Navier-Stokes Equation in Super-Critical Spaces $E^s_{p,q}$ (1904.01797v2)
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 $Es_{p,q}$ with exponentially decaying weights $(s<0, \ 1<p,q<\infty)$ for which the norms are defined by $$ |f|{Es{p,q}} = \left(\sum_{k\in \mathbb{Z}d} 2{s|k|q}|\mathscr{F}{-1} \chi_{k+[0,1]d}\mathscr{F} f|q_p \right){1/q}. $$ The space $Es_{p,q}$ is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding $H{\sigma}\subset Es_{2,1}$ 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 $ Es_{2,1}$ ($s<0$) is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to $Es_{2,1}$ ($s<0$) 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 $Es_{r,1}$ with $2< r \leq d$. Our results imply that NS has a unique global solution if the initial value $u_0$ is in $L2$ with ${\rm supp} \, \widehat{u}_0 \, \subset \mathbb{R}d_I$.