Long-Time Stabilization of Solutions to the Ginzburg-Landau Equations of Superconductivity

Monatsh. Math. 133, 197±221 (2001)

Long-Time Stabilization of Solutions to the Ginzburg±LandauEquations of Superconductivity

By

Eduard Feireisl1;� and Peter TakaÂcÏ2;y1Academy of Sciences of the Czech Republic, Praha, Czech Republic

2 UniversitaÈt Rostock, Germany

(Received 30 June 2000; in revised form 30 December 2000)

Abstract. The long-time dynamical properties of solutions ��;A� to the time-dependent Ginzburg±Landau (TDGL) equations of superconductivity are investigated. The applied magnetic ®eld varieswith time, but it is assumed to approach a long-time asymptotic limit. Suf®cient conditions (in terms ofthe time rate of change of the applied magnetic ®eld) are given which guarantee that the dynamicalprocess de®ned by the TDGL equations is asymptotically autonomous, i.e., it approaches a dynamicalsystem as time goes to in®nity. Analyticity of an energy functional is used to show that every solutionof the TDGL equations asymptotically approaches a (single) stationary solution of the (time-independent) Ginzburg±Landau equations. The standard `̀ � � ÿr � A'' gauge is chosen.

2000 Mathematics Subject Classi®cation: 35K55, 82D55, 35B40, 47H12Key words: Ginzburg±Landau equations, superconductivity, asymptotically autonomous dynami-

cal process, gauge, global attractor, stabilization

1. Introduction

The time-dependent Ginzburg±Landau (TDGL) equations of superconductivitydescribe the dynamical properties of a superconductor near the critical temperatureTc. Continuing discoveries of various kinds of materials that are superconductingbelow a relatively high critical temperature (the so-called high-Tc superconductors)justify recent investigations of qualitative properties of the TDGL equations, cf.Tinkham [31, Chap. 9]. Although certain properties of stationary solutions (such asthe Abrikosov vortex lattice [1] or similar vortex patterns) are the most importanttopics of current research, the time-dependent formation of a vortex patternbecomes increasingly important [31, Chap. 11]. When the applied magnetic ®eld isconstant in time, very fast transition to a stationary vortex pattern is observed intype-II superconductors both experimentally and numerically. In the theory ofdynamical systems, this phenomenon is called long-time stabilization (orconvergence) of solutions. In this article, we are concerned with the stabilizationof every mild solution to the time-dependent Ginzburg±Landau (TDGL) equationsof superconductivity. We are able to show stabilization not only for the case when

�Partially supported by Grants 201/98/1450 of GA CÏ R and A1019703 of GA AV CÏ R.yPartially supported by Deutsche Forschungsgemeinschaft (Germany).

the applied magnetic ®eld is constant in time, but also when it varies with time andapproaches a stationary limit fast enough as time goes to in®nity (with the time rateof change being absolutely integrable for all times). This generalization isimportant for physical experiments where the applied magnetic ®eld is notimmediately constant in time, but it is stabilized during the experiment as timegrows large. In mathematical terms, the TDGL equations de®ne a dynamicalsystem when the applied magnetic ®eld is constant in time, and a dynamicalprocess when the ®eld varies with time [11, 21].

Our main tools are the dynamical process generated by the TDGL equationsconstructed in Fleckinger, Kaper and TakaÂcÏ [11] and the method of provingstabilization developed in L. Simon [27]. Simon's method takes advantage of theanalyticity of an energy functional playing the role of the Liapunov functional forthe dynamical process. Finally, we make use of the fact that our dynamical processis asymptotically autonomous which is proved in Kaper and TakaÂcÏ [21]. Thisenables us to apply a re®ned version of Simon's results, due to Feireisl andSimondon [8, 9] and Jendoubi [19], in order to establish the long-time convergenceto a stationary solution. Our present result, convergence to a single stationarysolution as t!1 in Theorem 4.1, is much stronger than an earlier result in [21,Theorem 2(iv)] where it is shown that every (time-dependent) solution approaches a(nonempty compact connected) set of stationary solutions as t!1. To show thatthis set is a singleton presents a central problem in the theory of gradient systems;see Hale [16, Sect. 3.8, p. 49].

The stabilization of every solution to the TDGL equations was alreadyestablished in F.-H. Lin and Q. Du [23, Theorem 2.1] within the so-called zeroelectric potential gauge, `̀ � � 0'', under quite different, very restrictive conditionsimposed on the applied magnetic ®eld H. Moreover, their proof of an extension ofLojasiewicz' theorem, which is a crucial step, is somewhat incomplete ([23,Lemma 2.3]). This step corresponds to Proposition 6.1 in [9] or Proposition 1.3 in[19]. We carry out a full proof thereof in Section 6 (Proposition 6.1). Besides, thegauge `̀ � � 0'' does not guarantee suf®cient smoothness of solutions and therefore,another gauge, `̀ � � ÿ!�r � A�'', has to be employed in their proof of stabili-zation. Last but not least, the right-hand sides of Equations (4), �72� and �322� in[23] are missing the curl of the applied magnetic ®eld, r�H. This restrictionmeans a curl-free applied magnetic ®eld, r�H � 0, which excludes the presenceof externally imposed currents (transport currents); see [31, Sect. 5.4]. As aresult, the hypothesis H�x; t� � H � const, a constant vector in both space andtime, seems to be made. In our present work, we do not need to impose suchrestrictions.

While emphasizing the formal mathematical aspects of the TDGL equations,we comply with the physical nature of the problem. We work consistently in the`̀ � � ÿ!�r � A�'' gauge introduced in [10] and [28] with any ®xed constant !> 0.This gauge enables us to rigorously establish the long-time asymptotic behaviorand make the connection with solutions of the time-independent GL equations ofsuperconductivity [20]. An important advantage of our gauge choice is that thegauged TDGL equations form a semilinear, strongly parabolic system to whichwell-known methods for parabolic systems can be applied [11, 17].

198 E. Feireisl and P. TakaÂcÏ

1.1. Ginzburg±Landau model of superconductivity. In the Ginzburg±Landautheory of phase transitions [13], the state of a superconducting material near thecritical temperature is described by a complex-valued order parameter , a realvector-valued vector potential A and, when the system changes with time, a real-valued scalar potential �. However, we will eliminate the unknown function � by®xing a suitable gauge. The evolution of , A and � is governed by the followingsemilinear system of weakly coupled parabolic equations:

�@

@t� i��

� � � ÿ i

�r� A

� �2

� �1ÿ j j2� ; �1�

@A

@t�r� � ÿr�r� A� Js� ;A� � r �H: �2�

Here, the supercurrent density Js is a nonlinear function of and A,

Js � Js� ;A� � 1

2i�� r ÿ r �� ÿ j j2A

� ÿRe � i

�r� A

� �

� �: �3�

The system of Eqs. (1)±(3) must be satis®ed at every point x 2 , the regionoccupied by the superconducting material, and at all times t> 0. The boundaryconditions associated with the partial differential equations (1)±(3) are

n � i�r� Aÿ �

� i� � 0 and

n� �r� AÿH� � 0 on @;

(�4�

where @ is the boundary of and n the outer unit normal to @. They must besatis®ed at all times t> 0. Henceforth, the term `̀ TDGL equations'' refers to thesystem of Eqs. (1)±(4).

