Lukas Bol | Searcharxiv

arXiv2026

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

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.

math.PR↗

arXiv2024

Two-dimensional signal-dependent parabolic-elliptic Keller-Segel system and its mean-field derivation

In this paper, the well-posedness of two-dimensional signal-dependent Keller-Segel system and its mean-field derivation from a interacting particle system on the whole space are investigated. The signal dependence effect is reflected by the fact that the diffusion coefficient in the particle system depends non-linearly on the interactions between the individuals. Therefore, the mathematical challenge in studying the well-posedness of this system lies in the possible degeneracy and the aggregation effect when the concentration of signal becomes unbounded. The well-established method on bounded domains, to obtain the appropriate estimates for the signal concentration, is invalid for the whole space case. Motivated by the entropy minimization method and Onofri's inequality, which has been successfully applied for the parabolic-parabolic Keller-Segel system, we establish a complete entropy estimate benefited from the linear diffusion term, which plays an important role in obtaining the L^p estimates for the solution. Furthermore, the upper bound for the concentration of signal is obtained. Based on the estimates we obtained for the density of bacteria, the rigorous mean-field derivation is proved by introducing an intermediate particle system with a mollified interaction potential with logarithmic scaling. By using this mollification, we obtain the convergence of the particle trajectories in expectation, which implies the weak propagation of chaos. Additionally, under a regularity assumption of the initial data, we infer higher regularity for the solutions, which allows us to use the relative entropy method to derive the strong L^1 convergence for the propagation of chaos.

math.AP↗