| A new construction of regular solutions to the three dimensional Navier-Stokes equations is introduced and applied to the study of their asymptotic expansions. This construction and other Phragmen-Linderlof type estimates are used to establish sufficient conditions for the convergence of those expansions. The construction also defines a system of inhomogeneous differential 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 infinite system of differential 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. |
