Uniform stability and uniform-in-time mean-field limit of the thermodynamic Kuramoto model

We consider the thermodynamic Kuramoto model proposed in \cite{H-P-R-S}. For each oscillator in thermodynamic Kuramoto model, there is a coupling effect between the phase and the temperature field. For such a model, we study a uniform stability and uniform-in-time mean-field limit to the corresponding kinetic equation. For this, we first derive a uniform ℓ p \ell ^p -stability of the thermodynamic Kuramoto model with respect to initial data by directly estimating the temporal evolution of ℓ p \ell ^p -distance between two admissible solutions to the particle thermodynamic Kuramoto model. In a large-oscillator limit, the Vlasov type mean-field equation can be rigorously derived using the BBGKY hierarchy, uniform stability estimate, and particle-in-cell method. We construct unique global-in-time measure-valued solutions to the derived kinetic equation and also derive a uniform-in-time stability estimate and emergent estimates.