Abstract
The dynamic Maxwell equations with a conductivity term are considered. Conditions for the exponential and strong stability of an initial-boundary value problem are given. The permeability and the permittivity are assumed to be \begin{document}$ 3\times 3 $\end{document} symmetric, positive definite tensors. A result concerning solutions of higher regularity is obtained along the way.
Highlights
The dynamic Maxwell equations with a conductivity term are considered
Consider the evolution of the electromagnetic field (e, h) in a bounded medium Ω ⊂ R3 with electric permittivity ε, magnetic permeability μ, and conductivity σ, in the absence of external currents, that is the system of partial differential equations,t − ∇ × h + σe = 0t + ∇ × e = 0 in Q := R+ × Ω, (1)
The H1-regularity of the classical solutions to this initial-boundary value problems with initial data in H1 seems to be an interesting result in its own right
Summary
The dynamic Maxwell equations with a conductivity term are considered. Conditions for the exponential and strong stability of an initialboundary value problem are given. Given u0 = (e0, h0) ∈ H , there exists a unique mild solution S(·)u0 ∈ C([0, ∞), H ) to the initial-boundary value problem (1) - (3). For u0 ∈ D(A ) there exists a unique classical solution S(·)u ∈ C([0, ∞), D(A )) ∩ C1([0, ∞), H ) to the initial-boundary value problem (1) - (3).
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