Quantitative estimates in almost periodic homogenization of parabolic systems

We consider a family of second-order parabolic operators ${\mathrm{\partial }}_t+{\mathcal{L}}_{\epsilon }$ in divergence form with rapidly oscillating, time-dependent and almost-periodic coefficients. We establish uniform interior and boundary Hölder and Lipschitz estimates as well as convergence rate. The estimates of fundamental solution and Green’s function are also established. In contrast to periodic case, the main difficulty is that the corrector equation $({\mathrm{\partial }}_s+{\mathcal{L}}_1)({\chi }_j^{\beta })=-{\mathcal{L}}_1(P_j^{\beta })$ in ${\mathbb{R}}^{d+1}$ may not be solvable in the almost periodic setting for linear functions $P(y)$ and ${\mathrm{\partial }}_t{\chi }_S$ may not in $B^2({\mathbb{R}}^{d+1})$. Our results are new even in the case of time-independent coefficients.