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The aim of this book is to present some mathematical results describing the transition from kinetic theory, and, more precisely, from the Boltzmann equation for perfect gases to hydrodynamics.

Different fluid asymptotics will be investigated, starting always from solutions of the Boltzmann equation which are only assumed to satisfy the estimates coming from physics, namely some bounds on mass, energy and entropy.

This paper is concerned with a complete asymptotic analysis as $E o 0$ of the Munk equation $partial _xpsi -E Delta ^2 psi = au $ in a domain $Omega subset mathbf R^2$, supplemented with boundary conditions for $psi $ and $partial _n psi $. This equation is a simple model for the circulation of currents in closed basins, the variables $x$ and $y$ being respectively the longitude and the latitude. A crude analysis shows that as $E o 0$, the weak limit of $psi $ satisfies the so-called Sverdrup transport equation inside the domain, namely $partial _x psi ^0= au $, while boundary layers...