Well-Posedness and Exponential Stability of Nonlinear Maxwell Equations for Dispersive Materials with Interface

In this paper we consider an abstract Cauchy problem for a Maxwell system modelling electromagnetic fields in the presence of an interface between optical media. The electric polarization is in general time-delayed and nonlinear, turning the macroscopic Maxwell equations into a system of nonlinear integro-differential equations. Within the framework of evolutionary equations, we obtain well-posedness in function spaces exponentially weighted in time and of different spatial regularity and formulate various conditions on the material functions, leading to exponential stability on a bounded domain.


Introduction
The macroscopic Maxwell equations governing electromagnetic fields E = E(t, x), H = H(t, x) (t ∈ R, x ∈ Ω ⊆ R 3 ) in matter are given by where J and ρ are the current and charge densities.The auxiliary fields (the material response) D, B are induced by E, H through the constitutive relations with a real constant ǫ 0 > 0 and a symmetric matrix-valued function µ : Ω → R 3×3 .In nonlinear optics an analytic expansion P (E) = P (1) (E) + P (2) (E) + P (3) (E) + • • • is usually assumed, where each P (n) is given by the time-delayed action of a tensor with rank (n + 1), see [Boy08].We thus consider throughout variants of the model with k ∈ N, where the kernel χ is causal in the sense that χ(s 1 , . . ., s k ) = 0 whenever s j < 0 for some j ∈ {1, . . ., k}.Candidates for the vector function q are subjected to various conditions involving Lipschitz-continuity. Due to the convolution in (1.2) the dielectric constant, obtained from the Fourier transform (in time) of χ, is frequency dependent.In other words, the material is dispersive.In the linear case, i.e., k = 1 and q(E) = E, a standard model of χ is the Drude-Lorentz model, see Appendix A.
Our aim is to study Maxwell's equations in nonlinear optics for the interface problem depicted in Figure 1, where Ω = Ω 1 ⊔Γ⊔Ω 2 (⊔ denotes disjoint union) consists of two domains Ω 1 and Ω 2 , each with their own material response, separated by the interface Γ.In this case, the two systems resulting from (1.1) (on Ω i respectively) have to be supplemented by transmission conditions: If Γ is a 2-dimensional C 1 -manifold with a normal vector field n, these are given by ( [DL90a]) and are due to the absence of surface charges and currents.Here [f ] Γ := (f 1 − f 2 )| Γ denotes the jump of f across Γ (in the sense of traces), where 3) is equivalent to the continuity of the tangential components of E, H and transversal components of D, B. Γ : Schematic depiction of the electric field of a travelling surface wave induced by charge oscillations at an interface Γ between two media Ω 1 and Ω 2 .
A possible application of this interface setting lies in the modelling of surface waves such as surface plasmon polaritons (SPPs).These are evanescent electromagnetic fields resulting from charge excitation (achieved through coupling mechanisms) at an interface; see also Figure 1.SPPs exist at metal-dielectric interfaces: in the linear case, one can obtain the existence of travelling surface waves which satisfy a highly nonlinear dispersion relation due to frequency-dependent material response (see [Rae88]).In addition, as metals are intrinsically lossy, SPPs experience exponentially fast damping in time.In the theory this is reflected by the so-called exponential stability.
For nonlinear Maxwell systems with interface, a local well-posedness theory is available for materials without memory, see [SS18] and the references therein.An application of the latter to surface waves can be found in [DST22].The Maxwell problem with memory can be reformulated as an instantaneous system if the susceptibilities satisfy certain assumptions, see [SU03].
There are several aims of this article.First, we provide a well-posedness theory (global in time) for Maxwell systems with interface and with a nonlinear material response given by nonlocal models.The Maxwell system is formulated within the framework of evolutionary equations in the sense of Picard [MP02,Pic09] (we also refer to [STW22] as a general reference).As this theory works in Bochner spaces in space-time, memory effects can be treated more naturally (see also [SW17] for similar nonlocal models in SPDEs).
Second, the formulation in spaces of higher spatial regularity allows for a wider class of nonlinearities and complements previous work on spatial regularity for evolutionary equations, see [PTW17] and also [TW21].
Third, conditions for exponential stability of linear and nonlinear systems are provided; this is based on work in [Tro13,Tro18].
Finally, treating the paradigmatic Maxwell case may open up similar strategies for general evolutionary equations.This article is structured as follows.Section 2 is concerned with the well-posedness of the Maxwell system in the functional analytic framework of evolutionary equations.Since here we work exclusively with (weighted) L 2 -spaces, no regularity of the interface or the boundary is needed.Subsection 2.5 deals with higher regularity in space, which requires some regularity of the interface and the boundary as well as the boundedness of Ω (whereas the time regularity is covered by the general theory and is not Maxwell-specific).
In Section 3 we examine exponential stability for the electric and magnetic field.The result for the linear second-order formulation (wave equation for the electric field) is Theorem 3.10.The proof uses the theory in [Tro18] and the Picard-Weber-Weck selection theorem (B.1) as a key ingredient.The latter restricts Ω to a bounded (weak Lipschitz) domain.Imposing suitable Lipschitz-continuity on the nonlinearity, a corresponding result is obtained for the nonlinear system by a fixed-point argument.
Using similar methods as in Theorem 3.10, we establish exponential stability for the full Maxwell system, for materials of a different class, in Theorem 3.15 for fields in L 2 (Ω) 3 , and in Theorem 3.17 for fields in H 2 (Ω 1 ) 3 ⊕ H 2 (Ω 2 ) 3 .Theorem 3.17 allows for a wider class of nonlinearities, which are considered in Theorem 3.18.
The theory is accompanied by several examples.An overview of these applications is found in Appendix A.3.

