Boundary problems for the Ginzburg Landau equation

David Chiron

Description

Boundary problems for the Ginzburg-Landau equation David CHIRON Laboratoire Jacques-Louis LIONS, Universite Pierre et Marie Curie Paris VI, 4, place Jussieu BC 187, 75252 Paris, France E-Mail : Abstract We provide a study at the boundary for a class of equation including the Ginzburg- Landau equation as well as the equation of travelling waves for the Gross-Pitaevskii model. We prove Clearing-Out results and an orthogonal anchoring condition of the vortex on the boundary for the Ginzburg-Landau equation with magnetic field. 1 Introduction This paper is devoted to the study at the boundary for the equation for the complex-valued function u in a bounded regular domain ? ? RN , N ≥ 2, i|log ?|~c(x) · ?u = ∆u+ 1?2u(1? |u| 2) ? |log ?|2d(x)u, (1) where ~c : ? ? RN is a bounded lipschitz vector field, d : ? ? R+ is a lipschitz non negative bounded function and ? > 0 is a small parameter. For instance, the Ginzburg-Landau equation with magnetic field (?? i ~A/2)2u = 1?2u(1? |u| 2) (2) is of the type considered.

• ginzburg-landau equation


• mz ≤


• neumann condition


• dirichlet boundary


• boundary


• gross-pitaevskii equation


• clearing-out theorem


• c? ?

Sujets

• Chiron
• Ginzburg?Landau theory
• Boundary
• Gross?Pitaevskii equation

Informations

• Publié par: pefav
• Nombre de lectures: 23
• Langue: English

Extrait

1

Boundary

problems for the Ginzburg-Landau equation

David CHIRON

Laboratoire Jacques-Louis LIONS, UniversitePierreetMarieCurieParisVI, 4, place Jussieu BC 187, 75252 Paris, France E-Mail :[e-mail]

Abstract We provide a study at the boundary for a class of equation including the Ginzburg- Landau equation as well as the equation of travelling waves for the Gross-Pitaevskii model. We prove Clearing-Out results and an orthogonal anchoring condition of the vortex on the boundary for the Ginzburg-Landau equation with magnetic eld.

Introduction

This paper is devoted to the study at the boundary for the equation for the complex-valued functionuin a bounded regular domain R N ,N2, i|logε|~c(x)
ru=
u+ ε 1 2 u(1 |u| 2 ) |logε| 2 d(x)u,(1) wherec~: →R N erotcevztihcspi,lddedlbounisad: →R + is a lipschitz non negative bounded function andε >0 is a small parameter. For instance, the Ginzburg-Landau equation with magnetic eld

(r ~ i/A2) 2 u= ε 1 2 u(1 |u| 2 ) (2) is of the type considered. Another problem that can be written like equation (1) is the equation for the travelling waves for the Gross-Pitaevskii equation. This equation writes i ∂ ∂   t +
 + (1 | | 2 ) = 0,(3) where :RR N →C waves solutions to this equation are solutions of the form. Travelling (possibly rotating the axis)

Equation (3) reads now onU

 (t, x) =U(x 1  Ct, x 2 , . . . , x N ). iC ∂U = ∂x 1
U+U(1 |U| 2 ).

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In dimensionN

3, if the propagation speed is small, it is convenient to perform the scaling u(x) :=U  ε x  , c:= ε|lo C gε|

(in dimensionN= 2, the scaling for the speed isC=ε), and the equation becomes then ic|logε| x∂ u∂ 1 =
u+ ε 1 2 u(1 |u| 2 ) and we expectc This equation is of the type (1) withto be of order one.d0 and~c=c~e 1 . If N= 2, the equation is i ∂ ∂ x u 1 =
u+ ε 1 2 u(1 |u| 2 ), which is also of the considered type withd0 and~c= |logε| . We will be interested in (1) in the asymptoticε→0 with

and we supplement this equation with

eitherthe Dirichlet condition

div~0 c=,

(4)

u=g ε on∂ ,(5) eithergauge and the homogeneous Neumann conditionthe Coulomb ∂ ∂ u n = 0 and~c
n on= 0∂ .(6) Furthermore, we will assume that there exists a constant  0 >0 independent ofεsuch that |~c| 2 L ∞ ( ) +|rc~| 2 L ∞ ( ) +|d| 2 L ∞ ( ) +|rd| 2 L ∞ ( )  0 2 .(7) Finally, we may assume 0< εε 0 ( 0 )1/2 small enough so that ε 1/2 |logε| 2  1 2 .(8)  0 To this problem, is associated the energy E ε (u 2 1 =:) Z |ru| 2 ( + a ε (x 2 )  ε 2 |u| 2 ) 2 = Z e ε (u), where a ε (x) := 1 d(x)ε 2 |logε| 2 .

