Parabolic homogenization with an interface
Yiping Zhang · Discrete and Continuous Dynamical Systems · 2023
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 (i.e., composed of two different periodic media separated by the interface). 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} $ (for $ d\geq 2 $) with $ \varepsilon $-smoothing method and the uniform interior Lipschitz estimates via compactness argument.