Stephan Problem Book

	Preface to the English edition
	Preface
	Introduction	1
Ch. I	Preliminaries
1	Problem statement	8
2	Assumed notation. Auxiliary notation	18
2.1	Notation	18
2.2	Basic function spaces	18
2.3	Auxiliary inequalities and embedding theorems	20
2.4	Auxiliary facts from analysis	22
2.5	Properties of solutions of differential equations	23
2.6	The Cauchy problem for the heat equation over smooth unbounded manifolds in the classes [reproduction of symbols]	25
3	Existence and uniqueness of the generalized solution to the Stefan problem	26
Ch. II	Classical solution of the multidimensional Stefan problem
1	The one-phase Stefan problem. Main result	37
2	The simplest problem setting	39
3	Construction of approximate solutions to the one-phase Stefan problem over a small time interval	47
4	A lower bound on the existence interval of the solution. Passage to the limit	50
5	The two-phase Stefan problem	60
Ch. III	Existence of the classical solution to the multidimensional Stefan problem on an arbitrary time interval
1	The one-phase Stefan problem	65
2	The two-phase Stefan problem. Stability of the stationary solution	79
2.1	Problem statement. Main result	79
2.2	Formulation of the equivalent boundary value problem	80
2.3	Construction of approximate solutions	81
2.4	A lower bound for the constant [reproduction of symbols]	84
2.5	Proof of the main result	87
Ch. IV	Lagrange variables in the multidimensional one-phase Stefan problem
1	Formulation of the problem in Lagrange variables	90
2	Linearization	91
3	Correctness of the linear model	94
Ch. V	Classical solution of the one-dimensional Stefan problem for the homogeneous heat equation
1	The one-phase Stefan problem. Existence of the solution	99
2	Asymptotic behaviour of the solution of the one-phase Stefan problem	107
3	The two-phase Stefan problem	112
4	Special cases: one-phase initial state, violation of compatibility conditions, unbounded domains	122
5	The two-phase multi-front Stefan problem	127
6	Filtration of a viscid compressible liquid in a vertical porous layer	130
6.1	Problem statement. The main result	130
6.2	An equivalent boundary value problem in a fixed domain	132
6.3	A comparison lemma	133
6.4	The case [reproduction of symbols]	134
6.5	The case [reproduction of symbols]	135
6.6	The case [reproduction of symbols]	136
6.7	Asymptotic behaviour of the solution, as [reproduction of symbols]	139
Ch. 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	142
2	The homogeneous heat equation. Dynamic interactions between the mushy phase and the solid/liquid phases	149
3	The homogeneous heat equation. Coexistence of different phases	158
4	The case of an arbitrary initial distribution of specific internal energy	162
Ch. VII	Time-periodic solutions of the one-dimensional Stefan problem
1	Construction of the generalized solution	174
2	Structure of the mushy phase for temperature on the boundary of [reproduction of symbols] with constant sign	177
3	The case of [reproduction of symbols] with variable sign	181
Ch. VIII	Approximate approaches to the two-phase Stefan problem
1	Problem statement. Formulation of the results	191
2	Existence and uniqueness of the generalized solution to Problem [reproduction of symbols]	199
3	Existence of the classical solution to Problem [reproduction of symbols]	203
3.1	Auxiliary Problem [reproduction of symbols]	205
3.2	Differential properties of the solutions to Problem [reproduction of symbols]	208
3.3	Proof of Theorem 2	211
3.4	Proof of Lemma 5	213
4	The quasi-steady one-dimensional Stefan Problem (C)	215
	Appendix: Modelling of binary alloy crystallization	222
	References	231
	Supplementary references	243
	Index	245