The Kuramoto model is a kinetic model of oscillators for studying synchronisation behaviours. Like the Vlasov-Poisson equation it has a stability mechanism through phase mixing in weak topologies, which is the Landau damping. Compared to the Vlasov-Poisson equation, the interaction in the Kuramoto model is simpler allowing more results. In particular, we can study the stability of spatially inhomogeneous states.

In this talk, I will introduce (i) the Kuramoto model, (ii) the Landau damping and (iii) the stability of inhomogeneous states.