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March  2021, 10(1): 155-198. doi: 10.3934/eect.2020061

## Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions

 1 Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany 2 Department of Mathematics, University of Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

* Corresponding author: Roland Schnaubelt

Received  December 2018 Revised  March 2020 Published  June 2020

Fund Project: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173

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

Citation: Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061
##### References:
 [1] S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations, The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar [2] K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili and M. Wegener, Periodic nanostructures for photonics, Phys. Reports, 444 (2007), 101-202.  doi: 10.1016/j.physrep.2007.02.011.  Google Scholar [3] J. Cagnol and M. Eller, Boundary regularity for Maxwell's equations with applications to shape optimization, J. Differential Equations, 250 (2011), 1114-1136.  doi: 10.1016/j.jde.2010.08.004.  Google Scholar [4] J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations., North-Holland Publishing Co., Amsterdam-New York, 1982.  Google Scholar [5] P. D'Ancona, S. Nicaise and R. Schnaubelt, Blow-up for nonlinear Maxwell equations, Electron. J. Differential Equations, (2018), paper No. 73, 9 pp.  Google Scholar [6] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Spectral Theory and Applications, Springer-Verlag, Berlin, 2000.  Google Scholar [7] M. Eller, On symmetric hyperbolic boundary problems with nonhomogeneous conservative boundary conditions., SIAM J. Math. Anal., 44 (2012), 1925-1949.  doi: 10.1137/110834652.  Google Scholar [8] M. Eller, J. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comput. Appl. Math., 21 (2002), 135-165.   Google Scholar [9] M. Fabrizio and A. Morro, Electromagnetism of Continuous Media, Oxford University Press, Oxford, 2003.  doi: 10.1093/acprof:oso/9780198527008.001.0001.  Google Scholar [10] O. Gués, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations, 15 (1990), 595-645.  doi: 10.1080/03605309908820701.  Google Scholar [11] L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976.  Google Scholar [12] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.  Google Scholar [13] I. Lasiecka, M. Pokojovy and R. Schnaubelt, Exponential decay of quasilinear Maxwell equations with interior conductivity, NoDEA Nonlinear Differential Equations Appl., 26 (2019), no. 6, Paper No. 51, 34 pp. doi: 10.1007/s00030-019-0595-1.  Google Scholar [14] A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675.  doi: 10.1002/cpa.3160280504.  Google Scholar [15] R. H. Picard and W. M. Zajaczkowski, Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations, Math. Methods Appl. Sci., 18 (1995), 169-199.  doi: 10.1002/mma.1670180302.  Google Scholar [16] M. Pokojovy and R. Schnaubelt, Boundary stabilization of quasilinear Maxwell equations, J. Differential Equations, 268 (2020), 784-812.  doi: 10.1016/j.jde.2019.08.032.  Google Scholar [17] J. Rauch, $\mathcal{L}_2$ is a continuable initial condition for Kreiss' mixed problems, Comm. Pure Appl. Math., 25 (1972), 265-285.  doi: 10.1002/cpa.3160250305.  Google Scholar [18] J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.  doi: 10.1090/S0002-9947-1985-0797053-4.  Google Scholar [19] R. Schnaubelt and M. Spitz, Local wellposedness of quasilinear Maxwell equations with conservative interface conditions, preprint, 2018, arXiv: 1811.08714. Google Scholar [20] P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems, Arch. Rational Mech. Anal., 134 (1996), 155-197.  doi: 10.1007/BF00379552.  Google Scholar [21] M. Spitz, Local wellposedness of nonlinear Maxwell equations, Ph.D. thesis, Karlsruhe Institute of Technology, 2017. https://publikationen.bibliothek.kit.edu/1000078030. Google Scholar [22] M. Spitz, Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions, preprint, arXiv: 1805.00671. Google Scholar [23] M. Spitz, Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions, J. Differential Equations, 266 (2019), 5012-5063.  doi: 10.1016/j.jde.2018.10.019.  Google Scholar

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##### References:
 [1] S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations, The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar [2] K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili and M. Wegener, Periodic nanostructures for photonics, Phys. Reports, 444 (2007), 101-202.  doi: 10.1016/j.physrep.2007.02.011.  Google Scholar [3] J. Cagnol and M. Eller, Boundary regularity for Maxwell's equations with applications to shape optimization, J. Differential Equations, 250 (2011), 1114-1136.  doi: 10.1016/j.jde.2010.08.004.  Google Scholar [4] J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations., North-Holland Publishing Co., Amsterdam-New York, 1982.  Google Scholar [5] P. D'Ancona, S. Nicaise and R. Schnaubelt, Blow-up for nonlinear Maxwell equations, Electron. J. Differential Equations, (2018), paper No. 73, 9 pp.  Google Scholar [6] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Spectral Theory and Applications, Springer-Verlag, Berlin, 2000.  Google Scholar [7] M. Eller, On symmetric hyperbolic boundary problems with nonhomogeneous conservative boundary conditions., SIAM J. Math. Anal., 44 (2012), 1925-1949.  doi: 10.1137/110834652.  Google Scholar [8] M. Eller, J. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comput. Appl. Math., 21 (2002), 135-165.   Google Scholar [9] M. Fabrizio and A. Morro, Electromagnetism of Continuous Media, Oxford University Press, Oxford, 2003.  doi: 10.1093/acprof:oso/9780198527008.001.0001.  Google Scholar [10] O. Gués, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations, 15 (1990), 595-645.  doi: 10.1080/03605309908820701.  Google Scholar [11] L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976.  Google Scholar [12] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.  Google Scholar [13] I. Lasiecka, M. Pokojovy and R. Schnaubelt, Exponential decay of quasilinear Maxwell equations with interior conductivity, NoDEA Nonlinear Differential Equations Appl., 26 (2019), no. 6, Paper No. 51, 34 pp. doi: 10.1007/s00030-019-0595-1.  Google Scholar [14] A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675.  doi: 10.1002/cpa.3160280504.  Google Scholar [15] R. H. Picard and W. M. Zajaczkowski, Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations, Math. Methods Appl. Sci., 18 (1995), 169-199.  doi: 10.1002/mma.1670180302.  Google Scholar [16] M. Pokojovy and R. Schnaubelt, Boundary stabilization of quasilinear Maxwell equations, J. Differential Equations, 268 (2020), 784-812.  doi: 10.1016/j.jde.2019.08.032.  Google Scholar [17] J. Rauch, $\mathcal{L}_2$ is a continuable initial condition for Kreiss' mixed problems, Comm. Pure Appl. Math., 25 (1972), 265-285.  doi: 10.1002/cpa.3160250305.  Google Scholar [18] J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.  doi: 10.1090/S0002-9947-1985-0797053-4.  Google Scholar [19] R. Schnaubelt and M. Spitz, Local wellposedness of quasilinear Maxwell equations with conservative interface conditions, preprint, 2018, arXiv: 1811.08714. Google Scholar [20] P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems, Arch. Rational Mech. Anal., 134 (1996), 155-197.  doi: 10.1007/BF00379552.  Google Scholar [21] M. Spitz, Local wellposedness of nonlinear Maxwell equations, Ph.D. thesis, Karlsruhe Institute of Technology, 2017. https://publikationen.bibliothek.kit.edu/1000078030. Google Scholar [22] M. Spitz, Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions, preprint, arXiv: 1805.00671. Google Scholar [23] M. Spitz, Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions, J. Differential Equations, 266 (2019), 5012-5063.  doi: 10.1016/j.jde.2018.10.019.  Google Scholar
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