344 research outputs found
Ginzburg-Landau vortex dynamics with pinning and strong applied currents
We study a mixed heat and Schr\"odinger 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 \ep
\to 0, when the intensity of the electric current and applied magnetic field
on the boundary scale like \lep. 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.Comment: 48 pages; v2: minor errors and typos correcte
Vortex density models for superconductivity and superfluidity
We study some functionals that describe the density of vortex lines in
superconductors subject to an applied magnetic field, and in Bose-Einstein
condensates subject to rotational forcing, in quite general domains in 3
dimensions. These functionals are derived from more basic models via
Gamma-convergence, here and in a companion paper. In our main results, we use
these functionals to obtain descriptions of the critical applied magnetic field
(for superconductors) and forcing (for Bose-Einstein), above which ground
states exhibit nontrivial vorticity, as well as a characterization of the
vortex density in terms of a non local vector-valued generalization of the
classical obstacle problem.Comment: 34 page
Convergence of Ginzburg-Landau functionals in 3-d superconductivity
In this paper we consider the asymptotic behavior of the Ginzburg- Landau
model for superconductivity in 3-d, in various energy regimes. We rigorously
derive, through an analysis via {\Gamma}-convergence, a reduced model for the
vortex density, and we deduce a curvature equation for the vortex lines. In a
companion paper, we describe further applications to superconductivity and
superfluidity, such as general expressions for the first critical magnetic
field H_{c1}, and the critical angular velocity of rotating Bose-Einstein
condensates.Comment: 45 page
The Effects of Negative Legacies on the Adjustment of Parentally Bereaved Children and Adolescents
This is a report of a qualitative analysis of a sample of bereaved families in which one parent died and in which children scored in the clinical range on the Child Behavior Check List. The purpose of this analysis was to learn more about the lives of these children. They were considered to be at risk of developing emotional and behavioral problems associated with the death. We discovered that many of these “high risk” children had a continuing bond with the deceased that was primarily negative and troubling for them in contrast to a comparison group of children not at risk from the same study. Five types of legacies, not mutually exclusive, were identified: health related, role related, personal qualities, legacy of blame, and an emotional legacy. Coping behavior on the part of the surviving parent seemed to make a difference in whether or not a legacy was experienced as negative
Analysis of Nematic Liquid Crystals with Disclination Lines
We investigate the structure of nematic liquid crystal thin films described
by the Landau--de Gennes tensor-valued order parameter with Dirichlet boundary
conditions of nonzero degree. We prove that as the elasticity constant goes to
zero a limiting uniaxial texture forms with disclination lines corresponding to
a finite number of defects, all of degree 1/2 or all of degree -1/2. We also
state a result on the limiting behavior of minimizers of the Chern-Simons-Higgs
model without magnetic field that follows from a similar proof.Comment: 40 pages, 1 figur
The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate
We study the Gross-Pitaevskii (GP) energy functional for a fast rotating
Bose-Einstein condensate on the unit disc in two dimensions. Writing the
coupling parameter as 1 / \eps^2 we consider the asymptotic regime \eps
\to 0 with the angular velocity proportional to
(\eps^2|\log\eps|)^{-1} . We prove that if \Omega = \Omega_0
(\eps^2|\log\eps|)^{-1} and then a minimizer of
the GP energy functional has no zeros in an annulus at the boundary of the disc
that contains the bulk of the mass. The vorticity resides in a complementary
`hole' around the center where the density is vanishingly small. Moreover, we
prove a lower bound to the ground state energy that matches, up to small
errors, the upper bound obtained from an optimal giant vortex trial function,
and also that the winding number of a GP minimizer around the disc is in accord
with the phase of this trial function.Comment: 52 pages, PDFLaTex. Minor corrections, sign convention modified. To
be published in Commun. Math. Phy
Ginzburg-Landau model with small pinning domains
We consider a Ginzburg-Landau type energy with a piecewise constant pinning
term in the potential . The function is different from
1 only on finitely many disjoint domains, called the {\it pinning domains}.
