The Navier-Stokes equations. An elementary functional analytic approach. (English) Zbl 0983.35004

Sohr, Hermann

Birkhäuser Advanced Texts. Basel: Birkhäuser. x, 367 p. (2001).

The book is an elementary and selfcontained approach to the mathematical theory of viscous incompressible fluid in a completely general domain \(\Omega\subseteq \mathbb{R}^n\) \((n\geq 2)\).
The monograph has five chapters. The first two chapters have an introductory character and contain, without proofs, some fundamental properties of spaces, distributions, operators, inequalities, embeddings, etc. In the third chapter, the stationary Navier-Stokes equations are investigated in the linearized and nonlinear cases, respectively. The nonstationary linearized theory is presented in chapter four, the full nonlinear case in chapter five.
The book is written in a well arranged way, easy to survey and with utilization of the newest results. For its study, it is necessary to know only the basic functional analytic tools in Hilbert and Banach spaces. It is determined for an extensive circle of readers, from students up to experts in science and also for specialists in the field.

Reviewer: Rudolf Kodnár (Bratislava)

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
76-02	Research exposition (monographs, survey articles) pertaining to fluid mechanics
76D05	Navier-Stokes equations for incompressible viscous fluids
76D06	Statistical solutions of Navier-Stokes and related equations
76D07	Stokes and related (Oseen, etc.) flows
35Q35	PDEs in connection with fluid mechanics

Keywords:

Navier-Stokes equations; viscous incompressible fluid; functional-analytic approach