Dynamics of the Ginzburg-Landau equations of superconductivity

Abstract

Abstract. This article is concerned with the dynamical properties of solutions of the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity. It is shown that the TDGL equations define a dynamical process when the applied magnetic field varies with time, and a dynamical system when the applied magnetic field is stationary. The dynamical system describes the large-time asymptotic behavior: Every solution of the TDGL equations is attracted to a set of stationary solutions, which are .divergence free. These results are obtained in the ''4 = -w(V A)" gauge, which reduces to the standard %qj = -V -A" gauge if w = 1 and to the zero-electric potential gauge if w = 0; the treatment captures both in a unified framework. This gauge forces the London gauge, V -A = 0, for any stationary solution of the TDGL equations.

The frozen-field approximation and the Ginzburg-Landau equations of superconductivity

“…This choice is realized by identifying the gauge χ with a solution of the linear parabolic equation 10) subject to the condition n · ∇χ = −n · A on the boundary. In [9], it was shown that the TDGL equations, subject to the constraint (2.9), define a dynamical system under suitable regularity conditions on H. (In the more general case, where H varies not only in space but also in time, the TDGL equations define a dynamical process.) This dynamical system has a global attractor, which consists of the stationary points of the dynamical system and the heteroclinic orbits connecting such stationary points.…”

“…see [9]. The space W 1+α,2 (Ω) is continuously imbedded in W 1,2 (Ω) ∩ L ∞ (Ω), so ψ and A are bounded and differentiable with square-integrable (generalized) derivatives.…”

“…A good introduction to the mathematics of the GL equations is [6]. The dynamics of the GL equations have been studied by several authors; see [7,8,9] and the references cited therein. The present investigation is closely related to the work of Du and Gray [10].…”

Asymptotic behaviour of time-dependent Ginzburg-Landau equations of superconductivity

“…In di!erent gauge choices, that is, for di!erent choices of the gauge function , the existence, uniqueness and regularity of solutions for TDGL equations have been studied by many authors, such as Chen et al [2], Du [3], Tang [15], Liang and Tang [12], Tang and Wang [16], TakaH c\ [14], Fleckinger-PelleH et al [6], Wang and Wang [19], Wang and Zhan [20].…”

“…In [6], the generalized Lorentz gauge, for which, "! div A with *0, was used by Fleckinger-PelleH et al Their results can be summarized as follows.…”