  We study the emergent collective behaviors for an ensemble of identical
Kuramoto oscillators under the effect of inertia. In the absence of inertial
effects, it is well known that the generic initial Kuramoto ensemble relaxes to
the phase-locked states asymptotically (emergence of complete synchronization)
in a large coupling regime. Similarly, even for the presence of inertial
effects, similar collective behaviors are observed numerically for generic
initial configurations in a large coupling strength regime. However, this
phenomenon has not been verified analytically in full generality yet, although
there are several partial results in some restricted set of initial
configurations. In this paper, we present several improved complete
synchronization estimates for the Kuramoto ensemble with inertia in two
frameworks for a finite system. Our improved frameworks describe the emergence
of phase-locked states and its structure. Additionally, we show that as the
number of oscillators tends to infinity, the Kuramoto ensemble with infinite
size can be approximated by the corresponding kinetic mean-field model
uniformly in time. Moreover, we also establish the global existence of
measure-valued solutions for the Kuramoto equation and its large-time
asymptotics.
