Rigorous derivation of the degenerate parabolic-elliptic Keller-Segel System from a moderately interacting stochastic particle system

Gvozdik, Veniamin

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URN:	urn:nbn:de:bsz:180-madoc-635369
Document Type:	Doctoral dissertation
Year of publication:	2022
Place of publication:	Mannheim
University:	Universität Mannheim
Evaluator:	Chen, Li
Date of oral examination:	28 November 2022
Publication language:	English
Institution:	School of Business Informatics and Mathematics > Applied Analysis (Chen 2014-)
Subject:	510 Mathematics
Keywords (English):	moderately interacting particle systems , stochastic particle systems , mean-field limit , chemotaxis , Keller–Segel model , degenerate parabolic-elliptic system , propagation of chaos
Abstract:	The main goal of this thesis is a rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system of porous medium type on the whole space Rd from a moderately interacting stochastic particle system. After we review some existing results on this topic and introduce the setting of the problem as well as the main results of this thesis, we establish the classical solution theory of the degenerate parabolic-elliptic Keller-Segel system and its non-local version. This classical solution theory is used later to obtain required estimates on the particle level. Because of the non-linearity in diffusion and the singularity in aggregation we perform an approximation of the stochastic moderately interacting particle system using the cut-offed potential. The stochastic effect is introduced as a parabolic regularization of the system. Then we compare this new system with another cut-offed system of mean-field type. We present the propagation of chaos result with two different types of cut-off scaling, namely logarithmic and algebraic scaling. For the logarithmic scaling we prove the convergence of trajectories in expectation. For the algebraic scaling we obtain it in the sense of probability. Consequently, the propagation of chaos follows directly from these convergence results and the vanishing viscosity of the system.