In $L_2(\mathbb{ R}^d;\mathbb{ C}^n)$, we consider a matrix elliptic second order differential operator $B_\varepsilon >0$. Coefficients of the operator $B_\varepsilon$ are periodic with respect to some lattice in $\mathbb{ R}^d$ and depend on $\mathbf{ x}/\varepsilon$. We study the quantitative homogenization for the solutions of the hyperbolic system $\partial _t^2\mathbf{ u}_\varepsilon =-B_\varepsilon\mathbf{ u}_\varepsilon$. In operator terms, we are interested in approximations of the operators $\cos (tB_\varepsilon ^{ 1/2})$ and $B_\varepsilon ^{ -1/2}\sin (tB_\varepsilon ^{ 1/2})$ in suitable operator norms. Approximations for the resolvent $B_\varepsilon ^{ -1}$ have been already obtained by T.~A.~Suslina. So, we rewrite hyperbolic equation as a system for the vector with components $\mathbf{ u}_\varepsilon $ and $\partial _t\mathbf{ u}_\varepsilon$, and consider the corresponding unitary group. For this group, we adapt the proof of the Trotter-Kato theorem by introduction of some correction term and derive hyperbolic results from elliptic ones.
Categories: math.AP