Uniform estimates for the parabolic Ginzburg-Landau equation

Bethuel, F.; Orlandi, G.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 219-238.

We consider complex-valued solutions $u_{\epsilon }$ of the Ginzburg-Landau equation on a smooth bounded simply connected domain $\Omega$ of ${ℝ}^N$, $N\ge 2$, where $\epsilon >0$ is a small parameter. We assume that the Ginzburg-Landau energy $E_{\epsilon }\left(u_{\epsilon }\right)$ verifies the bound (natural in the context) $E_{\epsilon }\left(u_{\epsilon }\right)\le M_0|log\epsilon |$, where $M_0$ is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of $u_{\epsilon }$, as $\epsilon \to 0$, is to establish uniform $L^p$ 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.

Classification : 35K55, 35J60, 58E50, 49J10
Mots clés : Ginzburg-Landau, parabolic equations, Hodge-de Rham decomposition, jacobians

[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. | Zbl

[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. | Numdam | MR | Zbl

[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. | Zbl

[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). | Zbl

[7] F. Bethuel, J. Bourgain, H. Brezis and G. Orlandi, $W^{1,p}$ estimates for solutions to the Ginzburg-Landau equation with boundary data in $H^{1/2}$. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 1069-1076. | Zbl

[8] F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional. Calc. Var. Partial Differential Equations 1 (1993) 123-148. | Zbl

[9] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices. Birkhäuser, Boston (1994). | Zbl

[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. | Zbl

[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). | Zbl

[12] F. Bethuel and T. Rivière, Vortices for a variational problem related to superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 243-303. | Numdam | MR | Zbl

[13] J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces. J. Anal. 80 (2000) 37-86. | MR | Zbl

[14] J. Bourgain, H. Brezis and P. Mironescu, On the structure of the Sobolev space $H^{1/2}$ with values into the circle. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 119-124. | MR | Zbl

[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. | Zbl

[16] H. Federer, Geometric Measure Theory. Springer, Berlin (1969). | MR | Zbl

[17] Z.C. Han and I. Shafrir, Lower bounds for the energy of $S^1$-valued maps in perforated domains. J. Anal. Math. 66 (1995) 295-305. | MR | Zbl

[18] R. Hardt and F.H. Lin, Mappings minimizing the $L^p$-norm of the gradient. Comm. Pure Appl. Math. 40 (1987) 555-588. | MR | Zbl

[19] R. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals. SIAM J. Math. Anal. 30 (1999) 721-746. | Zbl

[20] R. Jerrard and H.M. Soner, Dynamics of Ginzburg-Landau vortices. Arch. Rational Mech. Anal. 142 (1998) 99-125. | Zbl

[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 Linéaire 16 (1999) 423-466. | Numdam | Zbl

[22] R. Jerrard and H.M. Soner, The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differential Equations (to appear). | Zbl

[23] F.H. Lin, Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math. 49 (1996) 323-359. | Zbl

[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 | Zbl

[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). | Numdam | Zbl

[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. | Zbl

[27] F.H. Lin and T. Rivière, A quantization property for static Ginzburg-Landau vortices. Comm. Pure Appl. Math. 54 (2001) 206-228. | Zbl

[28] F.H. Lin and T. Rivière, A quantization property for moving line vortices. Comm. Pure Appl. Math. 54 (2001) 826-850. | MR | Zbl

[29] L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123-142. | MR | Zbl

[30] L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285-299. | MR | Zbl

[31] T. Rivière, Line vortices in the $U\left(1\right)$-Higgs model. ESAIM: COCV 1 (1996) 77-167. | Numdam | MR | Zbl

[32] T. Rivière, Dense subsets of $H^{1/2}(S^2,S^1)$. Ann. Global Anal. Geom. 18 (2000) 517-528. | MR | Zbl

[33] T. Rivière, Asymptotic analysis for the Ginzburg-Landau Equation. Boll. Un. Mat. Ital. B 8 (1999) 537-575. | Zbl

[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. | MR | Zbl

[35] L. Simon, Lectures on Geometric Measure Theory, in Proc. of the Centre for Math. Analysis. Australian Nat. Univ., Canberra (1983). | MR | Zbl

[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. | Zbl