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Exponential decay for soft potentials near Maxwellian. (English) Zbl 1130.76069
Summary: We consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches zero at the rate of e -λt p for some λ>0 and 0<p<1. Our method is based on an unified energy estimate with appropriate exponential velocity weight. Our results extend the classical result [R. E. Caflisch, Commun. Math. Phys. 74, 97–109 (1980; Zbl 0434.76066)] to the case of very soft potential and Coulomb interactions, and also improve the recent “almost exponential” decay results by L. Desvilletes and C. Villani [Invent. Math. 159, No. 2, 245–316 (2005; Zbl 1162.82316)] and R. M. Strain and Y. Guo [Commun. Partial Differ. Equations 31, 417–429 (2006)].
76P05Rarefied gas flows, Boltzmann equation
82B40Kinetic theory of gases (equilibrium statistical mechanics)
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[9]Guo Yan. (2002) The Landau equation in a periodic box. Comm. Math. Phys. 231: 391–434 · Zbl 1042.76053 · doi:10.1007/s00220-002-0729-9
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