Incompressible limits of Patlak-Keller-Segel model and its stationary state

TL;DR

This paper analyzes the incompressible limit of the Patlak-Keller-Segel model, deriving a free boundary problem, establishing new estimates, and proving the uniqueness of solutions, including the stationary state behavior.

Contribution

It introduces a novel framework for the incompressible limit of the PKS model, with new estimates and proof of solution uniqueness, extending previous results significantly.

Findings

Derived weak form of Hele-Shaw type limit problem

Established uniform $L^3$ pressure gradient estimate

Proved uniqueness of the Hele-Shaw problem solutions

Abstract

We complete previous results about the incompressible limit of both the $n$-dimensional $(n\geq3)$ compressible Patlak-Keller-Segel (PKS) model and its stationary state. As in previous works, in this limit, we derive the weak form of a geometric free boundary problem of Hele-Shaw type, also called congested flow. In particular, we are able to take into account the unsaturated zone, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation. Not only our analysis uses a completely different framework than previous approaches, but we also establish a novel uniform $L^3$ estimate of the pressure gradient, regularity \`a la Aronson-B\'enilan, and a uniform $L^1$ estimate for the time derivative of the pressure. Furthermore, for the Hele-Shaw problem, we prove the uniqueness of solutions, meaning that the incompressible limit of the PKS model is unique. In addition, we establish the corresponding incompressible limit of the stationary state for the PKS model with a given mass, where, different from the case of PKS model, we obtain the uniform bound of pressure and the uniformly bounded support of density.