A density-constrained model for chemotaxis

Abstract

We consider a model of congestion dynamics with chemotaxis: the density of cells follows a chemical signal it generates, while subject to an incompressibility constraint. The incompressibility constraint results in the formation of patches, describing regions where the maximal density has been reached. The dynamics of these patches can be described by either Hele-Shaw or Richards equation type flow (depending on whether we consider the model with diffusion or the model with pure advection). Our focus in this paper is on the construction of weak solutions for this problem via a variational discrete time scheme of JKO type. We also establish the uniqueness of these solutions. In addition, we make more rigorous the connection between this incompressible chemotaxis model and the free boundary problems describing the motion of the patches in terms of the density and associated pressure variable. In particular, we obtain new results characterising the pressure variable as the solution of an obstacle problem and prove that in the pure advection case the dynamic preserves patches.

Incompressible Limits of the Patlak-Keller-Segel Model and Its Stationary State

“…System (3) with ( 5) is a weak and global form of the geometric Hele-Shaw free boundary problem (see [38,27]).…”

“…Because the geometric form of the Hele-Shaw problem uses the set Ω (t) = {ρ ∞ (t, •) = 1}, several authors have studied the question to know, when the initial data is the indicator function of Ω (0), if this property is propagated even if Ω (t) has little regularity. For this question it is convenient to work on time integrated variables, which leads to the obstacle problem, we refer to [38,6,24,27]. The regularity and stability of such patches is studied in [25,5].…”

“…This problem is studied in [40] and in the conservative case [27]. There is still a set Ω (t) where p ∞ > 0, but ρ ∞ is smooth and has a positive tail because the equation is parabolic non-degenerate.…”