Spectral and phase space analysis of the linearized non-cutoff Kac collision operator

Details Spectral and phase space analysis of the linearized non-cutoff Kac collision operator Spectral and phase space analysis of the linearized non-cutoff Kac collision operator

2011-11-02

arxiv.org/abs/1111.0423v2

Mathematics

Analysis of PDEs

39 pages

Scientific paper

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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