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Mean field theories and dual variation. (English) Zbl 1247.35001

Atlantis Studies in Mathematics for Engineering and Science 2. Hackensack, NJ: World Scientific; Amsterdam: Atlantis Press (ISBN 978-90-78677-14-7/hbk). x, 288 p. (2008).
This is a monograph on macroscopic phenomena starting from the microscopic description of the fields.
The book consists of two chapters. Chapter 1 contains 5 sections: Chemotaxis, Toland duality, Phenomenology, Thermodynamics, Phase fields. Chapter 2 contains 4 sections: Self-interacting continuum, Particle kinetics, Gauge fields, Higher-dimensional blowup. The bibliography contains 356 entries.
Mean field approximation is used widely in various branches of science, including kinetic theory, fluid mechanics, electrodynamics of continuous medium, wave scattering by many small particles, homogenization theories, to mention a few. Blowup phenomena are studied in this book for several model equations.
Description of the sets, on which blowup of solutions occurs, is given for some model nonlinear equations. The duality, used in the book, is the one related to the Legendre transform. The dual variation is illustrated, for example, by the system of chemotaxis equations associated with the entropy functional. Mathematical modelling of phase transition phenomena are discussed. The quantized blowup mechanism is studied by the scaling method. In the section on particle kinetics the ideas of statistical physics are discussed. One of the aspects of the turbulence problem is illustrated by a model of fluid in which many vortex points are distributed. The book deals with a wide variety of problems which arise in various branches of science and engineering.
Many proofs of the theorems formulated in this book are omitted, and the reader is referred to the papers and books cited in the bibliography.
The book is a survey of many results and ideas related to statistical physics and thermodynamics.
There are quite a few misprints (e.g., on pp. 7–10, 27, 28, 35, 40, 143, 232, 282, etc). In many cases the parentheses are omitted under the signs of integrals (e.g., on pp. 74, 75). Some formulas are not explained in the book (e.g., on p. 74, the coarea formula of H. Federer, from geometric measure theory, is used without any explanation; in this formula the symbol \(H^{n-1}(\{\varphi=s\})\) is not defined). The reader familiar with the formula can understand that the Hausdorff measure is meant, but the lack of explanations in many places makes the reader’s job more difficult than it would have been if the presentation had been more reader-friendly.
The problems discussed in this book are of interest. There is a large literature on the basic topic of this book: derivation of a macroscopic description of a large complex system in terms of a few quantities, which have physical meaning. For example, how can one derive the Navier-Stokes equations starting from kinetic theory for many small particles? An approach to this important problem, different from the one taken in the book under review, is presented in the book by A. N. Gorban and I. V. Karlin [Invariant manifolds for physical and chemical kinetics. Lecture Notes in Physics 660. Berlin: Springer (2005; Zbl 1086.82009)] (see also [A. G. Ramm, J. Math. Phys., 50, No. 4, 042701 (2009; Zbl 1214.37052)]).
The book will be of interest to specialists in nonlinear equations, formation of singularities of solutions to these equations, blowup phenomena, and mathematical problems of statistical physics.

MSC:

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35Jxx Elliptic equations and elliptic systems
35Kxx Parabolic equations and parabolic systems
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