Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling

Details Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling

2006-06-19

arxiv.org/abs/nlin/0606048v2

SIAM J. Appl. Math. 68, 292-310 (2007)

Nonlinear Sciences

Pattern Formation and Solitons

19 pages, 7 figures, accepted for SIAM J. Appl. Math

Scientific paper

10.1137/070683908

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.

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