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Models for phase separation and their mathematics. (English) Zbl 0957.35062

It is a very fortunate circumstance that this paper, written ten years ago and meant for a proceedings volume that never appeared, founds its way to a mathematical journal. Despite the considerable amount of work on the subject performed during the last decade, and shortly summarized by the author in the last section of his re-born paper, this is still a fine and useful overview, whose intentions go beyond the target of a mere survey. Indeed, not only the basic concepts are illustrated, along with the relevant literature, but the effort is made to conciliate apparently distant viewpoints on the basis of arguments that take the reader smoothly from the mathematical structure of the various models to their physical implications. In such a way differences and analogies emerge in a natural way.
Large space is devoted to the Cahn-Hilliard model with emphasis on the origin of spinodal decomposition and of nucleation. This discussion occupies the first half of the paper. The second half deals extensively with phase field models, again commenting their derivation on rigorous a thermodynamical basis. A particularly interesting comparison with the Cahn-Hilliard model is concerned with phase separation mechanisms. The paper is a very recommendable reading, revealing the richness of this research field, and it is written in a plain way, so to be accessible to non-specialists.

MSC:

35K55 Nonlinear parabolic equations
80A22 Stefan problems, phase changes, etc.
80M35 Asymptotic analysis for problems in thermodynamics and heat transfer
74A25 Molecular, statistical, and kinetic theories in solid mechanics
74N25 Transformations involving diffusion in solids
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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