On Moving Ginzburg-Landau Vortices

Transcription

1 communications in analysis and geometry Volume, Number 5, 85-99, 004 On Moving Ginzburg-Landau Vortices Changyou Wang In this note, we establish a quantization property for the heat equation of Ginzburg-Landau functional in R 4 which models moving vortices of surface types. It asserts that if the energy is sufficiently small on a parabolic ball in R 4 R + then there is no vortice in the parabolic ball of radius. This extends a recent result of Lin- Rivière [LR3] in R 3.. Introduction. For n andɛ > 0, the heat equation for the Ginzburg-Landau functional on R n is u ɛ t u ɛ = ɛ ( u ɛ )u ɛ, (x, t) R n R +, (.) u ɛ (x, 0) = g ɛ (x), x R n. Here g ɛ : R n R are given smooth maps. Notice that (.) is the negative gradient flow for the Ginzburg-Landau functional: E ɛ (v) = R n Dv + 4ɛ ( v ). (.) Asymptotic behaviors for minimizers of E ɛ in dimension two was first studied by Bethuel-Brezis-Hélein [BBH] (see also Struwe [S] and recent important works by Pacard-Rivière [PR] on steady solutions to (.)). Moreover, such static theories were developed by Rivière [R] [R] and Lin-Rivière [LR] in higher dimensions in connection with codimension two area minimizing currents. In particular, a crucial quantization property for steady solutions to the equation (.) was proved by Lin-Rivière [LR] for n =3. Theasympototics for the equation (.) in dimension two was initiated by Lin [L][L] and also by Jerrard-Soner [JS]. To study the limiting behavior for a sequence of u ɛ which are either static or time-dependent solutions to (.), one encounters the main difficulty that u ɛ may vanish on sets, called Ginzburg-Landau The author is partially supported by NSF. 85

2 86 Changyou Wang vortices, where the equation (.) degenerates and uɛ u ɛ gives nontrivial topological obstruction. On the other hand, it is well-known that the existence of vortices requires the Ginzburg-Landau energy at least of the order log ɛ. Since g ɛ is usually assumed to have E ɛ (g ɛ )=O(log( ɛ )), we have E ɛ (u ɛ (,t)) O(log ). (.3) ɛ From the analytic point of view, the size estimate for the bad set, B ɛ = {(x, t) R n R + : u ɛ (x, t) }, plays a critical role for W,p compactness for p (, ) (see [BBH] and [PR]). To obtain sharp size estimates of B ɛ, one needs the so-called η-compactness property for u ɛ which roughly says that if E ɛ (u ɛ )isoforderηlog ɛ for sufficiently small η>0then there is no interior bad points for u ɛ. This has been established for (i) minimizers of E ɛ by Rivière [R] [R] for n = 3 and by Lin-Rivière [LR] for n 3; and (ii) critical points of E ɛ by Lin-Rivière [LR] for n =3. Moreover,such η-compactness property was also proved for solutions to the equation (.) by Lin-Rivière [LR3] in the case n = 3. It was believed that their result still holds for R n with n 4. In this note, we confirm such a belief in the case that n = 4. More precisely, we prove Theorem A. For n =4and ɛ>0, letu ɛ : R 4 R + R be solutions to the equation (.) satisfying u ɛ and Du ɛ C 0 ɛ. Then there exist ɛ 0 > 0 and η>0 depending only on C 0 such that if, for (x 0,t 0 ) R 4 R +, 0 <ρ< t 0,andɛ ɛ 0, it holds then ρ 4 t0 t 0 ρ R4 ( Du ɛ + ( u ɛ ) 4ɛ u ɛ (x 0,t 0 ). )e x x0 4(t t 0) η log ρ ɛ (.4) We would like to remark that the idea developed by Lin-Riviere [LR] [LR3] was to interpolate between the Lorentz spaces L, and L, on generic two dimensional slices which therefore work very well in R 3, but it seems unclear how to extend the idea of [LR] to R n with n 4. On the other hand, there is the interpolation technique between L and L developed by Bethuel-Brezis-Orlandi [BBO] for the static case in R n for all n 3, where they made very clever use of the energy monotonicity formula for static solutions to the equation (.). Our method starts with the observation that there exists an energy monotonicity inequality for all time slice R n {t} when n = 4, which enables us to adapt the main ideas from [BBO] and

