Homogenization and kinetic models in extended phase-space
Received: 6 December 2010
Accepted: 11 April 2011
Mathematics Subject Classification (2000): Primary 82C70, 35B27; Secondary 82C40, 60K05.
Keywords: Linear Boltzmann equation, Periodic homogenization, extended phase-space, renewal equation.
Abstract: This paper reviews recent results obtained in collaboration with E. Bernard and E. Caglioti on the homogenization problem for the linear Boltzmann equation for a monokinetic population of particles set in a periodically perforated domain, assuming that particles are absorbed by the holes. We distinguish a critical scale for the hole radius in terms of the distance between neighboring holes, derive the homogenized equation under this scaling assumption, and study the asymptotic mass loss rate in the long time limit. The homogenized equation so obtained is set on an extended phase space as it involves an extra time variable, which is the time since the last jump in the stochastic process driving the linear Boltzmann equation. The present paper proposes a new proof of exponential decay for the mass which is based on a priori estimates on the homogenized equation instead of the renewal theorem used in [Bernard-Caglioti-Golse, SIAM J. Math. Anal. 42 (2010), 2082-2113].