# American Institute of Mathematical Sciences

December  2016, 36(12): 6623-6643. doi: 10.3934/dcds.2016087

## The influence of magnetic steps on bulk superconductivity

 1 Lund University, Department of Mathematics, Box 118, SE-22100, Lund, Sweden 2 Lebanese University, Department of Mathematics, Hadath, Lebanon

Received  January 2016 Revised  March 2016 Published  October 2016

We study the distribution of bulk superconductivity in presence of an applied magnetic field, supposed to be a step function, modeled by the Ginzburg-Landau theory. Our results are valid for the minimizers of the two-dimensional Ginzburg-Landau functional with a large Ginzburg-Landau parameter and with an applied magnetic field of intensity comparable with the Ginzburg-Landau parameter.
Citation: Wafaa Assaad, Ayman Kachmar. The influence of magnetic steps on bulk superconductivity. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6623-6643. doi: 10.3934/dcds.2016087
##### References:
 [1] Y. Almog, B. Helffer and X. B. Pan, Mixed normal-superconducting states in the presence of strong electric currents, Arch. Rational Mech. Anal. (2016). doi:10.1007/s00205-016-1037-4 [2] K. Attar, The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field, Ann. I.H.Poincaré-AN, 32 (2015), 325-345. doi: 10.1016/j.anihpc.2013.12.002. [3] K. Attar, Energy and vorticity of the Ginzburg-Landau model with variable magnetic field, Asympt. Anal., 93 (2015), 75-114. doi: 10.3233/ASY-151286. [4] K. Attar, Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model, Non-Linear Analysis: TMA., 139 (2016), 1-54. doi: 10.1016/j.na.2016.02.002. [5] V. Bonnaillie-Noël and S. Fournais, Superconductivity in domains with corners, Rev. Math. Phys., 19 (2007), 607-637. doi: 10.1142/S0129055X07003061. [6] S. J. Chapman, Q. Du and M. D. Gunzburger, A Ginzburg-Landau type model of superconducting/normal junctions including Josephson junctions, European Journal of Applied Mathematics, 6 (1995), 97-114. doi: 10.1017/S0956792500001716. [7] A. Contreras and X. Lamy, Persistence of superconductivity in thin shells beyond Hc1, Commun. Contemp. Math., 18 (2016),1550047, 21pp. doi: 10.1142/S0219199715500479. [8] S. Fournais and B. Helffer, Spectral Methods in Surface Superconductivity, Progress in Nonlinear Differential Equations and their Applications. Vol. 77, Birkhäuser, Boston, 2010. [9] S. Fournais and A. Kachmar, The ground state energy of the three dimensional Ginzburg-Landau functional. Part I: Bulk regime, Commun. Part. Diff. Equations, 38 (2013), 339-383. doi: 10.1080/03605302.2012.717156. [10] S. Fournais and A. Kachmar, Nucleation of bulk superconductivity close to critical magnetic field, Adv. Math., 226 (2011), 1213-1258. doi: 10.1016/j.aim.2010.08.004. [11] S. Fournais and A. Kachmar, On the transition to the normal phase for superconductors surrounded by normal conductors, J. Differential Equations, 247 (2009), 1637-1672. doi: 10.1016/j.jde.2009.04.012. [12] T. Giorgi and D. Phillips, The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model, SIAM J. Math. Anal., 30 (1999), 341-359. doi: 10.1137/S0036141097323163. [13] B. Helffer and A. Mohamed, Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal., 138 (1996), 40-81. doi: 10.1006/jfan.1996.0056. [14] B. Helffer and A. Kachmar, The ginzburg-landau functional with vanishing magnetic field, Arch. Rational Mech. Anal., 218 (2015), 55-122. doi: 10.1007/s00205-015-0856-z. [15] P. D. Hislop, N. Popoff, N. Raymond and M. P. Sundqvist, Band functions in presence of magnetic steps, Mathematical Models and Methods in Applied Sciences, 26 (2016), 161-184. doi: 10.1142/S0218202516500056. [16] A. Kachmar, The ground state energy of the three-dimensional Ginzburg-Landau model in the mixed phase, J. Funct. Anal., 261 (2011), 3328-3344. doi: 10.1016/j.jfa.2011.08.002. [17] A. Kachmar, On the perfect superconducting solution for a generalized Ginzburg-Landau equation, Asymptot. Anal., 54 (2007), 125-164. [18] K. Lu and X. B. Pan, Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity, Physica D, 127 (1999), 73-104. doi: 10.1016/S0167-2789(98)00246-2. [19] K. Lu and X. B. Pan, Surface nucleation of superconductivity in 3-dimension, J. Differential Equations, 168 (2000), 386-452. doi: 10.1006/jdeq.2000.3892. [20] X. B. Pan and K. H. Kwek, Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains, Tran. Amer. Math. Soc., 354 (2002), 4201-4227. doi: 10.1090/S0002-9947-02-03033-7. [21] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Vol. 343. American Mathematical Soc., 2001. doi: 10.1090/chel/343. [22] E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress in Nonlinear Partial Differential Equations and Their Applications. Vol. 70, Birkhäuser, 2007. [23] E. Sandier and S. Serfaty, The decrease of bulk superconductivity close to the second critical field in the Ginzburg-Landau model, SIAM J. Math. Anal., 34 (2003), 939-956. doi: 10.1137/S0036141002406084.

