Homogenization of transport equations

This article is devoted to the mathematical analysis of various formulas gi ving the equivalent absorption and scattering cross section for mixed mater ials in linear transport theory. We begin with a general result on the trea tment of high-frequency oscillations in linear transport equations which is partly based upon the velocity averaging results and is the analogue, for transport equations, of the compensated compactness class of results. The c ase of periodic inhomogeneities is then studied in detail; in particular we show the essential difference with periodic homogenization of diffusion eq uations, due to small divisor problems. These results were announced in [F. Golse, C. R. Acad. Sci. Paris Ser. I Math., 305 (1987), pp. 801-804; F. Go lse, Mathematical Aspects of Fluid and Plasma Dynamics, G. Toscani, V. Boff i, and S. Rionero, eds., Lecture Notes in Math 1460, Springer-Verlag, Berli n, New York, 1991, pp. 152-169]. Finally, we treat a case of stochastic inh omogeneities in linear transport theory inspired from results due to Papani colaou-Varadhan on the homogenization of diffusion processes.