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We show a global in time Strichartz estimate for the isotropic Maxwell system with divergence free data. On the scalar permittivity and permeability we impose decay assumptions as $|x|\rightarrow\infty$ and a non-trapping condition. The proof is based on smoothing estmates in weighted $L^2$ spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem.

We describe the spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators A = ∑ i , j = 1 n q i j D i j + ∑ i , j = 1 n b i j x j D i in L p ( R n ) , 1 ≤ p + ∞ , and in C 0 ( R n ) . We show that the spectrum of A is the sum of ( − ∞ , 0 ] and the spectrum of the drift term. Our result gives a complete picture of the spectral properties of Ornstein-Uhlenbeck operators in L p spaces.

We investigate an alternating direction implicit scheme for the time-integration of the Maxwell equations with currents, charges, and conductivity. This method is unconditionally stable, numericall...

In this paper we investigate an alternating direction implicit (ADI) time integration scheme for the linear Maxwell equations with currents, charges and conductivity. We show its stability and efficiency. The main results establish that the scheme converges in a space similar to \begin{document}$H^{-1}$\end{document} with order two to the solution of the Maxwell system. Moreover, the divergence conditions in the system are preserved in \begin{document}$H^{-1}$\end{document} with order one.

We propose and analyze a Strang splitting method for a cubic semilinear Schrodinger equation with forcing and damping terms and subject to periodic boundary conditions. The nonlinear part is solved analytically, whereas the linear part – space derivatives, damping and forcing – is approximated by the exponential trapezoidal rule. The necessary operator exponentials and ϕ -functions can be computed efficiently by fast Fourier transforms if space is discretized by spectral collocation. Under natural regularity assumptions, we first show global existence of the problem in H 4 ( T ) and establish global bounds depending on properties of the forcing. The main result of our error analysis is first-order convergence in H 1 ( T ) and second-order convergence in L 2 ( T ) on bounded time-intervals.

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, the lack of solenoidality, and the fact that only partial dissipation is imposed on the system. 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.

We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions.

We investigate the Lie and the Strang splitting for the cubic nonlinear Schrodinger equation on the full space and on the torus in up to three spatial dimensions. We prove that the Strang splitting converges in L 2 with order 1 + θ for initial values in H 2 + 2 θ with θ ∈ ( 0 , 1 ) and that the Lie splitting converges with order one for initial values in H 2 .

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data., Comment: minor modifications, to appear in Nonlinearity

We study a quasilinear wave equation on a domain arising in nonlinear optics. We show local well-posedness for strong solutions using Kato’s approach to quasilinear evolution equations.

We investigate sectoriality and maximal regularity in L p – L q -Sobolev spaces for the structurally damped plate equation with Dirichlet–Neumann (clamped) boundary conditions. We obtain unique solutions with optimal regularity for the inhomogeneous problem in the whole space, in the half-space, and in bounded domains of class C 4 . It turns out that the first-order system related to the scalar equation on R n is sectorial only after a shift in the operator. On the half-space one has to include zero boundary conditions in the underlying function space in order to obtain sectoriality of the shifted operator and maximal regularity for the case of homogeneous boundary conditions. We further show that the semigroup solving the problem on bounded domains is exponentially stable.

A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau type approximation., Comment: Minor modifications, to appear in Proc. Amer. Math. Soc