arXiv subjects

Robert L. Jerrard

Publications and source records attributed to Robert L. Jerrard.

28 records · Page 2Linked to original sources

arXiv2011

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.

math-ph↗

arXiv2011

On the motion of a curve by its binormal curvature

We propose a weak formulation for the binormal curvature flow of curves in $\R^3.$ This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence theorem in that framework.

math.DG↗

arXiv2011

On Schrödinger maps from $T^1$ to $S^2$

We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes its quotient by the continuous group action of $T^1$ given by translations. We also prove that without taking this quotient, for any $t>0$ the flow map at time $t$ is discontinuous as a map from $\mathcal{C}^\infty(T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to the space of distributions $(\mathcal{C}^\infty(T^1,\R^3))^*.$

math.AP↗

arXiv2011

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 Γ-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.

math-ph↗

arXiv2010

Non existence of vortices in the small density region of a condensate

In this paper, we answer a question raised by Len Pitaevskii and prove that the ground state of the Gross-Pitaevskii energy describing a Bose Einstein condensate at low rotation does not have vortices in the low density region. Therefore, the first ground state with vortices has its vortices in the bulk. This is obtained by proving that for small rotational velocities, the ground state is multiple of the ground state with zero rotation. We rely on sharp bounds of the decay of the wave function combined with weighted jacobian estimates.

math-ph↗

arXiv2009

Defects in semilinear wave equations and timelike minimal surfaces in Minkowski space

We study semilinear wave equations with Ginzburg-Landau type nonlinearities multiplied by a factor $ε^{-2}$, where $ε>0$ is a small parameter. We prove that for suitable initial data, solutions exhibit energy concentration sets that evolve approximately via the equation for timelike Minkowski minimal surfaces, as long as the minimal surface remains smooth. This gives a proof of predictions made, on the basis of formal asymptotics and other heuristic arguments, by cosmologists studying cosmic strings and domain walls, as well as by applied mathematicians.

math.AP↗

arXiv2008

On multiwell Liouville theorems in higher dimension

We consider certain subsets of the space of $n\times n$ matrices of the form $K = \cup_{i=1}^m SO(n)A_i$, and we prove that for $p>1, q \geq 1$ and for connected $Ω'\subset\subsetΩ\subset \R^n$, there exists positive constant $a<1$ depending on $n,p,q, Ω, Ω'$ such that for $ \veps=\| {dist}(Du, K)\|_{L^p(Ω)}^p$ we have $\inf_{R\in K}\|Du-R\|^p_{L^p(Ω')}\leq M\veps^{1/p}$ provided $u$ satisfies the inequality $\| D^2 u\|_{L^q(Ω)}^q\leq a\veps^{1-q}$. Our main result holds whenever $m=2$, and also for {\em generic} $m\le n$ in every dimension $n\ge 3$, as long as the wells $SO(n)A_1,..., SO(n)A_m$ satisfy a certain connectivity condition. These conclusions are mostly known when $n=2$, and they are new for $n\ge 3$.

math.CA↗

arXiv2007

Nonclassical rotational inertia for a supersolid under rotation

As proposed by Leggett [4], the supersolidity of a crystal is characterized by the Non Classical Rotational Inertia (NCRI) property. Using a model of quantum crystal introduced by Josserand, Pomeau and Rica [5], we prove that NCRI occurs. This is done by analyzing the ground state of the aforementioned model, which is related to a sphere packing problem, and then deriving a theoretical formula for the inertia momentum. We infer a lower estimate for the NCRI fraction, which is a landmark of supersolidity.

cond-mat.other↗

arXiv2002

On the shape of vortices for a rotating Bose Einstein condensate

For a Bose-Einstein condensate placed in a rotating trap, we study the simplified energy of a vortex line derived in Aftalion-Riviere Phys. Rev. A 64, 043611 (2001) in order to determine the shape of the vortex line according to the rotational velocity and the elongation of the condensate. The energy reflects the competition between the length of the vortex which needs to be minimized taking into account the anisotropy of the trap and the rotation term which pushes the vortex along the z axis. We prove that if the condensate has the shape of a pancake, the vortex stays straight along the z axis while in the case of a cigar, the vortex is bent.

cond-mat.mes-hall↗

arXiv1997

Vortex Dynamics for the Ginzburg-Landau-Schrödinger Equation

The initial value problem for the Ginzburg-Landau-Schrödinger equation is examined in the $ε\rightarrow 0$ limit under two main assumptions on the initial data $ϕ^ε$. The first assumption is that $ϕ^ε$ exhibits $m$ distinct vortices of degree $\pm 1$; these are described as points of concentration of the Jacobian $[Jϕ^ε]$ of $ϕ^ε$. Second, we assume energy bounds consistent with vortices at the points of concentration. Under these assumptions, we identify ``vortex structures'' in the $ε\rightarrow 0$ limit of $ϕ^ε$ and show that these structures persist in the solution $u^ε(t)$ of $GLS_ε$. We derive ordinary differential equations which govern the motion of the vortices in the $ε\rightarrow 0$ limit. The limiting system of ordinary differential equations is a Hamitonian flow governed by the renormalized energy of Bethuel, Brezis and Hélein. Our arguments rely on results about the structural stability of vortices which are proved in a separate paper.

math-ph↗