Title and Abstract
Title and Abstract
Solitary waves of the Vlasov-Poisson system (Masahiro Suzuki)
We consider the Vlasov-Poisson system describing a two-species plasma with spatial dimension 1 and the velocity variable in $\mathbb{R}^n$. We find the necessary and sufficient conditions for the existence of solitary waves of the system. To this end, we need to investigate the distribution of ions trapped by the electrostatic potential. Furthermore, we classify completely in all possible cases whether or not the solitary wave is unique, when we exclude the variant caused by translation. There are both cases that it is unique and nonunique.
See the invisible (Jia-Yuan Dai)
Unstable orbits can be mathematically predicted, but they are invisible in experiments or simulations due to lack of robustness under environmental errors. "To see the invisible”, in 1992, Kestutis Pyragas introduced a method of noninvasive feedback control that utilizes time delays to stabilize periodic orbits of ODEs. In 2016, Isabelle Schneider extended this control to PDEs using spatio-temporal delays. As a concrete application, in this talk I will stabilize Ginzburg–Landau spiral waves in circular and spherical geometry. This talk serves as my invitation for mathematicians and scientists to consider using this technique to stabilize many other unstable orbits, including traveling waves, in various models.
Numerical reconstruction of sources in transport equations and their applications (Hiroshi Fujiwara)
We consider a feasibility of computerized tomography (CT) by the use of transport equations. X-ray tomography has been conventionally modeled with an integral equation known as the Radon transform. On the other hand, in this talk, we treat it as an inverse source problem of the transport equation, and realize imaging from partial measurement data. A-analyticity in the sense of Bukhgeim and a singular integral equation with the Cauchy kernel play essential roles in the proposed method. Stability which is one of the most important concepts in numerical reconstruction for x-ray CT is discussed by spectral analysis of a Cauchy-type singular integral equation. Quantitative stability estimate is also shown. The talk is based on joint projects with A.Tamasan and K.Sadiq.
On steady states for the Vlasov-Schrödinger-Poisson system (Sangdon Jin)
The Vlasov-Schrödinger-Poisson system is a kinetic-quantum hybrid model describing quasi-lower dimensional electron gases. In this talk, we discuss the derivation of kinetic quantum hybrid models using partial confinement. Also, for this system, we study the construction of a large class of 2D kinetic/1D quantum steady states in a bounded domain as generalized free energy minimizers, and we show their finite subband structure, monotonicity, uniqueness, and conditional dynamical stability. This talk is based on joint work with Younghun Hong (Chung-Ang University).
Convergence rates of elliptic/parabolic regularizations applied to advection equations with boundary conditions (Masaki Imagawa)
One method for obtaining approximate solutions to first-order partial differential equations is to consider approximate equations obtained by adding a second-order elliptic operator with small parameters to the original equation. It is called the elliptic or parabolic regularization depending on the stationary or time-dependent problem, and also known as the vanishing viscosity method. In this talk, we mainly focus on linear advection equations with boundary conditions, and introduce convergence estimates and numerical examples of approximate solutions mentioned above. This is joint work with Dr. Daisuke Kawagoe.
Stability-instability of vortex solutions in incompressible Euler equations (Deokwoo Lim)
In this talk, we consider the vorticity form of the 2D incompressible Euler equations either in the whole plane $\mathbb{R}^2$ or in the half cylinder $S_+:=\mathbb{R}_{+}\times\mathbb{T} $. It is well-known that in $\mathbb{R}^2$, any radial vorticity is a steady solution of the vorticity form, such as circular vortex patches. We will discuss about ideas which shows stability of circular patches in some norm involving the angular impulse. Then we adapt these ideas to $S_+$, and construct a vortex patch in this domain that shows infinite perimeter growth for infinite time. This is a joint work with Kyudong Choi(UNIST) and In-Jee Jeong(SNU).
On the continuity and nowhere differentiability of the Brownian motions (Chun-Hsiung Hsia)
In this talk, we shall use the Schauder functions to construct the Brownian motions and we shall introduce the Kolmogorov test to show the Holder continuity of the Brownian motions. On the other hand, we shall show that the Brownian motion is almost nowhere differentiable.
Complete and Partial Synchronization for Two-Group and Three-Group Kuramoto Oscillators (Shih-Hsin Chen)
Synchronization has been investigated in diverse subjects including circadian rhythms, flash of fireflies, neural networks, biological oscillators, clocks, power systems, etc. To demonstrate such a phenomenon, the Kuramoto model has received a lot of attention and has been widely applied in various disciplines. Recently, the synchronization for the multi-group Kuramoto model has been studied by many authors. In this talk, I shall introduce the two-group and three-group Kuramoto models and give the main results. If time permits, the numerical simulations are also shown in this presentation.