Spectral and phase space analysis of the linearized non-cutoff Kac collision operator |
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| Authors: | N Lerner Y Morimoto K Pravda-Starov C-J Xu |
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| Institution: | 1. Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris VI), 4 Place Jussieu, 75252 Paris cedex 05, France;2. Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan;3. Université de Cergy-Pontoise, CNRS UMR 8088, Département de Mathématiques, 95000 Cergy-Pontoise, France;4. School of Mathematics, Wuhan University, 430072 Wuhan, PR China;5. Université de Rouen, CNRS UMR 6085, Département de Mathématiques, 76801 Saint-Etienne du Rouvray, France |
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| Abstract: | The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized non-cutoff radially symmetric Boltzmann operator are also proven. |
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| Keywords: | 35Q20 35S05 76P05 82B40 35R11 |
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