2015

**May 7**

When : May 7, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Shih-Hsien Yu(National University of Singapore)**

**
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**Title:** Greens function for initial-boundary value problem

**Abstract :**

In this talk, we will present an approach to construct the Greens function for an initial boundary value problem with precise pointwise structure in the space-time domain. This approach is given in terms of transform variable and physical variable. The mixed algebraic structure in the transform variable gives the large time asymptotic nature of the differential equation. One also uses the algebraic structure in the fundamental solution. Finally, the hybrid of transform variable and physics variable to a precise pointwise space-time structure of the solution of an initial boundary value problem.

**April 15**

When : April 15, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Han Taek Bae(UNIST)**

**Title:** Global existence for some transport equations with nonlocal velocity

**Abstract :**

In this talk, we consider nonlocal and quadratically nonlinear transport equations. Prototypical examples are the surface quasi-geostrophic equation, the incompressible porous medium equation, Stokes equations, magnetogeostrophic equation and their variants. Among them, we address the global existence of weak solutions of an 1D model of the quasi-geostrophic equation and the full 2D dissipative quasi-geostrophic equations. To this end, we carefully choose dissipative quantities to minimize conditions of initial data using entropies.

**April 10**

When : April 10, 15:00 ~ 16:00

Where : bldg. 104 / room 701

**Prof. Jin Hae Park****(Chngnam National University)**

**Title:** Landau-de Gennes theory for Liquid Crystals with singularities

**Abstract :**

Landau-de Gennes theory is very useful to describe many phenomena which can be explained by the classical Oseen-Frank theory for nematic liquid crystals.

In this talk, I shall give a brief introduction to Landau-de Gennes theory which has been paid much attention these days. I also plan to discuss recent developments of Landau-de Gennes theory and address some interesting mathematical questions.

**April 2**

When : April 2, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Jin Keun Seo****(YONSEI University)**

**Title:** Compactness and Dirichlet's Principle

**Abstract :**

In this talk, we explore the emergence of the notion of compactness within its historical beginning through rigor versus intuition modes in the treatment of Dirichlet`s principle. We emphasize on the intuition in Riemann`s statement on the principle criticized by Weierstrass`requirement of rigor followed by Hilbert`s restatement again criticized by Hadamard, which pushed the ascension of the notion of compactness in the analysis of PDEs. A brief overview of some techniques and problems involving compactness is presented illustrating the importance of this notion.

Compactness is discussed here to raise educational issues regarding rigor vs intuition in mathematical studies. The concept of compactness advanced rapidly after Weierstrass's famous criticism of Riemann's use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded mathematicians to big pictures. Fortunately, Poincare and Hilbert defended Riemann's use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor. Rigor (highly detailed examination with toy models) and intuition (broader view with real models) are essentially complementary to each other.

**March 20**

When : March 20, 16:00 ~ 17:00

Where : bldg. 104 / room 701

**Prof. Wilfrid Gangbo(Georgia Institute of Technology, USA)**

**Title:** The mass transportation theory and its applications

**Abstract :**

We introduce the Wasserstein distance on the set of probability measure of bounded second moments. We describe its geodesics and study several constrained variational problems in the Wasserstein metric for which the set of probability densities satisfying the constraint is not closed. We also analyze the induced geometry of the set of densities satisfying the constraint on the variance and means, and we determine all of the geodesics on it. Some of the problems solved here arose in a study of a variational approach to constructing and studying solutions of the non–linear kinetic Fokker–Planck equation.

**March 18**

When : March 18, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Oleg Yu. Imanouvilov(Colorado State University, USA)**

**
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**Title:** Remark on Calderon's problem for the system of elliptic equations

**Abstract :**

We consider the Calderon problem in the case of partial Dirichlet-to-Neumann map for the system of elliptic equations in a bounded two

dimensional domain. The main result of the manuscript is as follows:

If two systems of elliptic operators generate the same partial Dirichlet-to-Neumann map the coefficients can be uniquely determined up to the gauge equivalence.

