Filippov flows and mean-field limits in the kinetic singular Kuramoto model

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.