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The Stefan Problem : Kirchen der Welt - Anvarbek M. Meirmanov

The Stefan Problem

Kirchen der Welt

By: Anvarbek M. Meirmanov, Marek Niezgodka (Translator), Anna Crowley (Translator)

Hardcover Published: 1st March 1992
ISBN: 9783110114799
Number Of Pages: 255
For Ages: 22+ years old

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The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject.

Editorial Board

Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yakov G. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 6 (2018)
Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Botjan Gabrovek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Preface to the English edition
Preface
Introductionp. 1
Preliminaries
Problem statementp. 8
Assumed notation. Auxiliary notationp. 18
Notationp. 18
Basic function spacesp. 18
Auxiliary inequalities and embedding theoremsp. 20
Auxiliary facts from analysisp. 22
Properties of solutions of differential equationsp. 23
The Cauchy problem for the heat equation over smooth unbounded manifolds in the classes [reproduction of symbols]p. 25
Existence and uniqueness of the generalized solution to the Stefan problemp. 26
Classical solution of the multidimensional Stefan problem
The one-phase Stefan problem. Main resultp. 37
The simplest problem settingp. 39
Construction of approximate solutions to the one-phase Stefan problem over a small time intervalp. 47
A lower bound on the existence interval of the solution. Passage to the limitp. 50
The two-phase Stefan problemp. 60
Existence of the classical solution to the multidimensional Stefan problem on an arbitrary time interval
The one-phase Stefan problemp. 65
The two-phase Stefan problem. Stability of the stationary solutionp. 79
Problem statement. Main resultp. 79
Formulation of the equivalent boundary value problemp. 80
Construction of approximate solutionsp. 81
A lower bound for the constant [reproduction of symbols]p. 84
Proof of the main resultp. 87
Lagrange variables in the multidimensional one-phase Stefan problem
Formulation of the problem in Lagrange variablesp. 90
Linearizationp. 91
Correctness of the linear modelp. 94
Classical solution of the one-dimensional Stefan problem for the homogeneous heat equation
The one-phase Stefan problem. Existence of the solutionp. 99
Asymptotic behaviour of the solution of the one-phase Stefan problemp. 107
The two-phase Stefan problemp. 112
Special cases: one-phase initial state, violation of compatibility conditions, unbounded domainsp. 122
The two-phase multi-front Stefan problemp. 127
Filtration of a viscid compressible liquid in a vertical porous layerp. 130
Problem statement. The main resultp. 130
An equivalent boundary value problem in a fixed domainp. 132
A comparison lemmap. 133
The case [reproduction of symbols]p. 134
The case [reproduction of symbols]p. 135
The case [reproduction of symbols]p. 136
Asymptotic behaviour of the solution, as [reproduction of symbols]p. 139
Structure of the generalized solution to the one-phase Stefan problem. Existence of a mushy region
The inhomogeneous heat equation. Formation of the mushy regionp. 142
The homogeneous heat equation. Dynamic interactions between the mushy phase and the solid/liquid phasesp. 149
The homogeneous heat equation. Coexistence of different phasesp. 158
The case of an arbitrary initial distribution of specific internal energyp. 162
Time-periodic solutions of the one-dimensional Stefan problem
Construction of the generalized solutionp. 174
Structure of the mushy phase for temperature on the boundary of [reproduction of symbols] with constant signp. 177
The case of [reproduction of symbols] with variable signp. 181
Approximate approaches to the two-phase Stefan problem
Problem statement. Formulation of the resultsp. 191
Existence and uniqueness of the generalized solution to Problem [reproduction of symbols]p. 199
Existence of the classical solution to Problem [reproduction of symbols]p. 203
Auxiliary Problem [reproduction of symbols]p. 205
Differential properties of the solutions to Problem [reproduction of symbols]p. 208
Proof of Theorem 2p. 211
Proof of Lemma 5p. 213
The quasi-steady one-dimensional Stefan Problem (C)p. 215
Appendix: Modelling of binary alloy crystallizationp. 222
Referencesp. 231
Supplementary referencesp. 243
Indexp. 245
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783110114799
ISBN-10: 3110114798
Series: Kirchen der Welt
Audience: Professional
For Ages: 22+ years old
Format: Hardcover
Language: English
Number Of Pages: 255
Published: 1st March 1992
Country of Publication: DE
Dimensions (cm): 24.33 x 17.63  x 1.78
Weight (kg): 0.55

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