  We consider a general class of axion models, including the QCD and string
axion, in which the PQ symmetry is broken before or during inflation. Assuming
the axion is the dominant component of the dark matter, we discuss axion star
formation in virialized dark minihalos around primordial black holes through
gravitational Bose-Einstein condensation. We determine the conditions for
minihalos to kinetically produce axion stars before galaxy formation. Today, we
expect up to $\sim 10^{17}$ ($\sim 10^9$) axion stars in a radius of 100
parsecs around the Sun for the case of the QCD (string) axion.
