Existence of Time Periodic Solutions for the Ginzburg-Landau Equations. model of superconductivity

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1 Journal of Mahemacal Analyss and Applcaons 3, Arcle ID jmaa , avalable onlne a hp: on Exsence of me Perodc Soluons for he Gnzburg-Landau Equaons of Superconducvy Bxang Wang* Deparmen of Appled Mahemacs, snghua Unersy, Bejng 00084, P.R. Chna Submed by R. E. Showaler Receved March 6, 998 hs paper deals wh he me dependen Gnzburg-Landau equaons of superconducvy wh an appled magnec feld. he exsence of me perodc soluons s esablshed when he appled magnec feld s me perodc. 999 Academc Press Key Words: Superconducvy; exsence; Gnzburg-Landau equaons; perodc soluon.. INRODUCION In he presen paper, we show he exsence of me perodc soluons for he me-dependen Gnzburg-Landau Ž DGL. model of superconducvy wh an appled feld. he DGL equaons characerze he behavour of superconducvy maerals and have been used o sudy, boh analycally and numercally, he superconducvy phenomena; see Gorkov and Elashberg ; Frahm, Ullah, and Dorsey ; Machda and Kaburak 3 ; Chapman, owson, and Ockendon 4 ; E 5 ; nkham 6. he DGL *Correspondence o: Bxang Wang, Deparmen of Mahemacs, Brgham Young Unversy, Provo, Uah 8460, USA X99 $30.00 Copyrgh 999 by Academc Press All rghs of reproducon n any form reserved. 394

2 PERIODIC SOLUIONS FOR DGL MODEL 395 model akes he form: h es h e s a b grad A 0, m D h m c s 4 curl A curl ž A grad c c / s Ž.. 4 eh s es Ž *grad grad *. A c m mc s s Ž.. where R d, d or 3, represens he regon occuped by he superconducvy maerals. hs model has hree unknown funcons: he complex valued funcon : C s he order parameer; he vecor valued funcon A: R d s he magnec poenal; and he scalar valued funcon : R s he elecrc poenal. he consan h s he Planck consan, es and ms are he charge and mass of a Cooper par, s he conducvy of he normal phase, D s he dffuson coeffcen, c s he speed of lgh, s he appled magnec feld, * s he complex conjugae of, a and b are coeffcens. Wh proper scalng, sysem.. can be rewren as: grad A Ž. 0, Ž.3. A grad curl A Ž *grad grad *. A curl 0, Ž.4. where s he Gnzburg-Landau parameer, he nondmensonal dffusvy. Obvously, he me dependen Gnzburg-Landau equaons gven above are no well-posed n he sense ha her soluons lack unqueness. owever hs model does possess a gauge nvaran propery n ha f Ž, A,. s a soluon of he model, hen so s Ž, A,., where e, A A grad,, Ž.5. for any gven funcon. And hence he Gnzburg-Landau equaons can be nvesgaed under a fxed gauge.

3 396 BIXIANG WANG Varous properes of soluons for sysem Ž.3. Ž.4. have been nvesgaed by many auhors, such as Wang and Zhan 7 ; Chen, offmann, and Lang 8 ; Du 9 ; ang 0 ; ang and Wang ; Kaper, Wang, and Wang ; Wang and Su 3. he remander of hs arcle consss of 3 secons. In Sec., we sae he man resuls. Secon 3 s devoed o a pror esmaes on soluons. In parcular, we shall show ha he evoluon operaor defned here s a compac operaor. In Sec. 4, we presen he proof of man resuls by Schauder fxed pon heorem.. MAIN RESULS In hs secon, we consder he Gnzburg-Landau equaon wh a fxed gauge. ere we adop he Lorenz gauge dv A Žsee 8 or. 9, where any soluon Ž, A,. of sysem Ž.3. Ž.4. sasfes he consrans: dv A 0, n Ž 0,., A n 0, on Ž 0,.. Under he above consrans, Ž.3. and Ž.4. reduce o a sysem of equaons for and A: grad A dv A Ž., n Ž 0,.,. A A Ž *grad grad *. A curl, n Ž 0,... he sysem.. s supplemened wh he boundary condons: and 0, n A n 0, curl A n n, on Ž 0,., 3. Ž x,. Ž x,., AŽ x,. AŽ x,., x, R. 4. hroughou hs paper, we assume ha R. In he sequel, we denoe s, p by W he sandard Sobolev spaces of real scalar or vecor valued

