学术会议
【2025.04.11-04.13 北京】色散方程及相关问题青年研讨会
发布时间：2025-04-10
| 2025年4月12日 | ||
| 时间 | 报告人 | 报告题目 |
| 9:00-10:00 | 沈舜麟 | $l^{2}$-decoupling and the unconditional uniqueness for the Boltzmann equation |
| 10:00-10:30 | 茶歇&自由讨论 | |
| 10:30-11:30 | 申佳 | Global theory of nonlinear Schrödinger equations in the weighted space |
| 午餐 | ||
| 14:00-15:00 | 岳海天 | Invariant Gibbs measure for 3D cubic NLW |
| 15:00-16:00 | 苏一鸣 | On blow-up solutions to the nonlinear Schrödinger equations |
| 16:00-16:30 | 茶歇&自由讨论 | |
| 16:30-17:30 | 骆泳铭 | Scattering of the focusing energy-critical NLS on waveguide manifold |
| 2025年4月13日 | ||
| 9:00-10:00 | 兰洋 | Asymptotic dynamics near ground state for mass critical Zakharov-Kuznetsov equations in dimension two |
| 10:00-10:30 | 茶歇&自由讨论 | |
| 10:30-11:30 | 赵泽华 | NLS on waveguide manifolds and related problems |
| 午餐 | ||
| 14:00-15:00 | 单敏捷 | Global asymptotic behavior of solutions to the generalized derivative nonlinear Schrödinger equation |
| 15:00-16:00 | 张城 | Strichartz estimates for orthonormal systems on compact manifolds |
| 16:00-16:30 | 茶歇&自由讨论 | |
| 16:30-17:30 | 向圣权 | Global control of geometric equations |
报告信息
| 1 | 题目：$l^{2}$-decoupling and the unconditional uniqueness for the Boltzmann equation |
| 报告人：沈舜麟，中国科学技术大学 | |
| 摘要：We broaden the application of the $l^{2}$-decoupling theorem to the Boltzmann equation. We prove full-range Strichartz estimates for the linear problem in the $\mathbb{T}^d$ setting. We establish space-time bilinear estimates, and hence the unconditional uniqueness of solutions to the $\mathbb{R}^d$ and $\mathbb{T}^d$ Boltzmann equation for the Maxwellian particle and soft potential with an angular cutoff, adopting a unified hierarchy scheme originally developed for the nonlinear Schr\"{o}dinger equation. This talk is based on a joint work with Xuwen Chen and Zhifei Zhang. | |
| 2 | 题目：Global theory of nonlinear Schrödinger equations in the weighted space |
| 报告人：申佳，南开大学 | |
| 摘要：I will report some recent progress of the global well-posedness and scattering for the defocusing nonlinear Schrodinger equations (NLS) in the weighted space, which is based on the joint work with Prof. Yifei Wu. We will first give the global well-posedness of 3D quadratic NLS with radial data in the critical weighted space. Previously, Killip, Masaki, Murphy, and Visan proved its conditional global well-posedness and scattering in such space. Our result removes the a priori assumption for the global well-posedness part. Next, we will consider the scattering of mass subcritical NLS. Previously, it is shown by Tsutsumi-Ogawa that the scattering holds in the first-order weighted space, and by Lee that the continuity of the scattering operator breaks down in L^2. We extend the scattering result below the first-order weight, and give the scattering with a large class of L^2-data based on probabilistic method. | |
| 3 | 题目：Invariant Gibbs measure for 3D cubic NLW |
| 报告人：岳海天，上海科技大学 | |
| 摘要： In this talk, we'll present our results about invariant Gibbs measures for the periodic cubic nonlinear wave equation (NLW) in 3D. The interest in this result stems from connections to several areas of mathematical research. At its core, the result concerns a refined understanding of how randomness gets transported by the flow of a nonlinear equation which involves probability theory and partial differential equations. This is joint work with Bjoern Bringmann (Princeton), Yu Deng (UChicago) and Andrea Nahmod (UMass Amherst). | |
| 4 | 题目：On blow-up solutions to the nonlinear Schrödinger equations |
| 报告人：苏一鸣，杭州师范大学 | |
| 摘要： In this talk we will talk about the asymptotic behaviors of solutions to the nonlinear Schrodinger equations. First, we will provide a short survey on the well-posedness theory, the finite time singularity formation and the solitary wave theory. Then, we shall also introduce some of our recent process on this topic, focusing on the construction and classification of blow-up solutions. | |
| 5 | 题目：Scattering of the focusing energy-critical NLS on waveguide manifold |
| 报告人：骆泳铭，深圳北理莫斯科大学 | |
