Navier-Stokes equations. (English) Zbl 0687.35071

Constantin, Peter; Foias, Ciprian

Chicago Lectures in Mathematics. Chicago, IL etc.: University of Chicago Press (ISBN 978-0-226-11549-8; 978-0-226-76432-0/ebook). ix, 190 p. (1988).

These lecture notes are about the exciting subject of the relation between the solutions of the Navier-Stokes equations and finite-dimensional phenomena. The authors describe the results in this field, results which have been obtained recently and which for a large part are due to the authors and a.o. to Roger Temam.
Although the text is not intended as a complete course on the Navier-Stokes equations, it presents a lot of general material. We mention: existence and uniqueness of weak solutions, regularity (an adaptation of the classical \(L^ 2\)-theory), inequalities, in fact the whole technical machinery one needs to study these problems. This also includes vanishing viscosity limits, analyticity and backward uniqueness.
The heart of the matter lies in the last 3 chapters: exponential decay of volume elements, global Lyapunov exponents, Hausdorff and fractal dimension of the universal attractor, inertial manifolds.
There has been recent progress in lowering the bounds for the dimension of the universal attractor for 2D Navier-Stokes equations. This is discussed in chapter 14 together with upper bounds for bounded invariant sets in the 3D case.
This is an interesting and well-written text on a fascinating subject.

Reviewer: F.Verhulst

MSC:

35Q30	Navier-Stokes equations
35-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35B41	Attractors
35Dxx	Generalized solutions to partial differential equations
35B45	A priori estimates in context of PDEs
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

vanishing viscosity limits; analyticity; exponential decay; attractor; invariant sets