4 PERIODIC SOLUIONS FOR DGL MODEL 397 s, funcons defned on, and as usual, W s denoed by s Ž.. s, Sobolev spaces of complex valued funcons are denoed by W p and s wh callgraphc leers. We use and Ž,. for he usual norm and nner produc of L respecvely; p, denoe by p he norm of L p Ž.. X denoes he norm of any Banach space X. For laer purpose, we recall he followng nequales. and u C u, p, u Ž., 5. p u C u u, u Ž ere and afer, we denoe by C 0 any posve consans. We also noce ha Ždv A curl A. defnes he norm of wh A n 0 whch s equvalen o he sandard -norm, see Adams 4 or Graul and Ravar 5. So here exss a consan C such ha: A C dv A curl A, A Ž., A n Our man resul s saed as follows. Ž. Ž EOREM.. Assume ha L 0, ; L L 0, ; Ž.., L Ž0, ; L Ž.., and s -perodc, ha s Ž., R. hen sysem. 4. possesses a leas one soluon Ž Ž., A.. We remark ha f Ž Ž., A. s a soluon of problem. 4., hen for any me -perodc smooh funcon, Ž, A,. s a -perodc soluon of sysem Ž.3. Ž.4., where Ž, A,. s gven by Ž.5. wh dv A. 3. A PRIORI ESIMAES In hs secon, we derve a pror esmaes on soluons of DGL equaons. We here consder problem. 3. wh he nal condons: Ž x,0. 0Ž x., AŽ x,0. A0Ž x., n. Ž 3.. hen follows from 7 ha for every L and A L Ž. 0 0, he nal boundary problem. 3. wh Ž 3.. possesses a unque soluon Ž Ž., A. such ha loc Ž. C 0,., L L Ž 0,;., loc AŽ. C 0,., L L Ž 0,;..

5 398 BIXIANG WANG Ž. So here exss a semgroup S 0: L L L L such ha S Ž, A. Ž Ž., A., he soluon of. 3. wh Ž In parcular, he operaor S wh gven n heorem. s called he evoluon operaor. In nex secon, we shall see ha he exsence of -perodc soluons of problem. 4. s equvalen o he exsence of fxed pons of S. he exsence of fxed pons wll be obaned from he Schauder fxed pon heorem. o apply he fxed pon prncple, we need o show ha S maps a suable bounded se no self and S s compac. We begn wh an a pror bound on he order parameer. LEMMA 3.. here exss a consan R dependng on, and such ha for eery L 3 wh R, he soluon Ž Ž., A of. 3. and Ž 3.. sasfes Ž. 3 R. Proof. n, akng he real par of he nner produc of. wh n n n L, we fnd ha n d Ž n n. dx 6n d n n Re grad A grad A * dx Ž n n. Re dv Adx Ž n n. Re dx. Ž 3.. Noe ha n n Re grad A grad A * dx ž / ž n n Re grad A grad * A n n */

6 PERIODIC SOLUIONS FOR DGL MODEL 399 n n Re grad A grad * ž n n n n *grad A */ ž n n Re grad A grad A * n n *grad / n n ž / n n grad A Re grad A *grad n n grad A Re *grad grad n n ž / n n Re A grad n n Re *grad grad ž / ž / n n n n grad A grad A n n n Re grad grad * ž n n Ž n n. Ž *. grad n / n n grad A n n n n grad Re n n

7 400 BIXIANG WANG Ž n n. Ž *. Ž grad. n n grad A dx. Ž 3.3. he las nequaly s obaned by n n Ž n n. grad Re Ž *. Ž grad. 0. And hen follows from 3. and 3.3 ha n d Ž n n. Ž n n. dx grad A dx 6n d Due o 4Ž n n. Ž n n.. Ž 3.4. Ž n n. we ge ha Ž n n. 6n 4n Ž n n.ž0n46n. 0n 4 0n 4 6n 4n 4Ž n n., 0n 4 0n 4 0n 4 4n 4Ž n n. Ž n n.. 6n 6n And hen by 3.4 we see ha n d Ž n n. n n dx grad A dx 6n d 4n 4n Ž n n. dx. Ž n 6n hs mples ha d Ž n n. Ž n n. dx dx. d

