Sigmund Selberg
Published 2006-11-23Version 1
We prove global well-posedness below the charge norm (i.e., the $L^2$ norm of the Dirac spinor) for the Dirac-Klein-Gordon system of equations (DKG) in one space dimension. Adapting a method due to Bourgain, we split off the high frequency part of the initial data for the spinor, and exploit nonlinear smoothing effects to control the evolution of the high frequency part. To prove the nonlinear smoothing we rely on the null structure of the DKG system, and bilinear estimates in Bourgain-Klainerman-Machedon spaces.
Local well-posedness below the charge norm for the Dirac-Klein-Gordon system in two space dimensions
Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case
Global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical Hartree equation in $\mathbb{R}^d$
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