广州数学大讲坛第二期
第十四讲——上海师范大学王荣年教授学术报告
题目:The Invariant Manifold Approach Applied to the Study of Hyperdissipative Navier-Stokes Equations
时间：2022年5月20日（周五）下午15：00-16：30
地点：腾讯会议（会议ID：348460541）
报告人：王荣年
摘要：In this talk we consider the incompressible hyperdissipative Navier-Stokes equations
\begin{equation*}
\left\{\begin{array}{ll}
u_t+\epsilon(-\Delta)^\alpha u+(u\cdot\nabla)u+\nabla p=f,\\
\nabla\cdot u=0
\end{array}
\right.
\end{equation*}
on a 2D or 3D periodic torus, where the power $\alpha\geq {3}/{2}$ and the forcing function $f$ is time-dependent. We intend to reveal how the fractional dissipation and the time-dependent force affect long-time dynamics of weak solutions. More precisely,
we prove that with certain conditions on $f$, there exists a finite-dimensional Lipschitz manifold in the $L^2$-space of divergent-free vector fields with zero mean. The manifold is locally forward invariant and pullback exponentially attracting. Moreover, the compact uniform attractor is contained in the union of all fibers of the manifold. In our result, 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.
报告人简介：
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""J. 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。曾获聘广东省高等学校“千百十人才工程”省级培养对象、江西省高校中青年骨干教师等。近年来先后访问罗马尼亚科学院和雅西大学、奥地利克拉根福特大学、美国杨百翰大学和佐治亚理工学院等。