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1.1 Anchoring condition at the boundary

Our rst result is about the anchoring condition of the vortex on the boundary for the Ginzburg- Landau equation with Neumann condition. Assuming the upper bound

E ε (u)M|logε|, for the functionuwe expect that the energy of, uconcentrates at its vortices, which are curves  in dimensionN We therefore introduce the measure= 3. e ε (u) dx  ε := |logε| , the mass of which is bounded byM Weby hypothesis. may then assume, up to a subsequence, that asε→0,  ε *   weakly as measures. Moreover,wedenetheN 2-dimensional density of 

and the geometrical support of

(x) := lim inf   (B r (x)) r→0 r N 2

:={x∈ ,  (x)>0}.

From Theorem 3 in [BOS], we know that    is closed in  countably ( andN 2)-rectiable. Let us assume that the magnetic eldH=|logε|curl~cobeys the London equation ~  
H+H= 2 .    We may then describe further    in anear the boundary. In this regime of energy, consists  nitenumberofcurvesofnitelength.Therefore,fromLondonequation,weexpectHto be of order one, that is |logε|
 |curl~c|=|H| '1, and thus, since~0∂ , c
n= on |c~| →0 ifε→0.

Our result is concerned with the anchoring of   at the boundary, under the only hypothesis  c~ ε →0 inC 0  as ( ) ε→0.(9) We note that by hypothesis,~c ε is bounded inC 0,1 (   ), thus we may assume for a subsequence that~c ε →cinC 0 ( ).  We then only assumec~= 0. ~ In the case of the Neumann boundary condition (6), we will use the re ection principle. There exists >0 such that the nearest point projection map

: (∂ )  →∂

iswell-denedinthe-neighborhood (∂ )  of∂ and point a smooth bration. Ax∈(∂ )  may therefore be described by the couple (y, t), wherey= (x) is its projection on∂ and

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t=dist(x, ∂ ) =kx (x)k, the sign dene the re ection map

being + ifx andis inside   thenotherwise. We

:W   := c ∩(∂ )  →V:= ∩(∂ )

,

where(x) is the point described by the couple (y, t) ifxis described by (y, t dene the). We ˜ ˜ ˜ ˜ varifoldVbyV:=Vin  andV:= ] VinW, that isVconsists inVunion its re ection with respect to the boundary∂ W.teehcnnoisedtrmahefonildM:=  endowed with the smooth riemannian metricgbyedndeg=g 0 in    andg=  (g 0 ) inW, whereg 0 is the euclidian metric on .

Theorem 1.Assume(4)and(7). Letu ε be a family of solutions of(1)-(6)satisfying the energy bound E ε (u)M|logε| foravectoreldc ε satisfying ~

c~ ε →0inC 0  () asε→0. ˜ Then, the varifoldV(   ,  )is stationary in . Moreover,Vis a stationary varifold in(M, g). Remark 1.In the case where  (locally) the half-plane isR + N =R  + R N 1 , then the theorem ˜ states thatVis a stationary varifold in (locally )R N for the usual metric.

This Theorem says that, in some weak sense, the union of the varifoldVand its symmetric with respect to the boundary is “smooth”, that isVmust meet the boundary∂ orthogonally. SinceVcurve, we may only use a weak formulation of this orthogonality.is not in general a smooth However, ifVis a smooth curve up to the boundary, then Theorem 1 states that, denoting~the tangent unit vector toV, ~ ∂ non . =

The fact that the vortex must meet the boundary orthogonally can be found in the literature. For instance, in [CH], Chapman and Heron considered a domain which is the half-plane (inR 3 ) {z <0} dened , byand a straight line vortexy= 0,x=mz0 for a 0m <+∞, meeting the boundary{z= 0} the London equation and the boundary conditionsat 0. Using for the magnetic eld, they proved, computing the propagation speed of the vortex at 0, that the coecien tm However,must be zero, for otherwise, the propagation speed would be innite. theircomputationdoesnotexcludethecaseoftwovortices,denedbyy= 0,x=mz0 andy= 0,x= mz0, since in that case, the propagation is, due to the symmetry, zero. Our Theorem 1 states that there can not be another possibility involving two such coplanar straightlinesvortices,thatisvorticesdenedbyy= 0,x=mz0 andy= 0,x= m 0 z0 with 0m, m 0 <+∞andm6=m 0  Our Theorem even states that if we havecan not hold. two straight line vortices in the half plane{z <0}meeting at 0, then they must be in a plane orthogonal to{z= 0} the opposite of [CH], our approach is based on equation (1) only,. At whereas the London equation is the limit equation for the current (see (22) below), which is the second equation of the Ginzburg-Landau equation with magnetic eld.

Remark 2.In the case~c ε →c~6= 0 asε→0, by Theorem 3 in [BOS], we know that the varifold Vsatises inside the domain  the curvature equation J  H=?  c~∧( d d ?)  ,(10) 

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