Knots and links in steady solutions of the Euler equation

Abstract

Given any possibly unbounded, locally finite link, we show that there exists a smooth diffeomorphism transforming this link into a set of stream (or vortex) lines of a vector field that solves the steady incompressible Euler equation in $\mathbb{R}^3$. Furthermore, the diffeomorphism can be chosen arbitrarily close to the identity in any $C^r$ norm.

  • [Ar65] 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.
    @article {Ar65, MRKEY = {0202082},
      AUTHOR = {Arnold, V.},
      TITLE = {Sur la géométrie différentielle des groupes de {L}ie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {16},
      YEAR = {1966},
      NUMBER = {fasc. 1},
      PAGES = {319--361},
      ISSN = {0373-0956},
      MRCLASS = {57.50 (57.55)},
      MRNUMBER = {0202082},
      MRREVIEWER = {R. Hermann},
      ZBLNUMBER={0148.45301},
      }
  • [BG88] 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.
    @article {BG88, MRKEY = {0963157},
      AUTHOR = {Bagby, Thomas and Gauthier, P. M.},
      TITLE = {Approximation by harmonic functions on closed subsets of {R}iemann surfaces},
      JOURNAL = {J. Analyse Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {51},
      YEAR = {1988},
      PAGES = {259--284},
      ISSN = {0021-7670},
      CODEN = {JOAMAV},
      MRCLASS = {30F15 (30E10)},
      MRNUMBER = {0963157},
      MRREVIEWER = {Helmut K{ö}ditz},
      ZBLNUMBER = {0671.30037},
      }
  • [BR96] M. A. Berger and R. L. Ricca, "Topological ideas and fluid mechanics," Phys. Today, vol. 49, pp. 28-34, 1996.
    @article{BR96,
      author={Berger, M. A. and Ricca, R. L.},
      TITLE={Topological ideas and fluid mechanics},
      JOURNAL={Phys. Today},
      VOLUME={49},
      YEAR={1996},
      PAGES={28--34},
      }
  • [BCR98] J. Bochnak, M. Coste, and M. Roy, Real Algebraic Geometry, New York: Springer-Verlag, 1998, vol. 36.
    @book {BCR98, MRKEY = {1659509},
      AUTHOR = {Bochnak, Jacek and Coste, Michel and Roy, Marie-Fran{ç}oise},
      TITLE = {Real Algebraic Geometry},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {36},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1998},
      PAGES = {x+430},
      ISBN = {3-540-64663-9},
      MRCLASS = {14Pxx (11E25 32C05 58A07)},
      MRNUMBER = {1659509},
      MRREVIEWER = {A. Tognoli},
      ZBLNUMBER = {0912.14023},
      }
  • [EG99] 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.
    @incollection {EG99, MRKEY = {1732374},
      AUTHOR = {Etnyre, John B. and Ghrist, Robert W.},
      TITLE = {Stratified integrals and unknots in inviscid flows},
      BOOKTITLE = {Geometry and Topology in Dynamics},
      VENUE={{W}inston-{S}alem, {NC},
      1998/{S}an {A}ntonio, {TX},
      1999},
      SERIES = {Contemp. Math.},
      VOLUME = {246},
      PAGES = {99--111},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1999},
      MRCLASS = {37J35 (37N10 53D10 57M25 76B03)},
      MRNUMBER = {1732374},
      ZBLNUMBER = {0989.76010},
      }
  • [EG00] Go to document 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.
    @article {EG00, MRKEY = {1781279},
      AUTHOR = {Etnyre, John B. and Ghrist, Robert W.},
      TITLE = {Contact topology and hydrodynamics. {III}. {K}notted orbits},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {352},
      YEAR = {2000},
      NUMBER = {12},
      PAGES = {5781--5794},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {37J55 (37C27 57M25 76B47)},
      MRNUMBER = {1781279},
      MRREVIEWER = {Hansj{ö}rg Geiges},
      DOI = {10.1090/S0002-9947-00-02651-9},
      ZBLNUMBER={0960.76020},
      }
  • [FH91] Go to document 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.
    @article {FH91, MRKEY = {1114611},
      AUTHOR = {Freedman, Michael H. and He, Zheng-Xu},
      TITLE = {Divergence-free fields: energy and asymptotic crossing number},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {134},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {189--229},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58E15 (57M25)},
      MRNUMBER = {1114611},
      MRREVIEWER = {Anders Linn{é}r},
      DOI = {10.2307/2944336},
      ZBLNUMBER = {0746.57011},
      }
  • [Ha82] P. Hartman, Ordinary Differential Equations, second ed., Mass.: Birkhäuser, 1982.
