2020 Fall (Abstracts)
Time: Sep 9 (Wed) 4-5PM
Speaker: Sanghyuk Lee (Seoul National Univ)
Title: Sharp estimates for the Hermite spectral projection
Abstract: This talk primarily concerns the sharp spectral projection estimate for the Hermite operator in the Lebesgue spaces. Compared with the spectral projection associated with the Laplacian, the sharp boundedness of the Hermite spectral projection is not so well understood. We consider the spectral Hermite projection estimate in a general framework and obtain various new sharp estimates in an extended range. Especially, we provide a complete characterization of the local estimate and prove the endpoint $L^2$--$L^{2(d+3)/(d+1)}$ estimate for $d\ge 5$ which has been left open since the work of Koch and Tataru. We also discuss application of the projection estimate to related problems, such as the resolvent estimate for the Hermite operator and Carleman estimate for the heat operator. This talk is based on the recent joint work with Eunhee Jeong and Jaehyeon Ryu.
Time: Sep 23 (Wed) 4-5PM
Speaker: Bongsuk Kwon (UNIST)
Title: Small Debye length limit for the Euler-Poisson system
Abstract: We discuss existence, time-asymptotic behavior, and quasi-neutral limit for the Euler-Poisson equations. Specifically, under the Bohm criterion, we construct the global-in-time solution near the stationary solution of plasma sheath, and also investigate its time-asymptotic behavior and small Debye length limit. If time permits, some key features of the proof and related problems will be discussed. This is joint work with C.-Y. Jung (UNIST) and M. Suzuki (Nagoya Tech.).
Time: Oct 14 (Wed) 4-5PM
Speaker: Haewon Yoon (Chung-Ang Univ)
Title: Global well-posedness for the mass-critical nonlinear resonant Schrödinger system
Abstract: In this talk, we discuss abouit the global well-posedness and scattering belowe the threshold of the cubic focusing resonant Schrödinger system on $\mathbb{R}^2$ in $L^2h^1(\mathbb{R}^2\times\mathbb{Z})$. We first establish the variational characterization of the ground state, and give the threshold of the global well-posedness and scattering. Then we reduce the global well-posedness and scattering below the threshold to the exclusion of minimal counter examples known as almost periodic solutions. This is joint work with Xing Cheng (Hohai Univ.) and Gyeongha Hwang (Yeungnam Univ.).
Slide
Time: Oct 28 (Wed) 4-5PM
Speaker: Moon-Jin Kang (KAIST)
Title: Riemann problem of two shocks for 1D compressible Euler system
Abstract: I will present my recent work on uniqueness of Riemann problem consisting of two shocks for 1D isentropic Euler system. The uniqueness is guaranteed in the class of vanishing viscosity limits of solutions to the associate Navier-Stokes system, as the Bianchini-Bressan conjecture. The main idea to achieve this issue is to get a uniform stability of any large perturbations from a composite wave of two viscous shocks to the Navier-Stokes. Especially, I will explain about this main idea in a simpler context, that is, in the case of a single shock. This is based on joint works with Alexis Vasseur.
Time: Nov 11 (Wed) 4-5PM
Speaker: Changhun Yang (Chungbuk National University)
Title: On the Korteweg–de Vries limit for the Fermi-Pasta-Ulam system
Abstract: The Fermi-Pasta-Ulam system (FPU) is a simple nonlinear dynamical lattice model describing a one-dimensional chain of vibrating strings with nearest neighbor interactions. This model was introduced by Fermi, Pasta and Ulam in 1955. It was anticipated at that time that chaotic nonlinear interactions would lead to thermalization. Surprisingly however, numerical simulations showed the opposite behavior – it exhibited quasi-periodic motions. This phenomena is known as the FPU paradox. This puzzle has been solved by Zabusky and Kruskal by discovering a formal convergence of FPU to the Kortewegde-de Vries equation. Later, the convergence has been rigorously justified. We revisit this convergence problem, and show how to put it into the dispersive PDE framework. This talk is based on joint work with Younghun Hong and Chulkwang Kwak.
Time: Nov 25 (Wed) 4-5PM
Speaker: Yonggeun Cho (Jeonbuk National University)
Title: Small data scattering for 2d Hartree type Dirac equations
Abstract: In this talk, we discuss the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity $c(|\cdot|^{-\gamma} * \langle \psi, \beta \psi\rangle)\beta\psi$ with $c\in \mathbb R\setminus\{0\} $, $0 < \gamma < 2$. The aim is to introduce recent research and their proof shortly on the small data global well-posedness and scattering in $H^s$ for $s > \gamma-1$ and $1 < \gamma < 2$. The main difficulty of scattering stems from the singularity of the low-frequency part $|\xi|^{-(2-\gamma)}\chi_{\{|\xi|\le 1\}}$ of potential. To overcome it we adopt $U^p-V^p$ space argument and bilinear estimates of C. Yang(2019) and A. Tesfahun(2020) based on the null structure. These argument can be applied to the scattering in $L_x^2$ for the 2d Dirac equation with Yukawa potential. We also study a non-existence result of scattering in the long-range case $0 < \gamma \le 1$.
Time: Dec 9 (Wed) 4-5PM
Speaker: Jinmyoung Seok (Gyonggi University)
Title: Dynamical stability of uniformly rotating stars
Abstract: In this talk, we are concerned with dynamically stability of uniformly rotating binary stars, which are represented as stationary solutions to Euler-Poisson equations. These solutions were constructed as minimizers of suitable variational problems by McCann in which some kind of structural stability on them is discussed. This talk focuses on the nonlinear dynamical stability of them, based on Cazenave-Lions type arguments exploiting variational characterization of stationary solutions. We will see that the uniqueness of a minimizer, which is one of the main results of our work, plays an indispensable role in analysis. This talk is based on a joint work with Prof. Juhi Jang at USC.