A moving boundary problem motivated by electric breakdown: I. Spectrum of linear perturbations

TL;DR

This paper analyzes the linear stability of circular interfaces in a Laplacian growth model with kinetic undercooling, revealing that perturbations generally decay exponentially but can exhibit strong intermediate growth.

Contribution

It provides a detailed spectral analysis of the stability operator for the model, including asymptotic behavior of eigenvalues as regularization vanishes.

Findings

Eigenvalues are mostly negative, indicating stability.

Eigenvalues approach zero in a singular manner as regularization vanishes.

Potential for strong intermediate growth in perturbations.

Abstract

An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written as a Laplacian growth model regularized by a `kinetic undercooling' boundary condition. We study the linear stability of uniformly translating circles that solve the problem in two dimensions. In a space of smooth perturbations of the circular shape, the stability operator is found to have a pure point spectrum. Except for the zero eigenvalue for infinitesimal translations, all eigenvalues are shown to have negative real part. Therefore perturbations decay exponentially in time. We calculate the spectrum through a combination of asymptotic and series evaluation. In the limit of vanishing regularization parameter, all eigenvalues are found to approach zero in a singular fashion, and this asymptotic behavior is worked out in detail. A consideration of the eigenfunctions indicates that a strong intermediate growth may occur for generic initial perturbations. Both the linear and the nonlinear initial value problem are considered in a second paper.