J. Evol. Equ. © 2018 Springer International Publishing AG, part of Springer Nature https://doi.org/10.1007/s00028-018-0448-0
Journal of Evolution Equations
Strong solutions to a nonlinear stochastic Maxwell equation with a retarded material law Luca Hornung
Abstract. We study the Cauchy problem for a semilinear stochastic Maxwell equation with Kerr-type nonlinearity and a retarded material law. We show existence and uniqueness of strong solutions using a refined Faedo–Galerkin method and spectral multiplier theorems for the Hodge–Laplacian. We also make use of a rescaling transformation that reduces the problem to an equation with additive noise to get an appropriate a priori estimate for the solution.
1. Introduction In this article, we consider the semilinear stochastic Maxwell equation du(t) = Mu(t) − |u(t)|q u(t) + (G ∗ u)(t) + J (t) dt +[b(t) + B(t, u(t))]dW (t), (1.1) u(0) = u 0
in L 2 (D)6 = L 2 (D)3 × L 2 (D)3 driven by a cylindrical Brownian motion W (t) with the retarded material law t (G ∗ u)(t) = G(t − s)u(s) ds 0
and the perfect conductor boundary condition u 1 × ν = 0 on ∂ D. Here, the Maxwell operator is given by curl u 2 u1 = M u2 − curl u 1 for 3d vector fields u 1 and u 2 . We allow D to be a bounded domain or D might also be the full-space R3 (in this case the boundary condition drops). Mathematics Subject Classification: 35Q61, 35R60, 60H15, 34L05, 32A70, 60H30, 76M35 Keywords: Stochastic Maxwell equations, Kerr-type nonlinearity, Retarded material law, Monotone coefficients, Weak solutions, Strong solutions, Generalised Gaussian bounds, Spectral multiplier theorems, Hodge–Laplacian, Rescaling transformation.
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This equation is a model for a stochastic electromagnetic system in weakly nonlinear chiral media and was derived in [28] in Chapter 2. It originally comes from the deterministic Maxwell system ∂t (Lu(t)) = Mu(t) + J (t), t ∈ [0, T ] u(0) = u 0 with constitutive relation t Lu(t, x) = κ(x)u(t, x) + K 1 (t − s, x)u(s, x) ds 0 t + K 2 (t − s, x)|u(s, x)|q u(s, x) ds . 0
This material law consists of an instantaneous part κu with a hermitian, uniformly positive definite and uniformly bounded matrix κ : D → C6×6 , a linear dispersive part K 1 ∗ u and of a nonlinear dispersive part K 2 ∗ |u|q u. This power-type nonlinearity is motivated by the Kerr–Debye model. Note that in applications, one would only take the nonlinearity |u 1 |q u 1 or |u 2 |q u 2 to model a nonlinear polarisation or magnetisation. We take the two quantities together to study both phenomena at once. Using the product rule, we end up with κu = Mu − K 1 (0)u − K 2 (0)|u|q u − (∂t K 1 ) ∗ u − (∂t K 2 ) ∗ |u|q u + J u(0) = u 0 At this point, we introduce additional simplifications. We assume that the term (∂t K 2 )∗ |u|q u can be neglected. This is typical for a weakly nonlinear medium since one assumes that both the dispersion and the nonlinear effects are weak, so that the combination then satisfies (∂t K 2 ) ∗ |u|q u << K 2 (0)|u|q u. Usually one demands K 1 (0) : D → C6×6 to be bounded and positive semi-definite and K 2 (0) : D → C6×6 to be bounded and uniformly positive definite. But for sake of simplicity, we choose K 1 (0) ≡ 0 and K 2 (0) ≡ I. We just note that the results are unchanged by this simplification and the proofs could be adjusted easily. Moreover, we choose κ = I . We must admit that this simplification is necessary at this point since our methods cannot deal with coefficients so far. The problems one has to overcome if κ = I are discussed in Sect. 6 in detail. Setting G := −∂t K 1 , we get a deterministic version of (1.1). In many applications, there is some uncertainty concerning the external sources or the precise behaviour of the medium itself. In these cases, it is useful to model u as random variables on a probability space and to add a stochastic noise perturbation. Here one distinguishes between the additive noise b perturbing J and the multiplicative noise B(u) perturbing the medium. A linear stochastic version of (1.1) was already discussed in [28, chapter 12]. Moreover, in [8], the authors show that typical conservation laws of linear electromagnetic system are preserved under additive noise perturbation. The authors in [16] also treat linear stochastic Maxwell equations
Strong solutions to a nonlinear stochastic Maxwell equation
numerically with energy-conserving methods. More about the application of random media in scattering, wave propagation and in the theory of composites can be found in [2,12,13,23]. However, as far as we know, there are no known results about a nonlinear stochastic Maxwell equation. One reason might be that in the absence of Strichartz estimates for (et M )t∈R , even local solvability is a tricky issue. Moreover, there is no compact embedding D(M) → L p that helps to control the nonlinearity. Even the deterministic version of (1.1) has not been treated rigorously so far. In [28], the authors profess to prove well-posedness, but their argument ignores some severe complications. Since they claim to have better deterministic results than ours, we discuss their approach in Sect. 6 in detail. Now, we briefly sketch our strategy. At first, we show in Sect. 4 that (1.1) has a unique weak solution u ∈ L 2 (; C(0, T ; L 2 (D)))6 ∩ L q+2 ( × [0, T ] × D)6 .
(1.2)
This is done in two steps. First, we use a version of the Galerkin method from Röcker and Prévot (see [27]) to solve (1.1) in the special case G ≡ 0 and make use of the monotone structure of our nonlinearity. As this is approach is well known, we just discuss the different steps and concentrate on how to deal with the additional term Mu, despite the fact that u ∈ / D(M). Afterwards, we inflict the retarded material law with Banach’s fixed point theorem. The proof of the existence and uniqueness of a strong solution that additionally satisfies q+2
Mu ∈ L 2 (; L ∞ (0, T ; L 2 (D)))6 + L q+1 ( × [0, T ] × D)6 is more tricky. Again, we start with G ≡ 0 and we add a nontrivial G at the very end. In a deterministic setting, one would try to estimate u (t) L 2 (D)6 and then use (1.2) to control Mu. However, solutions of stochastic differential equations are not differentiable in time. The first idea was to derive an estimate for Mu(t)−|u(t)|q u(t)+ J (t) 2L 2 (D)6 with Gronwall’s Lemma, but we failed since the Itô formula for this quantity contains the term
Dvv (|v|q v)(u(t)) B(u(t)), B(u(t) 2L 2 (D)6 , we could not estimate properly. Hence, we choose the noise β j (t) and use the rescaling transform y(t) = u(t)e−i
N j=1
N j=1
b j (t) + i B j u(t) d
B j β j (t)
to get rid of the multiplicative noise in the same way as Barbu and Röckner in [4,5] (see also [6,7]). The difference to our approach is that the authors have natural a priori estimates before transforming the equation and they solely transform to solve the new
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equation with purely deterministic techniques. Moreover, they only use multiplicative noise. We use the transformation to get better a priori estimates and we consider an equation that also has additive noise. The arising equation has the form
N dy(t) = [M y(t) − |y(t)|q y(t) + A(t)y(t) + J (t)] dt + i=1 bi (t) dβi (t), (TSEE) u(0) = u 0 , with a nonautonomous operator A(t) having random coefficients. We truncate (TSEE) with a refined Faedo–Galerkin approach, i.e. we solve ⎧ q ⎪ (t)] dt ⎪ n M yn (t) − Pn |yn (t)| yn (t) + Pn A(t)yn (t) + Pn J ⎨dyn (t) = [P
N + i=1 Sn−1 bi (t) dβi (t), ⎪ ⎪ ⎩ y (0) = S u . n
n−1 0
Here, Pn = 1[0,2n ] (− H ) and Sn = ψ(−2−n H ) for some ψ ∈ Cc∞ (D) with supp ψ ⊂ [0, 2] and ψ = 1 on [0, 1] are spectral multipliers with respect to the Hodge–Laplacian H on L p that is the component-wise Laplacian with domain
1, p
(u 1 , u 2 ) ∈ L p (D)6 : curl u 1 , curl u 2 , curl curl u 1 , curl curl u 2 ∈ L p (D)3 , div u 1 ∈ W0 (D), div u 2 ∈ W 1, p (D), u 1 × ν = 0, u 2 · ν = 0, curl u 2 × ν = 0 on ∂ D .
We show that Pn , Sn are self-adjoint on L 2 (D)6 and commute with both H and M. Further, we have Sn u L p (D)6 ≤ C u L p (D)6 with a constant C > 0 depending on p, but not on u and n. Note that such an estimate is not available for Pn in a general situation. This remarkable uniform L p -boundedness is a consequence of [20], together with generalised Gaussian bounds for the Hodge–Laplacian (see [19,25]). The deep connection between H and M is a consequence of the formula − H = curl curl − grad div, which implies H = M 2 in the range of the Helmholtz projection PH and M 2 = 0 in the range of (I − PH ). This interplay will be examined in detail in Sect. 3. Pn and Sn reduce the problem to an ordinary stochastic differential equation that can be solved easily. Afterwards, we estimate
Pn M yn (t) − Pn |yn (t)|q yn (t) + Pn A(t)yn (t) + Pn J (t) 2L 2 (D)6 using Itô’s formula, the monotone structure of the equation and the properties of Pn , Sn . This yields the desired estimate for M yn uniformly in n. Finally, we pass to the limit again using the monotonicity of the nonlinearity and undo the transformation. The idea to use spectral multiplier results in such a way was firstly used by Brzezniak et al. [3]. In Sect. 6, we explain how the result changes if one strengthens some of the assumptions and we discuss interesting special cases, such as the deterministic version of (1.1), b ≡ 0 or a constant B. Moreover, we sketch a program to extend this approach to non-constant coefficients κ = I .
Strong solutions to a nonlinear stochastic Maxwell equation
2. Preliminaries The purpose of this section is to provide a short overview over the basic tools used in this paper. For most of the proofs and further details, we give references to the literature. Throughout this paper, let (, F, F = (Ft )t≥0 , P) be a filtered probability space that satisfies the usual conditions, i.e., F0 contains all P-null sets and the filtration is right-continuous. Moreover, given normed spaces X and Y, B(X, Y ) denotes the set of all linear and bounded operators from X to Y and if X = Y we shortly write B(X ). Further, we write C(a, b; X ) for the space of uniformly continuous functions on [a, b] with values in X equipped with its usual norm and L 2 (H1 , H2 ) for the space of Hilbert– Schmidt operators between the complex Hilbert spaces H1 and H2 . Throughout this article, D ⊂ R3 will either be a bounded C 1 -domain or D = R3 . If we evaluate a function on ∂ D, this always corresponds to the first case and has no meaning in the second case. Last but not least, we denote with L p (D) the space of all complex valued functions on D, whose p-th moment is integrable with the obvious variation in the case p = ∞. 2.1. The operators curl and div First, we give a short introduction into vector calculus. To motivate the definition of functions with vanishing tangential component or normal component on the boundary, we make the following calculation with smooth functions f, g : D → R3 . Using vector calculus and the Divergence theorem, we obtain f · (g × ν) dσ = ν · ( f × g) dσ = div( f × g)(x) dx ∂D ∂D D curl g(x) · f (x) dx − g(x) · curl f (x) dx . = D
D
Similarly, we get y(z · ν) dσ = div(y · z)(x) dx ∂D D ∇ y(x) · z(x) dx + y(x) div z(x) dx . = D
D
R3 .
Hence, we can define vanishing tangential for smooth y : D → R and z : D → and normal components on the boundary in a natural way. DEFINITION 2.1. Let D ⊂ R3 be bounded C 1 -domain with boundary ∂ D and p ∈ [1, ∞). (a) Let g ∈ L p (D)3 with curl g ∈ L p (D)3 . We say g × ν = 0 on ∂ D, if curl φ(x) · g(x) dx = φ(x) · curl g(x) dx D
for every φ ∈
C ∞ (D)3 .
D
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(b) Let z ∈ L p (D)3 with div z ∈ L p (D). We say z · ν = 0 on ∂ D, if ∇φ(x) · z(x) dx = − φ(x) div z(x) dx D
D
for every φ ∈ C ∞ (D). Next, we introduce the subspaces of L 2 (D)3 associated with curl and div . DEFINITION 2.2. We set (a) H (curl)(D) := u∈ L 2 (D)3 : curl u ∈ L 2 (D)3 . (b) H (curl, 0)(D) := u ∈ H (curl)(D) : u × ν = 0 on ∂ D . (c) H (div)(D) := u ∈ L 2 (D)3 : div u ∈ L 2 (D) . (d) H (div, 0)(D) := u ∈ H (div)(D) : u · ν = 0 on ∂ D . We define the Maxwell operator M with perfect conductor boundary condition by curl u 2 u1 = M u2 − curl u 1 on the domain D(M) = H (curl, 0)(D) × H (curl)(D). PROPOSITION 2.3. The Maxwell operator M is skew-adjoint on L 2 (D)6 , i.e., we have M y(x) · z(x) dx = − y(x) · M z(x) dx D
D
for every y, z ∈ D(M) and D(M) = D(M ∗ ). Proof. This result is well known. See, e.g., [15, section 3].
