动力系统与微分方程的最新进展

研讨会

苏州大学数学科学学院

2025年11月21日- 11月24日

主办单位：苏州大学

会务组：赵云臧运涛

联系人：赵  云  13914027601 zhaoyun@suda.edu.cn

臧运涛  15295672001ytzang@ustc.edu.cn

报到地点：苏州市南林饭店（苏州市十全街滚绣坊20号）

会议地点：苏州大学文辉楼2楼报告厅

苏州大学精正楼1楼报告厅

会议日程

报告题目和摘要

Asymptotic behavior of solutions to a perturbed second-order

Hamilton–Jacobi equation

陈秦波（南京大学）

Abstract: In this talk, we discuss a perturbation problem for degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians, and we will characterize the asymptotic limit of the solutions using stochastic Mather measures.

Trajectories Asymptotic to KAM Tori

程崇庆（南京大学）

Abstract: Such orbits are constructed by variational method. A consequence is that KAM tori are dynamically unstable.

Polish空间上的度量粘性解

崔小军（南京大学）

Abstract: 我们将在非紧 Polish（完备可分）测地空间上，主要是Wasserstein空间上，讨论度量粘性解的弱KAM理论，特别关注的是（几乎）负梯度线的存在性。同时，我们也将讨论粘性解集的紧化边界性质和幂等半模性质。

On Generalized Namioka Spaces and Joint Continuity of Functions on the Product of Two Spaces

代雄平（南京大学）

覆盖定理在动力系统中的应用

窦斗（南京大学）

Abstract: 覆盖定理是几何测度论和调和分析中的重要工具，这里我们将介绍覆盖定理在动力系统维数理论以及顺从群作用动力系统中的应用。

Slow-fast Traveling Waves in Localized Turbulent Pipe Flow with Transitional Reynolds Number

刘兴波（华东师范大学）

Abstract: Localized turbulence serves as a crucial transitional phase between laminar flow and fully turbulent flow. In this talk, we show the existence of traveling waves in localized turbulence for an experimentally validated model (Barkleyet al. (Nature, 2015)), which describes the onset of turbulent pipe flow through a reaction-diffusion system. By geometric singular perturbation theory and the generalized rotated vector field method, we demonstrate the existence and wave speed of slow-fast traveling pulses for transitional Reynolds numbers, which correspond to the homoclinic orbits connecting the unique equilibrium. Additionally, we investigate the existence of periodic wave trains in the traveling wave system. Furthermore, the (non-)coexistence and (non-)coexisting geometric structures of periodic and homoclinic orbits can be demonstrated for the transitional Reynolds number.

Orbit Structure of Chaotic Dynamical Systems

田学廷（复旦大学）

Abstract: There are many progress on understanding the complexity of various structures of orbits and measures from topological or probabilistic perspectives. In this talk we focus primarily on the orbital structures of chaotic dynamical systems, including saturation, recurrence and non-recurrence, same Birkhoff average, irregular and completely irregular, Lyapunov irregular and emergent behavior, non-physical behavior, shrinking targets, qualitative recurrence, etc. Results will be analyzed from the perspectives of existence, density, entropy, distributional chaos and its variants etc.

C^r-perturbation Lemmas for Systems with Weak Hyperbolicity

王晓东（上海交通大学）

Abstract: Perturbation lemmas play key roles in the study of most dynamics in differentiable dynamical systems, for instance in the exploration of stability conjecture and Palis density conjecture. Compared to the case of C^l-topology, the C^r-perturbation theory are far from being accomplished when r≥2. In this talk, we will present some recent progresses in C^r-perturbation techniques for certain systems with weak hyperbolicity. We prove for every r≥2: (i) the C^r-chain closing lemma for a class of partially hyperbolic diffeomorphisms; (ii) a C^r-connecting lemma for Lorenz attractors to get homo/heteroclinic orbits.

