发布单位：科学技术研究处 [2017-05-19 14:53:46] 打印此信息
题 目：On the Navier-Stokes equations in scaling-invariant spaces in any dimension
内容简介：Whether the solution to the systems of equations in fluid mechanics, such as those of Navier-Stokes and the magneto-hydrodynamics, remain smooth for all time in a three-dimensional space remains a challenging open problem. In 1962 Serrin provided a certain space-time integrability condition for smoothness in a scaling-invariant norms for the weak solution to the Navier-Stokes system, which is a three-dimensional velocity vector field. We discuss recent developments in the research direction in effort to improve such integrability conditions so that we only have to impose the condition on ``only one of the three'' velocity vector field components, instead of all of three, as well as its extension to the magneto-hydrodynamics system. The proof crucially relies on a key identity which is a consequence of the divergence-free property, and techniques from anisotropic Littlewood-Paley theory that consists of anisotropic Bernstein's inequality, anisotropic Bony paraproducts and anisotropic Besov and Sobolev spaces. Moreover, except only a very few recent results by the speaker, all such results have been limited to the three-dimensional case; we will also discuss the difficulty of extending to dimension such as four and beyond.
报告人：美国罗切斯特大学(University of Rochester) Kazuo Yamazaki 教授
报告人简介：Kazuo Yamazaki received his Ph.D. in Mathematics from Oklahoma State University under the advisory of Prof. Jiahong Wu. Now he is a Professor at University of Rochester His research direction is fluid dynamics PDE using harmonic and stochastic analysis, and mathematical biology and have published 35 papers.