  We study the collective excitation and stability of superfluid Fermi gases
flowing with a constant velocity in three-dimensional free space. In
particular, we investigate a possible gapless superfluid state induced by the
superflow using the mean-field theory and the generalized random-phase
approximation (GRPA). For weak attractive interactions, we find that the
mean-field superfluid order parameter can take a nonzero value even after the
superflow velocity exceeds the threshold for the onset of Bogoliubov
quasiparticle excitations. Since the Cooper pairs are only partially broken by
the quasiparticle excitations, a gapless superfluid state can be formed over a
certain range of superflow velocity above the pair-breaking onset. In addition
to the usual quasiparticle-pair continuum and the Anderson-Bogoliubov
collective mode, the GRPA excitation spectrum of the gapless superfluid state
has a quasiparticle-quasihole continuum and a second collective mode. We find
that the long-wavelength excitations of the second collective mode eventually
cause dynamical instability of the system when the superflow velocity
increases. However, the gapless superfluid state still remains stable in a
narrow but finite range of superflow velocity.
