Quantitative homogenization of first-order ODEs

TL;DR

This paper advances the understanding of homogenization in first-order ODEs by establishing sharp convergence rates, extending to multi-scale and quasi-periodic cases, and applying findings to PDEs and gradient systems.

Contribution

It provides new quantitative convergence rates for various classes of first-order ODEs, including multi-scale, quasi-periodic, higher-dimensional, and weakly coupled systems.

Findings

Sharp $O(\varepsilon)$ convergence rate for scalar single-scale ODEs.

Improved short-time error bounds in multi-scale homogenization.

First convergence rate results for weakly coupled systems with fast switching.

Abstract

This paper investigates the quantitative homogenization of first-order ODEs. For single-scale scalar ODEs, we obtain a sharp $O(\varepsilon)$ convergence rate and characterize the effective constant. In the multi-scale setting, our results match those of \cite{IM} for long times but improve the short-time error to $O(\varepsilon)$. We also initiate the study of quasi-periodic homogenization in this context. The scalar framework is further extended to higher dimensions under a boundedness assumption on trajectories. For weakly coupled systems with fast switching rates, we obtain for the first time a convergence rate of order $O(\varepsilon)$. These results have applications to linear transport equations and broader connections to PDEs and gradient systems.