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The long-time behavior of the transient Ginzburg-Landau model for superconductivity II
Appl. Math. Lett. Vol. 9, No. 5, pp. 107-109, 1996 Published by Elsevier Science Ltd. Printed in Great Britain 0893-9659/96 $15.00 + 0.00 PII: S0893-9659(96)00082-1
Pergamon
T h e L o n g - T i m e Behavior of the Transient Ginzburg-Landau Model for S u p e r c o n d u c t i v i t y II JISHAN FAN* Department of Mathematics, Suzhou University Suzhou 215006, P.R. China
(Received and accepted September 1995) A b s t r a c t - - I n this paper we prove that the global existence, uniqueness of the solution of a Ginzburg-Landau superconductivity model with the assumptions that the initialdata (¢0, ~40) E £2(12) x L2(ft) only. Under suitable choice of gauge, say, the Lorentz gauge or the Coulomb gauge, we prove that the solutions of the evolutionary superconductivity model must subconverge strongly in ~2(~) x H2(i-/) to one of the solutions of the stationary problem in the Coulomb gauge as time goes to infinity.Because we know littleabout the number of solutionsof the corresponding stationary problem, we can only prove subconvergence in time. However, we can also prove the existence of a maximal attractor in £:2(~t) x L2(f~) and of an inertial set under the Lorentz gauge.
K e y w o r d s - - S u p e r c o n d u c t i v i t y , Long-time behavior, Maximal attractor, Inertial sets, GinzburgLandau equations.
1. I N T R O D U C T I O N A f t e r p r o p e r n o n d i m e n s i o n a l i z a t i o n , t h e e v o l u n t i o n a r y G i n z b u r g - L a n d a u m o d e l m a y be g i v en by t h e following p a r a b o l i c system:
nh-f + i ~ k ¢ ¢ +
OA
-~v
i
-~- + V(I) + curl 2 A + ~ ( ¢
- A
, V~b -
¢ - ~ +
1~,1~¢ =
0
¢ V ¢ * ) + 1¢I2A = c u r l H
in n C R 2,
(1.1)
in ~ C_ R 2,
(1.2)
with the natural boundary conditions
(~V¢+A ¢ ) . n = - i ' ~ ¢ curiA x n = H × n
on 0g/,
(1.3)
on 0~,
(1.4)
and the usual initial conditions ¢(x,O) = Co(x),
A(x,O) =
Ao(x)
in ~ C_ R 2
(1.5)
I am indebted to L. Jiang under whose direction this work has been written as part of the author's doctoral thesis, to J. R. Ockendon for his encouragements, and to Q. Tang and Z. Chen for many helpful suggestions. *Current addresss: Basic Courses Division, Nanjing Forestry University, Nanjing 210 037, P.R. China. Typeset by ~42vtS-TEX
107
108
J. FAN
for the unknowns 10, A, and & denoting the complex-valued order parameter, the vector magnetic potential and the scalar electric potential, respectively. The parameters y and k are positive constants and i -- ~ZT. 7 _> 0 is very small for insulators and very large for magnetic materials, with nonmagnetic metals lying in between. H is the applied magnetic field. ¢* is the complex conjugate of 10 and [¢[2 = 10¢. represents the density of the superconducting carriers. ¢ = 0 corresponds to normal state and in a perfect superconducting state [10[ -- 1. gt C R 2 denotes the bounded and connected region occupied by the superconducting sample with smooth boundary 0 ~ whose unit outward normal vector is n. Under the assumptions 100 E ~1(~), A0 E ~ ( d i v , 12) and [[100[[£~(f~)C0 > 0 (arbitrary constant Co > 0), various results of global existence and uniqueness of solutions of (1.1)-(1.5) have been given by Du [1], Chen, Hoffmann and Liang [2], and Chen, Elliot and Tang [3]. The asymptotic behavior was discussed by Liang and Tang [4] with strict assumptions on the data and the choice of gauge or the regularity of the solutions and then this result was generalized greatly by Fan [5]. Under the assumptions 100 e ~/l(f~), A0 E ~/~(div, ~) the global existence, uniqueness and the existence of a maximal attractor with the choice of Coulomb gauge have been given very recently by Tang and Wang [6]. In this paper, we prove the global existence, uniqueness of the solution of a Ginzburg-Landau superconductivity model with the assumptions that the initial data (¢0,A0) E £:2(f~) x L2(D) only. Under suitable choice of gauge, say, the Lorentz gauge or the Coulomb gauge, we prove that the solutions of the evolutionary superconductivity model must subconverge strongly in 7-/2(f~) x H2(Q) to one of the solutions of the stationary problem in the Coulomb gauge as time goes to infinity. We also prove that the existence of a maximal attractor in/:2(f~) x L2(f/) and of an inertial set under the Lorentz gauge.
