Exponential decay of quasilinear Maxwell equations with interior conductivity

Abstract

We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a bounded smooth domain of R 3 with a strictly positive conductivity subject to the boundary conditions of a perfect conductor. Under appropriate regularity conditions, adopting a classical L 2 -Sobolev solution framework, a nonlinear energy barrier estimate is established for local-in-time H 3 -solutions to the Maxwell system by a proper combination of higher-order energy and observability-type estimates under a smallness assumption on the initial data. Technical complications due to quasilinearity, anisotropy and the lack of solenoidality, etc., are addressed. Finally, provided the initial data are small, the barrier method is applied to prove that local solutions exist globally and exhibit an exponential decay rate.In (2.1) and (2.2) to follow we impose symmetry assumptions on ε and µ under which (1.1) becomes a symmetric quasilinear hyperbolic system. For such systems in the full space case Ω = R 3 , one has a well-developed local well-posedness theory of H 3 -valued solutions due to Kato [12]. However, the above problem has a characteristic boundary which could lead to a loss of regularity in the normal direction. The available general existence results work in Sobolev spaces of very high order and with weights vanishing at the boundary, see [11,26] as well as [24] for tangential regularity. For absorbing boundary conditions, local existence results in H 3 were given in [22]. However, in our case a local well-posedness theory in H 3 was established only recently in the papers [28,29,30]. We will strongly rely on these results.In our main Theorem 2.2 we show that local solutions are indeed global and exhibit exponential decay rates in H 3 , provided that the initial fields are small and that the conductivity is strictly positive. In Remark 2 of [6], it is explained that certain solutions do not decay to 0 if one drops the simple connectedness of Ω or the magnetic compatibility conditions in (2.5), even for linear ε and µ. It should be emphasized that, while decay rates for the Maxwell system have been studied in a number of works (viz. [1, 6, 8, 9, 13, 14, 20, 21] and references therein), the cited studies only allow for linear permittivity and permeability and partly deal with constant isotropic coefficients. Stabilization results for general hyperbolic systems typically concern damping mechanism acting on all components of the solutions, see [25], whereas in our Maxwell system the dissipation via conductivity only affects the electric field. For Ω = R m the paper [2] allows for partial damping even in the quasilinear case, but its assumptions exclude the Maxwell equations. For the quasilinear Maxwell system we are only aware of a few results on the full space Ω = R 3 that establish global existence and decay for small and smooth solutions, see [16,23,27]. These works rely on dispersive estimates which are not available on bounded domains. On the other hand, it is known that blowup in W 1,∞ or H(curl) can occur in various cases, see [4]...

Boundary stabilization of quasilinear Maxwell equations

“…This will result in a uniform stabilizability inequality (4.2) stated in Proposition 4.2 below. Combining Corollary 3.5 and Proposition 4.1, we derive (4.2) in a fashion vaguely reminiscent of the proof of Proposition 4.1 of [30].…”

“…There we show an "elliptic" regularity result (essentially due to [12]), which allows us to reconstruct the full H 1 (Ω)norm of (E, H) from the L 2 (Ω)-norms of curl(E, H) and div(εE, µH) as well as the H 1/2 (Γ)-norm of the boundary data. In contrast to our earlier work [30], due to the absence of the electric resistance term, solenoidality properties are available both for E and H, which facilitates the application of this fact. However, higher-order normal derivatives cannot be controlled in this way since they destroy the boundary condition.…”

“…Using Young's inequality to estimate z 3/2 (τ ) ≤ 1 2 z(τ ) + z 2 (τ ) , the estimate in (4.3) yields the desired inequality with an appropriate positive constant C. Theorem 2.2 can now easily be proved, cf. [30] or [31].…”

“…However, higher-order normal derivatives cannot be controlled in this way since they destroy the boundary condition. Instead, the required normal regularity has to be deduced from the evolution and divergence equations in (1.1), using ideas as in [30]. Appropriate localization techniques in a boundary collar also need to be employed.…”

“…Then our main result from Theorem 2.2 is a direct consequence of the standard barrier method (cf. [30,31,32,38,42], etc. ).…”