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An introduction to Navier-Stokes equation and oceanography. (English) Zbl 1194.86001
Lecture Notes of the Unione Matematica Italiana 1. Berlin: Springer (ISBN 3-540-35743-2/pbk). xxvii, 245 p. (2006).
The central theme of this book is functional analytic treatment of the Navier-Stokes equations with particular focusing on issues in connection with mathematical models used in Oceanography. The book is an adapted and slightly enlarged and revised version of lecture notes from a graduate course that the author held at Carnegie Mellon University in 1999, initially under the title “Partial Differential Equations Models in Oceanography”. The author’s mathematical approach to the subject reflects the spirit of the pioneering works of Jean Leray and Olga Ladyzhenskaya, the two famous mathematicians in this field who the author had known personally and to whom too he dedicates his book in their memory and in admiration of their achievements.
The book consists of 44 lectures, completed with preface, introduction, detailed description of the lectures, bibliographical information, abbreviations and mathematical notation, references, and index. The author describes the general techniques for nonlinear partial differential equations that the he had developed, comprising, among others, the useful topics homogenization, compensated compactness and H-measures. Five lectures are explicitly devoted to or contain material connected with Sobolev spaces and the Sobolev embedding theorem. Explicitly stated theorems and lemmas are provided with proofs.
The book reads as a scientific autobiography of the writer. It is written in a vivid (sometimes a little too argumentative, sometimes seeming a little boasting) first-person writing style, not only in the preface but also throughout all the chapters of the book. Though writing scientific literature in the first-person writing style is a controversial and nowadays often disputed question, this book is, in the reviewer’s opinion, a very good exposition of the topic it is dealing with.
Another attractive feature of this book is the inclusion of short biographical data (given in footnotes) for all persons quoted in the text. Those of them, who the author had met or had/has collaborated with, are mentioned with their first names in the text, and it is impressive seeing how many they have been. (A note: One of them has been Robert Dautray (mentioned on page XIII), but his name has not been included (presumably accidently) into the list under “Biographical Information” at the end of the book.)
The course had been intended for mathematicians in the first place, in the present book form, however, it will be a welcome reading, in its larger part, also for hydrodynamicists and other researchers in the field with less specialization in functional analysis.

86-02 Research exposition (monographs, survey articles) pertaining to geophysics
86A05 Hydrology, hydrography, oceanography
76D05 Navier-Stokes equations for incompressible viscous fluids
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations