  We explore the evolution of supermassive black holes (SMBH) centered in a
circumnuclear disk (CND) as a function of the mass supply from the host galaxy
and considering different star formation laws, which may give rise to a
self-regulation via the injection of supernova-driven turbulence. A system of
equations describing star formation, black hole accretion and angular momentum
transport was solved for an axisymmetric disk in which the gravitational
potential includes contributions from the black hole, the disk and the hosting
galaxy. Our model extends the framework provided by Kawakatu et al. (2008) by
separately considering the inner and outer part of the disk, and by introducing
a potentially non-linear dependence of the star formation rate on the gas
surface density and the turbulent velocity. The star formation recipes are
calibrated using observational data for NGC 1097, while the accretion model is
based on turbulent viscosity as a source of angular momentum transport in a
thin viscous accretion disk. We find that current data provide no strong
constraint on the star formation recipe, and can in particular not distinguish
between models entirely regulated by the surface density, and models including
a dependence on the turbulent velocity. The evolution of the black hole mass,
on the other hand, strongly depends on the applied star formation law, as well
as the mass supply from the host galaxy. We suggest to explore the star
formation process in local AGN with high-resolution ALMA observations to break
the degeneracy between different star formation models.
