A periodic parabolic Cauchy problem: Homogenization with corrector
Elizaveta Vasilevskaya · St Petersburg Mathematical Journal · 2009
A wide class of matrix elliptic second-order differential operators $\mathcal {A}=\mathcal {A}(\mathbf {x},\mathbf {D})$ with periodic coefficients, acting in $L_2(\mathbb {R}^d;\mathbb {C}^n)$, is studied. The operator $\mathcal {A}$ is assumed to admit a factorization of the form $\mathcal {A}=\mathcal {X}^*\mathcal {X}$, where $\mathcal {X}$ is a homogeneous first-order differential operator. An approximation for the operator exponential $e^{-\mathcal {A}\tau }$ as $\tau \rightarrow \infty$ in the $(L_2(\mathbb {R}^d;\mathbb {C}^n))$-operator norm is obtained, with error estimate of the order of $\tau ^{-1}$. In the approximation, a corrector is taken into account. The result is applied to the study of homogenization for solutions of the Cauchy problem $\partial _\tau \mathbf {u}_\varepsilon = -\mathcal {A}_\varepsilon \mathbf {u}_\varepsilon$, where $\mathcal {A}_\varepsilon =\mathcal {A}(\mathbf {x}/\varepsilon ,\mathbf {D})$. An approximation with corrector for $\mathbf {u}_\varepsilon$ in the $(L_2(\mathbb {R}^d;\mathbb {C}^n))$-norm is obtained for fixed $\tau >0$, with error estimate of the order of $\varepsilon ^2$.