3 On Moving Ginzburg-Landau Vortices 87 some of those ideas from [LR3]. Since one can always view solutions to the equation (.) in R 3 R + as solutions to the equation (.) in R 4 R + which are independent of the fourth spatial variable, we also gives a different proof of the main theorem of [LR3]. As an important consequence of the η-compactness theorem, it can be shown that the vortice is moving by its mean curvature in the generalized sense. The paper is organized as follows. In, we derive the needed elliptic type energy inequality in R 4 {t}. In 3, we recall the parabolic type energy monotonicity inequalities by Struwe [S] and Lin-Rivière [LR3] and extract a good time slice. In 4, we illustrate the main estimate by performing an intrinsic Hodge decomposition on good time slices and prove theorem A. Added in Proof. After the paper has been accepted for publication, the author received a preprint by Bethuel-Orlandi-Smets [BOS] where theorem A is proved for any dimension n 5 through a much more delicate method.. Euclidean monotonicity at time slice for n =4. This section is devoted to the slice monotonicity inequality (.) for u ɛ : R n R + R satisfying (.) for n =4. Forn 4, x R n, r>0, and t>0, we denote E ɛ (x, r) := B r(x) ( Du ɛ + n( u ɛ ) )(y) dy. 4(n )ɛ Then we have Lemma.. For n 4 and ɛ>0, letu ɛ : R n R + R be a solution to (.). Then, for any (x, t) R n R + and r>0, we have d dr (r n E ɛ (x, r)+ r3 n ɛ 3 n B r(x) t u ɛ r ) r n ɛ r + ( u ɛ ) (n )ɛ + r3 n 3 n B r(x) B r(x) ɛ t u ɛ. (.) r Proof. We assume that x = 0 and denote u for u ɛ. Multiplying (.) by

4 88 Changyou Wang x Du, integrating over B r, and using integration by parts, we obtain u t x Du = ux Du B r B r 4ɛ x D( u ) = D (Dux Du) Du D(x Du) x D Br ( u ) 4ɛ = r B r r Du B r x D( Br Du + ( u ) 4ɛ ) = r ( B r r Du ( u ) 4ɛ ) +(n ) Br ( Du + n( u ) 4(n )ɛ ). This yields (n )E ɛ (0,r)= u t x Du + r ( B r B r Du r + ( u ) 4ɛ ). Therefore d dr (r n E ɛ (0,r)) = ( n)r n E ɛ (0,r)+r Br n ( Du + n( u ) 4(n )ɛ ) = r n u t x Du + r n ( B r B r r + ( u ) (n )ɛ ). Observe that r n u t x Du r n u t B r B r r = d dr ( r3 n 3 n B r u t r r3 n )+ u t 3 n B r r. Hence d dr (r n E ɛ (0,r)+ r3 n u t 3 n B r r ) r n ( B r r + ( u ) r3 n )+ (n )ɛ 3 n B r t u r.

5 On Moving Ginzburg-Landau Vortices 89 This completes the proof of (.). Now we have the following slice energy monotonicty inequality for n = 4. Proposition.. For ɛ>0, letu ɛ : R 4 R + R be a solution to (.). Then, for any (x, t) R 4 R + and 0 r R<, we have R r dr E ɛ (x, r)+ r r R E ɛ (x, R)+ B r(x) B R (x) ( u ɛ r +(4ɛ ) ( u ɛ ) ɛ t. (.) In particular, y x ( u ɛ(y) ) ɛ B R (x) 8R E ɛ (x, R)+8 ɛ B R (x) t. (.3) Proof. Write u for u ɛ. It is clear that (.), with r tending to zero, gives (.3). Therefore, it suffices to prove (.). For n = 4, integrating (.) from r to R, wehave R E ɛ (x, R) r E ɛ (x, r)+r + R r s B R (x) t u r r ( r + ( u ) 4ɛ ) For n = 4, we have the following estimate: r B r(x) t u r r B r(x) r B r(x) B r(x) R r s r + r + t u r B r(x) B R (x) Applying the Young inequality, we also have, for r s R, s t u r s r + so that R r s t u r R r s t u r. t t. t r + B R (x) t.

