2nd Mini Workshop on PDEs

27th - 28th April 2018, Department of Mathematics, Inha University

This is the continuation of the workshop on PDEs, which was first held in September 2017. The purpose of this workshop is to bring together young researchers in PDEs from Suwon and Incheon so that they can report their recent research results about different aspects of PDEs and share their scientific experiences. Graduate students and postdocs are particularly encouraged to participate in. Please send an email to ypchoi at inha.ac.kr if you need further information.

Invited speakers:

Sun-Ho Choi(Kyunghee University, Suwon)
Woocheol Choi(Incheon National University, Incheon)
Young-Pil Choi(Inha University, Incheon)
Xiaofei Li(Inha University, Incheon)
Byungsoo Moon(Incheon National University, Incheon)
Jihoon Ok(Kyunghee University, Suwon)
Ihyeok Seo(Sungkyunkwan University, Suwon)
Hyowon Seo(Kyunghee University, Suwon)
Jinmyoung Seok(Kyonggi University, Suwon)
Seok-Bae Yun(Sungkyunkwan University, Suwon)

Talk title/abstract and Schedule:

Chair: Young-Pil Choi

April 27, 2018 (14:00-14:40, 5E210A)

On a derivation of the polyatomic Vlasov equation with vibratory and rotational motions

Sun-Ho Choi
Department of Applied Mathematics, Kyunghee University

Abstract: In this talk, we discuss a mathematical theory for polyatomic gas. Polyatomic gas is common in our environment. For examples, hydrogen, nitrogen, oxygen. However, in the mathematical theory of gas, one assumes that each particle of gas is monoatomic for simplicity. Here, we derive a polyatomic Vlasov equation with the self-consistence Poisson force field and prove the global existence of the solution to the polyatomic Vlasov-Poisson equation with vibratory and rotational motions.

April 27, 2018 (14:40-15:20, 5E210A)

On the wave-breaking phenomena and global existence for the periodic rotation-two-component Camassa-Holm system

Byungsoo Moon
Deparment of Mathematics, Incheon National University

Abstract: In this talk, we discuss the periodic rotation-two-component Camassa-Holm system, which can be derived from the f-plane governing equations for the geophysical water waves with a constant underlying current. The nonlocal nonlinearities on blow-up criteria and wave-breaking phenomena are established. Finally, a sufficient condition for global solutions is obtained by using a method of the Lyapunov function.

Break(15:20 - 15:30)

Chair: Jinmyoung Seok

April 27, 2018 (15:30-16:10, 5E210A)

Strichartz and smoothing estimates for dispersive equations

Ihyeok Seo
Deparment of Mathematics, Sungkyunkwan University

Abstract: In this talk, we discuss Strichartz and smoothing estimates for Schrodinger and wave equations, which particularly give an answer to an open question posed by J. A. Barcelo, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela. This is a joint work with Youngwoo Koh.

April 27, 2018 (16:10-16:50, 5E210A)

Quantum BGK model near a global Fermi-Dirac distribution

Seok-Bae Yun
Deparment of Mathematics, Sungkyunkwan University

Abstract: In this talk, we consider the existence and asymptotic behavior of fermionic quantum BGK model, which is the relaxation model of the quantum Boltzmann equation for fermions, in the case when the initial data starts sufficiently close to a global Fermi-Dirac distribution. Two unexpected features, among others, unobserved in the study of classical or relativistic problems arise. First, the existence of the equilibrium parameters should be established through a set of nonlinear equations in each iteration step. Secondly, it is observed that the momentum weight imposed on the perturbation to guarantee the self-adjointness and the coercivity of the linearized relaxation operator must take a different form from that of the classical or relativistic case.

Break(16:50 - 17:00)

Chair: Seok-Bae Yun

April 27, 2018 (17:00-17:40, 5E210A)

Anisotropic diffusion for the vineyard

Hyowon Seo
Department of Applied Mathematics, Kyunghee University

Abstract: In this talk, we will propose an anisotropic diffusion model for aligned environments. To derive the anisotropic diffusion model we only consider two factors, namely, wind distribution and heterogeneous environment. From this model, we can calculate the spreading speed of pathogen using traveling wave solutions and observe the interesting phenomenon which is called pushed front.

