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=3[0,2π] with potential body forces. Let R⊆H13(Ω) denote the set of all initial data that lead to regular solutions. Our main result is to construct a suitable Banach space such that the normalization map is continuous, and such that the normal form of the Navier–Stokes equations is a well-posed system in all of . We also show that 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⋆.