Int. Journal of Math. Analyss, Vol. 6, 212, no. 22, 195-114 Unqueness of Weak Solutons to the 3D Gnzburg- Landau Model for Superconductvty Jshan Fan Department of Appled Mathematcs Nanjng Forestry Unversty Nanjng 2137, P.R. Chna fanjshan@njfu.com.cn Tohru Ozawa Department of Appled Physcs Waseda Unversty, Tokyo, 169-8555, Japan txozawa@waseda.jp Abstract We prove the unqueness for weak solutons of the tme-dependent Gnzburg-Landau model for superconductvty wth L 2 ntal data n the case of Coulomb gauge under the regularty hypothess on the solutons that ψ, A C([,T]; L 3 ). We also prove the unqueness of the 3-D radally symmetrc soluton wth the choce of Lorentz gauge and L 2 ntal data. Keywords: Unqueness, Gnzburg-Landau model, Superconductvty, Coulomb gauge, Lorentz gauge 1 Introducton We consder the unqueness problem for the Gnzburg-Landau model for superconductvty: ( ) 2 ηψ t + ηkφψ + k + A ψ +( ψ 2 1)ψ =, (1.1.1) {( ) } A t + φ + curl 2 A +Re ψ + ψa ψ = curl H, (1.1.2) k n Q T =(,T) Ω R R d,d=2, 3, wth boundary and ntal condtons ψ ν =,A ν =, curl A ν = H ν on (,T) Ω, (1.1.3) (ψ, A) t= =(ψ,a ) n Ω. (1.1.4)
196 Jshan Fan and Tohru Ozawa Here the unknowns ψ, A, and φ are C-valued, R d -valued, and R-valued functons respectvely, and they stand for the order parameter, the magnetc potental, and the electrc potental, respectvely. H := H(t, x) s the appled magnetc feld, η and k are Gnzburg-Landau postve constants, and = 1. Ω s a bounded doman wth smooth boundary Ω and ν s the outward normal to Ω. ψ denotes the complex conjugate of ψ, Reψ := (ψ + ψ)/2, ψ 2 := ψψ s the densty of superconductng carrers. T s any gven postve constant. It s well known that the Gnzburg-Landau equatons are gauge nvarant, that s, f (ψ, A, φ) s a soluton of (1.1.1)-(1.1.4), then for any real-valued smooth functon χ, (ψe kχ,a+ χ, φ χ t ) s also a soluton of (1.1.1)-(1.1.4). So n order to obtan the well-posedness of the problem, we need to mpose the gauge condton. From physcal pont of vew, one usually has three types of the gauge condton: Coulomb gauge: dv A = n Ω and φdx =. Ω Lorentz gauge: φ = dv A n Ω. Temporal gauge: φ = n Ω. For the ntal data ψ H 1 (Ω),A H 1 (Ω), Tang and Wang [7] obtaned the exstence and unqueness of global strong solutons and the exstence of the maxmal attractors n the case of the Coulomb Gauge when d = 3. They also proved the exstence of global weak solutons when ψ,a L 2 and d =2. Fan and Jang [2] showed the exstence of global weak solutons when ψ,a L 2 and d = 3. Zaouch [8] proved the exstence of tme-perodc solutons when the appled magnetc feld H s tme perodc and d = 3. Phllps-Shn [6], Chen-Hoffman [1] studed the well-posedness of classcal solutons to the nonsothermal models for superconductvty when d = 2, 3. When d = 3, the unqueness problem to (1.1.1)-(1.1.4) wth L 3 ntal data s stll open. Recently, Fan-Gao [3] and