https://neckpinch.com/authors/antoine-mellet/New arXiv papers by Antoine Mellet, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2503.20014/arxiv:2503.20014Tue, 25 Mar 2025 00:00:00 GMTJiwoong Jang, Antoine Mellet — The Patlak-Keller-Segel system of equations (PKS) is a classical example of 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 dimension 2 and 3 towards 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 Brakke type formulation of the mean-curvature flow and a reinterpretation of the problem as an Allen-Cahn equation with a nonlocal forcing term.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flowsFree boundary & capillaryPhase fields & multiphase flows https://neckpinch.com/papers/2408.14309/arxiv:2408.14309Mon, 26 Aug 2024 00:00:00 GMTAntoine Mellet, Michael Rozowski — 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.Mean curvature flowVolume-preserving & constrained flowsFree boundary & capillary