Global classical solution of quasi-stationary Stefan free boundary problem
type: Item
instance of: scholarly article
title: Global classical solution of quasi-stationary Stefan free boundary problem
zbMATH Open document ID: 1075.35120
published in: Applied Mathematics and Computation
publication date: 2005-02-22T00:00:00Z
review text: The quasi-stationary Stefan problem (i.e. the Stefan problem with zero specific heat) is also known as the Hele-Shaw problem. The author proves its classical solvability, also discussing the long time behaviour of the solutions. For simplicity of exposition the paper refers to the one-phase, two-dimensional problem, but the results are true in general. The only restriction adopted is that the initial and boundary data are small enough perturbation of the data of a one-dimensional problem. The problem to be solved is the following: find the pair \((u(x,y,t),\rho (x,t))\) satisfying \[ \begin{aligned} \Delta u=0,\quad & x\in\mathbb R,\quad 0<y<\rho(x,t),\quad t>0,\\ u(x,y,t)=0,\quad & x\in\mathbb R,\quad y=\rho(x,t),\quad t>0,\\ V_n=-\dfrac{\partial u}{\partial n},\quad & x\in\mathbb R,\quad y=\rho(x,t),\quad t>0\\ u(x,0,t)=g(x,t),\quad & x\in\mathbb R,\quad t>0,\\ \rho(x,0)=\rho_0(x),\quad & x\in\mathbb R,\end{aligned} \] and the solution is looked for in the form \[ u(x,y,t)=u_0(y,t)+\epsilon u_1(x,y,t), \] \[ \rho(x,t)=s(t)+\epsilon\sigma (x,t). \] Perturbations \(u_1, \sigma\) satisfy the system \[ \begin{aligned} \Delta u_1=0,\quad & x\in\mathbb R,\quad 0<y<\rho(x,t),\quad t>0,\\ u_1=\dfrac{g(t)}{s(t)}\sigma(x,t),\quad & x\in\mathbb R,\quad y=\rho(x,t),\quad t>0,\\ \partial_t\sigma+\partial_y u_1-\epsilon\partial_x\sigma\partial_x u_1=0,\quad & x\in\mathbb R,\quad y=\rho(x,t),\quad t>0,\\ u_1(x,0,t)=g_1(x,t),\quad & x\in\mathbb R,\quad t>0,\\ \sigma(x,0)=\sigma_0(x),\quad & x\in\mathbb R \end{aligned} \] which is shown to have a classical solution provided \(\epsilon\) is small enough. A problem more general than the latter is in fact studied. A fixed point technique is used, going through the determination of delicate estimates. Finally it is shown that when the perturbations of the data belong to a suitable class, then \(u_1,\sigma\) decay asymptotically at least as \((1+t)^{-1}, (1+t)^{-\frac{1}{2}}\), respectively.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 76D27
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2136867
zbMATH Keywords: free boundary problems
zbMATH Keywords: Hele-Shaw flows
zbMATH Keywords: Stefan problem with zero specific heat
zbMATH Keywords: classical solvability
zbMATH Keywords: long time behaviour
author: Fahuai Yi
reviewed by: Antonio Fasano
MaRDI profile type: Publication
full work available at URL: j.amc.2003.12.002
OpenAlex ID: W1986263965
cites work: Q3745373
cites work: Classical solution of quasi-stationary Stefan problem
cites work: Quasi-stationary Stefan problem as limit case of Mullins-Sekerka problem
cites work: Classical Solutions of Multidimensional Hele--Shaw Models
cites work: On one proof of the classical solvability of the Hele-Shaw problem with free boundary
cites work: The existence and uniqueness of analytic solutions for a moving boundary problem for Hele-Shaw flows in the plane
cites work: A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane
cites work: Existence of the Global Classical Solution for a Two-Phase Stefan Problem
cites work: Existance uniqueness and regularity of classical solutions of the mullins—sekerka problem
cites work: Q4941416
cites work: On Hele-Shaw models with surface tension
cites work: A center manifold analysis for the Mullins-Sekerka model
cites work: Q5696748
cites work: Analytic semigroups and optimal regularity in parabolic problems
cites work: Compact sets in the space \(L^ p(0,T;B)\)
cites work: Q4016476
cites work: Q4040640
DOI: 10.1016/J.AMC.2003.12.002
description: scientific article; zbMATH DE number 2136867