On travelling-wave solutions for a moving boundary problem of Hele-Shaw type

We discuss a 2D moving boundary problem for the Laplacian with Robin boundary conditions in an exterior domain. It arises as a model for Hele–Shaw flow of a bubble with kinetic undercooling regularization and is also discussed in the context of models for electrical streamer discharges. The corresponding evolution equation is given by a degenerate, non-linear transport problem with non-local lower-order dependence. We identify the local structure of the set of travelling-wave solutions in the vicinity of trivial (circular) ones. We find that there is a unique non-trivial travelling wave for each velocity near the trivial one. Therefore, the trivial solutions are unstable in a comoving frame. The degeneracy of our problem is reflected in a loss of regularity in the estimates for the linearization. Moreover, there is an upper bound for the regularity of its solutions. To prove our results, we use a quasi-linearization by differentiation, index results for degenerate ordinary differential operators on the circle and perturbation arguments for unbounded Fredholm operators.