Asymptotics of solutions of the stationary and nonstationary Maxwell systems in a domain with small cavities

Abstract:

The nonstationary Maxwell system is considered, for all times $t\in \mathbb {R}$, in a bounded domain $\Omega (\varepsilon )\subset \mathbb {R}^3$ with finitely many small cavities; the cavity diameters are proportional to a small parameter $\varepsilon$. The perfect conductivity conditions or the impedance conditions are prescribed on the boundary of $\Omega (\varepsilon )$. The asymptotics of solutions are derived as $\varepsilon$ tends to zero. The cavities are “singular” perturbations of the domain $\Omega (0)$: they are collapsing into points as $\varepsilon \to 0$. The presented mathematical model describes the electromagnetic field behavior inside a conductive resonator with metallic inclusions of small size. This model can be of use for the diagnostics of plasma filling a resonator and containing such inclusions.

To describe the asymptotics of solutions, the method of compound asymptotic expansions is employed. The asymptotics in question is formed by solutions of the “limit” problems independent of $\varepsilon$. One of such problems turns out to be a non-stationary problem in a bounded domain with singular points on the boundary. The other limit problems are stationary in the complements of bounded domains. The method of compound asymptotics makes it possible to study the behavior of waves with length greater than the diameters of cavities. It is shown that the contribution of the short waves into the asymptotics of solutions is negligible, due to the smoothness of the right-hand side of the Maxwell system in time.