Acoustic and Stokes limits for the Boltzmann equation

doi:10.1016/S0764-4442(98)80154-7

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics

Volume 327, Issue 3, August 1998, Pages 323-328

Comptes Rendus de l'...

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Abstract

The Boltzmann equation is considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L¹) to a unique limit governed by a solution of the acoustic or Stokes equations, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L² initial data of the acoustic or Stokes equations. The associated conservation laws are recovered in the limit.

Résumé

On considère l'équation de Boltzmann dans un domaine périodique et pour des noyaux de collision bornés. Après des changements d'échelle convenables, les fluctuations autour de l'équilibre de familles de solutions renormalisées (de DiPerna-Lions) convergent au sens entropique (et donc fortement dans L¹) vers une unique limite gouvernée par le système de l'acoustique ou par l'équation de Stokes, sous l'hypothèse que la famille des fluctuations initiales converge au sens entropique vers une limite canoniquement associée à toute donnée initiale L² pour le système de l'acoustique ou l'équation de Stokes. Les lois de conservation correspondantes sont obtenues après passage à la limite.

References (6)

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Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation
Commun. Pure Appl. Math.
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Bardos C., Golse F., Levermore C.D., (in...

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We consider the hydrodynamic limit of a stationary Boltzmann equation in a unit plate with in-flow boundary. The classical theory claims that the solution can be approximated by the sum of interior solution which satisfies steady incompressible Navier–Stokes–Fourier system, and boundary layer derived from Milne problem. In this paper, we construct counterexamples to disprove such formulation in $L^{\infty}$ both for its proof and result. Also, we show the hydrodynamic limit with a different boundary layer expansion with geometric correction.
Chapter 3 Examples of singular limits in hydrodynamics
2007, Handbook of Differential Equations: Evolutionary Equations
Chapter 1 From particles to fluids
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In particular, the conservation laws (4.12) are only formal in this framework. This program was started in [6,7] and progressive improvements have been obtained in a long list of papers by several authors [8–10,49,50,66–68] till the very recent work [51] which provides a complete proof under some restrictions on the scattering cross section. One can try to apply the above ideas to Hamiltonian particle systems.
From the BGK model to the Navier-Stokes equations
2003, Annales Scientifiques de l'Ecole Normale Superieure
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.
On donne ici une dérivation complète des équations de Navier–Stokes–Fourier à partir d'une équation cinétique collisionnelle modèle, l'équation de BGK. Bien que non conforme à la physique, ce modèle présente des propriétés très semblables à celles de l'équation de Boltzmann.
Le programme développé par Bardos, Golse et Levermore [Fluid dynamic limits of kinetic equations II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46 (5) (1993) 667–753] pour étudier les limites hydrodynamiques de l'équation de Boltzmann stationnaire, complété par Lions et Masmoudi [From Boltzmann equations to Navier–Stokes equations I, Archive Rat. Mech. Anal. 158 (2001) 173–193] dans le cas dépendant du temps, peut donc être adapté ici : il donne le résultat de convergence attendu pourvu que la densité de particules f satisfasse une hypothèse d'intégrabilité.
Plusieurs nouvelles idées permettent ici d'établir des estimations a priori précisées et de supprimer cette hypothèse. La plus importante consiste à décomposer f=(f−M_f)+M_f où M_f est la Maxwellienne locale associée à f. Le premier terme est alors estimé au moyen de la dissipation d'entropie, et l'autre est régulier en v. Une propriété de mélange de l'opérateur (ε∂_t+v.∇_x) permet de transférer une partie de cette régularité sur la variable x.
Chapter 2A General presentation
2002, Handbook of Mathematical Fluid Dynamics
Citation Excerpt :
As an important application of the theory of renormalized solutions, Levermore [302] proved the validity of the linearization approximation if the initial datum is very close to a global Maxwellian. Also the hydrodynamical transition towards some models of fluid mechanics can be justified without assumption of smoothness of the limit hydrodynamic equations: see, in particular, Bardos, Golse and Levermore [57,55,54,53], Golse [239], Golse et al. [241], Golse and Levermore [240], Lions and Masmoudi [317], Golse and Saint-Raymond [245], Saint- Raymond [401]. The high point of this program is certainly the rigorous limit from the DiPerna–Lions renormalized solutions to Leray’s weak solutions of the incompressible Navier–Stokes equation, which was performed very recently in [245]; see [441] for a review.
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2002, Handbook of Mathematical Fluid Dynamics

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Acoustic and Stokes limits for the Boltzmann equationLes limites acoustique et de Stokes de l'équation de Boltzmann

Abstract

Résumé

Fluid dynamic limits of kinetic equations. I. Formal derivations

J. Statist. Phys.

Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation

Commun. Pure Appl. Math.