The next technical lemma will be needed later on. We state it for functions in the sum of L p -spaces for technical reasons. LEMMA 2.4. Let D be a bounded C 1 - domain or D = R3 , y ∈ L 2 (D)6 and p ∈ [1, ∞). If there exists z ∈ L 2 (D)6 + L p (D)6 , such that y(x) · Mφ(x) dx = − z(x) · φ(x) dx (2.1) D
D p p−1
for every φ ∈ C ∞ (D)6 ∩ L 2 (D)6 ∩ L (D)6 with Mφ ∈ L 2 (D)6 and φ1 × ν = 0 on ∂ D, we have M y = z in the sense of distributions and y1 × ν = 0 on ∂ D. Proof. By inserting φ = (φ1 , 0) and φ = (0, φ2 ) into (2.1), we derive y2 (x) · curl φ1 (x) dx = z 1 (x) · φ1 (x) dx D D y1 (x) · curl φ2 (x) dx = − z 2 (x) · φ2 (x) dx D
D
Strong solutions to a nonlinear stochastic Maxwell equation
for any smooth φ1 with φ1 × ν = 0 on ∂ D and for any smooth φ2 . Inserting φ1 , φ2 ∈ Cc∞ (D)3 yields curl y2 = z 1 and curl y1 = −z 2 in the sense of distributions, i.e., M y = z in the sense of distributions. If D = Rd , we have to show the claimed boundary condition. The second identity implies y1 (x) · curl ψ(x) dx + curl y1 (x) · ψ(x) dx = 0 D
D
for every ψ ∈ C ∞ (D)3 and hence, y1 ×ν = 0 on ∂ D in the sense of Definition 2.1. 2.2. The power nonlinearity |u|q u In this subsection, we mention the basic properties of nonlinearity u → F(u) = q+2
|u|q u as a mapping from L q+2 (D)6 to L q+1 (D)6 with q > 0. We start with its monotonicity. LEMMA 2.5. F satisfies the estimate q+2 ReF(v)(x) − F(u)(x), u(x) − v(x)C6 dx ≤ −C u − v L q+2 (D)6
(2.2)
D
for some C > 0 and for all u, v ∈ L q+2 (D)6 . Proof. Clearly, F(u)
q+2 L q+1 (D)6
= u L q+2 (D)6 and therefore F has the claimed
mapping properties. The estimate (2.2) is a direct consequence of Lemma 4.4 in [9]. Since we often use Itô’s formula, we need to know the differentiability properties of F. q+2
LEMMA 2.6. The nonlinearity F : L q+2 (D)6 → L q+1 (D)6 , u → |u|q u is real continuously Fréchet differentiable with ReF (u)v, v L 2 (D)6 ≥ 0 and |F (u)v(x)| |u(x)|q |v(x)| for all u, v ∈ L q+2 (D)6 and x ∈ D. In particular, it is locally Lipschitz continuous, i.e.
F(u) − F(v)
q+2 L q+1 (D)6
q q u L q+2 (D)6 + v L q+2 (D)6 u − v L q+2 (D)6 .
Moreover, if q ∈ (1, ∞), it is twice real continuously differentiable with F (u)(v, v)(x) |u(x)|q−1 |v(x)|2 for all u, v ∈ L q+2 (D)6 and all x ∈ D.
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Proof. It is well known that F : L q+2 (D)6 → L q+1 (D)6 is real continuously Fréchet differentiable with F (u)v = q|u|q−2 Reu, vC6 u + |u|q v for every u, v ∈ L q+2 (D)6 (see, e.g., given [17, Corollary 9.3]). Consequently, we also have 2 q|u(x)|q−2 Reu(x), v(x)C6 + |u(x)|q |v(x)|2 dx ≥ 0. Re F (u)v, v L 2 (D)6 = D
Moreover, we estimate F (u)v(x) ≤ C|u(x)|q |v(x)| for some C > 0. For the second derivative, we start with a formal calculation for F and get F (u)(v, w) = q|u|q−2 (q − 2)|u|−2 Reu, w L 2 (D)6 Reu, v L 2 (D)6 u + Rew, v L 2 (D)6 u + Reu, w L 2 (D)6 v + Reu, v L 2 (D)6 w . q+2
For sake of readability, we do not rigorously show that F : L q+2 (D)6 → L q+1 (D)6 is twice Fréchet differentiable with this derivative. However, to give an impression how to prove this, we check that last term in F (u)v, namely q+2
u → [v → |u|q v] : L q+2 (D)6 → B(L q+2 (D), L q+1 (D)6 ), is Fréchet differentiable with derivative G(u)(v, w) = q|u|q−2 Reu, wC6 v. Let u, v, w ∈ L q+2 (D)6 with v, w = 0. Then Hölder’s inequality together with the mean value theorem yields q |u| v − |u + w|q v − G(u)(v, w) q+2 L q+1 (D)6 q q q−2 ≤ |u| − |u + w| − q|u| Reu, wC6 q+2
v L q+2 (D)6
L
0 1
|u + θ w|q−2 (u + θ w) − |u|q−2 u
≤
q
(D)6
Re|u + θ w|q−2 (u + θ w) − |u|q−2 u, wC6 dθ
1
0
q+2
L q−1 (D)6
0
|u + θ w|q−2 (u + θ w) − |u|q−2 u
1
q+2
q+2 q (D)6
v L q+2 (D)6
dθ w L q+2 (D)6 v L q+2 (D)6 .
Hence, we showed v → |u|q v − |u + w|q v − G(u)(v, w)
w −1 L q+2 (D)6
L
L q−1 (D)6
q+2
B(L q+2 (D)6 ,L q+1 (D)6 )
dθ
(2.3)
Strong solutions to a nonlinear stochastic Maxwell equation
for all u, w ∈ L q+2 (D)6 with w = 0. It remains to prove that this quantity tends to 0 as w → 0 in L q+2 (D)6 . Let (wn )n be a sequence in L q+2 (D)6 with wn → 0 as n → ∞ and let (wn k )k be an arbitrary subsequence. Hence, there exists another subsequence, still denoted with (wn k )k such that wn k → 0 almost everywhere for k → ∞ and such that |wn k | ≤ g for some g ∈ L q+2 (D)6 . We also have |u + θ wn k |q−2 (u + θ wn k ) − |u|q−2 u → 0 almost everywhere as k → ∞. Together with the bound |u + θ wn k |q−2 (u + θ wn k ) − |u|q−2 u ≤ |u|q−1 + |wn k |q−1 ≤ |u|q−1 + g q−1 , for θ ∈ [0, 1] and the fact that u ∈ L q+2 (D)6 , we get 1 |u + θ wn |q−2 (u + θ wn ) − |u|q−2 u k k 0
q+2
L q−1 (D)6
dθ → 0
as k → ∞. All in all, this shows that the left-hand side of (2.3) tends to 0 as w → 0 and we established the Fréchet differentiability of u → [v → |u|q v] with derivative G(u). The claimed estimate for F (u)(v, v)(x) is immediate. This closes the proof. 3. The Hodge–Laplacian on a bounded C 1 -domain and its spectral multipliers In this section, we provide the spectral theory basics for our well-posedness proofs. We discuss spectral multipliers of the Hodge–Laplacian H which is the componentwise Laplace operator on L p (D)6 with boundary conditions comparable to the boundary conditions contained in D(M 2 ). The method of finite-dimensional approximation with a sequence of orthogonal projections (Pn )n is well known in the literature about stochastic and deterministic partial differential equations. However, it turned out that we not only need Pn x → x in L 2 (D)6 for n → ∞ for all x ∈ L 2 (D)6 , but also a comparable convergence property in L p (D)6 , p = 2. A sequence of orthogonal projections on L 2 that also approximates the identity in L p can only be found in very special situations, e.g., the Fourier cut-off on the torus. Hence, we have to construct another sequence of operators (Sn )n that has the necessary convergence property in L p (D)6 , p ∈ (1, ∞), and that is not far away from being an orthogonal projection, i.e. R(Sn−1 ) ⊂ R(Pn ) ⊂ R(Sn ) and Pn Sn−1 = Sn−1 for all n ∈ N. In our construction, we make use of the spectral multiplier theorems of Kunstmann and Uhl (see [20]) that work on L p (D)6 . They require that the semigroup generated by H satisfies generalised Gaussian bounds. Here, we benefit from a work of Mitrea and Monniaux who already showed a version of these tricky estimates in [25]. At first, we precisely introduce H . We consider the bilinear form a(u, v) = (curl u)(x) · (curl v)(x) dx + (div u)(x)(div v)(x) dx D
D
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J. Evol. Equ.
with form domain D(a) either given by V (1) := W 2 (curl, 0)(D) ∩ W 2 (div)(D) or by V (2) := W 2 (curl)(D) ∩ W 2 (div, 0)(D) equipped with the norm
u 2V (i) := curl u 2L 2 (D) + div u 2L 2 (D) + u 2L 3 (D) for i = 1, 2. In both cases the form a is bilinear, symmetric and bounded. Moreover, a is coercive in the sense that a(u, u) = u 2V (i) − u 2L 2 (D) for all u ∈ V (i) , i = 1, 2. Setting D(A(1) ) = {u ∈ V (1) : curl curl u ∈ L 2 (D)3 , div u ∈ W01,2 (D)}, D(A(2) ) = {u ∈ V (2) : curl curl u ∈ L 2 (D)3 , curl u × ν = 0 on ∂ D, div u ∈ W 1,2 (D)}, it turns out that a with D(a) = V (1) is associated with the operator A(1) = curl curl − grad div = − on the domain D(A(1) ), whereas a with D(a) = V (2) is associated with the operator A(2) = curl curl − grad div = − on the domain D(A(2) ). To see this, use integration by parts for curl and div and exploit the respective boundary conditions. In a more general setting, this can be found in [25, (3.17) and (3.18)]. By the coercivity of the corresponding forms, the operators I + A(i) , i = 1, 2, are strictly positive. Moreover, the symmetry implies that they are self-adjoint on L 2 (D)3 (see, e.g., [26, Proposition 1.24.]). Since the embeddings V (i) → L 2 (D)3 are compact (see [1, Theorem 2.8]), the embeddings D(A(i) ) → L 2 (D)3 are also compact for i = 1, 2. To simplify the notation in what follows, we combine A(1) and A(2) to a self-adjoint operator − H (u 1 , u 2 ) := (A(1) u 1 , A(2) u 2 ) for (u 1 , u 2 ) ∈ D(A(1) ) × D(A(2) ) =: D( H ). In particular, the embedding D( H ) → L 2 (D)6 is compact and I − H is positive. Hence, there exists an orthonormal basis of eigenvectors (h j ) j∈N to the positive eigenvalues (λ j ) j∈N of I − H with λ j → ∞ for j → ∞. The next proposition shows that the semigroups generated by −A(i) and H satisfy generalised Gaussian estimates. We add an additional spectral shift since some of the theorems we apply in what follows require strictly positive operators. PROPOSITION 3.1. The semigroups generated by −(I + A(1) ), −(I + A(2) ) and −I + H satisfy generalised Gaussian (2, q) estimates for every q ∈ [2, ∞), i.e. for every q ∈ [2, ∞) there exist C, b > 0 such that
1
1
B(x,t
1 2)
B(x,t
1 2)
e−t (I +A
(i) )
1
e−t (I − H ) 1
B(L 2 (D)3 ,L q (D)3 ) ≤ Ct
− 23 ( 21 − q1 ) − b|x−y| t
B(L 2 (D)6 ,L q (D)6 ) ≤ Ct
− 23 ( 21 − q1 ) − b|x−y| t
B(y,t
1 2)
B(y,t
1 2)
2
e
e
2
, i = 1, 2,
Strong solutions to a nonlinear stochastic Maxwell equation
for all t > 0 and all x, y ∈ D. Proof. In [19], the authors argue on p. 239 that the semigroups generated by −A(1) and −A(2) satisfy generalised Gaussian (2, q)-bounds for every q ∈ [2, q D ). Here, q D ∈ [2, ∞) denotes the supremum over all indexes p for which the boundary value problems ⎧ ⎪ ⎪ ⎨u = f in D, curl u, curl curl u ∈ L p (D)3 , div(u) ∈ W 1, p (D), ⎪ ⎪ ⎩u · ν = 0, curl(u) × ν = 0 on ∂ D
and
⎧ ⎪ ⎪ ⎨u = f in D,
1, p
curl u, curl curl u ∈ L p (D)3 , div(u) ∈ W0 (D), ⎪ ⎪ ⎩u × ν = 0 on ∂ D have a unique solution for given f ∈ L p (D)3 . This argument heavily makes use of iterative resolvent estimate for the Hodge–Laplacian (see [25, section 5 and 6]). By [24, Theorem 1.2 and 1.3], we know that q D = ∞ since D is a C 1 -domain in R3 . Gaussian estimates are preserved under negative spectral shifts in the generator of the semigroup. Hence, also the semigroups generated by −(I + A(1) ) and −(I + A(2) ) satisfy generalised Gaussian (2, q)-bounds. Last but not least, we remark that by e−t (I +A(1) ) e−t (I − H ) = e−t (I +A(2) ) these estimates also hold true for the semigroup generated by −(I − H ). For more details about these operators, we refer to [25], where they are discussed in a more general differential geometric setting. We define spectral multipliers with the functional calculus for self-adjoint operators on a Hilbert space that have a basis of eigenvectors. Let ∈ Cc∞ (R) with supp() ⊂
[ 21 , 2] and l∈Z (2−l x) = 1 for all x > 0. The operators Pn : L 2 (D)6 → L 2 (D)6 and Sn : L 2 (D)6 → L 2 (D)6 are defined by x, h k L 2 (D)6 h k , Pn (u)x = 1[0,2n ] (I − H )x = k:λk ≤2n
Sn (u)x =
n l=−∞
−l
(2 (I − H ))x =
∞ n
(2−l λk )x, h k L 2 (D)6 h k
k=1 l=−∞
for x ∈ L 2 (D)6 and n ∈ N. Note that the last sum is in fact finite since only finitely many eigenvalues of (I − H ) are smaller than 2n+1 and hence, (2−l λk ) = 0 for all but finitely many l ∈ Z and k ∈ N. The next proposition summarises the most important properties of Sn and Pn as operators on L 2 (D)6 .
