Mean-field derivation of a two-dimensional signal-dependent parabolic-elliptic Keller-Segel system in algebraic scaling

TL;DR

This paper derives a two-dimensional Keller-Segel system using algebraic scaling in a particle model, proving convergence of particle trajectories and densities under certain conditions, with a novel focus on diffusive interaction.

Contribution

It introduces a new algebraic scaling approach where moderate interaction occurs in the diffusive term, advancing the mean-field derivation of the Keller-Segel system.

Findings

Proves convergence in probability of particle trajectories.

Shows convergence of densities in L1 norm for short times.

Establishes equivalence of convergence in probability and maximum norm.

Abstract

This paper continues our survey about the mean-field derivation of the two-dimensional signal-dependent Keller-Segel system studied in [1]. Therefore, we consider the same system of moderately interacting particles as before. The difference lies in the scaling. Since logarithmic scaling was treated in [1], we now consider algebraic scaling to obtain propagation of chaos in the weak sense. We prove convergence in probability for the particle trajectories. Moreover, for short times and regularity assumptions on the initial data we show the convergence of the densities in the L1 norm. The novelty of this paper is the treatment of a particle model (with algebraic scaling) where the moderate interaction completely takes place in the diffusive term. This structure with algebraic scaling makes tremendous difference from the propagation of chaos discussion when the interaction appears in the drift terms with singular kernel in [8]. The argument for convergence in probability proceeds by defining a stopping time based on a power mean of the sample paths. Furthermore, we prove that the convergence in probability with this power-mean is equivalent to the convergence with maximum norm on trajectories.