Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions

In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in \begin{document}$ {\mathcal{H}}^m $\end{document} for \begin{document}$ m \geq 3 $\end{document} . 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.

[1]  Matthias Eller,et al.  On Symmetric Hyperbolic Boundary Problems with Nonhomogeneous Conservative Boundary Conditions , 2012, SIAM J. Math. Anal..

[2]  Andrew J. Majda,et al.  Initial‐boundary value problems for hyperbolic equations with uniformly characteristic boundary , 1975 .

[3]  O. Gués.  Probleme Mixte Hyperbolique Quasi-Lineaire Caracteristique , 1990 .

[4]  A. Morro,et al.  Electromagnetism of Continuous Media , 2003 .

[5]  C. DeWitt-Morette,et al.  Mathematical Analysis and Numerical Methods for Science and Technology , 1990 .

[6]  Martin Spitz.  Local Wellposedness of Nonlinear Maxwell Equations , 2017 .

[7]  R. Schnaubelt,et al.  Local wellposedness of quasilinear Maxwell equations with conservative interface conditions , 2018, Communications in Mathematical Sciences.

[8]  Martin Spitz.  Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions , 2018, Journal of Differential Equations.

[9]  I. Lasiecka,et al.  Exponential decay of quasilinear Maxwell equations with interior conductivity , 2018, Nonlinear Differential Equations and Applications NoDEA.

[10]  Martin Spitz,et al.  Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions , 2018, Journal of Mathematical Analysis and Applications.

[11]  S. Nicaise,et al.  BLOW-UP FOR NONLINEAR MAXWELL EQUATIONS , 2018 .

[12]  Jacques Chazarain,et al.  Introduction to the theory of linear partial differential equations , 1982 .

[13]  Jeffrey Rauch,et al.  Symmetric positive systems with boundary characteristic of constant multiplicity , 1985 .

[14]  R. Picard,et al.  Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations , 1995 .

[15]  Tosio Kato,et al.  The Cauchy problem for quasi-linear symmetric hyperbolic systems , 1975 .

[16]  M. Wegener,et al.  Periodic nanostructures for photonics , 2007 .

[17]  Paolo Secchi,et al.  Well-posedness of characteristic symmetric hyperbolic systems , 1996 .

[18]  L. Hörmander.  Linear Partial Differential Operators , 1963 .

[19]  Jeffrey Rauch,et al.  L2 is a continuable initial condition for kreiss' mixed problems , 1972 .

[20]  Sylvie Benzoni-Gavage,et al.  Multidimensional hyperbolic partial differential equations : first-order systems and applications , 2006 .

[21]  John Cagnol,et al.  Boundary regularity for Maxwell's equations with applications to shape optimization , 2011 .