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摘要
This paper concerns the diffusion-homogenization of transport equations when both the adimensionalized scale of the heterogeneities alpha and the adimensionalized mean-free path epsilon converge to 0. When alpha = epsilon, it is well known that the heterogeneous transport solution converges to a homogenized diffusion solution. We are interested here in the situation where 0 %26lt; epsilon %26lt;%26lt; alpha %26lt;%26lt; 1 and in the respective rates of convergences to the homogenized limit and to the diffusive limit. Our main result is an approximation to the transport solution with an error term that is negligible compared to the maximum of alpha and epsilon/alpha via the Hilbert expansion methodology, which builds on and generalizes the corrector theory developed in [Ben Abdallah, Puel, and Vogelius, Diffusion and Homogenization Limits with separate scales]. %26lt;br%26gt;Our regime of interest involves singular perturbations in the small parameter eta = epsilon/alpha of the equation giving the global equilibrium. After establishing the diffusion-homogenization limit to the transport solution, we show that the corrector is dominated by an error to homogenization when alpha(2) %26lt;%26lt; epsilon and by an an error to diffusion when epsilon %26lt;%26lt; alpha(2). This can be done when sufficient no-drift conditions are satisfied.
- 出版日期2012
