Title & Abstract

Alexander Kiselev (Duke Univ.)

Title : I - Small scale and singularity formation in fluid mechanics

Abstract : The Euler equation describing motion of ideal fluid goes back to 1755.  The analysis of the equation is challenging since it is nonlinear and nonlocal. Its  solutions are often unstable and spontaneously generate small scales. The  fundamental question of global regularity vs finite time singularity formation  remains open for the Euler equation in three spatial dimensions. In this lecture, I will review the history of this question and its connection  with the arguably greatest unsolved problem of classical physics, turbulence. Recent results on small scale and singularity formation in two  dimensions and for a number of related models will also be presented.

Title : II -The flow of polynomial roots under differentiation

Abstract : The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas,  Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that  should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking  resemblance to the models used in mathematical biology to describe collective behavior and  flocking of various species - such as fish, birds or ants. I will discuss joint work with Changhui  Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its  solutions and evolution of roots under differentiation for a class of trigonometric polynomials.

Bongsuk Kwon (UNIST)

Title : Euler-Poisson system of plasma and related problems

Abstract : Various plasma phenomena are mathematically studied using a fundamental fluid model for plasmas, called the Euler-Poisson system. Among them, plasma solitary waves are of our interest, for which existence, stability, and the time-asymptotic behavior of the solitary wave will be discussed in Junsik Bae's talk. On the other hand, to study nonlinear stability, a question of existence of smooth global solution naturally arises, which is completely open, to the best of our knowledge. We introduce the finite-time blow-up results for the Euler-Poisson system, and discuss the related open questions. This talk is based on joint work with Junsik Bae at UNIST.

Hantaek BAE (UNIST)

Title : Local and Global Existence of Solutions to Some MHD Models

Abstract : In this talk, we provide some MHD models and explain how to show the local and global existence of unique solutions of these models in 2D.

Youngae Lee (UNIST)

Title : Non-Abelian Chern-Simons-Higgs system with indefinite functional

Abstract : In this talk, we are concerned with the general non-Abelian Chern-Simons-Higgs models of rank two. The corresponding self-dual equations can be reduced to a nonlinear elliptic system, and the form is determined by a non-degenerate matrix K. One of the major questions is how the matrix K affects the structure of the solutions to the self-dual equations. There have been some existence results of the solutions to the self-dual equations when det(K) > 0. However, the solvability for the case det(K) < 0 is not fully understood in spite of its physical importance. In contrast to det(K) > 0, one major difficulty for the case det(K) < 0 is that the energy functional associated with the elliptic system is usually indefinite. The direct variational method thus fails. We overcome this obstacle and obtain a partially positive answer for the solvability when det(K) < 0 by controlling the indefinite functional with a suitable constraint. 

Junsik BAE (UNIST)

Title : Solitary waves of the Euler-Poisson system

Abstract : The ion in a fully ionized electrostatic plasma is described by the Euler-Poisson system with the Boltzmann relation. It is well known that the 1d model has traveling solitary waves. We briefly review some results on the 1d solitary waves and discuss (non)existence of localized stationary solutions of the multidimensional models. 

Ky Ho (UNIST)

Title : Profile of blow-up solutions to the self-dual Chern-Simons-Higgs vortex equation

Abstract : In this talk,  we are interested to analyze the blow-up behaviour of a sequence of solutions to the self-dual Chern-Simons-Higgs vortex equation on a flat torus around a blow-up point, in the sense of Brezis-Merle, when the Chern–Simons constant tends to zero. We obtain the blow-up profile of a sequence of solutions and in particular, provide pointwise estimates for the profile of a sequence of solutions in the both situations where "simple blow-up property" and "non-simple blow-up property” occur.                                                                                                                                                                                                                    This is based on a joint work with Prof. Youngae Lee  (UNIST).

Tania Biswas (UNIST)

Title : Optimal boundary control for the Cahn-Hilliard-Navier-Stokes system in 2D

Abstract : The Cahn-Hilliard-Navier-Stokes system describes the evolution of an incompressible isothermal mixture of two immiscible fluids. It consists of the incompressible Navier-Stokes equation for the average velocity and a Cahn-Hilliard equation for the relative concentration. In this talk, I will consider a boundary optimal control problem related to the Cahn-Hilliard-Navier-Stokes system in a bounded domain where the control is taken on the boundary of the navier-stokes equation. I will discuss about the existence of an optimal control and the first-order necessary conditions of optimality i.e. how to characterize the optimal control.