English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

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