## Publicacions Matemàtiques

### Convergence of an Entropic Semi-discretization for Nonlinear Fokker-Planck Equations in ${\mathbb R}^d$

#### Abstract

A nonlinear degenerate Fokker-Planck equation in the whole space is analyzed. The existence of solutions to the corresponding implicit Euler scheme is proved, and it is shown that the semi-discrete solution converges to a solution of the continuous problem. Furthermore, the discrete entropy decays monotonically in time and the solution to the continuous problem is unique. The nonlinearity is assumed to be of porous-medium type. For the (given) potential, either a less than quadratic growth condition at infinity is supposed or the initial datum is assumed to be compactly supported. The existence proof is based on regularization and maximum principle arguments. Upper bounds for the tail behavior in space at infinity are also derived in the at-most-quadratic growth case.

#### Article information

Source
Publ. Mat. Volume 52, Number 2 (2008), 413-433.

Dates
First available in Project Euclid: 5 August 2008

https://projecteuclid.org/euclid.pm/1217964240

Mathematical Reviews number (MathSciNet)
MR2436732

Zentralblatt MATH identifier
1152.35317

#### Citation

Carrillo, J. A.; Gualdani, M. P.; Jüngel, A. Convergence of an Entropic Semi-discretization for Nonlinear Fokker-Planck Equations in ${\mathbb R}^d$. Publ. Mat. 52 (2008), no. 2, 413--433. https://projecteuclid.org/euclid.pm/1217964240.

#### References

• 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.
• P. Bénilan and M. G. Crandall, The continuous dependence on $\varphi$ of solutions of $u\sb{t}-\Delta \varphi (u)=0$, Indiana Univ. Math. J. 30(2) (1981), 161\Ndash177.
• 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\Ndash883.
• 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\Ndash6) (2001), 521\Ndash536.
• 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\Ndash53.
• 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\Ndash82.
• 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\Ndash1018.
• P.-H. Chavanis, Generalized thermodynamics and Fokker-Planck equations: Applications to stellar dynamics and two-dynamical turbulence, Phys. Rev. E 68 (2003), 036108.
• 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.
• 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\Ndash336.
• W. Fang and K. Ito, Solutions to a nonlinear drift-diffusion model for semiconductors, Electron. J. Differential Equations 1999(15) (1999), 1\Ndash38 (electronic).
• J. Fiestas, R. Spurzem, and E. Kim, 2D Fokker-Planck models of rotating clusters, Monthly Notices Roy. Astronom. Soc. (to appear).
• T. ED. Frank, “Nonlinear Fokker-Planck equations”, Fundamentals and applications, Springer Series in Synergetics, Springer-Verlag, Berlin, 2005.
• 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\Ndash703.
• A. Jüngel, “Quasi-hydrodynamic semiconductor equations”, Progress in Nonlinear Differential Equations and their Applications 41, Birkhäuser Verlag, Basel, 2001.
• 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.
• P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, “Semiconductor equations”, Springer-Verlag, Vienna, 1990.
• J. Nieto, Hydrodynamical limit for a drift-diffusion system modeling large-population dynamics, J. Math. Anal. Appl. 291(2) (2004), 716\Ndash726.
• F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26(1–2) (2001), 101\Ndash174.
• 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.
• J. Rulla, Weak solutions to Stefan problems with prescribed convection, SIAM J. Math. Anal. 18(6) (1987), 1784\Ndash1800.
• J. Simon, Compact sets in the space $L\sp p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65\Ndash96.
• 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\Ndash118.
• J. L. Vázquez, “The porous medium equation”, Mathematical theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
• C. Villani, “Topics in optimal transportation”, Graduate Studies in Mathematics 58, American Mathematical Society, Providence, RI, 2003.