We consider two- and three-dimensional problems (N � 2 and N � 3, re-spectively). The vector potential A takes its values in RN . The vector H representsthe (externally) applied magnetic ®eld, which is a given function of position andtime valued in RN . The function prescribed on @ is nonnegative. The parametersin the TDGL equations are �, a (dimensionless) friction coef®cient, and �, the(dimensionless) Ginzburg±Landau parameter. The former measures the temporalrate of change and the latter the spatial rate of change of the order parameterrelative to the vector potential. As usual, @t � @=@t; r � grad; r� � curl;r� � div, and r2 � r � r � r; i is the imaginary unit, and a superscript �denotes complex conjugation.

The order parameter can be thought of as the wave function of the center-of-mass motion of the `̀ superelectrons'' (Cooper pairs), whose density is ns � j j2and whose ¯ux is Js. The vector potential A determines the electromagnetic ®eld;E � ÿ@tAÿr� is the electric ®eld and B � r� A the magnetic induction.Equation (2) is essentially AmpeÁre's law, r� B � J, where the total currentJ � Jn � Js � Jt sums up a `̀ normal'' current Jn � E, the supercurrent Js, and thetransport current Jt � r�H. The trivial solution � � 0;B � H;E � 0� re-presents the normal state having no superconducting properties.

Long-Time Stabilization of Solutions to the Ginzburg±Landau Equations 199

The TDGL equations generalize the original GL equations to the time-dependent case. The generalization, ®rst proposed by Schmid [26], was analyzed byGor'kov and Eliashberg [15] in the context of the microscopic Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity [31, Chap. 3]. We refer the reader tothe physics literature [1, 4, 31] for further details.

1.2. Previous results and outline of present work. The TDGL equations havebeen the object of several recent mathematical studies, see e.g. Chen, Hoffmann,and Liang [3], Du [6], Elliott and Tang [7], Liang and Tang [22], Tang [29], andTang and Wang [30]. Recently, Fleckinger, Kaper and TakaÂcÏ [11] have given asimple rigorous construction of the semi¯ow determined by the TDGL equationsbased on standard methods from the existence-and-uniqueness theory for semi-linear systems of weakly coupled parabolic equations [17]. When the appliedmagnetic ®eld H is independent from time, they show also the existence of a globalattractor and the long-time convergence of every solution to a set of stationarysolutions. Their method hinges on ®xing the `̀ � � ÿ!�r � A�'' gauge. Finally, inKaper and TakaÂcÏ [21] these results are generalized to comprise the case of anapplied magnetic ®eld H that varies with time while approaching a stationary limitfast enough as t!1 (with @tH being absolutely integrable for all times).

An important feature of the TDGL equations in the `̀ � � ÿ!�r � A�'' gauge isthat the corresponding energy functional is real-analytic and it has all properties ofa Liapunov functional. Following Simon [27], we exploit this fact to show thatevery solution of the TDGL equations converges to a single stationary solution ast!1. In order to establish this stabilization result, we make use of a re®nedversion of Simon's results due to Feireisl and Simondon [8, 9] and Jendoubi [19].The stabilization results obtained in [8, 9, 19, 27] cannot be applied directly to theTDGL equations for the following three reasons: First, since !> 0, the TDGLequations do not de®ne a gradient system in the restrictive sense required in [8, 9,19, 27]; hence, some relevant identities for our energy functional are different fromthose used in previous studies. When the applied magnetic ®eld H is constant intime, the TDGL equations de®ne a gradient-like system as described in [16, Def.3.8.1, p. 49]. Notice that, if ! � 0, they do not constitute a strongly parabolicsystem, whence suf®cient smoothness of its solutions is not guaranteed (Lin and Du[23]). Second, the TDGL equations do not have the special form studied in [8, 9].Third, due to the boundary conditions, the solutions of the TDGL equations do notsatisfy the smoothness hypotheses required in [19, 27]. This means that we have towork in a Sobolev space determined by an energy functional whose dual space is aspace of distributions. Similar approach was used by Hoffmann and Rybka in orderto prove long-time stabilization of solutions to an equation arising in viscoelasticity[18] and to the Cahn±Hilliard equation [25].

This article is organized as follows. Section 2 contains mathematical preli-minaries. We introduce the `̀ � � ÿ!�r � A�'' gauge (Section 2.1) and give anestimate that follows from an energy-type functional (Section 2.1). Section 3 givesthe formulation of the TDGL equations as an abstract initial-value problem in aHilbert space. We ®rst introduce the notation (Section 3.1), homogenize theboundary conditions by means of the applied vector potential (Section 3.2), and


de®ne the abstract initial-value problem (Section 3.3). Section 4 summarizes ourmain results, Theorem 4.1 and Corollary 4.2. The former gives long-time stabil-ization of every solution in the `̀ � � ÿ!�r � A�'' gauge, whereas the latter givesan analogous result in the `̀ � � 0'' gauge. Our proofs of the main results, given inSection 7, are based on some results proved in Section 5 (analyticity) and Section 6(Lojasiewicz' inequality).

2. Preliminaries

In this section we introduce the gauge choice and de®ne an energy-typefunctional for the TDGL equations. We assume that is a bounded domain in RN

with a boundary @ of class C1;1. That is, is an open and connected set whoseboundary @ is a compact �N ÿ 1�-manifold described by once differentiablecharts whose all ®rst-order partial derivatives are Lipschitz continuous. Thefunction : @! �0;1� is assumed to be Lipschitz continuous.

2.1. Gauge choice. The TDGL equations are invariant under the gauge trans-formation

G� : � ;A; �� 7! � ei��;A�r�; �ÿ @t��: �5�The gauge � can be any (suf®ciently smooth) real scalar-valued function ofposition and time. In order to turn the TDGL equations into a strongly parabolicsystem (having suf®ciently smooth solutions), we adopt the `̀ � � ÿ!�r � A�''gauge, where !> 0 is an arbitrary, but ®xed number. This gauge, introduced in[10] and [28], is determined by taking � � �!�x; t� as the (unique) solution of theNeumann boundary-value problem

�@t ÿ !�� !�r � A� in � �0;1�; �6�n � r� � ÿn � A on @� �0;1�; �7�

subject to a suitable initial condition, ��; 0� � �0 in .In the `̀ � � ÿ!�r � A�'' gauge, at all times t5 0, we have the identities

�� !�r � A� � 0 in ; n � A � 0 on @: �8�Consequently, the differential equations (1) and (2) reduce to

�@

@t� ÿ i

�r� A

� �2

� i��! �r � A� � �1ÿ j j2�

in � �0;1�;�9�

@A

@t� ÿr�r� A� !r�r � A� � Js� ;A� � r �H

in � �0;1�;�10�

where Js is again given by Eq. (3). Likewise, the boundary conditions (4) reduce to

n � r � � 0; n � A � 0 and n� �r� AÿH� � 0

on @� �0;1�: �11�

Henceforth, the term `̀ gauged TDGL equations'' refers to the TDGL equations inthe `̀ � � ÿ!�r � A�'' gauge, given by the system of Eqs. (9)±(11).


The gauged TDGL equations govern the evolution of the pair � ;A� startingwith the prescribed initial data

� 0 and A � A0 on � f0g: �12�The boundary-value problem (9)±(11) is strongly parabolic because !> 0. (Itbecomes degenerate for ! � 0.) In the present work we allow ! to be arbitrarypositive, although when performing numerical simulations, it turns out to be veryuseful to make either of the following two choices: ! � 1=��2 which completelyeliminates the term r � A from Eq. (9); or ! � 1 which simpli®es the ellipticoperator in Eq. (10) to the Laplacian �.

