Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling
| dc.creator | Ebert, Ute | |
| dc.creator | Meulenbroek, Bernard | |
| dc.creator | Schaefer, Lothar | |
| dc.date | 2006-06-19 | |
| dc.date | 2007-07-24 | |
| dc.date.accessioned | 2026-07-07T08:41:08Z | |
| dc.date.available | 2026-07-07T08:41:08Z | |
| dc.description | We study the shape stability of disks moving in an external Laplacian field in two dimensions. The problem is motivated by the motion of ionization fronts in streamer-type electric breakdown. It is mathematically equivalent to the motion of a small bubble in a Hele-Shaw cell with a regularization of kinetic undercooling type, namely a mixed Dirichlet-Neumann boundary condition for the Laplacian field on the moving boundary. Using conformal mapping techniques, linear stability analysis of the uniformly translating disk is recast into a single PDE which is exactly solvable for certain values of the regularization parameter. We concentrate on the physically most interesting exactly solvable and non-trivial case. We show that the circular solutions are linearly stable against smooth initial perturbations. In the transformation of the PDE to its normal hyperbolic form, a semigroup of automorphisms of the unit disk plays a central role. It mediates the convection of perturbations to the back of the circle where they decay. Exponential convergence to the unperturbed circle occurs along a unique slow manifold as time $t\to\infty$. Smooth temporal eigenfunctions cannot be constructed, but excluding the far back part of the circle, a discrete set of eigenfunctions does span the function space of perturbations. We believe that the observed behaviour of a convectively stabilized circle for a certain value of the regularization parameter is generic for other shapes and parameter values. Our analytical results are illustrated by figures of some typical solutions. | |
| dc.description | 19 pages, 7 figures, accepted for SIAM J. Appl. Math | |
| dc.identifier | https://arxiv.org/abs/nlin/0606048 | |
| dc.identifier | http://arxiv.org/abs/nlin/0606048 | |
| dc.identifier | SIAM J. Appl. Math. 68, 292-310 (2007) | |
| dc.identifier | doi:10.1137/070683908 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/141591 | |
| dc.subject | Pattern Formation and Solitons | |
| dc.subject | Exactly Solvable and Integrable Systems | |
| dc.title | Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling | |
| dc.type | text |