Convergence of the parabolic ginzburg-landau equation to motion by mean curvature

For the complex parabolic Ginzburg-Landau equation, we prove that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke?s weak formulation. The only assumption is a natural energy bound on the initial data. In some cases, we also prove convergence to enhanced motion in the sense of Ilmanen.

[1]  Tom Ilmanen,et al.  Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature , 1993 .

[2]  Halil Mete Soner,et al.  Dynamics of Ginzburg‐Landau Vortices , 1998 .

[3]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[4]  G. Huisken.  Flow by mean curvature of convex surfaces into spheres , 1984 .

[5]  Thierry Gallouët,et al.  Nonlinear Schrödinger evolution equations , 1980 .

[6]  Giovanni Alberti,et al.  Variational convergence for functionals of Ginzburg-Landau type. , 2005 .

[7]  Fanghua Lin,et al.  Some Dynamical Properties of Ginzburg-Landau Vortices , 1996 .

[8]  William K. Allard,et al.  On the first variation of a varifold , 1972 .

[9]  Xinfu Chen,et al.  Generation and propagation of interfaces for reaction-diffusion equations , 1992 .

[10]  Y. Giga,et al.  Asymptotically self‐similar blow‐up of semilinear heat equations , 1985 .

[11]  W. Ziemer.  Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation , 1989 .

[12]  F. Almgren,et al.  Optimal isoperimetric inequalities , 1985 .

[13]  H. Fédérer.  Geometric Measure Theory , 1969 .

[14]  H. Soner.  Ginzburg-Landau equation and motion by mean curvature, II: Development of the initial interface , 1993 .

[15]  Richard Schoen,et al.  Analytic Aspects of the Harmonic Map Problem , 1984 .

[16]  F. Lin.  Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds , 1998 .

[17]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[18]  F. Lin,et al.  Harmonic and quasi-harmonic spheres, part III. Rectifiablity of the parabolic defect measure and generalized varifold flows , 2002 .

[19]  Giandomenico Orlandi,et al.  Asymptotics for the Ginzburg–Landau Equation in Arbitrary Dimensions , 2001 .

[20]  M. Schatzman,et al.  Development of interfaces in ℝN , 1990, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[21]  Patricia Bauman,et al.  Vortex annihilation in nonlinear heat flow for Ginzburg–Landau systems , 1995, European Journal of Applied Mathematics.

[22]  F. Béthuel,et al.  Uniform estimates for the parabolic Ginzburg–Landau equation , 2002 .

[23]  L. Bronsard,et al.  Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics , 1991 .

[24]  Michael Struwe,et al.  Existence and partial regularity results for the heat flow for harmonic maps , 1989 .

[25]  J. Bourgain,et al.  On the structure of the Sobolev space H1/2 with values into the circle , 2000 .

[26]  Mariano Giaquinta,et al.  Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105 , 1984 .

[27]  David Preiss,et al.  Geometry of measures in $\mathbf{R}^n$: Distribution, rectifiability, and densities , 1987 .

[28]  Leon Simon,et al.  Lectures on Geometric Measure Theory , 1984 .

[29]  Haim Brezis,et al.  A note on limiting cases of sobolev embeddings and convolution inequalities , 1980 .

[30]  P. Souganidis,et al.  Phase Transitions and Generalized Motion by Mean Curvature , 1992 .

[31]  J. Bourgain,et al.  W1,p estimates for solutions to the Ginzburg–Landau equation with boundary data in H1/2 , 2001 .

[32]  Michael Struwe,et al.  On the evolution of harmonic maps in higher dimensions , 1988 .

[33]  Changyou Wang.  On moving Ginzburg-Landau filament vortices , 2002 .

[34]  Halil Mete Soner,et al.  A measure theoretic approach to higher codimension mean curvature flows , 1997 .

[35]  Michael Struwe,et al.  Geometric evolution problems , 1995 .

[36]  Giandomenico Orlandi,et al.  Vortex rings for the Gross-Pitaevskii equation , 2004 .

[37]  Kenneth A. Brakke,et al.  The motion of a surface by its mean curvature , 2015 .

[38]  F. Béthuel,et al.  Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics , 2005 .

[39]  Tristan Rivière,et al.  Line vortices in the U(1) Higgs model , 1996 .

[40]  Jacob Rubinstein,et al.  Motion of Vortex lines in the Ginzburg-Landau model , 1991 .

[41]  F. Lin,et al.  Harmonic and quasi-harmonic spheres , 1999 .