Homogenization and diffusion approximation of the Vlasov–Poisson–Fokker–Planck system: A relative entropy approach

Article type: Research Article

Authors: Addala, Lanoir^a | El Ghani, Najoua^{b; *} | Tayeb, Mohamed Lazhar^c

Affiliations: [a] Université de Tunis Carthage, Faculté des Sciences de Bizerte, Systèmes dynamiques et leurs applications, UR17ES21, Tunisia. E-mail: [email protected] | [b] Université de Sousse, Institut Supérieur d’Informatique et des Technologies de Communication, Département de Réseaux et Multimédia, Rue G.P.1 Hammam Sousse-4011, Tunisia. E-mails: [email protected], [email protected] | [c] Université de Tunis El Manar, Faculté des Sciences de Tunis, Départment de Mathématiques, El Manar 2092, Tunisia. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We are concerned with the analysis of the approximation by diffusion and homogenization of a Vlasov–Poisson–Fokker–Planck system. Here we generalize the convergence result of (Comm. Math. Sci. 8 (2010), 463–479) where the same problem is treated without the oscillating electrostatic potential and we extend the one dimensional result of (Ann. Henri Poincaré 17 (2016), 2529–2553) to the case of several space dimensions. An averaging lemma and two scale convergence techniques are used to prove rigorously the convergence of the scaled Vlasov–Poisson–Fokker–Planck system to a homogenized Drift-Diffusion-Poisson system.

Keywords: Vlasov–Poisson–Fokker–Planck system, Drift-Diffusion equation, diffusion approximation, homogenization, renormalized solution, two scale convergence, averaging lemma

DOI: 10.3233/ASY-201608

Journal: Asymptotic Analysis, vol. 121, no. 3-4, pp. 401-422, 2021