Stability of the vortex in micromagnetics and related models

Communications on Pure and Applied Analysis, 2009

We study vortex nucleation for minimizers of a Ginzburg-Landau energy with discontinuous constraint. For applied magnetic fields comparable with the first critical field of vortex nucleation, we determine the limiting vorticities.

Asymptotic Analysis, 2015

We consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. The applied magnetic field varies smoothly and is allowed to vanish non-degenerately along a curve. Assuming that the strength of the applied magnetic field varies between two characteristic scales, and the Ginzburg-Landau parameter tends to +∞, we determine an accurate asymptotic formula for the minimizing energy and show that the energy minimizers have vortices. The new aspect in the presence of a variable magnetic field is that the density of vortices in the sample is not uniform.

Communications on Pure and Applied Mathematics, 2006

We consider the two-dimensional Ginzburg-Landau model with magnetic field for a superconductor with a multiply connected cross section. We study energy minimizers in the London limit as the Ginzburg-Landau parameter κ = 1/ǫ → ∞ to determine the number and asymptotic location of vortices. We show that the holes act as pinning sites, acquiring nonzero winding for bounded fields and attracting all vortices away from the interior for fields up to a critical value h ex = O(|ln ǫ|). At the critical level the pinning effect breaks down, and vortices appear in the interior of the superconductor at locations that we identify explicitly in terms of the solutions of an elliptic boundary value problem. The method involves sharp upper and lower energy estimates, and a careful analysis of the limiting problem that captures the interaction between the vortices and the holes.

Journal d'Analyse Mathématique, 2011

Eprint Arxiv 0906 4862, 2009

We study the variational convergence of a family of two-dimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity, as the Ginzburg-Landau parameter epsilon tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that, suitably normalized, the energy functionals Gamma-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.

Communications in Mathematical Physics, 2004

We establish the existence of locally minimizing vortex solutions to the full Ginzburg-Landau energy in three dimensional simply-connected domains with or without the presence of an applied magnetic field. The approach is based upon the theory of weak Jacobians and applies to nonconvex sample geometries for which there exists a configuration of locally shortest line segments with endpoints on the boundary.

2000

We study the linearized stability of n-vortex (n ∈ Z) solutions of the magnetic Ginzburg-Landau (or Abelian Higgs) equations. We prove that the fundamental vortices (n = ±1) are stable for all values of the coupling constant, λ, and we prove that the higher-degree vortices (|n| ≥ 2) are stable for λ < 1, and unstable for λ > 1. This resolves a long-standing conjecture (see, eg, [JT]).

Journal of Nonlinear Science, 2005

We study a gradient-flow version of the Ginzburg-Landau equations with an addition of a compactly supported potential term. We consider initial data close to a magnetic vortex solution of the Ginzburg-Landau equations and find the dynamical law governing the motion of the vortex center in the presence of the potential.

1999

We study the linearized stability of n-vortex solutions of the magnetic Ginzburg-Landau (or Abelian-Higgs) equations. We prove that the fundamental vortices (n=1,-1) are stable for all values of the coupling constant, k, and we prove that the higher-degree vortices (|n| > 1) are stable for k 1. This resolves a long-standing conjecture.

2001

We study the linearized stability of n-vortex (n ∈ Z) solutions of the magnetic Ginzburg-Landau (or Abelian Higgs) equations. We prove that the fundamental vortices (n = ±1) are stable for all values of the coupling constant, λ, and we prove that the higher-degree vortices (|n| ≥ 2) are stable for λ < 1, and unstable for λ > 1. This resolves a long-standing conjecture (see, eg, [JT]).