  Nuclear rings are sites of intense star formation at the centers of barred
galaxies. To understand what determines the structure and star formation rate
(SFR; $\dot{M}_{\rm SF}$) of nuclear rings, we run semi-global, hydrodynamic
simulations of nuclear rings subject to constant mass inflow rates
$\dot{M}_{\rm in}$. We adopt the TIGRESS framework of Kim \& Ostriker to handle
radiative heating and cooling, star formation, and related supernova (SN)
feedback. We find that the SN feedback is never strong enough to destroy the
ring or quench star formation everywhere in the ring. Under the constant
$\dot{M}_{\rm in}$, the ring star formation is very steady and persistent, with
the SFR exhibiting only mild temporal fluctuations. The ring SFR is tightly
correlated with the inflow rate as $\dot{M}_{\rm SF}\approx 0.8\dot{M}_{\rm
in}$, for a range of $\dot{M}_{\rm in}=0.125-8\,M_\odot\,{\rm yr}^{-1}$. Within
the ring, vertical dynamical equilibrium is maintained, with the midplane
pressure (powered by SN feedback) balancing the weight of the overlying gas.
The SFR surface density is correlated nearly linearly with the midplane
pressure, as predicted by the pressure-regulated, feedback-modulated star
formation theory. Based on our results, we argue that the ring SFR is causally
controlled by $\dot{M}_\text{in}$, while the ring gas mass adapts to the SFR to
maintain the vertical dynamical equilibrium under the gravitational field
arising from both gas and stars.
