Abstract
In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in \({\mathbb{R}^{3}}\), we prove that the curve evolves to leading order by binormal curvature flow. Our approach combines new estimates on the distance of the corresponding Hamiltonian-Poisson structures with stability estimates recently developed in Jerrard and Smets (J Eur Math Soc (JEMS) 17(6):1487–1515, 2015).
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References
Ambrosetti A., Struwe M.: Existence of steady vortex rings in an ideal fluid. Arch. Ration. Mech. Anal. 108(2), 97–109 (1989)
Benedetto D., Caglioti E., Marchioro C.: On the motion of a vortex ring with a sharply concentrated vorticity. Math. Methods Appl. Sci. 23(2), 147–168 (2000)
Buckmaster T., De Lellis C., Isett P., Székelyhidi J.L.: Anomalous dissipation for 1/5-Hölder Euler flows. Ann. Math. (2) 182(1), 127–172 (2015)
Da Rios L.: Sul moto d’un liquido indefinito con unfiletto vorticoso di forma qualunque. Rendiconti del CircoloMatematico di Palermo (1884-1940 22(1), 117–135 (1906)
de la Hoz F., Vega L.: Vortex filament equation for a regular polygon. Nonlinearity 27(12), 3031–3057 (2014)
De Lellis C., Székelyhidi J. L.: The Eulerequations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436 (2009)
De Lellis C., Székelyhidi J.L.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010)
Enciso A., Peralta-Salas D.: Existence of knotted vortex tubes in steady Euler flows. Acta Math. 214(1), 61–134 (2015)
Enciso A., Peralta-Salas D.: Knotted vortex lines and vortex tubes in stationary fluid flows. Eur. Math. Soc. Newsl. 96, 26–33 (2015)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1992
Federer, H.: Geometric measure theory. DieGrundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969
Fraenkel L.E., Berger M.S.: A global theory of steady vortex rings in an ideal fluid. Acta Math. 132, 13–51 (1974)
Jerrard R. L.: Vortex filament dynamics for Gross-Pitaevskytype equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1(4), 733–768 (2002)
Jerrard, R.L., Smets, D.: On Schrödinger maps fromT 1 to S 2. Ann. Sci. Éc. Norm. Supér. (4), (45)(4) (2012), 637–680 (2013)
Jerrard R.L., Smets D.: On the motion of a curve by its binormal curvature. J. Eur. Math. Soc. (JEMS) 17(6), 1487–1515 (2015)
Keener J.P.: Knotted vortex filaments in an ideal fluid. J. Fluid Mech. 211(2), 629–651 (1990)
Khesin, B.: Symplectic structures and dynamics on vortexmembranes. Mosc. Math. J. 12(2), 413–434, 461–462 (2012)
Levi-Civita T.: Attrazione newtoniana dei tubi sottili evortici filiformi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 1((1–2)), 1–33 (1932)
Levi-Civita T.: Attrazione newtoniana dei tubi sottili evortici filiformi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 1(3), 229–250 (1932)
Majda, A.J.; Bertozzi, A.L.: Vorticity and Incompressible Flow, vol. 27. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002
Marchioro C., Pulvirenti M.: Euler evolution for singular initial data and vortex theory. Commun. Math. Phys. 91(4), 563–572 (1983)
Marsden, J., Weinstein, A.: Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Phys. D 7(1–3), 305–323 (1983). Order in chaos (Los Alamos, N.M., 1982)
Ricca R.L.: Rediscovery of Da Rios equations. Nature 352(6336), 561–562 (1991)
Rudin, W.: Functional analysis, 2nd edn. International Series in Pure and Applied Mathematics. McGraw-Hill Inc., New York, 1991
Saffman, P.G.: Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1992
Scheffer V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3(4), 343–401 (1993)
Shashikanth, B.N.: Vortex dynamics in \({\mathbb{R}^{4}}\). J. Math. Phys. 53(1), 013103, 21 (2012)
Shnirelman A.: On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50(12), 1261–1286 (1997)
Shnirelman A.: Weak solutions with decreasing energy of incompressible Euler equations. Commun. Math. Phys. 210(3), 541–603 (2000)
Sogge, C.D.: Fourier integrals in classical analysis, vol. 105. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1993
Székelyhidi L., Wiedemann E.: Young measures generated by ideal incompressible fluid flows. Arch. Ration. Mech. Anal. 206(1), 333–366 (2012)
Thomson(Lord Kelvin) W.: Vortex statics. Proc. R. Soc. Edinb. 9, 59–73 (1875)
Wiedemann E.: Existence of weak solutions for the incompressible Euler equations. Ann. Inst. H. PoincaréAnal. Non Linéaire 28(5), 727–730 (2011)
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Jerrard, R.L., Seis, C. On the Vortex Filament Conjecture for Euler Flows. Arch Rational Mech Anal 224, 135–172 (2017). https://doi.org/10.1007/s00205-016-1070-3
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DOI: https://doi.org/10.1007/s00205-016-1070-3