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PROPOSITION 3.2. Pn and Sn satisfy the following properties. (i) Pn is a projection, i.e. we have Pn2 = Pn for all n ∈ N. (ii) The operators Pn , Sn are self-adjoint with Pn B(L 2 (D)6 ) = Sn B(L 2 (D)6 ) = 1 for every n ∈ N. (iii) Pn and Sm commute for every n, m ∈ N. (iv) The ranges of Pn and Sn are finite dimensional. Moreover, we have R(Pn ), R(Sn ) ⊂ D(M) for every n ∈ N. (v) We have R(Sn−1 ) ⊂ R(Pn ) ⊂ R(Sn ), Sn Pn = Pn and Pn Sn−1 = Sn−1 for every n ∈ N. (vi) We have limn→∞ Pn x = limn→∞ Sn x = x for every x ∈ L 2 (D)6 . Proof. For this proof, we just need the properties of the functional calculus for the self-adjoint and positive operator I − H on the Hilbert space L 2 (D)6 . It remains to show iv) and v).Pn and Sn have a finite-dimensional range, since only finitely many eigenvalues of I − H are smaller than 2n+1 . Moreover, let y = (y1 , y2 ) be in the range
n (2−l (I − H )). By functional calculus, of 1[0,2n ] (I − H ) and in the range of l=−∞ we have yi ∈ D( H ) and particularly yi ∈ V (i) for i = 1, 2. Thus, curl yi ∈ L 2 (D)3 for i = 1, 2 and y1 × ν = 0 on ∂ D, which shows (y1 , y2 ) ∈ D(M). Last but not least, we note that v) follows by n
(2−l ·) = 1(0,2n ) + ψ(2−n ·)1[2n ,2n+1 ) .
l=−∞
This closes the proof.
Moreover, the operators Sn have the following property that will be crucial in what follows. LEMMA 3.3. For every p ∈ (1, ∞), the operators Sn are bounded from L p (D)6 to p, but not on n ∈ N. Moreover, we have Sn f → f in L p (D)6 as n → ∞ for all f ∈ L p (D)6 . L p (D)6 with a bound depending on
Proof. The first statement follows from the spectral multiplier theorem 5.4 in [20] as a consequence of the generalised Gaussian bounds for the semigroup generated by I − H . One could also argue with the more general Theorem 7.1 in [21]. The claimed convergence property is then a special case from [22, Theorem 4.1]. To apply this theorem the 0-sectoriality of I − H and the boundedness of a Mikhlin functional calculus Mα in L p (D)6 for some α > 0 are needed. The first property is checked in [25, Theorem 6.1], whereas the second holds true with α > 4 by the generalised Gaussian bounds (see [22, Lemma 6.1, (3)]). Next, we introduce two different Helmholtz projections on L 2 (D)3 . The proof for the following statement is well known and can be found amongst others in [18, section 4.1.3].
Strong solutions to a nonlinear stochastic Maxwell equation
PROPOSITION 3.4. Let D ⊂ R3 be a bounded Lipschitz domain. Given u ∈ L 2 (D)3 , the following decompositions hold true. (1) There exists a unique p ∈ W01,2 (D) and u ∈ W 2 (div)(D) with div u = 0 such (1) 2 3 2 that u = u + ∇ p. The corresponding operator PH : L (D) → L (D)3 , u → u is an orthogonal projection. u ∈ W 2 (div, 0)(D) (2) There exists a unique p ∈ W 1,2 (D) with D p(x) dx = 0 and (2) with div u = 0 such that u = u + ∇ p. The corresponding operator PH : 2 3 2 3 u is an orthogonal projection. L (D) → L (D) , u → (1) (2) In particular, PH (u 1 , u 2 ) := PH u 1 , PH u 2 for u 1 , u 2 ∈ L 2 (D)3 defines an orthogonal projection on L 2 (D)6 . The Helmholtz projection PH is closely related to both M and H . For example, (i) due to div PH = 0, one calculates H PH =
(1)
(1)
− curl curl PH + grad div PH
− curl curl
PH(2)
+ grad div
PH(2)
=
(1)
− curl curl PH
− curl curl
PH(2)
= M 2 PH , (3.1)
which implies that M 2 = H on D(M) ∩ PH L 2 (D)6 ). We use this connection to show some powerful commutation identities. LEMMA 3.5. We have PH H = H PH on D( H ), M PH = PH M on D(M) and Pn M = M Pn , Sn M = M Sn on D(M). (i)
Proof. From [25, section 3] or from [19, Lemma 5.4] we know that PH A(i) = A(i) PH(i) for i = 1, 2. This shows PH H = H PH on D( H ) and by the properties of the functional calculus, we also have Sn PH = PH Sn and Pn PH = PH Pn . For the second statement, we first show that Mu = PH Mu for all u = (u 1 , u 2 ) ∈ (2) (1) D(M), i.e. we have to show PH curl u 1 = curl u 1 and PH curl u 2 = curl u 2 . Due to div curl u i = 0 for i = 1, 2, we just have to show curl u 1 · ν = 0 on ∂ D for u 1 ∈ W 2 (curl, 0)(D). The definition of u 1 × ν = 0 from Definition 2.1 a) together with curl ∇ = 0 and div curl = 0 yield
D
∇φ(x) · curl u 1 (x) dx =
D
curl ∇φ(x) · u 1 (x) dx = 0 =
D
φ(x) div curl u 1 (x) dx
for every φ ∈ C ∞ (D), which implies curl u 1 · ν = 0 according to Definition 2.1 b). As a consequence of curl ∇ = 0, we know M(I − PH ) = 0. All in all, we get M PH − PH M = M PH − M = M(PH − I ) = 0. Finally, by using M 2 = H on D(M) ∩ PH (L 2 (D)6 ) together with M = M PH , we get M Pn = M PH 1[0,2n ] (I − H ) = M1[0,2n ] (I − M 2 )PH = 1[0,2n ] (I − M 2 )PH M PH = 1[0,2n ] (I − H )M = Pn M
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on D(M). For Sn M = M Sn , one may argue analogously.
As a consequence, we get the following density relations.
COROLLARY 3.6. (1, ∞).
∞
n=1
R(Pn ) is dense in D(M) and in L p (D)6 for any p ∈
Proof. Let u ∈ D(M). Using the commutation property of Pn by Lemma 3.5 and (vi) from Proposition 3.2, we get n→∞
Mu − M Pn u L 2 (D)6 = Mu − Pn Mu L 2 (D)6 −−−→ 0. If on the other hand u ∈ L p (D)6 , we get Sn u → u in L p (D)6 as n → ∞ from Lemma 3.3. This together with Proposition 3.2 v) proves the claimed result. We also consider the nonlinear Maxwell equation with retarded material law (1.1) on the full space R3 , and hence, we need an analogue to the Pn and Sn in this different situation. However, in the absence of boundary conditions, things are well known and far more easy. We define Pn f = Sn f := F −1 ξ → 1[−2n ,2n ] (ξ1 )1[−2n ,2n ] (ξ2 )1[−2n ,2n ] (ξ3 ) fˆ(ξ ) for f ∈ L 2 (D)6 . As M is a differential operator, it commutes with this frequency cut-off. Moreover, Pn and Sn satisfy the same properties as in Proposition 3.2 expect iv). Further, as a consequence of the boundedness of the Hilbert transform on L p (R3 ), they are bounded on L p (R3 )6 . This finally results in an analogue to Lemma 3.3 and Corollary 3.6. For details, we refer to [14, Chapter 6.1.3]. We end this section with a lemma showing the mapping properties of the projection Pn as operator from L 2 (D)6 to L p (D)6 and as an operator from L 2 (R3 )6 to L p (R3 )6 . LEMMA 3.7. Let either D be a bounded C 1 -domain or D = R3 . For fixed n ∈ N, p ∈ [2, ∞) and q ∈ (1, 2] the operator Pn : L q (D)6 → L 2 (D)6 and Pn : L 2 (D)6 → L p (D)6 is linear and bounded with norm depending on n. Proof. This statement is trivial if D is bounded, since all norms on a finite-dimensional space are equivalent. In the other case, it is sufficient to show that Pn : L q (R3 )6 → L 2 (R3 )6 is bounded. The rest follows by duality. The Hölder and the Hausdorff–Young inequality yield
Pn f L 2 (R3 )6 = ξ → 1[−2n ,2n ] (ξ1 )1[−2n ,2n ] (ξ2 )1[−2n ,2n ] (ξ3 ) fˆ(ξ ) L 2 (R3 )6 n fˆ
q
L q−1 (R3 )6
≤ f L q (R3 )6 ,
which finishes the proof.
Strong solutions to a nonlinear stochastic Maxwell equation
4. Existence and uniqueness of a weak solution In this section, we will prove existence and uniqueness of a weak solution in the sense of partial differential equations of du(t) = Mu(t) − |u(t)|q u(t) + (G ∗ u)(t) + J (t) dt + B(t, u(t)) dW (t), (WSEE) u(0) = u 0 t for any q > 0. Here, we use (G ∗ u)(t) = 0 G(t − s)u(s) ds. For sake of readability, we sometimes write F(u) := |u|q u. Before we start, we explain our solution concept. As described in the preliminaries, we look for complex solutions in the complex space L 2 (D)6 . DEFINITION 4.1. We say that an adapted process u : × [0, T ] → L 2 (D)6 with u ∈ L 2 (; C(0, T ; L 2 (D)))6 ∩ L q+2 ( × [0, T ] × D)6 is a weak solution of (WSEE), if u(t) − u 0 , φ L 2 (D)6 =
t 0
− u(s), Mφ L 2 (D)6 + − |u|q u + J (s) + (G ∗ u)(s), φ L 2 (D)6 ds t
+
B(s, u(s)), φ
0
L 2 (D)6
dW (s)
holds almost surely for all t ∈ [0, T ] and for all φ ∈ D(M) ∩ L q+2 (D)6 . Moreover, we call a weak solution u unique if for any other weak solution v, there exists N ⊂ with P(N ) = 0, such that u(ω, t) = v(ω, t) for all ω ∈ \ N and all t ∈ [0, T ]. We make the following assumptions. [W1] Let D ⊂ R3 be a bounded C 1 -domain or D = R3 . [W2] The initial value u 0 : → L 2 (D)6 is strongly F0 -measurable. [W3] Let G : × [0, T ] → B(L 2 (D)6 ) such that x → G(t)x is for all x ∈ L 2 (D)6 strongly measurable and F-adapted. Moreover, we assume T ess supω∈
G(ω, t) B(L 2 (D)6 ) dt < ∞. 0
[W4] Let U be a separable Hilbert space and W a U -cylindrical Brownian motion. Moreover, let B : × [0, T ] × L 2 (D)6 → L 2 (U, L 2 (D)6 ) be strongly measurable such that ω → B(ω, t, u) is strongly Ft -measurable for almost all t ∈ [0, T ] and all u ∈ L 2 (D)6 . Furthermore, there exists a constant C > 0 such that B is of linear growth, i.e.,
B(t, u) L 2 (U ;L 2 (D)6 ) ≤ C 1 + u L 2 (D)6 and Lipschitz continuous, i.e.,
B(t, u) − B(t, v) L 2 (U ;L 2 (D)6 ) ≤ C u − v L 2 (D)6 almost surely for almost all t ∈ [0, T ] and all u, v ∈ L 2 (D)6 .
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[W5] J : × [0, T ] → L 2 (D)6 is strongly measurable, F-adapted and we assume J ∈ L 2 ( × [0, T ] × D)6 . At first, we need an Itô formula that is appropriate to deal with weak solutions. Our result is a version of [27, Theorem 4.2.5] that additionally allows a skew-adjoint operator M inspite of the fact that our weak solution is not in D(M). Our proof relies on a more straightforward regularisation technique than the original using the spectral multipliers Sn from section 2.2. q+2
LEMMA 4.2. Let X 0 ∈ L 2 ( × D)6 and Y ∈ L q+1 ( × [0, T ] × D)6 + L 2 ( × [0, T ] × D)6 and Z ∈ L 2 ( × [0, T ]; L 2 (U ; L 2 (D)6 )) be F-adapted. If
t
−X (s), Mφ L 2 (D)6 + Y (s), φ L 2 (D)6 ds X (t), φ L 2 (D)6 = X 0 , φ L 2 (D)6 + 0 t + Z (s), φ dW (s) L 2 (D)6 (4.1) 0
almost surely for all t ∈ [0, T ] and all φ ∈ D(M) ∩ L q+2 (D)6 and if we additionally have the regularity X ∈ L q+2 ( × [0, T ] × D)6 ∩ L 2 ( × [0, T ] × D)6 , then the Itô formula
X (t2 ) 2L 2 (D)6 − X (t1 ) 2L 2 (D)6 t2 = 2 ReX (s), Y (s) L 2 (D)6 + Z (s) 2L 2 (U ;L 2 (D)6 ) ds t1
t2
+2
t1
Re X (s), Z (s) dW (s) L 2 (D)6
(4.2)
holds almost surely for all 0 ≤ t1 ≤ t2 ≤ T. Moreover, we get X ∈ L 2 (; C(0, T ; L 2 (D)))6 . Proof. Let 0 ≤ t1 ≤ t2 ≤ T. We plug in φ = Sn for ∈ Cc∞ (D)6 into the equation X (t2 ), φ L 2 (D)6 = X (t1 ), φ L 2 (D)6 + +
t2 t1
t2
t1
−X (s), Mφ L 2 (D)6 + Y (s), φ L 2 (D)6 ds
Z (s), φ dW (s) L 2 (D)6 .
