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

2024/08/26 by Antoine Mellet, Mellet, Antoine, Michael Rozowski +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Stochastic processes and statistical mechanics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2408.14309

openalex publication_date 2024/08/26 · openalex created_date 2024/09/21 · openalex updated_date 2026/07/28

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 ρ∇ ρ^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.

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• Constrained hydrodynamic flocking models in the limit of large attraction-repulsion interactions

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