Volume-preserving mean-curvature flow as a singular limit of a diffusion-aggregation equation

TL;DR

This paper demonstrates that in a specific chemotaxis model, the interface between high and low density regions evolves according to a volume-preserving mean-curvature flow, linking phase separation to geometric flow limits.

Contribution

It establishes that the phase separation in the elliptic-parabolic PKS model asymptotically follows a volume-preserving mean-curvature flow, extending previous results from the Hele-Shaw problem.

Findings

Phase separation occurs in the elliptic-parabolic PKS model.

The interface evolution converges to a volume-preserving mean-curvature flow.

The result connects chemotaxis models to geometric flow limits.

Abstract

The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation in which the repulsive effect of diffusion is in competition with the attractive chemotaxis term. Recent work on the Parabolic-Elliptic PKS model have shown that when the repulsion is modeled by a nonlinear diffusion term $\rho \nabla \rho^{m-1}$ with $m>2$, this competition leads to phase separation phenomena. Furthermore, in some asymptotic regime corresponding to a large population observed over a long enough time, the interface separating regions of high and low density evolves according to the Hele-Shaw free boundary problem with surface tension. In the present paper, we consider the counterpart of that model, namely the Elliptic-Parabolic PKS model and we prove that the same phase separation phenomena occurs, but the motion of the interface is now described (asymptotically) by a volume-preserving mean-curvature flow.