Project A5 • Qualitative behavior of nonlinear Maxwell equations

Principal investigators

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. On the other hand, they pose many challenging mathematical problems, and so far various basic questions have been settled only partially at most. The equations contain material laws for polarization, magnetization and conductivity. If the constitutive relations for polarization and magnetization are linear, there exists a satisfactory theory.

However, nonlinear expressions for the polarization or the magnetization occur in many applications. A well-known example for instantaneous material laws is the Kerr nonlinearity for the polarization in nonlinear optics. Similar power-type laws for the magnetization are used for ferromagnetic materials. Other classes of material laws include retardations or additional evolution equations for the polarization or the magnetization. For the corresponding quasilinear Maxwell equations, even the wellposedness theory is much less complete than in the case of linear relations for polarization and magnetization. So far the results have been restricted to special cases or require highly regular data.

A further challenge is presented by stochastic Maxwell equations modeling electromagnetic fields in a random environment. As yet, there are rigorous results only for linear Maxwell equations with retarded constitutive relations for chiral media or with the Drude–Born–Fedorov material laws, perturbed by additive or multiplicative Gaussian noise.

In this project we study basic wellposedness questions and treat the qualitative behavior of solutions of nonlinear Maxwell equations. It will also provide the 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 Ampére's and Faraday's laws

\begin{align*} \partial_t \mathbf{D} &= \operatorname{curl} \mathbf{H} -\mathbf{J},\\ \partial_t \mathbf{B} &= -\operatorname{curl} \mathbf{E}. \end{align*}

These equations have to be complemented by constitutive relations, as mentioned above. In the first funding period we have focused on instantaneous nonlinear material laws of the form \[ (\mathbf{D}, \mathbf{B})= \theta(x,\mathbf{E}, \mathbf{H}) \qquad \text{and} \qquad \mathbf{J}=\sigma(x,\mathbf{E},\mathbf{H})\mathbf{E} + \mathbf{J}_0,\] where \(\sigma\) is the conductivity and \(\mathbf{J}_0\) a given current. A well-known isotropic example is the Kerr law \(\mathbf{D}=\mathbf{E}+\chi|\mathbf{E}|^2\mathbf{E}\) and \(\mathbf{B}=\mathbf{H}\) with \(\mathbf{J}=0\).

Local wellposedness for instantaneous material laws

On domains \(G\neq \mathbb{R}^3\) with compact smooth boundary, one has to add boundary conditions to the Maxwell system. One typically chooses those of a perfect conductor \[ \mathbf{E}\times \nu= 0 \qquad \text{and} \qquad \mathbf{B}\cdot\nu=0 \] or absorbing ones \[ \mathbf{H} \times \nu +(\zeta (\mathbf{E}\times \nu)(\mathbf{E}\times \nu)) \times \nu =0\] on the compact smooth boundary \(\partial G\). Here \(\nu\) is the outer unit normal and \(\zeta>0\) the boundary conductivity. In composite materials one has to study the corresponding interface problems.

For instantaneous material laws, no satisfying local wellposedness results were known in such cases. Assuming that the derivative of \(\theta\) with respect to \((\mathbf{E}, \mathbf{H})\) is symmetric and positive definite, we established a comprehensive local wellposedness and regularity theory in Sobolev spaces \(H^m(G)\) with \(m\geq 3\). In our arguments we used linearization arguments and a detailed regularity theory for non-autonomous linear systems of Maxwell type. Here the main challenge was to show full normal regularity despite the characteristic boundary.

Our papers: [Spi19], [Spi18], [RS18a], [RS18b].

Decay and blow-up

Conductivities \(\sigma\) or \(\zeta\) in the interior or at the boundary exhibit a damping of the electric fields. For strictly positive \(\sigma\) or \(\zeta\) we were able to show that for small initial fields the solution exists for all times and decays exponentially. These investigations combine results and methods of our local wellposedness theory with dissipation and observability-type estimates. For linear material laws we have shown exponential decay for all data under semilinear boundary damping including delays.

On the other hand, we have constructed fields which blow up in \(H(\operatorname{curl})\) in finite time. This was achieved for a large class of nonlinearities including those of Kerr-type and also saturated ones. For the blow-up example we employed periodic boundary conditions.

Our papers: [AP19], [DNS18], [LPS19], [PS20].

