The Stefan Problem

✍ Scribed by A.M. Meirmanov; Marek Niezgodka; Anna Crowley

• Publisher: De Gruyter
• Year: 1992
• Tongue: English
• Leaves: 256
• Series: De Gruyter Expositions in Mathematics; 3
• Edition: Reprint 2011
• Category: Library

✦ Synopsis

"This book, written by an outstanding researcher, is long overdue. The choice of material is restricted to author’s area of interest, but this is a positive feature, making the book readable and authoritative. The author deliberately avoided details of numerical techniques, control theoretic questions and problems with fractional order derivatives. This is a well-written monograph presenting results that are scattered through dozens of journals in one volume." Zentralblatt für Mathematik

✦ Table of Contents

Preface to the English edition
Preface
Introduction
Chapter I. Preliminaries
1. Problem statement
2. Assumed notation. Auxiliary notation
2.1. Notation
2.2. Basic function spaces
2.3. Auxiliary inequalities and embedding theorems
2.4. Auxiliary facts from analysis
2.5. Properties of solutions of differential equations
2.6. The Cauchy problem for the heat equation over smooth unbounded manifolds in the classes Hl+2,(l+2)/2(ST)
3. Existence and uniqueness of the generalized solution to the Stefan problem
Chapter II. Classical solution of the multidimensional Stefan problem
1. The one-phase Stefan problem. Main result
2. The simplest problem setting
3. Construction of approximate solutions to the one-phase Stefan problem over a small time interval
4. A lower bound on the existence interval of the solution. Passage to the limit
5. The two-phase Stefan problem
Chapter III. Existence of the classical solution to the multidimensional Stefan problem on an arbitrary time interval
1. The one-phase Stefan problem
2. The two-phase Stefan problem. Stability of the stationary solution
2.1. Problem statement. Main result
2.2. Formulation of the equivalent boundary value problem
2.3. Construction of approximate solutions
2.4. A lower bound for the constant δ3
2.5. Proof of the main result
Chapter IV. Lagrange variables in the multidimensional one-phase Stefan problem
1. Formulation of the problem in Lagrange variables
2. Linearization
3. Correctness of the linear model
Chapter V. Classical solution of the one-dimensional Stefan problem for the homogeneous heat equation
1. The one-phase Stefan problem. Existence of the solution
2. Asymptotic behaviour of the solution of the one-phase Stefan problem
3. The two-phase Stefan problem
4. Special cases: one-phase initial state, violation of compatibility conditions, unbounded domains
5. The two-phase multi-front Stefan problem
6. Filtration of a viscid compressible liquid in a vertical porous layer
6.1. Problem statement. The main result
6.2. An equivalent boundary value problem in a fixed domain
6.3. A comparison lemma
6.4. The case ∫1∞1/f(p)dp = ∞
6.5. The case f(p) = exp(p — 1)
6.6. The case f(p) = pγ, γ ≥ 1
6.7. Asymptotic behaviour of the solution, as t→∞
Chapter VI. Structure of the generalized solution to the one-phase Stefan problem. Existence of a mushy region
1. The inhomogeneous heat equation. Formation of the mushy region
2. The homogeneous heat equation. Dynamic interactions between the mushy phase and the solid/liquid phases
3. The homogeneous heat equation. Coexistence of different phases
4. The case of an arbitrary initial distribution of specific internal energy
Chapter VII. Time-periodic solutions of the one-dimensional Stefan problem
1. Construction of the generalized solution
2. Structure of the mushy phase for temperature on the boundary of Ω∞ with constant sign
3. The case of υ+(t) with variable sign
Chapter VIII. Approximate approaches to the two-phase Stefan problem
1. Problem statement. Formulation of the results
2. Existence and uniqueness of the generalized solution to Problem (A0)
3. Existence of the classical solution to Problem (A0)
3.1. Auxiliary Problem (Cr)
3.2. Differential properties of the solutions to Problem (Cr)
3.3. Proof of Theorem 2
3.4. Proof of Lemma 5
4. The quasi-steady one-dimensional Stefan Problem (C)
Appendix
Modelling of binary alloy crystallization
References
Supplementary references
Index