Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia

Abstract

We present the global existence and long-time behavior of measurevalued solutions to the kinetic Kuramoto-Daido model with inertia. For the global existence of measure-valued solutions, we employ a Neunzert's meanfield approach for the Vlasov equation to construct approximate solutions. The approximate solutions are empirical measures generated by the solution to the Kuramoto-Daido model with inertia, and we also provide an a priori local-in-time stability estimate for measure-valued solutions in terms of a bounded Lipschitz distance. For the asymptotic frequency synchronization, we adopt two frameworks depending on the relative strength of inertia and show that the diameter of the projected frequency support of the measure-valued solutions exponentially converge to zero.

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

“…In this setting, the Kuramoto model consists in a first approach towards a mathematical description of neuronal synchronization 195,210,214,215 , that is known to rule many cognitive process of the brain that are activated when a specific group of neurons fire together forming a cluster. Of course, this model can be made more realistic by adding coupling weights governing the plasticity of connections via learning mechanisms 82,123,176,183,198 , inertia terms and delays in time 70,71,72,73 , noise or many other features like singular couplings 178,187 . We will review some of this associated models later on.…”

“…Related to the last technique in Subsection 5.3, the classical Kuramoto model with inertia has been analyzed at the microscopic and kinetic scales in 70,71,72,73 :…”

Filippov flows and mean-field limits in the kinetic singular Kuramoto model

“…To cover such supercritical case, we shall develop an alternative method that is valid for all α ∈ (0, 1) (at least for identical oscillators g = δ 0 ). We will augment the first order singular Kuramoto system into an auxiliary second order regularized system with inertia, frequency damping and diffusion, see [13,14,15,16]. Under an appropriate scaling depending on a parameter ε ց 0, the inertia and noise terms will vanish and singularity of the coupling function will be recovered.…”

Invariance of velocity angles and flocking in the Inertial Spin model

“…with prescribed initial data given by (15). Here m i , γ i are the inertia and damping parameters of the i'th oscillator respectively, k is the coupling strength, and the Ω i 's are the natural frequencies of the oscillators which are independent to one another and each one follows a common distribution g(Ω).…”