Fan-Ozawa [4] proved a condtonal unqueness result when Ω s a bounded doman or Ω = R 3, respectvely. Theorem 1.1. ([3]) Let ψ,a L 2,H L 2 (Q T ) and d =3. Assume that ψ, A L r (,T; L p (Ω)) wth 2 r + 3 p =1, 3 <p. (1.1.5) Then there exsts at most one weak soluton (ψ, A) to the problem (1.1.1)- (1.1.4) satsfyng ψ, A V 2 (Q T ):=L (,T; L 2 ) L 2 (,T; H 1 ) wth the Lorentz or Coulomb gauge. The condton (1.1.5) comes from scalng for (1.1.1) and (1.1.2). Let us be more precse: If (ψ(t, x),a(t, x)) s a soluton of (1.1.1)-(1.1.2) assocated
3D Gnzburg-Landau model for superconductvty 197 wth the ntal value (ψ (x),a (x)) wthout lnear lower order term ψ and H, then (λψ(λ 2 t, λ(x x )),λa(λ 2 t, λ(x x ))) s also a soluton assocated wth (λψ (λ(x x )),λa (λ(x x ))) for any λ> and x R 3. A Banach space B of dstrbutons on R R 3 s a crtcal space f ts norm verfes for any λ and any u B, u B = λu(λ 2,λ( x )) B. If we choose B as L r (, ; L p (R 3 )), then (r, p) should satsfy We wll prove 2 r + 3 p =1. Theorem 1.2. Let ψ,a L 2,H L 2 (,T; L 2 ) and assume that ψ, A C([,T]; L 3 ). (1.1.6) Then there exsts at most one weak soluton (ψ, A, φ) to the problem (1.1.1)- (1.1.4) satsfyng ψ, A V 2 (Q T ) and φ L 2 (,T; W 1,6/5 ). In ths paper, we wll also be nterested n sphercally symmetrc weak solutons to (1.1.1)-(1.1.4) n the case of the Lorentz gauge. We study the unqueness of solutons to (1.1.1)-(1.1.4) of the form ψ(t, x) =ψ(t, r),a(t, x) =A(t, r) x r, wth r := x. Hence, H must satsfy H and (ψ, A)(t, r) should satsfy the followng system: ηψ t 1 (ψ k 2 rr + 2 ) ( ) 1 r ψ r + k ηk ψ (A r + 2r ) A + 2 k Aψ r + A 2 ψ +( ψ 2 1)ψ =, (1.1.7) A t (A r + 2r ) {( ) } A +Re r k ψ r + ψa ψ =, (1.1.8) (ψ r,a r ) r= =(ψ, A) r=a =, (1.1.9) (ψ, A) t= =(ψ (r),a (r)),r Ω:=B(,a). (1.1.1) Our next theorem s
198 Jshan Fan and Tohru Ozawa Theorem 1.3. Let ψ,a L 2 and d =3. Then, there exsts a unque soluton (ψ, A) to problem (1.1.7)-(1.1.1) wth the choce of Lorentz gauge. Remark 1.1. A smlar result s proved n [5] for the annulus Ω = {x R 3 ;<a< x <b}. In the followng proof, we wll use the followng well-known property: Lemma 1.1. [2]. Let ψ, A V 2 (Q T ) C([,T]; L 3 ), then φ L 2 (,T; W 1,6/5 ) satsfes {( Δφ = dv Re ψ + ψa k ) } ψ n Q T, (1.1.11) φ ν = on (,T) Ω. (1.1.12) 2 Proof of Theorem 1.2 The exstence of global weak solutons has been proved n [2], we only need to prove the unqueness. Frst, we recall that for any w C([,T]; L 3 ), there exst two functons w 3 and w satsfyng w = w 3 + w and w 3 L (,T ;L 3 ) ɛ, w L (Q T ) C, (2.2.1) for any postve ɛ. Here and later on we wll use ψ = ψ 3 + ψ or A = A 3 + A to denote ths decomposton. Let (ψ,a,φ )( =1, 2) be the two weak solutons to (1.1.1)-(1.1.4), we set ψ := ψ 1 ψ 2,A:= A 1 A 2,φ:= φ 1 φ 2. We defne ξ satsfyng Then Δξ = ψ 2 ψ n Q T, (2.2.2) ξ = on (,T) Ω. (2.2.3) ξ H 2 C ψ 2 ψ L 2 (2.2.4)
