# Project A5 • Qualitative behavior of nonlinear Maxwell equations

### Principal investigators

 Prof. Dr. Roland Schnaubelt (7/2015 - ) Prof. Dr. Lutz Weis (7/2015 - 6/2019)

### Project summary

Maxwell equations are the fundamental laws governing electromagnetic theory, and they are one of the main building blocks for coupled systems involving electromagnetic fields. The equations contain material laws describing the response of the material to the field. Here nonlinear expressions occur in many applications. For the resulting quasilinear or retarded Maxwell systems, little had been known even about the wellposedness theory. In this project we study basic wellposedness questions and treat the qualitative behavior of solutions of nonlinear Maxwell equations. It also provides background and methods for the error analysis in other projects, in particular A4.

The Maxwell system connects the electric fields $\mathbf E$ and $\mathbf D$, the magnetic fields $\mathbf B$ and $\mathbf H$, and the current $\mathbf J$ through Ampère's and Faraday's laws $\partial_t\mathbf D =\text{curl }\mathbf H-\mathbf J, \qquad \partial_t\mathbf B=-\text{curl }\mathbf E, \qquad t\ge0, \ x\in G,\tag{1}$ on a (smooth) spatial domain $G\subseteq \mathbb{R}^3$. These equations have to be complemented by constitutive relations like $\mathbf D= \mathbf E+ \mathbf P(\mathbf E)$ and $\mathbf H=\mathbf B-\mathbf M(\mathbf B)$ for the polarization $\mathbf P$ and the magnetization $\mathbf M$. So far we have focused on instantaneous nonlinear material laws of the form $\mathbf D=\varepsilon(x,\mathbf E)\mathbf E, \qquad \mathbf B=\mu(x,\mathbf H)\mathbf H, \qquad \mathbf J=\sigma(x,\mathbf E)\mathbf E+\mathbf J_0\tag{2}$ for a given current $\mathbf J_0$ and the permittivity $\varepsilon$, permeability $\mu$, and conductivity $\sigma$, which can take values in positive definite matrices in the anisotropic case. A well-known isotropic example is the Kerr law $\mathbf D=\mathbf E+\chi |\mathbf E|^2\mathbf E$ and $\mathbf B=\mathbf H$ for a scalar function $\chi$. Anisotropic laws occur if the crystal structure changes the orientation of the fields. Matrix-valued coefficients also occur in isotropic quasilinear problems after differentiating (1) in $t$ or $x$, as needed to control derivatives of $(\mathbf E,\mathbf H)$.

In nonlinear optics one often investigates retarded laws, for instance of the form $\mathbf B=\mathbf H$ and \begin{align*} \mathbf D(t,x) & = \varepsilon(x)\mathbf E(t,x)+\int_{-\infty}^t \chi_1(t-s,x)\mathbf E(s,x)\,\mathrm{d}s \tag{3}\\ & \quad + \int_{-\infty}^t \int_{-\infty}^t \chi_2(t-s_1,t-s_2,x)\bigl(\mathbf E(s_1,x), \mathbf E(s_2,x)\bigr) \, \mathrm{d}s_1\,\mathrm{d}s_2 \ + \ \mathrm{h.o.t.},\notag \end{align*} where the susceptibilities $\chi_j(\tau_1,\dots, \tau_j,x):\mathbb{R}^{3j}\to\mathbb{R}^3$ are multilinear. If $\chi_j$ are differentiable in $t$, $\partial_t\mathbf D$ only depends on $\mathbf E$ in a nonlinear way, so that the Maxwell equations (1) become a semilinear system which is nonlocal in time.

## Local wellposedness for instantaneous material laws

On domains $G\neq \mathbb{R}^3$ one has to add boundary conditions to the Maxwell system (1). One typically chooses those of a perfect conductor $\mathbf E\times\nu=0\qquad \text{and} \qquad \mathbf B\cdot\nu=0 \tag{4}$ or absorbing ones $\mathbf H\times \nu+ (\zeta(\mathbf E\times \nu)(\mathbf E\times\nu))\times \nu=0 \tag{5}$ on the boundary $\partial G$. Here $\nu$ is the outer unit normal and $\zeta>0$ the boundary conductivity. Composite materials are modeled by corresponding interface problems.

For instantaneous material laws, in the first funding period we established a comprehensive local wellposedness theory in Sobolev spaces $H^m(G)$ with $m\ge3$, assuming that the derivative of $(\varepsilon(\mathbf E)\mathbf E,\mu(\mathbf H)\mathbf H)$ with respect to $(\mathbf E,\mathbf H)$ is symmetric and positive definite, see [SSp22, Spi22, SSp21, Spi19, Spi17]. In our arguments we used linearization arguments and established a detailed regularity theory for non-autonomous linear Maxwell systems by energy methods. Here the main challenge was to show full regularity in normal direction despite the characteristic boundary. On the other hand, in [DNS18] we constructed initial fields which blow up in $H(\text{curl})$ in finite time for, e.g., (saturated) Kerr-type laws.

