Lemniscates do not survive Laplacian growth

TL;DR

This paper proves that lemniscates, a specific class of plane boundaries, cannot persist under Laplacian growth processes, as they are either instantly destroyed or cease to belong to this class immediately.

Contribution

The paper demonstrates that lemniscates are not stable solutions in Laplacian growth, showing they cannot survive or remain in the class during evolution.

Findings

Lemniscates are instantly destroyed by Laplacian growth.

Lemniscates do not remain in their class during evolution.

Laplacian growth destroys lemniscates immediately.

Abstract

The large class of moving boundary processes in the plane modeled by the so-called Laplacian growth, which describes, e.g., solidification, electrodeposition, viscous fingering, bacterial growth, etc., is known to be integrable and to exhibit a large number of exact solutions. In this work, the boundaries are assumed to be in the class of lemniscates with all zeros inside the bounded component of the complex plane. We prove that for any initial boundary taken from this class, the evolving boundary instantly stops being in the class, or else Laplacian growth destroys lemniscates instantly.