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arXiv · 1412.5534

A Stefan problem on an evolving surface

Amal Alphonse·Charles M. Elliott
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Abstract

We formulate a Stefan problem on an evolving hypersurface and study the well-posedness of weak solutions given $L^1$ data. To do this, we first develop function spaces and results to handle equations on evolving surfaces in order to give a natural treatment of the problem. Then we consider the existence of solutions for $L^\infty$ data; this is done by regularisation of the nonlinearity. The regularised problem is solved by a fixed point theorem and then uniform estimates are obtained in order to pass to the limit. By using a duality method we show continuous dependence which allows us to extend the results to $L^1$ data.

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Amal Alphonse, Charles M. Elliott. 2015-07-10. A Stefan problem on an evolving surface. https://doi.org/10.1098/rsta.2014.0279

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