6 90 Changyou Wang Putting these inequality together, we obtain R E ɛ (x, R) r E ɛ (x, r) R + s This implies (.). r B R (x) t ( u r + ( u ) 4ɛ ). 3. Parabolic monotonicity and extracting a good time. In this section, we gather together two more parabolic energy monotonicty inequalities by Struwe [S] and by Lin-Rivière [LR3]. The formula is valid for all n. Lemma 3. (Energy monotonicity). Let u ɛ : R n R + R be solutions to (.) and (x 0,t 0 ) R n R +. Then, for any 0 <ρ t 0, we have d dρ [ t0 ρ n = ρ n+ t 0 ρ t0 t 0 ρ Rn ( Du ɛ + ( u ɛ ) R n [ ( u ɛ ) ɛ 4ɛ )e x x0 4(t t 0) ] (t 0 t) (x x 0) Du ɛ +(t t 0 ) u ɛ t ]e x x0 4(t t 0). (3.) Proof. It follows exactly the same lines of the proof of [LR3] Lemma.. The next identity indicates how the energy decays along the spatial infinity. Lemma 3.. Under the same notations as Lemma 3.. For any t 0 > 0 and 0 <ρ t 0, the following holds: t0 Rn {( + x t 0 ρ 4(t 0 t) )( Du ɛ + ( u ɛ ) 4ɛ ) 4(t 0 t) x Du ɛ +(t t 0 ) u ɛ t }e x 4(t t 0 ) ρ [ R n R n {t 0 ρ } Du ɛ + ( u ɛ ) 4ɛ ]e x 4ρ t0 x + t 0 ρ 4(t 0 t) Du ɛ [x Du ɛ +(t t 0 ) u x ɛ 4(t t ]e 0). (3.) t

7 On Moving Ginzburg-Landau Vortices 9 Proof. It again follows from the same argument as that of [LR3] Lemma.. Now we describe the extraction of a good time slice as follows. We follow [LR3]. closely and the reader may refer to [LR3] for the detail. For simplicity, we assume that (x 0,t 0 )=(0, 0) and (.) holds in R 4 R. Assume that (.4) holds for some ρ>0. Then, by integrating (3.) from ɛ to ρ and using the Fubini s theorem, there exists a ρ = ρ ɛ (ɛ, ρ) such that 0 ρ 4 j ɛ (u ɛ )e x 4t η. (3.3) ρ R 4 Here j ɛ (u ɛ ) t x Du ɛ +t u ɛ t + ( u ɛ ) ɛ (3.4) so that ρ inf j ɛ (u ɛ )e x 4ρ η. (3.5) ρ ( ρ,ρ ) R 4 { ρ } Denote E = 0 ρ 4 e ɛ (u ɛ )e x 4t (3.6) R 4 where Then (3.) implies E inf ρ ρ ρ ρ 4 inf ρ ρ ρ ρ 4 inf ρ ρ ρ ρ 4 ρ e ɛ (u ɛ ) ( Du ɛ + ( u ɛ ) 4ɛ ). 0 ρ 0 ρ As in [LR3], we may assume so that inf ρ ρ ρ ρ 4 e ɛ (u ɛ )e x 4t R 4 + e ɛ (u ɛ )e x 4t R ρ 4 e ɛ (u ɛ )e x 4t R 4 x +4η. e ɛ (u ɛ )e 4ρ E inf R 4 ρ ρ 0 ρ 5 ρ j ɛ (u ɛ )e x 4t R 4 j ɛ (u ɛ )e x 4t R 4 E>>Cη (3.7) ρ ρ ρ ρ 4 x e ɛ (u ɛ )e 4ρ. (3.8) R 4

