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Diffusion-aggregation equations and volume-preserving mean curvature flows

Jiwoong Jang, Antoine Mellet

TL;DR

The paper addresses the sharp-interface limit of a diffusion-aggregation PKS system in the elliptic-parabolic regime, demonstrating unconditional convergence in dimensions 2 and 3 to a volume-preserving mean-curvature flow by recasting the problem as a nonlocal Allen-Cahn equation with a forcing term from density constraints. It develops an L^2-flow/varifold framework to handle potential energy losses during phase separation and proves a crucial uniform L^2 bound on the forcing term via a bound on the Lagrange multiplier, enabling passage to the limit without energy-convergence assumptions. The limit interface evolves with velocity v = h − (Λ/Θ) ν, where h is the generalized mean curvature, Λ ∈ L^2_loc(0,∞) is a weak multiplier, and Θ is an integer-valued multiplicity, reflecting Brakke-type convergence with possible multiplicities. This unconditional result extends prior conditional findings (MR24) by integrating nonlocal Allen-Cahn perspectives and Brakke-type flow concepts to obtain robust convergence to volume-preserving mean-curvature flow for PKS-type systems.

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 $2$ 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.

Diffusion-aggregation equations and volume-preserving mean curvature flows

TL;DR

The paper addresses the sharp-interface limit of a diffusion-aggregation PKS system in the elliptic-parabolic regime, demonstrating unconditional convergence in dimensions 2 and 3 to a volume-preserving mean-curvature flow by recasting the problem as a nonlocal Allen-Cahn equation with a forcing term from density constraints. It develops an L^2-flow/varifold framework to handle potential energy losses during phase separation and proves a crucial uniform L^2 bound on the forcing term via a bound on the Lagrange multiplier, enabling passage to the limit without energy-convergence assumptions. The limit interface evolves with velocity v = h − (Λ/Θ) ν, where h is the generalized mean curvature, Λ ∈ L^2_loc(0,∞) is a weak multiplier, and Θ is an integer-valued multiplicity, reflecting Brakke-type convergence with possible multiplicities. This unconditional result extends prior conditional findings (MR24) by integrating nonlocal Allen-Cahn perspectives and Brakke-type flow concepts to obtain robust convergence to volume-preserving mean-curvature flow for PKS-type systems.

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.

Paper Structure

This paper contains 14 sections, 11 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Patlak-Keller-Segel model
  3. A non-local Allen-Cahn equation
  4. $\Gamma$-convergence and conditional convergence result of MR24
  5. Preliminaries and main results
  6. Well-posedness
  7. Basic definitions
  8. Main results
  9. A brief review of the literature
  10. Proof of Theorem \ref{['thm:convergence-to-limit-flow']}
  11. Phase separation
  12. $L^2$-estimate on the forcing term
  13. End of the proof of Theorem \ref{['thm:convergence-to-limit-flow']}
  14. Proof of Proposition \ref{['prop:AC']}

Key Result

Proposition 1

The function $\overline W$ is a $C^{1,1}$ double-well potential (see Figure fig:double-well) satisfying The energy eq:energy2 can be written as follows: For all $\phi\in L^2(\Omega)$, the solution of the minimization problem eq:rhophi is given by $\rho_\phi = {g^\ast}'(\phi+\ell) .$ for some Lagrange multiplier $\ell=\ell(\phi)\in \mathbb{R}$. Equation eq:phiepsilon can be rewritten as where $\

Figures (1)

  • Figure 1: The double-well potentials $W(\rho)$ (solid line) and $\overline W(\phi)$ (dashed line).

Theorems & Definitions (19)

  • Proposition 1
  • Proposition 2: Theorem 2.3 in MR24
  • Proposition 3
  • Proposition 4
  • Definition 1: $L^2$-flow MR08
  • Theorem 1
  • Proposition 5
  • Proposition 6
  • proof
  • Proposition 7
  • ...and 9 more