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

TL;DR

This paper improves the understanding of the convergence rate in the Hele-Shaw limit for nonlinear degenerate diffusion equations with external potentials, especially in the presence of confining potentials, relevant for modeling tissue growth and crowd motion.

Contribution

It provides new, improved estimates of the convergence rate in the 2-Wasserstein distance for the Hele-Shaw limit with external drifts, leveraging convexity properties.

Findings

Enhanced convergence rate estimates in the Hele-Shaw limit.

Global-in-time results due to contractivity with convex potentials.

Applicability to models of tissue growth and crowd motion.

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.