Geometric and Functional Analysis

, Volume 27, Issue 5, pp 1161–1230 | Cite as

Nearly Parallel Vortex Filaments in the 3D Ginzburg–Landau Equations

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Abstract

We introduce a framework to study the occurrence of vortex filament concentration in 3D Ginzburg–Landau theory. We derive a functional that describes the free-energy of a collection of nearly-parallel quantized vortex filaments in a cylindrical 3-dimensional domain, in certain scaling limits; it is shown to arise as the \({\Gamma}\)-limit of a sequence of scaled Ginzburg–Landau functionals. Our main result establishes for the first time a long believed connection between the Ginzburg–Landau functional and the energy of nearly parallel filaments that applies to many mathematically and physically relevant situations where clustering of filaments is expected. In this setting it also constitutes a higher-order asymptotic expansion of the Ginzburg–Landau energy, a refinement over the arclength functional approximation. Our description of the vorticity region significantly improves on previous studies and enables us to rigorously distinguish a collection of multiplicity one vortex filaments from an ensemble of fewer higher multiplicity ones. As an application, we prove the existence of solutions of the Ginzburg–Landau equation that exhibit clusters of vortex filaments whose small-scale structure is governed by the limiting free-energy functional.

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References

  1. AR01.
    Aftalion A., Rivière T.: “Vortex energy and vortex bending for a rotating Bose-Einstein condensate. Phys Rev A., 64, 043611 (2001)CrossRefGoogle Scholar
  2. ABO05.
    Alberti G., Baldo S., Orlandi G.: “Variational convergence for functionals of Ginzburg–Landau type”. Indiana Univ. Math. J., 54(5), 1411–1472 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. BFT08.
    V. Barutello, D. Ferrario and S. Terracini, “On the singularities of generalized solutions to n-body-type problems”. Int. Math. Res. Not. IMRN (2008), Art. ID rnn 069, 78 pp.Google Scholar
  4. BBH94.
    F. Bethuel, H. Brezis and F. Hélein, “Ginzburg–Landau Vortices”, Progress in Nonlinear Differential Equations and their Applications 13, Birkhäuser Boston, Boston, MA, (1994)Google Scholar
  5. BBO01.
    Bethuel F., Brezis H., Orlandi G.: “Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions”, J. Funct. Anal. 186(2), 432–520 (2001)CrossRefMATHMathSciNetGoogle Scholar
  6. BBM04.
    J. Bourgain, H. Brezis, and P. Mironescu, “\({H^{1/2}}\) maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation”, Publ. Math. Inst. Hautes tudes Sci. (99) (2004), 1–115Google Scholar
  7. Che08.
    Chen K.: “Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses”. Ann. of Math. (2) 167(2), 325–348 (2008)CrossRefMATHMathSciNetGoogle Scholar
  8. CM99.
    Chenciner A., Montgomery R.: “A remarkable periodic solution of the three body problem in the case of equal masses”. Ann. of Math. 152, 881–901 (1999)CrossRefMATHMathSciNetGoogle Scholar
  9. Con11.
    Contreras A.: “On the First critical field in Ginzburg–Landau theory for thin shells and manifolds”. Archive for Rational Mechanics and Analysis, 200(2), 563–611 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. PK08.
    del Pino M., Kowalczyk M.: “Renormalized energy of interacting Ginzburg–Landau vortex filaments”. J. Lond. Math. Soc. (2) 77(3), 647–665 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. PKPW10.
    del Pino M., Kowalczyk M., Pacard F., Wei J: “The Toda system and multiple-end solutions of autonomous planar elliptic problems”. Adv. Math. 224(4), 1462–1516 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. PKW08.
    del Pino M., Kowalczyk M., Wei J: “The Toda system and clustering interfaces in the Allen–Cahn equation”. Arch. Ration. Mech. Anal. 190(1), 141–187 (2008)CrossRefMATHMathSciNetGoogle Scholar
  13. PKWY10.
