Abstract
We investigate the properties of the collision operator Q 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 Q in an Hilbert space setting, generalizing results from T. Carleman (Publications Scientifiques de l’Institut Mittag-Leffler, vol. 2, 1957) to granular gases. In the same way, we obtain from this integral representation of Q + that the semigroup in L 1(ℝ3×ℝ3,dx ⊗ dv) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from Arlotti (Acta Appl. Math. 23:129–144, 1991).
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Arlotti, L., Lods, B. Integral Representation of the Linear Boltzmann Operator for Granular Gas Dynamics with Applications. J Stat Phys 129, 517–536 (2007). https://doi.org/10.1007/s10955-007-9402-1
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DOI: https://doi.org/10.1007/s10955-007-9402-1