Diffusion limit of the Vlasov-Poisson-Fokker-Planck system. (English) Zbl 1193.35228

El Ghani, Najoua; Masmoudi, Nader

Commun. Math. Sci. 8, No. 2, 463-479 (2010).

Summary: We study the diffusion limit of the Vlasov-Poisson-Fokker-Planck System. Here, we generalize the local in time results and the two dimensional results of Poupaud-Soler and of Goudon to the case of several space dimensions. Renormalization techniques, the method of moments and a velocity averaging lemma are used to prove the convergence of free energy solutions (renormalized solutions) to the Vlasov-Poisson-Fokker- Planck system towards a global weak solution of the Drift-Diffusion-Poisson model.

MSC:

35Q83	Vlasov equations
35B40	Asymptotic behavior of solutions to PDEs
35B25	Singular perturbations in context of PDEs
45K05	Integro-partial differential equations

Keywords:

hydrodynamic limit; Vlasov-Poisson-Fokker-Planck system; drift-diffusion-Poisson model; moment method; velocity averaging Lemma; renormalized solutions