Quantitative Partial Regularity of the Navier-Stokes Equations and Applications
Time: 2022-10-31
Published By: He Liu
Speaker(s): Xiao REN (Shanghai Center for Mathematical Sciences)
Time: 10:30-11:30 November 1, 2022
Venue: Room 78301, Jingchunyuan 78, BICMR
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.
