  The agent-based singular Kuramoto model was proposed in [60] as a singular
version of the Kuramoto model of coupled oscillators that is consistent with
Hebb's rule of neuroscience. In such paper, the authors studied its
well-posedness via the concept of Filippov solutions. Interestingly, they found
some new emergent phenomena in the paradigm of Kuramoto model: clustering into
subgroups and emergence of global phase synchronization taking place at finite
time.
  This paper aims at introducing the associated kinetic singular Kuramoto model
along $\mathbb{T}\times\mathbb{R}$, that is reminiscent of the classical
Kuramoto-Sakaguchi equation. Our main goal is to propose a well-posedness
theory of measure-valued solutions that remains valid after eventual phase
collisions. The results will depend upon the specific regime of singularity:
subcritical, critical and supercritical. The cornerstone is the existence of
Filippov characteristic flows for interaction kernels with jump
discontinuities. Our second goal is to study stability with respect to initial
data, that in particular will provide quantitative estimates for the mean-field
limit of the agent-based model towards the kinetic equation in quadratic
Wasserstein-type distances. Finally, we will recover global phase
synchronization at the macroscopic scale under appropriate generic assumptions
on the initial data. The most singular regime will be tackled separately with
an alternative method, namely, a singular hyperbolic limit on a regularized
second order model with inertia. Note that we will work within the manifold
$\mathbb{T}\times\mathbb{R}$ and will avoid resorting on Euclidean
approximations. Also, no gradient-type structure will be needed, as opposed to
some preceding literature.
