  In the standard model of core-collapse supernova (CCSN), all neutrinos are
assumed to be in pure flavor eigenstates in CCSN cores, but the assumption
becomes invalid if neutrino distributions are unstable to flavor conversions.
In this paper, we present a study of the occurrences of two representative
neutrino-flavor instabilities, fast- and collisional flavor instabilities, in
the cooling phase of proto-neutron star (PNS) from 1- to 50 seconds. We follow
the long-term evolution of a PNS under spherically symmetric and quasi-static
approximations, in which the matter profile is determined by solving the
Tolman-Oppenheimer-Volkoff equation with neutrino feedback under the treatment
of multi-group flux limited diffusion. For the stability analysis of neutrino
flavor conversions, we recompute neutrino distributions using Monte Carlo
transport in order to obtain the full angular distribution needed to compute
the dispersion relations. We find no signs of flavor conversions in our models;
the physical reason is thoroughly investigated. We also argue that the negative
conclusion in flavor conversions could be changed qualitatively if
multi-dimensional effects are included, as similar to cases in the earlier
phase of CCSN.
