Abstract
We study the variational convergence of a family of twodimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity as the Ginzburg-Landau parameter ε tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that if suitably normalized, the energy functionals Γ-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.
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References
A. Aftalion, S. Alama, and L. Bronsard, Giant vortex and the breakdown of strong pinning in a rotating Bose-Einstein condensate, Arch. Ration. Mech. Anal. 178 (2005), 247–286.
S. Alama and L. Bronsard, Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains, Comm. Pure Appl. Math. 59 (2006), 36–70.
S. Alama and L. Bronsard, Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions, J. Math. Phys. 46 (2005), 095102.
S. Alama, L. Bronsard, and B. Galvão-Sousa, Thin film limits for Ginzburg-Landau with strong applied magnetic fields, SIAM J. Math. Anal. 42 (2010), 97–124.
L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.
H. Aydi, Lines of vortices for solutions of the Ginzburg-Landau equations, J. Math. Pure Appl. 89 (2008), 49–69.
F. Bethuel, H. Brezis, and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser Boston, Boston, MA, 1994.
L. Bers, F. John, and M. Schechter, Partial Differential Equations. AMS, Providence, RI, 1979.
S. J. Chapman, Q. Du, and M. D. Gunzburger, On the Lawrence-Donaich and anisotropic Ginzburg-Landau models for layered superconductors, SIAM J. Appl. Math. 55 (1995), 156–174.
G. Dal Maso, An Introduction to Γ-convergence, Birkhaüser Boston, Boston, MA, 1993.
R. Ignat and V. Millot, The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate, J. Funct. Anal. 233 (2006), 260–306.
R. Ignat and V. Millot, Energy expansion and vortex location for a two-dimensional rotating Bose-Einstein condensate, Rev. Math. Phys. 18 (2006), 119–162.
R. L. Jerrard and H. M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. Partial Differential Equations 14 (2002), 151–191.
A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint, ESAIM Control Optim. Calc. Var. 16 (2010), 545–580.
E. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, 1997.
E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. École Norm. Sup. (4) 33 (2000), 561–592.
E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Birkhäuser Boston, Boston, MA, 2007.
E. Sandier and M. Soret, S 1 -valued harmonic maps with high topological degree: asymptotic behavior of the singular set, Potential Anal. 13 (2000), 169–184.
S. Serfaty, On a model of rotating superfluids, ESAIM Control Optim. Calc. Var. 6 (2001), 201–238.
W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New-York, 1989.
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Alama, S., Bronsard, L. & Millot, V. Γ-Convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves. JAMA 114, 341–391 (2011). https://doi.org/10.1007/s11854-011-0020-0
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DOI: https://doi.org/10.1007/s11854-011-0020-0