Spectral analysis of transport equations with bounce-back boundary conditions.

We investigate the spectral properties of the time-dependent inear transport equation with bounce-back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from Sbihi (J. Evol. Equations 2007; 7:689–711), we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all L^p-spaces with 1<p<\infty. Application to the linear Boltzmann equation for granular gases is provided.