    del Pino M., Kowalczyk M., Wei J., Yang J.: “Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature.”. Geom. Funct. Anal. 20(4), 918–957 (2010)CrossRefMATHMathSciNetGoogle Scholar
  14. Fed69.
    H. Federer, “Geometric Measure Theory” Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, New York (1969)Google Scholar
  15. FT04.
    Ferrario D., Terracini S.: “On the existence of collisionless equivariant minimizers for the classical n-boby problem”. Invent. Math. 155, 305–362 (2004)CrossRefMATHMathSciNetGoogle Scholar
  16. Jer99a.
    Jerrard R.: “Lower bounds for generalized Ginzburg–Landau functionals”. SIAM Math. Anal. 30(4), 721–746 (1999a)CrossRefMATHMathSciNetGoogle Scholar
  17. Jer99b.
    Jerrard R.: “Vortex dynamics for the Ginzburg–Landau wave equation”. Calc. Var. and PDE 9, 1–30 (1999b)CrossRefMATHMathSciNetGoogle Scholar
  18. JS02.
    Jerrard R., Soner H.: “The Jacobian and the Ginzburg–Landau energy”. Calc. Var and PDE 14, 151–191 (2002)CrossRefMATHMathSciNetGoogle Scholar
  19. JS07.
    JerrardR. Spirn D.: “Refined Jacobian estimates for Ginzburg–Landau functionals”. Indiana Univ. Math. Jour. 56, 135–186 (2007)CrossRefMATHMathSciNetGoogle Scholar
  20. JS08.
    Jerrard R., Spirn D.: “Refined Jacobian estimates and Gross–Pitaevsky vortex dynamics”. Arch. Rat. Mech. Anal. 190, 425–475 (2008)CrossRefMATHMathSciNetGoogle Scholar
  21. JS09.
    Jerrard R., Sternberg P.: “Critical points via Gamma-convergence: general theory and applications”. Jour. Eur. Math. Soc. 11(4), 705–753 (2009)CrossRefMATHGoogle Scholar
  22. KPV03.
    Kenig C., Ponce G., Vega L.: “On the interaction of nearly parallel vortex filaments”. Comm. Math. Phys. 243, 471–483 (2003)CrossRefMATHMathSciNetGoogle Scholar
  23. KMD95.
    Klein R., Majda A., Damodaran K.: “Simplified equations for the interaction of nearly parallel vortex filaments”. J. Fluid Mech. 228, 201–248 (1995)CrossRefMATHMathSciNetGoogle Scholar
  24. Li99.
    F. H. Lin, “Vortex dynamics for the nonlinear wave equation”, Comm. Pure Appl. Math. (6)52 (1999), 737?761Google Scholar
  25. LR01.
    Lin F.-H., Rivir̀e T.: “A quantization property for static Ginzburg–Landau vortices”. Comm. Pure Appl. Math. 54(2), 206–228 (2001)CrossRefMathSciNetGoogle Scholar
  26. LM2000.
    P. L. Lions and A. Majda. “Equilibrium Statistical Theory for Nearly Parallel Vortex Filaments ”, Communications on Pure and Applied Mathematics, LIII (2000), 0076–0142Google Scholar
  27. MSZ04.
    Montero J. A., Sternberg P., Ziemer W.P.: “Local minimizers with vortices in the Ginzburg–Landau system in three dimensions”. Comm. Pure Appl. Math. 57(1), 99–125, (2004)CrossRefMATHMathSciNetGoogle Scholar
  28. PR2000.
    F. Pacard and T. Rivière, “Linear and nonlinear aspects of vortice. The Ginzburg–Landau model”, Progress in Nonlinear Differential Equations and their Applications 39, Birkhäuser Boston, Boston, MA, (2000)Google Scholar
  29. Riv96.
    T. Rivière, ‘ Line vortices in the \({U(1)}\)-Higgs model”, ESAIM Contrôle Optim. Calc. Var. 1 (1995/96), 77–167Google Scholar
  30. San98.
    Sandier E.: “Lower bounds for the energy of unit vector fields and applications”. J. Funct. Anal. 152, 379–403 (1998)CrossRefMATHMathSciNetGoogle Scholar
  31. San01.
    Sandier E.: “ Ginzburg–Landau minimizers from \({\mathbb{R}^{n+1}}\) to \({\mathbb{R}^n}\) and minimal connections”. Indiana Univ. Math. J. 50(4), 1807–1844 (2001)CrossRefMATHMathSciNetGoogle Scholar
  32. SS04.
    Sandier E., Serfaty S.: “ A product-estimate for Ginzburg–Landau and corollaries”. J. Funct. Anal. 211(1), 219–244 (2004)CrossRefMATHMathSciNetGoogle Scholar
  33. SS07.
    E. Sandier and S. Serfaty. “Vortices in the magnetic Ginzburg–Landau model”. Progress in Nonlinear Differential Equations and their Applications, 70. Birkhäuser Boston, Inc., Boston, MA, (2007)Google Scholar
  34. Sol84.
    Solomon B.: “A new proof of the closure theorem for integral currents”. Indiana Univ. Math. J. 33(3), 393–418 (1984)CrossRefMATHMathSciNetGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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