A Stefan problem on an evolving surface

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 exi...

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Bibliographic Details
Published in:arXiv.org 2015-07
Main Authors: Amal Alphonse, Elliott, Charles M
Format: Article
Language:English
Subjects:
Dependence
Evolution
Fixed points (mathematics)
Function space
Hyperspaces
Regularization
Well posed problems
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1412.5534