Maxwell operator and boundary conditions
We introduce the functional analytic setup in which we treat the Maxwell system (1.1) together with the transmission conditions.Let again Ω = Ω 1 ⊔ Γ ⊔ Ω 2 and set We denote by curl 0 the closure of the operator where ϕ 1/2 .We further set curl := curl * 0 .It is then easy to derive dom(curl) = H(curl, Ω) := {u ∈ H : curl u ∈ H}.
Remark 2.1.For the application of the well-posedness in Sections 2.3 and 2.4 the domains Ω 1 , Ω 2 may be quite general; in particular no regularity of the boundary is needed, and Ω 1 , Ω 2 may be unbounded (if Ω = R 3 , then curl = curl 0 ).However, to deal with higher regularity and exponential stability in Sections 2.5 and 3 they need to be bounded with more regularity of the boundary.
The above choice of dom(A) = H 0 (curl, Ω) × H(curl, Ω) and the skew-selfadjointness of A encode the interface conditions and the boundary condition of a perfect conductor, if the boundaries of Ω 1 , Ω 2 are sufficiently regular: Assume that Ω, Ω 1 , Ω 2 have Lipschitz boundaries and denote their outward normal fields by n.Let D ⊆ dom(A) be a subset consisting of functions that are smooth in Ω 1 and Ω 2 and fix (u E , u H ) ∈ D. Using the divergence theorem on Ω 1 and Ω 2 separately, we have for all and similarly, By skew-selfadjointness of A, the left-hand sides must vanish for arbitrary v E , v H . Therefore, The latter identity is the boundary condition of a perfect conductor.Using the traces in H(curl, Ω) and H 0 (curl, Ω), equations (2.1) can be shown to hold for u E ∈ H 0 (curl, Ω), u H ∈ H(curl, Ω) in the sense of traces, see [Lei86,BDPW22].In absence of this regularity of ∂Ω and Γ, conditions (2.1) are interpreted in a generalized sense.
We consider throughout a Cauchy problem for the Maxwell system (1.1), formulated as an evolutionary problem in (positive) time: 6 , where is a given history (this is necessary since the material function (D, B) is in general dependent on past values of its argument).The solution should meet the condition (E(t), H(t)) ∈ dom(A) for all t, in order for the jump conditions of E, H to be fulfilled.For the divergence equations for D, B one finds that they are largely redundant, in the sense that they follow from (2.2) and suitable initial values.Indeed (cf.[SS18]), applying div to the first line in (2.2) and integrating, it follows that div D(t) = ̺(t) holds for t ≥ 0 if and only if ̺ and J are related by Similarly, it follows from second line in (2.2) that div B is constant for all t > 0, so if div B(0) = 0, then div B(t) = 0 holds for all t > 0.
Regarding the jump conditions it suffices that they are fulfilled at time t = 0; then (2.3) follows for all t > 0 by taking derivatives in time and using the structure of the remaining equations (see again [SS18]).More generally, the jump conditions (2.3) are a property of the domain of both the operators div : H(div, Ω) ⊆ L 2 (Ω) 3 → L 2 (Ω) and div 0 : H 0 (div, Ω) ⊆ L 2 (Ω) 3 → L 2 (Ω), and can thus be interpreted as a regularity condition, see Sections 2.4, 2.5.Here div = − grad * 0 , div 0 = − grad * are defined similarly to curl, curl 0 in terms of the usual weak gradient grad : and the weak gradient with zero boundary condition, grad 0 : The interface setting will play no role in the solution theory established in the next sections, as the transmission conditions are naturally embedded into the domain dom(A), and is thus independent of any inhomogeneities in the material.Only in the context of higher regularity in space (Section 2.5, and Section 3.2.1 for exponential stability in those spaces) will we need to take the interface into account.

Linear evolutionary equations
In the following, we provide a short overview of the theory of evolutionary equations.For details, see [STW22].We will first consider a purely linear material function (for example taking k = 1 in (1.2) and q(u) = u, see Example 2.9 below).In this case, the linear Maxwell system fits into the category of abstract evolutionary equations of the form with given data g, understood as an operator equation in the weighted Hilbert space Here X is a Hilbert space and A : dom(A) ⊆ X → X a densely defined and closed operator, extended to a subset of is understood in the weak sense and is a densely defined and closed operator, where In most cases M (∂ t ) will denote a convolution operator, but more generally it is a linear material law: Definition 2.3.A linear material law is an analytic mapping M : dom(M ) ⊆ C → B(X) into the space B(X) of bounded linear operators on X (with norm denoted by • ), which is uniformly bounded on a right half-plane, i.e., The operator M (∂ t ) is defined by the composition is the unitary extension of the Fourier-Laplace transform The solution theory for (2.4) is established by Theorem 2.5 and is closely tied to the concept of causality.

By a consequence of the Paley
It is a key observation that ∂ t is boundedly invertible for ̺ = 0, and causally invertible for ̺ > 0. In the latter case ).We use the symbol ∂ −1 t exclusively to denote the causal map given by (2.5).
Theorem 2.5 (Picard's Theorem, see e.g.[STW22, Theorem 6.2.1]).Let A : dom(A) ⊆ X → X be skew-selfadjoint and M a linear material law, for which zM (z) is strictly accretive on a half-plane satisfy the assumptions of Theorem 2.5, the solution u of (2.4) is explicitly given using the spectral representation of the time-derivative: As such, Theorem 2.5 provides sufficient conditions for the operator to have a bounded and analytic extension on C Re>̺ 0 .In this case, we say that the problem (2.4) is well-posed in the range of spaces L 2 ̺ (R, X), ̺ > ̺ 0 , or simply well-posed, implicitly presuming the existence of such ̺ 0 ∈ R.