2014

**December 18**

When : Decemver 18, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Jung-ho Yoon(Ewha U.)**

**Title:** Data approximation methods and their applications

**Abstract :**

In this talk, we discuss (linear and non-linear) data approximation methods and their applications to CAGD and image interpolation. First, some approximation theories by radially symmetric functions and moving least squares methods are presented. Second, a nonlinear image interpolation algorithm is presented. The suggested method is based on the moving least squares projection technique, but introduces a fundamental modification. On the purpose of overcoming the drawbacks of linear methods, we introduce an adapted method using `non-local' penalty function such that the weights are determined in a way of depending local data similarilies.

**November**** 25**

When : November 25, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Oleg Yu. Imanuvilov(Colorado State University, USA)**

**Title **: Carleman estimates and Applications to Calderon's Problem

**Abstract** :

We present some recent results for the Calderon's problem for the systems of partial differential equations. Proof is based on construction of Complex Geometric Optics solutions and Carleman estimates.

**October 30**

When : October 30, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Shouhong Wang(Indiana University at Bloomington, USA)**

**Title **: Dynamic Transitions in Geophysical Fluid Dynamics

**Abstract** :

First, we present a brief introduction on dynamics transition theory, including in particular a general principle that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic and random. Then we shall address dynamic transitions in geophysical fluid dynamics and climate dynamics. Finally, we present the dynamic transitions of the Boussinesq equations associated with the thermohaline circulations and the El Nino Southern Oscillation. This is joint work with Tian Ma.

**October 28**

When : October 28, 16:00 ~ 18:00

Where : bldg. 302 / room 507

**Prof. Shouhong Wang(Indiana University at Bloomington, USA)**

**Title **: Unified Field Theory of Fundamental Interactions

**Abstract** :

**October 13-17**

When : October 13, 10:00 ~ 12:00

Where : bldg. 102 / room 106

When : October 14, 10:00 ~ 12:00

Where : bldg. 104 / room 109

When : October 14, 15:00 ~ 17:00

Where : bldg. 102 / room 106

When : October 15, 10:00 ~ 12:00

Where : bldg. 102 / room 809

When : October 17, 15:00 ~ 17:00

Where : bldg. 102 / room 106

**Prof. Jörg Wolf(Humboldt University of Berlin, Germany)**

** **

**Title :** On the Caffarelli-Kohn-Nirenberg theorem

**Abstract : **

By the following series of lectures we will present a detailed proof of the Caﬀarelli-Kohn-Nirenberg theorem, regarding the partial regularity of the Navier-Stokes equations. Among the models of incompressible viscous ﬂuids, perhaps the Navier-Stokes system is one of the most popular and studied system. Since the pioneering work by Leray [7] the theory of the Navier-Stokes equations has been widely developed, where fundamental problems such as existence of weak solutions, conditions for global and local regularity or asymptotical behaviour have been solved. However, despite strong eﬀorts one of the most important question, the existence of a unique classical global solution for general date has been left open.

Concerning the local regularity, the ﬁrst result is due to Scheﬀer [9], who introduced the notion of a suitable weak solutions to the Navier-Stokes equations, obeying the local energy inequality. In 1982, based on Scheﬀers notion, Caﬀarelli, Kohn and Nirenberg [1] obtained an optimal result of partial regularity of suitable weak solutions, by showing that the one-dimensional parabolic Haussdorﬀ measure of the singular set is zero (for alterative proofs cf. [8, 6, 12, 13]). The proof of the Caﬀarelli-Kohn-Nirenberg theorem rests on decay estimates, derived from the local energy inequality. Unlike the Leray-Hopf weak solutions, a suitable weak solution cant be obtained by using a Galerkin scheme. Further-more, for constructing such solutions the existence of the pressure has played an important role (cf. [11, 3, 10]). However, in the recent paper [15], by using a new local pressure representation, the proof of the Caﬀarelli-Kohn-Nirenberg theorem can be carried out for any domain. The ﬁnal result will be achieved by using the well-known regularity theory of the heat equation.

**September 24**

When : September 24, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Hideyuki Miura(Tokyo Institute of technology)**

**Title** : Remark on the Helmholtz decomposition in domains with noncompact boundary

**Abstract**:

**September 17**

When : September 17, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Post doc. So-yeun Jung(KIAS)**

**Title** : Pointwise asymptotic behavior of modulated periodic reaction-diﬀusion waves

**Abstract**:

We discuss, under standard spectral stability assumptions, pointwise nonlinear stability and asymptotic behavior of perturbations with detailed rates of decay of spatially periodic traveling waves u(x,t) = ¯u(x − at) of systems of reaction-diﬀusion equations. We ﬁrst estimate pointwise bounds on the Green function of the linearized operator about ¯u by working with the periodic resolvent kernel and Bloch decomposition. With our linearized estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, we show the perturbations of periodic traveling reaction-diﬀusion waves converge to the heat kernel under small initial perturbations with the Gaussian decay and the algebraic decay, respectively. Here, we emphasize again that it is the pointwise description that is the main new aspect of our research.

**September 15**

When : September 15, 16:00 ~ 16:50, 17:00 ~ 17:50

Where : bldg. 102 / room 105

**Prof. Nam-kwon Kim(Chosun U.)**

**Title** : Spectral decomposition and Serrin's condition in Navier-stokes

**Abstract**:

Spectral decomposition of solutions of Navier-stokes equations has been useful in analyzing qualitative properties of solutions of N-S. We consider those spectral decomposition and variants of Serrin's condition in terms of these decomposition.

**Prof. Jong-min Han(Chosun U.)**

**Title** : Bifurcation analysis of the damped Kuramoto-Sivashinsky equation

**Abstract**:

In this talk, we consider bifuracation of the damped Kuramoto-Sivashinsky equation on a odd periodic interval which has two bifurcation parameters: control parameter and the period. When one of two parameters is fixed, we present how the equation bifurcates to invariant sets as the other parameter varies. Using the center manifold analysis, we verify the structure of the bifurcated invariant sets.

**July 30**

When : July 30, 16:00 ~ 17:00

Where : bldg. 104 / room 701

**Prof. Alexis Vasseur**

**Title** : Shocks and layers in compressible fluid mechanics

**Abstract**:

In this talk, we will consider discontinuous solutions for inviscid compressible mechanics known as shocks. Usually, those shock solutions can be derived from refined models, as kinetic equations for instance. This involves the production of layers. We will study the interactions between those shocks, and the associated layers, on simplified models.

**June 12**

When : June 12, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Sung-Ik Sohn (GWNU)**

**Title** : Regularization Models for Vortex Sheet Motion

**Abstract**:

A vortex sheet is a sharp interface in an incompressible fluid across which the tangential velocity is discontinuous, and is a model for the interface of a parallel shear flow. It is well known that evolving vortex sheets generally form singularities in finite time. In this talk, we discuss regularization models for the vortex sheet motion, mainly on two blob-regularizations: the Krasny model and the Beale-Majda model, and also on the Euler-alpha regularization. The non-uniqueness of weak solutions of a vortex sheet has been recently established, and important questions are whether the solutions of different regularization models converge to distinct or the same weak solution, and what the weak solutions are. We thoroughly compare differences and similarities of the models including the linear stability and the limiting behavior of the models, and examine dependence of weak limits on the choice of regularization. Numerical results for sprial turns of vortex sheets are presented to investigate these isssues and explore the structure of weak solutions.

**June 10**

When : June 10, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Jerry Bona (University**** of Illinois at Chicago)**

**Title** : Solitary Waves and other Long wave Phenomena

**Abstract**:

Solitary waves in nature were first observed in the first part of the 19th century. For many years, they were a curiosity in fluid mechanics. In recent decades, their pervasive role in the evolution of general disturbances in nonlinear, dispersive systems has brought them to the forefront of modern research in a variety of subjects. The lecture will begin with a short historical account followed by a discussion of existence and stability for such wave motion. More complex phenomena will then be introduced, leading to a variety of open problems in the subject.

**May 16**

When : May 16, 17:00 ~ 18:00

Where : bldg. 104 / room 701

**Prof. Yong-Hoon Lee (Pusan National Univ.) **

**Title** : Solution operator for generalized Laplacian systems with a sign-changing weight

**Abstract**:

**April 18**

When : April 18, 15:00 ~ 16:00

Where : bldg. 102 / room 105

**Prof. Sun-Ho Choi (KAIST) **

**Title** : Dynamic behaviors of the Lohe oscillators under attractive and repulsive couplings

**Abstract**:

We present dynamic behaviors of the Lohe oscillators on the unit sphere under the attractive and repulsive couplings. For the dynamics of the ensemble of Lohe oscillators, we introduce an order parameter representing the degree of synchronization as the modulus of centroid as in mean-field theory of statistical physics, and study the dynamics of the order parameter. This order parameter also completely characterize the equilibria up to constant motion for identical oscillators. For the identical oscillators, we show that order parameter starting from non-zero values evolves exponentially toward the unit value, which corresponds to the completely synchronized state for attractive couplings. In contrast, the order parameter approaches to zero exponentially fast for repulsive couplings. For non-identical oscillators, we show that the ensemble has a practical synchronization as the coupling strength goes to infinity so that we can push the configuration close to the completely synchronized state asymptotically.

**April 18**

When : April 18, 14:00 ~ 15:00

Where : bldg. 102 / room 105

**Prof. Hwakil Kim (Seoul National University) **

**Title** : Optimal Mass Transportation and its applications to PDEs

**Abstract**:

Since Jordan-Kinderlehrer-Otto gave a link between Optimal Transport and PEDs, there has been an wide application of OT theory to PDEs. Among these, in this talk, we will briefly present on the following types of PDEs; Hamiltonian flows, Hamilton-Jacobi equations and Coupled system of fluids and Polymers.

**April 9**

Sanghyuk Lee (Seoul National University)

Title : Sharp bounds for Stein's square functions

Abstract: In this talk we consider sharp bounds for Stein's square functions in Lebesgue spaces. The problem is regarded as an extension of Bochner-Riesz conjecture, and thanks to square average the bounds have various applications. We obtain an improved range of boundedness and discuss related problems

Paul Smith (U.C. Berkeley)

Title: Low regularity local wellposedness of the Chern-Simons-Schroedinger System

Abstract: The Chern-Simons-Schroedinger model in two spatial dimensions is a covariant NLS-type problem and is L^2 critical. We prove that, with respect to the heat gauge, this problem is locally well-posed for initial data that is small in H^s, s > 0. This work is joint with Baoping Liu and Daniel Tataru.

Kwon, Soonsik (KAIST)

Title: Normal form reduction for unconditional well-posedness of canonical dispersive equations.

Abstract: Normal form method is a classical ODE technique begun by H. Poincare. Via a suitable transformation one reduce a differential equation to a simpler form, where most of nonresonant terms are cancelled. In this talk, I begin to explain the notion of resonance and the normal form method in ODE setting and Hamiltonian systems. Afterward, I will present how we apply the method to nonlinear dispersive equations such as KdV, NLS to obtain unconditional well-posedness for low regularity data.

When : April 9, 14:00 ~ 18:00

Where : bldg. 104 / room 701

**April 7**

Paul Smith (U.C. Berkeley)

Title: Global wellposedness of the equivariant Chern-Simons-Schroedinger System

Abstract: The Chern-Simons-Schroedinger model in two spatial dimensions is a covariant NLS-type problem and is L^2 critical. Establishing global wellposedness at the critical regularity given small initial data is an open problem. However, under a certain symmetry assumption, namely, the equivariance ansatz, which includes the radial setting as a special case, it is possible to say more. In particular, joint work of the speaker with Baoping Liu establishes global wellposedness of the equivariant Chern-Simons-Schroedinger system for large data in the defocusing setting and in the focusing setting for data with charge less than that of the ground state.

When : April 7, 16:00 ~ 18:00

Where : bldg. 104 / room 701

**March 27**

S. Machihara (Saitama University)

M. Okamoto (Kyoto University)

Title : On the Cauchy problem of the generalized Thirring equations

abstract : We study the Cauchy problem of the generalized Thirring equations in Sobolev space. According to structure of nonlinear terms, the needed regularity for well-posedness varies drastically. While S. Machihara shows well-posedness part, M. Okamoto explains ill-posedness issue.

