Abstract
The approximation by diffusion and homogenization of the initial-boundary value problem of the Vlasov–Poisson–Fokker–Planck model is studied for a given velocity field with spatial macroscopic and microscopic variations. The L1-contraction property of the Fokker–Planck operator and a two-scale Hybrid-Hilbert expansion are used to prove the convergence towards a homogenized Drift–Diffusion equation and to exhibit a rate of convergence.
Article PDF
Similar content being viewed by others
References
Allaire G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)
Banasiak J., Mika J.R.: Diffusion limit for the linear Boltzmann equation of the neutron transport theory. M2AS 17, 1071–1087 (1994)
Bardos C., Sentos R., Sentis R.: Diffusion approximation and computation of the critical size. Trans. Am. Math. Soc. 284(2), 617–649 (1984)
Ben Abdallah N.: Weak solutions of the initial-boundary value problem for the Vlasov–Poisson system. Math. Methods Appl. Sci. 17, 451–476 (1994)
Ben Abdallah N., Chaker H., Schmeiser C.: The high field asymptotics for a fermionic Boltzmann equation: entropy solutions and kinetic shock profiles. J. Hyperbolic Differ. Equ. 4, 679–704 (2007)
Ben Abdallah N., Degond P.: On a hierarchy of macroscopic models for semiconductors. J. Math. Phys. 37(7), 3306–3333 (1996)
Ben Abdallah N., Dolbeault J.: Relative entropies for kinetic equations in bounded domains (irreversibility, stationary solutions, uniqueness). Arch. Ration. Mech. Anal. 168(4), 253–298 (2003)
Ben Abdallah N., Escobedo M., Mischler S.: Convergence to the equilibrium for the Pauli equation without detailed balance condition. C. R. Math. Acad. Sci. Paris 341(1), 5–10 (2005)
Ben Abdallah N., Tayeb M.-L.: Diffusion approximation for the one dimensional Boltzmann–Poisson system. DCDS-B 4, 1129–1142 (2004)
Ben Abdallah N., Tayeb M.-L.: Diffusion Approximation and homogenization of the semiconductor Boltzmann equation. Multiscale Model Simul. 4(3), 896–914 (2005)
Dolbeault J.: Free energy and solutions of the Vlasov–Poisson–Fokker–Planck system: external potential and confinement (large time behavior and steady states). J. Math. Pures Appl. 78(2), 121–157 (1999)
Dolbeault J.: Kinetic models and quantum effects: a modified Boltzmann equation for Fermi–Dirac particles. Arch. Ration. Mech. Anal. 127(2), 101–131 (1994)
El-Ghani, N., Tayeb, M.-L.: Diffusion approximation and homogenization of a Vlasov–Poisson–Fokker–Planck system: a relative Entropy approach (in progress)
Frenod E., Raviart P.A., Sonnendrücker E.: Two scale expansion of a singularly perturbed convection equation. J. Maths. Pures Appl. 80(8), 815–843 (2001)
Golse F., Poupaud F.: Limite fluide des équations de Boltzmann des semiconducteurs pour une statistique de Fermi–Dirac. Asymptot. Anal. 6, 135–169 (1992)
Goudon T.: Hydrodynamic limit for the Vlasov–Poisson–Fokker–Planck system: analysis of two dimensional case. Math. Models Methods Appl. Sci. 15, 737–752 (2005)
Goudon T., Mellet A.: Homogenization and diffusion asymptotics of the linear Boltzmann. ESAIM Control Optim. Calc. Var. 9, 371–398 (2003)
Goudon T., Nieto J., Poupaud F., Soler J.: Multidimensional high-field limit of the electrostatic Vlasov–Poisson–Fokker–Planck system. J. Differ. Equ. 213, 418–442 (2005)
Goudon T., Poupaud F.: Approximation by homogeneization and diffusion of kinetic equations. Commun. Partial Differ. Equ. 26, 537–569 (2001)
Hairer M., Pavliotis G.A.: Periodic homogenization for hypoelliptic diffusions. J. Stat. Phys. 117(1–2), 261–279 (2004)
Masmoudi N., Tayeb M.-L.: Diffusion limit of a semiconductor Boltzmann–Poisson system. SIAM J. Math. Anal. 38, 1788–1807 (2007)
Masmoudi N., Tayeb M.-L.: Diffusion and homogenization approximation for semiconductor Boltzmann–Poisson system. J. Hyperbolic Differ. Equ. 5(1), 65–84 (2008)
Masmoudi N., Tayeb M.-L.: On the diffusion limit of a semiconductor Boltzmann–Poisson system without micro-reversible process. Commun. Partial Differ. Equ. 35(7), 1163–1175 (2010)
Michel P., Mischler S., Perthame B.: General relative entropy inequality: an illustration on growth models. J. Math. Pures Appl. (9) 84(9), 1235–1260 (2005)
Poupaud F.: Diffusion approximation of the linear semiconductor Boltzmann equation: analysis of boundary layers. Asymptot. Anal. 4(4), 293–317 (1991)
Poupaud F., Schmeiser S.: Charge transport in semiconductors with degeneracy effects. Math. Methods Appl. Sci. 14(5), 301–318 (1991)
Poupaud F., Soler J.: Parabolic limit and stability of the Vlasov–Fokker–Planck system. M3AS 10(7), 1027–1045 (2000)
Villani, C.: Entropy Production and Convergence to Equilibrium, Entropy Methods for the Boltzmann Equation. Lecture Notes in Mathematics, vol. 1916, pp. 1–70 (2008)
Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202(950) (2009)
Weinan E.: Homogenization of linear and nonlinear transport equations. Commun. Pure Appl. Math. 45(3), 301–326 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Nader Masmoudi.
Rights and permissions
About this article
Cite this article
Tayeb, M.L. Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model. Ann. Henri Poincaré 17, 2529–2553 (2016). https://doi.org/10.1007/s00023-016-0484-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-016-0484-7