Note that by Lemma 3.5, Sn and M commute. Moreover, R(Sn ) ⊂ D(M). Consequently, since Sn is self-adjoint and is chosen arbitrarily, we obtain Sn X (t2 ) − Sn X (t1 ) =
t2 t1
M Sn X (s) + Sn Y (s) ds +
t2
t1
Sn Z (s) dW (s)
Strong solutions to a nonlinear stochastic Maxwell equation
almost surely. Thus, we can apply the Itô formula for Hilbert space valued processes (see, e.g., [10, Theorem 4.32]) to the functional u → u 2L 2 (D)6 to get
Sn X (t2 ) 2L 2 (D)6 − Sn X (t1 ) 2L 2 (D)6 t2 = 2 ReSn X (s), M Sn X (s) L 2 (D)6 + 2 ReSn X (s), Sn Y (s) L 2 (D)6 t1
+ Sn Z (s) 2L 2 (U ;L 2 (D)6 ) ds + 2
t2 t1
Re Sn X (s), Sn Z (s) dW (s) L 2 (D)6
almost surely. Since M is skew-adjoint, the first term on the right hand side drops. In all the other terms, we can pass to the limit. Thereby, we need that Sn u → u for q+2
n → ∞ in L q+2 (D)6 and L q+1 (D)6 (see Lemma 3.3). This finally yields
X (t2 ) 2L 2 (D)6 − X (t1 ) 2L 2 (D)6 t2 = 2 ReX (s), Y (s) L 2 (D)6 + Z (s) 2L 2 (U ;L 2 (D)6 ) ds t1
+2
t2
t1
Re X (s), Z (s) dW (s) L 2 (D)6
(4.3)
almost surely. Together with X ∈ L q+2 ( × [0, T ] × D)6 ∩ L 2 ( × [0, T ] × D)6 , this identity implies u ∈ L 2 (; L ∞ (0, T ; L 2 (D)))6 by a classical Gronwall argument. It remains to show the almost sure continuity in time. From (4.1), we know that ⊂ with P( ) = 1 such that t → X (t), φ L 2 (D)6 is continuous on there exists for every φ ∈ D(M) ∩ L q+2 (D)6 . In particular, t → X (t) ∈ L 2 (D)6 is weakly such . On the other hand, by (4.2), there exists another set 2 ⊂ continuous on 2 . Let t ∈ [0, T ] and (tn )n ⊂ [0, T ] with tn → t that t → u(t) 2L 2 is continuous on 2 as n → ∞. As argued before, we both have X (tn ) → X (t) weakly in L 2 (D)6 on 2 as n → ∞. This implies and X (tn ) L 2 (D)6 → X (t) L 2 (D)6 on
X (tn ) − X (t) 2L 2 (D)6 = X (tn ) 2L 2 (D)6 + X (t) 2L 2 (D)6 − 2 Re X (tn ), X (t) L 2 (D)6 n→∞ −−−→ X (t) 2L 2 (D)6 + X (t) 2L 2 (D)6 − 2 Re X (t), X (t) L 2 (D)6 = 0
2 , which proves the desired continuity. on At first, we assume G ≡ 0 and solve (WSEE) without retarded material law. The reason for this simplification is that we make use of the monotone structure of the rest of the equation. We start with a Galerkin approximation with the spectral projection Pn , we defined in Sect. 2. We investigate the truncated equation du n = Pn Mu n − Pn F(u n (t)) + Pn J dt +Pn B(t, u n (t))dW (t), (4.4) u n (0) = Pn u 0
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in the range of Pn . This is a stochastic ordinary differential equation in R(Pn ) ⊂ L 2 (D)6 with a locally Lipschitz nonlinearity (see Lemma 2.6). Hence, there exists an (m) (m) increasing sequence of stopping times (τn )m∈N with 0 < τn ≤ T almost surely, a (m) stopping time τn = limm→∞ τn and an adapted process u n : × [0, τ ) → R(Pn ) with continuous paths that solves (4.4). Moreover, we have the blow-up alternative P τn < T, sup u n (t) L 2 (D)6 < ∞ = 0. (4.5) t∈[0,τ )
The next result shows τn = T for every n ∈ N and a uniform estimate for u n . PROPOSITION 4.3. We have τn = T almost surely for every n ∈ N and u n additionally satisfies T |u n (t, x)|q+2 dx dt < ∞. sup E sup u n (t) 2L 2 (D)6 + sup E n∈N
t∈[0,T ]
n∈N
0
D
Proof. Lemma 4.2 applied to u n , the self-adjointness of Pn and Pn2 = Pn yield
u n (s) 2L 2 (D)6 − Pn u 0 2L 2 (D)6 s =2 Re u n (r ), −|u n (r )|q u n (r ) + J (r ) L 2 (D)6 dr 0 s s +2 Re u n (r ), B(s, u n (r )) dW (r ) L 2 (D)6 +
Pn B(r, u n (r )) 2L 2 (U ;L 2 (D)6 ) dr 0
0
almost surely for every s ∈
(m) [0, τn ].
s
This expression simplifies to
u n (s) 2L 2 (D)6 + 2 |u n (s, x)|q+2 dx dt − Pn u 0 2L 2 (D)6 D 0 s ≤ 2 Re u n (r ), J (r ) L 2 (D)6 + B(r, u n (r )) 2L 2 (U ;L 2 (D)6 ) dr 0 s +2 Re u n (r ), B(s, u n (r )) dW (r ) L 2 (D)6
(4.6)
0
(m)
almost surely for every s ∈ [0, τn ]. Since the second term on the left-hand side is (m) positive, we can drop it for a moment. We first take the supremum over [0, τn ∧t] for t ∈ [0, T ] and afterwards the expectation value and estimate the remaining quantities term by term. We start with the deterministic part using [W4] and [W5]. s 2 Re u n (r ), J (r ) L 2 (D)6 + B(r, u n (r )) 2L 2 (U ;L 2 (D)6 ) dr E sup (m)
s∈[0,τn ∧t] t
1+
t 0
E1s≤τ (m) u n (s) L 2 (D)6 J (s) L 2 (D)6 + u n (s) 2L 2 (D)6 ds n
0
1+
0
E
sup (m) r ∈[0,s∧τn ]
u n (r ) 2L 2 (D)6 ds + J 2L 2 (×[0,T ]×D)6 .
Strong solutions to a nonlinear stochastic Maxwell equation
The stochastic part can be estimated with the Burkholder–Davies–Gundy inequality. s E sup Reu n (s), B(s, u n (s)) dW (s) L 2 (D)6 (m)
0
s∈[0,t∧τn ]
≤ CE
(m) τn ∧t
0
≤ CE
1/2 u n (s), B(s, u n (s) L 2 (U,L 2 (D)6 ) 2 ds
sup (m)
s∈[0,t∧τn ]
≤
u n (s) L 2 (D)6 1 +
(m)
t∧τn
0
1 2 1 + E sup
u n (s) 2L 2 (D)6 + C 4 s∈[0,t∧τ (m) ]
u(t) 2L 2 (D)6 dt
t
0
n
E
sup (m)
r ∈[0,s∧τn ]
1 2
u(r ) 2L 2 (D)6 ds .
Thereby, we used ab ≤ 41 a 2 + b2 for all a, b ≥ 0 in the last step. Putting these estimates together, we get E
sup (m) s∈[0,t∧τn ]
u n (s) 2L 2 (D)6
1 + u 0 2L 2 (D)6 + J 2L 2 (×[0,T ]×D)6 +
0
t
E
sup (m) r ∈[0,s∧τn ]
u n (r ) 2L 2 (D)6 ds
for all t ∈ [0, T ]. Consequently, Gronwall yields E
sup u n (t) 2L 2 (D)6 1 + J 2L 2 (×[0,T ]×D)6 + u 0 2L 2 (D)6 (m)
t∈[0,τn ]
with an estimate that is independent of n ∈ N. Now, we can go back to (4.6) and deal with the term we dropped at first. The estimate of E supt∈[0,τ (m) ] u n (t) 2L 2 (D) implies n
(m) τn
E 0
D
|u n (s, x)|q+2 dx dt 1 + J 2L 2 (×[0,T ]×D)6 + u 0 2L 2 (D)6
We use Fatou’s Lemma to pass to the limit m → ∞ in these estimates. Note that one can interchange sup and lim inf in an upper estimate, since lim inf can be written in the form sup inf and supremums can be interchanged, whereas sup inf ≤ inf sup. Hence, we have τn |u n (s, x)|q+2 dx dt E sup u n (t) 2L 2 (D)6 + E t∈[0,τn )
1 + J 2L 2 (×[0,T ]×D)6
0
D
+ u 0 2L 2 (D)6 .
(4.7)
Consequently, we also have surely. Indeed, there exists N ⊂ with τn = T almost P(N ) = 0 such that \ N ∪ {τn = T } can be decomposed into disjoint sets τn < T, sup u n (t) 2L 2 (D)6 < ∞ , τn < T, sup u n (t) 2L 2 (D)6 = ∞ . t∈[0,τn )
t∈[0,τn )
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J. Evol. Equ.
The first set has measure zero by (4.5) and the second one has measure zero since (4.7) implies supt∈[0,τn ) u n (t) L 2 (D)6 < ∞ almost surely. Pathwise uniform continuity on [0, T ] follows from Lemma 4.2. This closes the proof. In Proposition 4.3, we derived uniform estimates for u n . As a consequence, Lemma 2.2 q+2
yields the uniform boundedness of F(u n ) in L q+1 (×[0, T ]× D)6 . Thus, by Banach– q+2
Alaoglu, there exists u ∈ L 2 (; L ∞ (0, T ; L 2 (D)6 )), N ∈ L q+1 ( × [0, T ] × B ∈ L 2 ( × [0, T ]; L 2 (U ; L 2 (D)))6 and subsequences, still indexed with n, D)6 , such that (a) u n → u for n → ∞ in the weak ∗ sense in L 2 (; L ∞ (0, T ; L 2 (D)))6 . (b) u n → u for n → ∞ in the weak sense in L q+2 ( × [0, T ] × D)6 . q+2
(c) F(u n ) → N for n → ∞ in the weak sense in L q+1 ( × [0, T ] × D)6 . B for n → ∞ in the weak sense in L 2 (×[0, T ]; L 2 (U ; L 2 (D)))6 . (d) B(·, u n ) → Since u n is for every n ∈ N an adapted solution of the ordinary stochastic differential equation (4.4) in Pn L 2 (D)6 , we have u n ∈ L 2F ( × [0, T ] × D)6 . Consequently, since L 2F ( × [0, T ] × D)6 is a closed subspace of L 2 ( × [0, T ] × D)6 , it is also weakly closed. This implies u ∈ L 2F ( × [0, T ] × D)6 , which means that u is also adapted. Testing (4.4) with ρφ for arbitrary ρ ∈ L q+2 ( × [0, T ]) and φ ∈ ∞ n=1 R(Pn ), the symmetry of Pn and the skew-symmetry of M yield
T
E 0
u n (t) − u 0 , φ L 2 (D)6 ρ(t) dt
T
=E
0
t
0
T
+E
−u n (s), M Pn φ L 2 (D)6 + −F(u n (s)) + J (s), Pn φ L 2 (D)6 dsρ(t) dt
0
t
0
B(s, u n (s)), Pn φ L 2 (D)6 dW (s)ρ(t) dt.
By weak convergence, we can pass to the limit and obtain
T
E 0
u(t) − u 0 , φ L 2 (D)6 ρ(t) dt
T
=E
0
t 0
T
+E 0
−u(s), Mφ L 2 (D)6 + −N (s) + J (s), φ L 2 (D)6 dsρ(t) dt
t 0
B(s), φ L 2 (D)6 dW (s)ρ(t) dt.
Thereby, we used Pn φ = φ for n large enough since φ ∈ ∞ n=1 R(Pn ) and that linear and bounded operators are also weakly continuous. Since ρ was chosen arbitrarily, we finally get
Strong solutions to a nonlinear stochastic Maxwell equation
u(t) − u 0 , φ L 2 (D)6 =
t 0
−u(s), Mφ L 2 (D)6 + −N (s) + J (s), φ L 2 (D)6 ds t
+ 0
B(s), φ L 2 (D)6 dW (s)
(4.8)
almost surely for every t ∈ [0, T ]. Hence, by density (see Lemma 3.6), this holds true for every φ ∈ D(M) ∩ L q+2 (D)6 . To show that u is a weak solution of (WSEE) with G ≡ 0, it remains to show N = F(u) and B = B(·, u). This will be done by adapting a standard argument for stochastic evolution equations with monotone nonlinearies (see [27, proof of Theorem 4.2.4, page 86]) to our situation. To do this, we just need an Itô formula for / L 2 (D)6 . The rest follows the line of the origEe−K t u(t) 2L 2 (D)6 , although Mu(t) ∈ inal. LEMMA 4.4. For any K > 0, u n and u satisfy the Itô formulae E e−K t u(t) 2L 2 (D)6 − E u 0 2L 2 (D)6 t = E 2e−K s Re u(s), −N (s) + J (s) L q+2 + e−K s B(s) 2L 2 (U ;L 2 (D)6 ) ds 0 t −E K e−K s u(s) 2L 2 (D)6 ds 0
and E e−K t u n (t) 2L 2 (D)6 − E Pn u 0 2L 2 (D)6 t = E 2e−K s Re u n (s), −F(u n (s)) + J (s) L 2 (D)6 0 t −K s +e
Pn B(s, u n (s)) 2L 2 (U ;L 2 (D)6 ) ds − E K e−K s u n (s) 2L 2 (D)6 ds 0
almost surely for all t ∈ [0, T ]. Proof. These formulae are immediate by Lemma 4.2, the Itô product rule and the fact that the expectation of a stochastic integral is zero. All in all, we showed the following result. PROPOSITION 4.5. If we assume [W1] − [W5], the equation (WSEE) with G ≡ 0 has a unique weak solution u in the sense of Definition 4.1. Finally, we add a nontrivial the retarded material law G with a perturbation argument. THEOREM 4.6. If we assume [W1] − [W5], the equation (WSEE) has a unique weak solution u in the sense of Definition 4.1.
