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] with potential body forces. Let R ⊆ H(Ω) denote the set of all initial data that lead to regular solutions. Our main result is to construct a suitable Banach space S A such that the normalization map W : R → S A is continuous, and such that the normal form of the Navier–Stokes equations is a well-posed system in all of S A . We also show that S A 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 .