# The normal form of the Navier–Stokes equations in suitable normed spaces

### Eric Olson

Department of Mathematics, University of Nevada, Reno, NV 89503, USA### Mohammed Ziane

USC Department of Mathematics, Kaprielian Hall, Room 108, 3620 Vermont Avenue, Los Angeles, CA 90089-2532, USA### Ciprian Foias

Department of Mathematics, 3368 TAMU, Texas A&M University, College Station, TX 77843-3368, USA, Department of Mathematics, Indiana University, Bloomington, IN 47405, USA### Luan Hoang

Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX 79409-1042, USA

## Abstract

We consider solutions to the incompressible Navier–Stokes equations on the periodic domain $\Omega = [0,2\pi ]^{3}$ with potential body forces. Let $\mathscr{R} \subseteq H^{1}(\Omega )^{3}$ denote the set of all initial data that lead to regular solutions. Our main result is to construct a suitable Banach space $S_{A}^{ \star }$ such that the normalization map $W:\mathscr{R}\rightarrow S_{A}^{ \star }$ is continuous, and such that the normal form of the Navier–Stokes equations is a well-posed system in all of $S_{A}^{ \star }$. We also show that $S_{A}^{ \star }$ may be seen as a subset of a larger Banach space $V^{ \star }$ and that the extended Navier–Stokes equations, which are known to have global solutions, are well-posed in $V^{ \star }$.

## Cite this article

Eric Olson, Mohammed Ziane, Ciprian Foias, Luan Hoang, The normal form of the Navier–Stokes equations in suitable normed spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1635–1673

DOI 10.1016/J.ANIHPC.2008.09.003