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Improved convergence rates for the Hele-Shaw limit in the presence of confining potentials

Noemi David, Alpár R. Mészáros, Filippo Santambrogio

TL;DR

The paper addresses the incompressible (Hele-Shaw) limit for nonlinear degenerate diffusion with external drifts by establishing a global-in-time convergence rate in the 2-Wasserstein distance as the pressure exponent $m$ tends to infinity. The authors introduce a novel method based on differentiating the squared $W_2$-distance between solutions with different exponents and applying Grönwall, which yields a polynomial rate of $\mathcal{O}(m^{-1/2})$ that holds globally in time under convexity of the combined potential $V+W$. This framework unifies local and nonlocal drifts and covers both singular and power-law pressure laws, with auxiliary results for stationary states showing an $L^1$-rate of $\mathcal{O}(m^{-1})$ and a potentially faster $W_2$-rate under nested supports; the approach also relaxes several initial-data assumptions compared to prior works. Overall, the work extends the understanding of convergence in the Hele-Shaw limit to more general drift fields and interaction potentials, providing sharp quantitative rates with implications for modeling crowd motion and tissue growth. The results leverage optimal-transport tools, gradient-flow structure, and energy-dissipation identities to obtain robust, global convergence guarantees.

Abstract

Nowadays a vast literature is available on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. This problem has attracted a lot of attention due to its applications to tissue growth and crowd motion modelling as it constitutes a way to link soft congestion (or compressible) models to hard congestion (or incompressible) descriptions. In this paper, we address the question of estimating the rate of this asymptotics in the presence of external drifts. In particular, we provide improved results in the 2-Wasserstein distance which are global in time thanks to the contractivity property that holds for strictly convex potentials.

Improved convergence rates for the Hele-Shaw limit in the presence of confining potentials

TL;DR

The paper addresses the incompressible (Hele-Shaw) limit for nonlinear degenerate diffusion with external drifts by establishing a global-in-time convergence rate in the 2-Wasserstein distance as the pressure exponent tends to infinity. The authors introduce a novel method based on differentiating the squared -distance between solutions with different exponents and applying Grönwall, which yields a polynomial rate of that holds globally in time under convexity of the combined potential . This framework unifies local and nonlocal drifts and covers both singular and power-law pressure laws, with auxiliary results for stationary states showing an -rate of and a potentially faster -rate under nested supports; the approach also relaxes several initial-data assumptions compared to prior works. Overall, the work extends the understanding of convergence in the Hele-Shaw limit to more general drift fields and interaction potentials, providing sharp quantitative rates with implications for modeling crowd motion and tissue growth. The results leverage optimal-transport tools, gradient-flow structure, and energy-dissipation identities to obtain robust, global convergence guarantees.

Abstract

Nowadays a vast literature is available on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. This problem has attracted a lot of attention due to its applications to tissue growth and crowd motion modelling as it constitutes a way to link soft congestion (or compressible) models to hard congestion (or incompressible) descriptions. In this paper, we address the question of estimating the rate of this asymptotics in the presence of external drifts. In particular, we provide improved results in the 2-Wasserstein distance which are global in time thanks to the contractivity property that holds for strictly convex potentials.
Paper Structure (14 sections, 11 theorems, 142 equations)

This paper contains 14 sections, 11 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Previous results on convergence rates
  3. Summary of the strategy and main contributions
  4. Structure of the rest of the paper
  5. Assumptions and main results
  6. Results in the 2-Wasserstein distance
  7. Results for the stationary states
  8. Preliminaries and preparatory results
  9. Optimal transport toolbox
  10. Preparatory results
  11. Rate of convergence in the 2-Wasserstein distance
  12. Rate of convergence for stationary states
  13. Rate in the $L^1$ norm
  14. Improved $W_2$ rate

Key Result

Theorem 2.4

Let $\varrho_m$ be the solution of main tot coupled with either eq: pressure power law or singular pressure and endowed with initial data $\varrho_0\in \mathcal{P}_2(\mathbb{R}^d), \int_{\mathbb{R}^d} x \varrho_0 \mathop{}\!\mathup{d} x=0$, $\|\varrho_0\|_\infty \leq 1$. For all $T>0$, under Assumpt where $C$ is a uniform positive constant depending on the final time $T$. Moreover, if Assumption a

Theorems & Definitions (19)

  • Theorem 2.4: Rate of convergence in $W_2$
  • Theorem 2.6: Polynomial convergence rate in the $L^1$-norm for the stationary states
  • Theorem 2.7: Faster $W_2$-rate for some stationary solutions
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 9 more