Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions

2018/01/01 by Roland Schnaubelt, Schnaubelt, Roland, Martin Spitz +1
Engineering · Mathematics · #35L50 #35L60 #35Q61 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP #msc:35L50 #msc:35L60 #msc:35Q61

paper · pdf · doi:10.48550/arxiv.1812.03803

43 pages

arxiv created 2018/12/10 · openalex publication_date 2018/12/10 · arxiv updated 2018/12/11 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in H^m for m ≥ 3. The Maxwell equations are equipped with instantaneous nonlinear material laws leading to a quasilinear symmetric hyperbolic first order system. We consider both linear and nonlinear absorbing boundary conditions. We show existence and uniqueness of a local solution, provide a blow-up criterion in the Lipschitz norm, and prove the continuous dependence on the data. In the case of nonlinear boundary conditions we need a smallness assumption on the tangential trace of the solution. The proof is based on detailed apriori estimates and the regularity theory for the corresponding linear problem which we also develop here.

Citations

• The Cauchy problem for quasi-linear symmetric hyperbolic systems

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