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In this paper we develop a new way to study the global existence anduniqueness for the Navier-Stokes equation (NS) and consider the initial data ina 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 $$ \|f\|_{E^s_{p,q}}= \left(\sum_{k\in \mathbb{Z}^d} 2^{s|k|q}\|\mathscr{F}^{-1}\chi_{k+[0,1]^d}\mathscr{F} f\|^q_p \right)^{1/q}. $$ The space $E^s_{p,q}$ isa rather rough function space and cannot be treated as a subspace of tempereddistributions. For example, we have the embedding $H^{\sigma}\subset E^s_{2,1}$for all $\sigma<0$ and $s<0$. It is known that $H^\sigma$ ($\sigma<d/2-1$) is asuper-critical space of NS, it follows that $ E^s_{2,1}$ ($s<0$) is alsosuper-critical for NS. We show that NS has a unique global mild solution if theinitial data belong to $E^s_{2,1}$ ($s<0$) and their Fourier transforms aresupported 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 theinitial value $u_0$ is in $L^2$ with ${\rm supp} \, \widehat{u}_0 \, \subset\mathbb{R}^d_I$.

Navier-Stokes Equation in Super-Critical Spaces $E^s_{p,q}$