Local well-posedness for the Landau-Lifshitz equation with helicity term
aa r X i v : . [ m a t h . A P ] D ec Local well-posedness for the Landau-Lifshitzequation with helicity term ∗† Ikkei Shimizu
Abstract
We consider the initial value problem for the Landau-Lifshitz equa-tion with helicity term (chiral interaction term), which arises from theDzyaloshinskii-Moriya interaction. We prove that it is well-posed locally-in-time in the space ¯ k + H s for s ≥ s ∈ Z and ¯ k = t (0 , , k + H s for s > s ∈ R . Our proof is based on the analysis via the modified Schr¨odingermap equation. We consider the initial value problem for the Landau-Lifshitz equation withhelicity term: ( ∂ t u = u × ( − ∆ u + b ∇ × u ) on R × R u ( x,
0) = u ( x ) , (1.1)where u = u ( x, t ) is the unknown function from R × R to the sphere S = (cid:8) y ∈ R : | y | = 1 (cid:9) ⊂ R ,b ∈ R is a constant, × denotes the vector product in R , and the last term inthe right hand side, called the helicity term or chiral interaction term , is definedby ∇ × u = ∂ u − ∂ u ∂ u − ∂ u . In the physical context, (1.1) is considered as a mathematical model of theevolution of magnetization vectors in helimagnets, and the helicity term arisesfrom the physical effect called the Dzyaloshinskii-Moriya interaction. For thedetailed background of the equation, see for example [16, 18] and the referencestherein. ∗ † Keywords and phrases: Landau-Lifshitz equation, Helicity term, Well-posedness, ModifiedSchr¨odinger map equation, Magnetic potential E ( u ) := Z R |∇ u | + b u · ( ∇ × u ) dx. Another remarkable feature is that the first term of the left hand side of (1.1)does not contribute to the growth of (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L , where ¯ k = t (0 , , ddt Z R | u − ¯ k | dx = 2 b Z R ( u ∂ u + u ∂ u ) dx. (1.2)In other words, the helicity term breaks the L -conservation and the spatialsymmetry of the solutions. We note that the scaling symmetry is also destroyedby the helicity term.The corresponting energy-minimizing problem has been considered, for ex-ample, by [8, 16, 18], where some other additional terms are also taken intoaccount such as easy axis anisotropy. The energy minimizers with nontrivialhomotopy are called chiral skyrmion , which has been attracting a lot of inter-est. On the other hand, the initial value problem has little been investigated sofar, while D¨oring and Melcher [8] briefly mentions the local-in-time solvabilityin sufficient high regularities. The aim of the present paper is to show the localwell-posedness for (1.1) in low regularities.In the case when b = 0, in which (1.1) is especially called Schr¨odinger maps ,has been extensively studied from various aspects. The local well-posednesswith large data is shown in [17] (see also [25]). For small data, the global well-posedness is proved in critical regularities by [1, 2, 15, 24] (see also [12, 23]).Some results on global-in-time regularity for large data can be seen in [3, 9].The asymptotic behavior of the solutions is also explored by [2, 4, 11, 12, 13,14, 19, 21]. The Landau-Lifshitz equation which contains other terms arisingfrom physical effects, such as easy axis or easy plane anisotropy, is also studiedin [7], for example.Our first main theorem is the local well-posedness for (1.1) with initial datain the Sobolev classes. We define, for function spaces Y ,¯ k + Y := { u : R → S : u − ¯ k ∈ Y } , ¯ k = t (0 , , . Theorem 1.1. (i) (Existence) Let s ≥ be an integer. Then, for u ∈ ¯ k + H s , (1.1) has a (weak) solution u ∈ L ∞ ([0 , T ] : ¯ k + H s ) , where T = T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s , s ) > is a non-increasing function of (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s .(ii) (Uniqueness) For intervals I ⊂ R with ∈ I , the solutions to (1.1) areunique in C ( I : ¯ k + L ∩ L ∞ ) ∩ L ∞ ( I : ¯ k + H ) . (1.3) (iii) (Continuity) Let s ≥ be an integer, and let { u ( n )0 } ∞ n =1 be a boundedsequence in ¯ k + H s . Suppose that u ( n )0 → u in ¯ k + H s − ǫ for ǫ ∈ (0 , s − . Let u ( n ) , u be the solution in the class (1.3) with initial data u ( n )0 , u , respectively.Then, u ( n ) − u → in L ∞ t H s − ǫ . f : R → S zero-homotopic if f is homotopic to a constantmap. Then we can extend the regularities of local well-posedness into ¯ k + H s for s > s ∈ R . Theorem 1.2.
Let s > be a real number, and assume that u ∈ ¯ k + H s iscontinuous and zero-homotopic. Then the followings hold.(a) (Regularity) (1.1) has a solution in the class C t (¯ k + H s ) .(b) (Blow-up criterion) Let u ∈ C ([0 , T max ); ¯ k + H s ) be a solution to (1.1), where T max is the maximal existence time in this class. Suppose T max < ∞ . Then forany ǫ > , we have lim t → T max − (cid:13)(cid:13) u ( t ) − ¯ k (cid:13)(cid:13) H ǫ = ∞ . (c) (Bound) There exists T = T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s , s ) such that solutions u to (1.1)satisfy (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ ([0 ,T ]: H s ) ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s , s ) . (d) (Continuity) The solution map is continuous from H s to C t (¯ k + H s ) . Fur-thermore, the following statement is true: Let u (0) , u (1) be two solutions to (1.1)with u ( j ) | t =0 = u ( j )0 ∈ ¯ k + H s for j = 0 , . Then, there exists T = T ( M, s ) suchthat (cid:13)(cid:13)(cid:13) u (1) − u (0) (cid:13)(cid:13)(cid:13) L ∞ ([0 ,T ]: H s − ) ≤ C ( M, s ) (cid:13)(cid:13)(cid:13) u (1)0 − u (0)0 (cid:13)(cid:13)(cid:13) H s − , (1.4) where M = (cid:13)(cid:13)(cid:13) u (0)0 − ¯ k (cid:13)(cid:13)(cid:13) H s + (cid:13)(cid:13)(cid:13) u (1)0 − ¯ k (cid:13)(cid:13)(cid:13) H s . Remark 1.1. (i) Theorems 1.1 and 1.2 also hold true for negative direction intime. Indeed, if we consider the transform ˜ u ( t, x , x ) := u ( − t, x , x ) u ( − t, x , x ) u ( − t, x , x ) , ˜ u again satisfies (1.1) by switching the sign of b .(ii) We note that Theorem 1.2 allows s to be non-integer, while Theorem 1.1 isrestricted to integer exponents. We briefly mention the strategy of the proof. For the proof of Theorem 1.1(i), we employ hyperbolic-type regularization invented by McGahagan [17]. Indetail, we add second time-derivative term δ u × ∂ tt u to the equation, and thentake the limit δ →
0, which yields a solution. The advantage of this argumentcompared with the parabolic regularization is that we can cancel the secondderivative term ∆ u in the process of obtaining uniform bounds for perturbedsolutions in δ , which results in the reduction of regularities in the estimate (seethe estimate for I in Section 2, while its proof is referred to [17]). We provide asimplified version of proof from the original argument, where the δ -dependenceof the maximal existence time of solutions to (2.1) is obtained by a standardscaling argument.For the proof of Theorem 1.1 (ii), (iii), we follow the geometric energymethod due to McGahagan [17]. More precisely, we introduce a geodesics con-necting two solutions, and the difference of the derivatives between two solutions3s measured in the same tangent space via parallel transport. Our argument inthe present paper is rather based on the Yudovich argument [27] than the origi-nal paper, and it has already been applied in the case when b = 0 by the author[22].For the proof of Theorem 1.2, we introduce the method using orthonor-mal frames, which was first applied to the study of Schr¨odinger maps in [6].The remarkable advantage is that the problem (1.1) is reduced to a systemof semi-linear Schr¨odinger equations, called modified Schr¨odinger equation (see(4.5)). A remarkable difficulty here is that in (4.5), the helicity term appears asquadratic derivative nonlinearities, which are known to be difficult to control asa perturbation of the free Schr¨odinger equation. To overcome that, we exploita skew-adjoint structure of helicity term. More precisely, the bad part in thenonlinearity which contains derivative losses can be absorbed into the magneticterms, and then we can cancel them in the energy method based on the Sobolevspaces associated with new magnetic laplacian. Since the potential depends onthe solutions, we need to estimate the commutator between time differentiationand the associated differential operators, which forces us to assume the regu-larity of solutions bigger than 2. Our argument is reminiscent of the study ofMaxwell-Schr¨odinger system by [26].We make here some remarks on the method of orthonormal frames. As longas the author knows, the present work is the first study of Landau-Lifshitz equa-tion using the modified Schr¨odinger map equation in the case when other addi-tional terms than Schr¨odinger map term exist in the equation. We note that thezero-homotopic condition is required in the construction of orthonormal frameswith certain decay at spatial infinity, otherwise we have no quantitative boundfor the frames. We also note that the modified Schr¨odinger maps naturallyinduces magnetic potentials, arising from the geometry of the target, and theindefiniteness of orthonormal frames provides a gauge symmetry of equations.We adopt here the Coulomb gauge condition, which gives a simple relation be-tween differentiated fields and connection coefficients. A further discussion ofthis gauge can be seen in Remark 4.1.The organization of the present paper is as follows. Section 2 is devoted tothe construction of solutions (Theorem 1.1 (i)). We discuss the uniqueness andthe continuity of solutions with respect to the initial date in Section 3 (Theorem1.1 (ii) and (iii)). In Section 4, we construct the orthonormal frame and derivethe properties on them. The nonlinear analysis for modified Schr¨odinger mapequation is summarized in Section 5, which concludes Theorem 1.2 (a), (b) and(c). Section 6 is devoted to the proof of Theorem 1.2 (d).Here is the summary of notations. For a Banach space X and a function f : [0 , T ] → X with T >
0, we define k f k L pT X := ( R T k f ( t ) k X pdt ) /p . Wealso define C t X as the space of all continuous function from some interval I to X , where the interval will vary in each situations. We often put the symbol ofvariables in the right bottom of the space, like L px , in order to make clear whichvariable the integration is curried out in. For s ∈ R , we write the maximalinteger less than or equal to s as ⌊ s ⌋ . We write ∂ m for the differentiation withrespect to x m for m = 1 ,
2. We shall consider the time variable as 0-th variable,thus for example, ∂ stands for the differentiation with respect to t . We use C for representing a constant, whose value varies in each situations. If we make4lear that C depends on some quantity σ , we write it as C σ or C ( σ ). Theset of all Schwartz function from R to C is denoted by S ( R ). We definethe Fourier transform by F [ f ]( ξ ) := R R e − ix · ξ f ( x ) dx , and the inverse Fouriertransform by F − [ g ]( x ) := R R e ix · ξ g ( ξ ) dξ . Let |∇| σ denote the operator definedby |∇| σ f := (2 π ) − F − | · | σ F f for σ ∈ R . Let H sr = H sr ( R ) be the Sobolevspace H sr := { f : R → C | k f k H sr := (cid:13)(cid:13) (2 π ) − F − (1 + | · | ) s/ F f (cid:13)(cid:13) L r < ∞} . When r = 2, we especially write it as H s . In this section, we show the existence of weak solutions to (1.1), which is statedin Theorem 1.1 (i). We follow the hyperbolic regularizing argument of McGa-hagan [17], while we make some technical modifications to the original one. Wefirst consider the perturbed equation: − δ u × ∂ tt u + ∂ t u = u × ( − ∆ u + b ∇ × u ) (2.1)with δ >
0. (2.1) determines the evolution of maps from R to S . Now we showthat the initial-value problem for (2.1) is locally well-posed by using standardcontraction argument. Proposition 2.1.
