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Loss of embeddedness for the one-phase quasistationary Stefan problem in 2D

Friedrich Lippoth

TL;DR

This work analyzes loss of embeddedness in the 2D quasistationary one-phase Stefan problem with Gibbs–Thomson correction and kinetic undercooling. It introduces an artificial, continuous extension to immersed curves, accompanied by a trace-cut framework and a fixed-point argument to handle topology changes in a rigorous, local-in-time setting. The authors construct a critical initial state with a small bridge that leads to a true overlap under the generalized flow, and show that embedded geometries can be transported into nonembedded configurations, with precise compatibility and regularity controls. The approach illuminates how topological changes can occur in Stefan-type problems and provides a path toward extending the results to higher dimensions.

Abstract

We provide an example for a smooth and embedded initial state that looses embeddedness in finite time when evolving according to the quasistationary Stefan problem with Gibbs-Thomson correction and kinetic undercooling in 2D.

Loss of embeddedness for the one-phase quasistationary Stefan problem in 2D

TL;DR

This work analyzes loss of embeddedness in the 2D quasistationary one-phase Stefan problem with Gibbs–Thomson correction and kinetic undercooling. It introduces an artificial, continuous extension to immersed curves, accompanied by a trace-cut framework and a fixed-point argument to handle topology changes in a rigorous, local-in-time setting. The authors construct a critical initial state with a small bridge that leads to a true overlap under the generalized flow, and show that embedded geometries can be transported into nonembedded configurations, with precise compatibility and regularity controls. The approach illuminates how topological changes can occur in Stefan-type problems and provides a path toward extending the results to higher dimensions.

Abstract

We provide an example for a smooth and embedded initial state that looses embeddedness in finite time when evolving according to the quasistationary Stefan problem with Gibbs-Thomson correction and kinetic undercooling in 2D.

Paper Structure

This paper contains 10 sections, 6 theorems, 126 equations.

Table of Contents

  1. Introduction.
  2. The classical quasistationary one-phase Stefan problem in 2D (including the effects of surface tension and kinetic undercooling)...
  3. An artificial continuous extension of the Stefan problem to the case of truly immersed curves.
  4. Concept of a critical initial state.
  5. A trace mapping for curves with touching points.
  6. Cutting of overlaps.
  7. The fixed point argument
  8. Time trace zero
  9. A suitable initial state
  10. Appendix

Key Result

Lemma 2.2

The functions $\tilde{\rho}^\pm$ are uniformly Lipschitz continuous w.r.t. $\rho \in \mathbf{B}_\varepsilon^\ast$ and $\rho \mapsto \tilde{\rho}^\pm \in B(\mathbf{B}_\varepsilon^\ast,W_\infty^1(X^\pm, \mathbb{R}))$. In fact, there is $\varepsilon_0 > 0$ s.t. $\Vert \tilde{\rho}^\pm \Vert_{W_\infty^1

Theorems & Definitions (12)

  • Remark 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 3.1
  • Lemma 5.1
  • ...and 2 more