Density-constrained Chemotaxis and Hele-Shaw flow

TL;DR

This paper models congestion chemotaxis with an incompressibility constraint, showing that under certain conditions, cell density dynamics approximate a Hele-Shaw free boundary problem with surface tension, including boundary condition analysis.

Contribution

It establishes a rigorous connection between chemotaxis models with congestion and Hele-Shaw free boundary problems, including boundary condition implications.

Findings

Cell density converges to a characteristic function of a set.

The limit dynamics are described by a Hele-Shaw free boundary problem.

Robin boundary conditions lead to a contact angle condition for the interface.

Abstract

We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint. We show that when the chemical diffuses slowly and attracts the cells strongly, then the dynamics of the congested cells is well approximated by a surface-tension driven free boundary problem. More precisely, we show that in this limit the density of cell converges to the characteristic function of a set whose evolution is described by a Hele-Shaw free boundary problem with surface tension. Our problem is set in a bounded domain, which leads to an interesting analysis on the limiting boundary conditions for the density function. Namely, we prove that the assumption of Robin boundary conditions for the chemical potential leads to a contact angle condition for the free interface.