From the BGK model to the Navier–Stokes equations

Abstract

We give here a complete derivation of the Navier–Stokes–Fourier equations from a model collisional kinetic equation, the BGK model. Though physically unrealistic, this model shares some common features with more classical models such as the Boltzmann equation. Then the program developed by Bardos, Golse and Levermore [Fluid dynamic limits of kinetic equations II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46 (5) (1993) 667–753] to study hydrodynamic limits of the steady Boltzmann equation, and extended by Lions and Masmoudi [From Boltzmann equations to Navier–Stokes equations I, Archive Rat. Mech. Anal. 158 (2001) 173–193] in the time-dependent case, can be adapted here, and gives the expected convergence result provided that the particle density f satisfies some integrability assumption. The originality of the present work is to remove this assumption by establishing refined a priori estimates. The crucial idea is to decompose f as ( f− M f )+ M f where M f is the local Maxwellian associated with f. The first term is then estimated by means of the entropy dissipation, while the other is smooth in v. A mixing property of the operator ( ε∂ t + v.∇ x ) allows to transfer some of this extra-integrability on the variable x.