[Paper Review] Homogenization of hyperbolic equations with periodic coefficients in ${\mathbb R}^d$: sharpness of the results

Mark Dorodnyi, T. A. Suslina|arXiv (Cornell University)|Jul 26, 2020

Advanced Mathematical Modeling in Engineering18 references4 citations

TL;DR

This paper establishes sharp operator norm estimates for the homogenization of hyperbolic equations with periodic coefficients in $\mathbb{R}^d$, deriving approximations for $\cos(\mathcal{A}_\varepsilon^{1/2}\tau)$ and $\mathcal{A}_\varepsilon^{-1/2}\sin(\mathcal{A}_\varepsilon^{1/2}\tau)$ in $H^s \to L^2$ and $H^s \to H^1$ norms, with explicit error bounds in $\varepsilon$ and $\tau$. It proves the sharpness of these estimates in terms of operator type and $\tau$-dependence, applying them to the Cauchy problem for $\partial_\tau^2 \mathbf{u}_\varepsilon = -\mathcal{A}_\varepsilon \mathbf{u}_\varepsilon + \mathbf{F}$, including applications to acoustics and elasticity.

ABSTRACT

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, a selfadjoint strongly elliptic second order differential operator ${\mathcal A}_\varepsilon$ is considered. It is assumed that the coefficients of the operator ${\mathcal A}_\varepsilon$ are periodic and depend on ${\mathbf x}/\varepsilon$, where $\varepsilon >0$ is a small parameter. We find approximations for the operators $\cos ( {\mathcal A}_\varepsilon^{1/2}τ)$ and ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}τ)$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d)$ to $L_2({\mathbb R}^d)$ (with suitable $s$). We also find approximation with corrector for the operator ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}τ)$ in the $(H^s o H^1)$-norm. The question about the sharpness of the results with respect to the type of the operator norm and with respect to the dependence of estimates on $τ$ is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation $\partial_τ^2 {\mathbf u}_\varepsilon = - {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$.

Motivation & Objective

• To derive sharp operator norm estimates for the approximation of $\cos(\mathcal{A}_\varepsilon^{1/2}\tau)$ and $\mathcal{A}_\varepsilon^{-1/2}\sin(\mathcal{A}_\varepsilon^{1/2}\tau)$ in $H^s \to L^2$ and $H^s \to H^1$ operator norms.
• To analyze the sharpness of these estimates with respect to the type of operator norm and the dependence on the time parameter $\tau$.
• To apply the results to the Cauchy problem for second-order hyperbolic equations with periodic coefficients, establishing convergence rates for solutions.
• To extend the analysis to specific physical systems such as the acoustics equation and the elasticity system via factorized operator forms.

Proposed method

• Utilizes the spectral approach based on Floquet–Bloch theory and analytic perturbation theory for periodic differential operators.
• Employs a factorization framework: $\mathcal{A}_\varepsilon = f^\varepsilon^* b(\mathbf{D})^* g^\varepsilon b(\mathbf{D}) f^\varepsilon$, with $g^\varepsilon$ periodic and $f^\varepsilon$ bounded and invertible.
• Derives approximations for $\cos(\varepsilon^{-1}\tau A(t)^{1/2})P$ and $A(t)^{-1/2}\sin(\varepsilon^{-1}\tau A(t)^{1/2})P$ using operator-theoretic techniques.
• Introduces corrector terms to improve convergence rates in $H^1$-norm approximations, particularly for $\mathcal{A}_\varepsilon^{-1/2}\sin(\mathcal{A}_\varepsilon^{1/2}\tau)$.
• Applies the method to the effective operator $\mathcal{A}^0$ via the homogenized operator $\mathcal{A}^0 = b(\mathbf{D})^* g^0 b(\mathbf{D})$, with $g^0$ the harmonic mean of $g(\mathbf{x})$.
• Uses the method of stationary phase and spectral projections to derive sharp bounds in terms of $\varepsilon$ and $\tau$.

Experimental results

Research questions

• RQ1 What are the optimal convergence rates for approximating $\cos(\mathcal{A}_\varepsilon^{1/2}\tau)$ and $\mathcal{A}_\varepsilon^{-1/2}\sin(\mathcal{A}_\varepsilon^{1/2}\tau)$ in $H^s \to L^2$ and $H^s \to H^1$ operator norms?
• RQ2 How does the dependence on the time parameter $\tau$ affect the sharpness of the error estimates in the operator norm?
• RQ3 Are the derived error bounds sharp with respect to the type of operator norm (e.g., $L^2 \to L^2$ vs. $H^s \to H^1$)?
• RQ4 Can the results be extended to physical systems such as the acoustics and elasticity equations via the factorized operator structure?
• RQ5 What is the role of the corrector term in improving the convergence rate in the $H^1$-norm?

Key findings

• For $\mathcal{A}_\varepsilon^{-1/2}\sin(\mathcal{A}_\varepsilon^{1/2}\tau)$, the approximation in the $(H^s \to H^1)$-norm has an error bound of order $\varepsilon^{s/2}(1+|\tau|)^{s/2}$ for $0 \leq s \leq 2$, with $s=2$ yielding $O(\varepsilon)$ rate.
• The approximation of $\cos(\mathcal{A}_\varepsilon^{1/2}\tau)$ in the $(H^s \to L^2)$-norm has error $O(\varepsilon^{s/2}(1+|\tau|)^{s/2})$ for $0 \leq s \leq 2$.
• The sharpness of the estimates is confirmed: the $\varepsilon$-dependence and $\tau$-dependence in the bounds cannot be improved in general.
• For the Cauchy problem $\partial_\tau^2 \mathbf{u}_\varepsilon = -\mathcal{A}_\varepsilon \mathbf{u}_\varepsilon + \mathbf{F}$, the solution $\mathbf{u}_\varepsilon$ converges to the homogenized solution $\mathbf{u}_0$ in $L^2$ as $\varepsilon \to 0$ for all $\tau \in \mathbb{R}$.
• With a corrector term $\mathbf{v}_\varepsilon = \mathbf{u}_0 + \varepsilon \Lambda^\varepsilon b(\mathbf{D}) \Pi_\varepsilon \mathbf{u}_0$, the $H^1$-error is bounded by $C(1+|\tau|)\varepsilon \|\boldsymbol{\psi}_1\|_{H^2}$, showing $O(\varepsilon)$ convergence in $H^1$ under $H^2$ regularity.
• The results are sharp in the sense that no better $\varepsilon$-dependence is possible in the error bounds, even for smooth data, due to the oscillatory nature of the periodic coefficients.