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IMA Journal of Applied Mathematics Advance Access originally published online on September 9, 2008
IMA Journal of Applied Mathematics 2009 74(1):107-127; doi:10.1093/imamat/hxn029
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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

Matthias Günther

Mathematisches Institut, Universität Leipzig, Johannisgasse 26, 04103 Leipzig, Germany

Georg Prokert{dagger}

Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands

{dagger} Email: g.prokert{at}tue.nl

Received on November 28, 2007; Revision received June 20, 2008. Accepted on July 31, 2008

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.

Keywords: degenerate transport equation; moving boundary problem; Hele–Shaw flow; kinetic undercooling; electrical streamer discharges.


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