Convergence rates in homogenization of parabolic systems with locally periodic coefficients

TL;DR

This paper establishes quantitative homogenization error estimates for second-order parabolic systems with locally periodic coefficients, advancing understanding of convergence rates under minimal smoothness assumptions.

Contribution

It introduces a new flux corrector construction for parabolic systems and provides sharp estimates for temporal boundary layers, improving homogenization analysis.

Findings

Established $O(rac{1}{ ext{scale}})$ error estimates in $L^2$ norms

Developed a new parabolic flux corrector construction

Provided sharp estimates for temporal boundary layers

Abstract

In this paper we study the quantitative homogenization of second-order parabolic systems with locally periodic (in both space and time) coefficients. The $O(\varepsilon)$ scale-invariant error estimate in $L^2(0, T; L^{\frac{2d}{d-1}}(\Omega))$ is established in $C^{1, 1}$ cylinders under minimum smoothness conditions on the coefficients. This process relies on critical estimates of smoothing operators. We also develop a new construction of flux correctors in the parabolic manner and a sharp estimate for temporal boundary layers.