14 research outputs found
Global Strichartz estimates for an inhomogeneous Maxwell system
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 and a non-trapping condition. The proof is based on smoothing estmates in weighted spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem
Blow-up for nonlinear Maxwell equations
We construct classical solutions to the nonlinear Maxwell system with periodic boundary conditions which blow up in H(curl). A similar result is shown on the full space. Our construction is based on an analysis of a shock wave in one space dimension
On the flows associated to selfadjoint operators on metric measure spaces
Let X be a metric space with a doubling measure satisfying μ(B)≳rBn for any ball B with any radius rB> 0. Let L be a non negative selfadjoint operator on L2(X). We assume that e-tL satisfies a Gaussian upper bound and that the flow eitL satisfies a typical L1- L∞ dispersive estimate of the form
‖eitL‖L1→L∞≲|t|-n/2.
Then we prove a similar L1- L∞ dispersive estimate for a general class of flows eitϕ(L), with φ(r) of power type near 0 and near ∞. In the case of fractional powers φ(L) = Lν, ν∈ (0 , 1) , we deduce dispersive estimates for eitLν with data in Sobolev, Besov or Hardy spaces HLp with p∈ (0 , 1] , associated to the operator L
L^p boundedness of the wave operator for the one dimensional Schroedinger operator
Given a one dimensional perturbed Schroedinger operator H=-(d/dx)^2+V(x) we
consider the associated wave operators W_+, W_- defined as the strong L^2
limits as s-> \pm\infty of the operators e^{isH} e^{-isH_0} We prove that the
wave operators are bounded operators on L^p for all 1<p<\infty, provided
(1+|x|)^2 V(x) is integrable, or else (1+|x|)V(x) is integrable and 0 is not a
resonance. For p=\infty we obtain an estimate in terms of the Hilbert
transform. Some applications to dispersive estimates for equations with
variable rough coefficients are given.Comment: 26 page