Parabolic Homogenization with an Interface

TL;DR

This paper develops a homogenization theory for second-order parabolic equations with oscillating coefficients and an interface, providing effective equations, convergence rates, and regularity estimates.

Contribution

It extends homogenization results to parabolic equations with interfaces, including explicit convergence rates and interior regularity estimates.

Findings

Effective piecewise constant homogenized coefficients are derived.

Achieved $O( ext{ε})$ convergence rates in specific Lebesgue spaces.

Established uniform interior Lipschitz estimates for solutions.

Abstract

This paper considers a family of second-order parabolic equations in divergence form with rapidly oscillating and time-dependent periodic coefficients and an interface between two periodic structures. Following a framework initiated by Blanc, Le Bris and Lions and a generalized two-scale expansion in divergence form of elliptic homogenization with an interface by Josien, we can determine the effective (or homogenized) equation with the coefficient matrix being piecewise constant and discontinuous across the interface. Moreover, we obtain the $O(\varepsilon)$ convergence rates in $L^{2(d+2)/{d}}_{x,t}$ with $\varepsilon$-smoothing method and the uniform interior Lipschitz estimates via compactness argument.