2025年华罗庚青年论坛—分析方向

时间：2025年7月5-6日

7月5日

9-10点

报告人：邵城阳 （IHES）

Title：Paradifferential calculus and oscillatory motions

Abstract：Oscillatory motions appear in both classical Hamiltonian systems and dispersive PDE models. While often observed in simulations and experiments, rigorous mathematical justification of their existence is usually challenging due to "loss of regularity", which prohibits standard fixed-point methods. In this talk, I will explain how paradifferential calculus can effectively handle this issue. A modern tool from harmonic analysis, it provides clear control of regularity for nonlinear expressions. The method could (1) make long-time behavior more accessible in quasilinear dispersive PDEs (2) yield KAM type results without using traditional Nash-Moser type arguments (3) in certain cases, solve open problems that are out of reach by traditional methods. 

简介：Chengyang Shao, postdoctoral researcher at IHÉS starting from September 2025. Research interests include dispersive partial differential equations and dynamical systems.

10：15-11：15

报告人：孙长贞（CNRS)

Title: On the nonlinear transverse asymptotic stability of line solitary waves for the three-dimensional Euler-Poisson system.

Abstract: The ionic Euler–Poisson system serves as a hydrodynamic model for plasma, describing the motion of ions interacting with a background electron density.

It is known that this three-dimensional system admits a family of one-dimensional solitary waves that propagate in a single direction with exponential localization.

In this talk, we investigate the stability of these line solitary waves in the three-dimensional Euler–Poisson system.

Since perturbations are allowed to depend on the transverse variables, this is referred to as transverse stability.

We show that the line solitary waves are nonlinearly transversely stable: solutions starting close to a line solitary wave exist globally in time and converge,

in the L_x^{\infty} norm, to a modulated solitary wave as time tends to infinity.

This is joint work with Frédéric Rousset (Orsay, France). 

个人信息：Changzhen Sun received his Ph.D. in 2021 from University of  Paris-Saclay in France. From 2021 to 2023, he conducted postdoctoral research at the Toulouse Institute of Mathematics. Since October 2023, he has been working at  Laboratory of mathematics of Besançon and serves as a junior researcher in French national center of scientific research (CNRS). His research interests lie in partial differential equations, particularly in singular limits and the stability of non-constant equilibria in fluid mechanics.

7月6日

9-10点

报告人：于东晓  Vanderbilt University

Title: Sharp late-time asymptotics for scalar quasilinear wave equations satisfying the weak null condition

Abstract: We study the sharp late-time asymptotics for a class of quasilinear wave equations satisfying the weak null condition in three space dimensions. We prove that the asymptotics are very different from those for the equations satisfying the classical null condition. In particular, at leading order, the solution displays a continuous superposition of decay rates. Moreover, we show that any solution that decays faster than expected in a compact spatial region must vanish identically. The talk is based on joint work in progress with Jonathan Luk and Sung-Jin Oh.

 

个人简介：

于东晓，美国范德堡大学数学系博士后，主要研究方向为非线性波方程与色散方程。本科毕业于中山大学，博士毕业于加利福尼亚大学伯克利分校，师从Daniel Tataru教授。

10：15-11：15

报告人：黄晓琦（LSU）

Title: Strichartz estimates for the Schrödinger equation on the sphere

Abstract: We will discuss optimal space-time estimates in $L^q_{t,x}$ spaces for solutions to the Schrödinger equation on the standard round sphere, which is related to the results of Burq, Gérard and Tzvetkov (2004). The proof is based on the arithmetic properties of the spectrum of the Laplacian on the sphere, as well as local bilinear oscillatory integral estimates in harmonic analysis, which allow us to relate the problem to Strichartz estimate on one-dimensional tori. This is based on joint work with Christopher Sogge.

Bio: Xiaoqi Huang(黄晓琦) is currently an assistant professor at Louisiana State University. He obtained his Ph.D. at Johns Hopkins University in 2021 under the supervision of Christopher Sogge. His research focuses on harmonic analysis and partial differential equations.

11：30-12：30

报告人：马骁 （Umich）

Title: Recent Advances on Hilbert’s Sixth Problem

Abstract: In this talk, we present our recent progress on Hilbert’s sixth problem—deriving fluid equations from microscopic dynamics. We will provide the necessary physical background and introduce the key techniques developed in our work, including the Feynman diagram representation and the cutting algorithm.

个人介绍：马骁，本科毕业于中国科学技术大学。研究生毕业于普林斯顿大学，师从Alexandru Ionescu教授。目前在密歇根大学从事博士后工作。主要研究方向为偏微分方程与统计物理的交叉方向。