L. Hornung
J. Evol. Equ.
Proof. Let T0 ∈ (0, T ]. By Proposition 4.5 the equation du(t) = Mu(t) − F(u(t)) + (G ∗ v)(t) + J (t) dt + B(t, u(t)) dW (t), u(0) = u 0 has for v ∈ L 2 (; C(0, T0 ; L 2 (D)6 )) a unique solution u =: K v ∈ L 2 (; C(0, T0 ; L 2 (D)6 )). Indeed, by [W3], t G(t − s)u(s) ds ∈ L 2 ( × [0, T ] × D)6 t → 0
and thus G ∗ v satisfies [W5]. In the following, we will show that K is a contraction on X := L 2 (; C(0, T0 ; L 2 (D)6 )) if we choose T0 > 0 small enough. For given v, w ∈ X, we calculate with Lemma 4.2 that
K v(s) − K w(s) 2L 2 (D)6 s = 2 Re K v(r ) − K w(r ), F(K w(r )) − F(K v(r )) + (G ∗ (v − w))(r ) L 2 (D)6 0
+ B(r, K v(r )) − B(r, K w(r )) 2L 2 (U ;L 2 (D)6 ) dr t +2 Re K v(r ) − K w(r ), B(r, K v(r )) − B(r, K w(r )) dW (r ) L 2 (D)6 . 0
In the following estimates, we take the supremum over [0, t] for t ∈ [0, T0 ] and afterwards the expectation. We now estimate the occurring quantities term by term. s Re K v(r ) − K w(r ), (G ∗ (v − w))(r ) L 2 (D)6 ds 0 s r 2 1 1 2 ≤
K v(r ) − K w(r ) L 2 (D)6 + G(r − λ)(v(λ) − w(λ)) dλ L 2 (D)6 dr 2 2 0 0 s 1 sup K v(λ) − K w(λ) 2L 2 (D)6 dr ≤ 2 0 λ∈[0,r ] +
T0 G 2L 1 (0,T ;B(L 2 (D)6 )) 2
sup v(λ) − w(λ) 2L 2 (D)6
λ∈[0,T0 ]
for all s ∈ [0, T0 ]. We can drop the contribution of F, as
q+2 K v(r ) − K w(r ), F(K w(s)) − F(K v(s)) L 2 (D)6 ≤ −α K v(r ) − K w(r ) L q+2 (D)6
for all s ∈ [0, T0 ] and some α > 0 by Lemma 2.5. Moreover, by [W4], we have t
B(s, K v(s)) − B(s, K w(s)) 2L 2 (U ;L 2 (D)6 ) ds 0 t ≤ C2 sup K v(r ) − K w(r ) 2L 2 (D)6 ds. 0 r ∈[0,s]
Strong solutions to a nonlinear stochastic Maxwell equation
Last but not least, the Burkholder–Davies–Gundy inequality and [W4] yield E sup s∈[0,t]
Re K v(r ) − K w(r ), (B(r, K v(r )) − B(r, K w(r ))) dW (r ) L 2 (D)6
s
0
t 1/2 K v(r ) − K w(r ), B(r, K v(r )) − B(r, K w(r )) 2 6 2 2 dr ≤ CE L (D) L (U ) 0 1/2 t ≤ CE sup K v(s) − K w(s) L 2 (D)6
B(r, K v(r )) − B(r, K w(r )) 2L 2 (U ;L 2 (D)6 ) dr 0
s∈[0,t]
1 2 ≤ E sup K v(s) − K w(s) 2L 2 (D)6 + C 4 s∈[0,t]
t 0
E sup K v(r ) − K w(r ) 2L 2 (D)6 ds. r ∈[0,s]
All in all, we derived E sup K v(s) − K w(s) 2L 2 (D)6 s∈[0,t]
≤
t
2 + C 2 )E sup K v(λ) − K w(λ) 2 2 6 dr 2(1 + 2C L (D)
λ∈[0,r ] 2 + 2T0 G L ∞ (;L 1 (0,T ;B(L 2 (D)6 ))) E 0
sup v(λ) − w(λ) 2L 2 (D)6
λ∈[0,T0 ]
for every t ∈ [0, T0 ]. Hence, Gronwall implies E sup K v(s) − K w(s) 2L 2 (D)6 s∈[0,t]
2 2 ≤ 2T0 G 2L ∞ (;L 1 (0,T ;B(L 2 (D)6 ))) E sup v(λ) − w(λ) 2L 2 (D)6 e2(1+2C +C )T0 . λ∈[0,T0 ]
Now, we choose T0 > 0 small enough to ensure that K is a contraction. Then, by Banach’s fixed point theorem, there exists a u 1 ∈ L 2 (; C(0, T0 ; L 2 (D)6 )) solving (WSEE) on [0, T0 ] and from K u 1 = u 1 we deduce u 1 ∈ L q+2 ( × [0, T0 ] × D)6 . Clearly, by continuity in time, we have u 1 (T0 ) ∈ L 2 ( × D)6 and ω → u 1 (ω, T0 ) is strongly FT0 -measurable. Next, given v ∈ L 2 (; C(T0 , 2T0 ; L 2 (D)6 )), we consider the equation
· T dy = M y − F(y) + 0 0 G(· − s)u 1 (s) ds + T0 G(· − s)v(s) ds + J dt + B(·, y)dW, y(T0 ) = u 1 (T0 )
for t ∈ [T0 , 2T0 ]. By Proposition 4.5, we have a unique solution y := K 2 v. This defines an operator K 2 : L 2 (; C(T0 , 2T0 ; L 2 (D)6 )) → L 2 (; C(T0 , 2T0 ; L 2 (D)6 )). However, K 2 v − K w2 can be estimated in the very same way as above since the T additional term 0 0 G(· − s)u 1 (s) ds vanishes in this difference. As a consequence, K 2 is a contraction on L 2 (; C(T0 , 2T0 ; L 2 (D)6 )) and has a unique fixed point u 2 . Inductively, we construct u n ∈ L 2 (; C((n − 1)T0 , nT0 ; L 2 (D)6 )) solving t dy(t) = M y(t) − F(y(t)) + (n−1)T0 G(t − s)y(s) ds + f (t) dt + B(t, y) dW (t), y((n − 1)T0 ) = u n−1 ((n − 1)T0 )
L. Hornung
with f (t) = J (t) + the process u :=
J. Evol. Equ.
n−1 kT0
k=1 (k−1)T0 G(t − s)u k (s) ds and stop when nT0 ≥ T. Finally, T
T0 +1 u n 1[(n−1)T0 ,nT0 ) solves (WSEE) on [0, T ] and satisfies n=1
u ∈ L 2 (; C(0, T ; L 2 (D)6 )) ∩ L q+2 ( × [0, T ] × D)6 . By construction, u is unique on every interval [(n − 1)T0 , nT0 ), which implies uniqueness on [0, T ]. 5. Existence and uniqueness of a strong solution In this section, we will discuss the following stochastic Maxwell equation
N bn + i Bn u dβn (t), du = Mu − |u|q u + G ∗ u + J dt + n=1 (MSEE) u(0) = u 0 . on L 2 (D)6 with a monotone polynomial nonlinearity and a retarded material law and we derive existence and uniqueness of a strong solution in the sense of partial differential equations. For sake of readability, we sometimes write F(u) := |u|q u. Before we start, we explain our solution concept. DEFINITION 5.1. A weak solution u is called strong solution of (MSEE) if it additionally satisfies q+2
Mu ∈ L 2 (; L ∞ (0, T ; L 2 (D)))6 + L q+1 ( × [0, T ] × D)6 . Note that in case of a bounded domain D ⊂ R3 , this integrability property reduces q+2
Mu ∈ L q+1 ( × [0, T ] × D)6 . We make the following assumptions. [M1] Let q ∈ (1, 2] and D ⊂ R3 be a bounded C 1 - domain or D = R3 . [M2] Let u 0 be strongly F0 -measurable with 2(q+1)
E Mu 0 2L 2 (D)6 + E u 0 L 2(q+1) (D)6 < ∞. [M3] Let G ∈ L ∞ (; W 1,1 (0, T ; B(L 2 (D)6 ))), such that ω → G(t)x is for all x ∈ L 2 (D)6 and all t ∈ [0, T ] strongly Ft -measurable. [M4] Let J ∈ L 2 (; W 1,2 (0, T ; L 2 (D)6 )) be F-adapted. [M5] Let b j ∈ L 2 (; W 1,2 (0, T ; L 2 (D)6 )), j = 1, . . . , N , be F-adapted. If q ∈ 2(q+2)
(1, 2), we additionally assume b j ∈ L 2−q (×[0, T ]× D)6 and b j ∈ L ∞ (× n∈N [0, T ] × D)6 if q = 2. Moreover, if q = 2, we assume that there exists such that we have
N
N Pn b j e−i l=1 Bl βl = b j e−i l=1 Bl βl for all n > n . Here, we use the operator Pn defined in Sect. 3.
Strong solutions to a nonlinear stochastic Maxwell equation
[M6] Let B j ∈ W 1,∞ (D) for j = 1, . . . , N . At first, we assume G ≡ 0 and solve (MSEE) without retarded material law as in the last section. The reason for this simplification is that we make use of the monotone structure of the rest of the equation. As described in the introduction, we failed to derive an a priori estimate for Mu directly with Itô’s formula and Gronwall, since we could not control the terms F (u)(B j u(s), B j u(s) L 2 (D)6 and F (u)(B j u(s) L 2 (D)6 . Hence,
N
we start with a rescaling transformation of the form y(t) := e−i l=1 Bl βl (t) u(t), such that the multiplicative noise vanishes. We end up with
N bi (t) dβi (t), dy(t) = [M y(t) − |y(t)|q y(t) + A(t)y(t) + J (t)] dt + i=1 (TSEE) u(0) = u 0 ,
where A(t), J˜ and the new additive noise A(t, x)y(t, x):=
1 2
N
J˜(t, x):=
˜ dβ j are given by
j=1 b j
B j (x) y(t, x) + 2
j=1 N
N
N j=1
∇ B j (x) × y2 , iβ j (t) −∇ B j (x) × y1
N − ib j (t, x)B j (x) + J (t, x) e−i n=1 Bn (x)βn (t) ,
j=1
b˜i (t, x):= bi (t, x)e−i
N j=1
B j (x)β j (t)
for t ∈ [0, T ], x ∈ D and i = 1, . . . , N . Note that this is the point, where we make use of the fact that B is purely real-valued. This leads to |y(t)| = |u(t)| almost surely for all t ∈ [0, T ]. Otherwise, the equation (TSEE) would not have the monotone structure in the nonlinearity −|y(t)|q y(t) and this monotone structure is essentially needed for our a priori estimates. At first, we show that a solution of (TSEE) can be transformed to a solution of (MSEE). PROPOSITION 5.2. An adapted stochastic process u : × [0, T ] → L 2 (D) is a strong solution of (MSEE) with G ≡ 0 if and only if the adapted process y(t) :=
N −i B l=1 l βl (t) u(t) satisfies e T (i) E supt∈[0,T ] y(t) 2L 2 (D)6 + E 0 D |y(t, x)|q+2 dx dt < ∞, q+2 ∇ B j ×y2
q+1 (×[0, T ]×D)6 +L 2 (; L ∞ (0, T ; L 2 (D)6 )) (ii) M y+i Nj=1 β j −∇ ∈ L B ×y j
1
and solves the equation (TSEE). Proof. We assume that u is a solution of (MSEE) in the sense of Definition 5.1 with
the described regularity properties. At first, we calculate d(ei formula and obtain N N t
N
N i B j ei l=1 Bl βl (s) dβn (s) − 21 ei j=1 B j β j (t) − 1 = j=1 0
N j=1
j=1 0
t
B j β j (t)
B 2j ei
) with Itô’s
N l=1
Bl βl (s)
ds.
L. Hornung
J. Evol. Equ.
Therefore, Itô’s product rule yields
N
y(t), x L 2 (D)6 − u 0 , x L 2 (D)6 = u(t), ei l=1 Bl βl (t) x L 2 (D)6 − u 0 , x L 2 (D)6 N t
N
N = −u(s), 21 B 2j ei l=1 Bl βl (s) x L 2 (D)6 + b j (s) + i B j u(s), i B j ei l=1 Bl βl (s) x L 2 (D)6 ds j=1 0
+ +
t
Mu(s) − |u(s)|q u(s) + J (s), ei
0 N t
u(s), i B j ei
N l=1
N l=1
Bl βl (s)
x L 2 (D)6 ds
Bl βl (s)
x L 2 (D)6 + b j (s) + i B j u(s), ei
j=1 0
N
Bl βl (s)
x L 2 (D)6 dβn (s)
l=1
almost surely for every x ∈ Cc∞ (D) and for every t ∈ [0, T ]. As a consequence, we have t
N N y(t) − u 0 = e−i l=1 Bl βl (s) M ei l=1 Bl βl (s) y(s) − |y(s)|q y(s) ds 0
t
+
e
−i
N l=1
Bl βl (s)
J+
0
+
N
−i 1 2 2 B j y(s) − ib j (s)B j e
N l=1
Bl βl (s)
ds
j=1
N
t
bn (s)e−i
N l=1
Bl βl (s)
dβn (s)
(5.1)
n=1 0
almost surely for every t ∈ [0, T ]. Here, we used that u ∈ L q+2 ( × [0, T ] × D)6 implies |y|q y ∈ L q+2 ( × [0, T ] × D)6 . Since we want to derive an equation for y, we have to commute the exponential function with M. Therefore, we compute
N
M y(t) = M(e−i l=1 Bl βl (t) u(t))
N curl(e−i l=1 Bl βl (t) u 2 (t))
N = − curl(e−i l=1 Bl βl (t) u 1 (t))
N N −i l=1 N Bl βl (t) curl(u (t)) ∇ B j e−i l=1 Bl βl (t) × u 2 (t) e 2
N
N −iβ j (t) + = −e−i l=1 Bl βl (t) curl(u 1 (t)) −∇ B j e−i l=1 Bl βl (t) × u 1 (t) j=1 =
N
N −∇ B j × y2 (t) + e−i l=1 Bl βl (t) Mu(t). ∇ B j × y1 (t)
iβ j (t)
j=1
Together with y(t) := e−i e
−i
N l=1
Bl βl (t)
N l=1
Bl βl (t) u(t),
this implies
N −i N B β (t) l l l=1 M e y(t) = M y(t) + iβ j (t) j=1
∇ B j × y2 (t) . −∇ B j × y1 (t)
Inserting this into (5.1) finally proves that y solves (TSEE). The other direction follows the same lines.
Strong solutions to a nonlinear stochastic Maxwell equation
We solve (TSEE) by a refined Galerkin approximation of the skew-adjoint operator M. To do this, we truncated the equation with the spectral multipliers Pn and Sn , we defined in Sect. 2.1. We study
N bi (t) dβi (t), Sn−1 dyn (t) = [Pn M yn (t) − Pn F(yn (t)) + Pn A(t)yn (t) + Pn J (t)] dt + i=1 yn (0) = Sn−1 u 0 .