Well-posedness of the nonlinear Maxwell system
Nonlinear, (uniformly) Lipschitz-continuous perturbations of linear equations can be treated by a Banach fixed-point argument; see [SW17] for a similar argument.Precisely, inspired by [STW22, Section 4.2], we consider the following class of nonlinearities.
In this case we simply write f : Proposition 2.8.Let ̺ 0 , d ∈ R >0 , and let M be a linear material law satisfying (2.7) ) be causal and uniformly Lipschitz-continuous for all ̺ > ̺ 0 with lim sup Then there exists ̺ 1 ≥ ̺ 0 such that for all ̺ > ̺ 1 the problem ) for all ̺ > ̺ 0 , with S ̺ ≤ d/̺.By Theorem 2.5 and Lipschitz continuity of f we can estimate ) becomes a contraction for large ̺.By Theorem 2.5 (ii) and the assumption on f , the unique fixed point is independent of ̺.
From now on we apply this result to the general nonlinear Maxwell system (2.2) setting X = H × H.To this end, we isolate the linear part of the polarization and take where ǫ(•) is a linear material law (the linear permittivity) and the resulting nonlinear system takes the form (2.8) The assumptions of Proposition 2.8 are satisfied if: • ǫ satisfies Re zǫ(z) ≥ c ǫ Re z for Re z > ̺ 0 and some c ǫ > 0.
Example 2.9.In most cases we will consider ǫ(∂ t )E = ǫ 0 E + χ * E and µH = µ 0 H, where ǫ 0 , µ 0 > 0 are the vacuum permittivity and vacuum permeability, and χ * denotes the time convolution with the linear electric susceptibility tensor χ : R → B(H), with supp χ ⊆ [0, ∞) due to causality, such that The simplest case in an interface setting is given by with scalar-valued χ 1 , χ 2 supported in [0, ∞) and 1 Ω i being the characteristic function on Ω i .Let denote the frequency-dependent susceptibility.Since for ̺ ≥ ̺ 0 .Hence ǫ is a linear material law.The condition Re(zǫ(z)) ≥ c ǫ Re z ≥ c ǫ ̺ 0 above is satisfied for some ̺ 0 > 0 if z → |z χ(iz)| is bounded on C Re>̺ 1 for some ̺ 1 ∈ R (for example, this is the case for the Drude-Lorentz model in Appendix A).The extension to the case in which χ 1 (t), χ 2 (t) ∈ L ∞ (Ω) 3×3 are matrix-valued is straightforward; here we impose the condition χ For the nonlinear part we similarly assume that with κ : R → B(H), supp κ ⊆ [0, ∞), and q : H → H Lipschitz-continuous. Since ∂ t P nl (E) is the only term on the right-hand side of (2.8) depending on E and the needed Lipschitz-continuity in E is implied by conditions on κ and its derivative κ ′ .These conditions are provided next.
Lemma 2.10 yields the uniform Lipschitz-continuity of implying for ̺ > max{̺ κ , 0} that the solution operator is causal and Lipschitz-continuous for any Φ, Ψ ∈ L 2 ̺ (R, H), with Lipschitz constant at most where c is given by the condition (2.7) imposed on ǫ(•) and µ.
Example 2.11 (Saturable nonlinearity).Let k ∈ N ≥2 and τ > 0 and consider q : H → H given by (For k = 3 this is a saturable version of the Kerr-type nonlinearity E → |E| 2 E.) Since R 3 ∋ ξ → V (|ξ|)ξ is smooth and asymptotically linear, it is Lipschitz-continuous, hence q : H → H is Lipschitz-continuous. Thus, P nl defined as in (2.11) with κ as in Lemma 2.10, fulfills the necessary assumptions of the lemma.

Local well-posedness
The uniform Lipschitz-continuity in the range of spaces L 2 ̺ (R, X) imposed on the nonlinearity may seem restrictive (in particular, nonlinearities growing at a superlinear rate are excluded as candidates for q).In fact, this condition can be replaced by Lipschitz continuity on closed subsets in L 2 ̺ (R, X), which eventually (for large ̺) grow large enough to include given data.To illustrate this, we formulate the following refinement of Proposition 2.8.Proposition 2.12.Let A : dom(A) ⊂ X → X be skew-selfadjoint and M a linear material law with Re zM (z) Proof.Denote by Replacing r with 1 2 ̺ cd 1/α in the last inequality leads to the condition which is fulfilled by assumption on g for large ̺ > 0. This establishes S ̺ (f (•) + g) as a contraction on B r for some r < Consider now a quadratic nonlinearity of the form where K : R 2 → B(X) is an operator-valued kernel with supp K ⊆ [0, ∞) 2 (to ensure causality), and where q : X × X → X is a bounded bilinear map, i.e., q(u, v) X ≤ C q u X v X for some C q > 0. In analogy to (2.12) we impose the following integrability conditions, for some ̺ K ∈ R. Using the same strategy as in (2.13) we can show for all This computation makes it clear, however, that the mapping property f : this fact is relevant in combination with the notion of exponential stability, see Section 3).
Remark 2.14 (Nonlinearity with a cutoff in time).We now show how the fixed-point argument underlying Proposition 2.12 can still be applied to a modified version of the nonlinearity above to obtain a local well-posedness result for ̺ ≥ ̺ K > 0. In detail, we modify the kernel K by applying a cutoff in the t-variable.Suppose again K : R 2 → B(X) is causal map satisfying (2.17) and that, in addition, (2.18) For T > 0 we define K T : R 3 → B(X) and f T : and observe that for all ̺, t, T > 0. Now modifying the estimate above we obtain for Since by bilinearity of q we have (2.20) Hence, given ̺ ≥ ̺ K , the parameters T, r > 0 can be chosen small enough so that u → S ̺ (f T (u) + g) becomes a contraction on a closed ball with radius r in L 2 ̺ (R, X), provided that the data g ∈ L 2 ̺ (R, X) is small enough.
Example 2.15.As an application to the Maxwell system, let P nl = P (2) , where P (2) is a fully nonlocal quadratic polarization given by Here we assume that the spatial map q is defined via a tensor Λ = (Λ ijk ) i,j,k∈{1,2,3} with , . By the Cauchy-Schwarz inequality we have the pointwise estimate from which we obtain q(u, v) H ≤ C q u H v H with an appropriate constant C q .Hence, q is a bilinear, bounded map and satisfies (2.19) with X = L 2 (Ω) 3 .Furthermore, formally computing the derivative of P (2) (E) gives As such, K satisfies (2.17) and (2.18) for arbitrary ̺ K ∈ R. Thus, the cutoff version of ∂ t P (2) defined by The same principle applies to multilinear maps in general: Let n ∈ N ≥2 and q : (H) n → H be a bounded n-linear map.Let κ : R n → R n×n be supported in [0, ∞) n with κ(s 1 , . . ., s n ) = 0 whenever s j = 0 for some j ∈ {1, . . ., n}.Defining P (n) as in (1.2) by we find that, if κ satisfies an integrability condition similar to (2.17), then (2.21) Reasoning as above in (ii) we obtain the following result.