These pinning domains model small impurities in a homogeneous superconductor
and shrink to single points in the limit ; here, \v is the inverse of
the Ginzburg-Landau parameter. We study the energy minimization in a smooth
simply connected domain with Dirichlet boundary
condition on \d \O, with topological degree {\rm deg}_{\d \O} (g) = d
>0. Our main result is that, for small \v, minimizers have distinct
zeros (vortices) which are inside the pinning domains and they have a degree
equal to 1. The question of finding the locations of the pinning domains with
vortices is reduced to a discrete minimization problem for a finite-dimensional
functional of renormalized energy. We also find the position of the vortices
inside the pinning domains and show that, asymptotically, this position is
determined by {\it local renormalized energy} which does not depend on the
external boundary conditions.Comment: 39 page
Critical Rotational Speeds for Superfluids in Homogeneous Traps
We present an asymptotic analysis of the effects of rapid rotation on the
ground state properties of a superfluid confined in a two-dimensional trap. The
trapping potential is assumed to be radial and homogeneous of degree larger
than two in addition to a quadratic term. Three critical rotational velocities
are identified, marking respectively the first appearance of vortices, the
creation of a `hole' of low density within a vortex lattice, and the emergence
of a giant vortex state free of vortices in the bulk. These phenomena have
previously been established rigorously for a `flat' trap with fixed boundary
but the `soft' traps considered in the present paper exhibit some significant
differences, in particular the giant vortex regime, that necessitate a new
approach. These differences concern both the shape of the bulk profile and the
size of vortices relative to the width of the annulus where the bulk of the
superfluid resides. Close to the giant vortex transition the profile is of
Thomas-Fermi type in `flat' traps, whereas it is gaussian for soft traps, and
the `last' vortices to survive in the bulk before the giant vortex transition
are small relative to the width of the annulus in the former case but of
comparable size in the latter.Comment: To appear in J. Math. Phys, published versio
Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction
We study a singular-limit problem arising in the modelling of chemical
reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck
convection-diffusion equation with a double-well convection potential. This
potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the
solution concentrates onto the two wells, resulting into a limiting system that
is a pair of ordinary differential equations for the density at the two wells.
This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM
Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear
structure of the equation. In this paper we re-prove the result by using solely
the Wasserstein gradient-flow structure of the system. In particular we make no
use of the linearity, nor of the fact that it is a second-order system. The
first key step in this approach is a reformulation of the equation as the
minimization of an action functional that captures the property of being a
curve of maximal slope in an integrated form. The second important step is a
rescaling of space. Using only the Wasserstein gradient-flow structure, we
prove that the sequence of rescaled solutions is pre-compact in an appropriate
topology. We then prove a Gamma-convergence result for the functional in this
topology, and we identify the limiting functional and the differential equation
that it represents. A consequence of these results is that solutions of the
{\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference
Mappings of least Dirichlet energy and their Hopf differentials
The paper is concerned with mappings between planar domains having least
Dirichlet energy. The existence and uniqueness (up to a conformal change of
variables in the domain) of the energy-minimal mappings is established within
the class of strong limits of homeomorphisms in the
Sobolev space , a result of considerable interest in the
mathematical models of Nonlinear Elasticity. The inner variation leads to the
Hopf differential and its trajectories.
For a pair of doubly connected domains, in which has finite conformal
modulus, we establish the following principle:
A mapping is energy-minimal if and only if
its Hopf-differential is analytic in and real along the boundary of .
In general, the energy-minimal mappings may not be injective, in which case
one observes the occurrence of cracks in . Nevertheless, cracks are
triggered only by the points in the boundary of where fails to be
convex. The general law of formation of cracks reads as follows:
Cracks propagate along vertical trajectories of the Hopf differential from
the boundary of toward the interior of where they eventually terminate
before making a crosscut.Comment: 51 pages, 4 figure
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