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On the helicity in 3D-periodic Navier-Stokes equations. I: The non-statistical case. (English) Zbl 1109.76015

Summary: We consider the three-dimensional Navier-Stokes equations with potential forces and study the helicity of regular solutions which are periodic in space variables. We give a detailed description of the behavior of the helicity for large times. In particular, the following asymptotic dichotomy of the helicity is established: the helicity either is identically zero or is eventually non-zero and converges to zero as \(t^de^{-2h_0t}\) for time \(t\to \infty\). 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 asymptotic expansions of regular solutions of Navier-Stokes equations and its associated normalization map as well as on Phragmen-Lindelöf principle. The application of this principle is possible due to our proof that the domain of analyticity (in complexified time) of regular solutions contains (up to a logarithmic correction) a right half-plane.

MSC:

76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
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