Let s ≥ be an integer, and let ( u , v ) ∈ (¯ k + H s ) × H s − .Then there exists a unique solution u with the regularity u − ¯ k ∈ C ([0 , T ] : H s ) ∩ C ([0 , T ] : H s − ) for some T = T ( k u k H s , δ k v k H s − ) , where T ( · , · ) is apositive, non-increasing function with respect to both variables and independentof δ .Proof. We change the scaling of functions as follows: U ( x, t ) := u ( x, δt ) . We also write U ( x, t ) := u ( x, δt ) and V := v ( x, δt ). Then (2.1) can berewritten as ∂ tt U = ∆ U + F δ ( U, ∂ t U ) , where F δ ( U, V ) := δ − U × V + |∇ U | U − | V | U − b X j =1 U j · U × ∂ j U. By the Duhamel formula, (
U, V ) = (
U, ∂ t U ) can be written as U = Φ U ( U, V ) := cos( √− ∆ t )( U − ¯ k ) + sin( √− ∆ t ) √− ∆ V + ¯ k − Z t sin( √− ∆( t − τ )) √− ∆ F ( U, V ) dτ,V = Φ V ( U, V ) := − √− ∆ sin( √− ∆ t )( U − ¯ k ) + cos( √− ∆ t ) V − Z t cos( √− ∆( t − τ )) F ( U, V ) dτ. T, M >
0, we set the complete metric space W T,M := (cid:8) ( U, V ) ∈ C ([0 , T ] : ¯ k + H s ) × C ([0 , T ] : H s − ) |k U − U k L ∞ T H s + k V k L ∞ T H s − ≤ M o with the metric d W T,M (( U, V ) , ( ˜ U , ˜ V )) := (cid:13)(cid:13)(cid:13) U − ˜ U (cid:13)(cid:13)(cid:13) L ∞ T H sx + (cid:13)(cid:13)(cid:13) V − ˜ V (cid:13)(cid:13)(cid:13) L ∞ T H s − . Then by the standard argument, we can show that the map Φ := (Φ U , Φ V ) is acontraction on W T,M by setting T := C δ (cid:0)(cid:13)(cid:13) U − ¯ k (cid:13)(cid:13) H s + k V k H s − + 1 (cid:1) , M := C (cid:0)(cid:13)(cid:13) U − ¯ k (cid:13)(cid:13) H s + k V k H s − (cid:1) , for some proper choice of C , C >
0, which concludes the existence part of thelemma. The uniqueness part can be shown in the similar way.
Proposition 2.2.
Let s ≥ be an integer, and let u : R × [0 , T ] → R bea solution to (2.1), where T is as in Proposition 2.1. Then there exists ˜ T =˜ T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s , δ k v k H s − , k v k H s − ) > and C = C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s , δ k v k H s − , k v k H s − ) > such that k ∂ t u k L ∞ ˜ T H s − x ≤ C. Proof.
We first prepare some notations. We set I k := { , } k , and | γ | := k for γ ∈ I k . Let t ∈ [0 , T ]. For vector fields X = X ( x, t ) ∈ T u ( x,t ) S , the covariant derivativewith respect to α -th index is denoted by D j X := ∂ α X + ( X · ∂ α u ) u for α = 0 , , . For k ∈ N ∪ { } , we write D γ X := D γ · · · D γ k X for γ = ( γ , · · · γ k ) ∈ I k . Then the following inequalities hold true for k ∈ N : k X k H kx ≤ C k X γ ∈I k ′ ≤ k ′ ≤ k k D γ X k L x P k X γ ∈I k ′ ≤ k ′ ≤ k k D γ X k L x , (2.2) X γ ∈I k ′ ≤ k ′ ≤ k k D γ X k L x ≤ C k k X k H kx P k ( k X k H kx ) , (2.3)where P k is some polynomial depending only on k , and C k is a constant inde-pendent of X (see [17] for the detailed proof). Thus it suffices to show k D γ ∂ t u k L ∞ t L x ≤ C (cid:16) t k ∂ t u k L ∞ t H s − x (cid:17) (2.4)6or γ ∈ I k , 0 ≤ k ≤ s −
2, which completes the bootstrap with respect to k ∂ t u k L ∞ t H s − x .We set J := u × · . Then from (2.1), we have δ D γ D t ∂ t u + JD γ ∂ t u = X m =1 D γ D m ∂ m u − bD γ ( u J∂ u + u J∂ u ) . If we take L x -inner product with JD γ D t ∂ t u , by orthogonality we have h D γ ∂ t u, D γ D t ∂ t u i L x = X m =1 h D γ D m ∂ m u, JD γ D t ∂ t u i L x − b h D γ ( u J∂ u + u J∂ u ) , JD γ D t ∂ t u i L x , which is equivalent to12 ∂ t k D γ ∂ t u k L x = ∂ t ( A + A ) + X l =1 I l , where A := X m =1 h D γ D m ∂ m u, JD γ ∂ t u i L x ,A := − b X j =1 h D γ ( u j J∂ j u ) , JD γ ∂ t u i L x ,I = −h D γ ∂ t u, [ D γ , D t ] ∂ t u i L x ,I = − X m =1 h D γ D m ∂ m u, J [ D γ , D t ] ∂ t u i L x ,I = − X m =1 h D t D γ D m ∂ m u, JD γ ∂ t u i L x ,I = − b X j =1 h D γ ( u j ∂ j u ) , J [ D γ , D t ] ∂ t u i L x ,I = b X j =1 h D t D γ ( u j ∂ j u ) , JD γ ∂ t u i L x . Thus we have k D γ ∂ t u k L ∞ t L x ≤ k D γ ∂ t u k L x (cid:12)(cid:12)(cid:12) t =0 + 2( | A | + | A | ) | t = t t =0 + C X l =1 Z t | I l | dt. (2.5)Let ǫ > | A || t = t t =0 ≤ ǫ k D γ ∂ t u k L ∞ t L x + C ( ǫ, k v k H s − , (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H sx ) , t ( | I | + | I | + | I | ) dt ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H sx ) t k ∂ t u k L ∞ t H s − x P ( k ∂ t u k L ∞ t H s − x ) . We now show that A , I , I is also estimated as above. If we write ∂ γ f := ∂ γ · · · ∂ γ k f for f : R → C , γ = ( γ , · · · , γ k ) ∈ { , } k , we have | A || t = t t =0 ≤ | b | X | γ | + | γ | = k X j =1 , |h ∂ γ u j JD γ ∂ j u, JD γ ∂ t u i L || t = t + |h ∂ γ u j JD γ ∂ j u, JD γ ∂ t u i L || t =0 ≤ | b | X | γ | + | γ | = k X j =1 , k ∂ γ u j k L ∞ t L ∞ x k D γ ∂ j u k L ∞ t L x k D γ ∂ t u k L ∞ t L x ≤ ǫ k D γ ∂ t u k L ∞ t L x + C ( ǫ, (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H sx )by using the Sobolev inequality. For I , we use the commutator estimatesobtained in [17]: X m =1 , k [ D γ , D m ] X k L x ≤ C (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H sx P ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H sx ) k X k H s − x , (2.6) k [ D γ , D t ] X k L x ≤ C k ∂ t u k H s − x (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H sx P ( k ∂ t u k H s − x , (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H sx ) k X k H s − x . (2.7)By using (2.7), we have Z t | I | dt ≤ | b | Z t X | γ | + | γ | = k X j =1 , |h ∂ γ u j JD γ ∂ j u, J [ D γ , D t ] ∂ t u i L |≤ | b | t X | γ | + | γ | = k X j =1 , k ∂ γ u j k L ∞ t L ∞ x k D γ ∂ j u k L ∞ t L x k [ D γ , D t ] ∂ t u k L ∞ t L x ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H sx ) t k ∂ t u k L ∞ t H s − x P ( k ∂ t u k L ∞ t H s − x ) . For I , we have Z t | I | dt ≤ | b | X j =1 , Z t |h [ D t , D γ ] u j J∂ j u, JD γ ∂ t u i L | + |h D γ ( ∂ t u j J∂ j u ) , JD γ ∂ t u i L | + |h D γ ( u j JD j ∂ t u ) , JD γ ∂ t u i L | dt ≤ | b | X j =1 , Z t |h [ D t , D γ ] u j J∂ j u, JD γ ∂ t u i L | + X | γ | + | γ | = k |h D γ ∂ t u j JD γ ∂ j u, JD γ ∂ t u i L | + X | γ | + | γ | = k | γ |6 =0 |h D γ u j JD γ D j ∂ t u, JD γ ∂ t u i L | + |h u j J [ D γ , D j ] ∂ t u, JD γ ∂ t u i L | + |h u j JD j D γ ∂ t u, JD γ ∂ t u i L | dt =: X l =1 I l . I , (2.5), (2.6) and the same argument as above yields the followingestimate: X l =1 I l ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H sx ) t k ∂ t u k L ∞ t H s − x P ( k ∂ t u k L ∞ t H s − x ) . For I , integration by part gives I = | b | X j =1 , Z t (cid:12)(cid:12)(cid:12)(cid:12)Z R u j ∂ j | D γ ∂ t u | dx (cid:12)(cid:12)(cid:12)(cid:12) dt = | b | X j =1 , Z t (cid:12)(cid:12)(cid:12)(cid:12)Z R ∂ j u j | D γ ∂ t u | dx (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H sx ) t k ∂ t u k L ∞ t H s − x P ( k ∂ t u k L ∞ t H s − x ) . Note that a sort of skew-adjoint structure of helicity term plays an essential rolein the above estimate. Since the solution has the bound (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H sx ≤ C ( k u k H s , δ k v k H s − ) , we obtain (2.4) as claimed. Proof of Theorem 1.1 (i) . We take a sequence { v ( n )0 } ∞ n =1 ⊂ H s − satisfyingsup n ∈ N n (cid:13)(cid:13)(cid:13) v ( n )0 (cid:13)(cid:13)(cid:13) H s − < ∞ and sup n ∈ N (cid:13)(cid:13)(cid:13) v ( n )0 (cid:13)(cid:13)(cid:13) H s − < ∞ , and let u ( n ) be a solution to (2.1) with δ = n − and with u ( n ) | t =0 = u (0) , ∂ t u ( n ) | t =0 = v ( n )0 for each n . Then Proposition 2.1 implies that the maximal existence time for u ( n ) is uniformly bounded below by some positive number T , and so is ˜ T inProposition 2.2. Moreover, by Propositions 2.1 and 2.2, this sequence of solu-tions satisfiessup n ∈ N (cid:13)(cid:13)(cid:13) u ( n ) (cid:13)(cid:13)(cid:13) L ∞ T H s < ∞ and sup n ∈ N (cid:13)(cid:13)(cid:13) ∂ t u ( n ) (cid:13)(cid:13)(cid:13) L ∞ T H s − < ∞ . This especially implies that { u ( n ) } ∞ n =1 , if we take a subsequence if necessary,converges to some map u in the sense thatlim n →∞ (cid:13)(cid:13)(cid:13) u ( n ) − u (cid:13)(cid:13)(cid:13) L ∞ ˜ T L x ( K ) = 0 for every K ⋐ R ,u ( n ) − ¯ k ⇀ u − ¯ k in H s and ∂ t u ( n ) → ∂ t u in H s − , and hence u satisfies (1.1) (see [17] for the details).9 Uniqueness and Continuity
In this section, we prove Theorem 1.1 (ii) and (iii). We mainly follow theargument of [17, 22]. In this section, we shall write h f, g i := R R f · gdx for f, g : R → C .We first prepare the setting. We may only consider the positive direction intime, thus we set I = [0 , T ] for T >
0. Let u (0) , u (1) ∈ C ( I : L ∞ ) ∩ L ∞ ( I :˙ H ∩ ˙ H ) be two solutions. We may assume that (cid:13)(cid:13) u (0) | t =0 − u (1) | t =0 (cid:13)(cid:13) L ∞ x issufficiently small, and thus may choose T so that (cid:13)(cid:13) u (0) − u (1) (cid:13)(cid:13) L ∞ t,x < π .For ( x, t ) ∈ R × [0 , T ], we consider a minimal geodesic from u (0) ( x, t ) to u (1) ( x, t ). In detail, we take a map γ ( s, x, t ) : [0 , × R × [0 , → S such that • γ (0 , x, t ) = u (0) ( x, t ), γ (1 , x, t ) = u (1) ( x, t ) in ( x, t ) ∈ R × [0 , T ]. • ∂ ss γ + ( γ s · γ s ) γ = 0 in [0 , × R × [0 , T ].Next, we consider the parallel transport along the geodesics. We define theoperator X ( s, σ ) : T γ ( σ,x,t ) S → T γ ( s,x,t ) S such that for ξ ∈ T γ ( σ,x,t ) S , X ( s, σ ) ξ is the parallel transport of ξ along γ ( s ). X ( s, σ ) is, by definition, the resolutionoperator for the following ODE: D s F ≡ ∂ s F + ( F · γ s ) γ = 0 , F : R → R . We set G ( T ) := k q k L ∞ T L x + X m =1 , k V m k L ∞ T L x , where q = u (0) − u (1) , V m := X (1 , ∂ m u (0) − ∂ m u (1) for m = 1 , . In this section, we write | u max − ¯ k | := | u (0) − ¯ k | + | u (1) − ¯ k | for simplicity, and wefollow the same convention for other quantities. Then we claim the following: Proposition 3.1.
Let M = (cid:13)(cid:13) u max − ¯ k (cid:13)(cid:13) L ∞ T H x . Then for all p ∈ (2 , ∞ ) , we have G ( T ) ≤ ( G (0) p + C ( M ) T ) p (3.1) where C ( M ) is a positive constant independent of p . The key estimates for the proof are the following:
Proposition 3.2.
Let M as above. Then the following estimates are true forall p > .(i) (cid:13)(cid:13)(cid:13) ∂ t k q k L x (cid:13)(cid:13)(cid:13) L ∞ T ≤ C ( M ) k q k L ∞ T H x .(ii) k∇ q k L ∞ T L x ≤ P m =1 , k V m k L ∞ T L x + C ( M ) √ p k q k − /pL ∞ T L x .(iii) kh ∂ t V m , V m ik L ∞ T ≤ C ( M ) p G ( T ) − /p . We first show that these propositions imply Theorem 1.1 (ii) and (iii).
Proof of Theorem 1.1 (ii).
We assume that u (0) | t =0 = u (1) | t =0 . Then wehave G (0) = 0. Here we may suppose that T is sufficiently small such that10 ( M ) T <
1, , where C ( M ) is as in (3.1). Taking the limit p → ∞ , we have G ( T ) = 0, which completes the proof. Proof of Theorem 1.1 (iii).
Let T ( n )max > u ( n ) in the class (1.3). Then Theorem 1.1 (i) implies that thereexists T > • T ( n )max > T for all n ∈ N , • M := sup n ∈ N (cid:13)(cid:13) u ( n ) − ¯ k (cid:13)(cid:13) L ∞ T H s < ∞ .Especially by the Sobolev embedding, we may take T so that sup n ∈ N (cid:13)(cid:13) u ( n ) − ¯ k (cid:13)(cid:13) L ∞ t,x ([0 ,T ] × R d ) < ∞ . Then Proposition 3.1 yields G n ,n ( T ) ≤ ( G n ,n (0) /p + C ( M ) T ) p , where G n ,n ( T ) := (cid:13)(cid:13)(cid:13) u ( n ) − u ( n ) (cid:13)(cid:13)(cid:13) L ∞ T L x + d X m =1 (cid:13)(cid:13)(cid:13) X (1 , ∂ m u ( n ) − ∂ m u ( n ) (cid:13)(cid:13)(cid:13) L ∞ T L x . By taking the limit n , n → ∞ , we havelim sup n ,n →∞ G n ,n ( T ) ≤ ( C ( M ) T ) p . Here we may suppose that T is sufficiently small so that C ( M ) T <
1. Thus bytaking p → ∞ , we have lim sup n ,n →∞ G n ,n ( T ) = 0 , which implies lim n →∞ (cid:13)(cid:13)(cid:13) u ( n ) − u (cid:13)(cid:13)(cid:13) L ∞ T H = 0 . Consequently, by interpolation, we havelim n →∞ (cid:13)(cid:13)(cid:13) u ( n ) − u (cid:13)(cid:13)(cid:13) L ∞ T H s − ǫ = 0 . which completes the proof.We next observe that Proposition 3.2 implies Proposition 3.1. Proof of Proposition 3.1.
We first note that V m ∈ L ∞ ( I : H ) ∩ W , ∞ ( I :˙ H ). See [22] for the detailed proof.Let ǫ > ψ ∈ C ∞ ( R ) be a cut-off function, anddefine the operator P k := (2 π ) − F − ψ ( · / k ) F . Then for sufficiently large k uniformly in t , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k q k L + X m =1 , h V m , P k V m i + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > . P k V m ∈ C ( I : H ), we have ∂ t k q k L + X m =1 , h V m , P k V m i + ǫ ! /p = 1 p k q k L + X m =1 , h V m , P k V m i + ǫ ! /p − ∂ t k q k L x + 2 X m =1 , Re h ∂ t V m , P k V m i ! , for all t ∈ (0 , T ), and thus k q k L + X m =1 , h V m , P k V m i + ǫ ! /p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 = Z T p k q k L + X m =1 , h V m , P k V m i + ǫ ! /p − ∂ t k q k L x + 2 X m =1 , Re h ∂ t V m , P k V m i ! dt. By taking the limit k → ∞ , it follows that( G ( T ) + ǫ ) /p − ( G (0) + ǫ ) /p = Z T p ( G ( T ) + ǫ ) /p − Re ∂ t k q k L + X m =1 , h ∂ t V m , V m i ! dt ≤ C ( M ) T, where we used Proposition 3.2. By taking ǫ →
0, we obtain (3.1).
Proof of Proposition 3.2.