Initial values
In order to apply the well-posedness theory to system (2.2), it remains to discuss how the history of the electromagnetic field can be incorporated into the framework.First we mention the following result concerning regularity in time (see [PM11, Section 3.1] or [STW22, Section 6.3]).(We subsequently use the notation f g or equivalently g f to denote f ≤ Cg for some C > 0 independent of f, g.) Proposition 2.17.Let (∂ t M (∂ t ) + A) u = g be well-posed in the range of spaces , with continuous dependence on the data: In fact, by the Sobolev embedding theorem.
Consider a general Cauchy problem for the Maxwell equations, for a given history φ : (−∞, 0] → L 2 (Ω) 6 .For simplicity of this model problem we have set J = 0 and assume where M 0 is selfadjoint and uniformly positive definite, χ is causal, rapidly decaying, and smooth on [0, ∞), and q : L 2 (Ω) 6 → L 2 (Ω) 6 is Lipschitz-continuous with q(0) = 0. We want to convert (2.22) into a nonlinear evolutionary equation in L 2 ̺ (R, L 2 (Ω) 6 ) (we note however that the derivation below is not strictly tied to the Maxwell system).To this end, suppose U ∈ C(R, L 2 (Ω) 6 ) is a continuous solution of (2.22).Let θ + := θ + 0 denote multiplication with the Heaviside step function, then the projection separates the "unknown" solution u with supp u ⊆ [0, ∞) from the given history φ, which we extend trivially to the whole line, thus φ = (1 − θ + )φ.With U = u + φ we also have q(U (t)) = q(u(t)) + q(φ(t)) for all t ∈ R, and therefore in fact M(U )(t) = M(u)(t) + M(φ)(t).Interpreting now ∂ t in the distributional sense, we use the formula to extract from (2.22) an equation for u on the whole real line: where we used G(φ)(0 − ) = G(φ)(0 + ).The δ 0 -term can be removed by smoothing the jump of u at t = 0: Choose η ∈ C ∞ c (R) with η(0) = 1, and set Thus, using that φ + (0 + ) = φ(0 − ), (2.23) becomes Finally, the last identity can be written as where ).The wellposedness follows by Proposition 2.8 from the Lipschitz continuity of ũ → ∂ t G(ũ+φ + ).Since φ + = 0 on (−∞, 0], the causality of the solution operator and the fixed-point iteration implies ũ = 0 on (−∞, 0]. Remark 2.18.The initial value theory in [Tro18] for linear systems uses η ≡ 1, however, the present choice η ∈ C ∞ c (R) is more convenient since it also works in the context of exponential stability, i.e., if the system is well-posed for ), then solutions of (2.24) generate continuous solutions of (2.22).Indeed, in this case Proposition 2.17 justifies ũ ∈ H 1 ̺ (R, L 2 (Ω) 6 ), and since φ − φ + is continuous, ).In this case, since g φ = 0 on (−∞, 0], a necessary and sufficient condition for This, after a slight modification of φ + , can be interpreted as: φ must be a solution of the Maxwell system in t = 0. Indeed, under the assumption that φ is differentiable in t = 0, let Then, for g φ defined as above, (2.25) becomes (2.26) Example 2.20.Let us formulate the above transformation U → ũ in the original Maxwell variables E, H. Hence, consider To simplify the notation, we denote the zero extension of the history of the fields by E 0 , H 0 .Setting as before), we see that the resulting evolutionary system takes the form of (2.8), , and the role of g φ is played by (2.28) Thus, the history of H only enters the equation via the initial value H 0 (0 − ).To ensure div B(H) = div µH = 0 for t ∈ (0, ∞), it suffices that div µH 0 (0 − ) = 0.In general, we will assume at least that µH 0 (0 − ) ∈ H 0 (div, Ω).
In order to apply Proposition 2.8 or Proposition 2.12 to the Maxwell problem, we need ∂ t Pnl (0) = 0.This can, however, always be achieved by substituting Henceforth we shall drop the tilde and write E, H instead of Ẽ, H, as well as P nl instead of Pnl , and always assume that the system is given in the evolutionary form (2.27), where Φ, Ψ ∈ H 1 ̺ (R, H) are supported in [0, ∞).There is also no loss in assuming that J = 0, since a nonzero J (supported in [0, ∞) and J ∈ H 1 ̺ (R, H)) can be incorporated into the inhomogeneity Φ.