2.2. Energy-type functionals. Following [11] and [28], we introduce theenergy-type functional E! � E!� ;A� de®ned by

E!� ;A� ��

�� i

�r� A

� �

��2 � 1

2�1ÿ j j2�2 � 2!�r � A�2

� jr � AÿHj2�

dx��@

i

�

�� 2d��x�: �13�

If and A satisfy the gauged TDGL equations, the time derivative of E! is

dE!

dt�ÿ 2

�

�@

@tÿ i�! �r � A�

�� 2� @A

@t

�� 2�!2jr�r � A�j2" #

dx

ÿ 2

�

@H

@t� �r � AÿH�dx: �14�

If @tH � 0 (stationary applied magnetic ®eld), the expression on the right-handside is nonpositive, and so E!�t�4E!�s� for 04 s4 t<1. If the appliedmagnetic ®eld is not stationary, the following lemma taken from [11, Lemma 1]shows that E!�t� can still be estimated in terms of the seminorm

P�t� ��t

0

�

j@tH�x; s�j2dx

� �1=2

ds for t5 0:

Lemma 2.1. Let 0< T <1 and assume H 2 W1;1�0; T ; �L2��N�. Let the pair� ;A� be a weak solution of the gauged TDGL equations in � �0;1� having theregularity properties

2 W1;2�� 0; T�� \ L1�� 0; T��;A 2 �W1;2�� 0;T��N and r � A 2 L2�0; T ; W1;2��:

If E!�0� is ®nite then

E!�t� � 2

�t

0

�

�@

@tÿ i�! �r � A�

�� 2� @A

@t

�� 2�!2jr�r � A�j2" #

dx dt0

4 �E!�0��1=2 � P�t�� 2

; t 2 �0;T �:Such regularity of a weak solution is standard owing to the smoothing action of

the holomorphic semigroup generated by the linear part of the TDGL equations; see


[11, §3.4 and §4.1]. A standard approximation argument using smooth solutions tothe TDGL equations may be employed to derive the lemma.

3. Functional Formulation

In this section, we formulate the gauged TDGL equations as an abstractevolution equation in a Hilbert space. The notational conventions are established inSection 3.1, some auxiliary steps are presented in Section 3.2, and the functionalformulation of the gauged TDGL equations is given in Section 3.3.

3.1. Notation. All Banach spaces are real; the (strong) dual space of a Banachspace X is denoted by X0. We refer to [2, Chap. VII] for the Lebesgue, Sobolev andHoÈlder spaces, such as Lp�� with the norm k � kL p�14 p41� and the naturalinner product h�; �i in L2��; Ws;2�� with the norm k � kWs;2�04 s<1�; andC�� with the norm k � kC� �04 � <1�. These de®nitions extend to spaces ofvector-valued functions in the standard way, with the inner product in �L2��Nde®ned by hu; vi � � u � v dx. Complex-valued functions are interpreted as vector-valued functions with two real components.

Functions that vary in space and time are considered as mappings from the timedomain, which is a subinterval of �0;1�, into a space of complex- or vector-valuedfunctions de®ned in . If X � �X; k � kX� is a Banach space of functions de®ned in, then functions de®ned on � �0; T�, for T > 0, may be considered as elementsof Lp�0; T ; X� for 14 p41, or Wm;2�0;T ; X� for an integer m5 0, or C��0; T ; X�for �5 0; � � m� � with 04�< 1; see [17].

Function spaces of ordered pairs � ;A�, where : ! R2 stands for the orderparameter and A : ! RN (N � 2 or 3) for the vector potential, play a crucial rolein the study of the gauged TDGL equations. A suitable framework for thefunctional analysis is the Cartesian product W1��;2 � �W1��;2��2��W1��;2��Nwith 04�< 1. This space is continuously imbedded into L1 provided �> 1=2.Analogously, we write Lp � �Lp��2 � �Lp��N and so on.

We denote by �W1;2��Nn;0 the closed linear subspace of the Hilbert space�W1;2��N which consists of all vector-valued functions A : ! RN from�W1;2��N such that n � A � 0 on @ in the sense of traces.

The symbol X is reserved for the Hilbert space

X � �W1;2��2 � �W1;2��Nn;0 �W1;2:

Its (strong) dual space with respect to the natural inner product in L2 is denotedby X0. Notice that the imbeddings X,!L2 � �L2�0,!X0 are dense and compact,by Rellich's theorem. Finally, we denote by L�X;X0� the Banach space of allcontinuous linear mappings from X to X0; it is identi®ed with the Banach space ofall continuous bilinear forms on X�X. Therefore, we use the tensor product todenote the elements of X�X, i.e.,

�v;V� �w;W� � �v w;VW� � ��v;V�; �w;W�� 2 X�X:

Using the real and imaginary parts, we write � 1 � i 2 for the order parameter,with the corresponding vector notation ~ � � 1; 2� in R2. Similarly, we writeA � �A1;A2;A3� �or �A1;A2�� for the vector potential. Occasionally, we use alsoother lower-case letters in place of , and bold-face letters in place of A.


3.2. Reduction to homogeneous form. In order to reduce the boundaryconditions (11) to a homogeneous form, we ®x arbitrary time t5 0 and suppressexplicit dependence of our variables on t during the reduction process.

Assume H 2 �L2��N , and de®ne AH as a minimizer of the convex quadraticfunctional J! � J!�A�,

J!�A� ��

�!�r � A�2 � jr � AÿHj2�dx; �15�on the domain

D�J!� � �W1;2��Nn;0 � fA 2 �W1;2��N : n � A � 0 on @g:If !> 0, this minimizer is unique, and r � AH � 0 in . If ! � 0, we restrict theminimization to the closed linear subspace

D0�J0� � fA 2 D�J0� : r � A � 0 in gof D�J0� where J0 has a unique minimizer AH; see [11, Lemma 3]. In either case,AH is the unique solution of the boundary-value problem

r�r� AH � r�H and r � AH � 0 in ; �16�n � AH � 0 and n� �r � AH ÿH� � 0 on @: �17�

Thus, AH removes the inhomogeneity from the boundary conditions (11). Themapping H 7!AH, which is linear and time independent, is continuous from�W�;2��N to �W1��;2��N for 04 �4 1; see [11, Lemma 4].

