| Issue |
ESAIM: M2AN
Volume 60, Number 3, May-June 2026
|
|
|---|---|---|
| Page(s) | 1503 - 1524 | |
| DOI | https://doi.org/10.1051/m2an/2026013 | |
| Published online | 26 June 2026 | |
Traveling wave for a coupled incompressible Darcy's free boundary model with undercooling effect and surface tension
1
Université de Rennes, Inria, IRMAR UMR 6625, F-35000 Rennes, France
2
Université Évry Val d'Essonne, LaMME, CNRS UMR 8071, Évry, France
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
17
February
2025
Accepted:
27
January
2026
Abstract
We present a two-dimensional free-boundary Hele-Shaw model to describe certain aspects of eukaryotic cell motility on substrates. The key ingredients of this model are Darcy's law for the over-damping motion of the cytoplasm as a confined viscous droplet, with a particular boundary condition for the Young-Laplace equation. This particular condition describes the active force induced by the cytoskeleton, which can be generated in cells either by actin polymerization against the membrane or by actomyosin contraction of cortical filaments, which adhere to the membrane. In addition, in this model we take into account the friction exerted by the membrane on the substrate. First, we study the linear stability of the steady state and prove that above a threshold, the disk is linearly unstable. This analysis highlights the stabilizing effect of undercooling. Then, using a bifurcation argument, we prove the existence of traveling waves that describe a persistent motion in cell migration and justify the relevance of the model. We emphasize in particular that the friction exerted by the membrane on the substrate has a stabilizing effect on cell dynamics.
Mathematics Subject Classification: 35R35 / 35B32 / 35C07 / 92C17
Key words: Traveling waves / free boundary / cell migration and cell polarization
© The authors. Published by EDP Sciences, SMAI 2026
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