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An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I: Linearized steady problems. (English) Zbl 0949.35004
Springer Tracts in Natural Philosophy. 38. New York, NY: Springer-Verlag. xi, 450 p. (1994).
This book continues a series of monographs devoted to the mathematical investigation of Navier-Stokes equations. These monographs are O. A. Ladyzhenskaya [The mathematical theory of viscous incompressible flow. (Gordon and Breach (1969; Zbl 0184.52603)] and R. Temam [Navier-Stokes equations. (North-Holland (1977; Zbl 0383.35057)]. Galdi’s monograph reflects the state of the theory of steady problems at the beginning of the 1990’s.
It consists of two volumes. The linearized steady problems are presented in Volume I under review. The book consists of 7 chapters. In Chapter 1 the author formulates the problems analyzed in the book and some open problems. The basic information of function spaces (Sobolev spaces \(W^k_q\) in particular) is adduced in Chapter 2. Specific for hydrodynamics the function spaces \(H_q\) and Helmholtz decomposition are studied in Chapter 3.
The boundary value problem to the Stokes equations \[ \begin{aligned} &\Delta v-\nabla p=f,\qquad \text{div}v=0 \;\;\text{in} \;\;\Omega,\\ & \;\;\tag{1}\\ &v=\phi \qquad \text{on} \;\partial\Omega,\end{aligned} \] in a bounded domain \(\Omega\) is the subject of Chapter 4. It contains the detailed proofs of existence, uniqueness and estimates of the solution to the problem (1) in Sobolev spaces \(W^k_q\) and in the Hölder spaces.
The steady Stokes flow in exterior domains is investigated in Chapter 5. Let \(\Omega\) be the complement of a compact region. The subject of the chapter is the problem \[ \begin{aligned} &\Delta v-\nabla p=f,\qquad \text{div}v=0 \;\;\text{in} \;\;\Omega,\\ & \;\;\tag{2}\\ &v=\phi \qquad \text{on} \;\partial\Omega, \qquad \lim_{|x|\to\infty} =v_{\infty}, \end{aligned} \] where \(v_{\infty}\) is a prescribed constant vector. Existence and uniqueness of generalised and strong solutions are proved. The behaviour of solutions to the problem (2) at infinity is also discussed.
Chapter 6 is devoted to the analysis of a steady Stokes flow in domains with unbounded boundaries. The flow in unbounded channels with unbounded cross sections is investigated.
The subject of the last Chapter 7 is the problem of the Oseen system \[ \begin{aligned} &\Delta v-v_0\cdot\nabla v-\nabla p=f,\qquad \text{div}v=0 \;\;\text{in} \;\;\Omega,\\ & \;\;\tag{3} \\ &v=\phi \qquad \text{on} \;\partial\Omega, \qquad \lim_{|x|\to\infty} =v_{\infty}, \end{aligned} \] in an exterior domain \(\Omega\). Here \(v_0\) is a given constant vector. The existence, uniqueness and the validity of the corresponding estimates in Sobolev spaces \(D^k_q\) for the solutions to (3) are investigated.
Moreover, there are many interesting and useful exercises in the book.

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q30 Navier-Stokes equations