# Seminar on PDE and related topics in October 2018

1

Time: 12th October (Friday) 14:30 - 15:30
Speaker: Prof. Woocheol Choi (Incheon National University)
Place: #702, Natural Science Building, Hanyang University

Title: Introduction to the control theory with applications in mechanical engineerings.

Abstract:
제어이론은 수학과 공학의 접점에 있는 분야로 20세기를 거치며 광범위하게 발전해 왔다. 특히 기계공학에서의 필수적인 역할을 하고 있으며, 현대에는 에너지, 반도체 등의 기술 발전으로 드론과 다양한 로봇들이 상용화 되고 있고, 관련된 제어기술의 수요도 증가하고 있다. 이번 발표에서는 제어 이론의 몇가지 기본적인 아이디어들과 예들을 소개하고, 최근 몇가지 발전 동향을 살펴본다.

2

Time: 17th October (Wednesday) 16:00 - 17:00
Speaker: Seungyeon Cho (Sungkyunkwan University)
Place: #702, Natural Science Building, Hanyang University

Title: High order conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation

Abstract:
In this work, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. The classical semi-Lagrangian finite difference scheme for the BGK model performs stably for all the range of Knudsen number, but are not conservative. There are two source of such loss of the conservation property. First, the accuracy of the cancellation of the relaxation operator in the zeroth, first and second velocity moments depends heavily on the number of velocity grids and non-negligible errors may arise if the number of velocity grids is not sufficient. Secondly, since the scheme is not of the conservative form, the error may accumulate in the numerical computation of the transport term. To treat the first problem and ensure the machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced by Mieussens. The second difficulty is treated by implementing a conservative correction procedure based on the flux difference form. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

3

Time: 30th October (Tuesday) 14:30 - 16:00
Speaker: Prof. Kyung Hoon Han (University of Suwon)
Place: #208, Natural Science Building, Hanyang University

Title: 인공신경망의 기초

Abstract:
인공신경망 기법은 안면인식, 기계번역, 자율주행등에 응용되며 머신러닝의 핵심기법으로 부각되고 있다. 인공신경망은 그레디언트, 연쇄법칙과 같은 다변수 미분이론이 그 기초를 이루고 있다. 본 발표에서는 인공신경망의 원리와 수학적 기초를 설명하고 파이썬을 통해 시연해본다.

# PDE seminar in September 2018

1

Time: 14th September (Friday) 16:00 - 17:00
Speaker: Prof. Younghun Hong (Chung-Ang University)
Place: #701, Natural Science Building, Hanyang University

Title: Dynamics in the semi-classical limit

Abstract:

Quantum mechanics describes nature at the atomic/subatomic scale, while classical mechanics does at macroscopic scale. These two theories are connected via the (semi-)classical limit h→0. In this talk, we discuss the semi-classical limit focusing on analysis of orthogonal families of functions (by Lieb-Thirring, Frank-Lewin-Lieb-Seiringer and Frank-Sabin). This is a survey talk.

2

Time: 18th September (Tuesday) 14:30 - 15:30
Speaker: Prof. Hantaek Bae (UNIST)
Place: #208, Natural Science Building, Hanyang University

Title: Incompressible Navier-Stokes equations: regularity of the velocity field and the flow map

Abstract:
In this talk, I will introduce the notion of mild solution and its analyticity. After that, I will show Holder regularity of the associated flow map.

# 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.