中科院数学与系统科学研究院
数学研究所
偏微分方程研讨班
报告人：王保祥 教授 (北京大学)
题 目：Navier-Stokes Equation in Super-Critical Spaces
时 间：2020.12.10（星期四）, 15:00-16:00
地 点：腾讯会议，ID：802 816 070
摘 要：We develop a new way to study the global existence and uniqueness for the Navier-Stokes equation (NS) and consider the initial data in a class of modulation spaces $E^s_{p,q}$ with exponentially decaying weights $(s<0,\1<p,q<\infty)$ for which the norms are defined by< \INFTY)$&NBSP;FOR&NBSP;WHICH&NBSP;THE&NBSP;NORMS&NBSP;RE&NBSP; DEFINED&NBSP; BY $$\|f\|_{E^s_{p,q}}=\left(\sum_{k\in\mathbb{Z}^d}2^{s|k|q}\|\mathscr{F}^{-1} \chi_{k+[0,1]^d}\mathscr{F} f\|^q_p \right)^{1/q}.$$The space $E^s_{p,q}$ is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding $H^{\sigma}\subset E^s_{2,1}$ for any $\sigma<0$ and $s<0$. It is known that $H^\sigma$ ($\sigma<0$) is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to $E^s_{2,1}$ ($s<0$) and their Fourier transforms are supported in $ \mathbb{R}^d_I:= \{\xi\in \mathbb{R}^d: \ \xi_i \geq 0, \, i=1,...,d\}$. Similar results hold for the initial data in $E^s_{r,1}$ with $2< r \leq d$. Our results imply that NS has a unique global solution if the initial value $u_0$ is in $L^2$ with \, \widehat{u}_0 \, \subset \mathbb{R}^d_I$. This is a joint work with Professors H. Feichtinger, K. Gr\"ochenig and Dr. Kuijie Li.
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