For the proof of (ii), we refer the reader to [22].Thus we prove the rest part.(i) For a.a. t ∈ I , we have12 ddt k q k L x = h ∂ t q, q i L x = h− q × ∆ u (0) − u (1) × ∆ q, q i L x − b X j =1 h q j ∂ j u (0) + u (1) ∂ j q, q i L x = X j =1 h ∂ j ( u (0) + u (1) ) × ∂ j q, q i L x + b X j =1 (cid:18)Z R u (0) ∂ j ( | q | ) dx − h u (1) ∂ j q, q i L x (cid:19) ≤ C ( M ) k q k H . (3.2)(iii) We first define J by the complex structure of S , which can be explicitlywritten as Jξ := p × ξ for ξ ∈ T p S . From (1.1), V m satisfies the following equation: D t V m = − J X k =1 ( D k − b u (0) k J ) V m + X α =1 R α , R = [ D t , X ] ∂ m u (0) ,R = X k =1 J [ D k , X ] D k ∂ m u (0) ,R = X k =1 JD k [ D k , X ] ∂ m u (0) ,R = − J X k =1 n R ( X∂ m u (0) , X∂ k u (0) ) X∂ k u (0) − R ( ∂ m u (1) , ∂ k u (1) ) ∂ k u (1) o ,R = − b X k =1 u (0) j [ X, D j ] ∂ m u (0) ,R = − b X k =1 q j D j ∂ m u (1) ,R = X k =1 (cid:18) − b∂ m u (0) V k − b∂ m q∂ j u (1) + b ∂ k u (0) k V m − b u (0) k ) JV m (cid:19) It suffices to show the following estimate for m = 1 , t ∈ I : X α =1 |h R α , V m i| ≤ C ( M ) pG ( T ) − /p . (3.3)For α = 1 , · · · ,
4, we refer the reader to [22]. We only consider the case when α = 5 , ,
7. Denoting the distance between u (0) ( x, t ) and u (1) ( x, t ) on S by l ( x, t ), we have |h R , V m i| ≤ C Z R l |∇ u max | | V m | dx ≤ C k l k L ∞ T L p p − x k∇ u max k L ∞ T L px k V m k L ∞ T L p p − x ≤ Cp k l k − /pL ∞ T L x k l k /pL ∞ T L x k∇ u max k L ∞ T H k V m k − /pL ∞ T L x k V m k /pL ∞ T L x ≤ C ( M ) p k l k − /pL ∞ T H k V m k − /pL ∞ T L x , In the third inequality above, we used the inequality k f k L r ≤ C √ r k f k H for f ∈ H and r ∈ [2 , ∞ ) (see for example [20] for the proof). We also have |h R , V m i| ≤ C X j =1 k q k L ∞ T L px (cid:13)(cid:13)(cid:13) D j ∂ m u (0) (cid:13)(cid:13)(cid:13) L ∞ T L x k V m k L ∞ T L p p − x ≤ C ( M ) p k q k L ∞ T H k V m k − /pL ∞ T L x , h R , V m i| ≤ C Z R ( |∇ u max | + 1) | V m | dx + C Z R |∇ q ||∇ u max || V m | dx ≤ C k∇ u max k L ∞ T L px k V m k L ∞ T L p p − x + C k V m k L ∞ T L x + C k∇ q k L ∞ T L p p − x k∇ u max k L ∞ T L px k V m k L ∞ T L p p − x ≤ C ( M ) √ p k V m k − /pL ∞ T L x . Since | l | ≤ C | q | , the proof is completed. Our proof of Theorem 1.2 is based on the method using orthonormal frames. Inthis section, we prepare the setting for the argument.Let u = u ( x, t ) : R × [0 , T ] → S be a solution in the class C ([0 , T ] : ¯ k + H s )with s > T >
0. We further assume that u ( · , t ) is zero-homotopic for each t ∈ [0 , T ]. Then, we can construct an orthonormal frame of the tangent bundle u − T S with the following property: Lemma 4.1.
There exist v, w : R × [0 , T ] → R s.t.(i) v − ¯ k , w − ¯ k ∈ C ([0 , T ] : H ⌊ s ⌋ ) with ¯ k = t (1 , , , ¯ k = t (0 , , .(ii) { u ( x, t ) , v ( x, t ) , w ( x, t ) } is a positively-oriented orthonormal basis of R foreach ( x, t ) ∈ R × [0 , T ] .(iii) ∂ m v · w ∈ C ([0 , T ] : L ) .Proof. Our construction follows the argument in [1]. We divide the proof into2 steps.
Step 1 . We first note that lim | x |→∞ u ( x, t ) = ¯ k uniformly in t , and hense wecan consider u as a continuous map on S . In this step, we shall construct acontinuous map v ′ : R × [0 , T ] → S such that u · v ′ ≡
0, and lim | x |→∞ v ′ ( x, t ) =¯ k . By assumption, there is a continuous map H : S × [0 , → S such that H ( x,
0) = u ( x ) and H ( x,
1) = ¯ k for x ∈ S . By uniform continuity of H , thereexists n ∈ N such thatdist(( x , t , α ) , ( x , , t , α )) < n = ⇒ | H ( x , t , α ) − H ( x , t , α ) | < − . Now we set N ( U, V ) := U − ( U · V ) V | U − ( U · V ) V | for U, V ∈ S . This is well-defined when | U · V | < − . For ( x, t, α ) ∈ S × [0 , T ] × [ n − n , v ′ ( x, t, α ) := N (¯ k , H ( x, t, α )) . This is well-defined since | ¯ k · H ( x, t, α ) | < − , and satisfies | v | ≡ v ′ · H ≡ S × [0 , T ] × [ n − n , x, t, α ) ∈ S × [0 , T ] × [ n − n , n − n ],we define v ′ ( x, t, α ) = N ( v ′ ( x, t, n − n ) , H ( x, t, α )) . | v ′ ( x, t, n − n ) · H ( x, α ) | < − , and satisfies | v ′ | ≡ v ′ · H ≡ S × [0 , T ] × [ n − n , n − n ]. Repeating this procedure n times, we obtain a map v ′ : R × [0 , T ] × [0 , → S . By the above construction, v ′ ( x, t,
0) is exactly the desired map in this step.
Step 2 . We next regularize v ′ constructed above. Convolution with mollifiersand multiplication by smooth cut-off function yield v ′′ : R × [0 , T ] → R withthe following properties: • v ′′ : R → R , v ′′ ∈ C ∞ . • | v ′′ | ∈ [1 − − , − ]. • | v ′′ · u | ≤ − . • v ′′ ( x, t ) = ¯ k if | x | ≥ R for some R > t .Then we define v by v = N ( v ′′ , u ). Clearly we have | v | ≡ v · u ≡ R × [0 , T ]. The task is now to check the regularity of v . Let us first prove v ∈ C t (¯ k + L ). It suffices to prove ( v − ¯ k ) χ | x | >R ∈ C t L x . For | x | > R , wehave v − ¯ k = N (¯ k , u ) − ¯ k = u A (1 + A ) ¯ k − u A u, A = q − u , which leads to the claim. We next check that ∂v ∈ C t L for ∂ = ∂ , ∂ . It againsuffices to show ∂vχ | x | >R ∈ C t L x . For | x | > R , we have ∂v = (1 + 2 A ) u ∂u A (1 + A ) ¯ k + 2 u ∂u A (1 + A ) ¯ k − ∂u A u − u A ∂u, which leads to the claim. The above calculation also implies that ∂v · w ∈ C t L x where w = u × v . We can check the higher regularities in the similar way.Now we next define the differentiated field by ψ m := ∂ m u · v + i∂ m · w for m = 0 , , , and the connection coefficient by A m := ∂ m v · w = − v · ∂ m w for m = 0 , , . We also set ψ := t ( ψ , ψ ) and A := t ( A , A ) . Geometrically, ψ m is the representation of ∂ m u in terms of the complex coor-dinate with axis ( v, w ), and A m is the corresponding Christoffel symbol. Thesequantities are related to the original maps u, v, w via the following equation: ∂ m uvw = ψ m Im ψ m − Re ψ m A m − Im ψ m − A m uvw . (4.1)We next introduce the operator, called covariant derivative , as follows: D m := ∂ m + iA m for m = 0 , , . ψ , A satisfies the following relations: D m ψ l = D l ψ m for l, m = 0 , , . (4.2)[ D m , D l ] = i ( ∂ m A l − ∂ l A m ) = i Im( ψ m ψ l ) for l, m = 0 , , . (4.3)(4.2) represents the commutability of covariant derivatives and ordinary deriva-tives, and (4.3) represents the Ricci curvature tensor.The main advantage of our introduction of these quantities is the fact that ψ satisfies a system of nonlinear Schr¨odigner equations with magnetic potential A . Indeed, from (1.1), we have ψ = − i X l =1 D l ψ l − b X l =1 u l ψ l . (4.4)Hence by (4.2) and (4.3), for m = 1 ,
2, it follows that D ψ m = − i X l =1 D l D l ψ m + X l =1 Im( ψ m ψ l ) ψ l − b X l =1 D m ( u l ψ l ) . (4.5)In the case when b = 0, (4.5) is called modified Schr¨odinger map equation . Inthe present paper, we use (4.5) only in the case when u is smooth, and so theregularity does not cause any problem in the derivation of (4.5). However, wenote that our regularity assumption is sufficient to obtain (4.5) rigorously byusing the properties for ψ and A shown below. Checking this fact is left to thereader (see also [3, 23] for the related problem).Now we observe that we can retake { v, w } such that A satisfies the Coulombgauge condition:
Proposition 4.1.
There exist v, w : R → R satisfying the following proper-ties:(i) v − ¯ k , w − ¯ k ∈ C t ( L + L r ) ∩ C t ˙ H ∩ C t ˙ H ⌊ s ⌋ for all r ∈ (2 , ∞ ) , ¯ k , ¯ k ∈ S .(ii) { u ( x, t ) , v ( x, t ) , w ( x, t ) } is a positively-oriented orthonormal basis on R foreach ( x, t ) ∈ R × [0 , T ] .(iii) The following relation holds: ∂ A + ∂ A = 0 . (4.6) Proof.
Let { ˜ v, ˜ w } be an orthonormal frame as in Lemma 4.1. Then, any otherorthonormal frame { v, w } can be written as (cid:18) vw (cid:19) = (cid:18) cos χ sin χ − sin χ cos χ (cid:19) (cid:18) ˜ v ˜ w (cid:19) (4.7)for χ : R × [0 , T ] → R . It follows that { v, w } satisfies (4.6) if and only if∆ χ = − ∂ ˜ A − ∂ ˜ A (4.8)where ˜ A m is the connection coefficient corresponding to { ˜ v, ˜ w } . Since ˜ A ∈ C t ( L ∩ L ), a solution to (4.8) can be explicitly given by χ = (2 π ) − X m =1 F − (cid:20) | ξ | F (cid:16) R m ˜ A m (cid:17)(cid:21) ∈ C t ( L + L r ) , (2 < r ≤ ∞ ) , R m = (2 π ) − F − iξ m | ξ | F is the Riesz operator with respect to m -th index.(The integrability is shown by the same argument as the proof of Lemma 4.1.)We next check the regularity for v − ¯ k . (The case for w − ¯ k is similar.) Wefirst write v − ¯ k = (cos χ − v + (˜ v − ¯ k ) + (sin χ ) w, which implies v − ¯ k ∈ C t ( L + L r ). Since ∂ m v = (cos χ ) ∂ m ˜ v + (sin χ ) ∂ m ˜ w − ∂ m χ (sin χ ) v + ∂ m χ (cos χ ) w, we have ∂ m v ∈ C t L . The higher regularities can be similarly shown. Proposition 4.2.
