Asymptotic expansions of the regular solutions to the 3D Navier-Stokes equations and applications to the analysis of the helicity

A new construction of regular solutions to the three dimensional Navier{Stokes equa- tions is introduced and applied to the study of their asymptotic expansions. This construction and other Phragmen-Linderl??of type estimates are used to establish su??- cient conditions for the convergence of those expansions. The construction also de??nes a system of inhomogeneous di??erential equations, called the extended Navier{Stokes equations, which turns out to have global solutions in suitably constructed normed spaces. Moreover, in these spaces, the normal form of the Navier{Stokes equations associated with the terms of the asymptotic expansions is a well-behaved in??nite system of di??erential equations. An application of those asymptotic expansions of regular solutions is the analysis of the helicity for large times. The dichotomy of the helicity's asymptotic behavior is then established. Furthermore, the relations between the helicity and the energy in several cases are described.