Models of phase transitions.

*(English)*Zbl 0882.35004
Progress in Nonlinear Differential Equations and their Applications. 28. Boston: Birkhäuser. vii, 322 p. (1996).

This book provides a systematic presentation of the extended models of phase transition and of the mathematical investigation of the corresponding problems. The research in this field has experienced an enormous development during the last two decades with an impressive amount of papers published at an ever increasing rate, making the necessity of an overview and of a comparison of the various theories strongly being felt among the specialists (mathematicians, physicists and engineers). For such reason there was a strong expectation of a good book illustrating the various aspects of modelling phase change processes and analysing the incredible variety of the mathematical problems which have been proposed in the literature. Certainly the readers will not be disappointed by Visintin’s book, since with competence and expository skill the author leads the reader through a logic sequence of subjects of increasing physical and mathematical complexity.

The natural starting point is the Stefan problem and its different formulations, discussing at length the nature of mushy regions. Then generalizations such as kinetic undercooling, phase relaxation models, and phase change of hetereogeneous materials are dealt with.

Models based on a thermodynamical approach are illustrated in the chapter devoted to the Gibbs-Thomson law and containing the presentation of the phase field model. A further step is made considering phase change processes governed by nucleation and crystal growth, leading to the analysis of mean curvature flows, hysteresis phenomena, and other topics such as singular phase transitions, the Mullin-Sekerka problem, etc. An alternative nucleation model (non-adiabatic nucleation with small undercooling) is discussed in the last chapter with an application to the Landau-Lifshitz theory of ferromagnetism.

A remarkable feature of this book is the great attention devoted to the mathematical background. Indeed, the first part of the book contains an extensive summary of results about nonlinear PDEs of the kinds which appear most frequently in the sequel. The required non-elementary mathematical notions are explained in a long and detailed appendix.

Although devoted to phase change, the book analyses also problems of different origin (e.g. the Muskat problem, the Hele-Shaw problem, etc.) whose mathematical structure is closely related to phase change models.

The comments sections at the end of each chapter are very useful not only because they provide a synoptic view of the material but also for the guide to the quite extensive literature.

The natural starting point is the Stefan problem and its different formulations, discussing at length the nature of mushy regions. Then generalizations such as kinetic undercooling, phase relaxation models, and phase change of hetereogeneous materials are dealt with.

Models based on a thermodynamical approach are illustrated in the chapter devoted to the Gibbs-Thomson law and containing the presentation of the phase field model. A further step is made considering phase change processes governed by nucleation and crystal growth, leading to the analysis of mean curvature flows, hysteresis phenomena, and other topics such as singular phase transitions, the Mullin-Sekerka problem, etc. An alternative nucleation model (non-adiabatic nucleation with small undercooling) is discussed in the last chapter with an application to the Landau-Lifshitz theory of ferromagnetism.

A remarkable feature of this book is the great attention devoted to the mathematical background. Indeed, the first part of the book contains an extensive summary of results about nonlinear PDEs of the kinds which appear most frequently in the sequel. The required non-elementary mathematical notions are explained in a long and detailed appendix.

Although devoted to phase change, the book analyses also problems of different origin (e.g. the Muskat problem, the Hele-Shaw problem, etc.) whose mathematical structure is closely related to phase change models.

The comments sections at the end of each chapter are very useful not only because they provide a synoptic view of the material but also for the guide to the quite extensive literature.

Reviewer: A.Fasano (Firenze)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35R35 | Free boundary problems for PDEs |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

80A22 | Stefan problems, phase changes, etc. |

82B26 | Phase transitions (general) in equilibrium statistical mechanics |

82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |

35K55 | Nonlinear parabolic equations |