Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Abstract

In this article we develop the local wellposedness theory for quasilinear Maxwell equations in H m for all m ≥ 3 on domains with perfectly conducting boundary conditions. The macroscopic Maxwell equations with instantaneous material laws for the polarization and the magnetization lead to a quasilinear first order hyperbolic system whose wellposedness in H 3 is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in H m with m ≥ 3 of the corresponding initial boundary value problem if the material laws and the data are accordingly regular and compatible. We further characterize finite time blowup in terms of the Lipschitz norm and we show that the solutions depend continuously on their data. Finally, we establish the finite propagation speed of the solutions.2010 Mathematics Subject Classification. 35L50, 35L60, 35Q61. Key words and phrases. nonlinear Maxwell equations, perfectly conducting boundary conditions, quasilinear initial boundary value problem, hyperbolic system, local wellposedness, blow-up criterion, continuous dependance. 1 2 MARTIN SPITZThe macroscopic Maxwell equations in a domain G readfor all t ≥ t 0 . In (1.1) we consider the Maxwell system with the boundary conditions of a perfect conductor, where ν denotes the outer normal unit vector of G. System (1.1) has to be complemented by constitutive relations between the electric fields and the magnetic fields, the so called material laws. We choose E and H as state variables and express D and B in terms of E and H. The actual form of these material laws is a question of modelling and different kinds have been considered in the literature. The so called retarded material laws assume that the fields D and B depend also on the past of E and H, see [2] and [33] for instance. In dynamical material laws the material response is modelled by additional evolution equations for the polarization or magnetization, see e.g.[1], [11], [16], or [17]. In this work we concentrate on the instantaneous material laws, see [6] and [14]. Here the fields D and B are given as local functions of E and H, i.e., we assume that there are functions θ 1 , θ 2 : G × R 6 → R 3 such that D(t, x) = θ 1 (x, E(t, x), H(t, x)) and B(t, x) = θ 2 (x, E(t, x), H(t, x)). The most prominent example is the so called Kerr nonlinearity, wherewith ϑ : G → R 3×3 and the vacuum permittivity and permeability set equal to 1 for convenience. We further assume that the current density decomposes as J = J 0 + σ 1 (E, H)E, where J 0 is an external current density and σ 1 denotes the conductivity. If we insert these material laws into (1.1) and formally differentiate, we obtainfor the evolutionary part of (1.1). The resulting equation is a first order quasilinear hyperbolic system. In order to write (1.1) in the standard form of first order systems, we introduce the matrices J 1 =   0 0 0 0 0 −1 0 1 0   , J 2 =   0 0 1 0 0 0 −1 0 0   , J 3 =   0 −1 0 1 0 0 0 0 0   and A co j = 0 −J j J j 0 (1.3)

Exponential decay of quasilinear Maxwell equations with interior conductivity

“…The bound (2.8) is thus true on [0, T * ) =: J * . If T * < ∞, then the blow-up condition in Theorem 5.3 of [29] implies that T max > T * and hence z(T * ) := max k∈{0,1,2,3}…”

“…Afterwards one passes to the limit in the resulting variant of (3.4). In view of the available a priori estimates and regularity results from [7], [28] or [30], one has to approximate the data and the coefficients separately. The assertion is closely related to [7], but not stated there.…”

“…2) Theorem 1.1 of [30] yields functions (u n,m , v n,m ) in G 1 := C 1 (J, L 2 (Ω)) ∩ C(J, H 1 (Ω)) which solve the problem (3.3) with the coefficients and the data from step 1). We note that the required compatibility condition tr t u 0,n = χ n (0) is trivially satisfied.…”

“…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.…”

Boundary stabilization of quasilinear Maxwell equations

“…However, due to the characteristicity of the boundary symbol associated with (1.1) (cf. [58,60] for the case of a perfect conductor), a regularity loss in the normal direction possibly occurs making only few of these theories (e.g., [20,55]) applicable. The additional price to pay is that a cumbersome framework of weighted Sobolev spaces of very high order needs to be employed greatly reducing the practical applicability of these results.…”