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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

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 & 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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. Bonnaillie-Noël and S. Fournais, Superconductivity in domains with corners, Rev. Math. Phys., 19 (2007), 607-637. doi: 10.1142/S0129055X07003061.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Fournais and B. Helffer, Spectral Methods in Surface Superconductivity, Progress in Nonlinear Differential Equations and their Applications. Vol. 77, Birkhäuser, Boston, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Kachmar, On the perfect superconducting solution for a generalized Ginzburg-Landau equation, Asymptot. Anal., 54 (2007), 125-164.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Vol. 343. American Mathematical Soc., 2001. doi: 10.1090/chel/343.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. Bonnaillie-Noël and S. Fournais, Superconductivity in domains with corners, Rev. Math. Phys., 19 (2007), 607-637. doi: 10.1142/S0129055X07003061.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Fournais and B. Helffer, Spectral Methods in Surface Superconductivity, Progress in Nonlinear Differential Equations and their Applications. Vol. 77, Birkhäuser, Boston, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Kachmar, On the perfect superconducting solution for a generalized Ginzburg-Landau equation, Asymptot. Anal., 54 (2007), 125-164.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Vol. 343. American Mathematical Soc., 2001. doi: 10.1090/chel/343.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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