Under the Coulomb gauge condition, the following propertieshold true:(i) ψ ∈ C t H s − .(ii) A ∈ C t ( ˙ H ∩ ˙ H s ) .(iii) (cid:18) A A (cid:19) = (2 π ) − F − (cid:18) | ξ | − F ( R Im( ψ ψ )) | ξ | − F ( R Im( ψ ψ )) (cid:19) .Proof. (i) follows from Proposition 4.1 (i) and the Leibniz rule. By (4.3), wehave ∆ A = ∂ Im( ψ ψ ) , where we used (4.6). This leads to (iii). The case for A is similar. (ii) followsfrom (i) and (iii).We now show that ψ and the gradient for ( u, v, w ) are equivalent in thequantitative sense: Proposition 4.3.
Let u, v, w as above. Then, for σ > , there exists a polyno-mial P = P ⌊ σ ⌋ such that the followings hold for all t ∈ [0 , T ] . k∇ u k H σ + k∇ v k ˙ H σ + k∇ w k ˙ H σ ≤ k ψ k H σ P ( k ψ k H σ ) if σ ≤ , (4.9) k∇ u k H σ + k∇ v k ˙ H σ + k∇ w k ˙ H σ ≤ k ψ k H σ P ( k ψ k H σ − ) if σ > , (4.10) k ψ k H σ ≤ k∇ u k H σ P ( k∇ u k H σ ) if σ ≤ , (4.11) k ψ k H σ ≤ k∇ u k H σ P ( k∇ u k H σ − ) if σ > . (4.12)First, we observe the following lemma. Lemma 4.2. (i) For r , r ∈ (2 , ∞ ) with /r − / ∈ [0 , /r ) , we have k A k L r ≤ C ( k ψ k L + k ψ k L r ) . Especially, the above inequality holds for ( r , r ) = ( r, for r ∈ (2 , ∞ ) .(ii) For σ > and r ∈ (2 , ∞ ) with /r + σ < , we have k A k ˙ H σ ≤ C ( k ψ k L + k ψ k L r ) . roof. We only see the estimate for A . (i) Dividing A into low and highfrequencies, we have (cid:13)(cid:13) F − ( χ | ξ |≤ F A ) (cid:13)(cid:13) L r ≤ C (cid:13)(cid:13) χ | ξ |≤ | ξ | − F Im( ψ ψ ) (cid:13)(cid:13) L r ′ ≤ C (cid:13)(cid:13) | ξ | − χ | ξ |≤ (cid:13)(cid:13) L r ′ (cid:13)(cid:13) F Im( ψ ψ ) (cid:13)(cid:13) L ∞ ≤ C k ψ k L , (4.13) (cid:13)(cid:13) F − ( χ | ξ | > F A ) (cid:13)(cid:13) L r ≤ C (cid:13)(cid:13)(cid:13) F − ( | ξ | − /r χ | ξ | > F A ) (cid:13)(cid:13)(cid:13) L ≤ C (cid:13)(cid:13)(cid:13) | ξ | − /r χ | ξ | > F Im( ψ ψ ) (cid:13)(cid:13)(cid:13) L ≤ C (cid:13)(cid:13)(cid:13) | ξ | − /r χ | ξ | > (cid:13)(cid:13)(cid:13) L r − r (cid:13)(cid:13) F Im( ψ ψ ) (cid:13)(cid:13) L r r − ≤ C k ψ k L r , which completes the proof.(ii) We divide A into low and high frequencies again. Then we have (cid:13)(cid:13) F − ( χ | ξ |≤ F|∇| σ A ) (cid:13)(cid:13) L ≤ C (cid:13)(cid:13) χ | ξ |≤ | ξ | − σ F Im( ψ ψ ) (cid:13)(cid:13) L ≤ C (cid:13)(cid:13) | ξ | − σ χ | ξ |≤ (cid:13)(cid:13) L (cid:13)(cid:13) F Im( ψ ψ ) (cid:13)(cid:13) L ∞ ≤ C k ψ k L , (cid:13)(cid:13) F − ( χ | ξ | > F|∇| σ A ) (cid:13)(cid:13) L ≤ C (cid:13)(cid:13) | ξ | − σ χ | ξ | > F Im( ψ ψ ) (cid:13)(cid:13) L ≤ C (cid:13)(cid:13) | ξ | − σ χ | ξ | > (cid:13)(cid:13) L r − r (cid:13)(cid:13) F Im( ψ ψ ) (cid:13)(cid:13) L rr − ≤ C k ψ k L r , which completes the proof. Remark 4.1.
We note that k A k L x cannot be controlled due to the low frequencypart of A . Indeed, (cid:13)(cid:13) | ξ | − (cid:13)(cid:13) L r ′ in (4.13) is infinite when r = 2 . This kind ofobstacle does not appear when the spatial dimension is higher; it is specificproblem to D -case. This is an essential obstacle of the Coulomb gauge, whichcauses problems in the analysis of Schr¨odinger maps in critical regularities (see[1, 2] for the detailed discussions).Proof of Proposition 4.3. We first prove (4.9). For the control of k∇ u k H σ ,we divide the case into σ ∈ (0 , / σ ∈ (1 / ,
1) and σ = 1. When σ ∈ (0 , / k∇ u k ˙ H σ ≤ C ( k ψ k ˙ H σ + k ψ k L / (2 − σ ) k|∇| σ v k L /σ ) ≤ C ( k ψ k ˙ H σ + k ψ k ˙ H σ/ k∇ v k L / (2 − σ ) ) ≤ C k ψ k ˙ H σ + C k ψ k ˙ H σ/ ( k ψ k L / (2 − σ ) + k A k L / (2 − σ ) ) ≤ C k ψ k ˙ H σ + C k ψ k ˙ H σ/ ( k ψ k ˙ H σ/ + k ψ k L + k ψ k L / (1 − σ ) ) ≤ C k ψ k H σ (1 + k ψ k H σ ) , (4.14)where the condition σ ≤ / σ ∈ (1 / , k∇ u k ˙ H σ ≤ C k ψ k ˙ H σ + C k ψ k ˙ H σ/ ( k ψ k L / (2 − σ ) + k A k L / (2 − σ ) ) ≤ C k ψ k ˙ H σ + C k ψ k ˙ H σ/ ( k ψ k ˙ H σ/ + k ψ k L + k ψ k L ) ≤ C k ψ k H σ (1 + k ψ k H σ ) , σ = 1, it suffices to control k ∂ψ · v k L , k ψ · ∂v k L for ∂ = ∂ , ∂ . The former is controlled by k ψ k H . The latter is estimated asfollows: k ψ · ∂v k L ≤ k ψ k L k ∂v k L ≤ C k ψ k L ( k ψ k L + k A k L ) ≤ C k ψ k L ( k ψ k L + k ψ k L + k ψ k L ) , where we used (4.1) and Lemma 4.2. Hence (4.9) follows.We next prove the estimate for v , while that for w is shown in the same way.It suffices to control k|∇| σ ψ · u k L , k ψ · |∇| σ u k L , k|∇| σ A · w k L , k A · |∇| σ w k L . We first see the case when σ ∈ (0 , k ψ k ˙ H σ . For the second quantity, we have k ψ · |∇| σ u k L ≤ k ψ k L / (1 − σ ) k|∇| σ u k L /σ ≤ C k ψ k ˙ H σ k∇ u k L ≤ C k ψ k H σ . The third part is bounded by k|∇| σ A k L ≤ C ( k ψ k L + k ψ k L / (1 − σ ) ) ≤ C k ψ k H σ , where we used Lemma 4.2 (ii). For the forth quantity, when σ ∈ (0 , / k A · |∇| σ w k L ≤ C k A k L / (2 − σ ) k|∇| σ w k L /σ ≤ C (cid:16) k ψ k L + k ψ k L / (2 − σ ) (cid:17) (cid:16) k ψ k ˙ H σ/ + k ψ k L + k ψ k L / (2 − σ ) (cid:17) ≤ C k ψ k H σ (1 + k ψ k H σ ) , where we used the estimate for k∇ w k L / (2 − σ ) in (4.14). When σ ∈ (1 / , k A · |∇| σ w k L ≤ C k A k ˙ H σ/ k∇ w k L / (2 − σ ) ≤ C (cid:16) k ψ k L + k ψ k L / (2 − σ ) (cid:17) (cid:16) k ψ k ˙ H σ/ + k ψ k L + k ψ k L (cid:17) ≤ C k ψ k H σ (1 + k ψ k H σ ) . When σ = 1, it suffices to control k ∂ψ · u k L , k ψ · ∂u k L , k ∂A · w k L , k A · ∂w k L for ∂ = ∂ , ∂ . The control for the first two quantities is easy. The third termis bounded by k ∂A k L ≤ C k ψ · ψ k L = C k ψ k L . For the forth term, we have k A · ∂w k L ≤ k A k L k ∂w k L ≤ C (cid:16) k ψ k L + k ψ k L (cid:17) , I σ := k∇ u k H σ + k∇ v k H σ + k∇ w k H σ . Then for σ > ∂ = ∂ , ∂ , if we take a number ǫ ∈ (0 , min { , σ − } ), we have k|∇| σ ( ∂u ) k L ≤ C (cid:13)(cid:13) |∇| σ − ((Re ∂ψ ) v + (Re ψ ) ∂v + i (Re ∂ψ ) v + i (Re ψ ) ∂w ) (cid:13)(cid:13) L ≤ C k ψ k ˙ H σ + C k ∂ψ k L − ǫ (cid:13)(cid:13) |∇| σ − v (cid:13)(cid:13) L ǫ + C (cid:13)(cid:13) |∇| σ − ψ (cid:13)(cid:13) L ǫ (cid:16) k ∂v k L − ǫ + k ∂w k L − ǫ (cid:17) + C k ψ k L ∞ (cid:0)(cid:13)(cid:13) |∇| σ − ∂v (cid:13)(cid:13) L + (cid:13)(cid:13) |∇| σ − ∂w (cid:13)(cid:13) L (cid:1) ≤ C k ψ k H σ I σ − , k|∇| σ ( ∂v ) k L ≤ C (cid:13)(cid:13) |∇| σ − ( − (Re ∂ψ ) u − (Re ψ ) ∂u + ( ∂A ) w + A∂w ) (cid:13)(cid:13) L ≤ C k ψ k ˙ H σ + C k ∂ψ k L − ǫ (cid:13)(cid:13) |∇| σ − u (cid:13)(cid:13) L ǫ + C (cid:13)(cid:13) |∇| σ − ψ (cid:13)(cid:13) L ǫ k ∂u k L − ǫ + C k ψ k L ∞ (cid:13)(cid:13) |∇| σ − ∂u (cid:13)(cid:13) L + C (cid:13)(cid:13) |∇| σ − ∂A (cid:13)(cid:13) L + C k ∂ψ k L − ǫ (cid:13)(cid:13) |∇| σ − w (cid:13)(cid:13) L ǫ + C (cid:13)(cid:13) |∇| σ − A (cid:13)(cid:13) L ǫ k ∂w k L − ǫ + C k A k L − ǫ (cid:13)(cid:13) |∇| σ − ∂w (cid:13)(cid:13) L ǫ ≤ C k ψ k H σ ( I σ − + k ψ k H σ − ) , which yields I σ ≤ C k ψ k H σ ( I σ − + k ψ k H σ − ) . Hence, (4.10) follows from (4.9) by induction.We next prove (4.11). When σ ∈ (0 , k ψ m k ˙ H σ ≤ C ( k ∂ m u k ˙ H σ + k ∂ m u k L − σ k|∇| σ v k L /σ ) ≤ C ( k ∂ m u k ˙ H σ + k ∂ m u k ˙ H /σ k∇ v k L / (2 − σ ) ) ≤ C k ∂ m u k ˙ H σ (1 + k ψ k L / (2 − σ ) + k A k L / (2 − σ ) ) ≤ C k ∂ m u k ˙ H σ (1 + k ψ k L + k ψ k L / (2 − σ ) ) ≤ C k ∂ m u k ˙ H σ (1 + k∇ u k H σ/ ) , where we used (4.1) and Lemma 4.2. When σ = 1, it suffices to control (cid:13)(cid:13) ∂ u · v (cid:13)(cid:13) L , k ∂u · ∂v k L for ∂ = ∂ , ∂ . The former is controlled by k∇ u k H . The latter is estimated asfollows: k ∂u · ∂v k L ≤ k ∂u k L k ∂v k L ≤ C k∇ u k H (1 + k ψ k L + k A k L ) ≤ C k∇ u k H (1 + k∇ u k H ) , which leads to (4.11). 20t remains to prove (4.12). For σ > m = 1 , ∂ = ∂ , ∂ , if we takea number ǫ ∈ (0 , min { , σ − } ), we have (cid:13)(cid:13) |∇| σ − ∂ψ m (cid:13)(cid:13) L = (cid:13)(cid:13) |∇| σ − ( ∂∂ m u · v + ∂ m u · ∂v + i∂∂ m u · w + i∂ m u · ∂w ) (cid:13)(cid:13) L ≤ C k∇ u k H σ + C k ∂∂ m u k L − ǫ (cid:16)(cid:13)(cid:13) |∇| σ − v (cid:13)(cid:13) L ǫ + (cid:13)(cid:13) |∇| σ − v (cid:13)(cid:13) L ǫ (cid:17) + C (cid:13)(cid:13) |∇| σ − ∂ m u (cid:13)(cid:13) L ǫ (cid:16) k ∂v k L − ǫ + k ∂v k L − ǫ (cid:17) + C k ∂ m u k L ∞ (cid:0)(cid:13)(cid:13) |∇| σ − ∂v (cid:13)(cid:13) L + (cid:13)(cid:13) |∇| σ − ∂w (cid:13)(cid:13) L (cid:1) ≤ C k∇ u k H σ ( k∇ v k H σ − + k∇ w k H σ − ) ≤ k∇ u k H σ P ⌊ σ − ⌋ ( k ψ k H σ − ) , where we used (4.9) and (4.10). Thus (4.12) follows by induction.Finally we observe the properties concerning time derivatives. Proposition 4.4.
