Navier-Stokes Equation in Super-Critical Spaces $E_{p,q}^s$

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 $(s<0,1<p,q<\mathrm{\infty })$ for which the norms are defined by $$\mathrm{\parallel }f{\mathrm{\parallel }}_{E_{p,q}^s}={\left(\sum _{k\in {\mathbb{Z}}^d}{2}^{s\mathrm{\mid }k\mathrm{\mid }q}\mathrm{\parallel }{\mathcal{F}}^{-1}{\chi }_{k+[0,1{]}^d}\mathcal{F}f{\mathrm{\parallel }}_p^q\right)}^{1\mathrm{/}q}\mathrm{.}$$ 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 all $\sigma <0$ and $s<0$. It is known that $H^{\sigma }$ ($\sigma <d\mathrm{/}2-1$) 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 $ $\mathbb{R}$^d_I:= {\xi\in $\mathbb{R}$^d: \ \xi_i \geq 0, , i=1,…,d}$\mathrm{.}Similarresultsholdfortheinitialdatain$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 ${\rm supp} , \widehat{u}_0 , \subset $\mathbb{R}$^d_I$.