9 research outputs found
Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling
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 . 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.Comment: 19 pages, 7 figures, accepted for SIAM J. Appl. Mat