7R43. Foundations of Fluid Dynamics. - G Gallavotti (Dept di Fisica, INFN, Univ degli Studi di Roma “La Sapienza,” Piazzale Aldo Moro, 2, Roma, 00185, Italy). Springer-Verlag, Berlin. 2002. 513 pp. ISBN 3-540-41415-0. $59.95.
Reviewed by JH Lienhard V (Dept of Mech Eng, MIT, Rm 3-162, Cambridge MA 02139-4307).
During the past 100 years, a few textbooks on the fundamental aspects of fluid dynamics have stood out. Among these are the well-known books by Lamb, by Landau and Lifshitz, and by Batchelor. The last two, in particular, are noteworthy for their focus on the essential physical phenomena and the accompanying mathematical formulations, which are presented with a minimum of distracting development of minor particular cases. Yet, research in fluid dynamics has moved on since the last of these titles appeared many years ago.
Gallavotti’s remarkable new book, Foundations of Fluid Dynamics, takes a fresh look at the essential formulation and phenomenology of fluid dynamics as reflected in the research of the 70s, 80s, and 90s. He devotes significant attention to the mathematical properties of the Navier-Stokes equations, to chaos in fluid flows, and to turbulence. In this sense, the book may be regarded as complementary to, say, the Landau and Lifshitz book, which deals only very briefly with these subjects. The theoretical development in the text is supplemented by 400 expository exercises that take up many important and classical results.
Gallavotti begins with a chapter that summarizes the primary features of the continuum equations of motion. A central aim here is to lay out tools for use in subsequent chapters (the Rayleigh model for thermally driven convection, vector decompositions of the flow field, and so on), but the exercises within the chapter sketch a range of topics, such as Stokes formula, tidal theory, and acoustic radiation.
Gallavotti turns directly to the question of algorithms for the construction of solutions to the equations of motion. Euler algorithms, spectral algorithms, and vorticity algorithms are each considered. Note that this development is not done in pursuit of numerical techniques—its aim is to determine how (or whether) solutions to the phenomenological (Navier-Stokes) equations can, in general, be found. It comes as no surprise, then, that the third chapter deals entirely with the existence, uniqueness, and regularity of solutions to these equations.
The spectral algorithms enable the proof of such properties locally, which is to say for bounded time intervals. Global results are obtainable in two-dimensions. The three-dimensional case is more difficult. Gallavotti discusses LeRay’s theory in this context, eventually reaching the conclusion that existence and uniqueness are “likely.”
Two chapters on chaos and incipient turbulence follow. These include a careful development of the significance of reduced order models (such as the Lorenz equations), a discussion of bifurcation theory, and coverage of various scenarios for chaos, including the Ruell-Takens scheme. Chapter 5, titled Ordering Chaos, considers how chaotic behavior might be quantitatively studied or measured.
Strong turbulence—that with many degrees of freedom—is considered in Chapter 6, working primarily from Kolmogorov’s theory. The final chapter of the book presents original new results on the statistical properties of turbulence built upon the foundation of Ruell’s principle.
Professor Gallavotti is a mathematical physicist who uses concepts from analysis freely. This should pose no barrier to most graduate students in physics and applied mathematics, but many engineering students will lack the right preparation.
The book was typeset admirably by the author in LaTex. It is attractively bound and printed, and reasonably priced as well. Few figures are included, owing to the nature of the material. The book might well have benefited from a nomenclature list, but the only real flaw in the production is an advertisement for a Springer-Verlag website that defaces the closing page of the book (which will be on shelves for years after the website has gone the way of the 8-inch floppy disk).
Foundations of Fluid Dynamics is strongly recommended, by this reviewer, for acquisition by academic libraries and for the use of anyone with a serious interest in the mathematical foundations of fluid dynamics.