8 PERIODIC SOLUIONS FOR DGL MODEL 40 By he Gronwall lemma we have Ž. n n dx Ž n n. e Ž 0. dx, 0. Leng n, we nfer ha 3 3 Ž. Ž 0. e, 0, Ž and hus, when Ž. 0 R we see ha, 3 Ž3. 3 Ž3. 3 Ž. 3 Ž 0. 3e Re R. Ž 3.7. he las nequaly beng obaned by choosng R such ha R 3 Ž Ž 3. e., and hen Ž 3.7. concludes he proof of Lemma 3.. In he sequel, we always assume ha 0 3 R and R gven above. By Ž 3.6. we fnd ha and 3 3 R, 0, C, LEMMA 3.. Assume ha 0 3 R. hen here exss a consan C dependng on, and such ha 0 Ž. grad A dx d C, grad A dx d C, 0. Proof. akng he real par of he nner produc of. wh n L, we ge ha 4 d grad A dx dx d 4 dx dx,

9 40 BIXIANG WANG whch mples ha I follows ha d grad A dx, 0. d 0 grad A dx d Ž 0. C, grad A dx d Ž. C, 0, 4 Ž. and hus he proof s fnshed. LEMMA 3.3. Assume ha 0 3 R. hen here exss a consan R dependng on,, and such ha when A 0 R, he soluon Ž Ž., A. of. 3. and Ž 3.. sasfes A R. Proof. akng he nner produc of. wh A n L, we fnd ha d AŽ. curl A dv A d ž / ž / ž / Re grad A * Adx curl Adx grad A A curl A 3 6 C grad A A curl A by Ž 3.8. and 5. C grad A Ž curl A dv A. curl A by7. curl A dv A C grad A.

10 PERIODIC SOLUIONS FOR DGL MODEL 403 ha s d d A curl A dv A C grad A. By.7 we fnd ha here exss 0 such ha d d A A C grad A. From he Gronwall lemma we ge ha Ž s. 0 Ž 3.0. AŽ. AŽ 0. e C grad AŽ s. Ž s. e ds Ž s. Ž s. e ds 0 AŽ 0. e C Grad AŽ s. Ž s. ds 0 Ž s. ds, 0. Ž And hen, when A R, we fnd ha 0 Ž. Ž. A A 0 e C Re C R, 3. he las nequaly beng obaned by choosng R such ha CŽ Ž. e., and hus Ž 3.. concludes he proof of Lemma 3.3. In wha follows, we always assume ha A 0 R wh R gven above. hen Ž 3.. mples ha AŽ. C, 0. Ž 3.3. In order o show S s a compac operaor, we need o homogenze he boundary condons 3., whch s acheved by lookng for he soluon A of he problem: A curl 0, n, Ž 3.4. A n 0, curl A n n, on. Ž 3.5.

11 404 BIXIANG WANG 4 Le Q A, B be he blnear form on A : A n 0 such ha Q A, B curl A curl B dv A dv B for all A, B such ha A n 0 and B n 0. By.7 Q A, B s connuous and coercve, so by Lax-Mlgram heorem we clam ha here exss a unque funcon A wh A n 0 such ha curl A curl B dv A dv B curl B 0 for all B wh B n 0. I s easy o see ha A s a weak soluon of Ž 3.4. Ž We now nroduce Aˆ A A, hen he sysem. 3. wh Ž 3.. becomes: Ž Aˆ A. grad ˆ ˆ dv A A A A 0, Ž 3.6. A ˆ ˆ ˆ A A *grad grad * Ž A A. 0, Ž 3.7. grad n 0, curl Aˆ n 0, Aˆ n 0, on, Ž 3.8. Ž x,0. Ž x., Aˆ Ž x,0. A Ž x. A Ž 0. n, Ž We observe ha he above defnon of A consan C ndependen of such ha and mples ha here exss a A C, A C, 3.0 A A. Ž 3.. LEMMA 3.4. Assume ha R and A R. hen any soluon Ž Ž., Aˆ Ž.. of problem Ž 3.6. Ž 3.9. sasfes Ž. d Aˆ d C, 0, Ž. where C depends on,,, and.

12 Proof. PERIODIC SOLUIONS FOR DGL MODEL 405 Inegrang Ž 3.0. beween and, we fnd ha Ž. curl A dv A d Ž. Ž. AŽ. C grad A d Ž. d C, 0 Ž by Ž 3.3. and Lemma 3... So follows from 7. ha Ž. A d C, 0, Ž 3.. whch mples ha Ž. Ž. ˆ ž / A d A A d Ž. Ž. C AŽ. d C Ž. d by Ž 3.0. C, 0 Ž by Ž Ž 3.3. Due o we see ha ž / grad A grad Ž *grad grad *. Adx A dx, Ž. grad d Ž. grad A d Ž. Ž. A dx d Ž *grad grad *. Adxd

