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Cascade equation for the discontinuities in the Stefan problem with surface tension

Yucheng Guo, Sergey Nadtochiy, Mykhaylo Shkolnikov

Abstract

The Stefan problem with surface tension is well known to exhibit discontinuities in the associated moving aggregate (i.e., in the domain occupied by the solid), whose structure has only been understood under translational or radial symmetry so far. In this paper, we derive an auxiliary partial differential equation of second-order hyperbolic type, referred to as the cascade equation, that captures said discontinuities in the absence of any symmetry assumptions. Specializing to the one-phase setting, we introduce a novel (global) notion of weak solution to the cascade equation, which is defined as a limit of mean-field game equilibria. For the spatial dimension two, we show the existence of such a weak solution and prove a natural perimeter estimate on the associated moving aggregate.

Cascade equation for the discontinuities in the Stefan problem with surface tension

Abstract

The Stefan problem with surface tension is well known to exhibit discontinuities in the associated moving aggregate (i.e., in the domain occupied by the solid), whose structure has only been understood under translational or radial symmetry so far. In this paper, we derive an auxiliary partial differential equation of second-order hyperbolic type, referred to as the cascade equation, that captures said discontinuities in the absence of any symmetry assumptions. Specializing to the one-phase setting, we introduce a novel (global) notion of weak solution to the cascade equation, which is defined as a limit of mean-field game equilibria. For the spatial dimension two, we show the existence of such a weak solution and prove a natural perimeter estimate on the associated moving aggregate.

Paper Structure

This paper contains 15 sections, 30 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Derivation of the cascade equation
  3. PDE formulation
  4. Equilibrium formulation
  5. Properties of equilibria
  6. Arrival time function in the equilibrium formulation
  7. Perimeter of equilibrium aggregates
  8. Minimal solution to the cascade equation
  9. Existence of a minimal solution
  10. Perimeter of minimal aggregates
  11. Examples
  12. Example: One Dimension, One Interface
  13. Example: One Dimension, Two Interfaces
  14. Example: General Dimension with Radial Symmetry, Growing Ice Ball
  15. Example: General Dimension with Radial Symmetry, Shrinking Water Ball

Key Result

Theorem 1.4

Suppose that $d=2$, the set $\Gamma_{s-}$ has finite perimeter, $\int_{\partial_*\Gamma_{s-}} (\gamma+V_0)\,\mathrm{d}\mathcal{H}^{1} < \infty$, and $\int_{\mathbb{R}^d\setminus\overline{\Gamma}_{s-}} (1+u(s-,x))^-\,\mathrm{d}x<\infty$. Then, there exists a minimal solution to PDE:wave.

Theorems & Definitions (75)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 2.1: From PDE to equilibrium
  • Remark 2.2
  • proof
  • ...and 65 more