  We study a nonlinear partial differential equation that arises when
introducing inertial effects in the Kuramoto model. Based on the known theory
of degenerate Kolmogorov operators, we prove existence, uniqueness and a priori
estimates of the solution to the relevant Cauchy problem. Moreover, a stable
numerical operator, which is consistent with the degenerate Kolmogorov
operator, is introduced in order to produce numerical solutions. Finally,
numerical experiments show how the synchronization phenomena depend on the
parameters of the Kuramoto model with inertia.
