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Mathematical facets of fluid mechanics. Papers from the mathematical days X-UPS 2010. (Facettes mathématiques de la mécanique des fluides. Journées mathématiques X-UPS 2010.) (French) Zbl 1210.76002

Palaiseau: Les Éditions de l’École Polytechnique (ISBN 978-2-7302-1578-7/pbk). v, 111 p. (2010).
The book written in French consists of five lectures presenting some tools involved in the mathematical treatment of fluid mechanics problems. The purpose of these five lectures is to detail the corresponding fundamental concepts. The book starts with a short introduction mainly presenting historical aspects of fluid mechanics and focusing on the theories developed by d’Alembert, Euler, Navier and Stokes. This introduction ends with the solution of the Navier-Stokes equations for general initial data in the sense of well-posed problems. As is well known, this problem has been solved in the 2D case by Leray in 1934, but it remains still open in the 3D case.
The first lesson by J. Y. Chemin describes the derivation of the Euler equations for incompressible fluids. The author starts with the notion of perfect fluid defining the action
\[ \mathcal{A}(\psi )=\frac{1}{2} \int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{d}}|\partial _{t}\psi (t,x)|^{2}\,dt\,dx \]
for every smooth function \(\psi \) from \([ t_{0},t_{1}] \times \mathbb{T}^{d}\) to \(\mathbb{T}^{d}\) satisfying \(\psi (0)=Id\) and such that at each time \(t\), \(\psi \) is a diffeomorphism on the torus \(\mathbb{T}^{d}\). The author writes the Euler equations and defines an appropriate notion of weak solution through some variational formulations. An existence and uniqueness result is proved for smooth solutions which are local in time, assuming that the initial data are smooth. The author then discusses the links between the maximal time of existence of the solution and the \( L^{\infty }\)-norm of the curl of this solution, focusing on the 2D case. This first lecture ends with the paradox of d’Alembert for the solution of the Euler equations in the presence of an obstacle.
The second lecture by I. Gallagher presents some insights into the solution of the Cauchy problem for Navier-Stokes equations. The author first presents some properties of these equations. Then she defines three notions of solutions: a weak one, a turbulent one and a scale one, the two last kinds of solutions being weak solutions satisfying some additional properties. The author recalls the existence result for a turbulent solution if the initial data is divergence-free. Then she presents existence results for turbulent solutions in \(\mathbb{R}^{2}\) and for scale solutions in \(\mathbb{R }^{3}\). The lecture ends with a description of the solution behaviour for large times or close to the maximal time.
In the third lecture, D. Gérard-Varet considers the solution of a fluid flow in the presence of a solid obstacle. The definitions of turbulent and strong solutions are presented. The lecture ends with a description of the solution behaviour assuming that this solution belongs to \(C^{1,\alpha }\) for \(\alpha \in (0,1)\), and specially when \(\alpha <1/2\). The author also discusses the principle of Archimedes and the paradox of Cox and Brunner.
In the fourth lecture, D. Gérard-Varet explains how it is possible to pass from Navier-Stokes equations to Euler ones, considering high values of Reynolds number. The author distinguishes the cases where the problem is posed in \(\mathbb{R}^{d}\) or in a bounded domain. The lecture ends with the solution of Prandtl equations in the 2D case, obtained when considering an asymptotic expansion of the solution. The author examines here the well- or ill-posedness of Prandtl equations according to the functional space in which the problem is solved.
The final fifth lecture by J. C. Chemin describes some properties of geophysical fluids, namely the equation \(\partial _{t}u^{\varepsilon }-\Delta u^{\varepsilon }+\mathbb{P}^{\varepsilon }(u^{\varepsilon }\cdot \nabla u^{\varepsilon })+Lu^{\varepsilon }/\varepsilon =0\) with the solution \( u^{\varepsilon }\) starting from a divergence-free initial condition \(u_{0}\). Here \(\mathbb{P}^{\varepsilon }\) is the orthogonal projection operator on divergence-free functions, and \(L\) is the Coriolis operator. The author presents the notions of weak and turbulent solutions and recalls some existence results. The last part of the lecture describes the asymptotic behaviour of the solution when \(\varepsilon \) goes to 0, first studying the asymptotic problem.
As described in the preface, the book is mainly intended for teachers who want to get acquainted with recent developments in the solution of fluid mechanics problems. The main steps of the proofs of the results contained in the book are indicated, but for further details the authors refer to short lists of references at the end of each lecture.

MSC:

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
00B15 Collections of articles of miscellaneous specific interest
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