Electromagnetic scattering for time-domain Maxwell’s equations in an unbounded structure

Abstract

The goal of this work is to study the electromagnetic scattering problem of time-domain Maxwell's equations in an unbounded structure. An exact transparent boundary condition is developed to reformulate the scattering problem into an initial-boundary value problem in an infinite rectangular slab. The well-posedness and stability are established for the reduced problem. Our proof is based on the method of energy, the Lax-Milgram lemma, and the inversion theorem of the Laplace transform. Moreover, a priori estimates with explicit dependence on the time are achieved for the electric field by directly studying the time-domain Maxwell equations.

Analysis of time-domain Maxwell's equations in biperiodic structures

“…Theorem 3.2. For each m ∈ N, there exists a unique function W m , which is given in the form of (24) and satisfies (25)- (26).…”

“…Compared with the time-harmonic scattering problems, mathematical studies are much less done for the time-domain counterparts due to the challenge of the temporal dependence. The analysis can be found in [15,25,26,27,28,33] for the time-domain acoustic, elastic, and electromagnetic scattering problems in different structures including bounded obstacles, open cavities, and unbounded rough surfaces. In these works, the idea was to utilize the Laplace transform as a bridge between the time-domain and the frequency domain.…”

Analysis of transient acoustic scattering by an elastic obstacle

“…Compared with the time-harmonic scattering problems, the time-domain problems are less studied on their rigorous mathematical analysis due to the additional challenge of the temporal dependence. The analysis can be found in [6,19,20,32] for the time-domain acoustic and electromagnetic scattering problems in different structures including bounded obstacles, periodic surfaces, and unbounded rough surfaces. We refer to [21] and [22] for the mathematical analysis of the transient elastic scattering problems in an unbounded structure.…”

Unique solvability and stability of the time domain electromagnetic scattering problem with a locally perturbed perfectly conducting plate

“…For the time domain electromagnetic scattering problem, a similar "Laplace domain to time domain" analysis as the time domain acoustic scattering problem is involved to prove the well-posedness of the problem. Time domain electromagnetic scattering problems, such as the exterior scattering problem with a bounded scatterer [26], scattering by periodic structures [27] and rough surfaces [28], the open cavity problems [29][30][31][32], and scattering in chiral media [33], have been intensively analyzed. A brief survey of recent developments on analysis of the open cavity scattering problems is investigated in [29] for both the time harmonic case and the time domain case.…”