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

Luisa Arlotti (1) , Bertrand Lods (2)

1 Università degli Studi di Udine - University of Udine [Italie]
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

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.

Mots clés: Granular gas dynamics; linear Boltzmann equation; detailed balance law; spectral theory; $C_0$-semigroup

Domaines: Equations aux dérivées partielles [math.AP]

Luisa Arlotti, Bertrand Lods. Integral representation of the linear Boltzmann operator for granular gas dynamics with applications. Journal of Statistical Physics, 2007, 129 (3), pp.517-536. ⟨10.1007/s10955-007-9402-1⟩. ⟨hal-00128370v2⟩