The boundary conditions in the gauged TDGL equations become homogeneousif we formulate the equations in terms of and the reduced vector potentialA0;A0 � Aÿ AH. In fact, we may summarize the gauged TDGL equations (9)±(11)in the form

@

@tÿ 1

��2r � ' in � �0;1�; �18�

@A0

@t�r�r� A0 ÿ !r�r � A0� � F in � �0;1�; �19�

n � r � � 0; n � A0 � 0; n� �r � A0� � 0

on @� �0;1�; �20�

where ' and F are nonlinear functions of and A0,

' � '�x; t; ;A0� � 1

�

�ÿ 2i

��r � � �A0 � AH�

ÿ i

��1ÿ ��2!� �r � A0� ÿ jA0 � AHj2 � �1ÿ j j2� �;

�21�F � F�x; t; ;A0�� 1

2i�� r ÿ r �� ÿ j j2�A0 � AH� ÿ A@tH: �22�


These equations are supplemented by initial data, which follow from Eqs. (12) andA0 � Aÿ AH,

� 0 and A0 � A0 ÿ AH on � f0g: �23�3.3. Gauged TDGL equations. We recall our hypothesis !> 0. Let the vector

u : �0;1� !L2 represent the pair � ;A0�, i.e., u � � ;A0� � � ;Aÿ AH�, andlet A be the linear selfadjoint operator in L2 associated with the quadratic formQ! � Q!�u� de®ned by

Q!�u� ��

1

��2jr j2 � !�r � A0�2 � jr � A0j2

� �dx

��@

��2j j2d��x�

�24�

on the domain

D�Q!� � X � fu � � ;A0� 2W1;2 : n � A0 � 0 on @g:Since !> 0, given any constant c> 0;Q!� ;A0� � ck kL2�� is coercive on X forthe W1;2-norm. Hence, A is positive de®nite in L2, see [14, Chapt. I, Eq. (5.45),p. 92]. We use the same symbol A also for the restrictions A and A0A of A tothe respective linear subspaces �L2��2 � �L2��2 � f0g (for ) and �L2��N �f0g � �L2��N (for A0) of L2.

A weak solution of the boundary-value problem (18)±(20) that satis®es theinitial conditions (23) corresponds to a mild solution u 2L2 of the initial-valueproblem

du

dt�Au �F�t; u�t�� for t> 0; u�0� � u0; �25�

where F�t; u� � �';F�; ' and F are given by Eqs. (21) and (22), andu0 � � 0;A

00�. With 04�< 1 and u0 2W1��;2, a mild solution of (25) on

�0;T � is a continuous function u : �0; T � !W1��;2 such that

u�t� � eÿAtu0 ��t

0

eÿA�tÿs�F�s; u�s��ds for 04 t4 T : �26�Boundary-value problems of type Au � f in L2, where f � �';F� is any pairfrom L2, have been studied by Georgescu [12]. Applying his results, we concludethat D�A�, the domain of A, is a closed linear subspace of W2;2. Since A ispositive de®nite on L2, its fractional powers A� are well-de®ned for all � 2 R;they are unbounded for �> 0. Interpolation theory shows that D�A�� is a closedlinear subspace of W2�;2 for 0<�< 1.

4. Main Results

In this section we state our main results, Theorem 4.1 and Corollary 4.2. Theirproofs are given in Sections 5, 6 and 7. Unless otherwise mentioned, we assumethat the data satisfy the following hypotheses:

(H1) � RN (N � 2 or 3) is a bounded domain with a boundary @ of classC1;1;


(H2) : @! R is Lipschitz continuous with �x�5 0 for all x 2 @;(H3) !; � 2 R are constants such that 0<!<1 and 04�< 1;(H40) H 2 L1�0;1; �W�0;2��N� for some �0 2 ��; 1�; and

(H400) @tH 2 L1�0;1; �L2��N� \ L2�0;1; �L2��N�.We recall from an earlier article [11, Theorem 1] that the initial-value problem

(25) has a unique mild solution u 2 C��0; T �;W1��;2� for any u0 2 D�A�1��=2�and any T > 0. Strictly speaking, in this reference a stronger assumption1=2<�< 1 is made which renders a simpler proof. Nevertheless, also the case04�< 1 can be handled by a straight-forward application of parabolic regularitytheory [17]. The mild solutions of Eq. (25) generate a dynamical processU � fU�t; s� : 04 s4 t 4Tg on D�A�1��=2� by the de®nition

u�t� � U�t; s�u�s� for 04 s4 t4T : �27�The process U completely describes the dynamics of the TDGL equations. Here,we focus on the large-time asymptotic behavior of U�t; s� as t!1 (s ®xed,s5 0) in the special case when the applied magnetic ®eld is asymptoticallystationary. As a consequence of hypotheses �H40� and �H400�, the applied magnetic®eld H approaches a limit in �W�;2��N as t!1.

Indeed, for any 04 s4 t<1, we have

H�t� � H�s� ��t

s

@tH��; t0�dt0: �28�The integral exists as a Bochner integral in �L2��N . Hypothesis �H400� guaranteesthat the limit H1 � limt!1H�t� exists in �L2��N and is given by

H1 � H�s� ��1

s

@tH��; t0�dt0; s5 0: �29�By hypothesis �H40�, we obtain the same limiting relation in the weak topology on�W�0;2��N and, consequently, in the strong topology on �W�;2��N , because theimbedding W�0;2��,!W�;2�� is compact for �<�0, by Rellich's theorem andinterpolation.

4.1. Large-time asymptotic behavior. We compare the large-time asymptoticbehavior of the solution to the gauged TDGL equations, described by the dynamicalprocess U, with that of the gauged TDGL equations for a superconductor in thestationary applied magnetic ®eld H1 � limt!1H�t�,

�@

@t� ÿ i

�r� A

� �2

� i��! �r � A� � �1ÿ j j2�

in � �0;1�;�30�

@A

@t� ÿr�r� A� !r�r � A� � Js� ;A� � r �H1

in � �0;1�;�31�

n � r � � 0; n � A � 0; n� �r � AÿH1� � 0

on @� �0;1�: �32�


The quantity Js is given in terms of and A by the same expression (3) as in thetime-dependent ®eld case.

Equations (30)±(32) de®ne a dynamical system [11]. Before we can introducethis dynamical system, we homogenize the boundary conditions (32). Thehomogenization, described in Section 3.2, is achieved by reformulating Eqs.(30)±(32) in terms of and a reduced vector potential A0;A0 � Aÿ AH1 . Here,AH1 is the (unique) solution of the boundary-value problem (16) and (17) with thepair �AH1 ;H1� in place of �AH;H�. Equations (30)±(32) correspond to the abstractinitial-value problem

dv

dt�Av � G�v�t�� for t> 0; v�0� � v0; �33�

for a vector v : �0;1� !L2; v � � ;A0� � � ;Aÿ AH1�. The pair v0 �� 0;A0 ÿ AH1� is given, v0 2 D�A�1��=2�. The solutions of (33) generate adynamical system S � fS�t� : t 5 0g on D�A�1��=2� by the de®nition

v�t� � S�t�v0 for 04 t<1; �34�see [11, Corollary 2]. The system S completely describes the dynamics of theTDGL equations (30)±(32).

To guarantee stabilization of the solution as t!1, we impose our lasthypothesis on the applied magnetic ®eld:

(H5) There are constants � > 0 and c> 0 such that�1t

�

j@tH�x; t0�j2dx

� �1=2

dt04 ctÿ1ÿ� for all t 5 1: �35�

The following long-time stabilization theorem is our main result.

Theorem 4.1. Assume that all hypotheses (H1)±(H3), (H40), (H400) and (H5)are satis®ed. Then the dynamical process U de®ned in Eq. (27) and the dynamicalsystem S de®ned in Eq. (34) have the following properties:

(i) For each u0 2 D�A�1��=2�, we have

U�s� t; s�u0 ! ws �s fixed; s5 0� and S�t�u0 ! w as t!1;where the convergence takes place in W1��;2. Moreover, both ws and w aredivergence-free equilibria for S.

(ii) We have ws � w in (i) if and only if the following condition is satis®ed:There is a sequence ftkg1k�1 in �0;1� such that tk !1 and

U�s� tk; s�u0 ÿ S�tk�u0 ! 0 in W1��;2 as k !1:The proof of Theorem 4.1 is given in Sections 5, 6 and 7.

