# Navier–Stokes equation in super-critical spaces $E_{p,q}$

### Hans G. Feichtinger

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria### Karlheinz Gröchenig

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria### Kuijie Li

School of Mathematical Sciences, Fudan University, Shanghai, 200433, China### Baoxiang Wang

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

## 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 $E_{p,q}$ with exponentially decaying weights $(s<0,1<p,q<∞)$ for which the norms are defined by

The space $E_{p,q}$ is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding $H_{σ}⊂E_{2,1}$ for any $σ<0$ and $s<0$. It is known that $H_{σ}$ ($σ<d/2−1$) is a super-critical space of NS, it follows that $E_{2,1}$ ($s<0$) is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to $E_{2,1}$ ($s<0$) and their Fourier transforms are supported in $R_{I}:={ξ∈R_{d}:ξ_{i}⩾0,i=1,...,d}$. Similar results hold for the initial data in $E_{r,1}$ with $2<r⩽d$. Our results imply that NS has a unique global solution if the initial value $u_{0}$ is in $L_{2}$ with $suppu^_{0}⊂R_{I}$.

## Cite this article

Hans G. Feichtinger, Karlheinz Gröchenig, Kuijie Li, Baoxiang Wang, Navier–Stokes equation in super-critical spaces $E_{p,q}$. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 1, pp. 139–173

DOI 10.1016/J.ANIHPC.2020.06.002