April 27, 2018 (17:40-18:20, 5E210A)

Construction of weakly neutral inclusions of general shape by an imperfect interface

Xiaofei Li
Department of Mathematics and Institute of Applied Mathematics, Inha University

Abstract: We consider the problem of neutral inclusions to multiple uniform fields for two-dimensional conductivity. If an inclusion with a different material property is inserted into a medium with a uniform field, then the uniform field is perturbed in general. However, there are some inclusions which do not perturb the uniform field outside the inclusion, and such an inclusion is referred to as a neutral inclusion. There are two known ways of constructing neutral inclusions, a coated structure with perfect bonding interface and a single inclusion with an imperfect interface. It has been proved that for coated structure with perfect bonding interface, the only admissible shapes of neutral inclusions are concentric disks if the medium is isotropic and confocal ellipses if the medium is anisotropic. For neutral inclusions with imperfect interface, the only admissible shapes are disks (balls) with constant interface parameters if the medium is isotropic, and ellipses (ellipsoids) for the anisotropic case. In this paper we construct weakly neutral inclusions of general shape using imperfect interfaces. Weakly neutral inclusion is defined through vanishing of its first order polarization tensor. A weakly neutral inclusion can be seen vaguely by uniform fields. We prove that for any Lipschitz domain of a perfect conductor, there exists an imperfect bonding interface so that the domain together with the imperfect interface is weakly neutral to multiple uniform fields. Numerical results are also demonstrated to show weak neutrality. This is a joint work with Hyeonbae Kang.

Chair: Ihyeok Seo

April 28, 2018 (09:30-10:10, 5E210A)

Finite element approximation for the energy minimizing solution of the Lane-Emden equation

Woocheol Choi
Deparment of Mathematics Education, Incheon National University

Abstract: In this talk, we are concerned with the finite element method for the Lane-Emden equation, which is $-\Delta u = u^{p}$ , posed on a bounded domain with boundary zero condition. We consider the finite element approximation of the energy minimizing solutions to the equation, and obtain their sharp error estimates in $H^1$ and $L^2$ spaces. This is a joint-work with Younghun Hong and Jinmyoung Seok.

April 28, 2018 (10:10-10:50, 5E210A)

Regularity theory for equations with variable growth

Jihoon Ok
Department of Applied Mathematics, Kyunghee University

Abstract: We discuss about basic regularity theory for elliptic and parabolic equations with variable growth. The model equations is the $p(x)$ -Laplace equation:

$\mathrm{div}\left(|Du|^{p(x)-2}Du\right)=0.$

Here, the variable function $p(\cdot)$ satisfies that $1<\gamma_1\leq p(\cdot)\leq \gamma_2<\infty$ . Note that it is not enough to assume the plain continuity on $p(\cdot)$ in order to obtain regularity(even the boundedness) of the weak solution of the $p(x)$ -Laplace equation. In this talk, I will introduce relations between regularities of $p(\cdot)$ and the weak solution and a recently result in this direction.

Break(10:50 - 11:00)

Chair: Woocheol Choi

April 28, 2018 (11:00-11:40, 5E210A)

Consensus-based global optimization method

Young-Pil Choi
Department of Mathematics and Institute of Applied Mathematics, Inha University

Abstract: In this talk, we discuss an analytical framework for investigating the efficiency of a consensus based model for tackling global optimization problems. We study the optimization algorithm in the mean-field sense showing the convergence to the global minimizer for a large class of functions.

April 28, 2018 (11:40-12:20, 5E210A)

Ground states to 3-coupled Nonlinear Schrodinger equations

Jinmyoung Seok
Deparment of Mathematics, Kyonggi University

Abstract: The system of N-coupled nonlinear Schrodinger equations arises from optical communication and multispecies Bose-Einstein condensates. In this talk, we give a complete classification for existence of ground states and positive solutions to 3-coupled NLS. We will see that the necessary and sufficient condition for a ground state to have nonvanishing components is that 3 interaction coupling constants represent three sides of a triangle

Organizer:

Young-Pil Choi (Inha University)

This scientific event is sponsored by the BRL(Basic Research Lab) grant from Institute of Applied Mathematics at Department of Mathematics, Inha University.