  An important unresolved question in supersolid research is the degree to
which the non-classical rotational inertia (NCRI) phenomenon observed in the
torsional oscillator experiments of Kim and Chan, is evidence for a
Bose-condensed supersolid state with superfluid-like properties. In an open
annular geometry, Kim and Chan found that a fraction of the solid moment of
inertia is decoupled from the motion of the oscillator; however, when the
annulus is blocked by a partition, the decoupled supersolid fraction is locked
to the oscillator being accelerated by an AC pressure gradient generated by the
moving partition. These observations are in accord with superfluid
hydrodynamics. We apply a low frequency AC pressure gradient in order to search
for a superfluid-like response in a supersolid sample. Our results are
consistent with zero supersolid flow in response to the imposed low frequency
pressure gradient. A statistical analysis of our data sets a bound, at the 68%
confidence level, of 9.6$\times 10^{-4}$ nm/s for the mass transport velocity
carried by a possible supersolid flow. In terms of a simple model for the
supersolid, an upper bound of 3.3$\times 10^{-6}$ is set for the supersolid
fraction at 25 mK, at this same confidence level. These findings force the
conclusion that the NCRI observed in the torsional oscillator experiments is
not evidence for a frequency independent superfluid-like state. Supersolid
behavior is a frequency-dependent phenomenon, clearly evident in the frequency
range of the torsional oscillator experiments, but undetectably small at
frequencies approaching zero.
