Uniform estimates for the parabolic Ginzburg–Landau equation

F. Bethuel; G. Orlandi

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 219-238
  • ISSN: 1292-8119

Abstract

top
We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of ℝ N , N ≥ 2, where ε > 0 is a small parameter. We assume that the Ginzburg–Landau energy E ε ( u ε ) verifies the bound (natural in the context) E ε ( u ε ) ≤ M 0 | log ε | , where M0 is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of uE, as ε → 0, is to establish uniform Lp bounds for the gradient, for some p>1. We review some recent techniques developed in the elliptic case in [7], discuss some variants, and extend the methods to the associated parabolic equation.

How to cite

top

Bethuel, F., and Orlandi, G.. "Uniform estimates for the parabolic Ginzburg–Landau equation." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 219-238. <http://eudml.org/doc/90647>.

@article{Bethuel2010,
abstract = { We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of $\mathbb\{R\}^N$, N ≥ 2, where ε > 0 is a small parameter. We assume that the Ginzburg–Landau energy $E_\varepsilon(u_\varepsilon)$ verifies the bound (natural in the context) $E_\varepsilon(u_\varepsilon)\le M_0|\log\varepsilon|$, where M0 is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of uE, as ε → 0, is to establish uniform Lp bounds for the gradient, for some p>1. We review some recent techniques developed in the elliptic case in [7], discuss some variants, and extend the methods to the associated parabolic equation. },
author = {Bethuel, F., Orlandi, G.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Ginzburg–Landau; parabolic equations; Hodge–de Rham decomposition; Jacobians.; Hodge-de Rham decomposition; Jacobians},
language = {eng},
month = {3},
pages = {219-238},
publisher = {EDP Sciences},
title = {Uniform estimates for the parabolic Ginzburg–Landau equation},
url = {http://eudml.org/doc/90647},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Bethuel, F.
AU - Orlandi, G.
TI - Uniform estimates for the parabolic Ginzburg–Landau equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 219
EP - 238
AB - We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of $\mathbb{R}^N$, N ≥ 2, where ε > 0 is a small parameter. We assume that the Ginzburg–Landau energy $E_\varepsilon(u_\varepsilon)$ verifies the bound (natural in the context) $E_\varepsilon(u_\varepsilon)\le M_0|\log\varepsilon|$, where M0 is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of uE, as ε → 0, is to establish uniform Lp bounds for the gradient, for some p>1. We review some recent techniques developed in the elliptic case in [7], discuss some variants, and extend the methods to the associated parabolic equation.
LA - eng
KW - Ginzburg–Landau; parabolic equations; Hodge–de Rham decomposition; Jacobians.; Hodge-de Rham decomposition; Jacobians
UR - http://eudml.org/doc/90647
ER -

