International Conference: Long-time Dynamics of Nonlinear PDEs

Other Participants

• Junsik Bae Korea Advanced Institute of Science and Technology

• Xing Cheng Hohai University

• Hyungjin Huh Chung-Ang University

• Gyeongha Hwang YeoungNam University

• Bongsuk Kwon Ulsan National Institute of Science and Technology

• Jungheon Park Korea Advanced Institute of Science and Technology

• Changhun Yang Chungbuk National University

Lotte City Hotel Jeju Room Charlotte (6F)

83 Doryeong-ro, Jeju-si, Jeju-do +82 64 730 1000

• 9:30-10:30 Chenjie Fan Chinese Academy of Science
  On constructive blow up of NLS
  In this talk，we will discuss about the constructive blow up of NLS. We will highlight the influential works of Merle, Raphael and their collaborators and discuss some of our related works. 

• 11:00-12:00 Shiwu Yang Peking University
  On the motion of charged particles in a constant electromagnetic field
  In this talk, we present a rigorous mathematical derivation for the classical phenomenon in Maxwell's theory that a charged particle moves along a straight line in a constant electromagnetic field if the initial velocity is parallel to the constant electromagnetic field. The particle is modeled by scaled solitons to a class of nonlinear Klein-Gordon equations. We show that the soliton is stable up to any given time. This is based on joint work with Shuang Miao and Pin Yu. 

• 14:00-15:00 Kenji Nakanishi Kyoto University
  Global dynamics above the ground states for the Zakharov system in the 3D radial case

This is joint work with Zihua Guo (Monash). We study global behavior of solutions to the Zakharov system in three space dimensions under the radial symmetry and a mass-energy constraint slightly above the ground states. We give a classification of initial data into 9 sets by scattering, blow-up and trapping by the ground states in positive and negative time directions, similar to the nonlinear Schrodinger equation, but the proof is more involved as we need bilinear estimates to control the nonlinearity. We also derive a one-parameter family of virial identities for monotonicity away from the ground states.

• 15:30-16:30 Hayashi Masayuki Kyoto University
  Low regularity solutions to the logarithmic Schrödinger equation
  We consider the Cauchy problem for the logarithmic Schrödinger equation. The main difficulty for this equation is that the nonlinearity has a singularity at the origin and breaks the local Lipschitz continuity. On the other hand, this property of the nonlinearity leads to remarkable global behaviors of the solutions, which are quite different from the situation in the power-type NLS.

The first part of the talk will discuss a priori estimates of low regularity solutions and the extension of the flow map. This is based on the joint work with Rémi Carles and Tohru Ozawa. The second part, which may be the main focus, will discuss strong uniqueness properties of low regularity solutions. The proof of uniqueness is achieved by combining a nontrivial use of integral equations, local smoothing estimates, and quantitative estimates of the sublinear effect of the nonlinearity.

• 16:50-17:15 Taegyu Kim Korea Advanced Institute of Science and Technology
  Soliton resolution for Calogero-Moser derivative NLS
  We consider Calogero-Moser derivative NLS (CM-DNLS) equation which can be seen as a continuum version of completely integrable Calogero-Moser many-body systems in classical mechanics. Soliton resolution refers to the phenomenon where solutions asymptotically decompose into a sum of solitons and a dispersive radiation term as time progresses. Our work proves soliton resolution for both finite-time blow-up and global solutions without radial symmetry or size constraints. Although the equation exhibits integrability, our proof does not depend on this property, potentially providing insights applicable to other non-integrable models. This work is based on the joint work with Soonsik Kwon (KAIST). 

• 17:25-17:50 Junyeong Jang Chung-Ang University
  On the Benjamin-Bona-Mahony regularization of the Korteweg-de Vries equation
  The Benjamin-Bona-Mahony equation (BBM) was introduced as a regularization of the Korteweg-de Vries equation (KdV) for long water waves [T. B. Benjamin, J. L. Bona, and J. J. Mahony, Philos. Trans. Roy. Soc. London Ser. A 272(1220) (1972), pp. 47–78]. Also, the convergence from BBM equation to KdV equation has been established. In this talk, we revisit this convergence problem using Fourier analysis method and show this convergence for energy class solutions.  Furthemore, employing conservation laws, we extend the known temporal interval of validity for the BBM regularization. This talk is based on joint work with Younghun Hong and Changhun Yang.

