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Proceedings of the London Mathematical Society Advance Access originally published online on November 27, 2006
Proceedings of the London Mathematical Society 2007 94(1):53-90; doi:10.1112/plms/pdl003
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© 2006 London Mathematical Society

On the helicity in 3D-periodic Navier–Stokes equations I: the non-statistical case

Ciprian Foias1,2, Luan Hoang3 and Basil Nicolaenko4

1 Department of Mathematics
Indiana University
Rawles Hall Bloomington, IN 47405
USA
2 Department of Mathematics
TAMU 3368
Texas A&M University
College Station, TX 77843-3368
USA
3 School of Mathematics
University of Minnesota
127 Vincent Hall 206 Church St. S.E.
Minneapolis, MN 55455
USA
lthoang{at}math.umn.edu
4 Department of Mathematics
Arizona State University
Tempe, AZ 85287-1804
USA
byn{at}stokes.la.asu.edu

Received 11 May 2004. Revision received 30 June 2005.

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 tde–2h0t for time t -> {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 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.


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