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Boundary problems for the Ginzburg Landau equation

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46 pages
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? ?


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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 :chiron@ann.jussieu.fr
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 RN,N2, i|logε|~c(x) ru= u+ε12u(1 |u|2) |logε|2d(x)u,(1) wherec~: RNe 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)2u=ε12u(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 :RRNC waves solutions to this equation are solutions of the form. Travelling (possibly rotating the axis)
Equation (3) reads now onU
 (t, x) =U(x1 Ct, x2, . . . , xN). iCU= ∂x1U+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) :=Uεx, c:=ε|loCgε|
(in dimensionN= 2, the scaling for the speed isC=ε), and the equation becomes then ic|logε|xu1= u+ε12u(1 |u|2) and we expectc This equation is of the type (1) withto be of order one.d0 and~c=c~e1. If N= 2, the equation is ixu1= u+ε12u(1 |u|2), which is also of the considered type withd0 and~c=|logε|. We will be interested in (1) in the asymptoticε0 with
and we supplement this equation with
eitherthe Dirichlet condition
div~0 c=,
(4)
u=gεon ,(5) eithergauge and the homogeneous Neumann conditionthe Coulomb un= 0 and~cn on= 0 .(6) Furthermore, we will assume that there exists a constant 0>0 independent ofεsuch that |~c|2L( )+|rc~|2L( )+|d|2L( )+|rd|2L( )02.(7) Finally, we may assume 0< εε0(0)1/2 small enough so that ε1/2|logε|212.(8) 0 To this problem, is associated the energy Eε(u21=:)Z|ru|2(+aε(x2) ε2|u|2)2=Zeε(u), where aε(x) := 1 d(x)ε2|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
Eε(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. eε(u)dx ε:=|logε|, the mass of which is bounded byM Weby hypothesis. may then assume, up to a subsequence, that asε0, ε* weakly as measures. Moreover,wede netheN 2-dimensional density of
and the geometrical support of
(x) := lim inf(Br(x)) r0rN 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 , cn= on |c~| →0 ifε0.
Our result is concerned with the anchoring of at the boundary, under the only hypothesis  c~ε0 inC0 as( )ε0.(9) We note that by hypothesis,~cεis bounded inC0,1(  ), thus we may assume for a subsequence that~cεcinC0( ). 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=g0in   andg=(g0) inW, whereg0is the euclidian metric on .
Theorem 1.Assume(4)and(7). Letuεbe a family of solutions of(1)-(6)satisfying the energy bound Eε(u)M|logε| foravector eldcεsatisfying ~
c~ε0inC0() asε0. ˜ 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+RN 1, then the theorem ˜ states thatVis a stationary varifold in (locally )RNfor 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 (inR3) {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= m0z0 with 0m, m0<+andm6=m0 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~cεc~6= 0 asε0, by Theorem 3 in [BOS], we know that the varifold Vsatis es inside the domain  the curvature equation J  H=?c~(dd?),(10) 
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