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Hele-Shaw flow with surface tension and kinetic undercooling as a sharp interface limit of a fully parabolic Patlak-Keller-Segel system with nonlinear diffusion

Michael Rozowski

TL;DR

This work establishes a sharp-interface limit for a parabolic-parabolic Patlak-Keller-Segel system with nonlinear diffusion by proving Γ-convergence of the associated PKS energy to a weighted Modica–Mortola perimeter functional and showing that, under energy convergence, diffuse PKS solutions converge to BV-type solutions of a Hele-Shaw problem with surface tension and kinetic undercooling. The analysis reveals that phase separation is governed by a compatibility between the population pressure functional $f$ and the destruction kinetics $g$, and it identifies confinement effects when the chemo-destruction rate $c(x)$ is spatially varying. The limiting HS-STKU system comprises a harmonic pressure inside the occupied region, a Darcy-type velocity on the moving boundary, an anisotropic mean-curvature contribution, and a contact-angle condition with the fixed boundary, all derived from first variations and energy equipartition. The results connect diffuse-interface PKS models to classical free-boundary problems in fluid mechanics, providing a rigorous bridge between aggregation-diffusion dynamics and sharp-interface hydrodynamic limits with anisotropy and inhomogeneous media.

Abstract

A large population limit of the parabolic-parabolic Patlak-Keller-Segel (PKS) system with degenerate, nonlinear diffusion, e.g., of porous medium-type $-\frac{m}{m-1}\mathrm{div}(ρ\nabla ρ^{m-1})$, is studied. We show, asymptotically, a sharp interface develops separating a region containing organisms arranged in a constant-in-time, uniform density from a region without organisms. Under an energy convergence hypothesis, we prove the emergent interface evolves according to a Hele-Shaw free boundary problem with surface tension and kinetic undercooling, and the free boundary satisfies a contact angle-type condition with the fixed boundary. Further, we show that, for well-prepared initial data, phase separation in these systems is, roughly, the result of some compatibility between an antiderivative for the population pressure and the convex conjugate of an antiderivative of the chemical destruction kinetics. When compatible, an energy for which the parabolic-parabolic PKS system is a gradient flow is a penalized Modica-Mortola functional.

Hele-Shaw flow with surface tension and kinetic undercooling as a sharp interface limit of a fully parabolic Patlak-Keller-Segel system with nonlinear diffusion

TL;DR

This work establishes a sharp-interface limit for a parabolic-parabolic Patlak-Keller-Segel system with nonlinear diffusion by proving Γ-convergence of the associated PKS energy to a weighted Modica–Mortola perimeter functional and showing that, under energy convergence, diffuse PKS solutions converge to BV-type solutions of a Hele-Shaw problem with surface tension and kinetic undercooling. The analysis reveals that phase separation is governed by a compatibility between the population pressure functional and the destruction kinetics , and it identifies confinement effects when the chemo-destruction rate is spatially varying. The limiting HS-STKU system comprises a harmonic pressure inside the occupied region, a Darcy-type velocity on the moving boundary, an anisotropic mean-curvature contribution, and a contact-angle condition with the fixed boundary, all derived from first variations and energy equipartition. The results connect diffuse-interface PKS models to classical free-boundary problems in fluid mechanics, providing a rigorous bridge between aggregation-diffusion dynamics and sharp-interface hydrodynamic limits with anisotropy and inhomogeneous media.

Abstract

A large population limit of the parabolic-parabolic Patlak-Keller-Segel (PKS) system with degenerate, nonlinear diffusion, e.g., of porous medium-type , is studied. We show, asymptotically, a sharp interface develops separating a region containing organisms arranged in a constant-in-time, uniform density from a region without organisms. Under an energy convergence hypothesis, we prove the emergent interface evolves according to a Hele-Shaw free boundary problem with surface tension and kinetic undercooling, and the free boundary satisfies a contact angle-type condition with the fixed boundary. Further, we show that, for well-prepared initial data, phase separation in these systems is, roughly, the result of some compatibility between an antiderivative for the population pressure and the convex conjugate of an antiderivative of the chemical destruction kinetics. When compatible, an energy for which the parabolic-parabolic PKS system is a gradient flow is a penalized Modica-Mortola functional.
Paper Structure (32 sections, 21 theorems, 209 equations, 1 figure)

This paper contains 32 sections, 21 theorems, 209 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Large population limits of Patlak-Keller-Segel systems
  3. Intermediacy of \ref{['eq:HS-STKU']} to \ref{['eq:HS-ST']} and \ref{['eq:VPMCF']}
  4. Setting some constants to 1
  5. Outline of the paper
  6. Acknowledgements
  7. Main results
  8. Assumptions on coefficients and nonlinearities
  9. -convergence of the energy G_
  10. Weak solutions of PP-PKS_
  11. Singular limit of PP-PKS_ and derivation of HS-STKU, Hele-Shaw flow with surface tension and kinetic undercooling
  12. Relationship to prior work
  13. -convergence of the energy G_ (Proof of Theorem 2.1)
  14. The double-well potentials W, W_*, and W_*c.
  15. A compactness result
  16. ...and 17 more sections

Key Result

Theorem 2.1

Let $\Omega \subset \mathbb{R}^d$ ($d\ge 2$) be a bounded domain with Lipschitz boundary, make the standard assumptions hyp-standardAssumptions and suppose hyp:compatibilityConditionF&G, hyp-c:measure, and hyp-g:growthNear0-gamma. As $\varepsilon \to 0^+$, the functionals $\left\{ \mathscr{G}_\varep where $-\vec{\nu}$ is the Radon-Nikodym derivative of $\nabla\phi$ w.r.t. its total variation $\lef

Figures (1)

  • Figure 1: Double-well potentials constructed from differences of powers $f_m$ and $g_q^\ast$; see \ref{['eq:powerLawNonlinearities']}.

Theorems & Definitions (46)

  • Theorem 2.1: $\Gamma$-convergence of $\mathscr{G}_\varepsilon$ \ref{['eq:augmentedEnergy']}
  • Theorem 2.2
  • Lemma 3.1: the double-well potential $W$
  • proof
  • Lemma 3.2: the double-well potential ${W_\ast}$
  • proof
  • Lemma 3.3: the potential $W_{\ast\mathsf{c}}$
  • proof
  • Lemma 3.4: the transformation $F$
  • proof
  • ...and 36 more