Lin, Fang Hua Solutions of Ginzburg-Landau equations and critical points of the renormalized energy. (English) Zbl 0845.35052 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 12, No. 5, 599-622 (1995). The motion of vortices of solutions of the initial boundary value problem of the Ginzburg-Landau equation is the main concern of this paper, and the question is studied successfully. There are connections to the complete characterization of asymptotic behavior for vortices (governed by a certain energy functional), given in the book of F. Bethuel, H. Brezis and F. Hélein [Ginzburg-Landau vortices, Birkhäuser, Boston (1994; Zbl 0802.35142)]. Reviewer: A.Göpfert (Halle) Cited in 28 Documents MSC: 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs Keywords:motion of vortices; Ginzburg-Landau equation PDF BibTeX XML Cite \textit{F. H. Lin}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 12, No. 5, 599--622 (1995; Zbl 0845.35052) Full Text: DOI Numdam EuDML References: [1] Bethuel, F.; Brezis, H.; Helein, F., Ginzburg-Landau vertices, (1994), Birkhaüser Boston [2] Bethuel, F.; Brezis, H.; Helein, F., Asymptotics for the minimization of a Ginzburg-Landau functional, Cal. variations and P.D.E., Vol. 1#2, 123-148, (1993) · Zbl 0834.35014 [3] H. Brezis, F. Merle and T. Riviere, Quantization effects, for −Δu = u(1 − (u)^{2}) in ℝ^{2}, preprint. · Zbl 0809.35019 [4] Chen, Y. M.; Lin, F. H., Evaluation of harmonic maps with the Dirichlet boundary condition, Comm. in Analysis and Geometry, Vol. 1#3, 327-346, (1993) [5] Chen, Y. M.; Struwe, M., Existence and partial regularity for heat flow for harmonic maps, Math. Z, Vol. 201, 83-103, (1989) · Zbl 0652.58024 [6] R. Hardt and F. H. Lin, Singularities for p-energy minimizing unit vector fields on planar domains, to appear in Cal. variation and P. D. E. · Zbl 0828.58008 [7] E. Weinan, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, preprint. · Zbl 0814.34039 [8] O. A. Ladyzenskaya, N. A. Solonnikov and N. N. Uralseva, Linear and quasilinear equations of parabolic type, Translations of AMS, Mon. #23 (196X). [9] Neu, J., Vortex dynamics of complex scalar fields, Physics D, Vol. 43, 384-406, (1990) [10] Pismen, L.; Rubinstein, J., (Coron, J. M.; etal., Dynamics of defects, in nematics, mathematical and physical aspects, (1991), Kluwer Pubs) · Zbl 0850.76045 [11] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conf. Board of the Math. Sci., by AMS #65. · Zbl 0609.58002 [12] J. Rubinstein and P. Sternberg, On the Slow Motion of Vortices in the Ginzburg-Landau heat flow, preprint. · Zbl 0838.35102 [13] Simon, L., Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals of Math, Vol. 118, 527-571, (1983) · Zbl 0549.35071 [14] Struwe, M., On the asymptotic behavior of minimizers of the Ginsburg-Landau model in 2 dimensions, J. Diff. Int. Eqs, Vol. 7, (1994) [15] Struwe, M., On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv., Vol. 60, 558-581, (1985) · Zbl 0595.58013 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.