Equivalent formulations of the oxygen depletion problem, other implicit free boundary value problems, and implications for numerical approximation

TL;DR

This paper explores various mathematical formulations of the oxygen depletion free boundary problem, proves their equivalence, and discusses implications for numerical approximation, including convergence results and open challenges.

Contribution

It introduces a new gradient flow with constraint formulation and demonstrates its equivalence to existing models, advancing numerical methods for free boundary problems.

Findings

Proved equivalence of multiple formulations of the oxygen depletion problem.

Established convergence of a numerical approximation based on the new formulation.

Discussed extensions to more general free boundary problems.

Abstract

The Oxygen Depletion problem is an implicit free boundary value problem. The dynamics allow topological changes in the free boundary. We show several mathematical formulations of this model from the literature and give a new formulation based on a gradient flow with constraint. All formulations are shown to be equivalent. We explore the possibilities for the numerical approximation of the problem that arise from the different formulations. We show a convergence result for an approximation based on the gradient flow with constraint formulation that applies to the general dynamics including topological changes. More general (vector, higher order) implicit free boundary value problems are discussed. Several open problems are described.