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

Abstract

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 all $\sigma <0$ 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 ${\mathbb{R}}_I^d:=\{\xi \in {\mathbb{R}}^d:{\xi }_i\ge 0,i=1,...,d\}$. Similar results hold for the initial data in $E_{r,1}^s$ with $2<r\le d$. Our results imply that NS has a unique global solution if the initial value $u_0$ is in $L^2$ with $\mathrm{supp}{\hat{u}}_0\subset {\mathbb{R}}_I^d$.

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