집중 강연
곽철광 (이화여자대학교)
Well-posedness theory of Korteweg-de Vries equation
In this intensive lecture series, we are going to study Korteweg–de Vries(KdV) equation derived from incompressible, irrotational ideal fluid in the shallow water wave regime. We focus on the fundamental problem on PDEs, precisely, the well-posedness theory for KdV equation based on Harmonic analysis techniques will be taken into account. In particular, we are going to discuss dispersive properties of linear solutions, and will further discuss nonlinear problems.
Day 1: Preliminaries in Harmonic analysis
In the first time of the intensive lecture series, I am going to briefly introduce Korteweg-de Vries equation, and discuss some preliminary results in Harmonic analysis, which will be used in the rest of lectures.
Day 2: Linear theory for KdV
In the second class, we are going to discuss some basic properties of solutions to the linear KdV equation. Particularly, based on preliminaries in the previous lecture, we are going to observe dispersive properties, local and global smoothing effect of linear solutions.
Day 3: Introduction to nonlinear problems
In the last lecture, we are going to discuss the well-posedness theory for nonlinear dispersive equations. In particular, we are going to show the local well-posedness of KdV equation by using linear properties of solutions already observed in the previous lecture.
옥지훈 (서강대학교)
Introduction to the calculus of variations
In this intensive lecture series, we discuss on foundation of the calculus of variations and related partial differential equations.
Day 1: Basic concepts of the calculus of variations and the Euler-Lagrange equations
Day 2: Sobolev spaces
Day 3: Existence theory of minimizers
전공 강연
석진명 (서울대학교)
Why do we learn Calculus? Invitation to Differential equations and Analysis
Abstract:
Calculus is one of the main subjects in high school mathematics but it is also the subject the students mostly dislike and feel difficult. Why do we learn calculus? Who made it? When and for what? In this talk, we overview the development of calculus in a historical point of view. Then we shall be naturally led to the following subjects: Differential equations and Analysis. We will be introduced some wonderful things the calculus and differential equations have made during past 400 years and the contributions of mathematicians for making such a calculus mathematically complete and rigorous.
박향동 (고등과학원)
Shock and Contact discontinuity
Abstract:
In this talk, I will discuss the recent results on the existence and stability of shocks and contact discontinuities for the steady Euler system.
배기찬 (서울대학교)
On the kinetic equation: BGK model for two-component gases
Abstract:
In this talk, we introduce the kinetic equation. We especially establish the unique global-in-time classical solution to the two-component BGK model when the initial data is a small perturbation of global equilibrium. Furthermore, we prove that the higher momentum and the energy interchange rate lead to faster convergence to the global equilibrium.
황병훈 (상명대학교)
On a relativistic BGK model
Abstract:
The BGK model is the kinetic equation describing the time evolution of the velocity distribution function of particles based on the relaxation process toward equilibrium. For classical particles, the phase space usually consists of all possible values of position and velocity variables. But when it comes to particles whose speeds are comparable to the speed of light, another consideration is needed to take into account the relativistic effects. In this talk, we present a brief introduction to the relativistic setting and introduce the recent work on the relativistic BGK model.
이기연 (KAIST)
Introduction to nonlinear Dirac equations and space-time resonance space
Abstract:
In this talk, we give a survey of the nonlinear Dirac equation which has Hartree-type nonlinearity. Dirac equation is a model of relativistic quantum mechanics and this equation was first introduced by Paul Dirac, a British physicist, in 1928. It is consistent with both the principles of quantum mechanics and the special relativity theory. We describe the derivation of our main equations and the concept of local and global well-posedness and asymptotic behavior. Especially, we present a concept of the space-time resonance space and its application to Dirac equations.
이윤정 (부산대학교)
Weighted Hessian estimates in Orlicz spaces for nondivergence elliptic equations
Abstract :
To the scope of the Schrodinger operator, we are interested in nondivergence elliptic operator L:=-a_ij D_ij+V with V satisfying a reverse Holder condition and small-BMO coefficient a_ij. To investigate the solvability of Lu=f with an adequate integrability condition of f, Hessian estimates of the solution u play a crucial role. In this talk, we review the known results for such Lp-type estimates and we discuss a weighted version of Hessian estimates in Orlicz spaces which is a generalization of Lp-spaces.
고혜림 (서울대학교)
Smoothing estimate in Fourier analysis
Abstract:
In this talk, we overview recent development of the square function estimates and decoupling inequality for the cone. As an application, we discuss local smoothing estimates for the wave operator, and maximal estimates for averages over sphere and curve.
유재현 (고등과학원)
Almost everywhere convergence of Fourier series
Abstract:
The problem of classifying functions such that the Fourier series converges to its original function is of fundamental importance in diverse fields of analysis such as harmonic analysis, partial differential equations. There are various senses of convergence of the Fourier series including uniform convergence, pointwise convergence, and norm convergence, but we shall only study almost everywhere convergence of the Fourier series in this talk. As to this subject, we will discuss the problem of investigating the relationship between the Lebesgue exponent and the summability index for the Bochner-Riesz mean of the Fourier series to converge almost everywhere.