Sigmund Selberg, Piero D'Ancona
Published 2010-04-10, updated 2010-08-09Version 3
In recent work, Gr\"unrock and Pecher proved that the Dirac-Klein-Gordon system in 2d is globally well-posed in the charge class (data in $L^2$ for the spinor and in a suitable Sobolev space for the scalar field). Here we obtain the analogous result for the full Maxwell-Dirac system in 2d. Making use of the null structure of the system, found in earlier joint work with Damiano Foschi, we first prove local well-posedness in the charge class. To extend the solutions globally we build on an idea due to Colliander, Holmer and Tzirakis. For this we rely on the fact that MD is charge subcritical in two space dimensions, and make use of the null structure of the Maxwell part.
Comments: Submitted version. Filled gap in local to global argument. Data norm now depends implicitly on existence time, and there is a log. loss in the nonlinear growth estimate, hence the scheme of Colliander et al. no longer applies directly. But using a monotonicity property of our norm we nevertheless get the global result after a double iteration
Unconditional uniqueness in the charge class for the Dirac-Klein-Gordon equations in two space dimensions
Global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical Hartree equation in $\mathbb{R}^d$
Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case
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