Higher spatial regularity
While the interface setting can be safely ignored in the results established so far (as long as the material laws are bounded linear operators on L 2 (Ω) 3 ), the heterogeneity of the material plays a more important role if tools relying on spatial regularity are used.While spatial regularity is interesting in its own right, working in higher order Sobolev spaces also allows to control other types of nonlinearities for which Lipschitz-continuity fails in L 2 (compare in particular the local Lipschitz estimates in Theorem 3.13 and 3.18).
From now on we assume that Ω = Ω 1 ⊔ Γ ⊔ Ω 2 is a bounded domain with interface Γ (the boundedness of Ω is a necessary requirement of Proposition 2.21).We want to establish conditions that allow the solution E, H for k ∈ N to lie (pointwise almost everywhere in time) in the space (Note that for functions u 1 ∈ H k (Ω 1 ), u 2 ∈ H k (Ω 2 ) we identify the pair (u 1 , u 2 ) with the sum u 1 + u 2 of their zero extensions on Ω; in particular, H k is a subspace of We are in particular interested in k ≥ 2 since in this case H k (R d ) is a multiplication algebra in dimension d ≤ 3.By extension, this carries over to H k (Ω i ), i.e., for each bilinear map b : R . The subsequent analysis relies on the following spatial regularity result adapted from [Web81].
Proposition 2.23.Let Ω i and ǫ satisfy the assumptions of Proposition 2.21 for some k ∈ N and assume for ℓ Proof.In view of Remark 2.22 we can write Moreover, with grad H 1 0 (Ω) ⊆ ker(curl 0 ) and curl H(curl, Ω) ⊆ ker(div) we have curl 0 grad f = 0 and div curl A = 0. Hence, curl 0 u = curl 0 (u + grad f ) and div (ǫu + v) = div ǫ(u + grad f ), Since grad f ∈ H ℓ (Ω i ) 3 , also u ∈ H ℓ (Ω i ) 3 and the claim follows from Remark 2.24.To apply Proposition 2.21 to functions depending on time (or rather frequency in Fourier space), it is important to have uniform estimates, i.e., the dependence of the constants on ǫ should be removed.For simplicity, suppose that ǫ is constant on each Ω i and let u be given as in Proposition 2.21.In this case, a closer look at the proof in [Web81] holds, with C independent of ǫ.One then easily obtains the estimate Analogously, if, instead, u and v are chosen as in Proposition 2.23, then we have again with C independent of ǫ.
For simplicity, we assume for the rest of this section homogeneous materials in each Ω i , i ∈ {1, 2}, where ǫ(∂ t ) and µ satisfy the following condition.
Note in this case that ǫ(z) and µ satisfy the smoothness assumptions of Proposition 2.21 with k = ∞.
Under the condition (M1) on ǫ(∂ t ) and µ, we consider next the linearized system (in evolutionary form) For ℓ ∈ N 0 denote by Theorem 2.26 (H 2 -regularity for the linear Maxwell system).Impose the conditions on Ω of Proposition 2.21 with k = 2, and condition (M1) on ǫ and µ.Let ̺ 0 , c ∈ (0, ∞) be such that Re zǫ(z) ≥ c for Re z > ̺ 0 , making the system (2.33) well-posed in and where and with constants independent of E, H, Φ, Ψ, N, ̺.
Proof.As a result of the solution theory (Theorem 2.5) and the time-regularity (Proposition 2.17), since Φ, Ψ, N ∈ H 2 ̺ (R, H), the solution fulfills E, H ∈ H 2 ̺ (R, H) with continuous dependence on the data, ). Taking the Fourier-Laplace transform, we obtain from (2.36) pointwise for almost all ξ ∈ R, as identities in L 2 (Ω) 3 .Now Proposition 2.21 (with u = (L ̺ H)(ξ), using u ∈ H 0 (div) ∩ H(curl)) and Proposition 2.23 (with The same conclusion can be drawn for L ̺ (∂ t E), L ̺ (∂ t H); indeed, apply ∂ t to (2.36) and take the Fourier-Laplace transform to obtain the following identities in L 2 (Ω) 3 , again for almost all ξ ∈ R: follows by the same argument.This, together with the assumptions on the data Φ, N, Ψ implies that (2.37) are in fact identities in for almost all ξ ∈ R, once more by Proposition 2.21 and Proposition 2.23, together with the estimates (cf.Remark 2.24) where the norms are taken in Ω i .After integration, using the boundedness of µ −1 , ǫ(•) −1 and the Plancherel theorem, we have with constants independent of E, H, N, ̺.Now we can employ (2.36) and estimates of type (2.31) recursively to replace the curl-and div-terms on the right-hand side: and similarly, Finally estimating ∂ j t E ̺,0 , ∂ j t H ̺,0 with the help of (2.35) we obtain (2.34).
We now extend the regularity to the nonlinear case, replacing N by a map ∂ t P nl (•).
Corollary 2.27 (H 2 -regularity for the nonlinear Maxwell system I).Suppose that the domain Ω and the linear material law ǫ satisfy the conditions of Theorem 2.26 and impose the regularity assumptions on the data Φ, Ψ ∈ L 2 ̺ (R, L 2 (Ω) 3 ).Let P nl be a nonlinearity for which are uniformly Lipschitz-continuous and satisfy ∂ j t P nl (0) = 0 for j ∈ {1, 2, 3}, and Then, for ̺ 1 ≥ ̺ 0 large enough, the system (2.41) Then by Theorem 2.26, and from (2.34) we have the estimate with C independent of ̺.By assumption on L ̺ , the product CL ̺ is smaller than 1 for large ̺, thus, denoting by π 1 , π 2 : The solution to (2.40) is given by the unique fixed point The following is a variant of Corollary 2.27 in the spirit of Proposition 2.12.
Corollary 2.28 (H 2 for the nonlinear Maxwell system II).Suppose that the domain Ω and the linear material law ǫ satisfy the conditions of Theorem 2.26.Impose the regularity assumptions on the data Φ, Ψ ∈ L 2 ̺ (R, L 2 (Ω) 3 ) and suppose further that Φ, Ψ are supported in [0, ∞).Let P nl be a causal nonlinearity for which satisfy ∂ j t P nl (0) = 0 as well as the local Lipschitz estimate with C, α > 0.Then, for ̺ 1 ≥ ̺ 0 large enough, the system (2.40) admits a unique solution (E, H) with Proof.The proof follows a similar idea to that of Proposition 2.12.Note that since supp Φ, supp , where T ̺ is defined in (2.41) and π 1 denotes the projection π 1 (E, H) = E.
Then by (2.34) we have with c, d > 0. We conclude that F ̺ is a contraction on a closed ball B r ⊂ L 2 ̺ (R, H 2 ) of sufficiently small radius r > 0, if ̺ > 0 is sufficiently large.
Example 2.29.We adapt the quadratic model in Example 2.15 to the H 2 -setting.For simplicity we assume that κ ∈ C 3 ((0, ∞) 2 ) 3×3 is compactly supported, hence each time derivative of P (2) (E) is again of the form above: As for the spatial nonlinearity, by the algebra property of H 2 , each bilinear map q : R 3 × R 3 → R 3 extends to a bilinear and bounded map q : H 2 × H 2 → H 2 , and we have This can be generalized to x-dependent bilinear maps like with q as above and M of class H 2 on each Ω i (i ∈ {1, 2}), as well as to q like in Example 2.15, i.e., with the same regularity assumption on Λ = (Λ ijk ) i,j,k∈{1,2,3} .Now using a smooth cutoff in time, i.e.,

