A quasilinear transmission problem with application to Maxwell equations with a divergence-free $\mathcal D$-field
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arXiv
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| Format: | Preprint |
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2021
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| author | Dohnal, Tomáš Romani, Giulio Tietz, Daniel P. |
| author_facet | Dohnal, Tomáš Romani, Giulio Tietz, Daniel P. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_08513 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A quasilinear transmission problem with application to Maxwell equations with a divergence-free $\mathcal D$-field Dohnal, Tomáš Romani, Giulio Tietz, Daniel P. Analysis of PDEs 35J62 35R05 35B65 35B45 35Q61 78M45 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. |
| title | A quasilinear transmission problem with application to Maxwell equations with a divergence-free $\mathcal D$-field |
| topic | Analysis of PDEs 35J62 35R05 35B65 35B45 35Q61 78M45 |
| url | https://arxiv.org/abs/2109.08513 |