On the incompressible limit of Keller-Segel system with volume-filling effects

We consider the Keller-Segel system with a volume-filling effect and study its incompressible limit. Due to the presence of logistic-type sensitivity, $K=1$ is the critical threshold. When $K>1$, as the diffusion exponent tends to infinity, by supposing the weak limit of $u^2_m$, we prove that the limiting system becomes a Hele-Shaw type free boundary problem. For $K\le 1$, we justify that the stiff pressure effect ($\Delta P_\infty$) vanishes, resulting in the limiting system being a hyperbolic Keller-Segel system. Compared to previous studies, the new challenge arises from the stronger nonlinearity induced by the logistic chemotactic sensitivity. To address this, our first novel finding is the proof of strong convergence of the density on the support of the limiting pressure, thus confirming the validity of the \emph{complementarity relation} for all $K>0$. Furthermore, specifically for the case $K\le1$, by introducing the \emph{kinetic formulation}, we verify the strong limit of the density required to reach the incompressible limit.