Ginzburg-Landau Vortex Dynamics with Pinning and Strong Applied Currents

Sylvia Serfaty, Ian Tice

Research output: Contribution to journalArticlepeer-review


We study a mixed heat and Schrödinger Ginzburg-Landau evolution equation on a bounded two-dimensional domain with an electric current applied on the boundary and a pinning potential term. This is meant to model a superconductor subjected to an applied electric current and electromagnetic field and containing impurities. Such a current is expected to set the vortices in motion, while the pinning term drives them toward minima of the pinning potential and "pins" them there. We derive the limiting dynamics of a finite number of vortices in the limit of a large Ginzburg-Landau parameter, or e{open} → 0, when the intensity of the electric current and applied magnetic field on the boundary scale like {pipe}log e{open}{pipe}. We show that the limiting velocity of the vortices is the sum of a Lorentz force, due to the current, and a pinning force. We state an analogous result for a model Ginzburg-Landau equation without magnetic field but with forcing terms. Our proof provides a unified approach to various proofs of dynamics of Ginzburg-Landau vortices.

Original languageEnglish (US)
Pages (from-to)413-464
Number of pages52
JournalArchive for Rational Mechanics and Analysis
Issue number2
StatePublished - Aug 2011

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering


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