# zbMATH — the first resource for mathematics

Knots and links in steady solutions of the Euler equation. (English) Zbl 1238.35092
The incompressible Euler equation describes the motion of the ideal fluid. A steady solution of the Euler equation is given by a Beltrami field, which satisfies the equation $$\text{curl} \, u = \lambda u$$. The main theorem says that an arbitrary locally finite link in the space is a set of a stream lines (or a vortex lines) of a Beltrami field. This means that the given link is a periodic orbit of a Beltrami field in a tubular neighborhood of the link.

##### MSC:
 35Q31 Euler equations 37C27 Periodic orbits of vector fields and flows 76B47 Vortex flows for incompressible inviscid fluids
##### Keywords:
Euler equation; Beltrami fields; stream lines; vortex lines
Full Text:
##### References:
 [1] V. Arnold, ”Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits,” Ann. Inst. Fourier $$($$Grenoble$$)$$, vol. 16, iss. fasc. 1, pp. 319-361, 1966. · Zbl 0148.45301 · doi:10.5802/aif.233 · numdam:AIF_1966__16_1_319_0 [2] T. Bagby and P. M. Gauthier, ”Approximation by harmonic functions on closed subsets of Riemann surfaces,” J. Analyse Math., vol. 51, pp. 259-284, 1988. · Zbl 0671.30037 · doi:10.1007/BF02791126 [3] M. A. Berger and R. L. Ricca, ”Topological ideas and fluid mechanics,” Phys. Today, vol. 49, pp. 28-34, 1996. [4] J. Bochnak, M. Coste, and M. Roy, Real Algebraic Geometry, New York: Springer-Verlag, 1998, vol. 36. · Zbl 0912.14023 [5] J. B. Etnyre and R. W. Ghrist, ”Stratified integrals and unknots in inviscid flows,” in Geometry and Topology in Dynamics, Providence, RI: Amer. Math. Soc., 1999, vol. 246, pp. 99-111. · Zbl 0989.76010 · arxiv:math/9905009 [6] J. B. Etnyre and R. W. Ghrist, ”Contact topology and hydrodynamics. III. Knotted orbits,” Trans. Amer. Math. Soc., vol. 352, iss. 12, pp. 5781-5794, 2000. · Zbl 0960.76020 · doi:10.1090/S0002-9947-00-02651-9 · arxiv:math-ph/9906021 [7] M. H. Freedman and Z. He, ”Divergence-free fields: energy and asymptotic crossing number,” Ann. of Math., vol. 134, iss. 1, pp. 189-229, 1991. · Zbl 0746.57011 · doi:10.2307/2944336 [8] P. Hartman, Ordinary Differential Equations, second ed., Mass.: Birkhäuser, 1982. · Zbl 0476.34002 [9] M. Hénon, ”Sur la topologie des lignes de courant dans un cas particulier,” C. R. Acad. Sci. Paris, vol. 262, pp. 312-314, 1966. [10] M. W. Hirsch, Differential Topology, New York: Springer-Verlag, 1976, vol. 33. · Zbl 0356.57001 · doi:10.1007/978-1-4684-9449-5 [11] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant Manifolds, New York: Springer-Verlag, 1977, vol. 583. · Zbl 0355.58009 [12] B. Khesin, ”Topological fluid dynamics,” Notices Amer. Math. Soc., vol. 52, iss. 1, pp. 9-19, 2005. · Zbl 1078.37048 · www.ams.org [13] G. Koch, N. Nadirashvili, G. A. Seregin, and V. vSverák, ”Liouville theorems for the Navier-Stokes equations and applications,” Acta Math., vol. 203, iss. 1, pp. 83-105, 2009. · Zbl 1208.35104 · doi:10.1007/s11511-009-0039-6 [14] S. G. Krantz and H. R. Parks, ”Distance to $$C^k$$ hypersurfaces,” J. Differential Equations, vol. 40, iss. 1, pp. 116-120, 1981. · Zbl 0431.57009 · doi:10.1016/0022-0396(81)90013-9 [15] P. D. Lax, ”A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations,” Comm. Pure Appl. Math., vol. 9, pp. 747-766, 1956. · Zbl 0072.33004 · doi:10.1002/cpa.3160090407 [16] P. Laurence and E. W. Stredulinsky, ”Two-dimensional magnetohydrodynamic equilibria with prescribed topology,” Comm. Pure Appl. Math., vol. 53, iss. 9, pp. 1177-1200, 2000. · Zbl 1072.76689 · doi:10.1002/1097-0312(200009)53:9<1177::AID-CPA5>3.0.CO;2-A [17] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge: Cambridge Univ. Press, 2002, vol. 27. · Zbl 0983.76001 · doi:10.1017/CBO9780511613203 [18] B. Malgrange, ”Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution,” Ann. Inst. Fourier, Grenoble, vol. 6, pp. 271-355, 1955-1956. · Zbl 0071.09002 · doi:10.5802/aif.65 · numdam:AIF_1956__6__271_0 · eudml:73728 [19] W. S. Massey, ”On the normal bundle of a sphere imbedded in Euclidean space,” Proc. Amer. Math. Soc., vol. 10, pp. 959-964, 1959. · Zbl 0094.36002 · doi:10.2307/2033630 [20] H. K. Moffatt, ”Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. I. Fundamentals,” J. Fluid Mech., vol. 159, pp. 359-378, 1985. · Zbl 0616.76121 · doi:10.1017/S0022112085003251 [21] D. Rolfsen, Knots and Links, Houston, TX: Publish or Perish, 1990, vol. 7. · Zbl 0854.57002 [22] T. Vogel, ”On the asymptotic linking number,” Proc. Amer. Math. Soc., vol. 131, iss. 7, pp. 2289-2297, 2003. · Zbl 1015.57018 · doi:10.1090/S0002-9939-02-06792-8 · arxiv:math/0011159
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.