A quasilinear transmission problem with application to Maxwell equations with a divergence-free $\mathcal D$-field

TL;DR

This paper addresses a quasilinear transmission problem related to Maxwell equations, proposing a correction method for divergence constraints in nonlinear media, with theoretical proofs and numerical validation.

Contribution

It introduces a method to construct divergence-free corrections for nonlinear Maxwell problems with transmission interfaces, including existence proofs and regularity estimates.

Findings

Existence of correction terms for nonlinear divergence constraints.

Regularity estimates independent of the original ansatz's L2-norm.

Numerical experiments confirming theoretical results.

Abstract

Maxwell equations in the absence of free charges require initial data with a divergence free displacement field $\mathcal D$. In materials in which the dependence $\mathcal D=\mathcal D(\mathcal E)$ is nonlinear the quasilinear problem $\nabla\cdot\mathcal D(\mathcal E)=0$ is hence to be solved. In many applications, e.g. in the modelling of wave-packets, an approximative asymptotic ansatz of the electric field $\mathcal E$ is used, which satisfies this divergence condition at $t=0$ only up to a small residual. We search then for a small correction of the ansatz to enforce $\nabla\cdot\mathcal D(\mathcal E)=0$ at $t=0$ and choose this correction in the form of a gradient field. In the usual case of a power type nonlinearity in $\mathcal D(\mathcal E)$ this leads to the sum of the Laplace and $p$-Laplace operators. We also allow for the medium to consist of two different materials so that a transmission problem across an interface is produced. We prove the existence of the correction term for a general class of nonlinearities and provide regularity estimates for its derivatives, independent of the $L^2$-norm of the original ansatz. In this way, when applied to the wave-packet setting, the correction term is indeed asymptotically smaller than the original ansatz. We also provide numerical experiments to support our analysis.