On homogenization of the first initial-boundary value problem for periodic hyperbolic systems

TL;DR

This paper studies the homogenization of hyperbolic systems with periodic coefficients, providing operator approximations and correctors for the solutions of initial-boundary value problems in small period limits.

Contribution

It develops operator norm approximations and correctors for hyperbolic systems with periodic coefficients, advancing homogenization theory for such PDEs.

Findings

Operator approximations in Sobolev spaces for hyperbolic operators

Corrector results in H^1-norm for sine operators

Application to homogenization of hyperbolic initial-boundary value problems

Abstract

Let $\mathcal{O}\subset\mathbb{R}^d$ a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a self-adjoint matrix strongly elliptic second order differential operator $B_{D,\varepsilon}$, $0<\varepsilon \leqslant 1$, with the Dirichlet boundary condition. The coefficients of the operator $B_{D,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. We are interested in the behavior of the operators $\cos(tB_{D,\varepsilon}^{1/2})$ and $B_{D,\varepsilon} ^{-1/2}\sin (t B_{D,\varepsilon} ^{1/2})$, $t\in\mathbb{R}$, in the small period limit. For these operators, approximations in the norm of operators acting from some subspace $\mathcal{H}$ of the Sobolev space $H^4(\mathcal{O};\mathbb{C}^n)$ to $L_2(\mathcal{O};\mathbb{C}^n)$ are found. Moreover, for $B_{D,\varepsilon} ^{-1/2}\sin (t B_{D,\varepsilon} ^{1/2})$, the approximation with the corrector in the norm of operators acting from $\mathcal{H}\subset H^4(\mathcal{O};\mathbb{C}^n)$ to $H^1(\mathcal{O};\mathbb{C}^n)$ is obtained. The results are applied to homogenization for the solution of the first initial-boundary value problem for the hyperbolic equation $\partial ^2_t \mathbf{u}_\varepsilon =-B_{D,\varepsilon} \mathbf{u}_\varepsilon $.