4.2. Zero electric potential gauge. A result analogous to Theorem 4.1 is validalso in the zero electric potential gauge, `̀ � � 0'', which is determined by taking� � ��x; t� in Eqs. (5) as the (unique) solution of the initial-value problem

@t� � � in � �0;1�; �36�


subject to a suitable initial condition, ��; 0� � �0 in , which must satisfy theNeumann boundary condition

n � r� � ÿn � A on @� f0g: �37�This boundary condition then holds for every t5 0, which can be seen as follows.We apply the Neumann boundary operator n � r to Eq. (36) and combine the resultwith the ®rst equation in [11, Eq. (2.9), p. 651], thus arriving at

@t�n � r�� n � A� � 0 on @� �0;1�:In the `̀ � � 0'' gauge, at all times t 5 0, we have the identities

� � 0 in ; n � A � 0 on @: �38�Consequently, the differential equations (1) and (2) reduce to

�@

@t� ÿ i

�r� A

� �2

� �1ÿ j j2� in � �0;1�; �39�

@A

@t� ÿr�r� A� Js� ;A� � r �H in � �0;1�; �40�

where Js is again given by Eq. (3). Likewise, the boundary conditions (4) reduceto (11).

The existence and uniqueness of a weak solution � ;A� to the TDGL equationsin the `̀ � � 0'' gauge, Eqs. (39), (40) and (11) with the initial data (12), areestablished in Du [6] using a ®nite-element approach. Whereas the proof ofuniqueness is standard by showing that the squared L2-norm of the difference oftwo weak solutions must satisfy Gronwall's lemma, existence can be derived fromthe analogous result in the `̀ � � ÿ!�r � A�'' gauge shown in [11, Theorem 1].To this end, let a pair � ;A� be a mild solution of the gauged TDGL equations(9)±(11), and let � be the (unique) solution of the initial-value problem

@t� � ÿ!�r � A� in � �0;1�; �41�subject to the following initial condition at t � 0:

�� ÿ!�r � A� in ; n � r� � ÿn � A on @: �42�By [11, Theorem 1], for every T > 0, we have r � A 2 L2�0;T ; W1;2�� whence� 2 W1;2�0; T ; W1;2��. Now, according to Eqs. (5), de®ne

� � ei��; A� � A�r�; �� ÿ @t�:

Then �� 0 in � �0;1� by (8) and (41). Consequently, the pair ��;A�� is aweak solution of the TDGL equations in the `̀ � � 0'' gauge.

The following corollary of Theorem 4.1 describes long-time stabilization in the`̀ � � 0'' gauge.

Corollary 4.2 Let all hypotheses (H1)±(H3), (H40), (H400) and (H5) besatis®ed. Then every weak solution � ;A� of the TDGL equations in the `̀ � � 0''gauge converges to a pair � 1;A1� in L2 as t!1, which is a divergence-freeequilibrium for S:

k� ��; t�;A��; t�� ÿ � 1;A1�kL2 ! 0 as t!1 �43�and r � A1 � 0 in � �0;1�.

The proof of Corollary 4.2 is given in Section 7.


It follows from estimate (72) and the proof of Corollary 4.2 that the rate ofconvergence in (43) is faster than tÿ as t!1, where

�def 1

2�min

�

1ÿ � ;�

2� ��

depends on the constants � > 0 from (35) and � 2 0; 12

ÿ �from (52). The details will

be treated in our future work.

5. Energy Functional

Throughout this section we consider the energy-type functional (13) at ®xedtime t5 0,

E!�� 1; 2�;A�

��

1

�r 1 � A 2

�� 2� 1

�r 2 ÿ A 1

�� 2� 1

2�1ÿ j j2�2

" #dx

��

�2!�r � A�2 � jr � AÿHj2�dx��@

�2j j2d��x�: �44�

Hence, suppressing the time in H�t�, we assume the following

Hypothesis. N � 2 or 3, !> 0; 2 C0;1�@� is nonnegative, and H 2�W1;2��N .

The de®nition of an analytic mapping from one Banach space to another one,we always use, is taken from Deimling [5, Def. 15.1, p. 150].

Lemma 5.1. The functional E! is analytic on the space X.

Proof. Since N � 2 or 3, one has W1;2��,!L6��,!L4�� according to theSobolev embedding theorem. Furthermore, the trace mapping W1;2�� !W1=2;2�@�,!L2�@� is continuous. Inspecting all expressions in Eq. (44),which are polynomials of degree4 4 in all variables, we obtain the desiredconclusion. &

As an immediate consequence of the analyticity of E!, one obtains thefollowing two lemmas for the ®rst and second FreÂchet derivatives of E!:

Lemma 5.2. The FreÂchet derivative DE! of E! is an analytic mapping from Xto its dual X0. It takes the explicit form

hDE!� ;A�; �v;V�i � @E!

@ 1

; v1

� �� @E!

@ 2

; v2

� �� @E!

@A;V

� �for � ;A�; �v;V� 2 X, where

h@ 1E!�� 1; 2�;A�; v1i

� 2

�

1

�

1

�r 1 � A 2

� �� rv1 ÿ 1

�r 2 ÿ A 1

� �� Av1

� �dx

ÿ 2

�

�1ÿ j j2� 1v1dx� 2

�@

�2 1v1d��x�; �45�


h@ 2E!�� 1; 2�;A�; v2i

� 2

�

1

�

1

�r 2 ÿ A 1

� �� rv2 � 1

�r 1 � A 2

� �� Av2

� �dx

ÿ 2

�

�1ÿ j j2� 2v2dx� 2

�@

�2 2v2d��x�; �46�

and

h@AE!�� 1; 2�;A�;Vi

� 2

�

1

�r 1 � A 2

� �� V 2 ÿ 1

�r 2 ÿ A 1

� �� V 1

� �dx

� 4!

�

�r � A��r � V�dx� 2

�

�r � AÿH� � �r � V�dx

� 4!

�

�r � A��r � V�dx� 2

�

�r � AÿH� � �r � V�dx

� 2

�

ÿ 1

�� 1r 2 ÿ 2r 1� � Aj j2

� �� Vdx: �47�

Lemma 5.3. The second FreÂchet derivative of E! is an analytic mappingD2E! : X!L�X;X0� with values in the Banach space of all continuous bilinearforms on X�X. It takes the following form, for � ;A�; �v;V�; �w;W� 2 X,

hD2E!�~ ;A�; �~v;V� �~w;W�i

� 2

�

1

�2r~v � r~w�~v �~w

� �dx� 2

�@

�2~v �~wd��x�

� 4!

�

�r � V��r �W�dx� 2

�

�r � V� � �r �W�dx

� 2

�

�j j2 � jAj2 ÿ 2�~v �~wdx� 2

�

j j2V �Wdx

� 4

�

�~ �~v��~ �~w�dx

ÿ 2

�

1

��v1rw2 ÿ w2rv1 � w1rv2 ÿ v2rw1� � Adx

� 2

�

1

� 2�rv1 �W�rw1 � V�dx

ÿ 2

�

1

� 1�rv2 �W�rw2 � V�dx

ÿ 2

�

1

�r 2 ÿ 2A 1

� �� v1W� w1V�dx

� 2

�

1

�r 1 � 2A 2

� �� v2W� w2V�dx: �48�


Now de®ne a quadratic form Q : X�X! R by

Q��~v;V�; �~w;W��

�def2

�

1

�2r~v � r~w�~v �~w

� �dx� 2

�@

�2~v �~wd��x�

� 4!

