Rigorous Derivation of the Degenerate Parabolic-Elliptic Keller-Segel System from a Moderately Interacting Stochastic Particle System. Part I Partial Differential Equation

TL;DR

This paper rigorously analyzes the partial differential equations derived from a stochastic particle system, establishing the solution theory for the degenerate Keller-Segel system and its non-local version, which is foundational for understanding the mean-field limit.

Contribution

It provides the first comprehensive solution theory for the degenerate Keller-Segel PDEs derived from stochastic particle models, including existence and regularity results.

Findings

Established existence of solutions to the regularized system.

Proved uniform estimates and gradient bounds for approximate solutions.

Demonstrated well-posedness of the non-local Keller-Segel equation.

Abstract

The aim of this paper is to provide the analysis result for the partial differential equations arising from the rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. The rigorous derivation is divided into two articles. In this paper, we establish the solution theory of the degenerate parabolic-elliptic Keller-Segel problem and its non-local version, which will be used in the second paper for the discussion of the mean-field limit. A parabolic regularized system is introduced to bridge the stochastic particle model and the degenerate Keller-Segel system. We derive the existence of the solution to this regularized system by constructing approximate solutions, giving uniform estimates and taking the limits, where a crucial step is to obtain the L infty Bernstein type estimate for the gradient of the approximate solution. Based on this, we obtain the well-posedness of the corresponding non-local equation through perturbation method. Finally, the weak solution of the degenerate Keller-Segel system is obtained by using a nonlinear version of Aubin-Lions lemma.