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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 HĂ¶lder 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.

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 HĂ¶lder 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.

Reviewer: Il’ya Sh.Mogilevskij (Tver)