Continuum limit and emergent behaviors of continuum Kuramoto model on Euclidean spaces

We study approximation and emergent behaviors of the continuum Kuramoto model on the whole space Rd which corresponds to an integro-differential equation. The continuum Kuramoto model (in short, CKM) appears in the continuum limit from the Kuramoto lattice model on the spatially extended domain Rd. Compared to the model on the finite lattice in a bounded region, we use an infinite Kuramoto lattice model to construct a sequence of approximate solutions. The emergent dynamics of the CKM depends on the natural frequency function. For the constant natural frequency function, we derive existence of asymptotic frequency, exponential synchronization and quasi-stationary state under a suitable set of conditions for interaction kernel and initial datum, whereas for a nonconstant natural frequency, we show the emergence of practical synchronization that the phase diameter is proportional to the inverse of some measure of the interaction kernel.