Workshop on Nonlinear dispersive wave equations
August 18th-21st, 2022
Organizers: Zihua Guo, Rowan Killip, Monica Visan and Baoxiang Wang
Speakers：Justin Forlano，Thierry Laurens，Kuijie Li， Jason Murphy， Zhimeng Ouyang，Minjie Shan，Yuzhao Wang
Speaker: Jason Murphy
Talk Title: Sharp scattering results for the 3d cubic NLS
Talk Time: 2022-August-17 (Wednesday), 20:00-21:00 pm (Los Angeles Time)
2022-August-18 (Thursday), 11:00-12:00 am (Beijing Time)
Talk Place: Zoom
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.
Zoom Information: https://zoom.us/j/95108632348pwd=RzZKN1NaTlFTUkkvSXJlcG1WeUJHQT09
Conference ID: 951 0863 2348
Password: 009076
Speaker: Minjie Shan
Talk Title: Low regularity conservation laws for the Boussinesq equation
Talk Time: 2022-August-17 (Wednesday), 21:00-22:00 pm (Los Angeles Time)
2022-August-18 (Thursday), 12:00-13:00 pm (Beijing Time)
Talk Place: Zoom
Abstract: In this talk, we will show conservation laws at negative regularity for the Boussinesq equation both on the real line and on the circle. These conserved quantities control the $H^s\times H^{s-2}$ norm of the solution for $-\frac{1}{2}\leq s < 0$. The conservation laws are obtained by considering the perturbation determinant associated to the Lax pair of the equation. This method originates from Killip, Visan and Zhang.
Zoom Information: https://zoom.us/j/95108632348pwd=RzZKN1NaTlFTUkkvSXJlcG1WeUJHQT09
Conference ID: 951 0863 2348
Password: 009076
Speaker: Yuzhao Wang
Talk Title: Normalizability and non-normalizability of Gibbs measures
Talk Time: 2022-August-18 (Thursday), 20:00-21:00 pm (Los Angeles Time)
2022-August-19 (Friday), 11:00-12:00 am (Beijing Time)
Talk Place: Zoom
Abstract: The construction of Gibbs measures has been subject to extensive studies over several decades, including the significant progress in the 1970s on the constructive field theory. Since the 1990s, Bourgain revisited these problems from a harmonic analysis point of view, which led to the recent breakthrough on the sharp normalizability criterion on one-dimensional focusing Gibbs measure by Oh-Sosoe-Tolomeo. More recently, Barashkov-Gubinelli developed a variational approach to measure construction, which has seen tremendous development in these years. We shall give a gentle survey on this subject and discuss some new progress.
These are based on a series of joint works with Liang, Robert, Seong, and Tolomeo.
Zoom Information: https://zoom.us/j/94877907685pwd=YkxDbWxYSEdXQ2FmUEdhYVhNa2RFdz09
Conference ID: 948 7790 7685
Password: 139823
Speaker: Zhimeng Ouyang
Talk Title: Continuum Limit for the Ablowitz--Ladik System
Talk Time: 2022-August-18 (Thursday), 21:00-22:00 pm (Los Angeles Time)
2022-August-19 (Friday), 12:00-13:00 pm (Beijing Time)
Talk Place: Zoom
Abstract: We show that solutions to the Ablowitz--Ladik system converge to solutions of the cubic nonlinear Schrödinger equation for merely $L^2$ initial data. Furthermore, we consider initial data for this lattice model that excites Fourier modes near both critical points of the discrete dispersion relation and demonstrate convergence to a decoupled system of nonlinear Schrödinger equations. We achieve this through the introduction of a new method that synthesizes compactness and Strichartz-based techniques. This is a joint work with Rowan Killip, Monica Visan, and Lei Wu.
Zoom Information: https://zoom.us/j/94877907685pwd=YkxDbWxYSEdXQ2FmUEdhYVhNa2RFdz09
Conference ID: 948 7790 7685
Password: 139823
Speaker: Kuijie Li
Talk Title: Global behavior of small data solutions for the 2D Dirac-Klein-Gordon equations
Talk Time: 2022-August-19 (Friday), 20:00-21:00 pm (Los Angeles Time)
2022-August-20 (Saturday), 11:00-12:00 am (Beijing Time)
Talk Place: Zoom
Abstract: We are interested in the two-dimensional Dirac-Klein-Gordon system, which is a basic model in particle physics. We investigate the global behaviors of small data solutions to this system in the case of a massive scalar field and a massless Dirac field. More precisely, our main result is twofold:
1) we show sharp time decay for the pointwise estimates of the solutions which imply the asymptotic stability of this system;
2) we show the linear scattering result of this system which is a fundamental problem when it is viewed as dispersive equations. This is a joint work with S. Dong, Y. Ma and X. Yuan.
Zoom Information: https://zoom.us/j/95697272479pwd=NmdKcTRkYnU5OWVWc0hQTGxGcW9yUT09
Conference ID: 956 9727 2479
Password: 827249
Speaker: Thierry Laurens
Talk Title: KdV with exotic spatial asymptotics
Talk Time: 2022-August-19 (Friday), 21:00-22:00 pm (Los Angeles Time)
2022-August-20 (Saturday), 12:00-13:00 pm (Beijing Time)
Talk Place: Zoom
Abstract: Given a suitable solution V(t,x) to the Korteweg--de Vries equation on the real line, we will discuss global well-posedness for H^{-1}(\mathbb{R}) perturbations of V. Our conditions on V do include regularity but do not impose any assumptions on spatial asymptotics. We have shown that smooth periodic and step-like profiles V(0,x) satisfy our hypotheses, and we believe that this result can be applied to any other class amenable to complete integrability methods.
Zoom Information: https://zoom.us/j/95697272479pwd=NmdKcTRkYnU5OWVWc0hQTGxGcW9yUT09
Conference ID: 956 9727 2479
Password: 827249
Speaker: Justin Forlano
Talk Title: Global well-posedness and quasi-invariance of Gaussian measures for fractional nonlinear Schrödinger equations
Talk Time: 2022-August-20 (Saturday), 20:00-21:00 pm (Los Angeles Time)
2022-August-21 (Sunday), 11:00-12:00 pm (Beijing Time)
Talk Place: Zoom
Abstract: In this talk, we discuss the long-time dynamics and statistical properties of solutions to the cubic fractional nonlinear Schrödinger equation (FNLS) on the one-dimensional torus, with Gaussian initial data of negative regularity. We prove that FNLS is almost surely globally well-posed and the associated Gaussian measure is quasi-invariant under the flow. In lower-dispersion settings, the regularity of the initial data is below that amenable to the deterministic well-posedness theory. In our approach, inspired by the seminal work by DiPerna-Lions (1989), we shift attention from the flow of FNLS to controlling solutions to the infinite-dimensional Liouville equation of the transported Gaussian measure. We establish suitable bounds in this setting, which we then transfer back to the equation by adapting Bourgain’s invariant measure argument to quasi-invariant measures.
This is a joint work with Leonardo Tolomeo (Hausdorff Center for Mathematics).
Zoom Information: https://zoom.us/j/98002685844pwd=eHBEMUppVnFLVVJidWdJd3RIb0MwUT09
Conference ID: 980 0268 5844
Password: 322401
