Paper · Mar 25, 2025

Diffusion-aggregation equations and volume-preserving mean curvature flows

Jiwoong Jang, Antoine Mellet

math.APWider flowsarXiv:2503.20014
PDF Abstract page

As on arXiv

Abstract

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 and 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

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)6Takasao 2022 Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow (Milan Kroemer and Tim Laux, 2023)7Kroemer 2023 Volume-preserving mean-curvature flow as a singular limit of a diffusion-aggregation equation (Antoine Mellet and Michael Rozowski, 2024)1Mellet 2024 Existence of weak solutions to volume-preserving mean curvature flow with obstacles (Jiwoong Jang, 2025)2Jang 2025 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)4Chiesa 2025 Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness (Andrea Poiatti, 2025)8Poiatti 2025 Existence of flat flows for volume-preserving mean curvature flow with contact angle (Jiwoong Jang, 2025)3Jang 2025 Parabolic free boundary phase transition and mean curvature flow (Jingeon An-Lacroix and Kiichi Tashiro, 2025)5An-Lacroix 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Volume-preserving mean-curvature flow as a singular limit of a diffusion-aggregation equation

    Antoine Mellet and Michael RozowskiWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Same author Antoine Mellet · Shared terms “aggregation equation”, “diffusion aggregation”, “PKS model”

  2. Existence of weak solutions to volume-preserving mean curvature flow with obstacles

    Jiwoong JangWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows, Weak & singular flows · Same author Jiwoong Jang · Shared terms “Allen Cahn”, “Cahn equation”, “forcing term”

  3. Existence of flat flows for volume-preserving mean curvature flow with contact angle

    Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Weak & singular flows · Same author Jiwoong Jang

  4. Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

    Andrea Chiesa and Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows, Weak & singular flows · Shared terms “Brakke”

  5. Parabolic free boundary phase transition and mean curvature flow

    Jingeon An-Lacroix and Kiichi TashiroWider flows PDF

    Topics Phase fields & multiphase flows, Free boundary & capillary, Volume-preserving & constrained flows · Shared terms “Cahn equation”, “Allen”, “forcing term”

  6. Existence of weak solution to volume preserving mean curvature flow in higher dimensions

    Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows, Weak & singular flows · Shared terms “Cahn equation”, “Allen”

  7. Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow

    Milan Kroemer and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows, Weak & singular flows · Shared terms “Cahn equation”, “nonlocal”, “Allen”

  8. Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

    Andrea PoiattiWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows, Weak & singular flows · Shared terms “Allen Cahn”, “Cahn equation”, “unconditional”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.