北京 & 广州

2024. 4. 25

Schedule of Afternoon, 25th April

（4月25日下午报告安排）

题目：Multiple normalized solutions for first order Hamiltonian systems

报告人：余渊洋 (云南大学)

摘要： In this talk, we study the following first order Hamiltonian systems

where         ,arises as the Lagrange multiplier and          are         real matrices with        .Using the multiplicity theorem of Ljusternik-Schnirelmann together with variational methods, we show the existence of multiple normalized solutions for (HS). Moreover, we also obtain bifurcation results of this problem.

报告人介绍：余渊洋，中科院数学与系统科学研究院数学所博士，清华大学博士后。主要从事非线性泛函分析和偏微分方程问题的研究，对于Schrödinger方程和方程组，Dirac方程，利用强不定泛函的临界点理论，研究方程解的存在性、多解性、集中性、衰减性质以及正规化解的存在性等。已在 Calc. Var. Partial Differential Equations, Sci. China Math, J. Geom. Anal. 等知名期刊发表多篇学术论文，主持国家自然科学基金青年基金1项。

题目：Nonrelativistic Limit of Normalized Solutions to a class of nonlinear Dirac equations

报告人：郭琪 (中国人民大学)

摘要： In this talk, we discuss the nonrelativistic limit of normalized solutions to a nonlinear Dirac equation. Our research first confirms the presence of normalized solutions to Dirac equations under high-speed conditions. We then illustrate that these solutions converge to normalized ground states of nonlinear Schrödinger equations. This result not only aids in the study of normalized solutions of nonlinear Schrödinger equations, but also physically explains that the normalized ground states of high-speed particles and low-speed motion particles are consistent. This is a joint work with Pan Chen, Yanheng Ding and Hua-yang Wang.

报告人介绍：郭琪，中科院数学与系统科学研究院数学所博士，中国人民大学数学学院博士后，现任中国人民大学数学学院讲师。研究兴趣为变分法，临界点理论，随机图等，部分研究成果发表在Calc. Var. Partial Differential Equations, J. Differential Equations, SIAM J. Math. Anal., J. Math. Phys., Discrete Contin. Dyn. Syst. 等杂志上，已出版2本学术专著。主持或完成国家自然基金项目3项。

题目：Desingularization of 3D incompressible Euler equations with helical symmetry

报告人：万捷 (北京理工大学)

摘要： In this talk, I will introduce the vortex desingularization problem of 3D incompressible Euler equations. I will discuss some results about the existence and orbital stability of concentrated helical solutions to 3D incompressible Euler equation in infinite pipes, which tend asymptotically to a helix evolved the Binormal Curvature Flow.

报告人介绍：Wan Jie, Ph.D., graduated from the Institute of Mathematics, Chinese Academy of Sciences, and is now an assistant professor at School of Mathematics and Statistics, Beijing Institute of Technology. He is mainly engaged in the research of incompressible Euler equations and variational methods. Related research has been published in Math. Ann., JFA., SIAM J. Math. Anal., CVPDE, JDE and so on.

题目：Global exact controllability of the viscous and resistive MHD system in a rectangle

报告人：廖家江 (北京航空航天大学)

摘要： We consider the 2-D incompressible viscous and resistive magneto hydrodynamics (MHD) system in a rectangle, with controls on the lateral sides. The velocity satisfies Dirichlet boundary conditions, while the magnetic field follows perfectly conducting wall boundary conditions on the remaining, uncontrolled part of the boundary. We extend the small-time global exact null controllability result of Coron et al. in [Ann PDE 5(2):1-49, 2019] from Navier-Stokes equations to MHD equations, with a little help of distributed phantom forces, which can be chosen arbitrarily small in any given Sobolev spaces. Our analysis relies on Coron's return method, the well-prepared dissipation method, long-time nonlinear Cauchy-Kovalevskaya estimates and Badra's local exact controllability result.

报告人介绍：廖家江，北京航空航天大学副教授，2020年博士毕业于中科院数学与系统科学研究院。研究方向为流体力学方程组，包括边界层理论和控制问题。2022年获得了第四届全国PDE博士生论坛优秀论文奖。在ARMA、JDE、JMFM等国际著名期刊发表了多篇论文。主持国家自然科学青年基金项目1项。

题目：The existence of local regular solutions to Landau-Lifshitz equation with Neumann boundary condition

报告人：陈波 (华南理工大学)

摘要： In this talk, we show the existence of local regular solutions to the initial-Neumann boundary value problem of the Schrodinger flow for maps from smooth bounded domains in Euclidean spaces into the sphere S^2 (i.e., Landau-Lifshitz equation). This is a recent joint work with Prof. Wang Youde.

报告人介绍：陈波，博士毕业于中国科学院数学研究所，现为华南理工大学数学学院副教授，其主要从事具有物理背景的Yang-Mills-Higgs场和薛定谔流的研究，相关研究成果发表在CMP, IMRN, Transactions of AMS, Pacific J. Math.等期刊上。

The School of Mathematics at Renmin University of China and South China University of Technology are excited to host a small talk session focused on Variational Methods and Partial Differential Equations. This engaging event, organized by the Variational Methods Group, aims to provide participants with an in-depth understanding of these essential mathematical concepts and their applications.

The small talk, which will last between 30 minutes and 1 hour, will cover key topics such as Variational Problems, Fluid Mechanics, and their relevance in Modern Mathematical and Physical research. Additionally, participants will have the opportunity to discuss related questions and share their insights in a friendly and collaborative environment.