Overdamped Limits of the Vlasov-Poisson-Fokker-Planck equation in three dimensions

Voronine, Filip (2025) Overdamped Limits of the Vlasov-Poisson-Fokker-Planck equation in three dimensions. Master's Thesis / Essay, Mathematics.

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Abstract

We study the behavior of solutions to the damped Vlasov-Poisson-Fokker-Planck equation, characterizing the evolution of the phase-space density corresponding to an interacting many-body system in the presence of a thermal reservoir. This equation can be used to model the evolution of the distribution of ions interacting freely in a plasma subject to Coulomb forces or that of galaxies interacting in space subject to Newton's universal law of gravitation. In particular, we show that the particle densities of this system converge, in Wasserstein sense, to the unique solution to the corresponding Aggregation-Diffusion equation as the amount of damping introduced in the velocity variable increases. The result is obtained by applying a triangle inequality with an intermediate system inspired by the coarse-graining map regulating the influence of the velocity variable when computing the particle densities. Using tools from optimal transport and the theory of gradient flows, the Wasserstein distance is controlled by using Grönwall's lemma with a modified evolution-variational inequality characterizing the time-derivative of the Wasserstein distance along the evolution.

Item Type:	Thesis (Master's Thesis / Essay)
Supervisor name:	Peypouquet, J.G. and Seri, M.
Degree programme:	Mathematics
Thesis type:	Master's Thesis / Essay
Language:	English
Date Deposited:	22 Aug 2025 07:45
Last Modified:	22 Aug 2025 07:45
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/36803

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