Exponential Decay of Quasilinear Maxwell Equations with Interior Conductivity

We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a bounded smooth domain of $\mathbb{R}^{3}$ with a strictly positive conductivity subject to the boundary conditions of a perfect conductor. Under appropriate regularity conditions, adopting a classical $L^{2}$-Sobolev solution framework, a nonlinear energy barrier estimate is established for local-in-time $H^{3}$-solutions to the Maxwell system by a proper combination of higher-order energy and observability-type estimates under a smallness assumption on the initial data. Technical complications due to quasilinearity, anisotropy and the lack of solenoidality, etc., are addressed. Finally, provided the initial data are small, the barrier method is applied to prove that local solutions exist globally and exhibit an exponential decay rate.Comment: 24 page

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 $\mathcal{H}^m$ 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.Comment: 43 page

Local wellposedness of quasilinear Maxwell equations with conservative interface conditions

We establish a comprehensive local wellposedness theory for the quasilinear Maxwell system with interfaces in the space of piecewise $H^m$-functions for $m \geq 3$. The system is equipped with instantaneous and piecewise regular material laws and perfectly conducting interfaces and boundaries. We also provide a blow-up criterion in the Lipschitz norm and prove the continuous dependence on the data. The proof relies on precise a priori estimates and the regularity theory for the corresponding linear problem also shown here.Comment: 47 page

Exponential decay of quasilinear Maxwell equations with interior conductivity

We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a bounded smooth domain $\mathbb{R}$$^{3}$ with a strictly positive conductivity subject to the boundary conditions of a perfect conductor. Under appropriate regularity conditions, adopting a classical L$^{2}$-Sobolev solution framework, a nonlinear energy barrier estimate is established for local-in-time H$^{3}$-solutions to the Maxwell system by a proper combination of higher-order energy and observability-type estimates under a smallness assumption on the initial data. Technical complications due to quasilinearity, anisotropy and the lack of solenoidality, etc., are addressed. Finally, provided the initial data are small, the barrier method is applied to prove that local solutions exist globally and exhibit an exponential decay rate

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 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

Local wellposedness of quasilinear Maxwell equations with conservative interface conditions

We establish a comprehensive local wellposedness theory for the quasilinear Maxwell system with interfaces in the space of piecewise H$^{m}$- functions for $m$ $\geq\$3. The system is equipped with instantaneous and piecewise regular material laws and perfectly conducting interfaces and boundaries. We also provide a blow-up criterion in the Lipschitz norm and prove the continuous dependence on the data. The proof relies on precise a priori estimates and the regularity theory for the corresponding linear problem also shown here

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 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

Local Wellposedness of Nonlinear Maxwell Equations

In this work we establish a local wellposedness theory of macroscopic Maxwell equations with instantaneous material laws on domains with perfectly conducting boundary. These equations give rise to a quasilinear initial boundary value problem with characteristic boundary. We provide a priori estimates and a differentiability theorem in arbitrary regularity for the corresponding linear nonautonomous hyperbolic system of partial differential equations. A fixed point argument then yields a unique solution of the nonlinear problem in $H^m$ with $m \geq 3$. We further show a blow-up criterion in the Lipschitz-norm and the continuous dependance on the data