| 摘要：In this talk, we introduce how the framework of the semiviral-vanishing geometry can be applied for the focusing energy-critical NLS model on waveguide manifold, in order to establish large data scattering results. In particular, we reveal the interesting fact that despite the semiviral-vanishing geometry is of energy-subcritical nature at the first glance, it will indeed encode all the useful energy-critical features such as the Aubin-Talenti bubble solution. | |
| 6 | 题目：Asymptotic dynamics near ground state for mass critical Zakharov-Kuznetsov equations in dimension two |
| 报告人：兰洋，清华大学 | |
| 摘要：We consider the focusing mass critical Zakharov-Kuznetsov equation in 2D. We will provide a complete classification of the long time behavior of solutions with initial data near the ground and with a suitable decay on the first variable. We will show that only three behaviors are possible: 1. converging to a traveling wave, 2. blowing up in finite time, 3. linear behavior. We also prove the nonexistence of minimal mass blow-up solutions. Our result is an extention of the work of Martel-Merle-Raphael for mass critical gKdV equations. This work is joint with G. Chen and X. Yuan. | |
| 7 | 题目：NLS on waveguide manifolds and related problems |
| 报告人：赵泽华，北京理工大学 | |
| 摘要：In this talk, we will discuss the dynamics of NLS on waveguide manifolds (semi-periodic spaces) and related problems. We will give a brief survey on classical results and also discuss some recent progress from several different aspects. | |
| 8 | 题目：Global asymptotic behavior of solutions to the generalized derivative nonlinear Schrödinger equation |
| 报告人：单敏捷，中央民族大学 | |
| 摘要：This article is concerned with the global asymptotic behavior for the generalized derivative nonlinear Schrödinger (gDNLS) equation. When the nonlinear effect is not stronge, we show pointwise-in-time dispersive decay for solutions to the gDNLS equation with small initial data in $H^{\frac{1}{2}+}(\mathbb{R})$ utilizing crucially Lorentz-space improvements of the traditional Strichartz inequality. When the nonlinear effect is especially dominant, there exists a sequence of solitary waves that are arbitrary small in the energy space, which means the small data scattering is not true. However, there is evidence that it is not possible for the solitons to be localized in $L^{2}(\mathbb{R})$ and small in $H^{1}(\mathbb{R})$. With small and localized data assumption, we establish a dispersive estimate for solutions to the gDNLS equation globally in time by using vector field methods combined with the testing by wave packets method. | |
| 9 | 题目：Strichartz estimates for orthonormal systems on compact manifolds |
| 报告人：张城，清华大学 | |
| 摘要：We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the case of wave, Klein-Gordon and fractional Schr\"odinger equations. Our results generalize the classical (single-function) Strichartz estimates on compact manifolds by Kapitanski, Burq-G\'erard-Tzvetkov, Dinh, and extend the Euclidean orthonormal version by Frank-Lewin-Lieb-Seiringer, Frank-Sabin, Bez-Lee-Nakamura. On the flat torus, our new results cover prior work of Nakamura for the Schr\"odinger equation, which exploits the dispersive estimate of Kenig-Ponce-Vega. We achieve sharp results on compact manifolds by combining the frequency localized dispersive estimates for small time intervals with the duality principle due to Frank-Sabin. We observe a new phenomenon that the results in the supercritical regime are sensitive to the geometry of the manifold. Moreover, we establish sharp Strichartz estimates on the flat torus for the fractional Schr\"odinger equations by proving a new decoupling inequality for certain non-smooth hypersurfaces. As applications, we obtain the well-posedness of infinite systems of dispersive equations with Hartree-type nonlinearity. This is joint work with王兴（湖南大学），张安（北航）. | |
| 10 | 题目：Global control of geometric equations |
| 报告人：向圣权，北京大学 | |
| 摘要：Recently, together with Coron and Krieger, we initiated a topic on the global control of geometric equations, first on wave map equations and then on the harmnic map heat flow. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important control properties. Surprisingly, due to the geometric feature of the equation we also discover the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding problem for the nonlinear heat equations. |