# Seminar on particle and kinetic models describing collective behaviors.

Time: 18 June 2018, 14:30 - 16:45
Place: Department of Mathematics, Hanyang University
Organizer: Prof. Jinyeong Park (Hanyang University)

1. Large-time dynamics of the particle and kinetic Kuramoto models with frustration
Speaker: Dr. Dongnam Ko (Seoul National University) 14:30 - 15:10
Abstract:
The synchronous collective motion is widely studied for the practical reasons. The Kuramoto model is one of the simplified models which describes individuals and their interactions. In many real cases, the interaction has some phase shift error, which is called frustration. We look over the synchronization results of this model in the particle (ODE) and kinetic (PDE) descriptions. The main concern is to present a sufficient condition which leads to the synchronization phenomena. We will see how phase shift destroys the gradient structure and keeps the particle-path analysis.
2. Remarks on the Fokker-Planck type equations derived from the synchronization particle models
Speaker: Jaeseung Lee (Seoul National University) 15:10 - 15:30
Abstract:
In kinetic theory of many-body systems, it is well known that mean-field kinetic equations can effectively describe the large ODE systems. In this talk, we present two types of viscous kinetic mean-field equations: the Kuramoto-Sakaguchi-Fokker-Planck(KS-FP) equation and the swarming model on a unit sphere. For the KS-FP equation, we study the stability and instability of the incoherent state where the all oscillators are uniformly distributed on the phase space. The similar results are also sought with the swarming model on the unit sphere. These phenomena depend on the interplay between the diffusion and coupling strength between the oscillators.
3. Asymptotic behavior and stability problem for the Schrödinger-Lohe model
Speaker: Dohyun Kim (Seoul National University) 15:45 - 16:05
Abstract:
We present asymptotic behavior and stability problem for the Schrödinger-Lohe(S-L) system which was first introduced as a possible phenomenological model exhibiting quantum synchronization. We present several sufficient frameworks leading to the emergent behavior of the S-L system. Moreover, we show that there are only two possible asymptotic states: completely synchronized state or bi-polar state. On the other hand, we provide the standing wave solutions for the S-L model with the harmonic potential and discuss the stability for standing wave solutions.
4. Emergent behaviors of continuous and discrete thermomechanical Cucker-Smale models on general digraphs
Speaker: Do Heon Kim (Seoul National University) 16:05 - 16:25
Abstract:
We present emergent dynamics of continuous and discrete thermomechanical Cucker-Smale(TCS) models equipped with temperature as an extra observable on general digraph. In previous literature, the emergent behaviors of the TCS models were mainly studied on a complete graph or symmetric connected graphs. Under this symmetric setting, the total momentum is a conserved quantity. This determines the asymptotic velocity and temperature a priori using the initial data only. Moreover, this conservation law plays a crucial role in the flocking analysis based on the elementary $\ell_2$ energy estimates. In this paper, we consider a more general connection topology which is registered by a general digraph, and the weights between particles are given to be inversely proportional to the metric distance between them. Due to this possible symmetry breaking in communication, the total momentum is not a conserved quantity, and this lack of conservation law makes the asymptotic velocity and temperature depend on the whole history of solutions. To circumvent this lack of conservation laws, we instead employ some tools from matrix theory on the scrambling matrices and some detailed analysis on the state-transition matrices. We present two sufficient frameworks for the emergence of mono-cluster flockings on a digraph for the continuous and discrete models. Our sufficient frameworks are given in terms of system parameters and initial data.
5. Large-time behavior of Thermo-mechanical Cucker-Smale ensemble immersed in various fluids
Speaker: Jeongho Kim (Seoul National University) 16:25 - 16:45
Abstract:
In this presentation, we consider the kinetic Thermo-mechanical Cucker-Smale (TCS) equation coupled with compressible/incompressible fluid. In our model, particle ensemble and fluid are coupled via mechanical drag force. We first provide the well-posedness theory for a weak or strong solution using a standard technique. Then, we discuss the large-time behavior for TCS ensemble coupled with both types of fluids.