Program
Title and Abstract (download)
12/18 13:10 ~ 14:00
Kwan Woo (Ulsan National Institute of Science and Technology)
Title: Elliptic and parabolic equations with singular drifts
Abstract: We will review recent results on regularity properties of solutions to second-order elliptic and parabolic equations. In particular, we concentrate on the case where ‘drifts’ in Lebesgue space with or without divergence-free conditions.
12/18 14:00 ~ 14:50
Sungjin Lee (Yonsei University)
Title: Regularity for second-order elliptic equation and its application
Abstract: Let me introduce several results regarding the regularity of second-order elliptic equations. Additionally, I will present some results concerning the applications of the regularity theory to the estimates of Green’s functions and homogenization theory.
12/18 15:10 ~ 16:00
Juyoung Lee (Seoul National University)
Title: Recent progress in maximal functions
Abstract: Various types of maximal functions appear in many topics in analysis. They help to understand properties of functions, for example, differentiability. In this talk, we will focus on maximal averaging operators. The most famous and basic example is the Hardy-Littlewood maximal operator which is a maximal averaging operator over a family of balls. Since Stein's seminal work on the spherical maximal function in 1976, huge amount of literature was devoted to study maximal operators generated by given sub-manifold in the Euclidean space. We review some history in this area and see some new results.
12/18 16:00 ~ 16:50
Sewook Oh (Korea Institute for Advanced Study)
Title: Damping oscillatory integrals
Abstract: The Fourier transform of surface measure is a fundamental subject in harmonic analysis. The decay estimates of Fourier measures are strongly related to central problems in harmonic analysis such as restriction problems, Lp bounds for maximal averages, etc. When a hypersurface has nonvanishing Gaussian curvature, it is well known that optimal decay of the Fourier measure. If the curvature may vanish, it no longer holds. For compensating degeneracy, Sogge and Stein used oscillatory integral estimates with mitigating factors. In this talk, I will introduce basic tools for dealing with oscillatory integrals and some results for damping oscillatory integrals.
12/18 16:50 ~ 17:40
Jaehyeon Ryu (Korea Institute for Advanced Study)
Title: Lp estimate for the eigenfunctions and quasimodes
Abstract: Since the pioneered work of Sogge, the problem of estimating the Lp norm of eigenfunctions has been the subject of intense study. The problem has been considered in various settings of pseudodifferential operators on smooth manifolds including the Laplace-Beltrami operator on a smooth compact Riemannian manifold and Schrodinger operator on a differentiable manifold as representative examples. In this talk, we review some of the works that dealt with this subject.
12/19 10:30 ~ 11:20
Jaehoon Kang (Seoul National University)
Title: Estimates of density functions for symmetric jump processes
Abstract: First, we will discuss harmonic functions with respect to Markov processes. Then, results on estimates of density functions for symmetric jump processes (or non-local operators) will be introduced.
12/19 11:20 ~ 12:10
Jaehun Lee (Korea Institute for Advanced Study)
Title: Heat kernel estimates of non-symmetric jump process with mixed polynomial growth
Abstract: In this talk, we consider the heat kernel estimates of non-symmetric and non-local operators with Holder continuous jumping kernel. In the first part, we construct the heat kernel of non-symmetric operator. Next, we establish the upper and lower bound for the heat kernel in the alpha-stable like case. Lastly, we introduce the heat kernel estimates for symmetric and non-symmetric jump processes with mixed polynomial growth.
12/19 14:00 ~ 14:50
Dongkwang Kim (Ulsan National Institute of Science and Technology)
Title: Investigating existence and blow-up in Chemotaxis systems
Abstract: Chemotaxis, the directed movement of organisms in response to chemical substances, is a biological phenomenon observed across various organisms. This presentation introduces mathematical models that describe these chemotactic behaviors, with a specific focus on exploring the existence and occurrence of blow-up phenomena.
12/19 14:50 ~ 15:40
Junsik Bae (Ulsan National Institute of Science and Technology)
Title: Nonexistence of Multi-dimensional Solitary Waves in Unmagnetized Plasma
Abstract: We study the nonexistence of multi-dimensional solitary waves for the Euler-Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler-Poisson system has solitary waves that travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves for any traveling velocity and for general pressure laws. Our results provide theoretical evidence for the stability of line solitary waves in multi-dimensional Euler-Poisson flows. We derive some Pohozaev type identities associated with the energy and density integrals. This is a joint work with Daisuke Kawagoe (Kyoto University).
12/19 15:40 ~ 16:30
Sang-Hyuck Moon (Ulsan National Institute of Science and Technology)
Title: Nonlinear elliptic partial differential equations and systems
Abstract: Nonlinear elliptic partial differential equations arise in many fields like physics, biology, and geometry. In this talk, I will present some results for nonlinear elliptic PDE models. I will also introduce some basic tools.
12/19 16:30 ~ 17:20
Jaeyong Shin (Ulsan National Institute of Science and Technology)
Title: Beltrami structure in hydrodynamics and magneto-hydrodynamics
Abstract: In this talk, we study Beltrami structure in hydrodynamics and magneto-hydrodynamics (MHD). Beltrami fields, for which the velocity and the vorticity (curl of the velocity) are everywhere collinear, can exhibit chaotic behavior of trajectories in three dimension. Meanwhile, these fields arise in the incompressible Euler and Navier-Stokes equations, as well as in magnetic relaxation situation of the MHD equations. In addition, more complex structure of Beltrami fields can appear in the Hall-MHD equations. We study solutions having Beltrami structure in each aforementioned circumstance, and then show that near the specific solutions, the incompressible Navier-Stokes, MHD and Hall-MHD equations are well-posed globally in time (in the sense of Hadamard). This talk is based on a joint work with Hantaek Bae (UNIST) and Kyungkeun Kang (Yonsei University).
12/20 09:40 ~ 10:30
Daehan Park (Seoul National University)
Title: On the diffusion equations with non-local operators
Abstract: Diffusion equation is one of the main tools for describing various natural phenomena in diverse fields including mathematics, engineering, and biology. By involving non-local operators, such descriptions can be significantly improved. In this talk, we will consider diffusion equations with non-local operators $\partial_{t}^{\alpha}u-Lu=f$, where $\partial_{t}^{\alpha}$ is the Caputo fractional derivative of order $\alpha\in(0,1)$. The operator $L$ is a non-local operator related to certain stochastic processes. The existence, uniqueness, and estimation of solution in Sobolev space will be considered.
12/20 10:30 ~ 11:20
Kyeong Song (Korea Institute for Advanced Study)
Title: Regularity results for double phase problems involving nonlocal operators
Abstract: In this talk, we investigate the De Giorgi-Nash-Moser theory for double phase problems involving nonlocal operators.
We first recall known regularity results for nonlocal problems of fractional p-Laplacian type in the context of the calculus of variations. We then introduce two kinds of double phase problems; one is a nonlocal double phase problem, and the other is a mixed local-nonlocal double phase problem. We present our recent results about Hölder regularity and Harnack inequality for such problems under natural, possibly sharp, assumptions.
12/20 11:20 ~ 12:10
Jinsol Seo (Korea Institute for Advanced Study)
Title: Lp theory for the Poisson equation in non-smooth domains
Abstract: The Poisson equation (△u=f) is one of the most fundamental and classical PDEs, and its Lp theory is important for the regularity of solutions. In this presentation, we discuss a general Lp theory for the Poisson equation in non-smooth domains, together with its applications based on a relation between the Hardy inequality, superharmonic functions, and various domain conditions.