Quantitative Homogenization of a Cahn--Hilliard System with Source Term in Periodically Perforated Domains

Abstract

We study qualitative and quantitative homogenization for a Cahn--Hilliard system with a nonconservative source term in a periodically perforated domain. Using the periodic unfolding method, we derive uniform energy estimates and prove convergence to a homogenized Cahn--Hilliard system whose effective diffusion tensor is characterized by scalar Neumann cell problems on the pore cell. For the quantitative analysis, we construct first-order corrector approximations by means of a scale-splitting operator, so that the cell correctors are only required to belong to $H_{\mathrm{per}}^1(Y_p)$. Under $H^2$-regularity of the homogenized solution and well-prepared initial data, we obtain an order ${\epsilon }^{1/2}$ corrector estimate: the corrected order-parameter error is controlled in $L^2(0,T;H^1(\hat{I}{\text{c◯}}_p^{\epsilon }))$, while the uncorrected order parameter is controlled in $L^2(0,T;L^2(\hat{I}{\text{c◯}}_p^{\epsilon }))$. This improves the rate ${\epsilon }^{1/4}$ previously established for fourth-order phase-field equations in perforated media, and matches the natural rate for second-order elliptic problems in perforated domains. The rate reflects the boundary layer caused by incomplete cells near $\partial \hat{I}\text{c◯}$ and improves to order $\epsilon$ on the flat torus ${\mathbb{T}}^d$.

34 pages