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Exponential dephasing of oscillators in the Kinetic Kuramoto Model

We study the kinetic Kuramoto model for coupled oscillators with coupling constant below the synchronization threshold. We manage to prove that, for any analytic initial datum, if the interaction is small enough, the order parameter of the model vanishes exponentially fast, and the solution is asymptotically described by a free flow. This behavior is similar to the phenomenon of Landau damping in plasma physics. In the proof we use a combination of techniques from Landau damping and from abstract Cauchy-Kowalewskaya theorem.