**March 18**

Reinhard Farwig (Technische Universitat Darmstadt)

Title : How (fast) do solutions of the Boussinesq system decay?

**March 12**

Joerg Wolf (Humboldt University)

Title : On the local energy inequality of weak solutions to the generalized Navier-Stokes equations

**February 20**

Soyeong Choi (Dongguk University)

Title : Period functions and harmonic weak Maass forms

**February 18**

JINMYONG SEOK (KIAS)

Title : Existence results about the nonlinear Schrodinger-Poisson equations

2013

**December 19**

Minha Yoo (NIMS)

Title : The viscosity method for the homogenization of soft inclusions

**December 19**

Soojung Kim (NIMS)

Title : Asymptotic behavior in parabolic equations and its application to

elliptic eigenvalue problems

**December 9**

Sookkyung Lim (University of Cincinnati)

Title : DNA folding by Srochastic generalized IB method

Namhee Kim (Princeton University)

Title : Network Modeling of RNA Structure with Applications to RNA Structure Prediction

**December 6**

Peter Constantin (Princeton University)

Title : Regularity, long time behavior and absence of anomalous dissipation

in nonlocal evolution equations

**December 4**

Peter Constantin (Princeton University)

Title : Mathematical problems of incompressible fluids

**November 28**

Hyunseok Kim (Sogang University)

Title : On Leray's weak solutions of the stationary Navier-Stokes equations in exterior domains

**November 15**

Namkwon Kim (Chosun University)

Title : Nontopological solutions in Chern-Simons Theory

**November 13**

M. Okamoto (Kyoto University)

Title : Ill-posedness of the Cauchy problem for the Chern-Simons-Dirac system in one dimension

**November 12**

S. Machihara (Saitama University)

Title : Local and global well-posedness of the Cauchy problem for the Chern-Simons-Dirac system in

one dimension

**November 8**

Dohoon Choi (Korea Aerospace University)

Title : Arithmetic of Hecke fields

Abstract : The Hecke algebra plays a crucial role in the theory of modular forms. Richarithmetic properties of modular forms, including Ramanujan delta function, come from the structure of Hecke algebra. A Hecke field is a number field generated by the eigenvalues of a modular form for the action of the Hecke algebra. In this talk, I will talk about arithmetic of Hecke fiels and relations to other subjects.

Dohoon Choi (Korea Aerospace University)

Title : Arithmetic of Hecke fields

Abstract : The Hecke algebra plays a crucial role in the theory of modular forms. Richarithmetic properties of modular forms, including Ramanujan delta function, come from the structure of Hecke algebra. A Hecke field is a number field generated by the eigenvalues of a modular form for the action of the Hecke algebra. In this talk, I will talk about arithmetic of Hecke fiels and relations to other subjects.

Joonyeong Won (KIAS)

Title : Surveys on potential density

Abstract : This talk is about surveys on some recent results about potential density.

Zhouping Xin (Chinese University of Hong Kong)

Title : On Multi-Dimensional Compressible Navier-Stokes Equations

Abstract :

I will discuss some recent progress on the global in time well-posedness and asymptotic behavior of classical solutions for the multi-dimensional compressible Navier-Stokes system. The focus will be on solutions which are of possible large oscillations and may contain vacuum. First, I present blow-up of classical solutions for the full system for a large class of initial data containing vacuum. Then we concentrate on the global existence of smooth solutions to the ISENTROPIC flows in either 3 or 2 dimension for initial which have small energy but possible large oscillations and may contain vacuum. Our results, in particular, yield the uniqueness and regularity for the weak solutions of Lions-Feireisl provided that the initial total energy is suitably small. Furthermore, some results of the large time decay will also be discussed.

Ben Duan (Postech)

Title : Smooth transonic Euler-Poisson flow in nozzles

Abstract : In this talk, we consider the 2-D steady isentropic Euler-Poisson system. The aim is to find out a subsonic-sonic-supersonic smooth solution in a given flat nozzle. To compare the results of smooth transonic Euler flows, pioneer and very latest work will be discussed.