Let u, v, w as above. Then, for σ > , there exists a polyno-mial P = P ⌊ σ ⌋ such that the followings are true for all t ∈ [0 , T ] .(i) k ψ k H σ ≤ k ψ k H σ P ( k ψ k H σ ) .(ii) A = (2 π ) − P j =1 F − (cid:2) | ξ | − F R j Im( ψ ψ j ) (cid:3) .(iii) k∇ A k H σ ≤ C k ψ k H σ +1 P ( k ψ k H σ ) .Proof. By using (4.4), (i) follows by the same argument as in the proof ofProposition 4.3. (ii) follows from (4.3). By using (ii), we have k∇ A k H σ ≤ C d X j =1 k ψ j ψ k H σ ≤ C ( k ψ k H σ k ψ k L + k ψ k L k ψ k H σ ) ≤ k ψ k H σ P ( k ψ k H σ ) , which gives (iii). In this section, we prove Theorem 1.2 (a), (b) and (c). The key estimate is thefollowing:
Proposition 5.1.
Let L ≥ be an integer, and let ǫ be a number in [0 , when L ≥ , or in (0 , when L = 1 . When ǫ > , we also set δ ∈ (0 , ǫ ) . Let u ∈ C ([0 , T ] , ¯ k + H L + ǫ ) be a solution to (1.1), and set Y T := P | α | = L,m =1 , k ∂ αx ψ m k L ∞ T H ǫ .We further set M T := k ψ k L ∞ T H L − when L ≥ , or M T := k ψ k L ∞ T H δ when L = 1 . Then there exists K > , independent of L , ǫ , and δ , such that thefollowing inequality holds true. Y T ≤ C (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H L +1+ ǫ + T (1 + T ) K Y T P L ( M T ) , (5.1) where C = C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H max { L + ǫ, } , L ) > is a constant, and P L is a polyno-mial.
21e first show that Proposition 5.1 implies the conclusions.
Proof of Theorem 1.2 (a), (b) and (c).
We divide the proof into 3 steps:
Step 1 . We suppose here that u ∈ ¯ k + H ∞ . Theorem 1.1 implies that for L ≥ L ∈ Z , there exists a unique solution to (1.1) u ∈ L ∞ ([0 , T L max ) , ¯ k + H L )where T L max ( u ) is the maximal existence time in this class. Now we considerthe following proposition for s, s ′ ∈ R :( P ) s,s ′ : T s max = T s ′ max .In this step, we shall show that ( P ) s, is true for s ≥ , s ∈ Z by inductionwith respect to s . We assume that for some s ≥ , s ∈ Z , ( P ) s ′ , is true for all s ′ ∈ [3 , s ] ∩ Z . Suppose T s +1max < T and we will show the contradiction.We apply Proposition 5.1 with L = s −
1. Then Proposition 4.3 implies X | α | = s,m =1 , k ∂ αx ψ m k L ∞ T H ǫ ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s − ǫ ) (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s + ǫ for all T < T s +1max ∧ T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ Ts +1max H s ). On the other hand, (1.2) yields (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T L x ≤ (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L + CT (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T L x k∇ u k L ∞ T L x ≤ (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L + 12 (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T L x + CT k∇ u k L ∞ T L x , which implies (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T L x ≤ C (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L + CT k∇ u k L ∞ T L x . (5.2)Thus it follows that (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H s + ǫ ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s − ǫ ) (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s + ǫ . (5.3)for T < T s +1max ∧ T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ Ts +1max H s ). Repeating this procedure in finite timesyields the same bound for T = T s +1max with modifying the constant in the righthand side.Next we apply Proposition 5.1 for L = s , ǫ = 0. Then, the same argumentabove yields (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H s +1 ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s ) (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s +1 (5.4)for all T < T s +1max ∧ T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ Ts +1max H s ), where we used (5.3). If we use (5.4)repeatedly, we have the same bound for T = T s +1max with a certain change ofconstant, which contradicts to Theorem 1.1.As a consequence, for u ∈ ¯ k + H ∞ we have a unique solution u ∈ C ([0 , T ∞ max ) :¯ k + H ∞ ), where T ∞ max is the maximal existence time in this class, and we have T ∞ max = T . 22 tep 2 . Next, we prove Theorem 1.2 (a). For u ∈ ¯ k + H s with s > s ∈ R \ Z , we shall construct a unique solution u ∈ L ∞ T (¯ k + H s ) with some T = T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s ). By standard compactness argument, it suffices to show (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H s ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s ) (5.5)for u ∈ ¯ k + H ∞ with some T = T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s ).By Propositions 5.1 and 4.3, and by (5.2) we have (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H s ≤ C + CT (1 + T ) K (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H s P s ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T H s ) (5.6)for T < T ∞ max with C = C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s ). By bootstrapping argument, (5.6)yields (5.5) for all T < T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s ) ∧ T ∞ max ∧
1. Then the same argument asin Step 1 shows T ∞ max ≥ T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s ) ∧
1, which concludes the claim.The fact that the solution above is in C t (¯ k + H s ) can be shown by usingTheorem 1.2 (d), which will be proved in Section 6 independently of this fact. Step 3 . We finally show the rest part. (c) follows from (5.5) via standardapproximating argument. In particular, this bound implies that if the maximalexistence time T s max in the class L ∞ T H s is finite, thenlim t → T s max − (cid:13)(cid:13) u ( t ) − ¯ k (cid:13)(cid:13) H s = ∞ . (5.7)We can then show (b) by the same argument as in Step 1.The idea to prove Proposition 5.1 is as follows. We can rewrite the equation(4.5) as D ψ m = − i X l =1 e D l e D l ψ m + N m (5.8)for m = 1 ,
2, where e D l := ∂ l + i e A l , e A l := A l − bu l ( l = 1 , , N m := b X l =1 ( 12 ( ∂ l u l ) ψ m − ib u l ψ m − ( ∂ m u l ) ψ l ) + X l =1 ℑ ( ψ m ψ l ) ψ l ( m = 1 , . Note that i P l =1 e D l e D l is a skew-symmetric operator on L ( R ), and that N m includes no derivatives of ψ m . This structure allows us to apply the en-ergy method by introducing the differential operator associated with magneticpotential e A = t ( e A , e A ).Writing ∆ e A := P l =1 e D l e D l , we have the following lemma: Lemma 5.1.
