An introduction to the mathematical theory of the Navier-Stokes equations. Vol. II: Nonlinear steady problems.

*(English)*Zbl 0949.35005
Springer Tracts in Natural Philosophy. 39. New York: Springer-Verlag. xi, 323 p. (1994).

In the second volume of this monograph the steady Navier-Stokes flow is investigated both in bounded and unbounded domains. The numeration of the chapters continues those of the first volume (see the review Zbl 0949.35003 above).

The steady Navier-Stokes equations \[ \begin{aligned} &\nu\Delta v-v\cdot\nabla v-\nabla p=f,\qquad \text{div}v=0 \;\;\text{in} \;\;\Omega,\\ & \;\;\tag{1}\\ &v=\phi \qquad \text{at} \;\partial\Omega,\end{aligned} \] in a bounded domain \(\Omega\subset \mathbb{R}^n \;( n\geq 2)\) is studied in Chapter 8. Classical results of existence and uniqueness of generalized solutions to the problem (1) are proved. The regularity of the generalized solution is established in Sobolev spaces \(W^k_q\).

The problems in unbounded domains are analysed in Chapters 9-11.

The problem \[ \begin{aligned} &\nu\Delta v-v\cdot\nabla v-\nabla p=f,\qquad \text{div}v=0 \;\;= \text{in} \;\;\Omega,\\ & \;\;\tag{2}\\ &v=\phi \qquad \text{at} \;\partial\Omega, \qquad \lim_{|x|\to\infty} =v_{\infty}, \end{aligned} \] where \(v_{\infty}\) is a prescribed constant vector, \(\Omega\) is the exterior of \(\cup_{i=1}^{s}\Omega_i, \;\Omega_1,...\Omega_s\) are compact domains, is the subject of Chapter 9. Some important theorems of existence and uniqueness of the generalized solution to the problem (2) are proved in the chapter. The asymptotic structure of the solution at large distance is investigated in detail.

The steady Navier-Stokes flow in two-dimensional exterior domain \(\Omega\) is studied in the chapter 10. This problem has the form \[ \begin{aligned} &\Delta v-\mathcal R v\cdot\nabla v-\nabla p=\mathcal R f,\qquad \text{div }v=0 \;\;\text{in} \;\;\Omega,\\ & \;\;\tag{3} \\ &v=\phi \qquad \text{on} \;\partial\Omega, \qquad \lim_{|x|\to\infty} =v_{\infty}.\end{aligned} \] Here \(\Omega\) is an exterior domain in \(\mathbb{R}^2\), \(\mathcal R\) is the Reynolds number. The problems of existence, uniqueness and asymptotic behaviour of solutions to the problem (3) are discussed. Important open problems are also formulated.

Chapter 11 is devoted to the analysis of a steady Navier-Stokes flow in domains with unbounded boundaries in the 2- and 3-dimensional cases. The results of existence and uniqueness of solutions to Leray’s problem are obtained in the first part of the chapter. Existence, uniqueness and the asymptotic structure of generalised solutions for flow in an aperture domain are established in the second part.

The steady Navier-Stokes equations \[ \begin{aligned} &\nu\Delta v-v\cdot\nabla v-\nabla p=f,\qquad \text{div}v=0 \;\;\text{in} \;\;\Omega,\\ & \;\;\tag{1}\\ &v=\phi \qquad \text{at} \;\partial\Omega,\end{aligned} \] in a bounded domain \(\Omega\subset \mathbb{R}^n \;( n\geq 2)\) is studied in Chapter 8. Classical results of existence and uniqueness of generalized solutions to the problem (1) are proved. The regularity of the generalized solution is established in Sobolev spaces \(W^k_q\).

The problems in unbounded domains are analysed in Chapters 9-11.

The problem \[ \begin{aligned} &\nu\Delta v-v\cdot\nabla v-\nabla p=f,\qquad \text{div}v=0 \;\;= \text{in} \;\;\Omega,\\ & \;\;\tag{2}\\ &v=\phi \qquad \text{at} \;\partial\Omega, \qquad \lim_{|x|\to\infty} =v_{\infty}, \end{aligned} \] where \(v_{\infty}\) is a prescribed constant vector, \(\Omega\) is the exterior of \(\cup_{i=1}^{s}\Omega_i, \;\Omega_1,...\Omega_s\) are compact domains, is the subject of Chapter 9. Some important theorems of existence and uniqueness of the generalized solution to the problem (2) are proved in the chapter. The asymptotic structure of the solution at large distance is investigated in detail.

The steady Navier-Stokes flow in two-dimensional exterior domain \(\Omega\) is studied in the chapter 10. This problem has the form \[ \begin{aligned} &\Delta v-\mathcal R v\cdot\nabla v-\nabla p=\mathcal R f,\qquad \text{div }v=0 \;\;\text{in} \;\;\Omega,\\ & \;\;\tag{3} \\ &v=\phi \qquad \text{on} \;\partial\Omega, \qquad \lim_{|x|\to\infty} =v_{\infty}.\end{aligned} \] Here \(\Omega\) is an exterior domain in \(\mathbb{R}^2\), \(\mathcal R\) is the Reynolds number. The problems of existence, uniqueness and asymptotic behaviour of solutions to the problem (3) are discussed. Important open problems are also formulated.

Chapter 11 is devoted to the analysis of a steady Navier-Stokes flow in domains with unbounded boundaries in the 2- and 3-dimensional cases. The results of existence and uniqueness of solutions to Leray’s problem are obtained in the first part of the chapter. Existence, uniqueness and the asymptotic structure of generalised solutions for flow in an aperture domain are established in the second part.

Reviewer: Il’ya Sh.Mogilevskij (Tver)