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An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. 2nd ed. (English) Zbl 1245.35002

New York, NY: Springer (ISBN 978-0-387-09619-3/hbk; 978-0-387-09620-9/ebook). xiv, 1018 p. (2011).
This book continues a series of monographs devoted to the mathematical investigation of the Navier-Stokes equations [O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. New York - London - Paris: Gordon and Breach Science Publishers (1969; Zbl 0184.52603); R. Temam, Navier-Stokes equations. Theory and numerical analysis. Studies in Mathematics and its Applications. Vol. 2. Amsterdam - New York - Oxford: North-Holland Publ. Co. (1977; Zbl 0383.35057); H. Sohr, The Navier-Stokes equations. An elementary functional analytic approach. Birkhäuser Advanced Texts. Basel: Birkhäuser (1999; Zbl 0983.35004)]. Galdi’s monograph reflects the state of the theory of steady problems to the Navier-Stokes equations by the end of 2010.
The book is a new edition of the original two-volume book, under the same title, published in 1994 (cf. Zbl 0949.35004 and Zbl 0949.35005). In the book under review the author studies various boundary value problems for the Navier-Stokes equations \[ ({\mathbf v} \cdot {\mathbf {\nabla}}) \, {\mathbf v} - \nu \, {\mathbf {\nabla}}^2 \, {\mathbf v} + {\mathbf{\nabla}} p = {\mathbf f}, \qquad \qquad {\mathbf {\nabla}} \cdot {\mathbf v} = {\mathbf 0} \qquad \text{in} \quad \Omega, \]
\[ {\mathbf v} = {\mathbf g} \qquad \text{on} \quad \partial \Omega. \] and also for the linearized Stokes and Oseen equations. The corresponding flow domains \(\Omega \) can be 2- or 3-dimensional (or sometimes also of higher dimension). They are either bounded domains, exterior domains or domains with unbounded boundaries. Both the irrotational and the rotational cases are considered.
The book yields a comprehensive, detailed and self-contained study of the basic mathematical properties of different boundary-value problems related to the Navier-Stokes equations. These properties include existence, uniqueness and regularity of solutions. For unbounded domains the asymptotic behavior of solutions is also investigated. In comparison with the previous edition of 1994 two chapters have been added – namely the steady generalized Oseen flow in exterior domains and the steady Navier-Stokes flow in three-dimensional exterior domains in presence of rotation. Most of the proofs given in the former edition have been updated. The first chapter (Introduction) characterizes all important problems treated in the book and presents a number of significant and still open questions. Galdi’s monograph is written like a good textbook and is accessible also to non-specialists. All the necessary preliminaries like function spaces of hydrodynamics and several inequalities are presented in two separate chapters. Each chapter starts with a preliminary discussion of the main problem and ends with alternative procedures and historical notes.
The book contains more than 400 carefully chosen exercises at different levels of difficulty that will help the young researcher. The comprehensive bibliography contains more than 500 items. Galdi’s monograph can strongly be recommended to every mathematician (and theoretical physicist) interested in mathematical fluid mechanics or in PDEs.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
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