The influence of magnetic steps on bulk superconductivity

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

The influence of magnetic steps on bulk superconductivity

Wafaa Assaad ^1, and Ayman Kachmar ^2,

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

Abstract
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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.

Keywords: magnetic fields, Ginzburg-Landau equation, type II superconductors..

Mathematics Subject Classification: Primary: 35Q56; Secondary: 35J1.

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.

show all references

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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