8 9 Changyou Wang Therefore, there exists a ρ 0 [ ρ,ρ ] such that max{ ρ ρ 0 j ɛ (u ɛ )e x 4t, j ɛ (u ɛ )e x 4ρ 0 } Cη, (3.9) R 4 R 4 { ρ 0 } ρ 0 0 ρ 4 e ɛ (u ɛ )e x 4t E C 0 0 ρ 0 R 4 ρ 4 e ɛ (u ɛ )e x 4t, 0 ρ 0 R 4 (3.0) ρ e ɛ (u ɛ )e x 4ρ 0 E C 0 R 4 { ρ 0 } ρ e ɛ (u ɛ )e x 4ρ 0. 0 R 4 { ρ 0 } (3.) These inequalities, combined with Lemma 3., also yield ρ 0 R4 { ρ0} x Observe that (3.9) and (3.) also imply t e ɛ(u ɛ )e x 4t CE. (3.) ɛ R 4 { ρ 0 } t e x 4ρ 0 CE. (3.3) In particular, for any λ>> to be chosen later, one has B 4λρ0 { ρ 0 } ɛ t Ce 4λ E. (3.4) Hence, applying the monotonicity inequality (.3) for u ɛ at t = ρ 0,we obtain the following key inequality: Bλρ0 (x) { ρ0} y x ( u ɛ ) On the other hand, (3.9) also yields ρ 0 Notice that (3.) implies that ρ 0 B4λρ0 { ρ0} ( u ɛ ) ɛ Ce 4λ E, x B λρ0. (3.5) ɛ Ce 4λ η. (3.6) x e ɛ (u ɛ )e 4ρ 0 (R 4 \B λρ0 ) { ρ 0 } C E. λ (3.7)

9 On Moving Ginzburg-Landau Vortices 93 This, combined with suitable choice of λ>>according to the Fubini s theorem, gives e ɛ (u ɛ )e x 4ρ 0 C E, (3.8) ρ 0 B λρ0 λ3 x 4ρ 0 E 3. (3.9) ρ 0 B λρ0 { ρ 0 } e ɛ (u ɛ )e Together with the inequalities from (3.9) to (3.8), we can proceed estimating E by estimating the left hand side of (3.9) as in 4 below. 4. An intrinsic Hodge decomposition to estimate u ɛ du ɛ. This section is devoted to the proof of theorem A. The main techinical part is to obtain L -estimate of u ɛ du ɛ on B λρ0 { ρ 0 }. To do it, we need an intrinsic Hodge decompostion of u ɛ du ɛ at t = ρ 0. For this purpose, we adopt ideas from both [BBO] and [LR3]. In this section, we work on t = ρ 0 and denote u as u ɛ. First, define H : B λρ0 R by the auxillary Neumann problem: (e x i x 4ρ 0 H )= (e x i x i H r = u u r, on B λρ 0. (4.) x 4ρ 0 u u x i ), in B λρ0, (4.) Observe that (e x 4ρ 0 u u ) = e x 4ρ 0 ( ρ 0 u t + x Du) u x i x i ρ 0 e x ρ 4ρ 0 u t + x Du 0 ρ 0 ρ 0 x 4ρ e 0 (j ɛ (u ɛ )) so that we can establish the following estimate for DH. Lemma 4. Under the same notations as above. There exists a C λ > 0 such that ρ 0 B λρ0 DH e x 4ρ 0 C λ ρ 0 j ɛ (u ɛ )e x 4ρ 0 B λρ0 + Cλ ρ 0 B λρ0 r e x 4ρ 0. (4.3)