�

�r � V��r �W�dx� 2

�

�r � V� � �r �W�dx �49�

for all �v;V�; �w;W� 2 X. Using an estimate from Girault and Raviart [14, Chap. I,Eq. (5.45), p. 92], we conlude that

�v;V� 7! �Q��v;V�; �v;V��1=2

de®nes an equivalent norm on X. This fact combined with the Lax±Milgramtheorem yields the following result:

Lemma 5.4. The quadratic form Q in (49) de®nes an inner product on theHilbert space X which induces an equivalent norm. Accordingly, the equation

hB�v;V�; �w;W�i � Q��~v;V�; �~w;W�� for �v;V�; �w;W� 2 X �50�de®nes a linear operator B : X! X0 which is an isomorphism of X onto X0 (bothalgebraically and topologically) and it is also self-adjoint.

Notice that, having introduced the duality h�; �i between X and X0 determinedby the natural inner product in L2, we have also identi®ed X with the (strong) dualspace X00 of X0. Consequently, the adjoint operator B0 : X00 ! X0 to B : X! X0

coincides with B by the Lax±Milgram theorem.

Lemma 5.5. At every � ;A� 2 X ®xed, the second FreÂchet derivative of E!decomposes as

D2E!� ;A� � B� C� ;A�; �51�where C : X!L�X;X0� is an analytic mapping whose values are compact self-adjoint operators from X to X0.

Proof. The analyticity of C is an immediate consequence of Lemma 5.3,where C is determined by Formula (48). Self-adjointness follows from Lemma 5.4and compactness from that of the Sobolev embedding W1;2��,!Lq�� for14 q< 6. &

Since from now on we work with a single pair � ;A� 2 X which is ®xed(typically, a critical point of the functional E!), we leave it out from the argumentof the FreÂchet derivatives DE!; D2E! and other families of linear operatorsdepending on � ;A�, such as C, if no confusion can arise.

So let us consider any pair � ;A� 2 X of E!. By virtue of Lemma 5.5, at� ;A� 2 X we get that

Bÿ1�D2E!� � id�Bÿ1C

is a compact self-adjoint perturbation of the identity on X relative to the innerproduct (49) on X. Hence, it is a Fredholm operator on X. Consequently, the


kernel N � Ker�D2E!� of D2E!� ;A� is a ®nite-dimensional subspace of X.Finally, let P denote the L2-orthogonal projection of X �L2 � �L2��2�N

ontoN, and de®ne ~P : X! X0 by

h ~P�v;V�; �w;W�i ��

�P�v;V�� w;W�dx

for �v;V�; �w;W� 2 X. Clearly, � ~P�0 � ~P. Then we have

Lemma 5.6. For each � ;A� 2 X ®xed, the operator

D2E!� ;A� � ~P : X! X0

is an isomorphism of X onto X0 (both algebraically and topologically) and it isalso self-adjoint.

Proof. To begin with, we observe that also Bÿ1�D2E! � ~P� is a Fredholmoperator on X, because it is a compact perturbation (with a ®nite-dimensional range)of a Fredholm operator Bÿ1�D2E!� on X. Consequently, it suf®ces to show that

Ker�D2E! � ~P� � f0g:So take an arbitrary U 2 Ker�D2E! � ~P� and decompose it as an L2-orthogonalsum

U � V �W with V � PU and W � �idÿP�U:Hence, V 2N and W 2 X. We compute

0 � h�D2E! � ~P�U;Vi � h�D2E!�W ;Vi � h� ~P�V;Vi� h�D2E!�V;Wi � kVk2

L2 � kVk2L2 ;

which forces V � 0. Thus,

0 � �D2E! � ~P�U � �D2E!�W ;

which yields W 2N, and so W � 0. We have shown U � 0 as desired. &

Finally, Lemmas 5.2 and 5.6 show that all hypotheses in the analytic version ofthe inverse function theorem (see Deimling [5, Theorem 15.3, p. 151]) are ful®lledfor the mapping D2E! � ~P : X!L�X;X0� in a neighborhood of a given point� ;A� 2 X. Thus, we have the following result:

Corollary 5.7. Let � ;A� 2 X be a critical point of the functional E!, i.e.,

DE!� ;A� � 0 �the zero linear form on X�:Then there exist open neighborhoods U of � ;A� in X and V of ~P� ;A� in X0 suchthat DE! � ~P is an analytic diffeomorphism of U onto V.

6. Abstract Version of Lojasiewicz' Theorem

Next, we prove an auxiliary result of crucial importance which adapts aclassical result due to Lojasiewicz [24] in a ®nite-dimensional space. Alsothroughout this section we assume the hypothesis stated at the beginning of Section5 and keep the same notation as well.


Proposition 6.1. Let � ;A� 2 X be a critical point of the functional E!. Thenthere exist some constants �> 0; "> 0 and � 2 0; 1

2

ÿ �such that the inequality

jE!�w;W� ÿ E!� ;A�j1ÿ�4�kDE!�w;W�kX0 �52�holds for every �w;W� 2 X satisfying

k�w;W� ÿ � ;A�kX<": �53�Proof. With the notation from Corollary 5.7, we may apply the analytic version

of the inverse function theorem (see Deimling [5, Theorem 15.3, p. 151]) todeduce that the mapping �DE! � ~P�ÿ1 : V ! U is analytic. Moreover, U and Vmay be chosen small enough, so that both

L1 � supy2conv V

kD�DE! � ~P�ÿ1�y�kL�X0;X�n o

and

L2 � supx2conv U

kD2E!�x�kL�X;X0�n o

are ®nite, where conv U (conv V) denotes the convex hull of U (V, respectively).The convex hull is needed for the mean value theorem.

We identify the kernel N with a subspace ~N of X0 by setting ~N � ~PN. Now,since

~Px � ~PPx 2 ~N for x 2 X;

we can de®ne a mapping

� � �DE! � ~P�ÿ1j ~N : V \ ~N! U

and a function

G : V \ ~N! R1 by G�y� � E!��y��:By Lemma 5.1 and Corollary 5.7, G is a real analytic function on the open setV \ ~N in the ®nite-dimensional space ~N. The gradient of G is given by theformula

hrG�y�; zi � DE! �DE! � ~P�ÿ1�y��

D�DE! � ~P�ÿ1�y�z

� D�DE! � ~P�ÿ1�y�z;DE! �DE! � ~P�ÿ1�y�� D E

for z 2 ~N and y 2 V \ ~N: �54�G being a real analytic function on V \ ~N, by virtue of Lojasiewicz' theorem [24](the original version in a ®nite-dimensional space), one can see that the openneighborhood V of ~P� ;A� in X0 (not renamed for simplicity) can be chosen smallenough, so that

jrG�y�j5 jE!��DE! � ~P�ÿ1�y�� ÿ E!� ;A�j1ÿ�for all y 2 V \ ~N;

�55�

where � 2 0; 12

ÿ �is a constant.