接触型Hamilton-Jacobi方程解的Lyapunov稳定性

严军（复旦大学）

Abstract: 我们将给出接触型Hamilton-Jacobi方程粘性解的Lyapunov稳定性和不稳定性及解的唯一性，全局稳定性的充分条件，并给出具体的判别方法。

拟线性哈密顿偏微分方程的KAM理论

袁小平（复旦大学）

Abstract: 在处理可积偏微分方程（PDEs）的无界扰动时，KAM理论面临着显著困难。Kuksin 率先取得了处理此类无界扰动的突破性进展，他建立了Kuksin引理，并发展出实用的KAM定理，用于分析KdV方程有限隙解的持久性。随后，Kuksin引理被推广至极限情形，相关KAM定理也被构建用于研究导数非线性薛定谔方程（DNLS）和Benjamin-Ono方程。几乎同时，意大利学派为一类一维拟线性偏微分方程建立了KAM定理。本报告将介绍多维度拟线性偏微分方程KAM理论的若干最新研究进展。

Relating Topological and Measure-theoretic Entropies of Subsets

张国华（复旦大学）

Abstract:Motivated by the famous Brin-Katok formula, Feng and Huang introduced the concepts of measure-theoretical upper and lower entropies for each Borel probability measure, and established a variational principle relating packing and Bowen topological entropy of an analytic subset, respectively, to measure-theoretical upper and lower entropy. We show that, such a variational principle fails if we consider packing and Bowen topological entropies of a general subset. Note that Brin-Katok formula shows that, if the measure is invariant then the measure-theoretical entropies can be defined equivalently via Kolmogorov-Sinai's idea. We show that, if the measure is not invariant then the measure-theoretical upper and lower entropy, respectively, may differ from the Kolmogorov-Sinai type entropy. These results answer negatively to some questions raised by Professors Chen and Huang, respectively. This is a joint work with Xulei Wang.

Long Time Stability of Hamiltonian Derivative Nonlinear Schrodinger Equations Without Potential

张静（华东师范大学）

Abstract: In this talk, we give an abstract Birkhoff normal form theorem for some unbounded infinite dimensional Hamiltonian systems. Based on this result we obtain that the solution to Derivative Nonlinear Schrodinger equations under periodic boundary condition with typical small enough initial value remains small in Sobolev norm H^s ̃(T)over a long time interval. The length of the time interval equals to e^(|ln⁡R|^(1+γ)) with 0<γ<1/5 as the initial value is smaller than R≪1.

ESS in Advective Patchy Environments

周鹏（上海师范大学）

Abstract: In a recent work by Jiang, Lam and Lou [Bull. Math. Biol., 2020, Paper No. 131, 42pp], where, to discuss the evolution of dispersal, the authors considered the case of three patches, proposed three models by considering different topology of river network and found that the slower or faster diffuser may win,  or there may appear the evolutionarily singular strategy, depending on given parameters. However, the issue whether there is evolutionarily stable strategy (ESS, a central concept in evolution game theory) is unknown. In this paper, focusing on ``Model I" proposed by them, we give a confirmed answer to this unsolved problem. Our main approach is also useful to treat the other two models proposed by them.

On equilibrium states for certain partially hyperbolic systems

朱玉峻（厦门大学）

Abstract: In this talk, the equilibrium states for a C^2 partially hyperbolic endomorphism on a closed Riemannian manifold with one-dimensional center bundle are considered. Applying the criterion of Climenhaga-Thompson, we use the techniques of inverse limit to obtain the uniqueness and robustness of equilibrium states for the system which satisfies certain conditions about the unstable entropy, stable entropy and the potential function. The same topic in the setting of random perturbations is also considered. It is a joint work with Zhang Yifan.

Dynamics of C^1 systems with continuous invariant splittings

邹瑞（南京信息工程大学）

Abstract: For C^1 diffeomorphisms with continuous invariant splitting without domination, we prove the existence of (un)stable manifolds under the hyperbolicity of invariant measures. We also prove the shadowing lemma, existence of horseshoe, and the periodic approximation of Lyapunov exponents. This is joint work with Yongluo Cao and Zeya Mi.