Exponential stability of the Maxwell system on a bounded domain
In [Tro18], exponential stability for linear equations was investigated.Assuming (3.1) is well-posed in the range of spaces L 2 ̺ (R, X), ̺ > ̺ 0 , the equation is said to be exponentially stable with decay rate ν 0 > 0, if the implication holds.If s 0 (M, A) denotes the infimum over ̺ ∈ R such that the equation is well-posed in L 2 ̺ (R, X), then exponential stability with rate ν 0 is essentially equivalent (under some natural assumptions on the domain of M , [Tro18, Theorem 2.1.3])to s 0 (M, A) ≤ −ν 0 .This implies in particular the continuous dependence on the data in the There are two abstract criteria to ensure exponential stability of the linear system (2.4).The first requires strict and uniform accretivity of zM (z).(See also [STW22,Chapter 11].)Recall that a linear operator A : dom(A) ⊂ X → X on a Hilbert space X is called m-accretive, if Re Ax, x ≥ 0 for all x ∈ dom(A) and A + λ is onto for all λ ∈ C Re>0 .In particular, every skew-selfadjoint operator is m-accretive.

The latter condition can be relaxed if one assumes that
Proposition 3.2 ([Tro18, Theorem 2.1.6]).Let A m-accretive and boundedly invertible, let ν 0 > 0 be such that C Re>−ν 0 dom(M ) is discrete, and assume that for some δ > 0 Then the linear problem (3.1) is well-posed and exponentially stable with decay rate ν 0 .
These criteria can be applied to second-order equations of the form ) analytic and bounded.The strategy relies on the substitution which converts (3.3) into the first-order system where The accretivity properties of zM (z) are inherited by zM d (z), provided d > 0 is sufficiently small, and Proposition 3.1 resp.Proposition 3.2 are applicable, see [Tro18, §2.2], [STW22,Section 11.4].We mention explicitly the following result.
Theorem 3.3.Let M be given by M (z) := M 0 (z)+z −1 M 1 (z), where M 0 , M 1 : dom(M ) ⊆ C → B(X 0 ) are analytic and bounded, C Re>−ν dom(M ) is discrete for some ν > 0, and is met, then there exist d 0 , ν 0 > 0 such that system (3.4) with d = d 0 is exponentially stable with decay rate ν 0 .Remark 3.4.We note that v, q Remark 3.5.We will subsequently use Theorem 3.3 to formulate stability results for Maxwell's equations.As before, the presence of an interface will play no role at first, and only be of importance in the later Section 3.2.1 when higher spacial regularity is involved.