Now take any �w;W� 2 X with k�w;W� ÿ � ;A�kX<". Taking "> 0 smallenough, we have

�w;W� 2 U together with k ~P�w;W� ÿ ~P� ;A�kX0 <";i.e., y � ~P�w;W� 2 V \ ~N. Thus, we can use Eq. (54) to compute

jrG� ~P�w;W��j4 kDE!��DE! � ~P�ÿ1� ~P�w;W��kX0 kD�DE! � ~P�ÿ1� ~P�w;W��kL�X0;X�4L1kDE!��DE! � ~P�ÿ1� ~P�w;W��kX04L1�kDE!�w;W�kX0� kDE!��DE! � ~P�ÿ1� ~P�w;W�� ÿ DE!�w;W�kX0 �;

where the last norm can be estimated by the mean value theorem as

kDE!��DE! � ~P�ÿ1� ~P�w;W�� ÿ DE!��DE! � ~P�ÿ1�DE! � ~P��w;W��kX04L2kDE!�w;W�kX0 :

We infer that

jrG� ~P�w;W��j4CkDE!�w;W�kX0 whenever k�w;W� ÿ � ;A�kX<": �56�Here and in what follows, C 5 0 stands for a generic constant which is in-dependent from �w;W�. Combining (55) with (56), we deduce

jE!��DE! � ~P�ÿ1� ~P�w;W�� ÿ E!� ;A�j1ÿ�4CkDE!�w;W�kX0 provided k�w;W� ÿ � ;A�kX<": �57�

Finally, E! being C2-FreÂchet differentiable on X, we have

jE!��DE! � ~P�ÿ1� ~P�w;W�� ÿ E!�w;W�j4 kDE!�w;W�kX0 k�DE! � ~P�ÿ1� ~P�w;W�� ÿ �w;W�kX� ��k�DE! � ~P�ÿ1� ~P�w;W�� ÿ �w;W�kX�� k�DE! � ~P�ÿ1� ~P�w;W�� ÿ �w;W�k2

X; �58�where � : �0;1� ! �0;1� is a function which is bounded on bounded intervals.Here, the norm of the difference of arguments can be estimated by the mean valuetheorem,

k�DE! � ~P�ÿ1� ~P�w;W�� ÿ �w;W�kX� k�DE! � ~P�ÿ1� ~P�w;W�� ÿ �DE! � ~P�ÿ1�DE! � ~P��w;W��kX4L1kDE!�w;W�kX0 :

Combining this estimate with (58) above, we conclude that

jE!��DE! � ~P�ÿ1� ~P�w;W�� ÿ E!�w;W�j4CkDE!�w;W�k2

X0 provided k�w;W� ÿ � ;A�kX<": �59�


We ®nish our proof by combining the triangle inequality with the relations (57)and (59) to conclude that there exists a constant �> 0 such that the inequality (52)holds for every �w;W� 2 X satisfying (53). &

Corollary 6.2. Let � ;A� 2 X be a critical point of the functional E! andM 5 0 an arbitrary number. Let � 2 0; 1

2

ÿ �be the constant obtained in Proposition

6.1 above. Then there exist some constants � � ��M�> 0 and " � "�M�> 0depending on M such that the inequality

jE!�w;W� ÿ E!� ;A�j1ÿ�4�kDE!�w;W�kX0 �60�holds for every �w;W� 2 X satisfying

jE!�w;W� ÿ E!� ;A�j4M and k�w;W� ÿ � ;A�kL2 <": �61�Proof. Assume the contrary, i.e., there is a sequence f�wn;Wn�g1n�1 in X such

that

�wn;Wn� ! � ;A� in L2 � �L2��2�Nas n!1 �62�

and

nkDE!�wn;Wn�kX0 4 jE!�w;W� ÿ E!� ;A�j1ÿ�4M1ÿ� �63�for all n � 1; 2; . . . (nonexistence of � and "). It follows that f�wn;Wn�g1n�1 isbounded in X and DE!�wn;Wn� ! 0 in X0 as n!1. The Sobolev imbeddingX,!L2 being compact, (62) forces �wn;Wn�* � ;A� weakly in X as n!1.Hence, �wn;Wn� ! � ;A� strongly in Lq as n!1, for 14 q< 6. Finally, weapply the decomposition (51) to Formula (48) for the second FreÂchet derivative ofE! which appears in the difference

DE!�wm;Wm� ÿ DE!�wn;Wn�

��1

0

D2E!�s�wm;Wm� � �1ÿ s��wn;Wn��ds

� ��wm;Wm� ÿ �wn;Wn��

� B��wm;Wm� ÿ �wn;Wn��

��1

0

C�s�wm;Wm� � �1ÿ s��wn;Wn��ds

� ��wm;Wm� ÿ �wn;Wn��:

As m; n!1, both terms on the left-hand side as well as the second summand onthe right-hand side tend to zero and, therefore, also

B��wm;Wm� ÿ �wn;Wn�� ! 0 in X0 as m; n! 0:

Thus, we have �wn;Wn� ! � ;A� strongly in X as n!1, by Lemma 5.4. Thistogether with (63) contradicts Proposition 6.1. &

7. Convergence

Now we are ready to give proofs of our main results. Recall the energy-typefunctional E!;H�t�� ;A� de®ned by (13) and its analogue with H�t� replaced by H1.


Since H is time dependent, E!;H�t� is not a Liapunov functional for the dynamicalprocess U.

7.1. Proof of Theorem 4.1. In order to give a more transparent presentation ofSimon's method [27], we ®rst treat the simpler case when the applied magnetic ®eldH�t� � H : ! RN is time independent.

Consider a ®xed weak solution � ;A� : � �0;1� ! R of the gauged TDGLequations (9)±(11). This solution forms a trajectory � �t�;A�t�� t 5 0� in W1��;2

which is bounded [21, Theorem 2(iii)] and has an !-limit set !�� ;A�� with respectto the L2-topology. We know [21, Theorem 2(iv)] that this set is formed by criticalpoints of the functional E!;H of the same energy, say, E!;1, and

limt!1E!;H� �t�;A�t�� E!;1: �64�

Moreover, these critical points are divergence-free and the !-limit set is compactin L2 � �L2��2�N

. Applying Corollary 6.2 with M � 1 to any point � ;A� in the!-limit set, we conclude that there is an open neighborhood U" of � ;A� in L2

such that (60) holds provided � �t�;A�t�� belongs to U". The set !�� ;A�� beingcompact, there is a ®nite covering of it by fU"i

: i � 1; . . . ;mg. Consequently,taking

� � mini�1;...;m

�i and � � maxi�1;...;m

�i

we get

jE!;H� �t�;A�t�� ÿ E!;1j1ÿ�4�kDE!;H� �t�;A�t��kX0for all t 5 t0;

�65�

where t0 5 0 is a suf®ciently large constant. Since and A satisfy the gaugedTDGL equations, with H independent from t, the time derivative of E! is

d

dtE! �ÿ 2

�

�@ 1

@t� �! 2�r � A�

� �2

� @ 2

@tÿ �! 1�r � A�

� �2" #

dx

ÿ 2

�

@A

@t

�� 2�!2jr�r � A�j2" #

dx: �66�

Now, on one hand, for every t 5 0 one has

E!;H� �t�;A�t�� ÿ E!;1 � ÿ�1

t

dE!;H

dtdt

� 2

�1t

�

�@ 1

@t� �! 2�r � A�

� �2

� @ 2

@tÿ �! 1�r � A�

� �2" #

dx dt

� 2

�1t

�

@A

@t

�� 2�!2jr�r � A�j2" #

dx dt: �67�


On the other hand, the Ginzburg±Landau equations (9), (10) and (11) can berewritten in X0 as

�@ 1

@t� �! 2�r � A�

� �; �

@ 2

@tÿ �! 1�r � A�

� �;@A

@t� !r�r � A�

� �� ÿ 1

2DE!;H� 1�t�; 2�t�;A�t��: �68�

Introducing a function Z : �t0;1� ! �0;1� by

Z�t�2 �def�

@ 1

@t� �! 2�r � A�

2

L2�� @ 2

@tÿ �! 1�r � A�

2

L2��

!