References

top
  1. G. Alberti, S. Baldo and G. Orlandi, Variational convergence for functionals of Ginzburg-Landau type. Preprint (2001).  
  2. L. Almeida, S. Baldo, F. Bethuel and G. Orlandi (in preparation).  
  3. L. Almeida and F. Bethuel, Topological methods for the Ginzburg-Landau equation. J. Math. Pures Appl.11 (1998) 1-49.  
  4. L. Ambrosio and H.M. Soner, A measure theoretic approach to higher codimension mean curvature flow. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)25 (1997) 27-49.  
  5. P. Baumann, C.-N. Chen, D. Phillips and P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems. Eur. J. Appl. Math.6 (1995) 115-126.  
  6. F. Bethuel, Variational methods for Ginzburg-Landau equations, in Calculus of Variations and Geometric evolution problems, Cetraro 1996, edited by S. Hildebrandt and M. Struwe. Springer (1999).  
  7. F. Bethuel, J. Bourgain, H. Brezis and G. Orlandi, W1,pestimates for solutions to the Ginzburg-Landau equation with boundary data in H1/2. C. R. Acad. Sci. Paris Sér. I Math.333 (2001) 1069-1076.  
  8. F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional. Calc. Var. Partial Differential Equations1 (1993) 123-148.  
  9. F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices. Birkhäuser, Boston (1994).  
  10. F. Bethuel, H. Brezis and G. Orlandi, Small energy solutions to the Ginzburg-Landau equation. C. R. Acad. Sci. Paris Sér. I Math.331 (2000) 763-770.  
  11. F. Bethuel, H. Brezis and G. Orlandi, Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions. J. Funct. Anal.186 (2001) 432-520. Erratum (to appear).  
  12. F. Bethuel and T. Rivière, Vortices for a variational problem related to superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire12 (1995) 243-303.  
  13. J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces. J. Anal.80 (2000) 37-86.  
  14. J. Bourgain, H. Brezis and P. Mironescu, On the structure of the Sobolev space H1/2 with values into the circle. C. R. Acad. Sci. Paris Sér. I Math.331 (2000) 119-124.  
  15. H. Brezis and P. Mironescu, Sur une conjecture de E. De Giorgi relative à l'énergie de Ginzburg-Landau. C. R. Acad. Sci. Paris Sér. I Math.319 (1994) 167-170.  
  16. H. Federer, Geometric Measure Theory. Springer, Berlin (1969).  
  17. Z.C. Han and I. Shafrir, Lower bounds for the energy of S1-valued maps in perforated domains. J. Anal. Math.66 (1995) 295-305.  
  18. R. Hardt and F.H. Lin, Mappings minimizing the Lp-norm of the gradient. Comm. Pure Appl. Math. 40 (1987) 555-588.  
  19. R. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals. SIAM J. Math. Anal.30 (1999) 721-746.  
  20. R. Jerrard and H.M. Soner, Dynamics of Ginzburg-Landau vortices. Arch. Rational Mech. Anal.142 (1998) 99-125.  
  21. R. Jerrard and H.M. Soner, Scaling limits and regularity results for a class of Ginzburg-Landau systems. Ann. Inst. H. Poincaré Anal. Non Linéaire16 (1999) 423-466.  
  22. R. Jerrard and H.M. Soner, The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differential Equations (to appear).  
  23. F.H. Lin, Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math.49 (1996) 323-359.  
  24. F.H. Lin, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds. Comm. Pure Appl. Math. 51 (1998) 385-441  
  25. F.H. Lin, Rectifiability of defect measures, fundamental groups and density of Sobolev mappings, in Journées ``Équations aux Dérivées Partielles", Saint-Jean-de-Monts, 1996, Exp. No. XII. École Polytechnique, Palaiseau (1996).  
  26. F.H. Lin and T. Rivière, Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc.1 (1999) 237-311. Erratum, Ibid. 
  27. F.H. Lin and T. Rivière, A quantization property for static Ginzburg-Landau vortices. Comm. Pure Appl. Math.54 (2001) 206-228.  
  28. F.H. Lin and T. Rivière, A quantization property for moving line vortices. Comm. Pure Appl. Math.54 (2001) 826-850.  
  29. L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123-142.  
  30. L. Modica and S. Mortola, Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. B14 (1977) 285-299.  
  31. T. Rivière, Line vortices in the U(1)-Higgs model. ESAIM: COCV1 (1996) 77-167.  
  32. T. Rivière, Dense subsets of H1/2(S2,S1). Ann. Global Anal. Geom.18 (2000) 517-528.  
  33. T. Rivière, Asymptotic analysis for the Ginzburg-Landau Equation. Boll. Un. Mat. Ital. B8 (1999) 537-575.  
  34. E. Sandier, Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 152 (1997) 379-403; Erratum 171 (2000) 233.  
  35. L. Simon, Lectures on Geometric Measure Theory, in Proc. of the Centre for Math. Analysis. Australian Nat. Univ., Canberra (1983).  
  36. M. Struwe, On the asymptotic behavior of the Ginzburg-Landau model in 2 dimensions. J. Differential Equations 7 (1994) 1613-1624; Erratum 8 (1995) 224.  

NotesEmbed ?

top

You must be logged in to post comments.