Exponential stability of the E-field via the second-order formulation
A first observation is that the material law M (∂ t ) = ǫ(∂t) 0 0 µ associated with the linearized system does not fulfill any of the strict accretivity conditions above, since whenever Re z = 0, independently of ǫ.
Recall that well-posedness of (3.5) means that holds as an equation in L 2 ̺ (R, H) 2 for ̺ large enough.If the data satisfy the regularity assumption Φ, Ψ ∈ H 1 ̺ (R, H), then by Theorem 2.5 (i) we can drop the closure bar and (3.5) itself holds in L 2 ̺ (R, H) 2 .Applying ∂ t to the first line, we can insert the second line via We will impose the following conditions on the permittivity.
Remark 3.6.An example of a physically relevant ǫ(∂ t ) compatible with (M2) and (M3) is the one given by the Drude-Lorentz model, see Appendix A.
Conditions (M2) and (M3) are sufficient to obtain a notion of exponential stability for the secondorder system (3.6) on a bounded domain Ω, which is similar to that in (3.2).Note that (3.6) is not yet of the form (3.3), because curl, curl 0 are not invertible.We need two preparatory results, in which we follow a strategy akin to that in [TW14].
Lemma 3.7.Let K 0 , K 1 be Hilbert spaces and C : dom(C) ⊆ K 0 → K 1 a densely defined and closed operator with closed range.Let µ ∈ B(K 1 ) be selfadjoint and uniformly positive definite.Denote by ι k : ker(C) ⊥ ֒→ K 0 the canonical embedding.Then is selfadjoint, continuously invertible, and nonnegative.
Proof.Let ι r : ran(C) ֒→ K 1 be the canonical embedding.Then ι * r Cι k is injective, surjective, and closed, thus, by the closed graph theorem, continuously invertible.Note also that since ran(C) is closed, ran(C * ) is closed (see [Bre11,Theorem 2.19]).Now it is not difficult to see that Thus we obtain continuous invertibility and selfadjointness.The nonnegativity follows from the nonnegativity of µ.
Remark 3.8.In the situation of the previous lemma we have for some continuously invertible operator B : dom(B) ⊆ ker(C) ⊥ → ker(C) ⊥ .Indeed, this is a direct consequence of the lemma in conjunction with the spectral theorem for unbounded selfadjoint operators.Lemma 3.9.Let H be a Hilbert space and H 0 ⊂ H a closed subspace.Denote by ι 0 : H 0 ֒→ H and ι 1 : H ⊥ 0 ֒→ H the canonical embeddings.Let T ∈ B(H ) be a bounded linear operator and define If Re T ≥ d for some d > 0, then also In particular, T 11 is invertible.As an operator on H 0 ⊕ H ⊥ 0 we can identify and setting we have the factorization In the following, we apply this result to H = H = L 2 (Ω) 3 , where Ω is a bounded weak Lipschitz domain (a bounded domain with a local Lipschitz boundary), and H 0 = ran(curl).As a consequence of the Picard-Weber-Weck selection theorem (see Lemma B.2), H 0 is a closed subspace of H. Lemma 3.9 can be applied to the second-order formulation (3.6) if Φ, Ψ are regular enough, as we show next.Let Π 0 : H → ker(curl 0 ) denote the canonical projection.
Due to the regularity of the right-hand side we can drop the closure bar and obtain the second-order system (3.6) for E, thus The aim is to show that To this end, we set Denote by ι 0 : H 0 ֒→ H, ι 1 : H ⊥ 0 ֒→ H the canonical embeddings and define (note that ι * 1 = Π 0 , thus h = ∂ −2 t g 1 ).Then (3.8) can be written equivalently in the form Next we observe that, by assumption (M3), is a bounded linear operator for ν < ν 1 .Indeed, since ǫ is bounded on C Re>−ν 1 , so is ǫ 01 , and with Lemma 3.9 also ǫ 11 (•) −1 .Hence, is bounded on C Re>−ν 1 as desired.Now apply this operator to the second equation in (3.9) and subtract the result from the first equation to obtain Regarding the equation for E 0 , we have that ι * 0 curl µ −1 curl 0 ι 0 is selfadjoint, continuously invertible, and nonnegative on H 0 .In view of Remark 3.8 there exists a densely defined and boundedly invertible operator C such that We verify that ǫ satisfies the accretivity conditions of Theorem 3.3.Indeed, with Lemma 3.9 we have Re zǫ(z) ≥ c whenever Re zǫ(z) ≥ c, thus ǫ fulfills (M2).Furthermore, we find that ǫ where, due to (M3), M 0 (•), M 1 (•) are uniformly bounded on C Re>−ν 1 and lim z→0 M 1 (z) = 0. Consequently, there exists ν 0 ∈ (0, ν 1 ] such that Therefore, if we take ν < ν 0 , then with some K > 0 independent of g and E. Turning to the equation for E 1 in (3.10), we infer Since by assumption The assertion now follows due to the density of Since the proof relies on an application of Theorem 3.3, in view of Remark 3.4 in fact a stronger result is implied by Theorem 3.10.Corollary 3.11.Under the assumptions of Theorem 3.10, and with the notation in the proof, the following holds: Remark 3.12.Obtaining exponential decay for the H-field through a second-order system is not completely analogous, the problem being that the material law ǫ(∂ t ) is non-instantaneous and, due to jumps, in general does not commute with curl.We address this issue in Section 3.2.

Exponential stability of the nonlinear second-order system
We want to use the results in the linearized case to obtain exponential stability for the nonlinear system (2.8).However, the fixed-point argument we employed previously to obtain well-posedness in L 2 ̺ , for large ̺ > 0, cannot be repeated in L 2 −ν (for ν < ν 0 ).This can be seen in (2.14), where a large ̺ > 0 is needed to ensure the contraction property on L 2 ̺ .As we show next, this problem can be avoided if we restrict ourselves to small solutions.

