Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type

Details Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type

2005-05-22

arxiv.org/abs/math/0505456v3

Math. Phys. Anal. Geom. 10 (2007), no. 1, 43-64

Mathematics

Analysis of PDEs

18 pages; replaced with revised version; remark and reference on blow up added

Scientific paper

10.1007/s11040-007-9020-9

We prove local and global well-posedness for semi-relativistic, nonlinear Schr\"odinger equations $i \partial_t u = \sqrt{-\Delta + m^2} u + F(u)$ with initial data in $H^s(\mathbb{R}^3)$, $s \geq 1/2$. Here $F(u)$ is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type self-interactions. For focusing $F(u)$, which arise in the quantum theory of boson stars, we derive a sufficient condition for global-in-time existence in terms of a solitary wave ground state. Our proof of well-posedness does not rely on Strichartz type estimates, and it enables us to add external potentials of a general class.

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