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Existence of knotted vortex tubes in steady Euler flows. (English) Zbl 1317.35184
The purpose of this paper is the proof of a theorem which states that knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in \(\mathbb{R}^3\) exist. The proof consists of three steps, which are gradually improved in the paper.
The construction of a local Beltrami field, which satisfies the Beltrami equation \(\mathrm{curl}~v=\lambda v\), and has a set of certain invariant tori.
It is proved that these invariant tori are “robust” in a certain sense.
It is proved that the local Beltrami field can be approximated by a global field, which satisfies the Beltrami equation in \(\mathbb{R}^3\), and drops off at infinity in an optimal way.
The proofs use Lyapunov stability, Cauchy-Schwartz, Poincaré, Jensen and Sobolev inequalities, Hodge decomposition, Riesz representation theorem, Fredholm alternative, Poincaré map, Hahn-Banach theorem, Riesz-Markov theorem, Poissons equation and spherical Bessel functions. Finally, applications to Navier-Stokes equation are briefly discussed.

35Q31 Euler equations
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
57M25 Knots and links in the \(3\)-sphere (MSC2010)
35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
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[1] Angenent, S. B., A remark on the topological entropy and invariant circles of an area preserving twistmap, in Twist Mappings and their Applications, IMA Vol. Math. Appl., 44, pp. 1-5. Springer, New York, 1992. · Zbl 0769.58036
[2] Arnold, V. I., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. · Zbl 0148.45301
[3] Arnold, V. I., The asymptotic Hopf invariant and its applications, in Materials of the All-Union School on Differential Equations with Infinitely Many Independent Variables and on Dynamical Systems with Infinitely Many Degrees of Freedom. Dilijan, Erevan, 1973 (Russian); English translation in Selecta Math. Soviet., 5 (1986), 327-345. · Zbl 1015.57018
[4] Arnold, V. I. & Khesin, B.A., Topological Methods in Hydrodynamics. Appl. Math. Sci., 125. Springer, New York, 1998. · Zbl 0902.76001
[5] Bruce, J. W. & Giblin, P. J., Curves and Singularities. Cambridge University Press, Cambridge, 1984. · Zbl 0534.58008
[6] Choffrut, A.; Šverák, V., Local structure of the set of steady-state solutions to the 2D incompressible Euler equations, Geom. Funct. Anal.,, 22, 136-201, (2012) · Zbl 1256.35076
[7] Cordoba, D.; Fefferman, C., On the collapse of tubes carried by 3D incompressible flows, Comm. Math. Phys.,, 222, 293-298, (2001) · Zbl 0999.76020
[8] Deng, J.; Hou, T.Y.; Yu, X., Geometric properties and nonblowup of 3D incompressible Euler flow, Comm. Partial Differential Equations,, 30, 225-243, (2005) · Zbl 1142.35549
[9] Ebin, D.G.; Marsden, J., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math.,, 92, 102-163, (1970) · Zbl 0211.57401
[10] Enciso, A.; Peralta-Salas, D., Knots and links in steady solutions of the Euler equation, Ann. of Math.,, 175, 345-367, (2012) · Zbl 1238.35092
[11] Etnyre, J.B. & Ghrist, R. W., Stratified integrals and unknots in inviscid flows, in Geometry and Topology in Dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), Contemp. Math., 246, pp. 99-111. Amer. Math. Soc., Providence, RI, 1999. · Zbl 0989.76010
[12] Etnyre, J.B. & Ghrist, R. W., Contact topology and hydrodynamics. III. Knotted orbits. Trans. Amer. Math. Soc., 352 (2000), 5781-5794. · Zbl 0960.76020
[13] Fayad, B.; Krikorian, R., Herman’s last geometric theorem, Ann. Sci. Éc. Norm. Supér.,, 42, 193-219, (2009) · Zbl 1175.37062
[14] Foiaş, C. & Temam, R., Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation. Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (1978), 28-63. · Zbl 0384.35047
[15] Freedman, M. H. & He, Z. X., Divergence-free fields: energy and asymptotic crossing number. Ann. of Math., 134 (1991), 189-229. · Zbl 0746.57011
[16] Gambaudo, J. M. & Ghys, É., Signature asymptotique d’un champ de vecteurs en dimension 3. Duke Math. J., 106 (2001), 41-79. · Zbl 1010.37010
[17] González-Enrìquez, A. & de la Llave, R., Analytic smoothing of geometric maps with applications to KAM theory. J. Differential Equations, 245 (2008), 1243-1298. · Zbl 1160.37024
[18] Herman, M. R., Existence et non existence de tores invariants par des difféomorphismes symplectiques, in Séminaire sur les équations aux dérivées partielles 1987-1988, Exp. No. XIV. École Polytech., Palaiseau, 1988.
[19] Khesin, B. & Misio lek, G., Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math., 176 (2003), 116-144. · Zbl 1017.37039
[20] Kleckner, D.; Irvine, W.T.M., Creation and dynamics of knotted vortices, Nature Phys.,, 9, 253-258, (2013)
[21] Laurence, P. & Stredulinsky, E. W., Two-dimensional magnetohydrodynamic equilibria with prescribed topology. Comm. Pure Appl. Math., 53 (2000), 1177-1200. · Zbl 1072.76689
[22] Massey, W. S., On the normal bundle of a sphere imbedded in Euclidean space. Proc. Amer. Math. Soc., 10 (1959), 959-964. · Zbl 0094.36002
[23] Moffatt, H. K., Vortex- and magneto-dynamics—a topological perspective, in Mathematical Physics 2000, pp. 170-182. Imp. Coll. Press, London, 2000. · Zbl 0989.76093
[24] Morrey, C. B., Jr., On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. II. Analyticity at the boundary. Amer. J. Math., 80 (1958), 219-237. · Zbl 0081.09402
[25] Nadirashvili, N., On stationary solutions of two-dimensional Euler equation. Arch. Ration. Mech. Anal., 209 (2013), 729-745. · Zbl 1287.35062
[26] Pelz, R.B., Symmetry and the hydrodynamic blow-up problem. J. Fluid Mech., 444 (2001), 299-320. · Zbl 1002.76095
[27] Ricca, R. L., New developments in topological fluid mechanics: from Kelvin’s vortex knots to magnetic knots, in Ideal Knots, Ser. Knots Everything, 19, pp. 255-273. World Sci. Publ., River Edge, NJ, 1998. · Zbl 0976.76505
[28] Thomson, W. (Lord Kelvin), Vortex statics. Proc. R. Soc. Edinburgh, 9 (1875), 59- 73; reprinted in Mathematical and Physical Papers IV, pp. 115-128, Cambridge Univ. Press, Cambridge, 2011.
[29] Vogel, T., On the asymptotic linking number, Proc. Amer. Math. Soc.,, 131, 2289-2297, (2003) · Zbl 1015.57018
[30] Yoccoz, J. C., Analytic linearization of circle diffeomorphisms, in Dynamical Systems and Small Divisors (Cetraro, 1998), Lecture Notes in Math., 1784, pp. 125-173. Springer, Berlin-Heidelberg, 2002. · Zbl 1417.37153
[31] Yoshida, Z.; Giga, Y., Remarks on spectra of operator rot, Math. Z.,, 204, 235-245, (1990) · Zbl 0676.47012
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