Recent developments in the Navier-Stokes problem. (English) Zbl 1034.35093

Lemarié-Rieusset, P. G.

Chapman & Hall/CRC Research Notes in Mathematics 431. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-220-4/pbk). xii, 395 p. (2002).

From the author’s introduction: This book is a self-contained exposition of recent results on the Navier-Stokes equations, presented from the point of view of real harmonic analysis. A quarter of the book is an introduction to real harmonic analysis, where all the material needed in the book is introduced and proved.
The Navier-Stokes equations are examined in a very restricted setting: \[ \rho\delta_t\vec u=\mu \Delta\vec u-\rho(\vec u. \vec\Delta)\vec u-\vec\nabla p, \] \[ \vec\nabla\cdot\vec u=0, \] that is a viscous, homogeneous, incompressible fluid that fills the entire space and is not submitted to external forces. The contents of the book in more detail:
Part I is devoted to the recalling of some (presumably) well-known results of harmonic analysis on some special spaces of functions or distributions, and on some convolution operators (fractional integration, Calderón-Zygmund operators, Riesz transforms, etc.). In Parts 2 to 6, those tools are applied to the study of the Cauchy problem for the Navier-Stokes equations: Part 2 presents some general shift-invariant estimates for the Navier-Stokes equations; Part 3 reviews the classical existence results of Leray (weak solutions \(\vec u\) such that \(\vec u\in L^\infty((0,\infty)\), \((L^2)^d)\), \(\vec\nabla \otimes\vec u\in L^2((0,\infty), (L^2)^{d^2})\) and Kato and Fujita (mild solutions in \({\mathcal C}([0,T]\), \((H^s)^d)\), \(s\geq d/2-1\), or in \({\mathcal C}([0,T], (L^p)^d)\), \(p\geq d\); Part 4 and 5 describe some recent results on mild solutions (generalizations of Kato’s results), including the theorem of Koch and Tataru on the existence of solutions for data in \(BMO^{(-1)}\) and Cannone’s theory of self-similar solutions: Part 6 considers suitable solutions when \(d=3\), the main tool is the local energy inequality of Scheffer and the regularity criterion of Caffarelli, Kohn and Nirenberg, with applications to the study of weak solutions with infinite energy.

Reviewer: Josef Wloka (Kiel)

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
42B35	Function spaces arising in harmonic analysis

Keywords:

real harmonic analysis; viscous, homogeneous, incompressible fluid; fractional integration; Calderón-Zygmund operators; Riesz transforms; existence; weak solutions; mild solutions; self-similar solutions; energy inequality; regularity criterion