3D Gnzburg-Landau model for superconductvty 199 It s easy to nfer that ηψ t + ηkφ 1 ψ + ηkφψ 2 1 2 Δψ + k2 k A 1 ψ + 2 k A ψ 2 +A 2 1ψ 1 A 2 2ψ 2 + ψ 1 2 ψ 1 ψ 2 2 ψ 2 ψ =, (2.2.5) A t + φ + curl 2 A {( ) ( ) } +Re k ψ 1 + ψ 1 A 1 ψ 1 k ψ 2 + ψ 2 A 2 ψ 2 =, (2.2.6) φdx =, dv A =, (2.2.7) Ω Δφ = dv Re {( k ψ 1 + ψ 1 A 1 ) ψ 1 ( k ψ 2 + ψ 2 A 2 ) } ψ 2, (2.2.8) and ( ) ( ) k ψ 1 + ψ 1 A 1 ψ 1 k ψ 2 + ψ 2 A 2 ψ 2 ( ) ( ) = k ψ 1 + ψ 1 A 1 ψ + k ψ + A 1ψ + Aψ 2 ψ 2. (2.2.9) Multplyng (2.2.5) by ψ and ntegratng by parts, then takng the real part, we obtan η d ψ 2 dx + 1 ψ 2 dx 2 dt k 2 ηk φψ 2 ψdx + 2 A 1 ψ ψ dx + 2 ψ 2 A ψ dx k k + A 1 + A 2 ψ 2 A ψ dx + ψ 1 ψ 1 + ψ 2 ψ 2 dx + ψ 2 dx 5 =: I + ψ 2 dx. (2.2.1) =1
11 Jshan Fan and Tohru Ozawa Each term I can by bounded as follows I 2 2 k A 13 L 3 ψ L 2 ψ L 6 + 2 k A 1 L ψ L 2 ψ L 2 Cɛ ψ 2 H 1 + C ψ 2 L 2, I 3 2 k ψ 23 L 3 A L 6 ψ L 2 + 2 k ψ 2 L A L 2 ψ L 2 Cɛ A 2 H + 1 Cɛ ψ 2 L + 2 C A 2 L 2, I 4 ψ 23 L 3 A 1 + A 2 L 3 A L 6 ψ L 6 + ψ 2 L A 1 + A 2 L 3 A L 6 ψ L 2 Cɛ A 2 H + 1 Cɛ ψ 2 H + 1 C ψ 2 L 2, I 5 ψ 13 L 3 ψ 1 + ψ 2 L 3 ψ 2 L + ψ 1 6 L ψ 1 + ψ 2 L 3 ψ L 6 ψ L 2 Cɛ ψ 2 H + 1 C ψ 2 L 2, ( ) I 1 = ηk ξδφdx ηk ξ k ψ 1 + ψ 1 A 1 ψdx +ηk ξ k ψ + A 1ψ + Aψ 2 ψ 2 dx ηk k ψ 1ψΔξdx + ηk ξ ψ 1 ψ dx + ηk ξ ψ 1 A 1 ψ dx +ηk ξ k ψ + A 9 1ψ + Aψ 2 ψ 2 dx =: I. (2.2.11) Each term I ( =6, 7, 8) can be bounded as follows I 6 η ψ 1 ψ 2 ψ 2 dx η ψ 13 L 3 ψ 2 L 3 ψ 2 L 6 + η ψ 1 L ψ 2 L 3 ψ L 6 ψ L 2 Cɛ ψ 2 H + 1 C ψ 2 L 2, I 7 ηk ξ L 6 ψ 1 L 3 ψ L 2 C ξ L 6 ψ L 2 ɛ ψ 2 L + 2 C ψ 2 L 2, I 8 ηk ξ L 6 ψ 13 L 3 A 1 L 3 ψ L 6 + ηk ξ L 6 ψ 1 L A 1 L 3 ψ L 2 due to Cɛ ψ 2 H 1 + C ψ 2 L 2 ξ L 6 C ξ H 2 C ψ 23 L 3 ψ L 6 + C ψ 2 L ψ L 2. (2.2.12) The last term I 9 can be bounded as follows I 9 η ξ ψ ψ 2 dx + ηk ξ A 1 ψ ψ 2 dx +ηk ξ A ψ 2 2 dx =: I 1 + I 11 + I 12, (2.2.13) =6
3D Gnzburg-Landau model for superconductvty 111 where I 1 η ξ L 6 ψ L 2 ψ 2 L 3 C ξ H 2 ψ L 2 C ψ 2 ψ L 2 ψ L 2 C ψ 23 L 3 ψ L 6 ψ L 2 + C ψ 2 L ψ L 2 ψ L 2 Cɛ ψ 2 L + 2 C ψ 2 L 2, I 11 ηk ξ L 6 A 1 L 3 ψ 2 L 3 ψ L 6 C ξ H 2 ψ H 1 Cɛ ψ 2 L + 2 C ψ 2 L 2, I 12 ηk ξ L 6 A L 6 ψ 2 2 L 3 C ξ H 2 A L 6 Cɛ ψ 2 H 1 + Cɛ A 2 H 1 + C ψ 2 L 2. Insertng these estmates nto (2.2.1) and takng ɛ small enough, we get η d ψ 2 dx + 1 ψ 2 dx Cɛ A 2 dt k 2 H + 1 C( ψ 2 L + 2 A 2 L2). (2.2.14) Now testng (2.2.6) by A, we see that 1 d A 2 dx + curl A 2 dx 2 dt 1 k ψ 1 ψ A + ψ 1 A 1 ψ A + 1 k ψ 2 ψ A + A 1 ψ ψ 2 A dx 4 =: J. (2.2.15) =1 Each term J can be bounded as follows J 1 1 k ψ 13 L 3 ψ L 2 A L 6 + 1 k ψ 1 L ψ L 2 A L 2 Cɛ ψ 2 L + 2 Cɛ A 2 H + 1 C A 2 L 2, J 2 ψ 13 L 3 A 1 L 3 ψ L 6 A L 6 + ψ 1 L A 1 L 3 ψ L 6 A L 2 Cɛ ψ 2 H + 1 Cɛ A 2 H + 1 C A 2 L 2, J 3 1 k ψ 23 L 3 ψ L 2 A L 6 + 1 k ψ 2 L ψ L 2 A L 2 Cɛ ψ 2 L + 2 Cɛ A 2 H + 1 C A 2 L 2, J 4 A 13 L 3 ψ L 6 ψ 2 L 3 A L 6 + A 1 L ψ L 6 ψ 2 L 3 A L 2 Cɛ ψ 2 H + 1 Cɛ A 2 H + 1 C A 2 L 2.