2. M A I N
RESULTS
We just state our main results with the choice of Lorentz gauge, i.e., A • n = - d i v A in ~2.
-
0 on cof~,
THEOREM 2.1. For any (¢0, Ao) E £2(~) x L~(f~) and any constant T > O, the system (1.1)-(1.5) admits a unique solution (10, A ), which satisfies (10,A) e L °° (0,T; £2(1~)) n£: 2 (0,T; ~ / 1 ( ~ ) ) , L °° (0, T; L2(f~)) ML 2 ( 0 , T ; ~ / I ( ~ ) ) , (1or,At) e £1 (0,T; ( x l ( a ) ) *) x L 1 (0,T; (HI(a))*). THEOREM 2.2. Theorem 2.1 a11ows us to define a nonlinear semigroup S(t) : (¢0, A0) E £:2(~) × L2(f~) --* (¢, ,4) E £2(~) x L2(f~), which possesses in £2(f~) x L2(f~) a maximal absorbing set B and a maximal attractor .,4. With these results, using the same proof as that of [5], we get Theorems 2.3 and 2.4. THEOREM 2.3. Let (10o,Ao) E £2(f~) x L2(f~). When t -~ oo, the solutions (10,A) of (1.1)-(1.5) will subconverge strongly in 7-/2(f~) x N2(f~ ) to a solution of the corresponding stationary problem
(1.1)-(1.5). THEOREM 2.4. Let (!00, A0) E/22(f~) xL2(f~). Then there exists an inertial set M for ({S(t)}t>_o, X), where X = Ut>toS(t)B t'or to > 0 determined in Theorem 2.2.
REFERENCES 1. Q. Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, (Preprint), (1992). 2. Z. Chen, K.H. Hoffmann and J. Liang, On a non-stationaxy Ginzburgh-Landau superconductivity model, Math. Meth. Appl. Sci. 16, 855-875, (1993).
Ginzburg-Landau Model for Superconductivity II
109
3. Z. Chen, C. Elliot and Q. Tang, Justification of a two-dimensional evolutionary Ginzburg-Landau superconductivity model, (Preprint), (1994). 4. J. Liang and Q. Tang, Asymptotic behavior of the solutions of an evolutionary Ginzburg-Landau Superconductivity model, (Preprint), (1993). 5. J. Fan, On the long-time behavior of the transient Ginzburg-Landau model for superconductivity, CNS Preprint No. 28. 6. Q. Tang and S. Wang, Long time behavior of the Ginzburg-Landau superconductivity equations, Appl. Math. Lett. 8 (2), 31-34, (1995).
Pergamon
T h e L o n g - T i m e Behavior of the Transient Ginzburg-Landau Model for S u p e r c o n d u c t i v i t y II JISHAN FAN* Department of Mathematics, Suzhou University Suzhou 215006, P.R. China
(Received and accepted September 1995) A b s t r a c t - - I n this paper we prove that the global existence, uniqueness of the solution of a Ginzburg-Landau superconductivity model with the assumptions that the initialdata (¢0, ~40) E £2(12) x L2(ft) only. Under suitable choice of gauge, say, the Lorentz gauge or the Coulomb gauge, we prove that the solutions of the evolutionary superconductivity model must subconverge strongly in ~2(~) x H2(i-/) to one of the solutions of the stationary problem in the Coulomb gauge as time goes to infinity.Because we know littleabout the number of solutionsof the corresponding stationary problem, we can only prove subconvergence in time. However, we can also prove the existence of a maximal attractor in £:2(~t) x L2(f~) and of an inertial set under the Lorentz gauge.
K e y w o r d s - - S u p e r c o n d u c t i v i t y , Long-time behavior, Maximal attractor, Inertial sets, GinzburgLandau equations.
