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

Abstract

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 nonlinearly 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.