In the second funding period, jointly with project A6 we have shown global wellposedness for a one dimensional wave equation with a fully non-linear boundary condition that arises in the study of polarized travelling wave solutions to the Maxwell–Kerr system, see [ORS22]. Moreover, based on the analysis in [SSp22], in [DST22] we have constructed wave packets (so-called surface solitons) for the two dimensional Maxwell system which are located on an interface between two different Kerr materials and which are close to true solutions on large time scales.

## Strichartz estimates and improved local wellposedness

In the second phase we focused on dispersive properties for the linear Maxwell system in media with rough coefficients. These are quantified by Strichartz estimates which control mixed space-time Lebesgue norms of the solutions $(\mathbf E,\mathbf H)$. In known results for scalar wave equations, a loss of regularity occurs in the estimates if the coefficients have regularity below $C^2$, and the range of exponents for the Lebesgue norms is reduced if one works on domains $G$ with (Dirichlet or Neumann) boundary conditions. Moreover, global-in-time estimates require, e.g., non-trapping conditions to reduce back reflections that counteract dispersion.

In our proofs, by means of Littlewood–Paley decompositions, the FBI transform, and commutator estimates, we reduce the problem to solutions which are localized in a space-time unit cube and have frequencies $(\tau,\xi)$ near a number $\lambda\in 2^\mathbb{N}$. Here we adapt ideas from the wave case. For the Maxwell system, one needs regularity properties of the charges $(\text{div }\mathbf D, \text{div }\mathbf B)$ to control the large kernel of $\text{curl}$. To bound such localized pieces of $(\mathbf E,\mathbf H)$, one uses Fourier extension estimates for the characteristic surface $S=\{\ell(t,x,\tau,\xi)=0\}$ of the main symbol $\ell$ of the Maxwell system. In the isotropic case, the slice $S_\tau$ of $S$ with fixed $(t,x,\tau)$ is an ellipsoid as for the scalar wave equation. If $\varepsilon$ has two different eigenvalues (and $\mu=1$, for example), $S_\tau$ consists of two ellipsoids touching in two points. For three different eigenvalues, $S_\tau$ becomes the Fresnel surface having two sheets that are linked by four conical singularities surrounded by so-called Hamilton circles where one principle curvature vanishes, see Figure 1. This loss of curvature leads to a reduced range for the exponents in the Strichartz estimate.

For the two dimensional Maxwell system and in the isotropic 3D case, in [ScS22, BS22, Sch21] we obtained Strichartz inequalities which are analogous to those obtained for scalar wave equations on $\mathbb{R}^d$ by Tataru and on domains by Blair, Smith and Sogge. In the 2D case we proved the sharpness of the regularity loss caused by rough coefficients. The problem on domains is reduced to the full space by reflection arguments, which lower the regularity of coefficients, see [BS22]. In [DS22] we have established global-in-time results for isotropic laws on $\mathbb{R}^3$. A crucial ingredient of the proofs in [DS22] are estimates of the resolvent for the corresponding time-harmonic problem in weighted $L^2$-spaces.

For certain classes of anisotropic systems on $\mathbb{R}^3$ in [SSc22, Sch21] we showed Strichartz estimates with some regularity loss compared to the above results. In the fully anisotropic case we used a new Fourier extension result on the singular Fresnel surface, which is based on techniques from [MS22] for constant coefficients.

The Strichartz estimates allow us to improve the local wellposedness theory of certain quasilinear Maxwell equations compared to energy methods, since they provide additional control on derivatives of $(\mathbf E,\mathbf H)$. On domains we also need the wellposedness results from [Spi22, Spi19] of the first period as a starting point.

Closely related to the proof of Strichartz estimates are resolvent estimates for time-harmonic Maxwell equations in $L^p$-spaces. In various situations, we established the first $L^p$ – $L^q$ resolvent estimates and resulting limiting absorption principles, see [Sch22, CM21, MS22]. In the paper [CS22], Strichartz and resolvent estimates were proven for operators with generic two dimensional characteristic surfaces.

## Decay caused by conductivity

Conductivities $\sigma$ or $\zeta$ in the interior of $G$, see (2), or at the boundary, see (5), act as a damping on the electromagnetic fields. In the first phase, we showed for small initial fields that the solution exists for all times and decays exponentially, if $\sigma$ or $\zeta$ are strictly positive, see [LPS19] or [PS20], respectively. These investigations combine results and methods of our local wellposedness theory with dissipation and observability-type estimates. These were the first results of this kind for the Maxwell system. For linear material laws we have shown exponential decay for all data under nonlinear boundary damping with delays in [AP19].

During the second funding period, in [NS22] we have extended the decay result in [PS20] to nonlinear boundary damping and removed a severe geometrical restriction. The result is based on additional trace regularity for the solutions to the quasilinear problem with an absorbing boundary (5).

## Retarded material laws

In the first funding period our paper [Hor18] provided global existence for a simplified retarded material law in the presence of stochastic perturbations.

