On the helicity in 3d-periodic navier-stokes equations I: The non-statistical case

We consider the three-dimensional Navier-Stokes equations with potential forces and study the helicity of the regular solutions which are periodic in the space variables. We will give a detailed description of the behavior of the helicity for large times. In particular, the following asymptotic dichotomy of the helicity will be established: the helicity either is identically zero or is eventually non-zero and converges to zero as t^de^-2h ₀t for time t → ∞. The relation between the helicity and the energy is also investigated in connection with that between the energy and enstrophy. Our study relies on the theory of the asymptotic expansion of the regular solutions of the Navier-Stokes equations and its associated normalization map as well as a Phragmen-Linderlöf principle. The application of this principle is possible due to our proof that the domain of analyticity (in complexified time) of the regular solutions contains (up to a logarithmic correction) a right half plane.