A Stefan problem on an evolving surface

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.

21 pages

References in corpus (1)

• An abstract framework for parabolic PDEs on evolving spaces

Cited by in corpus (3)

• Free Boundary Problems: The Forefront of Current and Future Developments
• Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs
• An evolving space framework for Oseen equations on a moving domain