13 406 BIXIANG WANG Ž. grad A d Re A grad * dx d hs yelds 6 3 C C A grad d Ž by Lemma 3.. Ž. Ž. C C A grad d by 3.8 Ž. Ž. C grad d C A d Ž. C grad Ž. d Ž by Ž Ž. grad Ž. d C, 0. Ž 3.4. he esmaes 3.3 and 3.4 complee he proof. For laer purpose, we recall he unform Gronwall lemma. LEMMA 3.5. Le yž., g, Ž. hž. be hree pose negrable funcons n Ž,. such ha dyd s negrable n Ž,. and sasfy 0 0 wh r dy gy h,, 0 d r gž s. ds c, hž s. ds c, r y s ds c, r, 3 0 for some pose consans r, c, c, c 3 c ž 3 / 0 dependng on. hen yž. c expž c., r. r We observe ha f he consans c are ndependen of, hen Lemma 4. s also vald for. In ha case s called he unform Gronwall lemma Žsee 6..

14 PERIODIC SOLUIONS FOR DGL MODEL 407 LEMMA 3.6. Assume ha R and A R. hen any soluon Ž Ž., Aˆ Ž.. of problem Ž 3.6. Ž 3.9. sasfes Ž. Aˆ C,, where C depends on,,, and. Proof. akng he real par of he nner produc of Ž 3.6. wh n L, we oban ha d grad d ˆ ž / Re Ž Aˆ A. grad, Ž. Re dv A A,. Ž ˆ Ž.. Re A A, Re,. Ž 3.5. We now majorze each erm above as follows. Re Ž Aˆ A. grad, Ž. C A ˆ A grad dx Ž. C Aˆ A grad ˆ C A grad C A grad by 5. ˆ C A Aˆ grad grad Cgrad grad by 6. and Ž 3.0. C Aˆ Aˆ C 3 3 ˆ ˆ CŽ. A A C Ž 0. CŽ. A ˆ Ž. C by Ž 3.3. and A A A ˆ CŽ. A C CŽ.. Ž 3.6. ˆ 4 4

15 408 BIXIANG WANG Smlarly, ž / Re dvž A A., C dvž A A. C 4 dv A4 dv A 4 C dv A C A C ˆ ˆ dv A dv A C Ž by Ž ˆ C A Aˆ C ˆ C A Aˆ C Ž by Ž ˆ 4 ˆ 4 A C A Ž. Ž. CŽ.. Ž 3.7. Re Aˆ A, Ž. ˆ Ž ˆ. Ž ˆ. A A A A 3 C A 3 by 5. and Ž 3.0. CŽ. A 4 CŽ. by Ž Ž 3.8. We can also show ha Ž. 4 Re, C CŽ.. Ž 3.9. From we nfer ha d grad d 4 ˆ A C 4 ˆ 4 A CŽ.. Ž 3.30.

16 PERIODIC SOLUIONS FOR DGL MODEL 409 ˆ akng he nner produc of Ž 3.7. wh A n L, we fnd ha Noe ha d Ž curl Aˆ dv Aˆ. Aˆ d Ž *grad grad *., Aˆ ž / / A ˆ ˆ ž Ž A A, A, A ˆ. / ž. Ž 3.3. Ž *grad grad *, Aˆ. 4 4 C *grad Aˆ dx C grad Aˆ ˆ A C 4 Ž by smlar calculaon as above.. Ž 3.3. ž Ž A ˆ A., A ˆ / 3 Ž ˆ. Also we have A A Aˆ C ˆ A Aˆ Ž by Ž 3.8. and 5.. Ž. Aˆ C 4 ˆ 4 CŽ. A CŽ.. Ž A, A ˆ A A ˆ Ž by Ž 3... By we see ha d Ž curl Aˆ dv Aˆ. Aˆ d C Aˆ Aˆ CŽ.. Ž Aˆ 4 ˆ 4 C A CŽ.. Ž 3.35.

17 40 BIXIANG WANG I follows from 3.30 and 3.35 ha d ž grad curl Aˆ dv Aˆ / Aˆ d 4 3 Aˆ 4 ˆ 4 C A C, 0. Choosng small enough we nfer ha here exss C such ha d 4 4 ˆ ˆ ˆ d grad curl A dv A C A C 4 C grad curl Aˆ 4 dv Aˆ 4 C ž ˆ ˆ / Ž by Ž 3.9. and 7.. C grad curl A dv A C, 0. hen by Lemma 3.4 and Lemma 3.5 we clam ha grad Ž. curl AˆŽ. dv AˆŽ. C,, whch along wh 3.9 and 3.3 concludes he proof. Noe ha AŽ. Aˆ Ž. Ž. Ž by Ž C, Ž by Lemma 3.6., so we fnd ha here exss a consan C such ha Ž. A C,. Ž PROOF OF MAIN RESULS In hs secon, we presen he proof of heorem.. We shall apply Schauder fxed pon heorem o show he man resuls. Frs we recall he followng proposons.

