Quantitative Partial Regularity of the Navier-Stokes Equations and Applications
发布时间：2022年10月31日
浏览次数：3320
发布者: He Liu
主讲人： Xiao REN (Shanghai Center for Mathematical Sciences)
活动时间： 从 2022-11-01 10:30 到 11:30
场地： 北京国际数学研究中心，镜春园78号院（怀新园）78301室
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. Two applications in the axially symmetric case will be discussed. Based on joint work with Zhen Lei.