(5.2) In the next Proposition, we derive a priori estimates for the solution exploiting the structure of the equation. PROPOSITION 5.3. The truncated equation (5.2) has for every n ∈ N a unique, pathwise continuous solution yn : × [0, T ] → L 2 (D)6 that additionally satisfies T q+2
yn (t) L q+2 (D)6 dt E sup yn (t) 2L 2 (D)6 + E t∈[0,T ]
0
≤ C J 2L 2 (×[0,T ]×D) +
N
b j 2L 2 (×[0,T ]×D) + u 0 2L 2 (D)
(5.3)
j=1
for some constant C > 0 only depending on N , T and sup j=1,...,N B j L ∞ (D) , but not on n ∈ N. Proof. First, we define the stopping time τm := inf t ∈ [0, T ] : |βi (t)| > m for some i = 1, . . . , N and solve the equation (m)
N (m) (m) dyn = [Pn M yn − Pn F(yn ) + Pn A(m) ynm + Pn J ] dt + i=1 Sn−1 bi dβi , u(0) = Sn−1 u 0 , (5.4) where the truncated linear operator A(m) is given by N ∇ B j × y2 (t) (m) + B 2j y(t). iβ j (t ∧ τm ) A (t)y(t) := −∇ B j × y1 (t) j=1
By Lemmas 2.6 and 3.7, this is an ordinary stochastic differential equation in the closed subspace R(Pn ) ⊂ L 2 (D)6 with locally Lipschitz nonlinearity. The stopping time τm is necessary at this point since it ensures β j (· ∧ τm ) ∈ L ∞ ( × [0, T ]). We need this truncation to be able to apply the classical results for stochastic ordinary differential equations. There exists a stopping time τ (m,n) with 0 ≤ τ (m,n) ≤ T almost surely, an increasing (m,n) (m,n) )k with τk → τ (m,n) almost surely as k → ∞ sequence of stopping times (τk (m) and adapted processes yn : × [0, T ] → Pn L 2 (D)6 with yn(m) ∈ C(0, τk(m,n) ; L 2 (D)6 )
L. Hornung (m)
J. Evol. Equ.
(m,n)
almost surely such that yn solves (5.4) on [0, τk ]. Moreover, we have the blow-up alternative (5.5) P τ (m,n) < T, sup yn (t) L 2 (D)6 < ∞ = 0. t∈[0,τ (m,n) )
To show the a priori estimate, we use the Itô formula from Lemma 4.2 and get
yn(m) (t) 2L 2 (D)6 − Sn−1 u 0 2L 2 (D)6 t =2 Re yn(m) (s), −|yn(m) (s)|q yn(m) (s) + A(m) (s)yn(m) (s) + J (s) L 2 (D)6 ds 0
+2
N
t
Re
j=1 0
yn(m) (s), Sn−1 b j (s) L 2 (D)6
dβ j (s) +
N
t
j=1 0
Sn−1 b j (s) 2L 2 (D)6 ds.
Using the skew-symmetry of the cross-product and the fact that both B j and β j are real-valued, we calculate
yn(m) (s), iβ j (s
∧ τm )
(m) (s) ∇ B j × yn,2
(m)
−∇ B j × yn,1 (s) L 2 (D)6 (m) (s) ∇ B j × yn,2 (m) = − iβ j (s ∧ τm )yn (s), (m)(s) L 2 (D)6 −∇ B j × yn,1 (m) (s) ∇ B j × yn,2 (m) , y = iβ j (s ∧ τm ) (s) , n (m) L 2 (D)6 −∇ B j × yn,1 (s)
which implies Re
yn(m) (s), iβ j (s
∧ τm )
(m)
∇ B j × yn,2 (s)
(m) −∇ B j × yn,1 (s)
L 2 (D)6
=0
(n,m)
for all s ∈ [0, τk
yn(m) (t) 2L 2 (D)6
]. Hence, the expression from above simplifies to t +2 |yn(m) (s, x)|q+2 dx dt
= u 0 2L 2 (D)6 + 2 +2
N j=1 0
D
0
t
Re
t 0
N Re yn(m) (s), J (s) + B 2j yn(m) (s) L 2 (D)6 ds j=1
yn(m) (s), Sn−1 b j (s) L 2 (D)6
dβ j (s) +
N j=1 0
t
Sn−1 b j (s) 2L 2 (D)6 ds (5.6)
(m,n)
]. Since the second term on the left-hand side is almost surely for all t ∈ [0, τk positive, we can drop it for a moment. Afterwards, we take the supremum over time
Strong solutions to a nonlinear stochastic Maxwell equation
and then the expectation. We estimate the remaining quantities term by term and start with the deterministic part.
E
sup (m,n)
s∈[0,t∧τk
]
(m,n) t∧τk
≤E 0
+ ≤
N
0
N N 2 Re yn(m) (r ), J (r ) + B 2j yn(m) (r ) dr +
s
j=1 0
j=1
2 yn(m) (r ) L 2 (D)6 J (r ) L 2 (D)6 + 2N
sup j=1,...,N
b j (r ) 2L 2 (D)6 dr
Sn−1
B j 2L ∞ (D) yn(m) (r ) 2L 2 (D)6 dr
b j 2L 2 (×[0,T ]×D)6
j=1 t
0
+
s
E
sup (m,n) r ∈[0,s∧τk ]]
N
yn(m) (r ) 2L 2 (D)6 2N
sup j=1,...,N
B j 2L ∞ (D) + 1 ds + J 2L 2 (×[0,T ]×D)6
b j 2L 2 (×[0,T ]×D)6 .
j=1
The stochastic part can be estimated with the Burgholder–Davies–Gundy inequality. We have E
sup (m,n)
s∈[0,t∧τk
N
s
] j=1 0
Re yn(m) (s), Sn−1 b j (s) L 2 (D)6 dβ j (s)
N ≤ CE
(m,n)
t∧τk
j=1 0
≤ CE
sup (m,n)
s∈[0,t∧τk
]
2 1/2 Rey (m) (s), Sn−1 b j (s) L 2 (D)6 ds n
yn(m) (s) L 2 (D)6
N
Sn−1 b j 2L 2 ([0,T ]×D)
1/2
j=1
1 sup ≤ E
yn(m) (s) 2L 2 (D)6 + C 2 E
b j 2L 2 (×[0,T ]×D) . 4 s∈[0,t∧τ (m,n) ] N
j=1
k
Putting these estimates together, we get E
sup (m,n) s∈[0,t∧τk ]
yn(m) (s) 2L 2 (×D)
u 0 2L 2 (D)6 + J 2L 2 (×[0,T ]×D) +
+ N
sup j=1,...,N
B j 2L ∞ (D) + 1
0
N
b j 2L 2 (×[0,T ]×D)
j=1 t
E
sup (m,n) r ∈[0,s∧τk ]
yn(m) (r ) 2L 2 (D)6 ds.
L. Hornung
J. Evol. Equ.
Consequently, Gronwall yields
E
sup (m,n) s∈[0,t∧τk ]
yn(m) (s) 2L 2 (D)6
N B j ,N ,T J 2L 2 (×[0,T ]×D)6 +
b j 2L 2 (×[0,T ]×D)6 + u 0 2L 2 (×D) j=1
for every t ∈ [0, T ]. Next, we pass to the limit k → ∞ with Fatou’s Lemma and get E
sup t∈[0,τ (m,n) )
yn(m) (t) 2L 2 (D)6
≤ lim inf E k→∞
sup (m,n) t∈[0,τk ]
yn(m) (t) 2L 2 (D)6
N B j J 2L 2 (×[0,T ]×D) +
b j 2L 2 (×[0,T ]×D) + u 0 2L 2 (×D) .
(5.7)
j=1
Note that this bound is independent of m and n. In particular, it implies τ (m,n) = T almost surely. Indeed, there exists an N ⊂ with P(N ) = 0 such that \ N ∪ {τ (m,n) = T } can be decomposed into disjoint sets τ (m,n) < T, τ (m,n) < T,
sup t∈[0,τ (m,n) )
sup t∈[0,τ (m,n) )
yn(m) (t) L 2 (D)6 < ∞ ,
yn(m) (t) L 2 (D)6 = ∞ .
The first of these sets has measure zero by (5.5), whereas the second one has measure zero since (5.7) implies supt∈[0,τ (m,n) ) yn(m) (t) L 2 (D)6 < ∞ almost surely. As a consequence of (5.6), we also get
T
E 0
D
|yn(m) (s, x)|q+2 dx dt
N
b j 2L 2 (×[0,T ]×D)6 + u 0 2L 2 (D) . B j J 2L 2 (×[0,T ]×D)6 +
(5.8)
j=1 (m)
We already know that yn is almost surely continuous on [0, T ) as a function with values in L 2 (D)6 . The pathwise continuity up to T follows from Lemma 4.2.
Strong solutions to a nonlinear stochastic Maxwell equation (m)
(k)
It remains to take the limit m → ∞. By uniqueness, we have yn (ω, t) = yn (ω, t) for almost all ω ∈ , all t ∈ [0, τm ] and for every k ≥ m. Moreover, for almost all ω ∈ , there exists m(ω), such that τm(ω) (ω) = T. Hence, the limit yn = limm→∞ yn(m) is well defined, adapted and satisfies (5.4). Again using Fatou’s Lemma yields analogous estimates to (5.7) and (5.8) for yn . This closes the proof. To obtain strong solutions, we need an estimate for M yn that is uniform in n ∈ N. We do this in the following way. We derive an a priori estimate for N 2 ∇ B j × yn,2 (t) + Pn J (t) 2 6 B 2j yn (t) + Pn iβ j (t) Pn M yn (t) − Pn F(yn (t)) + Pn L (D) −∇ B j × yn,1 (t) j=1
and afterwards we use the estimates from Proposition 5.3 to get a bound for M yn . To do this, we have to show that the above quantity is an Itô process in Pn L 2 (D)6 . LEMMA 5.4. The stochastic process n (t) := Pn M yn (t) − Pn F(yn (t)) + Pn
N
∇ B j × yn,2 (t) Pn iβ j (t) −∇ B j × yn,1 (t)
B 2j yn (t) +
j=1
+Pn J (t) is an Itô process with N ∇ B j × n,2 iβ j dn = Pn Mn − F (yn )(n ) + + B 2j n −∇ B j × n,1 j=1
−
+
1 2
N j=1
N k=1
+
N
B 2j
N
N −ibk Bk + J e− l=1 Bl βl ·
k=1
N N 1 b j , Sn−1 b j ) dt −i∂t bk Bk + ∂t J e− l=1 Bl βl · − F (yn )(Sn−1 2 j=1
Pn M Sn−1 b j − F (yn )(Sn−1 b j ) +
j=1
+
N
N
iβk
k=1
Bk2 Sn−1 b j + i
k=1
∇ Bk × Sn−1 b j,2 b j,1 −∇ Bk × Sn−1
N N ∇ B j × yn,2 −ibk Bk + J e− l=1 Bl βl · dβ j − i Bj −∇ B j × yn,1 k=1
almost surely on [0, T ]. Proof. With Lemmas 2.6 and 3.7, one shows that Pn F(yn ) is an Itô process in Pn L 2 (D)6 with d(Pn F(yn )) = Pn F (yn )(n ) +
1 2
N
F (yn )(Sn−1 b j , Sn−1 b j ) dt
j=1
+
N j=1
Pn F (yn )Sn−1 b j dβ j .
L. Hornung
J. Evol. Equ.
Moreover, by the product rule, Pn J (t, x) = Pn
N
N −ib j (t, x)B j (x) + J (t, x) e−i l=1 Bl βl (t)
j=1
is an Itô process in L 2 (D)6 of the form N N 1 B 2j −ibk (t)Bk + J (t) d(Pn J )(t) = Pn − 2 j=1
+
N
k=1
N −i∂t bk (t)Bk + ∂t J (t) e−i l=1 Bl βl (t) dt
k=1
− Pn
N N N i Bj −ibk (t)Bk + J (t) e−i l=1 Bl βl (t) dβ j . j=1
k=1
The remaining expression n + Pn F(yn ) − Pn J is a function of the Itô processes dyn (t, x) = n (t)dt + Sn−1
N
b j dβ j (t)
j=1
and β j , j = 1, . . . , N . Hence, we can calculate d(n + Pn F(yn ) − Pn J ) with Itô’s formula. Thereby, it is crucial that all occurring terms depend only linearly on yn and β j and consequently the second derivatives vanish. This finally proves the claimed result. Now we can derive an a priori estimate for n that is uniform in n ∈ N. PROPOSITION 5.5. The process n satisfies the estimate 2q+2 E sup n (t) 2L 2 (D)6 ≤ C 1 + E Mu 0 2L 2 (D)6 + E u 0 2L 2 (D)6 + E u 0 L 2q+2 (D)6 t∈[0,T ]
with a constant C > 0 depending on J, b j and B j for j = 1, . . . , N , but not on n ∈ N. Proof. At first, we calculate n (t) 2L 2 (D)6 with the Itô formula from Lemma 4.2. We obtain
Strong solutions to a nonlinear stochastic Maxwell equation
n (t) 2L 2 (D)6 − n (0) 2L 2 (D)6 = 2
t
Re n (s), Mn (s) − F (yn (s))(n (s))
0
N ∇ B j × n,2 (s) + iβ j (s) + B 2j n (s) −∇ B j × n,1 (s) j=1
−
N N N 1 2 Bj −ibk (s)Bk + J (s) e−i l=1 Bl βl (s) 2 j=1
+
N
k=1
N −i∂t bk (s)Bk + ∂t J (s) e−i l=1 Bl βl (t)
k=1 N 1 b j (s), Sn−1 b j (s)) 2 6 ds F (yn )(Sn−1 L (D) 2 j=1 t + M Sn−1 b j (s) − F (yn )(Sn−1 b j (s))
−
0
+
N
b j,2 (s) ∇ Bk × Sn−1 −∇ Bk × Sn−1 b j,1 (s)
iβk (s)
k=1
+
N
Bk2 Sn−1 b j (s) + i
k=1
− i Bj
N
∇ B j × yn,2 (s) −∇ B j × yn,1 (s)
2 N −ibk (s)Bk + J (s) e−i l=1 Bl βl (s) 2
L (D)6
k=1
+2
N
t
ds
Re n (s), M Sn b j (s) − F (yn )(Sn−1 b j (s))
j=1 0
+
N
iβk (s)
k=1
+
N
Bk2 Sn−1 b j (s) + i
k=1
− i Bj
b j,2 (s) ∇ Bk × Sn−1 −∇ Bk × Sn−1 b j,1 (s)
N
∇ B j × yn,2 (s) −∇ B j × yn,1 (s)
N −ibk (s)Bk + J (s) e−i l=1 Bl βl (s)
k=1
L 2 (D)6
dβ j (s).
