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

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 eigenvalue lambda(0) = 0 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 A consideration of the eigenfunctions indicates that a strong intermediate growth may occur for in detail. generic initial perturbations. Both the linear and the nonlinear initial value problem are considered in a second paper. (C) 2009 Elsevier B.V. All rights reserved.