Spectral analysis of transport equations with bounce‐back boundary conditions

We investigate the spectral properties of the time‐dependent linear 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<∞. Application to the linear Boltzmann equation for granular gases is provided. Copyright © 2008 John Wiley & Sons, Ltd.