As we have seen before in the proof of Proposition 5.3, the term N ∇ B j × n,2 (s) iβ j (s) Re n (s), Mn (s) + −∇ B j × n,1 (s) L 2 (D)6 j=1
vanishes. Moreover, by Lemma 2.6, we have − Re n (s), F(yn (s)) n (s) L 2 (D)6 ≤ 0 almost surely for every s ∈ [0, T ] and we can drop this term in an upper estimate. We split the remaining expression into a deterministic integral Idet and a stochastic integral Istoch .
L. Hornung
J. Evol. Equ.
We take the supremum over time and afterwards the expectation. Further, we aim to control the left-hand side with Gronwall. We start with an estimate for the deterministic integral Idet . Using Cauchy–Schwartz and the assumptions on B j , ∇ B j , ∂t b j , J and ∂t J from [M4] − [M6], we get E sup |Idet (s)| s∈[0,t]
t
0
n (r ) 2L 2 (D)6 +
N
n (r ) L 2 (D)6 F (yn )(Sn−1 b j (r ), Sn−1 b j (r ) L 2 (D)6
j=1
N b j (r ) 2L 2 (D)6 + M Sn b j (r ) 2L 2 (D)6 + F (yn (r ))(Sn−1 b j (r )) 2L 2 (D)6 +
βk (r )Sn−1 k=1
+
N
Sn−1 b j (r ) 2L 2 (D) + yn (r ) 2L 2 (D) dr.
k=1
The growth estimates for F and F from Lemma 2.6 together with the uniform boundedness of Sn−1 on L 2 (D)6 yield
E sup |Idet (s)| s∈[0,t]
t
n (r ) 2L 2 (D)6
0
+
N
|yn (r )|q−1 |Sn−1 b j (r )|2 2L 2 (D)6 + M b j (r ) 2L 2 (D)6
j=1
+ b j (r ) 2L 2 (D) + |yn (r )|q Sn−1 b j (r ) 2L 2 (D)6 +
N
βk (r )2 b j (r ) 2L 2 (D)6
k=1
+ yn (r ) 2L 2 (D) dr. In the following estimate, we have to distinguish the cases q ∈ (1, 2) and q = 2. We start with the first one. Hölder’s inequality, the fact βk ∈ L α (; C(0, T )) for every α ∈ [2, ∞) and the boundedness of Sn−1 on L p (D)6 for every p ∈ (1, ∞) with norm independent of n yield E sup |Idet (s)| s∈[0,t]
t
E sup n (r ) 2L 2 (D)6 ds + yn L q+2 (×[0,T ]×D)6 b j 4 4(q+2) 2(q−1)
r ∈[0,s]
L 4−q (×[0,T ]×D)6 2q + M b j 2L 2 (×[0,T ]×D)6 + yn L q+2 (×[0,T ]×D)6 b j 2 2(q+2) L 2−q (×[0,T ]×D)6 2 2 + b j (s) L 2+ε (;L 2 ([0,T ]×D))6 + b j L 2 (×[0,T ]×D) + yn 2L 2 (×[0,T ]×D) 0
for any ε > 0. In the case q = 2, the same argument yields
Strong solutions to a nonlinear stochastic Maxwell equation
t
E sup |Idet (t)| 0
s∈[0,t]
E sup n (r ) 2L 2 (D)6 ds + yn 2L 4 (×[0,T ]×D)6 b j 4L 8 (×[0,T ]×D)6 r ∈[0,s]
+ M b j 2L 2 (×[0,T ]×D)6 + yn 4L 4 (×[0,T ]×D)6 Sn−1 b j 2L ∞ (×[0,T ]×D)6 + b j 2L 2+ε (;L 2 ([0,T ]×D))6 + b j 2L 2 (×[0,T ]×D) + yn 2L 2 (×[0,T ]×D)
for any ε > 0. b j = b j for large enough n from At this point, we need the requirement Sn−1 [M5] to get rid of Sn−1 . Note that we already bounded yn L q+2 (×[0,T ]×D)6 and
yn L 2 (×[0,T ]×D)6 in Proposition 5.3 uniformly in n. Hence, it remains to estimate the terms including b j . By the product rule for the curl operator, we have N N −∇ Bk × b j,2 −i N Bl βl (s) l=1 M b j (s) = Mb j (s) e−i l=1 Bl βl (s) + iβk (s) , e ∇ Bk × b j,1 k=1
which implies
M b j L 2 (×[0,T ]×D) N ,Bk Mb j L 2 (×[0,T ]×D) +
N
βk b j L 2 (×[0,T ]×D)
k=1
≤ Mb j L 2 (×[0,T ]×D) + b j L 2+ε (;L 2 ([0,T ]×D)) for any ε > 0. Here, we again used the fact βk ∈ L α (; C(0, T )) for every α ≥ 2 and Hölder’s inequality. It remains to bound b j L 2+ε (;L 2 ([0,T ]×D)) , but this is immediate by [M5], because we have both b j ∈ L 2 ( × [0, T ] × D)6 and b j ∈ L [0, T ] × D)6 . Altogether, we have t E sup |Idet (s)| 1 + E sup n (r ) 2L 2 (D)6 ds s∈[0,t]
0
2(q+2) 2−q
( ×
r ∈[0,s]
and the estimate only depends on B j , b j and J but not on n ∈ N. The stochastic term Istoch can be controlled in the same way as in the proof of Proposition 5.3 with the Burkholder–Davies–Gundy inequality and the assumptions on B j , b j and J together with the growth estimates for F and F . Thus, we end up with t E sup n (s) 2L 2 (D) 1 + E n (0) 2L 2 (D) + E sup n (r ) 2L 2 (D)6 ds. 0
s∈[0,t]
r ∈[0,s]
It remains to bound n (0) = Pn M Sn−1 u 0 − Pn F(Sn−1 u 0 ) + Pn
N j=1
B 2j Sn−1 u 0 − Pn
N
ib j (0)B j
j=1
+Pn J (0) in L 2 ( × D)6 independent of n ∈ N. Since both b j and J are in L 2 (; W 1,2 (0, T ; L 2 (D)6 )), the corresponding initial data b j (0) and J (0) are contained in L 2 (× D)6 .
L. Hornung
J. Evol. Equ.
As a consequence, the uniform boundedness of Sn−1 on L p (D)6 for every p ∈ (1, ∞) and of Pn on L 2 (D)6 yield E n (0) 2L 2 (D) 1 + E M Sn−1 u 0 2L 2 (D)6 + E |Sn−1 u 0 |q Sn−1 u 0 2L 2 (D)6 +
N
B j 2L ∞ (D) Sn−1 u 0 2L 2 (D)6
j=1
1 + E Mu 0 2L 2 (D)6 + E u 0 2(q+1) 2(q+1) L
(D)6
+ E u 0 2L 2 (D)6 .
Finally, an application of Gronwall’s Lemma proves the claimed result.
In Propositions 5.3 and 5.5, we derived uniform estimates for yn and n . As a consequence, we also get the uniform boundedness of F(yn ) since
F(yn )
q+2 L q+1 (×[0,T ]×D)
|yn |q+1
q+2 L q+1 (×[0,T ]×D)6
q+1
= yn L q+2 (×[0,T ]×D)6 . q+2
By Banach–Alaoglu, there exist y ∈ L 2 (; L ∞ (0, T ; L 2 (D)))6 , N ∈ L q+1 ( × [0, T ] × D)6 , ∈ L 2 (; L ∞ (0, T ; L 2 (D)))6 and subsequences, still indexed with n such that (a) yn → y for n → ∞ in the weak ∗ sense in L 2 (; L ∞ (0, T ; L 2 (D)))6 , (b) yn → y for n → ∞ in the weak sense in L 2 ( × [0, T ] × D)6 , q+2
(c) F(yn ) → N for n → ∞ in the weak sense in L q+1 ( × [0, T ] × D)6 , (d) n → for n → ∞ in the weak sense in L 2 ( × [0, T ] × D)6 , (e) n → for n → ∞ in the weak ∗ sense in L 2 (; L ∞ (0, T ; L 2 (D)))6 . Since yn is for every n ∈ N an adapted solution of the ordinary stochastic differential equation (5.4) in Pn L 2 (D)6 , we have yn ∈ L 2F ( × [0, T ] × D)6 . Consequently, since L 2F ( × [0, T ] × D)6 is a closed subspace of L 2 ( × [0, T ] × D)6 , it is also weakly closed. This implies y ∈ L 2F ( × [0, T ] × D)6 , which means that y is also adapted. In the next Lemma, we show that has the correct form that M y(t) exists in the sense of distributions and that we have y1 × ν = 0 on ∂ D. LEMMA 5.6. The process y : ×[0, T ] → L 2 (D)6 additionally satisfies y(ω, t)× ν = 0 on ∂ D for almost all ω ∈ and t ∈ [0, T ]. Moreover, we have
My +
N j=1
iβ j
∇ B j × y2 −∇ B j × y1
q+2
∈ L 2 (; L ∞ (0, T ; L 2 (D)6 )) + L q+1 ( × [0, T ] × D)6
and the identity = My − N +
N j=1
holds true.
B 2j y + iβ j
∇ B j × y2 + J −∇ B j × y1
Strong solutions to a nonlinear stochastic Maxwell equation
Proof. Let φ : × [0, T ] → ∪∞ n=1 R(Pn ) be a simple function. By weak convergence and the skew-adjointness of M, we obtain −y, Mφ L 2 (×[0,T ]×D)6 = − lim yn , Mφ L 2 (×[0,T ]×D)6 n→∞
= lim M yn , φ L 2 (×[0,T ]×D)6 n→∞ = lim n + Pn F(yn ) − Pn J n→∞
− Pn
N
B j yn − Pn
j=1
N
iβ j
j=1
= + N − J −
N
Bj y −
∇ B j × yn,2 , φ L 2 (×[0,T ]×D)6 −∇ B j × yn,1
N
j=1
iβ j
j=1
∇ B j × y2 , φ L 2 (×[0,T ]×D)6 . −∇ B j × y1
Here, we were able to drop the Pn since Pn φ = φ for large enough n. By density of p 6 simple functions and by the density of ∪∞ n=1 R(Pn ) in D(M) and in L (D) for every p ∈ (1, ∞) (see Corollary 3.6), we get −y(t), Mψ L 2 (D)6
∇ B j × y2 (t) , ψ L 2 (D)6 (5.9) −∇ B j × y1 (t)
N N = (t) + N (t) − J (t) − B j y(t) − iβ j (t) j=1
j=1
almost surely for almost every t ∈ [0, T ] and for every ψ ∈ D(M) ∩ L q+2 (D)6 . This holds true especially for all ψ ∈ Cc∞ (D)6 and hence the definition of the weak version of the curl operator in Chapter 2 yields M y(t) = (t) + N (t) −
N
∇ B j × y2 (t) − J (t) −∇ B j × y1 (t)
B 2j y(t) − iβ j (t)
j=1
almost surely for almost every t ∈ [0, T ]. This proves the claimed result in case that D = R3 since we then do not have boundary conditions. So we can assume D to be a bounded C 1 -domain for the rest of the proof. We show y1 × ν = 0 on ∂ D. Note that ψ = (0, φ) with φ ∈ C 1 (D)3 is contained in D(M) ∩ L q+2 (D). We insert this into (5.9) and get
−y1 (t), curl φ L 2 (D)3 = 2 (t) + N2 (t) − J 2 (t) −
N
B 2j y2 (t) + iβ j ∇ B j × y1 (t), φ L 2 (D)3
j=1
= − curl y1 (t), φ L 2 (D)3
almost surely for almost every t ∈ [0, T ] and for all φ ∈ C 1 (D)3 . By definition of the tangential trace in Definition 2.1, this shows y1 × ν = 0 on ∂ D almost surely for almost every t ∈ [0, T ].
L. Hornung
J. Evol. Equ.
Consequently,we pass to the limit weakly in (4.4) and obtain
N dy(t) = [M y(t) − N (t) + A(t)y(t) + J (t)] dt + i=1 bi (t) dβi (t),
(5.10)
yn (0) = u 0 .
as an equation in L 2 (; L ∞ (0, T ; L 2 (D)))6 . So far, we just showed y ∈ L 2 (; L ∞ (0, T ; L 2 (D)6 )). However, Lemma 4.2 implies pathwise continuity of t → y(t) ∈ L 2 (D)6 . It remains to show N (t) = F(y(t)). But this proof is step by step the same as in Proposition 4.5 and uses the monotonicity of the deterministic part of the equation. All in all, we showed that y ∈ L q+2 ( × [0, T ] × D)6 ∩ L 2 (; C(0, T ; L 2 (D)6 )) solves
N bi (t) dβi (t), dy(t) = [M y(t) − F(y(t)) + A(t)y(t) + J (t)] dt + i=1 (5.11) yn (0) = u 0 . as an equation in L 2 (; L ∞ ([0, T ]; L 2 (D)))6 . Transforming the equation backwards with Proposition 5.2, we get the following result. PROPOSITION 5.7. (MSEE) with G ≡ 0 has a unique strong solution u satisfying with u ∈ L q+2 ( × [0, T ] × D)6 ∩ L 2 (; C(0, T ; L 2 (D)))6 and q+2
Mu ∈ L q+1 ( × [0, T ] × D)6 + L 2 (; L ∞ (0, T ; L 2 (D)))6 . Proof. The product rule yields M(e
i
N j=1
Bjβj
y) = e
i
N j=1
Bjβj
M y + ie
i
N j=1
Bjβj
N j=1
βj
∇ B × y2 −∇ B × y1
and hence, we have M(ei
N j=1
Bjβj
q+2
y) ∈ L q+1 ( × [0, T ] × D)6 + L 2 (; L ∞ (0, T ; L 2 (D)))6
if and only if
My + i
N j=1
βj
∇ B × y2 −∇ B × y1
q+2
∈ L q+1 ( × [0, T ] × D)6 + L 2 (; L ∞ (0, T ; L 2 (D)))6 .
