  We explore the self-regulation of star formation using a large suite of high
resolution hydrodynamic simulations, focusing on molecule-dominated regions
(galactic centers and [U]LIRGS) where feedback from star formation drives
highly supersonic turbulence. In equilibrium the total midplane pressure,
dominated by turbulence, must balance the vertical weight of the ISM. Under
self-regulation, the momentum flux injected by feedback evolves until it
matches the vertical weight. We test this flux balance in simulations spanning
a range of parameters, including surface density $\Sigma$, momentum injected
per stellar mass formed ($p_*/m_*$), and angular velocity. The simulations are
2D radial-vertical slices, including both self-gravity and an external
potential that confines gas to the disk midplane. After the simulations reach a
steady state in all relevant quantities, including the star formation rate
$\Sigma_{SFR}$, there is remarkably good agreement between the vertical weight,
the turbulent pressure, and the momentum injection rate from supernovae. Gas
velocity dispersions and disk thicknesses increase with $p_*/m_*$. The
efficiency of star formation per free-fall time at the mid-plane density is
insensitive to the local conditions and to the star formation prescription in
very dense gas. We measure efficiencies $\sim$0.004-0.01, consistent with low
and approximately constant efficiencies inferred from observations. For
$\Sigma\in$(100--1000) \msunpc, we find $\Sigma_{SFR}\in$(0.1--4) \sfrunits,
generally following a $\Sigma_{SFR}\propto \Sigma^2$ relationship. The measured
relationships agree very well with vertical equilibrium and with turbulent
energy replenishment by feedback within a vertical crossing time. These
results, along with the observed $\Sigma_{SFR}-\Sigma$ relation in high density
environments, provide strong evidence for the self-regulation of star
formation.
