Cell-cell adhesion and multiphase Hele-Shaw problem as the singular limit of a Keller-Segel system
Abstract
We investigate a singular limit of a system of Patlak-Keller-Segel (PKS) equations modeling the evolution of multiple interacting species. Our primary motivation is the Differential Adhesion Hypothesis (DAH), introduced by Malcolm Steinberg in 1962, which posits that cell populations self-organize by minimizing adhesion energy, in a manner analogous to fluids minimizing surface tension. Our starting point is a continuum model describing the evolution of the density distributions of N distinct species, representing different cell types. These species interact through attractive nonlocal forces governed by interaction kernels of similar form but different strengths (the N× N matrix of interaction coefficients encodes the key properties of the system). These attractive forces are balanced by a nonlinear pressure, depending on the total density and strong enough to prevent concentration. In the limit of short-range interactions, we establish sufficient conditions on the interaction matrix for cell sorting to take place (i.e., the spontaneous separation of the different species). We then prove a general Γ-convergence result for the associated energy functional, showing convergence to an interfacial energy where the surface tension coefficients are determined by a geodesic problem. A detailed analysis of this problem allows us to identify regimes in which engulfment occurs (more adhesive species cluster together and are surrounded by less adhesive ones) which is a key feature of the DAH. In the second part of the paper, we analyze the asymptotic behavior of solutions to the PKS system in the combined long-time and short-range interaction limit. Under a standard energy convergence assumption, we prove the convergence to a multiphase Hele--Shaw problem with surface tension (and contact angle conditions at triple junctions).
Related
- The Keller problem for the lowest positive eigenvalue of Dirac operators with Lorentz scalar potentials
- A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent
- Well-posedness of 2D stochastic parabolic-parabolic Keller-Segel system with logistic source
- Existence of Finite Time Blow-Up in Keller-Segel System
- Boundedness of radial solutions to the parabolic-parabolic Keller--Segel system on the plane with subcritical mass
BibTeX
@misc{jang2026cellcell,
title = {{Cell-cell adhesion and multiphase Hele-Shaw problem as the singular limit of a Keller-Segel system}},
author = {Jiwoong Jang and Antoine Mellet},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.10205},
eprint = {2606.10205},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.10205},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cell-cell adhesion and multiphase Hele-Shaw problem as the singular limit of a Keller-Segel system
AU - Jiwoong Jang
AU - Antoine Mellet
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2606.10205
UR - https://arxiv.org/abs/2606.10205
AB - We investigate a singular limit of a system of Patlak-Keller-Segel (PKS) equations modeling the evolution of multiple interacting species. Our primary motivation is the Differential Adhesion Hypothesis (DAH), introduced by Malcolm Steinberg in 1962, which posits that cell populations self-organize by minimizing adhesion energy, in a manner analogous to fluids minimizing surface tension. Our starting point is a continuum model describing the evolution of the density distributions of N distinct species, representing different cell types. These species interact through attractive nonlocal forces governed by interaction kernels of similar form but different strengths (the N× N matrix of interaction coefficients encodes the key properties of the system). These attractive forces are balanced by a nonlinear pressure, depending on the total density and strong enough to prevent concentration. In the limit of short-range interactions, we establish sufficient conditions on the interaction matrix for cell sorting to take place (i.e., the spontaneous separation of the different species). We then prove a general Γ-convergence result for the associated energy functional, showing convergence to an interfacial energy where the surface tension coefficients are determined by a geodesic problem. A detailed analysis of this problem allows us to identify regimes in which engulfment occurs (more adhesive species cluster together and are surrounded by less adhesive ones) which is a key feature of the DAH. In the second part of the paper, we analyze the asymptotic behavior of solutions to the PKS system in the combined long-time and short-range interaction limit. Under a standard energy convergence assumption, we prove the convergence to a multiphase Hele--Shaw problem with surface tension (and contact angle conditions at triple junctions).
ER -