Navier-Stokes equations and nonlinear functional analysis. 2nd ed.

*(English)*Zbl 0833.35110
CBMS-NSF Regional Conference Series in Applied Mathematics. 66. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xiv, 141 p. (1995).

As the title of the book indicates, the reader is assumed to be familiar with some basic concepts of (linear) functional analysis. E.g. selfadjoint operators and spaces such as \(L^2(0, T; H^1(\Omega))\) etc. are steadily used; thus some familiarity with linear functional analysis such as contained in any classical textbook (Riesz-Nagy, Akhieser-Glasmann) is advisable. No knowledge of semigroup theory is required since this notion does not appear in the book. In fact, the book is to a certain extent selfcontained.

It starts with a reminder of some basic concepts such as Sobolev spaces, and definitions of those spaces are given which are relevant for Navier- Stokes (N-S) theory. There is a preference for the space periodic case which can be handled by Fourier series computations and does not require the full apparatus of Sobolev space theory on arbitrary domains.

This introduction is followed by a discussion of the linearized N-S equations (Stokes equations) which is again treated in the space periodic context via Fourier series. A digression then follows on the trilinear form \(b(u, v, w)= \langle w, (u\nabla) v\rangle\) and some basic inequalities involving this form are derived for the periodic case. Based on these inequalities and the definitions of weak and strong solutions, fundamental (classical) existence (and for strong solutions) uniqueness results are proved for dimension \(n= 2,3\). The reader is thereby immediately lead to the well-known unsolved problems in N-S theory which are still open even in the periodic case. As to nonperiodic domains, there are some remarks but in general the reader is referred to the author’s book [Navier-Stokes equations, North Holland (1979; Zbl 0522.35002)].

In addition to the above material, further information is given on energy inequalities and singularities of weak solutions. Despite of its small size (120 pages) the book contains an astonishing amount of information. Thus, in addition to the above basic existence and uniqueness questions, a series of further topics are addressed, which will be briefly indicated in terms of key words: Couette-Taylor experiment, stationary solutions, squeezing property, Hausdorff dimension of attractors. The book concludes with a digression on numerical questions.

The book is a highly recommendable introduction to the circle of questions associated with N-S theory, suited for everyone who wants to get acquainted with the mathematical structure of the field. Further texts must be consulted if the reader wants to pursue actual research.

It starts with a reminder of some basic concepts such as Sobolev spaces, and definitions of those spaces are given which are relevant for Navier- Stokes (N-S) theory. There is a preference for the space periodic case which can be handled by Fourier series computations and does not require the full apparatus of Sobolev space theory on arbitrary domains.

This introduction is followed by a discussion of the linearized N-S equations (Stokes equations) which is again treated in the space periodic context via Fourier series. A digression then follows on the trilinear form \(b(u, v, w)= \langle w, (u\nabla) v\rangle\) and some basic inequalities involving this form are derived for the periodic case. Based on these inequalities and the definitions of weak and strong solutions, fundamental (classical) existence (and for strong solutions) uniqueness results are proved for dimension \(n= 2,3\). The reader is thereby immediately lead to the well-known unsolved problems in N-S theory which are still open even in the periodic case. As to nonperiodic domains, there are some remarks but in general the reader is referred to the author’s book [Navier-Stokes equations, North Holland (1979; Zbl 0522.35002)].

In addition to the above material, further information is given on energy inequalities and singularities of weak solutions. Despite of its small size (120 pages) the book contains an astonishing amount of information. Thus, in addition to the above basic existence and uniqueness questions, a series of further topics are addressed, which will be briefly indicated in terms of key words: Couette-Taylor experiment, stationary solutions, squeezing property, Hausdorff dimension of attractors. The book concludes with a digression on numerical questions.

The book is a highly recommendable introduction to the circle of questions associated with N-S theory, suited for everyone who wants to get acquainted with the mathematical structure of the field. Further texts must be consulted if the reader wants to pursue actual research.

Reviewer: B.Scarpellini (Basel)

##### MSC:

35Q30 | Navier-Stokes equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

46N20 | Applications of functional analysis to differential and integral equations |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |