Density-constrained Chemotaxis and Hele-Shaw flow

2022/04/25 by Inwon Kim, Kim, Inwon, Antoine Mellet +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Cellular Mechanics and Interactions #FOS: Mathematics #Mathematical Biology Tumor Growth #Micro and Nano Robotics

paper · pdf · doi:10.48550/arxiv.2204.11917

openalex publication_date 2022/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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