A Stefan problem on an evolving surface

TL;DR

This paper formulates and analyzes a Stefan problem on an evolving surface, establishing well-posedness of solutions with minimal regularity data through advanced functional analysis and regularization techniques.

Contribution

It develops a framework for Stefan problems on evolving surfaces and proves existence, uniqueness, and continuous dependence of solutions with $L^1$ data, extending classical results to dynamic geometries.

Findings

Existence of weak solutions for $L^ ext{infty}$ data.

Extension of solutions to $L^1$ data via duality methods.

Development of function spaces on evolving surfaces.

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.