Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extinction rates for nonradial solutions to the Stefan problem

Gabriele Fioravanti, Xavier Ros-Oton, Clara Torres-Latorre

TL;DR

This work analyzes the one-phase Stefan problem and its free boundary near isolated singular points. By employing second-blow-up analysis, expansion at singular points, and parabolic barrier constructions, the authors derive sharp melting-rate bounds in 2D that closely mirror radial solutions and extend to higher dimensions with precise, stratified behavior. They establish global C^1 regularity of the free boundary away from the top stratum Σ_{n-1}, and show that Γ(x) = t is not C^1 on Σ_{n-1}, with optimal notations tied to extinction rates. The results provide a deep, quantitative description of the free boundary structure, including a thorough treatment of singular sets via self-similar caloric solutions and Whitney-type regularity, significantly advancing the understanding of the Stefan problem’s singularities and regularity theory.

Abstract

We consider the one-phase Stefan problem describing the evolution of melting ice. On the one hand, we focus on understanding the evolution of the free boundary near isolated singular points, and we establish for the first time upper and (more surprisingly) lower estimates for its evolution. In 2D, these bounds almost match the best known ones for radial solutions, but hold for all solutions to the Stefan problem, with no extra assumption on the initial or boundary data. On the other hand, as a consequence of our results, we also characterize the global regularity of the free boundary, as follows: it can be written as a graph $t = Γ(x)$, where $Γ$ is $C^1$ (and not $C^2$) near any singular points in the lower strata $Σ_m$, $m \leq n - 2$. Moreover, $Γ$ is not $C^1$ at singular points in $Σ_{n-1}$.

Extinction rates for nonradial solutions to the Stefan problem

TL;DR

This work analyzes the one-phase Stefan problem and its free boundary near isolated singular points. By employing second-blow-up analysis, expansion at singular points, and parabolic barrier constructions, the authors derive sharp melting-rate bounds in 2D that closely mirror radial solutions and extend to higher dimensions with precise, stratified behavior. They establish global C^1 regularity of the free boundary away from the top stratum Σ_{n-1}, and show that Γ(x) = t is not C^1 on Σ_{n-1}, with optimal notations tied to extinction rates. The results provide a deep, quantitative description of the free boundary structure, including a thorough treatment of singular sets via self-similar caloric solutions and Whitney-type regularity, significantly advancing the understanding of the Stefan problem’s singularities and regularity theory.

Abstract

We consider the one-phase Stefan problem describing the evolution of melting ice. On the one hand, we focus on understanding the evolution of the free boundary near isolated singular points, and we establish for the first time upper and (more surprisingly) lower estimates for its evolution. In 2D, these bounds almost match the best known ones for radial solutions, but hold for all solutions to the Stefan problem, with no extra assumption on the initial or boundary data. On the other hand, as a consequence of our results, we also characterize the global regularity of the free boundary, as follows: it can be written as a graph , where is (and not ) near any singular points in the lower strata , . Moreover, is not at singular points in .

Paper Structure

This paper contains 18 sections, 22 theorems, 179 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The 2D case
  3. Higher dimensions
  4. Global regularity of the free boundary
  5. Organization of the paper
  6. Acknowledgements
  7. Preliminaries
  8. Setting
  9. Estimates for the heat equation
  10. The parabolic obstacle problem
  11. The second blow-up
  12. Expansion of solutions at singular points
  13. A parabolic almost positivity property
  14. Self-similar solutions
  15. Extinction rate at singular points
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $u(x,t)$ be any solution of eq:obstacleProblem in $Q_1\subset\mathbb R^{2+1}$, and assume that $(0,0)$ is a singular free boundary point in $\Sigma_0$. Then, for any $\delta>0$, in a neighborhood of the origin we have for some positive constants $c_1,C_1$. In particular, for any $\delta>0$ the free boundary near the origin satisfies for some positive constants $c_1,C_1$.

Figures (1)

  • Figure 4.1: The complement of $D_{\eta,1}$ with $n = 2$. The orange surface is $\{|y| = \eta|t|^{1/2}\}$, and the blue one is $\{|y| = \eta|x|\}\cup\{|y|\leq\eta|x|,t=0\}$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3: Caf77, CF79
  • Lemma 2.4: FRS24, Corollary 6.2, Lemma 6.3
  • Proposition 2.5: FRS24, Lemma 5.8, Corollary 5.9, Proposition 6.7
  • ...and 31 more