� @A

@t

2

�L2��N�!2kr�r � A�k2

�L2��N

and inserting Eqs. (67) and (68) into Ineq. (65), we arrive at�1t

Z�s�2ds

� �1ÿ�4C Z�t� for all t5 t0; �69�

where � 2 0; 12

ÿ �and C> 0 are constants independent from t. Thus, we employ

Lemma 7.1 below, taken from Feireisl and Simondon [9, Lemma 7.1], to concludethat the integral

�1t0

Z�t�dt is convergent. Since j �x; t�j is bounded uniformly in

�x; t� 2 � �t0;1�, this yields�1t0

@

@t

L2�� @A

@t

�L2��N

!dt<1: �70�

The relation (70) implies strong convergence of the trajectory � �t�;A�t�� t 5 0�in L2.

Here is the auxiliary result due to [9, Lemma 7.1]:

Lemma 7.1. Let 0<�< 12. Assume that Z : R� ! R� is Lebesgue measurable,�1

0Z�s�2ds<1, and there are some constants c> 0 and t0 5 0 such that�1

t

Z�s�2ds4 cZ�t�1=�1ÿ�� for a:e: t5 t0: �71�Then also

�10

Z�s�ds<1 and

t�=2�1ÿ��1

t

Z�s�ds! 0 as t!1: �72�Now we allow the general case with H : � �0;1� ! RN depending on both

position and time. We highlight only the differences with the previous case. First ofall, in place of Eq. (64), we have

limt!1E!;H�t�� t�;A�t�� lim

t!1E!;H1� �t�;A�t�� E!;1:

Replacing H by H1 in (65), we further get

jE!;H1� �t�;A�t�� ÿ E!;1j1ÿ�4�kDE!;H1� �t�;A�t��kX0for all t5 t0;

�73�


where t0 5 0 is a suf®ciently large constant. Since now H depends on t, Eq. (66)has to be replaced by

d

dtE!;H�t� � ÿ 2

�

�@ 1

@t� �! 2�r � A�

� �2

� @ 2

@tÿ �! 1�r � A�

� �2" #

dx

ÿ 2

�

@A

@t

�� 2�!2jr�r � A�j2" #

dxÿ 2

�

@H

@t� �r � AÿH�dx:

Consequently, for every t5 0, Eq. (67) becomes

E!;H�t�� t�;A�t�� ÿ E!;1 � ÿ�1

t

dE!;H�t�dt

dt

� 2

�1t

�

�@ 1

@t� �! 2�r � A�

� �2

� @ 2

@tÿ �! 1�r � A�

� �2" #

dx dt

� 2

�1t

�

@A

@t

�� 2�!2jr�r � A�j2" #

dx dt

� 2

�1t

�

@H

@t� �r � AÿH�dx dt:

The Ginzburg±Landau equations (68) in X0 become

�@ 1

@t� �! 2�r � A�

� �; �

@ 2

@tÿ �! 1�r � A�

� �;@A

@t� !r�r � A�

� �� ÿ 1

2DE!;H�t�� 1�t�; 2�t�;A�t��: �75�

In order to obtain an analogue of Ineq. (69) for the function Z�t�, we use theCauchy±Schwarz inequality to estimate the expressions�

@H

@t� �r � AÿH�dx

�� 4 �E!;H�t�� t�;A�t��1=2 @H

@t

�L2��N

�76�

and

1

2DE!;H�t�� 1; 2;A� ÿ DE!;H1� 1; 2;A�

X0

� kr � �H�t� ÿH1�kX0 4 kH�t� ÿH1k�L2��N

4�1

t

k @H

@tk�L2��N ds: �77�

Here, the function

��t� �def�1

t

k@tHk�L2��N ds; t5 0;

is nonincreasing with ��t� & 0 as t!1.


Next, we combine the equations (74) and (75) with the inequalities (73), (76)and (77), thus arriving at�1

t

Z�s�2dsÿ c2��t�4 c1�Z�t� � ��t��1=�1ÿ�� for all t 5 t0;

where c1 and c2 are positive constants independent from t. Since ��t� & 0 ast!1, the last estimate simpli®es to�1

t

Z�s�2ds4 c01Z�t�1=�1ÿ�� c02��t� for all t 5 t0; �78�

with some other constants c01 > 0 and c02 > 0. Recalling hypothesis (H5), let usde®ne a number �� 2 0; 1

2

ÿ �by

�1ÿ 2��ÿ1 � 1� � > 1:

We replace � by minf�; ��g, denoting the minimum by � again; hence, we mayassume

1< �1ÿ 2��ÿ1 4 1� �:Then (35) implies

��t�4 z�t� �defc tÿ1=�1ÿ�� for all t5 1: �79�

A straight-forward computation yields�1t

z�s�2�1ÿ��ds4 c0z�t� for all t5 1; �80�

with some constant c0> 0. We now take t0 large enough, so that t0 5 1. Combiningthe inequalities (78), (79) and (80), we deduce that�1

t

�Z�s� � z�s�1ÿ��2ds4C�Z�t� � z�t�1ÿ��1=�1ÿ�� for all t 5 t0;

where C> 0 is a constant independent from t. The last inequality replaces (69) andthe rest of the proof remains unchanged.

We have proved Part (i) of Theorem 4.1; Part (ii) follows directly from thede®nition of an !-limit set.

7.2. Proof of Corollary 4.2. Recalling the gauge transformation (5) and Eq.(41), it suf®ces to show that there exists a function �1 2 W1;2�� such that

k��; t� ÿ �1kW1;2�� ! 0 as t!1:By Eq. (41), this convergence follows from�1

0

kr � AkW1;2��dt<1 �81�

which we now verify. Using�1

0Z�t�dt<1 from the proof of Theorem 4.1, we

infer that �10

kr�r � A�kL2��dt<1: �82�


Furthermore, we have �

�r � A�dx ��@

�n � A�d��x� � 0

as a consequence of the boundary condition in (38). Thus, according to PoincareÂ'sinequality, (82) forces (81) as desired. Corollary 4.2 is proved.

Acknowledgements. The work of E. Feireisl is supported in part by the Grant Agency of the CzechRepublic under Grant 201/98/1450 and the Grant Agency of the Academy of Sciences of the CzechRepublic under Grant A1019703. This author would also like to express his thanks to the Universityof Rostock for support during his visit. The work of P. TakaÂcÏ is supported in part by the DeutscheForschungsgemeinschaft (DFG, Germany).

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Authors' addresses: E. Feireisl, Institute of Mathematics, Academy of Sciences of the CzechRepublic, ZÏ itnaÂ 25, CZ-115 67 Praha 1, Czech Republic, e-mail: [email protected]; P. TakaÂcÏ,Fachbereich Mathematik, UniversitaÈt Rostock, UniversitaÈtsplatz 1, D-18055 Rostock, Germany,e-mail: [email protected]


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Long-Time Stabilization of Solutions to the Ginzburg-Landau Equations of Superconductivity

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