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Some recent results about the sixth problem of Hilbert. (English) Zbl 1291.35126

Calgaro, Catarina (ed.) et al., Analysis and simulation of fluid dynamics. Collected papers based on the presentations at the conference, Lille, France, June 2005. Basel: Birkhäuser (ISBN 3-7643-7741-0/hbk). Advances in Mathematical Fluid Mechanics, 183-199 (2007).
The paper, written by one of the leading experts in the field, reviews the most recent results concerning the so-called “sixth problem” proposed by David Hilbert on the occasion of the International Congress of Mathematicians, held in Paris in 1900.
This problem, which has kept the attention of many researchers since the time of its formulation, asks for a global understanding of the relationships between the different approaches in gas dynamics: the microscopic description (which treats the gas as a system of \(N\) interacting particles), the kinetic picture, introduced by Boltzmann (which considers the gas as a statistical collection described by the distribution of positions and velocities), and the fluid description (which studies the system by means of macroscopic quantities, such as pressure, temperature and velocity).
The paper describes the mathematical derivation of the incompressible limit (both in the inviscid and in the viscous case) in the framework of the renormalized solutions of the Boltzmann equation.
For each theorem, a concise sketch of the proof is provided.
The paper is written in a very clear style and should prove very useful also to non-specialists who want a better understanding of the state-of-the-art and the mathematical techniques currently in use in the field of hydrodynamic limits.
For the entire collection see [Zbl 1106.76005].

MSC:

35L60 First-order nonlinear hyperbolic equations
76N15 Gas dynamics (general theory)
35F20 Nonlinear first-order PDEs
76A02 Foundations of fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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