The elliptic–parabolic Patlak–Keller–Segel system and volume-preserving mean curvature flows

Abstract

The Patlak–Keller–Segel system of equations (PKS) is a classical example of an aggregation-diffusion equation. It describes the aggregation of some organisms via chemotaxis, limited by some nonlinear diffusion. It is known that for some choice of this nonlinear diffusion, the PKS model asymptotically leads to phase separation and mean-curvature-driven free boundary problems. In this paper, we focus on the elliptic–parabolic PKS model and we obtain the first unconditional convergence result in dimensions $2$ and $3$ toward the volume-preserving mean curvature flow. This work builds up on previous results that were obtained under the assumption that phase separation does not cause energy loss in the limit. In order to avoid this assumption, we rely on the Brakke-type formulation of the mean curvature flow and a reinterpretation of the problem as an Allen–Cahn equation with a nonlocal forcing term.