Navier-Stokes equations and turbulence.

*(English)*Zbl 0994.35002
Encyclopedia of Mathematics and Its Applications. 83. Cambridge: Cambridge University Press. xiv, 347 p. (2001).

The book presents and makes connections between the mathematical and physical theory of turbulence in a way which is accessible to mathematicians, phycisists and engineers. It is divided into five chapters: the first chapter mainly gives a derivation of the Navier-Stokes equations and provides an account of the Kolmogorov theory of turbulence. Chapter 2 recalls some elements of the classical mathematical theory of the Navier-Stokes equations. This includes the introduction of appropriate function spaces, results on existence and uniqueness of weak/strong solutions as well as the analyticity of solutions. The main theme of Chapter 3 is the finite dimensionality of flows. The authors discuss determining modes and determining nodes for various boundary conditions. They also study the global attractor and present estimates for both Hausdorff and fractal dimensions. The chapter concludes with a discussion on inertial manifolds. Chapter 4 introduces stationary statistical solutions and relates these to the limits of averages. Furthermore, the corresponding invariant measure is considered and related to the attractor that carries it. Finally, Chapter 5 extends the study of statistical solutions to the time-dependent case and establishes connections to the conventional theory of turbulence.

Reviewer: Klaus Deckelnick (Brighton)

##### MSC:

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

35Q30 | Navier-Stokes equations |

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

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

37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |

76D06 | Statistical solutions of Navier-Stokes and related equations |

76F60 | \(k\)-\(\varepsilon\) modeling in turbulence |