Ginzburg‐Landau vortex dynamics driven by an applied boundary current

Tice, Ian

doi:10.1002/cpa.20328

Cited by 15 publications

(21 citation statements)

References 32 publications

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“…Instead of considering the boundary conditions (1.1e,f,g), it is possible to use an equivalent boundary condition where we prescribe instead the magnetic field. As in [38] we note that by (1.1b,e,f), on each point on ∂Ω, except for the corners, we have…”

Section: Equivalent Boundary Conditionsmentioning

confidence: 88%

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Global Stability of the Normal State of Superconductors in the Presence of a Strong Electric Current

Almog

Helffer

2014

Commun. Math. Phys.

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We consider the time-dependent Ginzburg-Landau model of superconductivity in the presence of an electric current flowing through a two-dimensional wire. We show that when the current is sufficiently strong the solution converges in the long-time limit to the normal state. We provide two types of upper bounds for the critical current where such global stability is achieved: by using the principal eigenvalue of the magnetic Laplacian associated with the normal magnetic field, and through the norm of the resolvent of the linearized steady-state operator. In the latter case we estimate the resolvent norm in large domains by the norms of approximate operators defined on the plane and the half-plane. We also obtain a lower bound, in large domains, for the above critical current by obtaining the current for which the normal state looses its local stability.1991 Mathematics Subject Classification. 82D55, 35B25, 35B40, 35Q55.

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Section: Equivalent Boundary Conditionsmentioning

confidence: 88%

“…This result suggests, once again, that stability is being forced not only by the magnetic field that the current induces, but also by the potential term in (1.1a). We conclude the foregoing literature survey by mentioning a few works considering the motion of vortices under the action of an electric current [38,37,32].…”

Section: Introductionmentioning

confidence: 90%

Global Stability of the Normal State of Superconductors in the Presence of a Strong Electric Current

Almog

Helffer

2014

Commun. Math. Phys.

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“…This is due to the strength of the forcing terms (now in |log ε| instead of constant) and pinning terms, and it is a significant difference: while the η can be taken small enough (< π inf b) so that no additional vortex can appear, there can still be an energy of the same order as the vortex energy floating around. The usual proofs of dynamics mentioned above [Li1,Li2,JS,Sp,Ti,Mi,KMMS] use the fact that from a priori bounds, the energy density 1 |log ε| e ε (u ε ) can only concentrate at the vortex locations and converges in the sense of measures to π δ a i (t) where the a i (t) are the vortex centers. Here we cannot use this and have to accept the possibility of another term in the limiting energy measure, which is not necessarily concentrated at points, and with which we work until we eventually can prove it is zero by a Gronwall argument.…”

Section: Methods Of the Proof And Case Of The Ginzburg-landau Equationmentioning

confidence: 99%

“…This is the main force driving the dynamics (possibly supplemented with the forcing of Z) in the dynamics derived in [Ti]. It is in particular this computation in finite parts that makes the proofs [Li1,Li2,JS,Sp,Ti,Mi,KMMS] delicate.…”

Section: Methods Of the Proof And Case Of The Ginzburg-landau Equationmentioning

confidence: 99%

Ginzburg–Landau Vortex Dynamics with Pinning and Strong Applied Currents

Serfaty

Tice

2011

Arch Rational Mech Anal

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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 ε → 0, when the intensity of the electric current and applied magnetic field on the boundary scale like |log ε|. 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.Here Ω is the bounded two-dimensional domain representing the region occupied by the superconducting sample. The unknown functions are the triple (u, A, Φ), where u : Ω → C is the "order parameter," A : Ω → R 2 is a vector potential of the magnetic field, itself given by h := curl A, and Φ : Ω → R is the scalar potential associated to the electric field, itself given by E := −(∂ t A + ∇Φ). This is a gauge theory, i.e. (u, A, Φ) are only known up to gauge-transformations of the form uFor a vector X ∈ R 2 we write X ⊥ = (−X 2 , X 1 ), and for the perpendicular gradient we write ∇ ⊥ h = (∇h) ⊥ . Also, (·, ·) denotes the scalar product in C defined by (a, b) = R(a)R(b) + I(a)I(b), and (a, X) for a ∈ C and X ∈ C 2 stands for the vector in R 2 with components (a, X 1 ) and (a, X 2 ).The function b(x) is interpreted as a pinning potential. We assume that b :Ω → R is a smooth function satisfying( 1.2)The situation without pinning corresponds to the case b ≡ 1. Let us explain the meaning of the various parameters in the equation. We assume α > 0, σ > 0, and β ∈ R. When β = 0 and b ≡ 1, these equations are the Gorkov-Eliashberg system (see [GE]), which are the standard gauge-invariant heat flow version of the Ginzburg-Landau equation. The case α = 0, β > 0 would correspond to a pure gauge-invariant Schrödinger flow. Here, for the sake of generality, we consider α > 0 and β real, which corresponds to a mixed flow or "complex Ginzburg-Landau," also commonly considered in the modeling of superconductivity [Do, KIK].The parameter ε > 0 is a small parameter, equal to the inverse of κ, the "Ginzburg-Landau parameter" in superconductivity, which is a material constant defined as the ratio between two characteristic length scales. We will be interested in the asymptotic limit ε → 0, corresponding to "extreme type-II superconductors." The parameter σ is called the conductivity. Note that the parameters α, β, σ as well as the function b(x) are assumed to be independent of ε.The boundary conditions are what make th...

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“…As formally predicted by Chapman and Richardson [15], and first completely proven by [31,49] (see also [29,34] for the conservative case), in the asymptotic regime of a large Ginzburg-Landau parameter, this non-uniform density a translates at the level of isolated vortices into an effective "pinning potential" h = log a, indeed attracting the vortices to the minima of a. As shown in our companion paper [24], the mean-field equations (1.6)-(1.7) are then replaced by (1.1)-(1.2), where the forcing Ψ can be decomposed as Ψ := F ⊥ − ∇ ⊥ h in terms of the pinning force −∇h and of some vector field F : R 2 → R 2 related to the imposed electric current (see also [51,49]). In the conservative regime α = 0, β = 1, the incompressible model (1.1) takes the form of the following inhomogeneous version of the 2D Euler equation: using the identity v ⊥ curl v = (v · ∇)v − 1 2 ∇|v| 2 , and settingP := P − 1 2 |v| 2 ,…”

Section: Brief Discussion Of the Modelmentioning

confidence: 99%

Well-posedness for mean-field evolutions arising in superconductivity

Duerinckx¹,

Fischer²

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We establish the existence of a global solution for a new family of fluid-like equations, which are obtained in certain regimes in [24] as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.

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Ginzburg‐Landau vortex dynamics driven by an applied boundary current

Cited by 15 publications

References 32 publications

Global Stability of the Normal State of Superconductors in the Presence of a Strong Electric Current

Global Stability of the Normal State of Superconductors in the Presence of a Strong Electric Current

Ginzburg–Landau Vortex Dynamics with Pinning and Strong Applied Currents

Well-posedness for mean-field evolutions arising in superconductivity

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