Se Eun Noh

Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators

We study the dynamic interplay between inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators. When external fields and internal forces are exerted on a system of Kuramoto oscillators, each oscillator has its own distinct dynamics, so that there is no notion of collective dynamics in the ensemble, and complete synchronization is not observed in such systems. In this paper, we study a relaxed version of synchronization, namely the “practical synchronization”, of Kuramoto oscillators, emerging from the dynamic interplay between inertia and heterogeneous decoupled dynamics. We will show that for some class of initial configurations and parameters, the fluctuation of phases and frequencies around the average values will be proportional to the inverse of the coupling strength. We provide several numerical examples, and compare these with our analytical results.

Global existence and stability of mild solutions to the Boltzmann system for gas mixtures

We present the global existence and stability of mild solutions to the Boltzmann system with inverse power molecular interactions for a binary gas mixture, when initial data are sufficiently small and decay exponentially in phase space. For the existence and stability of mild solutions, we employ a modified Kaniel-Shinbrots scheme and a weighted nonlinear functional approach. Time-asymptotic convergence toward the free molecular motion is established using a weighted collision potential, and we show that the weighted L 1 -distance between two mild solutions is uniformly controlled by that of initial data.

Remarks on the stability of the frictionless Vlasov-Poisson-Fokker-Planck system

We present the uniform L1-stability estimate of classical solutions to the frictionless Vlasov-Poisson-Fokker-Planck system, when initial data are small and decay algebraically fast enough in phase space. The hypoelliptic structure of the system plays a key role in the stability analysis in low dimensions two and three, which are missing in the Vlasov-Poisson system [M. Chae and S.-Y. Ha, SIAM J. Math. Anal. 37, 1704–1731 (2006)].

Synchronization of Kuramoto Oscillators with Adaptive Couplings

We study the synchronization of Kuramoto oscillators with adaptive coupling in interacting networks. Network dynamics preserves the sum of all incoming pairwise coupling strengths and is designed to adaptively interact with system dynamics. For adaptive couplings, we use two adaptive coupling laws for the pairwise coupling strength. Kuramoto oscillators are assumed to be on the nodes of the networks. We present frameworks that guarantee the emergence of synchronization for various coupling feedback laws. Our results generalize earlier work on the synchronization of Kuramoto oscillators in fixed and symmetric networks.

On the relaxation dynamics of the Kuramoto oscillators with small inertia

For the Kuramoto oscillators with small inertia, we present several quantitative estimates on the relaxation dynamics and formational structure of a phase-locked state (PLS) for some classes of initial configurations. In a super-critical regime where the coupling strength is strictly larger than the diameter of natural frequencies, we present quantitative relaxation dynamics on the collision numbers and the structure of PLS. In a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies, we provide a sufficient condition for an asymptotically PLS solution. In particular, we show the existence of slow relaxation to a PLS, when there are exactly two natural frequencies. This generalizes the earlier results of Choi et al. [“Asymptotic formation and orbital stability of phase locked states for the Kuramoto model,” Physica D 241, 735–754 (2012)10.1016/j.physd.2011.11.011; Choi et al. “Complete synchronization of Kuramoto oscillators with finite inertia,” Physica D 240,...

GLOBAL EXISTENCE AND STABILITY OF MILD SOLUTIONS TO THE INELASTIC BOLTZMANN SYSTEM FOR GAS MIXTURES

We study the global existence and uniform stability estimate of mild solutions to the inelastic Boltzmann system for gas mixtures, when initial data are small and decay exponentially in phase space, and we also provide a general multi-dimensional Bony-type potential which yields a priori weighted two-point correlation estimates in phase space to the mild solutions with finite mass and energy without any smallness restriction.