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Discrete time Navier-Stokes limit for the BGK Boltzmann equation. (English) Zbl 1009.35071
A program enabling to study hydrodynamic limits of the Boltzmann equation was developed by C. Bardos, F. Golse and C. D. Levermore [Arch. Ration. Mech. Anal. 153, 177-204 (2000; Zbl 0973.76075)]. This paper completes this program for the BGK equation which is a simplified kinetic model still containing most of the basic properties of hydrodynamics. Moreover, its features make it likely that it has the same hydrodynamic limit as the Boltzmann equation.
The difficulty in the program arises from the limits of the nonlinear terms which require strong compactness in both time and space. The time compactness issue is removed by taking the discrete time equation. The heart of the paper is devoted to proving strong compactness in space. This is achieved thanks to two new ideas: first a control over large velocities is obtained by decomposing the solution of the BGK equation $$f$$ using the associated Maxwellian $$M_f$$ and controlling $$f- M_f$$ by means of entropy dissipation. Secondly an equiintegrability result is deduced from a new velocity averaging result.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
##### Keywords:
kinetic equation; BGK equation; hydrodynamic limits
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##### References:
 [1] DOI: 10.1103/PhysRev.94.511 · Zbl 0055.23609 [2] Bardos C., Ann. Inst. H. Poincaré Anal. Non Linéaire 2 pp 101– (1985) [3] DOI: 10.1007/BF01026608 [4] DOI: 10.1002/cpa.3160460503 · Zbl 0817.76002 [5] Bardos C., C. R. Acad. Sci. Paris 327 pp 323– (1998) · Zbl 0918.35109 [6] Cercignani C., The Mathematical Theory of Dilute Gases 106 (1994) · Zbl 0852.76081 [7] DOI: 10.1007/BF01218592 · Zbl 0688.76057 [8] DOI: 10.2307/1971423 · Zbl 0698.45010 [9] DOI: 10.1002/cpa.3160450102 · Zbl 0832.35020 [10] DOI: 10.1016/0022-1236(88)90051-1 · Zbl 0652.47031 [11] Lions P.L., C. R. Acad. Sci. Paris 329 pp 387– (1999) [12] DOI: 10.1016/0022-0396(89)90173-3 · Zbl 0694.35134 [13] DOI: 10.1016/0378-4371(81)90067-4 [14] Saint-Raymond L., Asympt. Anal. 19 pp 149– (1999)
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