2021 Fall (Abstracts)

Time: Oct 13 (Wed) 4-5PM

Speaker: Kiyeon Lee (Ewha Womans University)

Title: Low regularity well-posedness of Hartree type Dirac equations in 2,3-dimensions.

Abstract: We give a survey of small data scattering of 2,3 dimensional Dirac equation with Hartree type nonlinearity $c(|\cdot|^{-\gamma} * \langle \psi, \beta \psi\rangle)\beta\psi$. Since the nonlinear potentials of our equations are Coulomb potential, these equations can be a simplified physical model of Chern-Simons-Dirac equations and Dirac-Klein-Gordon equations. The results about local, global well-posedness, and small data scattering of Dirac equations can be proven to control singularities of the frequency of potential. To handle the frequencies of potential we need the short time Bourgain spaces, Up-Vp spaces argument, and localized bilinear estimates arising from the null structure. In this talk, I will discuss two well-posedness problems. The one is almost critical regularity of local well-posedness of Dirac equations. The other is the small initial data scattering for solution to Dirac equations in the case $1 < \gamma <2$ and nonexistence result for scattering in the case $0<\gamma \le 1$ and describe the details of estimates. This talk is based on co-work with Yonggeun Cho and Tohru Ozawa.

Time: Oct 27 (Wed) 4-5PM

Speaker: Byungsoo Moon (Incheon National University)

Title: On a shallow-water model with the Coriolis effect

Abstract: In this talk an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force is derived from the governing equation in two dimensional flows. The transport equation theory is then applied to investigate the local well-posedness and wave breaking phenomena for this model. The nonexistence of the Camassa-Holm-type peaked solution and classification of various traveling-wave solutions to the new system are also established. Moreover it is shown that all the symmetric waves to this model are traveling waves. This is a joint work with Ting Luo, Yue Liu and Yongsheng Mi.

Time: Nov 10 (Wed) 4-5PM

Speaker: Jaehyeon Ryu (Jeonbuk National University)

Title: Unique continuation for the heat operator with potentials in weak spaces

Abstract: For a given operator $P$, the strong unique continuation property (abbreviated to sucp in what follows) for $V$ means that a nontrivial solution $u$ to $Pu = 0$ cannot vanish to infinite order at any point. The sucp for second order parabolic operator has been studied by many authors. It was Poon and Chen who first considered the problem. They studied sucp for the parabolic operator $\partial_t+\Delta + V$ and obtained the results for bounded potentials $V$. In this talk, we prove sucp for the operator $\partial_t+\Delta + V$ with the potential $V$ contained in the weak space $L^\infty_t L^{\frac d2,\infty}_x$ and the assumption that the norm of $V$ is sufficiently small. It can be shown that the smallness assumption is necessary by utilizing Wolff's construction. This has been left open since the works of Escauriaza and Escauriaza-Vega. Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces. This talk is based on a recent joint work with Eunhee Jeong and Sanghyuk Lee.

Time: Nov 24 (Wed) 4-5PM

Speaker: Bora Moon (Hanyang University)

Title: Hydrodynamic limits of the nonlinear Schrodinger equation coupled with the Chern-Simons gauge fields

Abstract: In this talk, we present the hydrodynamic limit problem for the Chern-Simons-Schrodinger(CSS) system. We consider the Madelung transformation and two different scalings of the CSS system to derive the compressible and incompressible Euler systems coupled with the Chern-Simons equations and Poisson equation, respectively. Both cases are based on modulated energy estimates for rigorous derivation. Here we focus on the case of compressible limit to which the classical theory of relative entropy method can be applied. This talk is based on a recent joint work with Dr. Jeongho Kim.

Time: Dec 8 (Wed) 4-5PM

Speaker: Sanghyeon Yu (Korea University)

Title: Mathematical Analysis and Design of Meta-materials

Abstract: Meta-materials, which are materials made of artificially designed atoms, is an emerging field between physics and materials science. Meta-materials exhibit many exotic phenomena such as negative refractive index, topological edge states and invisibility cloaks. The rational design of meta-materials requires solving interesting problems involving PDEs, spectral geometry, operator theory and topology. In this talk, we discuss our recent works on plasmonic and acoustic meta-materials.

Time: Dec 22 (Wed) 4-5PM

Speaker: Junha Kim (Chung-Ang University)

Title: On the asymptotic stability of a stratified solution for the Boussinesq equations with a velocity damping term

Abstract: In this talk, we consider an initial value problem of the multi-dimensional Boussinesq equations with a velocity damping term for strongly stratified fluids. We prove the global-in-time existence of the classical solution with initial data near a stationary stratified solution. Then, we establish the temporal decay estimates for the solution by analyzing the linear operator in a way that is available for any dimension. This talk is based on recent joint work with Prof. Jihoon Lee.