In this paper 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 $E_{p,q}^s$ with exponentially decaying weights $(s<0,1<p,q<\infty )$ for which the norms are defined by${\Vert f\Vert }_{E_{p,q}^s}={\left(\sum _{k\in {\mathbb{Z}}^d}{2}^{s|k|q}{\Vert {\mathcal{F}}^{-1}{\chi }_{k+{[0,1]}^d}\mathcal{F}f\Vert }_p^q\right)}^{1/q}.$ The space $E_{p,q}^s$ is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding $H^{\sigma }\subset E_{2,1}^s$ for any $\sigma <0$ and $s<0$. It is known that $H^{\sigma }$ ($\sigma <d/2-1$) is a super-critical space of NS, it follows that $E_{2,1}^s$ ($s<0$) is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to $E_{2,1}^s$ ($s<0$) and their Fourier transforms are supported in ${\mathbb{R}}_I^d:=\{\xi \in {\mathbb{R}}^d:{\xi }_i⩾0,i=1,...,d\}$. Similar results hold for the initial data in $E_{r,1}^s$ with $2<r⩽d$. Our results imply that NS has a unique global solution if the initial value $u_0$ is in $L^2$ with $\mathrm{supp}{\hat{u}}_0\subset {\mathbb{R}}_I^d$.

在本文中，我们开发了一种新方法来研究Navier-Stokes方程（NS）的全局存在性和唯一性，并考虑了一类调制空间中的初始数据 ${Ë}_{p，q}^s$ 权重呈指数衰减 $（s<0，\mathrm{1个}<p，q<\infty ）$ 为其定义的规范${\Vert F\Vert }_{{Ë}_{p，q}^s}={（\sum _{ķ\in {\mathbb{ž}}^d}{2}^{s|ķ|q}{\Vert {\mathcal{F}}^{-\mathrm{1个}}{\chi }_{ķ+{[0，\mathrm{1个}]}^d}\mathcal{F}F\Vert }_p^q）}^{\mathrm{1个}/q}。$ 空间 ${Ë}_{p，q}^s$是一个相当粗糙的函数空间，不能将其视为调整分布的子空间。例如，我们有嵌入$H^{\sigma }\subset {Ë}_{2，\mathrm{1个}}^s$ 对于任何 $\sigma <0$ 和 $s<0$。众所周知$H^{\sigma }$ （$\sigma <d/2-\mathrm{1个}$）是NS的超临界空间，因此 ${Ë}_{2，\mathrm{1个}}^s$ （$s<0$）对NS也是至关重要的。我们证明，如果初始数据属于，则NS具有唯一的全局温和解${Ë}_{2，\mathrm{1个}}^s$ （$s<0$）及其傅里叶变换在 ${\mathbb{[R}}_{\mathrm{一世}}^d：=\{\xi \in {\mathbb{[R}}^d：{\xi }_{\mathrm{一世}}⩾0，\mathrm{一世}=\mathrm{1个}，。。。，d\}$。类似的结果适用于的初始数据${Ë}_{\mathrm{[R}，\mathrm{1个}}^s$ 与 $2<\mathrm{[R}⩽d$。我们的结果表明，如果初始值相等，则NS具有唯一的全局解决方案${ü}_0$ 在 ${\mathrm{大号}}^2$ 与 $\mathrm{支持}{\widehat{ü}}_0\subset {\mathbb{[R}}_{\mathrm{一世}}^d$。