112 Jshan Fan and Tohru Ozawa Insertng these estmates nto (2.2.15) and takng ɛ small enough, we have d dt A 2 dx + curl A 2 dx Cɛ ψ 2 H + 1 C A 2 L2. (2.2.16) Combnng (2.2.14) and (2.2.16) and takng ɛ small enough, we conclude that d dt ψ 2 + A 2 dx + C ψ 2 + curl A 2 dx C ψ 2 + A 2 dx whch mples ψ 2 + A 2 dx + T ψ 2 + curl A 2 dxdt = and hence ψ 1 = ψ 2,A 1 = A 2 and φ 1 = φ 2 due to (2.2.8) and Gronwall s nequalty. Ths completes the proof. 3 Proof of Theorem 1.3 In ths secton, we wll use Theorem 1.1 to prove Theorem 1.3. We only need to prove that ψ, A L 2 (,T; L (Ω)), (3.3.1) whch satsfes the condton (1.1.5). Testng (1.1.7) by ψ and takng real part, we see that η 2 + d dt a a whch gves ψ 2 dr + ψ 2 dr = a a a k ψ r + ψa dr = a, ψ 2 dr + 2 T a dr + 1 a ψ 2 2k 2 r dr + 2 a ( ψ 2 1) 2 dr k ψ 2 r + ψa drdt C. (3.3.2)
3D Gnzburg-Landau model for superconductvty 113 Testng (1.1.8) by A, usng (3.3.2), we fnd that whch yelds and thus a a a 1 d A 2 dr + A 2 A 2 2 dt rdr + r dr 2 a k ψ r + ψa ψa dr k ψ r + ψa ψ L 2 (,a) A L (,a) L 2 (,a) C k ψ r + ψa A r L 2 (,a) L 2 (,a) 1 2 2 A r 2 L 2 (,a) + C k ψ r + ψa, L 2 (,a) A L 2 (,T ;H 1 ) C, (3.3.3) A L 2 (,T ;L ) C, (3.3.4) ψa L 2 (,T ;L 2 ) ψ L (,T ;L 2 ) A L 2 (,T ;L ) C, ψ r L 2 (,T ;L 2 ) C, (3.3.5) ψ L 2 (,T ;L ) C. (3.3.6) Ths completes the proof. 4 Acknowledgments Ths paper s supported by NSFC (No. 11171154). References [1] Z. M. Chen and K. H. Hoffmann, Global classcal solutons to a nonsothermal dynamcal Gnzburg-Landau model n superconductvty, Numer. Funct. Anal. Optm., 18(1997), 91-92. [2] J. Fan and S. Jang, Global exstence of weak solutons of a tmedependent 3-D Gnburg-Landau model for superconductvty, Appl. Math. Lett., 16(23), 435-44.
114 Jshan Fan and Tohru Ozawa [3] J. Fan and H. Gao, Unqueness of weak solutons n crtcal spaces of the 3-D tme-dependent Gnzburg-Landau equatons for superconductvty, Math. Nachr., 283(21), 1134-1143. [4] J. Fan and T. Ozawa, Unqueness of weak solutons to the Cauchy problem for the 3-D tme-dependent Gnzburg-Landau model for superconductvty, Dfferental and Integral Equatons, 22(29), 27-34. [5] J. Fan and T. Ozawa, Unqueness of weak solutons to the Gnzburg- Landau model for superconductvty, Z. Angew. Math. Phys., (publshed onlne: 24 September 211) [6] D. Phllps and E. Shn, On the analyss of a non-sothermal model for superconductvty, Eur. J. Appl. Math., 15(24), 147-179. [7] Q. Tang and S. Wang, Tme dependent Gnzburg-Landau equaton of superconductvty, Physca D, 88(1995), 139-166. [8] F. Zaouch, Tme-perodc solutons of the tme-dependent Gnzburg- Landau equatons of surperconductvty, Z Angew Math. Phys., 54(23), 95-918. Receved: November, 211