18 PERIODIC SOLUIONS FOR DGL MODEL 4 LEMMA 4. Žsee 7.. Assume ha Ž x,. s -perodc n. hen problem. 4. has a -perodc soluon f and only f he eoluon operaor SŽ. has a fxed pon. LEMMA 4. Schauder Fxed Pon heorem. Le M be a nonempy, closed, bounded, conex subse of a Banach space X, and suppose G: M M s a compac operaor. hen G has a fxed pon. We are now n a poson o complee heorem.. Proof of heorem.. Le B L 3 Ž.: R 4 and B 3 A L Ž.: A R 4, where R and R are consans n Lemma 3. and Lemma 3.3, respecvely. Denoe by X L 3 L and M B B. hen follows from Lemma 3. and Lemma 3.3 ha he operaor S maps M no self. Snce he embeddng L 3 L s compac, by Ž we clam ha S: M M s a compac operaor. And hus follows from Lemma 4. ha S has a fxed pon, whch s a -perodc soluon of problem. 4. as saed n Lemma 4.. he proof s complee. We remark ha we only show here he exsence of perodc soluons for he wo dmensonal DGL equaons. I seems ha he mehod used n hs paper does no apply o he hree dmensonal DGL model. In fac, we even do no know wheher he hree dmensonal DGL equaons are well-posed for general L p Ž p. nal daa; see 3 or 7. ACKNOWLEDGMENS hs work s parally suppored by he Posdocoral Fellowshp of Mnsero de Educacon y Culura Span under SB96-A. REFERENCES. L. Gor kov and G. Elashberg, Generalzaon of he Gnzburg-Landau equaons for nonsaonary problems n he case of alloys wh paramagnec mpures, Soe Phys. JEP, 7 Ž 968., Frahm, S. Ullah, and A. Dorsey, Flux dynamcs and he growh of he superconducng phase, Phys. Re. Le., 66 Ž 99., M. Machda and. Kaburak, Drec smulaon of he me-dependen Gnzburg-Landau equaon for ype-ii superconducng hn flm: vorex dynamcs and V-I characerscs, Phys. Re. Le., 7 Ž 993., S. J. Chapman, S. D. owson, and J. R. Ockendon, Macroscopc models of superconducvy, SIAM Re., 34 Ž 99., E. Wenan, Dynamcs of vorces n Gnzburg-Landau heores wh applcaons o superconducvy, Phys. D, 77 Ž 994.,

19 4 BIXIANG WANG 6. M. nkham, Inroducon o Superconducvy, McGraw-ll, New York, S. Wang and M. Zhan, L p soluons o he me dependence Gnzburg-Landau equaons of superconducvy, he Insue for Scenfc Compung and Appled Mahemacs, Indana Unversy, preprn. 8. Z. M. Chen, K.. offmann, and J. Lang, On a non-saonary Gnzburg-Landau superconducvy model, Mah. Meh. Appl. Sc., 6 Ž 993., Q. Du, Global exsence and unqueness of soluons of he me-dependen Gnzburg- Landau model for superconducvy, Appl. Anal., 53 Ž 994., Q. ang, On an evoluonary sysem of Gnzburg-Landau equaons wh fxed oal magnec flux, Comm. Paral Dfferenal Equaons 0 Ž 995., 36.. Q. ang and S. Wang, me dependen Gnzburg-Landau equaons of superconducvy, Physca D, 88 Ž 995., G. Kaper, B. Wang, and S. Wang, Deermnng nodes for he Gnzburg-Landau equaons of superconducvy, Dscree Connuous Dynamcal Sysems, 4 Ž 998., B. Wang, N. Su, Global weak soluons for he Gnzburg-Landau equaons of superconducvy, Appl. Mah. Le. Ž 999., R. A. Adams, Sobolev Spaces, Academc Press, New York, V. Graul and P. A. Ravar, Fne Elemen Mehods for Naver-Sokes Equaons, Sprnger-Verlag, New York, R. emam, Infne Dmensonal Dynamcal Sysems n Mechancs and Physcs, Sprnger-Verlag, New York, Amann, Ordnary Dfferenal Equaons: An Inroducon o Nonlnear Analyss, Waler de Gruyer, Berln, 990.