×

Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics. (English) Zbl 1087.35008

The paper aims to establish a rigorous mathematical setting for the usual two-dimensional Ginzburg-Landau equation with real coefficients, in the form of \[ \varepsilon\left(u_t - \nabla^2u\right) = u\left(1 - | u| ^2\right), \] where \(\varepsilon\) is treated as a formal small parameter. An evolution problem is considered, initiated by a configuration containing several fundamental vortices with topological charges \(\pm 1\) (evolution of unstable vortices with a multiple topological charge is not considered in the paper) and a plane wave. The evolution includes collisions between vortices, and drift of individual vortices driven by the plane wave. The proofs of theorems about the existence of the evolution solutions are based on separation between dynamical effects taking place at different time scales. The existence of solutions is established both before and after collisions between the vortices. In particular, it is proven that the evolution of a configuration including equal numbers of the fundamental vortices and antivortices leads, in the generic case, to complete annihilation of the vortices.

MSC:

35B25 Singular perturbations in context of PDEs
35K55 Nonlinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. Alberti, S. Baldo, and G. Orlandi, Variational convergence for functionals of Ginzburg-Landau type , Indiana Univ. Math. J. 54 (2005), 1411–1472. · Zbl 1160.35013
[2] L. Ambrosio and H. M. Soner, A measure-theoretic approach to higher codimension mean curvature flows , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 27–49. · Zbl 1043.35136
[3] P. Bauman, C.-N. Chen, D. Phillips, and P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems , European. J. Appl. Math. 6 (1995), 115–126. · Zbl 0845.35042
[4] F. Bethuel, H. Brezis, and F. HéLein, Ginzburg-Landau Vortices , Progr. Nonlinear Differential Equations Appl. 13 , Birkhäuser, Boston, 1994.
[5] F. Bethuel, G. Orlandi, and D. Smets, Approximations with vorticity bounds for the Ginzburg-Landau functional , Commun. Contemp. Math. 6 (2004), 803–832. · Zbl 1129.35329
[6] -, Vortex rings for the Gross-Pitaevskii equation , J. Eur. Math. Soc. (JEMS) 6 (2004), 17–94. · Zbl 1091.35085
[7] -, Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature , to appear in Ann. of Math. (2) 163 (2006). · Zbl 1103.35038
[8] K. A. Brakke, The Motion of a Surface by Its Mean Curvature , Math. Notes 20 , Princeton Univ. Press, Princeton, 1978. · Zbl 0386.53047
[9] H. BréZis and T. Gallouet, Nonlinear Schrödinger evolution equations , Nonlinear Anal. 4 (1980), 677–681. · Zbl 0451.35023
[10] L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics , J. Differential Equations 90 (1991), 211–237. · Zbl 0735.35072
[11] G. Buttazzo, “Integral representation theory for some classes of local functions” in Optimization and Nonlinear Analysis (Haifa, Israel, 1990) , Pitman Res. Notes Math. Ser. 244 , Longman Sci. Tech., Harlow, England, 1992, 64–75. · Zbl 0772.47015
[12] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations , J. Differential Equations 96 (1992), 116–141. · Zbl 0765.35024
[13] E. De Giorgi, “Some conjectures on flow by mean curvature” in Methods of Real Analysis and Partial Differential Equations (Capri, Italy, 1990) , Quad. Accad. Pontaniana 14 , Accad. Pontaniana, Naples, 1992, 9–16.
[14] W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity , Phys. D 77 (1994), 383–404. · Zbl 0814.34039
[15] A. Friedman, Partial Differential Equations of Parabolic Type , Prentice-Hall, Englewood Cliffs, N.J., 1964. · Zbl 0144.34903
[16] R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation , Calc. Var. Partial Differential Equations 9 (1999), 1–30. · Zbl 0941.35099
[17] R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices , Arch. Rational Mech. Anal. 142 (1998), 99–125. · Zbl 0923.35167
[18] -, Scaling limits and regularity results for a class of Ginzburg-Landau systems , Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 423–466. · Zbl 0944.35006
[19] -, The Jacobian and the Ginzburg-Landau energy , Calc. Var. Partial Differential Equations 14 (2002), 151–191. · Zbl 1034.35025
[20] F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices , Comm. Pure Appl. Math. 49 (1996), 323–359. · Zbl 0853.35058
[21] -, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-\(2\) submanifolds , Comm. Pure Appl. Math. 51 (1998), 385–441. · Zbl 0932.35121
[22] J. C. Neu, Vortices in complex scalar fields , Phys. D 43 (1990), 385–406. · Zbl 0711.35024
[23] L. M. Pismen and J. Rubinstein, Motion of vortex lines in the Ginzburg-Landau model , Phys. D 47 (1991), 353–360. · Zbl 0728.35090
[24] Ju. G. RešEtnjak [Yu. G. Reshetnyak], The weak convergence of completely additive vector-valued set functions, Siberian Math. J. 9 (1968), 1039–1045.
[25] J. Rubinstein and P. Sternberg, On the slow motion of vortices in the Ginzburg-Landau heat flow , SIAM J. Math. Anal. 26 (1995), 1452–1466. · Zbl 0838.35102
[26] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau , Comm. Pure Appl. Math. 57 (2004), 1627–1672. · Zbl 1065.49011
[27] -, A product-estimate for Ginzburg-Landau and corollaries , J. Funct. Anal. 211 (2004), 219–244. · Zbl 1063.35144
[28] S. Serfaty, Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow , preprint, 2005, http://www.math.nyu.edu/faculty/serfaty/publis.html H. M. Soner, Ginzburg-Landau equation and motion by mean curvature, I: Convergence; II: Development of the initial interface , J. Geom. Anal. 7 (1997), 437–475.; 477–491. Mathematical Reviews (MathSciNet):
[29] D. Spirn, Vortex dynamics of the full time-dependent Ginzburg-Landau equations , Comm. Pure Appl. Math. 55 (2002), 537–581. · Zbl 1032.35163
[30] M. Struwe, On the evolution of harmonic maps in higher dimensions , J. Differential Geom. 28 (1988), 485–502. · Zbl 0631.58004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.