Operator error estimates in the homogenization problem for nonstationary periodic equations

Abstract:

Matrix periodic differential operators (DO’s) $\mathcal A=\mathcal A (\mathbf x,\mathbf D)$ in $L_2({\mathbb R}^d;{\mathbb C}^n)$ are considered. The operators are assumed to admit a factorization of the form ${\mathcal A}={\mathcal X}^*{\mathcal X}$, where $\mathcal X$ is a homogeneous first order DO. Let ${\mathcal A}_\varepsilon ={\mathcal A}(\varepsilon ^{-1}{\mathbf x},{\mathbf D})$, $\varepsilon >0$. The behavior of the solutions ${\mathbf u}_\varepsilon ({\mathbf x},\tau )$ of the Cauchy problem for the Schrödinger equation $i\partial _\tau {\mathbf u}_\varepsilon = {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon$, and also the behavior of those for the hyperbolic equation $\partial ^2_\tau {\mathbf u}_\varepsilon = - {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon$, is studied as $\varepsilon \to 0$. Let ${\mathbf u}_0$ be the solution of the corresponding homogenized problem. Estimates of order $\varepsilon$ are obtained for the $L_2({\mathbb R}^d;{\mathbb C}^n)$-norm of the difference ${\mathbf u}_\varepsilon - {\mathbf u}_0$ for a fixed $\tau \in {\mathbb R}$. The estimates are uniform with respect to the norm of initial data in the Sobolev space $H^s({\mathbb R}^d;{\mathbb C}^n)$, where $s=3$ in the case of the Schrödinger equation and $s=2$ in the case of the hyperbolic equation. The dependence of the constants in estimates on the time $\tau$ is traced, which makes it possible to obtain qualified error estimates for small $\varepsilon$ and large $|\tau | =O(\varepsilon ^{-\alpha })$ with appropriate $\alpha <1$.