Global existence for a stochastic problem

In applications often retarded material laws occur. So far we have looked at model problems such as \[ \mathbf{D}(t)= \varepsilon(x) \mathbf{E}(t) + \int_0^t \gamma_1(t-s) E(s)\,\text{d}s + \int_0^t \gamma_2(t-s) |E(s)|^\alpha E(s)\,\text{d}s \] for positve definite matrices \(\varepsilon(x)\) and some \(\alpha>0\). If the kernels \(\gamma_j\) are differentiable, the Maxwell system simplifies to a semilinear problem after applying the time derivative to the convolutions. Even under stochastic perturbations, we could show global existence and uniqueness using methods from monotone operators and functional calculi. (However, so far we had to neglect the resulting convolution with \(\gamma_2'\).)

Our paper: [Hor18].

Publications

1. M. Pokojovy and R. Schnaubelt. Boundary stabilization of quasilinear Maxwell equations. J. Differential Equations, 268(2):784–812, January 2020. URL https://doi.org/10.1016/j.jde.2019.08.032. [preprint] [bibtex]

2. H. Demchenko, A. Anikushyn, and M. Pokojovy. On a Kelvin–Voigt viscoelastic wave equation with strong delay. SIAM J. Math. Anal., 51(6):4382–4412, November 2019. URL https://doi.org/10.1137/18M1219308. [preprint] [bibtex]

3. I. Lasiecka, M. Pokojovy, and R. Schnaubelt. Exponential decay of quasilinear Maxwell equations with interior conductivity. NoDEA Nonlinear Differential Equations Appl., 26(6):Paper No. 51, October 2019. URL https://doi.org/10.1007/s00030-019-0595-1. Online first. [preprint] [bibtex]

4. I. Lasiecka, M. Pokojovy, and X. Wan. Long-time behavior of quasilinear thermoelastic Kirchhoff–Love plates with second sound. Nonlinear Anal., 186:219–258, September 2019. URL https://doi.org/10.1016/j.na.2019.02.019. [preprint] [bibtex]

5. A. Anikushyn and M. Pokojovy. 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 2019. URL https://doi.org/10.1016/j.jmaa.2019.02.042. [preprint] [bibtex]

6. M. Spitz. Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions. J. Differential Equations, 266(8):5012–5063, April 2019. URL https://doi.org/10.1016/j.jde.2018.10.019. [preprint] [bibtex]

7. L. Hornung. Strong solutions to a nonlinear stochastic Maxwell equation with a retarded material law. J. Evol. Equ., 18(3):1427–1469, September 2018. URL https://doi.org/10.1007/s00028-018-0448-0. [preprint] [bibtex]

8. P. D'Ancona, S. Nicaise, and R. Schnaubelt. Blow-up for nonlinear Maxwell equations. Electron. J. Differential Equations, Paper No. 73, 9, March 2018. URL https://ejde.math.txstate.edu/Volumes/2018/73/abstr.html. [preprint] [bibtex]

9. A. Anikushyn and M. Pokojovy. Multidimensional thermoelasticity for nonsimple materials—well-posedness and long-time behavior. Appl. Anal., 96(9):1561–1585, February 2017. URL https://doi.org/10.1080/00036811.2017.1295447. [preprint] [bibtex]

Preprints

10. R. Schippa. Resolvent estimates for the time-harmonic Maxwell's equations in the partially anisotropic case. CRC 1173 Preprint 2021/11, Karlsruhe Institute of Technology, March 2021. [bibtex]

11. P. D'Ancona and R. Schnaubelt. Global Strichartz estimates for an inhomogeneous Maxwell system. CRC 1173 Preprint 2020/28, Karlsruhe Institute of Technology, October 2020. [bibtex]

12. S. Kim, M. Pokojovy, and X. Wan. The taut string approach to statistical inverse problems: theory and applications. CRC 1173 Preprint 2020/19, Karlsruhe Institute of Technology, July 2020. [bibtex]

13. R. Schnaubelt and M. Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. CRC 1173 Preprint 2018/46, Karlsruhe Institute of Technology, December 2018. [bibtex]

14. R. Schnaubelt and M. Spitz. Local wellposedness of quasilinear Maxwell equations with conservative interface conditions. CRC 1173 Preprint 2018/35, Karlsruhe Institute of Technology, November 2018. [bibtex]

15. M. Spitz. Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions. CRC 1173 Preprint 2018/8, Karlsruhe Institute of Technology, May 2018. [bibtex]

Theses

16. L. Hornung. Semilinear and quasilinear stochastic evolution equations in Banach spaces. PhD thesis, Karlsruhe Institute of Technology (KIT), December 2017. [bibtex]

17. M. Spitz. Local wellposedness of nonlinear Maxwell equations. PhD thesis, Karlsruhe Institute of Technology (KIT), July 2017. [bibtex]