Competitive Hele-Shaw flow and quadratic differentials

TL;DR

This paper generalizes Hele-Shaw flow to multiple competing droplets, linking stationary solutions to quadratic differentials, and introduces a discrete model related to competitive erosion, with convergence conjectures.

Contribution

It introduces a novel multi-droplet Hele-Shaw model, connects stationary solutions to quadratic differentials, and proposes a lattice-based discrete model with convergence conjectures.

Findings

Stationary solutions correspond to critical trajectories of quadratic differentials.

Existence of solutions proved via an extremal electrostatic energy problem.

A discrete random model related to competitive erosion is introduced.

Abstract

We introduce and investigate a generalization of the Hele-Shaw flow with injection where several droplets compete for space as they try to expand due to internal pressure while still preserving their topology. Droplets are described by their closed non-crossing interface curves in $\mathbb{C}$ or more generally in a Riemann surface of finite type. Our main focus is on stationary solutions which we show correspond to the critical vertical trajectories of a particular quadratic differential with second order poles at the source points. The quadratic differentials that arise in this way have a simple description in terms of their associated half-translation surfaces. Existence of stationary solutions is proved in some generality by solving an extremal problem involving an electrostatic energy functional, generalizing a classic problem studied by Teichm\"uller, Jenkins, Strebel and others. We study several special cases, including stationary Jordan curves on the Riemann sphere. We also introduce a discrete random version of the dynamics closely related to Propp's competitive erosion model, and conjecture that realizations of the lattice model will converge towards a corresponding solution to the competitive Hele-Shaw problem as the mesh size tends to zero.