Well-posedness for Semi-relativistic Hartree Equations of Critical Type
Tóm tắt
We prove local and global well-posedness for semi-relativistic, nonlinear Schrödinger equations
$i \partial_t u = \sqrt{-\Delta + m^2} u + F(u)$
with initial data in H
s
(ℝ3),
$s \geqslant 1/2$
. 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 global-in-time existence for small initial data, where the smallness condition is expressed in terms of the L
2-norm of solitary wave ground states. Our proof of well-posedness does not rely on Strichartz type estimates. As a major benefit from this, our method enables us to consider external potentials of a quite general class.
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