10 94 Changyou Wang In particular, we have ρ 0 B λρ0 DH e x 4ρ 0 C λ η + CE λ. (4.4) Proof. First, notice that (4.4) is the consequence of (4.3) and the inequalities (3.9) and (3.8). Secondly, the proof of (4.3) can be obtained by copying lines of arguments of [LR3] Lemma.4. Observe that (4.) and (4.) can be rewritten as (e 4ρ 0 ( H u u )I Bλρ0 )=0 (4.5) x i x i x i x in the sense of distributions on R 4,hereI Bλρ0 denotes the characteristic function of the ball B λρ0. Define δ C (R +,R + )byδ(r) =r for 0 r λρ 0, δ(r) =(4λρ 0 ) for r 4λρ 0, and (λρ 0 ) δ(r) (4λρ 0 ) for r [λρ 0, 4λρ 0 ]. Let g ij (x) =e δ( x ) 4ρ 0 δ ij be the new conformal metric on R 4, which is readily seen to be bilipschitzly equivalent to the standard metric on R 4.Denoted g as the adjoint of d with respect to g and g d g d + dd g as the Laplace-Beltrami operator with respect to g. Noticethat(4.5) is equivalent to d g ((dh u du)i B λρ0 )=0, in R 4. (4.6) Therefore, by the classical Hodge decompostion theory (see, e.g., Iwaniec- Martin [IW]), there exists a -form α H g (R 4, Λ (R 4 )) such that d gα =(dh u du)i Bλρ0, dα =0, (4.7) Dα L g (R 4 ) C( Du L g (B λρ0 ) + DH L g (B λρ0 )). (4.8) Here Hg (or L g resp.) denotes H (or L resp.) with respect to g. Notice that Df L g (R4 ) = Df (x)e δ( x ) 4ρ 0. R 4 In order to estimate Dα in L g, we modify the approach of [BBO] as follows. Let β (0, be determined later, and f : R + [, β ]beasmooth function such that f(t) = t for t β, f(t) =fort β, and f 4. On R 4, define the function a such that a(x) =f ( u (x)) on B λρ0

11 On Moving Ginzburg-Landau Vortices 95 and a(x) = elsewhere, so that 0 a 4β holds on R 4.Observethat f ( u )u du = f( u )u d(f( u )u). Therefore, (4.7) implies d(ad gα) =I Bλρ0 d(f( u )u) d(f( u )u) + f( u )u du d x σ g B λρ0 d(i Bλρ0 adh) = ω + ω + ω 3 (4.9) where σ g B λρ0 denotes the surface measure of B λρ0 with respect to the metric g. Observethatif u β then d(f( u )u) d(f( u )u) =d( u u ) d( u u )= 0, otherwise we have β ( u ) so that ω (x) Cɛ Cβ ( u(x) ) ɛ, x B λρ0. (4.0) Using the fact that dα =0,weget g α = dd g α = d(ad g α)+d(( a)d g α)=ω +ω +ω 3 +d(( a)d gα) (4.) Denote G(x, y) =G( x y ) as the fundamental solution of g on R 4.Then it follows from the bilipschitz equivalence between g and the euclidean metric on R 4 that there exists a C>0such that Ce 4λ x y G(x, y) Ce 4λ x y, D y G(x, y) Ce 4λ x y 3. (4.) Let α i = G ω i for i 3. Then α 4 = α 3 i= α i solves g α 4 = d(( a)d gα). (4.3) Direct calculations, using a 4β and smallness of β, yield 3 Dα 4 L g (R4 ) Cβ Dα i L g (R4 ). (4.4) The main difficulty comes from the estimate of Dα which can be done as follows, due to the monotonicity inequality (3.5) and (3.6). Indeed, by the maximum principle, we have α L (R 4 ) = α L (B λρ0 ) and, by (4.0), (4.), and (3.5), α L (B λρ0 ) sup G(x y) ω (y) x B λρ0 B λρ0 C λ β sup x y Bλρ0 ( u(y) ) x B λρ0 i= C λ β E. (4.5) ɛ

12 96 Changyou Wang This, combined with (3.6), implies Dα L g (R4 ) ω L (R 4 ) α L (R 4 ) C λ β ρ 0ηE. (4.6) For α 3, using integration by parts and (4.4), we have Dα 3 L g (R4 ) C DH L g (B λρ 0 ) C ληρ 0 + Cρ 0 E λ. (4.7) For α, we can modify the Lemma A of appendix in [BBO] to conclude that Dα L g (R4 ) Cλρ 0 Du L g ( B λρ 0 ) (4.8) this, combined with (3.8), gives Dα L g (R4 ) Cρ 0 E. (4.9) λ Putting these estimates for α i for i 4 and Lemma 4. together, we then obtain ρ 0 B λρ0 u du e x 4ρ 0 C λ η + CE λ + C λβ ηe. (4.0) This, combined with the fact that 4 u du =4 u du + d u and the following estimate (see (.67) of [LR3] page 845) D u e x 4ρ 0 Cη 4 E + Cη (4.) B λρ0 ρ 0

