**FRANÇOIS GOLSE **and ** LAURE SAINT-RAYMOND**

*Hydrodynamic Limits for the Boltzmann Equation*

**Pages** 1-144

**Received:** 14 September 2005

**Mathematics Subject Classification (2000):** 35Q30 -35Q35 - 82C40 - 76P05

**Abstract**:
This article surveys recent mathematical results on the kinetic theory
of gases. Specifically, the following topics are discussed in some detail
- the global existence theory of R. DiPerna and P.-L. Lions for the
Boltzmann equation, and
- the derivation of the classical models of fluid mechanics (i.e. the
Euler or Navier-Stokes equations) from the Botzmann equation.
Among all existing results on these topics, we have chosen to discuss
mostly those bearing on solutions that are global in time and for
arbitrary initial data satisfying only a priori estimates with
intrinsic physical meaning --- typically, bounds on the total mass,
energy or entropy.
Consequently, the mathematical methods presented here are well adapted
to handling limits of sequences of functions with little or no uniform
regularity. In particular, we study compactness arguments in L^{p}
spaces
implied by controls on derivatives of solutions coming from the partial
differential equations satisfied by these solutions.

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