In the second phase we have shown local wellposedness in $H^2(G)$ for the boundary conditions (4) and retarded material laws (3) of 'scalar type' (where $\mathbf D$ inherits the boundary condition of $\mathbf E$), see [BS22]. This has been achieved by semigroup theory and perturbative methods. In the literature only the full space case had been treated.

## Publications

1. and . Local wellposedness of quasilinear Maxwell equations with conservative interface conditions. Commun. Math. Sci., 20(8):2265–2313, November . [preprint] [bibtex]

2. and . On quasilinear Maxwell equations in two dimensions. Pure Appl. Anal., 4(2):313–365, October . [preprint] [bibtex]

3. and . Fourier transform of surface-carried measures of two-dimensional generic surfaces and applications. Commun. Pure Appl. Anal., 21(9):2873–2889, September . [preprint] [bibtex]

4. and . Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner–Riesz estimates with negative index for non-elliptic surfaces. Ann. Henri Poincaré, 23(5):1831–1882, May . [preprint] [files] [bibtex]

5. and . Global Strichartz estimates for an inhomogeneous Maxwell system. Commun. Partial Differ. Equ., 47(3):630–675, March . [preprint] [bibtex]

6. . Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions. J. Math. Anal. Appl., 506(1):125646, February . [preprint] [bibtex]

7. . Resolvent estimates for time-harmonic Maxwell's equations in the partially anisotropic case. J. Fourier Anal. Appl., 28(2):16, February . [preprint] [bibtex]

8. and . A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equation. J. Funct. Anal., 281(11):109233, December . [preprint] [bibtex]

9. and . Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evol. Equ. Control Theory, 10(1):155–198, March . [preprint] [bibtex]

10. and . Boundary stabilization of quasilinear Maxwell equations. J. Differential Equations, 268(2):784–812, January . [preprint] [bibtex]

11. , , and . Exponential decay of quasilinear Maxwell equations with interior conductivity. Nonlinear Differ. Equ. Appl., 26(6):Paper No. 51, December . [preprint] [bibtex]

12. , , and . On a Kelvin–Voigt viscoelastic wave equation with strong delay. SIAM J. Math. Anal., 51(6):4382–4412, November . [preprint] [bibtex]

13. , , and . Long-time behavior of quasilinear thermoelastic Kirchhoff–Love plates with second sound. Nonlinear Anal., 186:219–258, September . [preprint] [bibtex]

14. and . Global well-posedness and exponential stability for heterogeneous anisotropic Maxwell's equations under a nonlinear boundary feedback with delay. J. Math. Anal. Appl., 475(1):278–312, July . [preprint] [bibtex]

15. . Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions. J. Differential Equations, 266(8):5012–5063, April . [preprint] [bibtex]

16. . Strong solutions to a nonlinear stochastic Maxwell equation with a retarded material law. J. Evol. Equ., 18(3):1427–1469, September . [preprint] [bibtex]

17. , , and . Blow-up for nonlinear Maxwell equations. Electron. J. Differ. Equ., Paper No. 73, 9, March . [preprint] [bibtex]

18. and . Multidimensional thermoelasticity for nonsimple materials—well-posedness and long-time behavior. Appl. Anal., 96(9):1561–1585, February . [preprint] [bibtex]

## Preprints

1. and . Strichartz estimates for Maxwell equations on domains with perfectly conducting boundary conditions. CRC 1173 Preprint 2022/78, Karlsruhe Institute of Technology, December . [bibtex]

2. and . Local wellposedness of Maxwell systems with scalar-type retarded material laws. CRC 1173 Preprint 2022/68, Karlsruhe Institute of Technology, November . [bibtex]

3. and . Strichartz estimates for Maxwell equations in media: the fully anisotropic case. CRC 1173 Preprint 2022/65, Karlsruhe Institute of Technology, November . [bibtex]

4. and . Normal trace inequalities and decay of solutions to the nonlinear Maxwell equations with absorbing boundary. CRC 1173 Preprint 2022/63, Karlsruhe Institute of Technology, November . [bibtex]

5. , , and . Wellposedness for a $(1+1)$-dimensional wave equation with quasilinear boundary conditions. CRC 1173 Preprint 2022/54, Karlsruhe Institute of Technology, October . [bibtex]

6. , , and . Rigorous envelope approximation for interface wave-packets in Maxwell's equations in 2D localization. CRC 1173 Preprint 2022/26, Karlsruhe Institute of Technology, June . [files] [bibtex]

7. . Strichartz estimates for Maxwell equations in media: the partially anisotropic case. CRC 1173 Preprint 2021/36, Karlsruhe Institute of Technology, August . Revised version from September 2022. [bibtex]

## Theses

1. . Semilinear and quasilinear stochastic evolution equations in Banach spaces. PhD thesis, Karlsruhe Institute of Technology (KIT), December . [bibtex]

2. . Local wellposedness of nonlinear Maxwell equations. PhD thesis, Karlsruhe Institute of Technology (KIT), July . [bibtex]