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

We consider solutions to the incompressible Navier–Stokes equations on the periodic domain Ω=[0,2π]3 with potential body forces. Let R⊆H1(Ω)3 denote the set of all initial data that lead to regular solutions. Our main result is to construct a suitable Banach space SA⋆ such that the normalization map W:R→SA⋆ is continuous, and such that the normal form of the Navier–Stokes equations is a well-posed system in all of SA⋆. We also show that SA⋆ may be seen as a subset of a larger Banach space V⋆ and that the extended Navier–Stokes equations, which are known to have global solutions, are well-posed in V⋆.