Local regularity of weak solutions to the hypodissipative Navier--Stokes equations
报告题目(Title)：Local regularity of weak solutions to the hypodissipative Navier--Stokes equations
报告人(Speaker)：Wojciech Ożański (University of Southern California)
地点(Place)：Zoom id: 63691904537
时间(Time)：2020年11月24日上午9:00-10:00.
邀请人(Inviter)：CALVIN KHOR; 许孝精
报告摘要
We will consider weak solutions to the 3D incompressible hypodissipative Navier--Stokes equations (HNSE), $\partial_t u + (- \Delta )^s u + (u\cdot \nabla )u +\nabla p =0$, $\mathrm{div}\, u=0$, where $(-\Delta )^s$ is the Fourier multiplier with symbol $|\xi |^{2s}$, and $s\in (3/4,1)$. We will discuss a new iteration scheme that allows one to study suitable weak solutions of HNSE locally in space-time. We will show how this can be used to obtain that $\nabla^k u \in L^{p,\infty }_{\mathrm{loc}}(\mathbb{R}^3\times(0,\infty))$ for every such solution $u$, where $p=\frac{2(3s-1)}{k+2s-1}$, $k=1,2$, and to improve the partial regularity result of Tang \& Yu (2015) as well provide an estimate on the box-counting dimension of the singular set $S$, $d_B(S\cap \{t\geq t_0 \} )\leq \frac13 (15-2s-8s^2) $ for every $t_0>0$. This is joint work with Hyunju Kwon (IAS).
