Applied analysis of the Navier-Stokes equations.

*(English)*Zbl 0838.76016
Cambridge Texts in Applied Mathematics. Cambridge: Cambridge Univ. Press. xiii, 217 p. £ 14.95; $ 24.95/pbk, £ 40.00; $ 59.95/hbk (1995).

This book, essentially mathematical in nature, tries to highlight for applied mathematicians and mathematical physicists some topics related to existence, uniqueness and regularity of the solution to the Navier-Stokes equations. In particular, it remains an open problem whether the three-dimensional solutions, even starting from very smooth and physically reasonable initial conditions, exhibit finite-time singularities. If so, then the Navier-Stokes equations can really describe the generation of turbulence, namely vortex stretching. Thus the above questions that often seem to applied scientists to be only mathematical formalities, are intimately tied up with the possibility to use the Navier-Stokes equations as an adequate model for fluid turbulence. Unfortunately, the investigations into the corresponding mathematical problems are often phrased in language of abstract functional analysis and thus remain closed for the community of applied specialists. One of the goals of this book is to present some of the techniques and results of these studies in a more familiar physical context, with possible explanations of all mathematical tools used.

The book contains ten chapters. The first four chapters (1. The equations of motion; 2. Dimensionless parameters and stability; 3. Turbulence; 4. Degrees of freedom, dynamical systems, and attractors) present the derivation of the basic equations (Navier-Stokes, Boussinesq) as well as some fundamental ideas about stability, turbulence, and dynamical systems in general. The remaining chapters (5. On the existence, uniqueness, and regularity of solutions; 6. Ladder results for the Navier-Stokes equations; 7. Regularity and length scales for the 2D and 3D Navier-Stokes equation; 8. Exponential decay of the Fourier power spectrum; 9. The attractor dimension for the Navier-Stokes equations; 10. Energy dissipation and rate estimates for boundary-driven flows) treat the related mathematical topics starting from the classical results on existence, uniqueness and regularity and leading to state-of-the-art developments in the problem.

The book is excellently written and is well suited for researchers and graduated students in applied mathematics and mathematical physics, especially as each chapter contains recommendations for further reading and exercises which should help the interested reader to utilize the material presented in the text.

The book contains ten chapters. The first four chapters (1. The equations of motion; 2. Dimensionless parameters and stability; 3. Turbulence; 4. Degrees of freedom, dynamical systems, and attractors) present the derivation of the basic equations (Navier-Stokes, Boussinesq) as well as some fundamental ideas about stability, turbulence, and dynamical systems in general. The remaining chapters (5. On the existence, uniqueness, and regularity of solutions; 6. Ladder results for the Navier-Stokes equations; 7. Regularity and length scales for the 2D and 3D Navier-Stokes equation; 8. Exponential decay of the Fourier power spectrum; 9. The attractor dimension for the Navier-Stokes equations; 10. Energy dissipation and rate estimates for boundary-driven flows) treat the related mathematical topics starting from the classical results on existence, uniqueness and regularity and leading to state-of-the-art developments in the problem.

The book is excellently written and is well suited for researchers and graduated students in applied mathematics and mathematical physics, especially as each chapter contains recommendations for further reading and exercises which should help the interested reader to utilize the material presented in the text.

Reviewer: O.Titow (Berlin)

##### MSC:

76D05 | Navier-Stokes equations for incompressible viscous fluids |

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

35Q30 | Navier-Stokes equations |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |