The 11th Korea PDE Winter School

0. 체크인

스쿨 행사는 1월6일 9시부터 시작하지만 참가자들은 유니스트 숙소에 5일 오후에 투숙합니다. 체크인시 숙소 키를 픽업 하셔야 해서 3시~6:30분에 유니스트에 도착할 수 있도록 일정을 준비해 주십시요. 

체크인, 키 픽업 장소 : 유니스트 경영관(114동) 로비 (지도 참조)

5일 저녁식사는 6시부터 경영관 로비에서 푸드박스로 제공합니다. 

부득이하게 6:30 이후에 도착하시는 참가자들은 미리 saarc@kaist.ac.kr  로 알려 주십시요. 

1. 숙소 안내

∙강연자 및 위원회: 게스트하우스(308동), 호실 및 비밀번호는 개별 이메일로 안내해 드릴 예정입니다.

∙참가자: 기숙사(306동, 307동)

체크인/체크아웃 시간

∙체크인: 15:00

∙체크아웃: 11:00 

기숙사 객실 키 수령/반납

∙수령: 5일(일) 15시부터 UNIST 경영관 로비에서 호실 확인 후 키 수령

∙반납: 행사장 로비에 키 반납박스 구비 예정, 키 반납 후 퇴실

*기숙사 구비 물품 및 세부 이용 안내는 첨부 문서 참고 바랍니다. 

*체크인 시 문제가 발생하면 현장 담당자에게 문의 주시기 바랍니다.

(담당자 경영관 로비에 상주 예정)

2. 행사장 안내

The 11th Korea PDE Winter School은 UNIST 경영관(114동) 111호에서 진행됩니다.

같은 건물의 소형 강의실을 토론 공간으로 대관하였으니 활용하시길 바랍니다. 

토론실 205호, 208호, 209호

오전 9시부터 강의장 로비에서 다과와 커피가 제공됩니다.

3. 명찰 및 식권 배부 안내

명찰과 식권은 경영관 로비에서 1월 5일(일) 체크인 시 배부 예정입니다. (미수령자는 6일 월요일 오전 9시부터 수령 가능)

4. 식사 안내

점심식사(바우처)와 저녁식사(케이터링)를 제공합니다. 아침식사는 개별적으로 기숙사식당에서 이용가능합니다.

원활한 식사 제공 분량 추정을 위하여 식사 설문에 답하지 않으신 분들은 답해 주세요. 

- 설문지 링크: https://forms.gle/fPptLjbaJ4TzkPAeA

∙아침식사: 기숙사식당(300동), 08:00-09:00

∙점심식사: 기숙사식당(300동), 11:30-13:30 or 퀴즈노스(203동), 11:30-13:30

∙저녁식사: 유니스파크(307동 1층) 18:00-20:00

*식사는 점심과 저녁만 제공됩니다. (1월 5일 일요일 저녁부터 제공)

*퀴즈노스 이용인원을 분산하고자 사용 가능 날짜를 기입하였으니 유의 바랍니다. 인당 1만원까지 사용 가능.

*결제는 오후 2시에 진행하오니, 정해진 시간 내에 이용해 주시기를 바랍니다.

*식사 여부 설문 미응답자는 식사 제공이 어려울 수 있습니다.

5. 뱅큇 안내

∙일시: 1월 6일 월요일 오후 6시~8시

∙장소: 유니스파크(307동 1층)

6. 유니스파크(UNISPARK)

기숙사 307동 1층 유니스파크를 전 행사기간동안 대관하여 저녁 식사와 이후 시간 토론에 활용할 예정입니다. 많은 이용 바랍니다. 

매일 오전 9시, 오후 2시에 커피와 다과가 제공됩니다.

Frank Merle  Soliton resolution for the radial critical wave equation in dimension d=3,5  

Abstract: Consider the energy-critical focusing wave equation in dimension d=3,5. The equation has a nonzero radial stationary solution W, which is unique up to scaling and sign change. Together with T. Duyckaerts and C. Kenig, we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated W, decoupled by the scaling, and a radiation term. 

The proof in dimension 3 follows from a exceptional property of W (true only in dimension 3,4), that any none zero radial initial data different from W will produce at the initial time in some sense a radiation.

The proof in dimension 5 essentially boils down to the fact that the equation does not have purely nonradiative multisoliton solutions. The proof overcomes the fundamental obstruction for the extension of the 3D case by reducing the study of a multisoliton solution to a finite dimensional system of ordinary differential equations on the modulation parameters. The key ingredient of the proof is to show that this system of equations creates some radiation, contradicting the existence of pure multisolitons. 

References 

https://arxiv.org/abs/math/0610801

https://arxiv.org/abs/1204.0031

https://arxiv.org/abs/1912.07664

Lecture 1:General presentation of the problem

Lecture 2,3; These two lecture will be devoted to the proof in dimension d=3.

Lecture 4: The first half of the lecture will be devoted to the conclusion of the proof in d=3. Presentation of the proof in dimension d=5

Lecture 5: outline of the proof in dimension 5

Kyeongsu Choi and Beomjun Choi Singularity formation in mean curvature flow

Lecture 1. Preliminaries I: overview              note 

Lecture 2. Preliminaries II: setting up PDEs        note

Lecture 3. Monotonicity formula and blow-up analysis  note

Lecture 4. Spectral theory around solitons  note

Lecture 5. Classification of ancient flows note   

Lecture 6. Applications and beyond  note

Kihyun Kim On classifying the rates of concentration of bubbles for the critical nonlinear heat equation within radial symmetry in high dimensions

In this two-hour talk, we will consider the critical nonlinear heat equation within radial symmetry. We first briefly introduce (the result of) soliton resolution, which roughly says that any ($\dot{H}^1$-)bounded solutions must decompose into finitely many scale-decoupled ground states and a radiation. Assuming soliton resolution, we turn to classify the behavior of scales and the signs. We begin with the one-bubble case. Then, we cover the two-bubble case, which already contains the essence of the proof for the multi-bubble case.

Seunghyeok Kim Compactness theorems of conformally invariant equations arising from conformal geometry  note1 note2

We consider whether the full solution set of conformally invariant equations on n-dimensional Riemannian manifolds M is L^\infty(M)-bounded. 

Establishing this compactness property is important because it allows us to understand the topological structure of the solution set via the topological (Leray–Schauder) degree or Morse theory.

In this talk, I will explain the argument of Khuri, Marques, and Schoen (2009) for the C^2(M)-compactness theorem of the Yamabe problem for 3  \le n \le 24. 

Additionally, I will briefly discuss my recent result with Liuwei Gong and Juncheng Wei (CUHK) on the C^4(M)-compactness theorem of the Q-curvature problem for 5 \le n \le 24.

For both problems, examples of Riemannian manifolds whose solution sets are unbounded in L^{\infty}(M) are known for n \ge 25.

We will appreciate a beautiful combination of bubbling analysis and algebraic arguments in the study of differential geometric objects.

Jaemin Park Absence of anomalous dissipation in the 2D Navier Stokes equations.

In this talk, we will discuss Leray-Hopf solutions to the two-dimensional Navier-Stokes equations with vanishing viscosity. We aim to demonstrate that when the initial vorticity is only integrable, the Leray-Hopf solutions in the vanishing viscosity limit do not exhibit anomalous dissipation. Moreover, we extend this result to the case where the initial vorticity is merely a Radon measure, assuming its singular part maintains a fixed sign. Our proof draws on several key. observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler. equation. This is a joint work with Luigi De Rosa (Gran Sasso Science Institute).

Joonhyun La Local well-posedness and smoothing of MMT kinetic wave equation

In this talk, I will prove local well-posedness of kinetic wave equation arising from MMT equation, which is introduced by Majda, Mclaughlin, and Tabak and is one of the standard toy models to study wave turbulence. Surprisingly, our result reveals a regularization effect of the collision operator, which resembles the situation of non-cutoff Boltzmann. This talk is based on a joint work with Pierre Germain (Imperial College London) and Katherine Zhiyuan Zhang (Northeastern).

Taehun Lee  Allen--Cahn Equations and Minimal Surfaces: A PDE Perspective

Minimal surfaces, as critical points of the area functional, have provided a major impetus for the development of both geometry and PDE theory. Interestingly, they can be approximated by the nodal sets of solutions to the Allen--Cahn equation, a nonlinear PDE arising from phase transition theory. In this talk, we will examine the relationship between the Allen–Cahn equation and minimal surfaces, focusing on three significant problems in these fields: De Giorgi conjecture, Bernstein problem, and Yau’s conjecture. This talk is based on joint work with Kyeongsu Choi (KIAS) and Sanghoon Lee (KIAS).