    @book {Ha82, MRKEY = {0658490},
      AUTHOR = {Hartman, Philip},
      TITLE = {Ordinary Differential Equations},
      EDITION = {second},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Mass.},
      YEAR = {1982},
      PAGES = {xv+612},
      ISBN = {3-7643-3068-6},
      MRCLASS = {34-01 (58Fxx)},
      MRNUMBER = {0658490},
      ZBLNUMBER = {0476.34002},
      }
  • [He66] M. Hénon, "Sur la topologie des lignes de courant dans un cas particulier," C. R. Acad. Sci. Paris, vol. 262, pp. 312-314, 1966.
    @article{He66,
      author={Hénon, M.},
      TITLE={Sur la topologie des lignes de courant dans un cas particulier},
      JOURNAL={C. R. Acad. Sci. Paris},
      VOLUME={262},
      YEAR={1966},
      PAGES={312--314},
      }
  • [Hi76] M. W. Hirsch, Differential Topology, New York: Springer-Verlag, 1976, vol. 33.
    @book {Hi76, MRKEY = {0448362},
      AUTHOR = {Hirsch, Morris W.},
      TITLE = {Differential Topology},
      SERIES = {Graduate Texts in Math.},
      VOLUME={33},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1976},
      PAGES = {x+221},
      MRCLASS = {57DXX (58-01)},
      MRNUMBER = {0448362},
      MRREVIEWER = {Ulrich Koschorke},
      ZBLNUMBER = {0356.57001},
      }
  • [HPS77] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant Manifolds, New York: Springer-Verlag, 1977, vol. 583.
    @book {HPS77, MRKEY = {0501173},
      AUTHOR = {Hirsch, M. W. and Pugh, C. C. and Shub, M.},
      TITLE = {Invariant Manifolds},
      SERIES = {Lecture Notes in Math.},
      VOLUME={583},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {ii+149},
      MRCLASS = {58F15 (58F10)},
      MRNUMBER = {0501173},
      MRREVIEWER = {M. C. Irwin},
      ZBLNUMBER = {0355.58009},
      }
  • [Kh05] B. Khesin, "Topological fluid dynamics," Notices Amer. Math. Soc., vol. 52, iss. 1, pp. 9-19, 2005.
    @article {Kh05, MRKEY = {2105567},
      AUTHOR = {Khesin, Boris},
      TITLE = {Topological fluid dynamics},
      JOURNAL = {Notices Amer. Math. Soc.},
      FJOURNAL = {Notices of the American Mathematical Society},
      VOLUME = {52},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {9--19},
      ISSN = {0002-9920},
      CODEN = {AMNOAN},
      MRCLASS = {37N10 (37K65 57M60 58D05 76B99)},
      MRNUMBER = {2105567},
      ZBLNUMBER = {1078.37048},
      }
  • [Acta] Go to document 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.
    @article {Acta, MRKEY = {2545826},
      AUTHOR = {Koch, Gabriel and Nadirashvili, Nikolai and Seregin, Gregory A. and {Š}ver{á}k, Vladimir},
      TITLE = {Liouville theorems for the {N}avier-{S}tokes equations and applications},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {203},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {83--105},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {35Q30 (35B53 76D05)},
      MRNUMBER = {2545826},
      MRREVIEWER = {Maria Specovius-Neugebauer},
      DOI = {10.1007/s11511-009-0039-6},
      ZBLNUMBER = {1208.35104},
      }
  • [KP81] Go to document S. G. Krantz and H. R. Parks, "Distance to $C^{k}$ hypersurfaces," J. Differential Equations, vol. 40, iss. 1, pp. 116-120, 1981.
    @article {KP81, MRKEY = {0614221},
      AUTHOR = {Krantz, Steven G. and Parks, Harold R.},
      TITLE = {Distance to {$C\sp{k}$} hypersurfaces},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {40},
      YEAR = {1981},
      NUMBER = {1},
      PAGES = {116--120},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {58C07 (53A07)},
      MRNUMBER = {0614221},
      MRREVIEWER = {Jean-Marie Morvan},
      DOI = {10.1016/0022-0396(81)90013-9},
      ZBLNUMBER = {0431.57009},
      }
  • [La56] Go to document 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.
    @article {La56, MRKEY = {0086991},
      AUTHOR = {Lax, P. D.},
      TITLE = {A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {9},
      YEAR = {1956},
      PAGES = {747--766},
      ISSN = {0010-3640},
      MRCLASS = {35.0X},
      MRNUMBER = {0086991},
      MRREVIEWER = {F. Browder},
      ZBLNUMBER = {0072.33004},
      DOI={10.1002/cpa.3160090407},
      }
  • [LS00] 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.
    @article {LS00, MRKEY = {1761413},
      AUTHOR = {Laurence, Peter and Stredulinsky, Edward W.},
