A corrector theory for diffusion-homogenization limits of linear transport equations

TL;DR

This paper develops a corrector theory for the diffusion-homogenization limits of linear transport equations, analyzing convergence rates and error estimates in regimes where heterogeneity scale and mean-free path vanish at different rates.

Contribution

It introduces a novel approximation with negligible error compared to key parameters and establishes conditions for reconnection of local and global equilibria in singular perturbation regimes.

Findings

Established diffusion-homogenization limit for transport solutions.

Derived error estimates for corrector approximations.

Identified conditions for reconnecting local and global equilibria.

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 $\eps$ converge to 0. When $\alpha=\eps$, it is well known that the heterogeneous transport solution converges to a homogenized diffusion solution. We are interested here in the situation where $0<\eps\ll\alpha\ll1$ 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 $\frac\eps\alpha$. After establishing the diffusion-homogenization limit to the transport solution, we show that the corrector is dominated by an error to homogenization when $\alpha^2\ll\eps$ and by an an error to diffusion when $\eps\ll\alpha^2$. Our regime of interest involves singular perturbations in the small parameter $\eta=\frac\eps\alpha$. Disconnected local equilibria at $\eta=0$ need to be reconnected to provide a global equilibrium on the cell of periodicity when $\eta>0$. This reconnection between local and global equilibria is shown to hold when sufficient {\em no-drift} conditions are satisfied. The Hilbert expansion methodology followed in this paper builds on corrector theories for the result developed in \cite{NBAPuVo}.