Contractivity of transport distances for the kinetic Kuramoto equation

We present synchronization and contractivity estimates for the kinetic Kuramoto model obtained from the Kuramoto phase model in the mean-field limit. For identical Kuramoto oscillators, we present an admissible class of initial data leading to time-asymptotic complete synchronization, that is, all measure valued solutions converge to the traveling Dirac measure concentrated on the initial averaged phase. In the case of nonidentical oscillators, we show that the velocity field converges to the average natural frequency proving that the oscillators move asymptotically with the same frequency under suitable assumptions on the initial configuration. If two initial Radon measures have the same natural frequency density function and strength of coupling, we show that the Wasserstein pJ. A. Carrillo Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom E-mail: [email protected] Y.-P. Choi Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom E-mail: [email protected] S.-Y. Ha Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Korea E-mail: [email protected] M.-J. Kang Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea E-mail: [email protected] Y. Kim Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea E-mail: [email protected]