### 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. 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].