The point scatterer approximation for wave dynamics
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909307148173312 |
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| author | Mantile, Andrea Posilicano, Andrea |
| author_facet | Mantile, Andrea Posilicano, Andrea |
| contents | Given an open, bounded and connected set $Ω\subset\mathbb{R}^{3}$ and its rescaling $Ω_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$ (\varepsilon^{-2}χ_{Ω_{\varepsilon}}+χ_{\mathbb{R}^{3}\backslashΩ_{\varepsilon}})\partial_{tt}u=Δu+f $$ with initial data and source supported outside $Ω_{\varepsilon}$; here, $χ_{S}$ denotes the characteristic function of a set $S$. We provide the first-order $\varepsilon$-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the $L^{\infty}((0,1/\varepsilon^τ),L^{2}(\mathbb{R}^{3})) $-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in $L^{2}(Ω)$ and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_17195 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The point scatterer approximation for wave dynamics Mantile, Andrea Posilicano, Andrea Mathematical Physics Analysis of PDEs Given an open, bounded and connected set $Ω\subset\mathbb{R}^{3}$ and its rescaling $Ω_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$ (\varepsilon^{-2}χ_{Ω_{\varepsilon}}+χ_{\mathbb{R}^{3}\backslashΩ_{\varepsilon}})\partial_{tt}u=Δu+f $$ with initial data and source supported outside $Ω_{\varepsilon}$; here, $χ_{S}$ denotes the characteristic function of a set $S$. We provide the first-order $\varepsilon$-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the $L^{\infty}((0,1/\varepsilon^τ),L^{2}(\mathbb{R}^{3})) $-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in $L^{2}(Ω)$ and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain. |
| title | The point scatterer approximation for wave dynamics |
| topic | Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2401.17195 |