMBROSIO, N. GiGLI, AND G. SAVARÉ, "Gradient flows in metric spaces and in the space of probability measures", Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.
2. t - Δψ (u) = 0, Indiana Univ. Math. J. 30(2) (1981), 161-177.
3. M. BERTSCH AND D. A. HILHORST, A density dependent diffusion equation in population dynamics: stabilization to equilibrium, SIAM J. Math. Anal. 17(4) (1986), 863-883.
4. P. BILER, J. DOLBEAULT, AND P. A. MARKOWICH, Large time asymptotics of nonlinear drift-diffusion systems with Poisson coupling, The Sixteenth International Conference on Transport Theory, Part II (Atlanta, GA, 1999), Transport Theory Statist. Phys. 30(4-6) (2001), 521-536.
5. J. A. CARRILLO, M. DI FRANCESCO, AND M. P. GUALDANI, Semidiscretization and long-time asymptotics of nonlinear diffusion equations, Commun. Math. Sci. 5, Supplement (2007), 21-53.
6. J. A. CARRILLO, A. JÜNGEL, P. A. MARKOWICH, G. TOSCANI, AND A. UNTERREITER, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133(1) (2001), 1-82.
7. J. A. CARRILLO, R. J. MCCANN, AND C. VILLANI, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana 19(3) (2003), 971-1018.
8. P.-H. CHAVANIS, Generalized thermodynamics and Fokker-Planck equations: Applications to stellar dynamics and two-dynamical turbulence, Phys. Rev. E 68 (2003), 036108.
9. G. CHAVENT AND J. JAFFRE, "Mathematical Models and Finite Elements for Reservoir Simulation-Single Phase, Multiphase and Multicomponent Flows through Porous Media", Studies in Mathematics and its Applications 17, North-Holland, Amsterdam, 1986.
10. J. I. DÍAZ, G. GALIANO, AND A. JÜNGEL, On a quasilinear degenerate system arising in semiconductors theory. I. Existence and uniqueness of solutions, Nonlinear Anal. Real World Appl. 2(3) (2001), 305 336.
11. W. FANG AND K. ITO, Solutions to a nonlinear drift-diffusion model for semiconductors, Electron. J. Differential Equations 1999(15) (1999), 1 38 (electronic).
12. J. FIESTAS, R. SPURZEM, AND E. KIM, 2D Fokker-Planck models of rotating clusters, Monthly Notices Roy. Astronom. Soc. (to appear).
13. T. D. FRANK, "Nonlinear Fokker-Planck equations", Fundamentals and applications, Springer Series in Synergetics, Springer-Verlag, Berlin, 2005.
14. A. JÜNGEL, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci. 4(5) (1994), 677-703.
15. A. JÜNGEL, "Quasi-hydrodynamic semiconductor equations", Progress in Nonlinear Differential Equations and their Applications 41, Birkhäuser Verlag, Basel, 2001.
16. O. A. LADYZHENSKAYA AND N. N. URAL'TSEVA, "Linear and quasilinear elliptic equations", Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968.
17. P. A. MARKOWICH, C. A. RINGHOFER, AND C. SCHMEISER, "Semiconductor equations", Springer-Verlag, Vienna, 1990.
18. J. NIETO, Hydrodynamical limit for a drift-diffusion system modeling large-population dynamics, J. Math. Anal. Appl. 291(2) (2004), 716-726.
19. F. OTTO, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26(1-2) (2001), 101-174.
20. N. ROSTOKER AND M. N. ROSENBLUTH, Fokker-Planck equation for a plasma with a constant magnetic field, J. Nucl. Energy. Part C: Plasma Physics 2 (1961), 195-205.
21. J. RULLA, Weak solutions to Stefan problems with prescribed convection, SIAM J. Math. Anal. 18(6) (1987), 1784-1800.
22. p(0, T;B), Ann. Mat. Pura Appl. (4) 146 (1987), 65-96.
23. J. L. Vázquez, Asymptotic beahviour for the porous medium equation posed in the whole space, Dedicated to Philippe Bénilan, J. Evol. Equ. 3(1) (2003), 67-118.
24. J. L. VAZQUEZ, "The porous medium equation", Mathematical theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
25. C. VILLANI, "Topics in optimal transportation", Graduate Studies in Mathematics 58, American Mathematical Society, Providence, RI, 2003.