Variations on the theme of the Trotter-Kato theorem for homogenization of periodic hyperbolic systems
Yu. M. Meshkova · Russian Journal of Mathematical Physics · 2023
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. DOI 10.1134/S106192082304012X