Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (I) A frequency dependent Lyapunov function approach

TL;DR

This paper investigates the long-term decay of electromagnetic energy in dissipative Lorentz materials using a frequency-dependent Lyapunov function, revealing polynomial decay rates and connecting mathematical analysis with physical energy balances.

Contribution

It introduces a novel Lyapunov-based method to analyze energy decay in unbounded dissipative media, extending previous bounded media results with a physically intuitive approach.

Findings

Energy decays polynomially over time in dissipative Lorentz media.

The Lyapunov approach provides a less abstract, physics-connected analysis.

Results extend prior bounded media studies to unbounded structures.

Abstract

It is well-known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude-Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude-Lorentz open structures. We wish to quantify the loss in such media in terms of the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency dependent Lyapounov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in [18] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances.