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Rosseland equation is one of the most popular models of theconduction-radiation coupled heat transfer in the thermal protection system.The well-posedness, the corresponding mathematical theory and the Multi-scaleanalysis method for the Rosseland-type equations with small periodicoscillating coefficients are concerned, which provides a theoretical basis forthe Multi-scale computation of the conduction-radiation coupled heat transferin an optically thick medium with a small periodic structure.  The global well-posedness of the Rosseland-type (parabolic) equations isgiven in the first part. The corresponding solving algorithms and theirconvergence analysis are presented in the second part. In the third part westudy the well-posedness and the second-order two-scale asymptotic expansion ofthe Rosseland-type (elliptic) equations with small periodic oscillatingcoefficients. The convergence analysis of the second-order two-scale asymptoticexpansion is studied in the last part.

Multi-scale Analysis for Rosseland Equation with Small Periodic Oscillating Coefficients