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

Abstract

We consider solutions to the incompressible Navier–Stokes equations on the periodic domain $\Omega =[0,2\pi {]}^3$ with potential body forces. Let $\mathcal{R}\subseteq H^1(\Omega {)}^3$ 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 $W:\mathcal{R}\to S_A^{\star }$ 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 $V^{\star }$.