      TITLE = {Two-dimensional magnetohydrodynamic equilibria with prescribed topology},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {53},
      YEAR = {2000},
      NUMBER = {9},
      PAGES = {1177--1200},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {76W05 (35Q35 47J30 76M30)},
      MRNUMBER = {1761413},
      MRREVIEWER = {Manuel N{ú}{ñ}ez Jim{é}nez},
      DOI = {10.1002/1097-0312(200009)53:9<1177::AID-CPA5>3.0.CO;2-A},
      ZBLNUMBER = {1072.76689},
      }
  • [MB02] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge: Cambridge Univ. Press, 2002, vol. 27.
    @book {MB02, MRKEY = {1867882},
      AUTHOR = {Majda, Andrew J. and Bertozzi, Andrea L.},
      TITLE = {Vorticity and Incompressible Flow},
      SERIES = {Cambridge Texts Appl. Math.},
      VOLUME = {27},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2002},
      PAGES = {xii+545},
      ISBN = {0-521-63057-6; 0-521-63948-4},
      MRCLASS = {76-02 (35Q30 35Q35 76B03 76D03 76D05)},
      MRNUMBER = {1867882},
      MRREVIEWER = {Yuxi Zheng},
      ZBLNUMBER = {0983.76001},
      }
  • [Ma56] 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.
    @article {Ma56, MRKEY = {0086990},
      AUTHOR = {Malgrange, Bernard},
      TITLE = {Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution},
      JOURNAL = {Ann. Inst. Fourier, Grenoble},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {6},
      YEAR = {1955--1956},
      PAGES = {271--355},
      ISSN = {0373-0956},
      MRCLASS = {35.0X},
      MRNUMBER = {0086990},
      MRREVIEWER = {R. C. Buck},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0071.09002},
      }
  • [Ma59] W. S. Massey, "On the normal bundle of a sphere imbedded in Euclidean space," Proc. Amer. Math. Soc., vol. 10, pp. 959-964, 1959.
    @article {Ma59, MRKEY = {0109351},
      AUTHOR = {Massey, W. S.},
      TITLE = {On the normal bundle of a sphere imbedded in {E}uclidean space},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {10},
      YEAR = {1959},
      PAGES = {959--964},
      ISSN = {0002-9939},
      MRCLASS = {57.00},
      MRNUMBER = {0109351},
      MRREVIEWER = {Sze-Tsen Hu},
      ZBLNUMBER = {0094.36002},
      }
  • [Mo85] Go to document H. K. Moffatt, "Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. I. Fundamentals," J. Fluid Mech., vol. 159, pp. 359-378, 1985.
    @article {Mo85, MRKEY = {0819398},
      AUTHOR = {Moffatt, H. K.},
      TITLE = {Magnetostatic equilibria and analogous {E}uler flows of arbitrarily complex topology. {I}. {F}undamentals},
      JOURNAL = {J. Fluid Mech.},
      FJOURNAL = {Journal of Fluid Mechanics},
      VOLUME = {159},
      YEAR = {1985},
      PAGES = {359--378},
      ISSN = {0022-1120},
      CODEN = {JFLSA7},
      MRCLASS = {76W05 (76C05)},
      MRNUMBER = {0819398},
      MRREVIEWER = {John Adam},
      DOI = {10.1017/S0022112085003251},
      ZBLNUMBER={0616.76121},
      }
  • [Ro90] D. Rolfsen, Knots and Links, Houston, TX: Publish or Perish, 1990, vol. 7.
    @book {Ro90, MRKEY = {1277811},
      AUTHOR = {Rolfsen, Dale},
      TITLE = {Knots and Links},
      SERIES = {Math. Lecture Series},
      VOLUME = {7},
      NOTE = {Corrected reprint of the 1976 original},
      PUBLISHER = {Publish or Perish},
      ADDRESS = {Houston, TX},
      YEAR = {1990},
      PAGES = {xiv+439},
      ISBN = {0-914098-16-0},
      MRCLASS = {57M25},
      MRNUMBER = {1277811},
      ZBLNUMBER = {0854.57002},
      }
  • [Vo03] Go to document T. Vogel, "On the asymptotic linking number," Proc. Amer. Math. Soc., vol. 131, iss. 7, pp. 2289-2297, 2003.
    @article {Vo03, MRKEY = {1963779},
      AUTHOR = {Vogel, Thomas},
      TITLE = {On the asymptotic linking number},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {131},
      YEAR = {2003},
      NUMBER = {7},
      PAGES = {2289--2297},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {57R25 (37C10 53A04 76W05)},
      MRNUMBER = {1963779},
      MRREVIEWER = {Nikolai K. Smolentsev},
      DOI = {10.1090/S0002-9939-02-06792-8},
      ZBLNUMBER = {1015.57018},
      }

Authors

Alberto Enciso

Instituto de Ciencias Matemáticas
CSIC-UAM-UC3M-UCM
C/ Serrano 123
28006 Madrid
Spain

Daniel Peralta-Salas

Instituto de Ciencias Matemáticas
CSIC-UAM-UC3M-UCM
C/ Serrano 123
28006 Madrid
Spain