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 $dot{M}_{rm SF}approx 0.8dot{M}_{rm in}$, 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.
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