Quantitative Partial Regularity of the Navier-Stokes Equations and Applications
数学专题报告
报告题目(Title)：Quantitative Partial Regularity of the Navier-Stokes Equations and Applications
报告人(Speaker)：任潇 （北京大学国际数学研究中心）
地点(Place)：后主楼1124
时间(Time)：2024年1月2日上午10:00-11:00
邀请人(Inviter)：许孝精
报告摘要
The classical Caffarelli-Kohn-Nirenberg theorem states that the 1d parabolic Hausdorff measure of the singular set of a suitable weak solution must vanish. Its proof relies on the absolute continuity of the dissipation energy, which is a non-quantitative fact. We develop a quantitative argument using the pigeonhole principle and improve the Caffarelli-Kohn-Nirenberg theorem by a logarithmic factor. This further improves a result of Choe and Lewis (2000). Based on the same method, for any suitable weak solution, we show the existence of intervals of regularity in one spatial direction with length depending only on the natural energies of the solution. A number of applications will be discussed, including a new regularity criterion concerning the direction of vorticity. Based on joint work with Prof. Zhen Lei and Prof. Gang Tian.
