Mean field limit of many particle system with non-Lipschitz force

Yin, Qitao

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URN: urn:nbn:de:bsz:180-madoc-513040
Document Type: Doctoral dissertation
Year of publication: 2019
Place of publication: Mannheim
University: Universität Mannheim
Evaluator: Chen, Li
Date of oral examination: 24 July 2019
Publication language: English
Institution: School of Business Informatics and Mathematics > Applied Analysis (Chen 2014-)
Subject: 510 Mathematics
Keywords (English): mean field limit , particle model
Abstract: In this thesis we consider the mean field limit of $N$-particle system induced both from social science application and from physical background. Based on establishing the ordinary differential equation of characteristics for transport equation, we handle the non-Lipschitz force in the non-linear partial differential equation, which is also a Vlasov-type equation. The first part of the thesis is to review different kinds of kinetic particle systems to their corresponding Vlasov-related equations. We will summerize the literatures to date and give a whole picture of what the mean field limit is all about through very concrete examples and models done so far. Followed by the review, we will in the chapters to come present some novel ideas and methods that we used to tackle those problems during the years of research. Further for the particle model, we investigate a two-dimensional pedestrian flow system and illustrate the probabilistic method in detailed steps to show that the $N$-particle pedestrian flow system can be represented by the one particle density function when $N$ approaches infinity, which is so-called mean field equation. As regard to the wellposedness of the mean field equation, the weak solution is also presented with a more general setting for the equation, or in other words, for the non-Lipschitz force. Last but not least, we focus on the well-known relativistic Vlasov-Maxwell and present both the non-relativistic limit and mean field limit for the corresponding particle relativistic Vlasov-Maxwell model, which converges to what we know as the Vlasov-Poisson equation, with all the analysis in the last chapter.

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