We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Incompressible Limits of the Patlak-Keller-Segel Model and Its Stationary State.
- Authors
He, Qingyou; Li, Hai-Liang; Perthame, Benoît
- Abstract
We complete previous results about the incompressible limit of both the n -dimensional (n ≥ 3) 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 two novel uniform estimates in L 3 of the pressure gradient and in L 1 for the time derivative of the pressure. We also prove regularity à la Aronson-Bénilan. 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.
- Subjects
ELLIPTIC equations; GEOMETRIC shapes; TIME pressure; LINEAR complementarity problem
- Publication
Acta Applicandae Mathematicae, 2023, Vol 188, Issue 1, p1
- ISSN
0167-8019
- Publication type
Article
- DOI
10.1007/s10440-023-00622-1