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

Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights

We investigate the properties of a class of weighted vector-valued $L_p$-spaces and the corresponding (an)isotropic Sobolev-Slobodetskii spaces. These spaces arise naturally in the context of maximal $L_p$-regularity for parabolic initial-boundary value problems. Our main tools are operators with a bounded \calH^\infty-calculus, interpolation theory, and operator sums.Comment: This is a preprint version. Published in Journal of Functional Analysis 262 (2012) 1200-122

Boundary Stabilization of Quasilinear Maxwell Equations

We investigate an initial-boundary value problem for a quasilinear nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary condition of Silver & M\"uller type in a smooth, bounded, strictly star-shaped domain of $\mathbb{R}^{3}$. Imposing usual smallness assumptions in addition to standard regularity and compatibility conditions, a nonlinear stabilizability inequality is obtained by showing nonlinear dissipativity and observability-like estimates enhanced by an intricate regularity analysis. With the stabilizability inequality at hand, the classic nonlinear barrier method is employed to prove that small initial data admit unique classical solutions that exist globally and decay to zero at an exponential rate. Our approach is based on a recently established local well-posedness theory in a class of $\mathcal{H}^{3}$-valued functions.Comment: 22 page

Stochastic equations with boundary noise

We study the wellposedness and pathwise regularity of semilinear non-autonomous parabolic evolution equations with boundary and interior noise in an $L^p$ setting. We obtain existence and uniqueness of mild and weak solutions. The boundary noise term is reformulated as a perturbation of a stochastic evolution equation with values in extrapolation spaces.Comment: submitte

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

Structurally damped plate and wave equations with random point force in arbitrary space dimensions

In this paper we consider structurally damped plate and wave equations with point and distributed random forces. In order to treat space dimensions more than one, we work in the setting of $L^q$--spaces with (possibly small) $q\in(1,2)$. We establish existence, uniqueness and regularity of mild and weak solutions to the stochastic equations employing recent theory for stochastic evolution equations in UMD Banach spaces.Comment: accepted for publication in Differential and Integral Equation

Stable foliations near a traveling front for reaction diffusion systems

We establish the existence of a stable foliation in the vicinity of a traveling front solution for systems of reaction diffusion equations in one space dimension that arise in the study of chemical reactions models and solid fuel combustion. In this way we complement the orbital stability results from earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential spectrum of the differential operator obtained by linearization at the front touches the imaginary axis. In spaces with exponential weights, one can shift the spectrum to the left. We study the nonlinear equation on the intersection of the unweighted and weighted spaces. Small translations of the front form a center unstable manifold. For each small translation we prove the existence of a stable manifold containing the translated front and show that the stable manifolds foliate a small ball centered at the front

Strong convergence of solutions to nonautonomous Kolmogorov equations

We study a class of nonautonomous, linear, parabolic equations with unbounded coefficients on $\mathbb R^{d}$ which admit an evolution system of measures. It is shown that the solutions of these equations converge to constant functions as $t\to+\infty$. We further establish the uniqueness of the tight evolution system of measures and treat the case of converging coefficients

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