**October 31**

Sunho Choi (National University of Singapore)

Title : Asymptotic behavior of the nonlinear Vlasov equation with a sef-consistent force

Abstract :

We present a critical threshold phenomenon on the L1-asymptotic completeness for the nonlinear Vlasov equation with a self-consistent force. For a long-ranged self-consistent force, we show that the nonlinear Vlasov equation has no L1-asymptotic completeness, which means that the nonlinear Vlasov flow cannot be approximated by the corresponding free flow in L1-norm time asymptotically.

In contrast, for a short-ranged force, the nonlinear Vlasov flow can be approximated by the free flow time-asymptotically. Our result corresponds to the kinetic analogue of scattering results to the Schrodinger-type equations in quantum mechanics.

**October 21**

Changheon Kim (HanYang University)

Title : On the infinite product exponents of meromorphic modular forms for certain arithmetic groups

Abstract :

In this talk I will obtain a formulae for the infinite product exponents of meromorphic modular forms for certain arithmetic groups which are determined by the divisors of the modular forms. As an application I will reprove the formula for the number of representations of a given integer as a sum of four squares. This is a joint work with Soyoung Choi.

**October 10**

Yeongpil Choi (Imperial College London)

Title : Global existence, large-time behavior, and hydrodynamic

Abstract :

In this talk, we present a particle-fluid model which describes the interactions between particles with local alignment force and incompressible viscous fluid. For the presented model, we provide global existence of weak solutions and a priori estimates for large-time behavior of solutions. We also study the hydrodynamic limit from the particle-fluid system to type of two-phase fluid system.

**September 27**

Daeyeol Jeon (Kong-Ju)

Title : Torsion subgroups of elliptic curves over number fields

Abstract : In this talk, we give a brief introduction to the recent results on the torsion subgroups of elliptic curves over number fields.

**September 12**

Shangkun Weng (Chung-Ang University)

Title : On Multi-dimensional Steady Subsonic Flows Determined by Physical Boundary Conditions

Abstract : This talk will concern an inflow-outflow problem for steady subsonic gas flows in a nozzle with finite length, aiming at finding physically acceptable boundary conditions on upstream and downstream. It contains the following three new issues. Firstly, we characterize a set of physically acceptable boundary conditions to ensure the existence of subsonic flows in 2-D finite long nozzles, both the irrotational and full Euler flows are considered. Secondly, we develop a new formulation for 3-D Euler system and discover a new conserved quantity and a system of new conservation laws and obtain the existence of subsonic Euler flows in a rectangular cylinder with physical boundary conditions. Finally, we establish the existence of a class of subsonic flows to the Euler-Poisson models.

**September 12**

Jiahong Wu (Chung-Ang University)

Title : Recent Developments on the 2D Magnetohydrodynamic Equations with Partial Dissipation

Abstract : The magnetohydrodynamic (MHD) equations model electrically conducting fluids such as plasmas and are important in understanding many natural phenomena such as solar flares and the formation of Northern Lights. Mathematically the MHD equations can be difficult to analyze due to the nonlinear coupling between the induction equation and the Naver-Stokes equations with the Lorentz force. One fundamental problem on the MHD equations is whether or not their solutions exist

for all time. This problem has attracted considerable interest recently for the 2D MHD equations with partial dissipation. This talk presents some very recent global regularity results for various partial dissipation cases.

**September 4**

Juhee Jang (UC Riverside)

Title : Stability theory of polytropic gaseous stars

Abstract : I'll discuss stability theory of Lane-Emden equilibrium stars under Euler-Poisson or Navier-Stokes-Poisson system. A linear stability can be characterized by the adiabatic exponent. A nonlinear instability will be also discussed.

**August 30**

Wanho Lee (NIMS)

Title : Computer Simulations of C. elegans and Glioma Invasion Models Using the Immersed Boundary Method

Abstract : In this talk, I will present numerical simulations of C. elegans movement and glioma invasion models using the immersed boundary (IB) method. The IB method is used to investigate problems of fluid-structure interaction type.

