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Ginzburg‐Landau vortex dynamics driven by an applied boundary current
Author(s) -
Tice Ian
Publication year - 2010
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20328
Subject(s) - vortex , bounded function , mathematics , current (fluid) , boundary (topology) , dynamics (music) , lorentz force , relaxation (psychology) , limit (mathematics) , domain (mathematical analysis) , lorentz transformation , scale (ratio) , statistical physics , classical mechanics , physics , mathematical analysis , mechanics , magnetic field , quantum mechanics , psychology , social psychology , acoustics , thermodynamics
Abstract In this paper we study the time‐dependent Ginzburg‐Landau equations on a smooth, bounded domain Ω ⊂ ℝ 2 , subject to an electrical current applied on the boundary. The dynamics with an applied current are nondissipative, but via the identification of a special structure in an interaction energy, we are able to derive a precise upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit ε → 0. We first consider the original time scale in which the vortices do not move and the solutions undergo a “phase relaxation.” Then we study an accelerated time scale in which the vortices move according to a derived dynamical law. In the dynamical law, we identify a novel Lorentz force term induced by the applied boundary current. © 2010 Wiley Periodicals, Inc.