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The grazing collision limit for the Boltzmann-Lorentz model. (English) Zbl 1067.82051

Summary: The Lorentz operators are derived from either Boltzmann or Fokker-Planck collisions operators when considering a mixture of species with disparate masses [P. Degond and B. Lucquin-Desreux, Math. Models Methods Appl. Sci. 6, No. 3, 405–436 (1996; Zbl 0853.76079)]. The Fokker-Planck operator is the so called ‘grazing collision limit’ of the Boltzmann operator as proved in [A.A. Arsenev and O.E. Buryak, Math. USSR, Sb. 69, No. 2, 465–478 (1991); translation from Mat. Sb. 181, No. 4, 435–446 (1990; Zbl 0724.35090), L. Desvillettes, Transp. Theory Stat. Phys. 21, No. 3, 259–276 (1992; Zbl 0769.76059), P. Degond and B. Lucquin-Desreux, Math. Models Methods Appl. Sci. 2, No. 2, 167-182 (1992; Zbl 0755.35091)]. In our simpler case, we improve the results by proving uniform in time convergence and by controling the speed of the trend to equilibrium. The results are based on a spectral analysis of the operators which share the same basis of eigenfunctions.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
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