讲座题目：Hyperdissipative Navier-Stokes Equations Driven by Time-dependent Forces: Invariant Manifolds

主办单位：长江大学信息与数学学院

报告专家：王荣年（上海师范大学 教授）

报告时间：2023年6月18（周日） 14:45-15:45

报告地点：东校区8-406

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、不变测度等问题的研究，完成的研究结果已被“Mathematische Annalen“、“Int Math Res Notices”、“SIAM Journal on Mathematical Analysis”“SIAM Journal on Applied Dynamical Systems”、“Journal of Functional Analysis”、“Journal of Differential Equations”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、6项省厅级基金项目。曾获聘广东省高等学校省级培养对象等。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。

摘要：We consider in this talk the incompressible hyperdissipative Navier-Stokes equations on a 2D or 3D periodic torus, where the hyperdissipative is considered 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.