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Stratified integrals and unknots in inviscid flows. (English) Zbl 0989.76010
Barge, Marcy (ed.) et al., Geometry and topology in dynamics. AMS special session on topology in dynamics, Winston-Salem, NC, USA, October 9-10, 1998 and the AMS-AWM special session on geometry in dynamics, San Antonio, TX, USA, January 13-16, 1999. Providence, RI: American Mathematical Society. Contemp. Math. 246, 99-111 (1999).
In view of nonsingular \(C^\omega\) vector fields on \(S^3\) without closed flow lines disproving the Seifert conjecture the following is a striking result: any time-independent nonsingular vector field on \(S^3\) solving the \(C^\omega\) Euler equations of fluid dynamics has a closed flow line representing a trivial knot. Here, the Euler equations model the velocity of an inviscid incompressible fluid without exterior force. As the authors show, for their result they have to deal with two cases: either the vector field has a nontrivial integral, or it is a nonzero section of the characteristic line field of a contact form. In both cases they heavily rely on deep results, by M. Wada [J. Math. Soc. Japan 41, No. 3, 405-413 (1989; Zbl 0672.58042)] on the characterization of nonsingular Morse-Smale flows on \(S^3\) in the first case, and by H. Hofer, K. Wysocki and E. Zehnder [Topol. Methods Nonlinear Anal. 7, No. 2, 219-244 (1996; Zbl 0898.58018)] on unknotted orbits in Reeb fields in the second case.
For the entire collection see [Zbl 0930.00046].

MSC:
76B47 Vortex flows for incompressible inviscid fluids
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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