标题： 一阶常微分方程的定量均质化 显示英文标题

标题： Quantitative homogenization of first-order ODEs

摘要： 本文研究了一阶常微分方程的定量均质化问题。 对于单尺度标量常微分方程，我们得到了一个精确的$O(\varepsilon)$收敛率并刻画了有效常数。 在多尺度情况下，我们的结果在长时间范围内与\cite{IM}的结果一致，但将短时间误差改进为$O(\varepsilon)$。 我们还在此背景下开启了准周期均质化的研究。 在轨迹有界性假设下，标量框架进一步扩展到高维情况。 对于快速切换速率的弱耦合系统，我们首次获得了阶为$O(\varepsilon)$的收敛率。 这些结果在线性输运方程中有应用，并与偏微分方程和梯度系统有更广泛的联系。 显示英文摘要

摘要： 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.