Navier-Stokes equation with very rough data
报告题目(Title)：Navier-Stokes equation with very rough data
报告人(Speaker)：Baoxiang Wang （Peking University）
地点(Place)：后主楼1124
时间(Time)：2019年6月19日16:00-17:00
报告摘要
We study the Cauchy problem for the incompressible
Navier-Stokes equation (NS):
\begin{align*}
u_t -\Delta u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div}\, u=0, \ \ u(0,x)= u_0.
\end{align*}
We consider a class of very rough initial data in $E^s_{2,2}$ for which the norm are defined by $\|u_0\|_{E^s} = \|2^{s|\xi|} \widehat{u}_0(\xi)\|_{L^2}, \ \ s<0$, and show that NS has a unique global solution if the initial value $u_0\in E^s$, $s<0$ and their Fourier transforms are supported in $ \mathbb{R}^d_I:= \{\xi\in \mathbb{R}^d: \ \xi_i \geq 0, \, i=1,...,d\}$. Our results imply that NS has a unique global solution if the initial value $u_0$ is in $L^2$ with ${\rm supp} \, \widehat{u}_0 \, \subset \mathbb{R}^d_I$. This is a joint work with Professors Feichtinger, Gr\"ohenig and Dr Li.
