On the dynamical law of the Ginzburg-Landau vortices on the plane

Fang-Hua Lin, J. X. Xin

Research output: Contribution to journalArticle

Abstract

We study the Ginzburg-Landau equation on the plane with initial data being the product of n well-separated +1 vortices and spatially decaying perturbations. If the separation distances are O(ε-1), ε ≪ 1, we prove that the n vortices do not move on the time scale O(ε-2λε), λε = o(log 1/ε); instead, they move on the time scale O(e-2 log 1/ε) according to the law ẋj = -∇xjW, W = -Σl≠jlog|xl - Xj|, xj = (ξj, ηj) ∈ ℝ2, the location of the jth vortex. The main ingredients of our proof consist of estimating the large space behavior of solutions, a monotonicity inequality for the energy density of solutions, and energy comparisons. Combining these, we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law.

Original languageEnglish (US)
Pages (from-to)1189-1212
Number of pages24
JournalCommunications on Pure and Applied Mathematics
Volume52
Issue number10
StatePublished - Oct 1999

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Ginzburg-Landau
Vortex
Vortex flow
Time Scales
Ginzburg-Landau Equation
Behavior of Solutions
Energy
Energy Density
Monotonicity
Perturbation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

On the dynamical law of the Ginzburg-Landau vortices on the plane. / Lin, Fang-Hua; Xin, J. X.

In: Communications on Pure and Applied Mathematics, Vol. 52, No. 10, 10.1999, p. 1189-1212.

Research output: Contribution to journalArticle

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