A density-constrained model for Chemotaxis

TL;DR

This paper develops a variational scheme to construct and analyze weak solutions for a chemotaxis model with congestion constraints, revealing patch formation and pressure dynamics akin to free boundary problems.

Contribution

It introduces a novel variational discrete scheme for weak solutions, proves their uniqueness, and rigorously links the model to free boundary problems and obstacle problems.

Findings

Patch formation occurs at maximal density regions.

Weak solutions are constructed via a JKO-type scheme.

Pressure variable characterized as an obstacle problem.

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 characterizing the pressure variable as the solution of an obstacle problem and prove that in the pure advection case the dynamic preserves patches.