Exponential stability of the first order system
To obtain exponential decay of the H-field, we now consider exponential stability for the first-order system.In fact, the next result is a refinement of Theorem 3.10, and also relies on the second-order formulation (3.6).Denote by Π 0 : H → ker(curl 0 ) and Π 1 : H → ker(curl) the canonical projections.
Theorem 3.15 (Exponential stability of the linear Maxwell equations).Let Ω be a bounded weak Lipschitz domain, let ǫ(∂ t ) = ǫ 0 +χ(∂ t ) be a linear material law satisfying (M2) and (M3).Furthermore, let µ be selfadjoint and uniformly positive definite.Then there exists ν > 0 such that if ν < ν and then the solution E, H of (3.13) Proof.We use projections similar to those in the proof of Theorem 3.10, now for the full system (3.12).Consider and observe that each of the maps τ * 1 curl 0 , ι * 1 curl, curl τ 1 , curl 0 ι 1 is zero on its corresponding domain.Now set for i, j ∈ {0, 1} Then (3.12) can be written in the form Here we note that the operator B := τ * 0 curl 0 ι 0 : dom(B) ⊆ H 0 → H 1 is injective, surjective, and closed, and thus continuously invertible by the closed graph theorem; the same is true for its adjoint B * = ι * 0 curl τ 0 .Solving the second and fourth equation in (3.14) for we obtain the system 11 , μ are bounded in L 2 −ν for ν < ν 1 by assumptions on ǫ, µ and Lemma 3.9.Moreover, Φ −ν,0 Φ −ν,0 and Ψ −ν,0 Ψ −ν,0 .The assertions for E follow from Theorem 3.10 (the notation E 0 , E 1 is the same as in the proof), since the data fulfills the necessary conditions.By Corollary 3.11 we even have for C given by B * µ −1 B = C * C, and the latter also implies BE 0 ∈ L 2 −ν (R, H 1 ) since µ is selfadjoint and boundedly invertible.Moreover, the estimates hold.Using (3.15) now provides and overall (3.17) To obtain the assertions for H, we first take the first line in (3.16) and have , by boundedness of (B * ) −1 , with and the second line gives The corresponding statement for H 1 follows using (3.15), since We obtain (3.18) Now (3.17) and (3.18) imply (3.13).
3.2.1.Exponential stability in H 2 for materials constant on each Ω i We now study exponential stability in the context of higher spatial regularity.This is motivated by the fact that the spaces H k , k ≥ 2, enjoy the algebra property which is useful when polynomial nonlinearities are present.We will combine the assertion of Theorem 3.15 with the results in Section 2.5.Thus, we will only consider material laws that satisfy (M1).Let us introduce a more succinct notation for admissible data Φ, Ψ .
By assumption on N and since Φ ∈ V −ν , Ψ ∈ W −ν , Theorem 3.15 applies (with Φ replaced by Φ − N ) and yields E, ∂ t E, H, ∂ t H ∈ L 2 −ν (R, H).Moreover, the additional assumptions on the regularity in time of the data imply that Theorem 3.15 can be applied once more, with From (3.13) we obtain the following estimates Collecting all the above terms, we observe that Next, using the identities in the Maxwell system and Propositions 2.21 and 2.23, we have Together with (3.21), we see that all terms can be controlled by the right-hand side of (3.19).The claim now follows with (3.20).
Now we can formulate an exponential stability result in the nonlinear H 2 -setting.We employ a fixed-point argument in L 2 −ν (R, H 2 ).
Theorem 3.18 (Small solutions of the nonlinear system in H 2 ).Let Ω, ǫ, µ, ν be given as in Theorem 3.17 and impose the regularity conditions of Theorem 2.26 on the data Φ, Ψ .Suppose now that the map P nl is such that for 0 < ν < ν and for all j ∈ {0, 1, 2, 3}, are causal and satisfy the local Lipschitz estimates for some α, C > 0 and all u, v ∈ L 2 −ν (R, H 2 ).Then there exist ε 0 , c 0 > 0 such that for all ε ∈ (0, ε 0 ], ν < ν, the following holds: Proof.Define the solution map S by then by Theorem 3.17, for ν < ν and u, v Thus, with the projection π 1 (E, H) := E, the map π 1 S is a contraction on the ball The above result applies in particular to the fully nonlocal nonlinearities in Example 3.14 (or the local H 2 -version in Example 2.29) after imposing suitable regularity on the spatial kernel Λ as well as the temporal kernel κ.We provide here an additional class of admissible nonlinearities.
Then, by an analogous estimate to that in (2.13) we can show that F fulfills making repeated use of the Cauchy-Schwarz inequality and of u 0,2 ≤ u −ν,2 for supp u ⊆ [0, ∞).It appears that this estimate cannot be generalized to the case when with α > 1, like the k-linear (k ≥ 2) map q considered at the end of Example 2.15.

Materials with conductivity
The established stability results are based on Theorem 3.3.Alternatively, one can impose a stricter material damping in the form of the following criterion.Then there exists d > 0 such that the evolutionary problem (3.4) is well-posed and exponentially stable.
Consider the following linear Maxwell system where the conductivity σ ∈ B(H) is selfadjoint and uniformly positive definite with σ ≥ c σ > 0. For ǫ(∂ t ) we assume that the following conditions hold for some δ > 0.

A. A class of permittivities of Drude-Lorentz-type and applications
Here we check the accretivity conditions in Picard's theorem and conditions (M2) and (M3) for a particular class of scalar permittivities.

Analytic correction to the material law
We have seen that the linear Maxwell system with the "standard" Drude-Lorentz susceptibility does not fulfill the criteria for exponential stability.However, these are still global criteria that assure exponential decay of the solution for rather general right-hand sides Φ, Ψ ∈ L 2 −ν .If the Fourier-Laplace transform of the right-hand side is localized around a certain "frequency" z = z 0 ∈ C, we can argue that the exact form of the solution operator plays no role for large |z|.

Figure 2 :
Figure 2: Schematic for the conversion of the Cauchy problem to an evolutionary equation.
.33)Remark 2.25.The term N acts purely as a placeholder for the nonlinearity ∂ t P nl (E), which is added in Corollary 2.27.The jump of the material law at the interface leads to a jump of the electric field E in normal direction, hence, if for D(E) = ǫ(∂ t )E + P nl (E) we impose div D = 0 and hence [n • D] Γ = 0, then, in general, the normal components of P nl (E) as well as N = ∂ t P nl (E) are also discontinuous at the interface.
In view of Section 2.4 we can assume that supp E, supp H ⊆ [0, ∞).In particular, estimate (3.11) holds for F = Π 0 P nl and F = ∂ 2 t P nl .