A Corrector Theory for Diffusion-Homogenization Limits of Linear Transport Equations

Abstract

This paper concerns the diffusion-homogenization of transport equations when both the adimensionalized scale of the heterogeneities $\alpha$ and the adimensionalized mean-free path $\varepsilon$ converge to $0$. When $\alpha=\varepsilon$, it is well known that the heterogeneous transport solution converges to a homogenized diffusion solution. We are interested here in the situation where $0<\varepsilon\ll\alpha\ll1$ 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 $\frac\varepsilon\alpha$ 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]. Our regime of interest involves singular perturbations in the small parameter $\eta=\frac\varepsilon\alpha$ 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 $\alpha^2\ll\varepsilon$ and by an an error to diffusion when $\varepsilon\ll\alpha^2$. This can be done when sufficient no-drift conditions are satisfied.