• 9:30-10:30 Beomjong Kwak Korea Advanced Institute of Science and Technology
  Global well-posedness of cubic nonlinear Schrödinger equation on \mathbb{T}^2
  In this talk, we present the global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension $d=2$ for large initial data in $H^s,s>0$. The result is based on a new inverse Strichartz inequality, which is proved by using incidence geometry and additive combinatorics, in particular the inverse theorems for Gowers uniformity norms by Green-Tao-Ziegler. In addition, we construct an approximate periodic solution showing ill-behavior of the flow map at the $L^2$ regularity. This is based on joint works with Sebastian Herr.

• 11:00-12:00 Ioan Bejenaru University of California, San Diego
  Dynamics of equivariant Schrödinger Maps near solitons
  I will review the current literature of the near soliton dynamics for Schrödinger Maps and its parabolic cousin Harmonic Map Heat Flow; we will discuss stability versus blow-up.  I will introduce some new results in the context of Schrödinger Maps. 

• 14:00-15:00 Baoping Liu Peking University
  Scattering for defocusing energy sub-critical wave equation with inverse square potential
  We consider the defocusing energy sub-critical nonlinear wave equation with inverse square potential, and prove global wellposedness and scattering for radial data lying in critical Sobolev space. The main ingredients for our proof include the Fourier truncation method, the hyperbolic coordinate transformation, and the radial endpoint Strichartz estimate for wave equations with potential of critical decay. 

• 15:30-16:30 Kihyun Kim Seoul National University
  On classification of global dynamics for energy-critical equivariant harmonic map heat flows and radial nonlinear heat equation
  We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices D≥3; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $\dot{H}^1$-bounded radial solutions to energy-critical heat equations in dimensions N≥7, building upon soliton resolution for such solutions. This is a joint work with Frank Merle (IHES and CY Cergy-Paris University).

• 16:50-17:15 Uihyeon Jeong Korea Advanced Institute of Science and Technology
  Quantized slow blow up dynamics for the energy-critical co-rotational wave maps problem
  In this talk, we consider the blow-up dynamics of co-rotational solutions for energy-critical wave maps with the 2-sphere target.  We briefly introduce the (2+1)-dimensional wave maps problem and its co-rotational symmetry, which reduces the full wave map to the (1+1)-dimensional semilinear wave equation. Under such symmetry, we see that this problem has a unique explicit stationary solution, so-called "harmonic map". Then we point out some of the works of analyzing the long-term dynamics of the flow near the harmonic map.  Among them, we focus on the smooth blow-up result that corresponds to the stable regime. In particular, the case where the homotopy index is one has a distinctive nature from the other cases, which allows us to exhibit the smooth blow-up with the quantized blow-up rates corresponding to the excited regime.

• 9:30-10:30 Junhyun La Korea Institute for Advanced Study
  Wave turbulence and some applications
  In this talk, we briefly introduce wave turbulence theory, a statistical theory of nonlinear dispersive systems in weakly turbulent regimes. Then we see applications of wave turbulence theory - on MMT equation, an one-dimensional toy model to understand wave turbulence, and on FPUT-experiment. The talk is based on a joint work with Pierre Germain (ICL) and Zhiyuan Zhang (Northeastern), and one with Pierre Germain and Angeliki Menegaki (ICL). 

• 11:00-12:00 Kiyeon Lee Korea Advanced Institute of Science and Technology
  Nonlinear scattering for the Maxwell-Dirac system in Lorenz and Coulomb gauges
  In this talk, I will present the nonlinear scattering results for the Maxwell-Dirac system in the Coulomb gauge. In the Lorenz gauge, we obtained the nonlinear scattering results in Cho–Lee (arXiv:2406.18887). Using this approach, we also prove nonlinear scattering in the Coulomb gauge. Although the Coulomb gauge lacks Lorentz invariance, it allows the elliptic nature of the scalar potential to be exploited. We introduce a spinorial null structure to overcome the lack of Lorentz invariance and to obtain the desired asymptotic behavior. Moreover, the vector field method combined with various resonance analyses adapted to the Coulomb gauge will be discussed.