1. I N T R O D U C T I O N A f t e r p r o p e r n o n d i m e n s i o n a l i z a t i o n , t h e e v o l u n t i o n a r y G i n z b u r g - L a n d a u m o d e l m a y be g i v en by t h e following p a r a b o l i c system:
nh-f + i ~ k ¢ ¢ +
OA
-~v
i
-~- + V(I) + curl 2 A + ~ ( ¢
- A
, V~b -
¢ - ~ +
1~,1~¢ =
0
¢ V ¢ * ) + 1¢I2A = c u r l H
in n C R 2,
(1.1)
in ~ C_ R 2,
(1.2)
with the natural boundary conditions
(~V¢+A ¢ ) . n = - i ' ~ ¢ curiA x n = H × n
on 0g/,
(1.3)
on 0~,
(1.4)
and the usual initial conditions ¢(x,O) = Co(x),
A(x,O) =
Ao(x)
in ~ C_ R 2
(1.5)
I am indebted to L. Jiang under whose direction this work has been written as part of the author's doctoral thesis, to J. R. Ockendon for his encouragements, and to Q. Tang and Z. Chen for many helpful suggestions. *Current addresss: Basic Courses Division, Nanjing Forestry University, Nanjing 210 037, P.R. China. Typeset by ~42vtS-TEX
107
108
J. FAN
for the unknowns 10, A, and & denoting the complex-valued order parameter, the vector magnetic potential and the scalar electric potential, respectively. The parameters y and k are positive constants and i -- ~ZT. 7 _> 0 is very small for insulators and very large for magnetic materials, with nonmagnetic metals lying in between. H is the applied magnetic field. ¢* is the complex conjugate of 10 and [¢[2 = 10¢. represents the density of the superconducting carriers. ¢ = 0 corresponds to normal state and in a perfect superconducting state [10[ -- 1. gt C R 2 denotes the bounded and connected region occupied by the superconducting sample with smooth boundary 0 ~ whose unit outward normal vector is n. Under the assumptions 100 E ~1(~), A0 E ~ ( d i v , 12) and [[100[[£~(f~)C0 > 0 (arbitrary constant Co > 0), various results of global existence and uniqueness of solutions of (1.1)-(1.5) have been given by Du [1], Chen, Hoffmann and Liang [2], and Chen, Elliot and Tang [3]. The asymptotic behavior was discussed by Liang and Tang [4] with strict assumptions on the data and the choice of gauge or the regularity of the solutions and then this result was generalized greatly by Fan [5]. Under the assumptions 100 e ~/l(f~), A0 E ~/~(div, ~) the global existence, uniqueness and the existence of a maximal attractor with the choice of Coulomb gauge have been given very recently by Tang and Wang [6]. In this paper, we prove the global existence, uniqueness of the solution of a Ginzburg-Landau superconductivity model with the assumptions that the initial data (¢0,A0) E £:2(f~) x L2(D) only. Under suitable choice of gauge, say, the Lorentz gauge or the Coulomb gauge, we prove that the solutions of the evolutionary superconductivity model must subconverge strongly in 7-/2(f~) x H2(Q) to one of the solutions of the stationary problem in the Coulomb gauge as time goes to infinity. We also prove that the existence of a maximal attractor in/:2(f~) x L2(f/) and of an inertial set under the Lorentz gauge.
2. M A I N
RESULTS
We just state our main results with the choice of Lorentz gauge, i.e., A • n = - d i v A in ~2.
-
0 on cof~,
THEOREM 2.1. For any (¢0, Ao) E £2(~) x L~(f~) and any constant T > O, the system (1.1)-(1.5) admits a unique solution (10, A ), which satisfies (10,A) e L °° (0,T; £2(1~)) n£: 2 (0,T; ~ / 1 ( ~ ) ) , L °° (0, T; L2(f~)) ML 2 ( 0 , T ; ~ / I ( ~ ) ) , (1or,At) e £1 (0,T; ( x l ( a ) ) *) x L 1 (0,T; (HI(a))*). THEOREM 2.2. Theorem 2.1 a11ows us to define a nonlinear semigroup S(t) : (¢0, A0) E £:2(~) × L2(f~) --* (¢, ,4) E £2(~) x L2(f~), which possesses in £2(f~) x L2(f~) a maximal absorbing set B and a maximal attractor .,4. With these results, using the same proof as that of [5], we get Theorems 2.3 and 2.4. THEOREM 2.3. Let (10o,Ao) E £2(f~) x L2(f~). When t -~ oo, the solutions (10,A) of (1.1)-(1.5) will subconverge strongly in 7-/2(f~) x N2(f~ ) to a solution of the corresponding stationary problem
(1.1)-(1.5). THEOREM 2.4. Let (!00, A0) E/22(f~) xL2(f~). Then there exists an inertial set M for ({S(t)}t>_o, X), where X = Ut>toS(t)B t'or to > 0 determined in Theorem 2.2.
REFERENCES 1. Q. Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, (Preprint), (1992). 2. Z. Chen, K.H. Hoffmann and J. Liang, On a non-stationaxy Ginzburgh-Landau superconductivity model, Math. Meth. Appl. Sci. 16, 855-875, (1993).
Ginzburg-Landau Model for Superconductivity II
109
3. Z. Chen, C. Elliot and Q. Tang, Justification of a two-dimensional evolutionary Ginzburg-Landau superconductivity model, (Preprint), (1994). 4. J. Liang and Q. Tang, Asymptotic behavior of the solutions of an evolutionary Ginzburg-Landau Superconductivity model, (Preprint), (1993). 5. J. Fan, On the long-time behavior of the transient Ginzburg-Landau model for superconductivity, CNS Preprint No. 28. 6. Q. Tang and S. Wang, Long time behavior of the Ginzburg-Landau superconductivity equations, Appl. Math. Lett. 8 (2), 31-34, (1995).