13 On Moving Ginzburg-Landau Vortices 97 implies ρ Du e x 4ρ 0 0 B λρ0 = ρ ( u ) Du e x 4ρ B λρ0 ρ u Du e 0 B λρ0 C ( u ) ρ Du e x 4ρ 0 0 B λρ0 ɛ + 4 ρ ( u du + D u )e x 4ρ 0 0 B λρ0 C λ ρ Bλρ0 ( u ) ɛ e x 4ρ 0 +(C λ η + Cη ) +(λ + Cλ + C λ β η + Cη 4 )E (λ + Cλ + C λ β η + Cη 4 )E +(C λ η + Cη ) (4.) Therefore, for any given δ>0, we can first choose a sufficiently large λ> and a sufficiently small β and then choose much smaller η so that x 4ρ 0 E Cδ (4.3) so that, using the monotonocity inequality (3.) again, 0 ɛ 6 ɛ B ɛ(0) ( u ɛ ) ɛ e x 4t δ. (4.4) This, combined with the fact that Du ɛ Cɛ, yields u ɛ (0, 0).Therefore, the proof of theorem A is complete. References. [BBH] F. Bethuel, H. Brezis, F. Hélein, Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications, 3. Birkhuser Boston, Inc., Boston, MA, 994. [BBO] F. Bethuel, H. Brezis, G. Orlandi, Small energy solutions to the Ginzburg-Landau equation. C. R. Acad. Sci. Paris Sr. I Math. 33 (000), no. 0,

14 98 Changyou Wang [BOS] F.Bethuel,G.Orlandi,D.Smets,Convergence of parabolic Ginzburg- Landau equation to motion by mean curvature. To appear in, Annal. Math. [IM] [JS] [L] [L] [LR] [LR] [LR3] [PR] [R] [R] T. Iwaniec, G. Martin, Quasiregular mappings in even dimensions. Acta Math. 70 (993), no., 9 8. R. Jerrard, H. Soner, Dynamics of Ginzburg-Landau vortices. Arch. Rational Mech. Anal. 4 (998), no., F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math. 49 (996), no. 4, F. H. Lin, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension- submanifolds. Comm. Pure Appl. Math. 5 (998), no. 4, F. H. Lin, T. Rivière, Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. (JEMS) (999), no. 3, F. H. Lin, T. Rivière, A quantization property for static Ginzburg- Landau vortices. Comm. Pure Appl. Math. 54 (00), no., F. H. Lin, T. Rivière, A quantization property for moving line vortices. Comm. Pure Appl. Math. 54 (00), no. 7, F. Pacard, T. Rivière, Linear and nonlinear aspects of vortices. The Ginzburg-Landau model. Progress in Nonlinear Differential Equations and their Applications, 39. Birkhuser Boston, Inc., Boston, MA, 000. x+34 pp. ISBN: T. Rivière, Lignes de tourbillon dans le modle ablien de Higgs. (French) [Vortex lines in the Higgs abelian model] C. R. Acad. Sci. Paris Sr. I Math. 3 (995), no., T. Rivière, Line vortices in the U()-Higgs model. ESAIM Contrle Optim. Calc. Var. (995/96), (electronic). [S] M. Struwe, On the asymptotic behavior of minimizers of the Ginzburg-Landau model in dimensions. Differential Integral Equations 7 (994), no. 5-6, [S] M. Struwe, On the evolution of harmonic maps in higher dimensions. J. Differential Geom. 8 (988), no. 3,

15 On Moving Ginzburg-Landau Vortices 99 Department of Mathematics University of Kentucky Lexington, KY Received September 9, 00.