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What does it mean for a 3D star-shaped scatterer to be small in the time domain?

Maryna Kachanovska, Adrian Savchuk

TL;DR

The paper tackles time-domain scattering by a small three-dimensional sound-soft star-shaped obstacle and shows that replacing the obstacle with a point scatterer can yield time-uniform errors, without introducing extra temporal scales. It develops a resolvent-based framework, including a Krein-type formula and Laplace-domain estimates, to obtain local energy decay bounds and to bound the error of asymptotic models in time, with explicit dependence on the small parameter ε. The main results establish a local energy decay bound for the scattered field in the far-field region that decays exponentially after the arrival time, and prove time-domain error bounds for the asymptotic single-particle model that do not worsen with time, together with a long-time convergence result for the sini_wang_yao model. The discussion also outlines extensions to 2D, multi-particle configurations, and resonant scenarios, highlighting the potential for broader applicability of time-uniform smallness concepts in wave scattering.

Abstract

In the frequency domain wave scattering problems, obstacles can be effectively replaced by point scatterers as soon as the wavelength of the incident wave exceeds significantly their diameter. The situation is less clear in the time domain, where recent works suggest the presence of an additional temporal scale that quantifies the smallness of the obstacle. In this paper we argue that this is not necessarily the case, and that it is possible to construct asymptotic models with an error that does not deteriorate in time, at least in the case of a sound-soft scattering problem by a star-shaped obstacle in 3D.

What does it mean for a 3D star-shaped scatterer to be small in the time domain?

TL;DR

The paper tackles time-domain scattering by a small three-dimensional sound-soft star-shaped obstacle and shows that replacing the obstacle with a point scatterer can yield time-uniform errors, without introducing extra temporal scales. It develops a resolvent-based framework, including a Krein-type formula and Laplace-domain estimates, to obtain local energy decay bounds and to bound the error of asymptotic models in time, with explicit dependence on the small parameter ε. The main results establish a local energy decay bound for the scattered field in the far-field region that decays exponentially after the arrival time, and prove time-domain error bounds for the asymptotic single-particle model that do not worsen with time, together with a long-time convergence result for the sini_wang_yao model. The discussion also outlines extensions to 2D, multi-particle configurations, and resonant scenarios, highlighting the potential for broader applicability of time-uniform smallness concepts in wave scattering.

Abstract

In the frequency domain wave scattering problems, obstacles can be effectively replaced by point scatterers as soon as the wavelength of the incident wave exceeds significantly their diameter. The situation is less clear in the time domain, where recent works suggest the presence of an additional temporal scale that quantifies the smallness of the obstacle. In this paper we argue that this is not necessarily the case, and that it is possible to construct asymptotic models with an error that does not deteriorate in time, at least in the case of a sound-soft scattering problem by a star-shaped obstacle in 3D.

Paper Structure

This paper contains 21 sections, 21 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction and problem setting
  2. Introduction
  3. Problem setting
  4. Questions studied in the paper
  5. Local energy decay estimates
  6. Strategy of the proof of Theorem \ref{['theorem:answer1']} and auxiliary notation
  7. Representation formula
  8. Convenient expressions of resolvents of $\mathbb{B}^{\varepsilon}$, $\mathbb{B}$
  9. A Krein-like resolvent formula for $\mathcal{D}^{\varepsilon}_D$
  10. Behavior of $\mathcal{D}^{\varepsilon}_D$ on $\mathbb{C}$
  11. Estimates involving the free resolvent
  12. Estimates on $\mathcal{N}^{\varepsilon}_D(z)$
  13. Bounds on the cut-off resolvent $\chi_{a\varepsilon}\mathcal{R}^{\varepsilon}_D(s)\chi_{b\varepsilon}$
  14. Proof of Proposition \ref{['prop:bounds']}
  15. Proof of Theorem \ref{['theorem:answer1']}
  16. ...and 6 more sections

Key Result

Theorem 1

Assume that $k_{reg}\geq 6$. There exists $\varepsilon_0>0$, s.t. for all $0<\varepsilon<\varepsilon_0$, for the problem described in Section sec:pb_setting, the following energy decay bounds on the scattered field hold true. With $\gamma_{\Omega}>0$ that depends on the obstacle only, the scattered where the constant $C_{\operatorname{ff}}$ depends on $R_{\operatorname{ff}}$, $r_{\operatorname{ff

Figures (1)

  • Figure 1: Left: support of $\chi_{\varepsilon}$ on the plane $z=0$. In dark gray we mark the sphere where $\chi_{\varepsilon}=1$, and in light gray where $\chi_{\varepsilon}$ varies between $0$ and $1$. Outside of the sphere of radius $2\varepsilon$, $\chi_{\varepsilon}=0$. Right: support of $\chi_{m\varepsilon}$, $m\geq 2$, vs the obstacle $\Omega^{\varepsilon}$, $z=0$.

Theorems & Definitions (39)

  • Theorem 1
  • Remark 1
  • Lemma 1: A convenient expression of the resolvent of $\mathbb{B}^{\varepsilon}$
  • Lemma 2: A convenient expression of the resolvent of $B$
  • Remark 2
  • Definition 1
  • Proposition 1: Krein-like formula for the resolvent
  • Proposition 2
  • Proposition 3: Bounds in $\Omega^{\varepsilon,c}$
  • proof
  • ...and 29 more