C. elegnas is a transparent nematode about 1mm in length that lives in temperate soil environments. In our model, three curved lines represent slender body of C. elegans as dorsal / ventral muscles and center skeleton and each curve consists of spring element. In particular, curves which are represented the dorsal and ventral muscles consist of two parts by active and passive elements. In order to simulate swimming motion of C. elegans, traveling wave signal is alternately applied in dorsal and ventral active elements. Our model can reproduce forwarding swimming motion and coiling motion of C. elegans via changing applied signal.

Glioma is the aggressive brain cancer with the poor survival rate. We develop two-dimensional hybrid immersed boundary method where glioma cells migrate under physical constraints in the given microenvironment in the brain. Experimentally it was known that glioma cells can use myosin II to migrate and invade surrounding brain tissue when injected into mouse brain. We investigated a mathematical model based on cell mechanics which predicts how glioma cells infiltrate the brain tissue under the complex biochemical and biomechanical signals. The model also predicts the myosin II plays an essential role in regulating the cell migration through fixed glial normal cells in the brain.

**August 12**

Dongwook Lee (FLASH Center for Computational Science, The University of Chicago)

Title: High-order accurate numerical methods for large-scale scientific computing

Abstract: Modeling diverse physical processes using mathematical algorithms has become a great success in modern science and engineering. The underlying mathematical models are carefully designed to perform large scale computer simulations that involve disparate scales of space and time. Such complexities often arise when incorporating various multiphysical components which are represented by classes of partial differential equations (PDE).

In this talk, I will show some of the key ideas and challenges of computational mathematics in the framework of the University of Chicago's FLASH code.

FLASH is a highly-capable, massively parallel, publicly available open source scientific code with a wide user base in the fields of astrophysics, cosmology, and high-energy-density physics.

In the first part, I will discuss fundamental components of mathematical algorithms to solve PDEs in order to construct numerical solutions of computational fluid dynamics, gas dynamics and plasma physics.

Mathematical algorithms are going to be described with special cares on two numerical approaches: first, the traditional high-order polynomial based formulation, and second, a new innovative exponentially converging formulation based on Gaussian Process.

In this part of my talk, I will show valuable importances of using high-order accurate numerical methods that will be cruicial for future high-performance (HPC) computing architectures.

In the second part, I will present state-of-art scientific simulations using the numerical algorithms introduced in the first part. They will include large scale computer simulations of astrophysics and high-energy-density plasma physics.

**July 24**

Hantaek Bae (UC Davis)

Title: Log-Lipschitz Regularity of Mild Solutions of the 3D Navier-Stokes Equations

Abstract: In this talk, we present how to obtain Log-Lipschitz regularity of solutions to the 3D Navier-Stokes equations with initial data in . As a corollary, we also obtain lder regularity of the flow map.

**July 19**

Kyungkeun Kang (Yonsei University)

Title: A 2D-model of cell sorting induced by propagation of chemical signals along spiral waves

Abstract: We study a model for cell sorting based in the presence of differential chemotactic sensitivities. The chemical waves which are responsible for the cell motion propagate along some spiral waves. We prove rigorously that cells with larger chemotactic sensitivity are trapped in a region close to the center of the spiral waves if these propagate along some archimedian spirals.

**June 24 **

1. Myeongju Chae (Hankyung Univ.)

Title: Stability of the Hamiltonian system in High sobolev space

Abstrac: We study a long time behavior of the Hamiltonian PDE on periodic domains.

2. Sung-Jin Oh (Princeton Univ.)

Title: Finite energy global well-posedness of the non-abelian Chern-Simons-Higgs equations using a Yang-Mills heat flow gauge

Abstract: In this talk, we will prove finite energy global well-posedness of the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space for general compact non-abelian gauge groups. The case of abelian gauge groups was recently established by [Selberg-Tesfahun, DCDS-A 2013] using the Lorenz gauge; in the non-abelian case, however, conventional gauges (such as Lorenz or Coulomb) become troublesome for large initial data. To address this difficulty, we will utilize the caloric-temporal gauge, introduced in [Oh, arXiv 2012] for the Yang-Mills equations, which is constructed using the Yang-Mills heat flow. This is despite the apparent lack of a naturally associated heat flow for the Chern-Simon-Higgs equations.

