Integral representation of the linear Boltzmann operator for granular gas dynamics with applications

Details Integral representation of the linear Boltzmann operator for granular gas dynamics with applications Integral representation of the linear Boltzmann operator for granular gas dynamics with applications

2007-01-31

arxiv.org/abs/math/0701933v2

Mathematics

Analysis of PDEs

19 pages, to appear in Journal of Statistical Physics

Scientific paper

10.1007/s10955-007-9402-1

We investigate the properties of the collision operator associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of the collision operator in an Hilbert space setting, generalizing results from T. Carleman to granular gases. In the same way, we obtain from this integral representation of the gain operator that the semigroup in $L^1(\R \times \R,\d \x \otimes \d\v)$ associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from the first author.

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