Interface growth in two dimensions: A Loewner-equation approach

TL;DR

This paper applies the Loewner-equation framework to analyze two-dimensional Laplacian growth, presenting new exact solutions, general models for interface growth, and discussing extensions involving evolving measures.

Contribution

It introduces a general class of interface growth models using Loewner equations and provides new exact solutions for multi-finger configurations.

Findings

New exact solution for three-finger growth configuration

Development of a general Loewner-based interface growth model

Discussion of growth models with evolving measures

Abstract

The problem of Laplacian growth in two dimensions is considered within the Loewner-equation framework. Initially the problem of fingered growth recently discussed by Gubiec and Szymczak [T. Gubiec and P. Szymczak, Phys. Rev. E 77, 041602 (2008)] is revisited and a new exact solution for a three-finger configuration is reported. Then a general class of growth models for an interface growing in the upper-half plane is introduced and the corresponding Loewner equation for the problem is derived. Several examples are given including interfaces with one or more tips as well as multiple growing interfaces. A generalization of our interface growth model in terms of ``Loewner domains,'' where the growth rule is specified by a time evolving measure, is briefly discussed.