The Stefan Problem

A.M. Meirmanov, Marek Niezgodka, Anna Crowley, Marek Niezgodka, Anna Crowley

• Publisher: De Gruyter
• Year: 2011
• Print ISBN: 9783110114799
• eBook ISBN: 9783110846720
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

No detailed description available for "The Stefan Problem".

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