  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 domain, 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 parabolic-parabolic
Keller-Segel system, we establish a complete entropy estimate benefited from
linear diffusion term, which plays important role in obtaining the Lp estimates
for the solution. Furthermore, the upper bound for the concentration of signal
is obtained. Based on estimates we obtained for the density of cells, 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 cell-density, we derive the
strong L1 convergence for the propagation of chaos by using relative entropy
method.
