讲座题目：The Invariant Manifold Approach Applied to the Study of Hyperdissipative Navier-Stokes Equations in High Dimensions

主办单位：三峡数学研究中心/理学院

报告专家：王荣年教授（上海师范大学）

报告时间：2021年12月8日（周三） 上午9:00

报告地点：腾讯会议（ID： 366 319 752）

专家简介： 王荣年, 博士, 上海师范大学教授, 博士生导师（应用数学）。目前主要从事非线性发展方程适定性、多值扰动及解集的拓扑正则性、不变流形理论等问题的研究, 完成的研究结果已被"Mathematische Annalen"、“Int. Math. Res. Notices 、"Journal of Functional Analysis"、"Journal of Differential Equations"" Journal of Phys. A: Math. Theo."等学术期刊发表. 主持承担了2项国家自然科学基金面上项目、国家自然科学基金青年项目、4项省自然科学基金项目和2项省教育厅基金项目。

报告摘要： We consider the incompressible hyperdissipative Navier-Stokes equations on a 2D or 3D periodic torus. We intend to reveal how the fractional dissipation affects long - time dynamics of weak solutions. More precisely, we prove that in the $L^2$-space of divergent-free vector fields with zero mean, there exists a finite-dimensional Lipschitz manifold being locally forward invariant and pullback exponentially attracting. Moreover, the compact uniform attractor is contained in the union of all fibers of the manifold. No large viscosity $\epsilon$ is assumed. It is also significant that in the 3D case the spectrum of the fractional Laplacian $(-\Delta)^{3/2}$ does not have arbitrarily large gaps.