This holds true by 4.4. Consequently, we can apply Proposition 5.2 and obtain a solution u of (MSEE) with G ≡ 0. Uniqueness is immediate by Proposition 4.5, since our solution is also a weak solution of the equation.
Strong solutions to a nonlinear stochastic Maxwell equation
Last but not least, we want to add the term (G ∗ u). This leads to the main result of this article. THEOREM 5.8. (MSEE) has a unique solution u satisfying with u ∈ L q+2 ( × [0, T ] × D)6 ∩ L 2 (; C(0, T ; L 2 (D)))6 and q+2
Mu ∈ L q+1 ( × [0, T ] × D)6 + L 2 (; L ∞ ([0, T ]; L 2 (D)))6 . Proof. Let u ∈ L q+2 (×[0, T ]× D)6 ∩ L 2 (; C(0, T ; L 2 (D)6 )) be the unique t weak solution of (MSEE) from Proposition (4.6). The expression (G ∗ u)(t) = 0 G(t − s)u(s) ds is differentiable in time with t G (t − s)u(s) ds . ∂t (G ∗ u)(t) = G(0)u(t) + 0
By [M5], both (G ∗ u) and ∂t (G ∗ u) are contained in L 2 ( × [0, T ] × D)6 . Hence, u is a solution of (MSEE) with the current G ∗ u + J satisfying [M4]. Consequently, u has the regularity properties from Proposition 5.7. This closes the proof. 6. Remarks and discussion In this section, we want to compare our results to the literature and we discuss some instructive special cases of our assumptions. First, we want to mention that Roach, Stratis and Yannacopoulus already treated our equation in the deterministic setting in [28]. They claim in Theorem 11.3.14 that u (t) = κ −1 Mu(t) − κ −1 |u(t)|q u(t) + κ −1 (G ∗ u)(t) + κ −1 J (t), t ∈ [0, T ], u(0) = u 0 q+2
has a unique strong solution u ∈ L q+2 ([0, T ] × D)6 with Mu ∈ L q+1 ([0, T ] × D)6 if D ⊂ R3 is a bounded Lipschitz domain and κ : D → R6×6 is a uniformly bounded and uniformly elliptic matrix with measurable dependence in space. Their idea is to make a Galerkin approximation with respect to an orthonormal basis (h n )n of W 2 (curl, 0)(D) × W 2 (curl)(D) that is also a basis of L 2 (D)6 . However, besides many inaccuracies, they make two mistakes which cannot be fixed in a direct way. Beginning from (11.12) on p. 239, they derive T T (G ∗ u n )(s), u n (s) L 2 (D)6 ds ≤ (G ∗ u)(s), u(s) L 2 (D)6 ds lim inf n→∞
0
0
as n → ∞ as a consequence of the weak convergences of G ∗ u n → G ∗ u and u n → u in L 2 ([0, T ] × D)6 as n → ∞. However, such an argument is not available
L. Hornung
J. Evol. Equ.
in the general situation they discuss. Maybe one can fix this with strong assumptions on the convolution kernel G (see, e.g., [11]). Moreover, in their a priori estimate for the approximating problem, they implicitly use n u 0 , h j L 2 (D)6 h j ≤ C u 0 L 2(q+1) (D)6 2(q+1) 6 j=1 (D)
L
with a constant independent of n ∈ N, which would mean in our notation that the norm of Pn : L 2(q+1) (D)6 → L 2(q+1) (D)6 could be estimated independent of n. However, this is not true in general. As far as we know, such a result is only known for the Fourier basis h n (x) = einx on the torus. This is the main reason why we had to use the operators Sn that are also bounded on L p (D)6 uniformly in n. Getting back to our result, we want to point out that the restriction to q ∈ (1, 2] only comes from the Hölder estimate
F (yn )Sn−1 b j L 2 (×[0,T ]×D)6 ≤ yn L q+2 (×[0,T ]×D)6 Sn−1 b j 2 2(q+2) 2q
L
2−q
(×[0,T ]×D)6
in the proof of Proposition 5.5. Hence, if one assumes b j ≡ 0 one gets the same result as in Theorem 5.8 for all q ∈ (1, ∞). In particular, this is true for the deterministic equation. Especially, we gave a proof for the theorem of Roach, Stratis and Yannacopoulus if κ ≡ I and D is a bounded C 1 -domain or D = R3 . Next, we want to comment on the odd-looking condition
N
N Pn bi (s)e−i j=1 B j β j (s) = bi (s)e−i j=1 B j β j (s)
from [M5] for all s ∈ [0, T ], i = 1, . . . , N and for n ∈ N large enough in case that q = 2. We need it in the proof of Proposition 5.5 for the estimate
N
N
Sn bi (s)e−i j=1 B j β j (s) L ∞ (D)6 ≤ C bi (s)e−i j=1 B j β j (s) L ∞ (D)6 with a constant independent of n ∈ N. It might be possible to get this inequality without our restrictive assumption in special cases. However, we want to point out that even in the case D = R3 the boundedness of Sn on L ∞ (D)6 is wrong since it would imply the boundedness of the Hilbert transform on L ∞ (D). If the B j are constant, the assumption reduces to Pn bi (s) = bi (s) for all s ∈ [0, T ]. If D = R3 , this means that the Fourier transform b!i (s) is compactly supported on a timely independent set. In case that D is a bounded C 1 -domain, this means that bi is of the form bi (s) =
M
(k)
bi (s)h k , s ∈ [0, T ]
k=1 (k)
for some scalar-valued bi : × [0, T ] → C. Here, (h k )k is the sequence of eigenvectors of the Hodge–Laplacian, we introduced in Sect. 3.
Strong solutions to a nonlinear stochastic Maxwell equation
Last but not least, we want to discuss why we did not treat coefficients in front of the Maxwell operator. Our approach is based on the interplay of M 2 , H and the Helmholtz projection PH . In fact, we showed M 2 = H on R(PH ) and M2 = 0 on grad div N (PH ) = N (M). One might say that we added a self-adjoint operator A = − − grad div with N (A) = R(PH ) to M 2 such that the sum, namely H , generates a semigroup having generalised Gaussian bounds. This was essential for the definition of (Sn )n and (Pn )n from Sect. 3. If we now replace M by u1 ε(x)−1 curl u 2 Mε,μ = u2 −μ(x)−1 curl u 1 with the same perfect conductor boundary condition u 1 × ν = 0 on ∂ D and with uniformly bounded, positive definite and Hermitian ε, μ : D → C3×3 . Hence, we have u1 −ε(x)−1 curl μ(x)−1 curl u 1 2 = Mε,μ u2 −μ(x)−1 curl ε(x)−1 curl u 2 with the boundary condition u 1 ×ν = 0 and ε−1 curl u 2 ×ν = 0 on ∂ D. The operator 2 is then positive and self-adjoint with respect to a weighted inner product on −Mε,μ 2 L (D)6 , namely ε(x)v1 (x) · w1 (x) dx + μ(x)v2 (x) · w2 (x) dx. v, wε,μ := D
D
To adapt the our strategy from the setting with ε, μ ≡ I , we would need a weighted version of the Helmholtz projection Pε,ν . We project orthogonally with respect to ·, ·ε,μ onto (u 1 , u 2 ) ∈ L 2 (D)6 : div(εu 1 ) = 0, div(μu 2 ) = 0 and (μu 2 ) · ν = 0 on ∂ D . Analogously to A from above, we define u1 − grad div(εu 1 ) = . Aε,μ u2 − grad div(μu 2 ) 2 , M2 + One calculates that Aε,μ is symmetric with respect to ·, ·ε,μ . Moreover, Mε,μ ε,μ Aε,μ and Pε,μ have the same relationship as their counterparts with ε = μ = I. Hence, to follow our proof strategy, one has to show that the semigroup generated 2 + A by Mε,μ ε,μ on the domain 1, p curl u 1 , curl u 2 , curl μ−1 curl u 1 , curl ε−1 curl u 2 ∈ L p (D)3 , div(εu 1 ) ∈ W0 (D), div(μu 2 ) ∈ W 1, p (D), u 1 × ν = 0, (μu 2 ) · ν = 0, (ε−1 curl u 2 ) × ν = 0 on ∂ D
satisfies generalised Gaussian bounds. However, even in case of smooth ε, μ and ∂ D such a result is unknown so far.
L. Hornung
J. Evol. Equ.
Acknowledgements I gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. Moreover, I thank my advisor Lutz Weis and Roland Schnaubelt for many useful discussions and for pointing out references on the subject. I am also grateful that Peer Kunstmann answered many questions about the Hodge Laplacian and about generalised Gaussian bounds. Last but not least, I want to mention the help of Fabian Hornung and Christine Grathwohl. They read the article carefully and gave many useful comments.
REFERENCES [1] [2] [3] [4] [5]
[6] [7] [8]
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
[19]
C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci., 21(9):823–864, 1998. Guillaume Bal. Lecture notes. waves in random media. http://www.columbia.edu/~gb2030/ COURSES/E6901-Waves/Lectures-Waves.pdf, Columbia, 2006. Z. Brzezniak, F. Hornung, and L. Weis. Martingale solutions for the NLS in the energie space. to appear, 2017. Viorel Barbu and Michael Röckner. On a random scaled porous media equation. J. Differential Equations, 251(9):2494–2514, 2011. Viorel Barbu and Michael Röckner. Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise. Arch. Ration. Mech. Anal., 209(3):797–834, 2013. Viorel Barbu, Michael Röckner, and Deng Zhang. Stochastic nonlinear schrödinger equations with linear multiplicative noise: Rescaling approach. Journal of Nonlinear Science, 24(3):383–409, 2014. Viorel Barbu, Michael Röckner, and Deng Zhang. Stochastic nonlinear Schrödinger equations. Nonlinear Anal., 136:168–194, 2016. Chuchu Chen, Jialin Hong, and Liying Zhang. Preservation of physical properties of stochastic maxwell equations with additive noise via stochastic multi-symplectic methods. Journal of Computational Physics, 306:500–519, 2016. Emmanuele DiBenedetto. Degenerate parabolic equations. Universitext. Springer, New York, 1993. Giuseppe Da Prato and Jerzy Zabczyk. Stochastic equations in infinite dimensions. Encyclopedia of mathematics and its applications ; 152. Cambridge Univ. Press, Cambridge, 2. ed. edition, 2014. P. Elbau. Sequential Lower Semi-Continuity of Non-Local Functionals. ArXiv e-prints, April 2011. T.R. Field. Electromagnetic Scattering from Random Media. International Series of Monographs on Physics. OUP Oxford, 2008. Jean-Pierre Fouque, editor. Wave propagation and time reversal in randomly layered media. Stochastic Modelling and Applied Probability; 56. Springer, New York, 2007. Loukas Grafakos. Classical fourier analysis. Graduate texts in mathematics ; 249. Springer, New York, 3. ed. edition, 2014. Marlis Hochbruck, Tobias Jahnke, and Roland Schnaubelt. Convergence of an ADI splitting for Maxwell’s equations. Numer. Math., 129(3):535–561, 2015. Jialin Hong, Lihai Ji, and Liying Zhang. A stochastic multi-symplectic scheme for stochastic maxwell equations with additive noise. Journal of Computational Physics, 268:255 – 268, 2014. Dirk Hundertmark, Martin Meyries, Lars Machinek, and Roland Schnaubelt. Operator semigroups and dispersive equations. Karlsruhe, 2013. Andreas Kirsch and Frank Hettlich. The mathematical theory of time-harmonic Maxwells equations: Expansion-, integral-, and variational methods. Applied mathematical sciences; 190. Springer, Cham, 2015. Peer Christian Kunstmann and Matthias Uhl. L p -spectral multipliers for some elliptic systems. Proc. Edinb. Math. Soc. (2), 58(1):231–253, 2015.
Strong solutions to a nonlinear stochastic Maxwell equation
[20] [21] [22] [23] [24] [25]
[26] [27] [28]
Peer Christian Kunstmann and Matthias Uhl. Spectral multiplier theorems of Hörmander type on Hardy and Lebesgue spaces. J. Operator Theory, 73(1):27–69, 2015. C. Kriegler and L. Weis. Spectral multiplier theorems via H ∞ calculus and R-bounds. ArXiv eprints, December 2016. Christoph Kriegler and Lutz Weis. Paley-Littlewood decomposition for sectorial operators and interpolation spaces. Math. Nachr., 289(11-12):1488–1525, 2016. Graeme Milton. The Theory of Composites, volume 6 of Cambridge monographs on applied and computational mathematics. Cambridge, Cambridge, 2002. Marius Mitrea. Sharp Hodge decompositions, Maxwell’s equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds. Duke Math. J., 125(3):467–547, 2004. Marius Mitrea and Sylvie Monniaux. On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds. Trans. Amer. Math. Soc., 361(6):3125–3157, 2009. El-Maati Ouhabaz. Analysis of heat equations on domains. London Mathematical Society monographs ; N.S., 31. Princeton University Press, Princeton, NJ [u.a.], 2005. Claudia Prévôt and Michael Röckner. A concise course on stochastic partial differential equations, volume 1905 of Lecture Notes in Mathematics. Springer, Berlin, 2007. G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos. Mathematical analysis of deterministic and stochastic problems in complex media electromagnetics. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2012. Appendix D by George Dassios. Luca Hornung Institute for Analysis Karlsruhe Institute of Technology 76128 Karlsruhe Germany E-mail:
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