Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On behavior of free boundaries to generalized two-phase Stefan problems for parabolic partial differential equation systems

Toyohiko Aiki, Hana Kakiuchi

TL;DR

The paper analyzes a generalized two-phase Stefan problem with a moving evaporation front modeling bread baking, where the front speed is governed by $l\,w\,e'(t)=k_l\,u_x(t,e(t)^-)-k_a\,u_x(t,e(t)^+)$ and the crust–crumb temperature and water content couple through transmission and boundary conditions. By leveraging improved regularity and mollified boundary data, the authors establish local-in-time existence and uniqueness of strong solutions for initial data with high regularity, and derive a maximal existence time dichotomy: either global existence or front-contact with $0$ or $1$. They introduce a fixed-point framework for the free boundary via a solution operator AP$(e)$ and prove uniform a priori estimates for $u$, $w$, and the boundary trace $u(\cdot,1)$, plus stability results for water content under perturbations of the front. The work provides a rigorous foundation for the bread-baking free boundary model and its long-time behavior, including convergence to limiting profiles as the maximal time is approached. Key contributions include the local well-posedness (existence/uniqueness) under precise regularity hypotheses, the contraction-based fixed-point construction for the free boundary, and the global-in-time or blow-up alternatives with boundary-limit characterizations.

Abstract

Recently, we have proposed a new free boundary problem representing the bread baking process in a hot oven. Unknown functions in this problem are the position of the evaporation front, the temperature field and the water content. For solving this problem we observed two difficulties that the growth rate of the free boundary depends on the water content and the boundary condition for the water content contains the temperature. In this paper, by improving the regularity of solutions, we overcome these difficulties and establish existence of a solution locally in time and its uniqueness. Moreover, under some sign conditions for initial data, we derive a result on the maximal interval of existence to solutions.

On behavior of free boundaries to generalized two-phase Stefan problems for parabolic partial differential equation systems

TL;DR

The paper analyzes a generalized two-phase Stefan problem with a moving evaporation front modeling bread baking, where the front speed is governed by and the crust–crumb temperature and water content couple through transmission and boundary conditions. By leveraging improved regularity and mollified boundary data, the authors establish local-in-time existence and uniqueness of strong solutions for initial data with high regularity, and derive a maximal existence time dichotomy: either global existence or front-contact with or . They introduce a fixed-point framework for the free boundary via a solution operator AP and prove uniform a priori estimates for , , and the boundary trace , plus stability results for water content under perturbations of the front. The work provides a rigorous foundation for the bread-baking free boundary model and its long-time behavior, including convergence to limiting profiles as the maximal time is approached. Key contributions include the local well-posedness (existence/uniqueness) under precise regularity hypotheses, the contraction-based fixed-point construction for the free boundary, and the global-in-time or blow-up alternatives with boundary-limit characterizations.

Abstract

Recently, we have proposed a new free boundary problem representing the bread baking process in a hot oven. Unknown functions in this problem are the position of the evaporation front, the temperature field and the water content. For solving this problem we observed two difficulties that the growth rate of the free boundary depends on the water content and the boundary condition for the water content contains the temperature. In this paper, by improving the regularity of solutions, we overcome these difficulties and establish existence of a solution locally in time and its uniqueness. Moreover, under some sign conditions for initial data, we derive a result on the maximal interval of existence to solutions.
Paper Structure (7 sections, 24 theorems, 150 equations)

This paper contains 7 sections, 24 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Known results
  4. Proof of Propoosition \ref{['prop1']}
  5. Estimates for water contents
  6. Proof of Theorem \ref{['th1']}
  7. Proof of Theorem \ref{['th2']}

Key Result

Theorem 2.1

Assume (A1) $p \in C^1({\mathbb R})$, $0 \leq p, p' \leq M_p$ on ${\mathbb R}$, where $M_p$ is a positive constant. (A2) $0 < e_0 < 1$, $u_0 \in W^{1,\infty}(0, 1)$, $u_0 \geq 0$ on $[e_0,1]$, $u_0 \leq 0$ on $[0,e_0]$. (A3) $w_0 \in W^{1,2}(0,1)$, $w_0(e_0) > 0$. (A4) $u_b \in W^{1,2}(0,T)$, $u_b

Theorems & Definitions (46)

  • Definition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.2
  • Proposition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 36 more