# Homogenization of Parabolic Equations with Non-self-similar Scales

## TL;DR

This paper develops quantitative homogenization techniques for second-order parabolic systems with rapidly oscillating periodic coefficients across different scales, providing new estimates and convergence rate results.

## Contribution

It introduces higher-order correctors to achieve large-scale Lipschitz and Hölder estimates, advancing the understanding of homogenization in multi-scale parabolic equations.

## Key findings

- Established large-scale interior and boundary Lipschitz estimates
- Derived interior $C^{1, eta}$ and $C^{2, eta}$ estimates using higher-order correctors
- Analyzed convergence rates for initial-boundary value problems

## Abstract

This paper is concerned with quantitative homogenization of second-order parabolic systems with periodic coefficients varying rapidly in space and time, in different scales. We obtain large-scale interior and boundary Lipschitz estimates as well as interior $C^{1, \alpha}$ and $C^{2, \alpha}$ estimates by utilizing higher-order correctors. We also investigate the problem of convergence rates for initial-boundary value problems.