Homogenization of Transport Equations
Laurent S. Dumas, François Golse · SIAM Journal on Applied Mathematics · 2000
This article is devoted to the mathematical analysis of various formulas giving the equivalent absorption and scattering cross section for mixed materials in linear transport theory. We begin with a general result on the treatment 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 case of periodic inhomogeneities is then studied in detail; in particular we show the essential difference with periodic homogenization of diffusion equations, due to small divisor problems. These results were announced in [F. Golse, C.R. Acad. Sci. Paris Sér. I Math., 305 (1987), pp. 801--804; F. Golse, Mathematical Aspects of Fluid and Plasma Dynamics, G. Toscani, V. Boffi, and S. Rionero, eds., Lecture Notes in Math 1460, Springer-Verlag, Berlin, New York, 1991, pp. 152--169]. Finally, we treat a case of stochastic inhomogeneities in linear transport theory inspired from results due to Papanicolaou--Varadhan on the homogenization of diffusion processes.