应兰州大学数学与统计学院杨璐教授邀请,俄罗斯科学院信息传输问题研究所首席科学研究员VladimirChepyzhov将于2024年9月1日至9月6日访问我校并作系列学术报告。
报告题目一：Dynamical semigroups and global attractors
时间：2024年9月2日（星期一）上午9: 30-10: 30
地点：大学生活动中心506报告厅
报告题目二：Global attractors：Application to evolution equations (ODEs and PDEs).
时间：2024年9月2日（星期一）上午10: 30-11: 30
地点：大学生活动中心506报告厅
报告题目三：Trajectory attractors for ODEs without uniqueness
时间：2024年9月3日（星期二）上午9: 30-10: 30
地点：大学生活动中心506报告厅
报告题目四：Theory of trajectory attractors and applications to PDEs (reaction-diffusion systems, 3D Navier-Stokes equations).
时间：2024年9月3日（星期二）上午10: 30-11: 30
地点：大学生活动中心506报告厅
报告摘要：These talks are devoted to the theory of trajectory attractors of dissipative partial differential equations (PDEs). Many important problems arising in mechanics and physics lead to the study of complicated evolution PDEs and especially to the study of their solutions as time tends to infinity. For the last 5 decades, the considerable progress in solving such problems has been achieved using the theory of infinite dimensional dynamical systems and their attractors. The classical approach suggests to consider the dynamical semigroup in the phase space of initial data of the PDE under the study, which is a Hilbert or Banach space. After that, one looks for a
global attractor of this semigroup. However, to construct such single-valued semigroup the involved Cauchy problem must be uniquely solvable on arbitrary long time interval. If the solution is not unique, or the uniqueness theorem is unknown, then the classical approach is not directly applicable. Recall, there are many important PDEs for which that is the case. For example, the famous 3D Navier-Stokes system is a bounded domain. To overcome this drawback of the classical theory, one can use the theory of so-called trajectory dynamical system and their trajectory attractors developed in the works of Mark Vishik and Vladimir Chepyzhov, presented in this mini-course. In two starting lectures we will shortly expose the classical approach with application to dissipative evolution equations (ODEs and PDEs), such as reaction-diffusion systems and 2D Navier-Stokes system. In lecture 3, we explain the method of trajectory dynamical systems using quite simple but substantial system of ODEs without uniqueness of their solutions. In lecture 4, we explain how this theory is applicable to more complicated reaction-diffusion systems and to inhomogeneous 3D Navier-Stokes system in a bounded domain, for which, as has been shown in the recent works, the long standing hypothesis of the uniqueness is closed to be refute. So, the trajectory attractors are very important in the study of long time behaviour for solutions of this and other PDEs without uniqueness.
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报告人简介
V. Chepyzhov研究员（科学博士）作为无穷维动力系统苏联学派著名数学家M.Vishik教授的代表性学生之一，是国际上无穷维动力系统领域的杰出学者和新一代的代表学者之一。现任俄罗斯科学院信息传输问题研究所首席科学研究员。主要从事无穷维动力系统吸引子理论的研究，特别是在一致吸引子和轨道吸引子的基础理论方面做出了奠基性以及深刻创新的工作，与M. Vishik教授共同撰写的专著是本领域的经典著作之一，到目前发表学术论文95篇（数据来源于MathSciNet数据库），被引用文献次数达2198次。其中多篇论文都发表在Comm. Pure Appl. Math.，J. Math. Pures Appl.，Indiana Univ. Math. J.，Russian Math. Surveys等国际顶尖学术期刊上。
甘肃应用数学中心
甘肃省高校应用数学与复杂系统省级重点实验室
数学与统计学院
萃英学院
2024年8月27日