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##### References:
 [1] Y. Almog, B. Helffer and X. B. Pan, Mixed normal-superconducting states in the presence of strong electric currents, Arch. Rational Mech. Anal. (2016). doi:10.1007/s00205-016-1037-4 [2] K. Attar, The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field, Ann. I.H.Poincaré-AN, 32 (2015), 325-345. doi: 10.1016/j.anihpc.2013.12.002. [3] K. Attar, Energy and vorticity of the Ginzburg-Landau model with variable magnetic field, Asympt. Anal., 93 (2015), 75-114. doi: 10.3233/ASY-151286. [4] K. Attar, Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model, Non-Linear Analysis: TMA., 139 (2016), 1-54. doi: 10.1016/j.na.2016.02.002. [5] V. Bonnaillie-Noël and S. Fournais, Superconductivity in domains with corners, Rev. Math. Phys., 19 (2007), 607-637. doi: 10.1142/S0129055X07003061. [6] S. J. Chapman, Q. Du and M. D. Gunzburger, A Ginzburg-Landau type model of superconducting/normal junctions including Josephson junctions, European Journal of Applied Mathematics, 6 (1995), 97-114. doi: 10.1017/S0956792500001716. [7] A. Contreras and X. Lamy, Persistence of superconductivity in thin shells beyond Hc1, Commun. Contemp. Math., 18 (2016),1550047, 21pp. doi: 10.1142/S0219199715500479. [8] S. Fournais and B. Helffer, Spectral Methods in Surface Superconductivity, Progress in Nonlinear Differential Equations and their Applications. Vol. 77, Birkhäuser, Boston, 2010. [9] S. Fournais and A. Kachmar, The ground state energy of the three dimensional Ginzburg-Landau functional. Part I: Bulk regime, Commun. Part. Diff. Equations, 38 (2013), 339-383. doi: 10.1080/03605302.2012.717156. [10] S. Fournais and A. Kachmar, Nucleation of bulk superconductivity close to critical magnetic field, Adv. Math., 226 (2011), 1213-1258. doi: 10.1016/j.aim.2010.08.004. [11] S. Fournais and A. Kachmar, On the transition to the normal phase for superconductors surrounded by normal conductors, J. Differential Equations, 247 (2009), 1637-1672. doi: 10.1016/j.jde.2009.04.012. [12] T. Giorgi and D. Phillips, The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model, SIAM J. Math. Anal., 30 (1999), 341-359. doi: 10.1137/S0036141097323163. [13] B. Helffer and A. Mohamed, Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal., 138 (1996), 40-81. doi: 10.1006/jfan.1996.0056. [14] B. Helffer and A. Kachmar, The ginzburg-landau functional with vanishing magnetic field, Arch. Rational Mech. Anal., 218 (2015), 55-122. doi: 10.1007/s00205-015-0856-z. [15] P. D. Hislop, N. Popoff, N. Raymond and M. P. Sundqvist, Band functions in presence of magnetic steps, Mathematical Models and Methods in Applied Sciences, 26 (2016), 161-184. doi: 10.1142/S0218202516500056. [16] A. Kachmar, The ground state energy of the three-dimensional Ginzburg-Landau model in the mixed phase, J. Funct. Anal., 261 (2011), 3328-3344. doi: 10.1016/j.jfa.2011.08.002. [17] A. Kachmar, On the perfect superconducting solution for a generalized Ginzburg-Landau equation, Asymptot. Anal., 54 (2007), 125-164. [18] K. Lu and X. B. Pan, Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity, Physica D, 127 (1999), 73-104. doi: 10.1016/S0167-2789(98)00246-2. [19] K. Lu and X. B. Pan, Surface nucleation of superconductivity in 3-dimension, J. Differential Equations, 168 (2000), 386-452. doi: 10.1006/jdeq.2000.3892. [20] X. B. Pan and K. H. Kwek, Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains, Tran. Amer. Math. Soc., 354 (2002), 4201-4227. doi: 10.1090/S0002-9947-02-03033-7. [21] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Vol. 343. American Mathematical Soc., 2001. doi: 10.1090/chel/343. [22] E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress in Nonlinear Partial Differential Equations and Their Applications. Vol. 70, Birkhäuser, 2007. [23] E. Sandier and S. Serfaty, The decrease of bulk superconductivity close to the second critical field in the Ginzburg-Landau model, SIAM J. Math. Anal., 34 (2003), 939-956. doi: 10.1137/S0036141002406084.
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