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.

翻译：氧气消耗问题是一个隐含的自由边界值问题。 动态允许自由边界的地形变化。 我们从文献中展示了该模型的若干数学公式, 并给出了基于渐变流的新的公式, 并附有限制。 所有公式都显示为等值。 我们探讨不同公式产生的问题的数值近似可能性。 我们显示了基于梯度流的近似趋同结果, 其约束公式适用于一般动态, 包括地形变化 。 我们讨论了更一般的( 矢量、 更高顺序) 隐含的自由边界值问题。 我们描述了几个开放的问题 。