**May 30**

1. Minkyu Kwak (Chonnam National University)

Title: Some remarks regarding to Navier-Stokes equations

Abstract: In this talk, we discuss some results regarding to Navier-Stokes equations. The first one is about the global regularity of 3D equations. We introduce a spectral decomposition and will discuss how much we can extend previous results. The second one is about the cone property of parabolic partial differential equations, which is an important question to 2D N-S equations but widely open problem for long times.

2. Hyeong-Ohk Bae (Ajou University)

Title: Strong Solutions for the Interaction of Particles and Fluid

Abstract: We presented a coupled kinetic-fluid model for the interactions between Cucker-Smale(C-S) flocking particles and incompressible fluid on the periodic spatial domain T^d. Our coupled system consists of the kinetic Cucker-Smale equation and the incompressible Navier-Stokes equations, and these two systems are coupled through the advection term and drag force. For the proposed model, we provide a strong solutions in the periodic domain and bounded domains. This is joint work with Young-Pil Choi, Seung-Yeal Ha and Moon-Jin Kang.

**May 3**

1. Speaker: Yong-Jung Kim (KAIST)

Title: Burgers equation with stationary point source

Abstract: Existence, uniqueness and regularity of the global weak solution to the Burgers equation with a reaction term is shown when the reaction term is given as a time independent point source and produces heat constantly. An explicit solution is obtained and used to show the long time asymptotic convergence of the solution to a steady state. For the heat equation case without any convection the solution diverges everywhere as time increases and hence it is the first order convection term that gives the compactness of the solution trajectory of the Burgers equation with reaction. This is a joint work with Jaywan Chung and Marshall Slemrod.

2. Speaker: Ki-Ahm Lee (Seoul National University)

Title: Curvature Flows

Abstract: In this talk, we are going to consider the evolution of surfaces with a speed proportional to curvatures. Regularity theory depends on the class operators that the flow satisfies. We will discuss the curvature estimate, pinching estimate, entropy estimate, monotonicity formulas to find the regularity and the analysis of singularities. And we also discuss free boundary problems in the curvature flows.

**April 11**

Speaker: Joonil Kim (Yonsei Univ.)

Title: Multiple Hilbert Transforms associated with polynomials

**April 4**

1. Speaker: Jaeyoung Byeon (KAIST)

Title: Variational construction of clustering standing waves for NLS

2. Speaker: Jongmin Han (Kyung Hee University)

Title: Bubbling solutions for the Chern-Simons gauged O(3) Sigma model in a plane

**March 18**

1. Speaker: Lee, Hojoo (KIAS)

Title: Twin surface equations

Abstract: We introduce various generalizations of Calabi's correspondence between the minimal surface equation in Euclidean three space and the maximal surface equation in Lorentz three space.

2. Speaker: Kang, Hyunsuk (KIAS)

Title: Anisotropic flow of convex hypersurfaces by the square root of the scalar

curvature

Abstract: We study the deformation of convex hypersurfaces governed by the scalar curvature in Euclidean space. The speed of deformation is given by the square root of the scalar curvature multiplied by a positive function which we call an anisotropic factor. We show that the existence of the flow and the smooth convergence to a round point given some condition on the anisotropic factor. In dimension two, we also obtain the limit equation satisfied by the rescaled flow. This is a joint work with Lami Kim and Ki-Ahm Lee.

**March 12**

Speaker: Tsuyoshi Yoneda (Hokkaido University)

Title: Mathematical consideration of separation phenomena on the two-dimensional Navier-Stokes equation.

Abstract: In general, before separating from a boundary, the flow moves toward reverse direction near the boundary against the laminar flow direction. In the non-stationary two-dimensional Navier-Stokes equation with initial flow having diffusing laminar structure,

topologically changing flow (inducing the reverse flow) must occur in finite time.