Youngae Lee    Degree counting formula for Toda systems of rank two

In this talk, we consider Toda systems of rank two on a compact Riemann surface M. We derive the relation between the topological property of M and the Leray Schauder degree of the Toda systems by computing the degree jump caused by bubbling phenomena of the Toda system. We also show that the bubbling phenomena can reduce the computations of this degree jump to the calculations of the topological degree for some mean field equation together with an additional condition, which is called the shadow system.

Youngsam Kwon(Dong-A University) Taylor-Couette flow with temperature fluctuations: Time periodic solutions

We consider the motion of a viscous compressible and heat conducting fluid confined in the gap between two rotating cylinders (Taylor-Couette flow). The temperature of the cylinders is fixed but not necessarily constant. We show that the problem admits a time--periodic solution as soon as the ratio of the angular velocities of the two cylinders is a rational number.

Minjun Jo(Duke University) Cusp formation of vortex patches

We prove instantaneous cusp formation of any vortex patches with acute corners, which was conjectured to occur in the numerical simulations by Cohen-Danchin (2000) and Carrillo-Soler (2000). This is a recent joint work with Tarek Elgindi.

Taehun Kim(Seoul National University) Strict condition for the L^2-wellposedness of fifth and sixth order dispersive equations

We provide a set of conditions that is necessary and sufficient for the $L^{2}$-wellposedness of the Cauchy problem for fifth and sixth order variable-coefficient linear dispersive equations. The necessity of these conditions had been presented by Tarama, and we scrutinized their proof to split the conditions into several parts so that an inductive argument is applicable. This inductive argument simplifies the engineering process of the appropriate pseudodifferential operator needed for the proof of $L^{2}$-wellposedness.

Sangdon Jin(Chungbuk National University) Legendre-Hardy-Sobolev inequality on bounded domains

We study the Legendre-Hardy-Sobolev inequality, which can be regarded as a nonlinear version of the Legendre-Hardy inequality. Further, we investigate whether the optimal constant is attained or not; the attainability is meaningful since the minimizer of the optimal constant is a least energy solution of a corresponding nonlinear elliptic equation satisfying a Neumann boundary condition in some typical cases.

Seongbin Park(POSTECH) Gaussian lower and upper bounds for the Boltzmann equation with Fermi-Dirac statistics

The Boltzmann equation with Fermi-Dirac statistics is a quantum modification of the classical Boltzmann equation, which describes mesoscopic dynamics of dilute gas system, for gas system following Fermi-Dirac statistics. In contrast to the classical Boltzmann equation, the highly non-linear terms in the collision operator make it more difficult to investigate its behavior. In this talk, we discuss uniform in time Gaussian lower bound for the solution of the Boltzmann equation with Fermi-Dirac statistics. Then, we will explore some upper bounds results, including the Gaussian upper bound of the solution.

Uihyeon Jeong(KAIST) Quantized blow-up dynamics for Calogero–Moser derivative nonlinear Schrödinger equation

In this talk, we consider the Calogero–Moser derivative nonlinear Schrödinger equation (CM-DNLS) which can be seen as a continuum version of completely integrable Calogero–Moser many-body systems in classical mechanics. CM-DNLS is a mass-critical nonlinear Schrödinger type equation enjoying a number of numerous structures, such as nonlocal nonlinearity, self-duality, pseudo-conformal symmetry, and complete integrability. Based on these properties, previous studies have established various results on the CM-DNLS, such as global well-posedness, blow-up phenomena and soliton resolution. In particular, we present a construction of smooth finite-time blow-up solutions to the CM-DNLS that exhibit a sequence of discrete blow-up rates, so-called quantized blow-up rates. Our approach is based on modulation analysis, and our main contribution is to simplify the analysis by utilizing the nonlinear adapted derivative suited to the Lax pair, along with the hierarchy of conservation laws inherent in this structure. This result highlights that the integrable structure remains a powerful tool, even in the presence of blow-up solutions. This is a joint work with Taegyu Kim (KAIST).

Beomjong Kwak(KAIST) Strichartz estimates and global well-posedness of the cubic NLS on 2D turus.

In this talk, we present an optimal $L^4$-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus $\mathbb{T}^2$. We first recall the previously known results and counterexamples on the Strichartz estimates on the torus. Then we present our new Strichartz estimate, which has an optimal amount of loss, and the small-data global well-posedness of (mass-critical) the cubic NLS in $H^s,s>0$ as its consequence. An intuition for the relation between them is then provided.

Our Strichartz estimate is based on a combinatorial proof. We introduce our key proposition, the Szemerédi-Trotter theorem, and explain the idea of the proof. This is a joint work with S. Herr.