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

## TL;DR

This paper establishes global existence and uniqueness of solutions to the Navier-Stokes equations for initial data in certain super-critical modulation spaces with exponential weights, expanding the class of initial conditions for which solutions are known.

## Contribution

The paper introduces a novel approach to analyze Navier-Stokes in super-critical spaces $E^s_{p,q}$, proving global solutions for initial data with specific Fourier support.

## Key findings

- Global mild solutions exist for initial data in $E^s_{2,1}$ with Fourier support in the positive orthant.
- Results extend to initial data in $E^s_{r,1}$ for $2< r extless d$, with similar global existence.
- Solutions are unique even for rough initial data in these super-critical spaces.

## 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 $E^s_{p,q}$ 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 $E^s_{p,q}$ is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding $H^{\sigma}\subset E^s_{2,1}$ 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 $ E^s_{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 $E^s_{2,1}$ ($s<0$) 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 $E^s_{r,1}$ 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 ${\rm supp} \, \widehat{u}_0 \, \subset \mathbb{R}^d_I$.