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The Vlasov-Poisson-Boltzmann system without angular cutoff. (English) Zbl 1285.35115
Authors’ abstract: This paper is concerned with the Vlasov-Poisson-Boltzmann system for plasma particles of two species in three space dimensions. The Boltzmann collision kernel is assumed to be angular non-cutoff with \(3 < \gamma < - 2s\) and 1/\(2 \leq s < 1\), where \(\gamma , s\) are two parameters describing the kinetic and angular singularities, respectively. We establish the global existence and convergence rates of classical solutions to the Cauchy problem when initial data is near Maxwellians. This extends the results in [R. Duan et al., J. Differ. Equations 252, No. 12, 6356–6386 (2012; Zbl 1247.35174); Math. Models Methods Appl. Sci. 23, No. 6, 979–1028 (2013; Zbl 1437.76056)] for the cutoff kernel with \(-2 \leq \gamma \leq 1\) to the case \(-3 < \gamma < - 2\) as long as the angular singularity exists instead and is strong enough, i.e., \(s\) is close to 1. The proof is based on the time-weighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in [P. T. Gressman and R. M. Strain, J. Am. Math. Soc. 24, No. 3, 771–847 (2011; Zbl 1248.35140)] and the Vlasov-Poisson-Landau system in [Y. Guo, J. Am. Math. Soc. 25, No. 3, 759–812 (2012; Zbl 1251.35167)].

MSC:
35Q60 PDEs in connection with optics and electromagnetic theory
35Q83 Vlasov equations
35Q20 Boltzmann equations
82D10 Statistical mechanics of plasmas
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