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We study the shape stability of disks moving in an external Laplacian fieldin two dimensions. The problem is motivated by the motion of ionization frontsin streamer-type electric breakdown. It is mathematically equivalent to themotion of a small bubble in a Hele-Shaw cell with a regularization of kineticundercooling type, namely a mixed Dirichlet-Neumann boundary condition for theLaplacian field on the moving boundary. Using conformal mapping techniques,linear stability analysis of the uniformly translating disk is recast into asingle PDE which is exactly solvable for certain values of the regularizationparameter. We concentrate on the physically most interesting exactly solvableand non-trivial case. We show that the circular solutions are linearly stableagainst smooth initial perturbations. In the transformation of the PDE to itsnormal hyperbolic form, a semigroup of automorphisms of the unit disk plays acentral role. It mediates the convection of perturbations to the back of thecircle where they decay. Exponential convergence to the unperturbed circleoccurs along a unique slow manifold as time $t\to\infty$. Smooth temporaleigenfunctions cannot be constructed, but excluding the far back part of thecircle, a discrete set of eigenfunctions does span the function space ofperturbations. We believe that the observed behaviour of a convectivelystabilized circle for a certain value of the regularization parameter isgeneric for other shapes and parameter values. Our analytical results areillustrated by figures of some typical solutions.Comment: 19 pages, 7 figures, accepted for SIAM J. Appl. Mat

Convective Stabilization of a Laplacian Moving Boundary Problem with Kinetic Undercooling