Let X = L ( R ) ∩ ˙ H ( R ) . Then the followings are true.(i) − ∆ e A is a self-adjoint, positive operator on L ( R ) for each t ∈ [0 , T ] withdomain H ( R ) . ii) Let Ω e A := (1 − ∆ e A ) / . Then there exist constants C, K > with K ∈ Z such that for all s ∈ [ − , and t ∈ [0 , T ] , we have C − h (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) X i − K k f k H s ≤ (cid:13)(cid:13)(cid:13) Ω s e A f (cid:13)(cid:13)(cid:13) L ≤ C h (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) X i K k f k H s . (iii) For all s ∈ [ − , and λ > , we have (cid:13)(cid:13)(cid:13) Ω s e A (Ω e A + λ ) − f (cid:13)(cid:13)(cid:13) L ≤ C h λ i − s k f k L . (iv) Let ǫ > and r ∈ (2 , ∞ ) . Then there exist C = C ǫ,r , K > with K ∈ Z such that for each t ∈ [0 , T ] , we have (cid:13)(cid:13)(cid:13) Ω ǫ e A [Ω − ǫ e A , D ] f (cid:13)(cid:13)(cid:13) L ≤ C h (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) X i K (cid:16)(cid:13)(cid:13)(cid:13) e F (cid:13)(cid:13)(cid:13) L r + (cid:13)(cid:13)(cid:13) e F (cid:13)(cid:13)(cid:13) L r (cid:17) k f k L , where e F k := ∂ e A k − ∂ k A = ℑ ( ψ ψ k ) − b∂ t u k .Proof. All the proof can be found in [26], while the only difference here is thatthe norm of A is measured by X , instead of H in [26]. Therefore, it suffices toshow the following inequality: k V f k L ≤ ǫ k (1 − ∆) f k L + Cǫ − hk A k X i k f k L , where V is the function defined by 1 − ∆ e A = 1 − ∆ + V , or explicitly V = P k =1 (cid:16) − i e A k ∂ k − i ( ∂ k e A k ) + e A k (cid:17) . However, using the Gagliardo-Nirenberg in-equality, we have k V f k L ≤ C X k =1 (cid:13)(cid:13)(cid:13) e A k (cid:13)(cid:13)(cid:13) L k ∂ k f k L + C X k =1 (cid:13)(cid:13)(cid:13) ∂ k e A k (cid:13)(cid:13)(cid:13) L k f k L ∞ + C ( X k =1 (cid:13)(cid:13)(cid:13) e A k (cid:13)(cid:13)(cid:13) L ) k f k L ∞ ≤ C (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) X k f k / L k ∆ f k / L + h (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) X i k f k / L k ∆ f k / L ≤ ǫ k (1 − ∆) f k L + Cǫ − h (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) X i k f k L , which is the conclusion. Proof of Proposition 5.1 . For simplicity of notation, let C e A stand for C h (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) L ∞ T L x i K for some constant C, K > K ∈ Z which are independent of u and ψ .Let α ∈ N with | α | ≤ L . Operating Ω ǫ e A ∂ α ψ m on both sides of (5.8), weobtain D Ω ǫ e A ∂ α ψ m = − i ∆ e A ψ m + X ν =1 R ν , (5.9)where R := [ D , Ω ǫ e A ] ∂ α ψ m ,R := − i X β + γ = α,β =0 α ! β ! γ ! Ω ǫ e A ( ∂ β A · ∂ γ ψ m ) , := 2 X β + γ = α,β =0 2 X l =1 α ! β ! γ ! Ω ǫ e A ( ∂ β e A l · ∇ ∂ γ ψ m ) ,R := i X β + γ = α,β =0 α ! β ! γ ! Ω ǫ e A ( ∂ β ( | e A | ) · ∂ γ ψ m ) ,R := Ω ǫ e A ∂ α N m − b ǫ e A ∂ α (( ∂ u + ∂ u ) ψ m ) + b ǫ e A (( ∂ u + ∂ u ) ∂ α ψ m ) . Then we consider the inner product of (5.9) and Ω ǫ e A ∂ α ψ m , which yields (cid:13)(cid:13)(cid:13) Ω ǫ e A ∂ α ψ m (cid:13)(cid:13)(cid:13) L ∞ T L x ≤ (cid:13)(cid:13)(cid:13) Ω ǫ e A ∂ α ψ m (cid:13)(cid:13)(cid:13) L x (cid:12)(cid:12)(cid:12)(cid:12) t =0 + C X ν =1 Z T k R ν k L x dt. (5.10)Now we claim that X ν =1 Z T k R ν k L x dt ≤ C e A T k ψ k H L + ǫ P L ( M T ) . (5.11)Let us first examine R . We may assume ǫ >
0, since R = 0 when ǫ = 0.Applying Lemma 5.1 (iv) with r = 2 / (1 − δ ) for δ ∈ (0 , ǫ ), we have k R k L x = (cid:13)(cid:13)(cid:13) Ω ǫ e A [Ω − ǫ e A , D ]Ω ǫ e A ∂ α ψ m (cid:13)(cid:13)(cid:13) L x ≤ C e A ( k F k L / (1 − δ ) x + k F k L / (1 − δ ) x ) (cid:13)(cid:13)(cid:13) Ω ǫ e A ∂ α ψ m (cid:13)(cid:13)(cid:13) L x . (5.12)For F k ( k = 1 , k F k k L / (1 − δ ) x ≤ C k ψ k L / (1 − δ ) x (cid:16) k ψ k k L ∞ x + 1 (cid:17) ≤ k ψ k H δ P ( k ψ k H δ ) . Thus (5.12) is bounded by C e A P ( k ψ k H δx ) (cid:13)(cid:13)(cid:13) Ω ǫ e A ∂ α ψ m (cid:13)(cid:13)(cid:13) L x . For R , we divide the proof into the case when L = 1 and L ≥
2. If L = 1,Proposition 4.4, Lemma 5.1 and the Leibniz rule yield k R k L x ≤ C e A k ( ∂ α A ) · ψ m k H ǫ ≤ C e A (cid:16) k ∂ α A k H ǫ k ψ k L ∞ + k ∂ α A k L / (1 − ǫ ) k ψ k H ǫ /ǫ (cid:17) ≤ C e A k ψ k H ǫ P ( k ψ k H δ ) . When L ≥
2, we have k R k L x ≤ C e A (cid:16) k ∂ α A k H ǫ k ψ k L ∞ x + k ∂ α A k L / (1 − ǫ ) x k ψ k H ǫ /ǫ (cid:17) + C e A X β + γ = α,β =0 ,α (cid:16)(cid:13)(cid:13) ∂ β A (cid:13)(cid:13) H ǫ k ∂ γ ψ k L x + (cid:13)(cid:13) ∂ β A (cid:13)(cid:13) L x k ∂ γ ψ k H ǫ (cid:17) ≤ C e A k ψ k H L + ǫ P L ( M T ) . L = 1, R is estimated as follows: k R k L x ≤ C e A X l =1 (cid:13)(cid:13)(cid:13) ∂ α e A l · ∇ ψ m (cid:13)(cid:13)(cid:13) H ǫ ≤ C e A (cid:13)(cid:13)(cid:13) ∂ α e A (cid:13)(cid:13)(cid:13) H ǫ /ǫ k∇ ψ k L / (1 − ǫ ) x + (cid:13)(cid:13)(cid:13) ∂ α e A (cid:13)(cid:13)(cid:13) L ∞ x k∇ ψ k H ǫ ! ≤ C e A k ψ k H ǫ P L ( k ψ k H δ ) . where we used Proposition 4.2 (iii) in the last inequality. If L ≥
2, we have k R k L x ≤ C e A X β + γ = α, | β | =1 (cid:13)(cid:13)(cid:13) ∂ β e A (cid:13)(cid:13)(cid:13) H ǫ /ǫ k∇ ∂ γ ψ k L / (1 − ǫ ) + (cid:13)(cid:13)(cid:13) ∂ β e A (cid:13)(cid:13)(cid:13) L ∞ k∇ ∂ γ ψ k H ǫ ! + C e A X β + γ = α, | β |≥ (cid:18)(cid:13)(cid:13)(cid:13) ∂ β e A (cid:13)(cid:13)(cid:13) H ǫ k∇ ∂ γ ψ k L + (cid:13)(cid:13)(cid:13) ∂ β e A (cid:13)(cid:13)(cid:13) L k∇ ∂ γ ψ k H ǫ (cid:19) ≤ C e A k ψ k H L + ǫ P L ( M T ) . For R , we obtain k R k L x ≤ C e A X β + γ = α,β =0 (cid:13)(cid:13)(cid:13) ∂ β ( | e A | ) ∂ γ ψ (cid:13)(cid:13)(cid:13) H ǫ ≤ C e A (cid:16)(cid:13)(cid:13)(cid:13) ∂ α ( | e A | ) (cid:13)(cid:13)(cid:13) H ǫ k ψ k L ∞ + (cid:13)(cid:13)(cid:13) ∂ α ( | e A | ) (cid:13)(cid:13)(cid:13) L / (1 − ǫ ) k ψ k H ǫ /ǫ (cid:17) + C e A X β + γ = α,β =0 ,α (cid:16)(cid:13)(cid:13)(cid:13) ∂ β ( | e A | ) (cid:13)(cid:13)(cid:13) L k ∂ γ ψ k L + (cid:13)(cid:13)(cid:13) ∂ β ( | e A | ) (cid:13)(cid:13)(cid:13) L ∞ k ∂ γ ψ k H ǫ (cid:17) . (5.13)Here we have (cid:13)(cid:13)(cid:13) ∂ α ( | e A | ) (cid:13)(cid:13)(cid:13) H ǫ ≤ X α + α = α X k =1 , (cid:13)(cid:13)(cid:13) ∂ α e A k (cid:13)(cid:13)(cid:13) H ǫ (cid:13)(cid:13)(cid:13) ∂ α e A k (cid:13)(cid:13)(cid:13) L ≤ CM T . Thus (5.13) is bounded by C e A k ψ k H L + ǫ P L ( M T ) . For R , we first estimate (cid:13)(cid:13)(cid:13) Ω ǫ e A ∂ α ψ m (cid:13)(cid:13)(cid:13) L x . To this end, we observe that N m is alinear combination of the followings:( ∂ j u k ) ψ m , u k ψ m , Im( ψ m ψ k ) ψ k . ( j, k = 1 , (cid:13)(cid:13)(cid:13) Ω ǫ e A ∂ α (( ∂ j u k ) ψ m ) (cid:13)(cid:13)(cid:13) L x ≤ C e A k ( ∂ j u k ) ψ m k H L + ǫ ≤ C e A (cid:16) k ∂ j u k k H L + ǫ k ψ k L ∞ x + k ∂ j u k k L ∞ x k ψ k H L + ǫ (cid:17) ≤ C e A k ψ k H L + ǫ P L ( M T ) . For the third one, the Leibniz rule yields (cid:13)(cid:13)(cid:13) Ω ǫ e A ∂ α (Im( ψ m ψ k ) ψ k ) (cid:13)(cid:13)(cid:13) L x ≤ C e A (cid:13)(cid:13) Im( ψ m ψ k ) ψ k (cid:13)(cid:13) H L + ǫx ≤ C e A k ψ k H L + ǫ M T . (cid:13)(cid:13)(cid:13) Ω ǫ e A ∂ α ( u k ψ m ) (cid:13)(cid:13)(cid:13) L x ≤ C e A (cid:13)(cid:13) u k ∂ α ψ m (cid:13)(cid:13) H ǫ + X α + α = α,α =0 k ∂ α u k · u k · ∂ α ψ m k H ǫ + X α + α + α = α,α ,α =0 k ∂ α u k · ∂ α u k · ∂ α ψ m k H ǫ =: R + R + R , Each of these is estimated as follows: R ≤ C e A (cid:16)(cid:13)(cid:13) u k ψ m (cid:13)(cid:13) L + (cid:13)(cid:13) |∇| ǫ ( u k ψ m ) (cid:13)(cid:13) L (cid:17) ≤ C e A (cid:16) k ψ k H L + k|∇| ǫ u k k L /ǫ k u k k L ∞ k ∂ α ψ m k L / (1 − ǫ ) + k u k k L ∞ k|∇| ǫ ∂ α ψ m k L (cid:17) ≤ C e A k ψ k H L + ǫ P L ( M T ) ,R ≤ C e A X α + α = α,α =0 (cid:16) k ∂ α u k · u k · ∂ α ψ m k L x + k|∇| ǫ ( ∂ α u k · u k · ∂ α ψ m ) k L x (cid:17) ≤ C e A X α + α = α,α =0 ( k ∂ α u k k L k ∂ α ψ k L + k|∇| ǫ ∂ α u k k L /ǫ k u k k L ∞ k ∂ α ψ k L / (1 − ǫ ) + k ∂ α u k k L ∞ k∇| ǫ u k L /ǫ k ∂ α ψ k L / (1 − ǫ ) + k ∂ α u k k L ∞ k u k k L ∞ k|∇| ǫ ∂ α ψ m k L ) ≤ C e A k ψ k H L + ǫ P L ( M T ) , R ≤ C e A k ψ k H L + ǫ P L ( M T ) . The other terms in R can be similarly estimated to the above. Hence (5.11) isproved.Next we estimate (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) L ∞ T X . Using Lemma 4.2 and (5.2), we have (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) L ∞ T X ≤ C (cid:16) k A k L ∞ T L x + k∇ A k L ∞ T L x + (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T L x + k∇ u k L ∞ T L x (cid:17) ≤ C (cid:16)(cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) L ∞ T L x + k ψ k L ∞ T L x + k ψ k L ∞ T L x + k ψ k L ∞ T L x (cid:17) . Here we observe that the following inequalities hold true: k ψ k L ∞ T L x ≤ C (cid:16) k∇ u k L x + T ( k ψ k L ∞ T L x + k ψ k L ∞ T L x ) (cid:17) , (5.14) k ψ k L ∞ T L x ≤ C (cid:16) k∇ u k L x + T ( M T + M / T ) (cid:17) , (5.15)Indeed, (5.14) immediately follows from (5.8), and (5.15) follows from ∂ t k ψ k L x = 4Re Z R | ψ m | ψ m D ψ m dx = 4Re X k =1 Z R ∂ k ( | ψ m | ) ψ m e D k ψ m dx + 4Re Z R | ψ m | ψ m N m dx. (cid:13)(cid:13)(cid:13) e A (cid:13)(cid:13)(cid:13) L ∞ T X ≤ C ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H / ) + C ( T / + T ) (cid:0) M T + M T (cid:1) . (5.16)Applying (5.11) and (5.16) to (5.10), we obtain the desired estimate. In this section, we prove Theorem 1.2 (d). Our strategy is based on [2]. Namely,we first take a smooth homotopy map connecting the two solutions. Then we canconsider the derivative with respect to the new parameter, thus can define theassociated differentiated field. We will show that this quantity has quantitativeequivalence to the difference of two solutions, and hence we can reduce theproblem to its estimate.Letting Ω := R \{ } , we begin by defining Π : Ω → S by Π( y ) := y | y | . ThenΠ is smooth, and Π( p ) = p for all p ∈ S .Let us first prove (1.4). Let u (0)0 , u (1)0 ∈ ¯ k + H s . Without loss of generality,we may assume that (1 − h ) u (0)0 + hu (1)0 ∈ Ω by taking (cid:13)(cid:13)(cid:13) u (1)0 − u (0)0 (cid:13)(cid:13)(cid:13) H s ≪ u (0) , u (1) ∈ ¯ k + H ∞ , since the generalcase follows from the continuity of solution map shown later.For h ∈ [0 , u ( h )0 := Π ◦ (cid:16) (1 − h ) u (0)0 + hu (1)0 (cid:17) . Note that this definition is consistent when h = 0 ,
1. Let u ( h ) ∈ C T (¯ k + H ∞ )be the solution with u ( h ) | t =0 = u ( h )0 . It follows that u ( h ) is defined in t ∈ [0 , T ]with T > h , which is clear from the consequence of Section 5.Then the following estimate holds. Proposition 6.1.
For σ ≥ , there exists a polynomial P = P ⌊ σ ⌋ such that thefollowing holds for all t ∈ [0 , T ] and h ∈ [0 , . (cid:13)(cid:13)(cid:13) ∂ h u ( h ) (cid:13)(cid:13)(cid:13) H σ ≤ (cid:13)(cid:13)(cid:13) u (1) − u (0) (cid:13)(cid:13)(cid:13) H σ P (cid:0)(cid:13)(cid:13) u max − ¯ k (cid:13)(cid:13) H max { σ, } (cid:1) , (6.1) where by u max we follow the same convention as in Section 3. Remark 6.1.
The differentiability with respect to h is justified in the followingway. Let h , h ∈ [0 , . Since both of u ( h ) , u ( h ) satisfy (1.1), a similarargument to (3.2) yields (cid:13)(cid:13)(cid:13) u ( h ) − u ( h ) (cid:13)(cid:13)(cid:13) L ∞ t,x ≤ C (cid:13)(cid:13)(cid:13) u ( h ) − u ( h ) (cid:13)(cid:13)(cid:13) L ∞ T H d ≤ C (cid:13)(cid:13)(cid:13) u ( h )0 − u ( h )0 (cid:13)(cid:13)(cid:13) H d , where C = C ( (cid:13)(cid:13)(cid:13) u ( h )0 − ¯ k (cid:13)(cid:13)(cid:13) H N , (cid:13)(cid:13)(cid:13) u ( h )0 − ¯ k (cid:13)(cid:13)(cid:13) H N ) for sufficiently large N . Hencethe claim follows from absolute continuity of initial data with respect to h . roof. When σ ∈ Z , (6.1) follows from the chain rule. If σ ∈ (0 , (cid:13)(cid:13)(cid:13) ∂ h u ( h ) (cid:13)(cid:13)(cid:13) ˙ H σ ≤ C (cid:13)(cid:13)(cid:13)(cid:16) ( ∇ Π) ◦ ((1 − h ) u (0) + hu (1) ) (cid:17) · ( u (1) − u (0) ) (cid:13)(cid:13)(cid:13) H σ ≤ C σ (cid:13)(cid:13)(cid:13) |∇| σ (cid:16)(cid:16) ∇ Π ◦ ((1 − h ) u (0) + hu (1) ) − ∇ Π(¯ k ) (cid:17) · ( u (1) − u (0) ) (cid:17)(cid:13)(cid:13)(cid:13) L x + C σ (cid:13)(cid:13)(cid:13) u (1) − u (0) (cid:13)(cid:13)(cid:13) H σ ≤ C σ (cid:13)(cid:13)(cid:13) |∇| σ (cid:16) ∇ Π ◦ ((1 − h ) u (0) + hu (1) ) − ∇ Π(¯ k ) (cid:17)(cid:13)(cid:13)(cid:13) L /σx (cid:13)(cid:13)(cid:13) u (1) − u (0) (cid:13)(cid:13)(cid:13) L / (1 − σ ) x + C σ (cid:13)(cid:13)(cid:13) ∇ Π ◦ ((1 − h ) u (0) + hu (1) ) − ∇ Π(¯ k ) (cid:13)(cid:13)(cid:13) L ∞ x (cid:13)(cid:13)(cid:13) ∇ σ ( u (1) − u (0) ) (cid:13)(cid:13)(cid:13) L x + C σ (cid:13)(cid:13)(cid:13) u (1) − u (0) (cid:13)(cid:13)(cid:13) H σ ≤ C σ (cid:16)(cid:13)(cid:13)(cid:13) ∇ (cid:16) ∇ Π ◦ ((1 − h ) u (0) + hu (1) ) − ∇ Π(¯ k ) (cid:17)(cid:13)(cid:13)(cid:13) L + 1 (cid:17) (cid:13)(cid:13)(cid:13) u (1) − u (0) (cid:13)(cid:13)(cid:13) H σ ≤ C σ (cid:13)(cid:13)(cid:13) u (1) − u (0) (cid:13)(cid:13)(cid:13) H σ P ( (cid:13)(cid:13) u max − ¯ k (cid:13)(cid:13) H max { σ, } ) . The other case can be proved similarly.Next, let v, w be an orthonormal frame of T u ( h ) S with v − ¯ k , w − ¯ k ∈ C t,h ( L + L r ) ∩ C t,h ∇ H ∞ , and with ∂ A + ∂ A = 0 for all ( x, t, h ) ∈ R × [0 , T ] × [0 , , where A m = A m ( x, t, h ) := ∂ j v ( h ) · w ( h ) for m = 0 , ,
2. (Such frame can beconstructed in the same manner as Section 4.) We also define ψ m and D m for m = 0 , , m = 3 to the h -variable,and define ψ , A in the same manner. Then it follows that (4.2), (4.3), (4.5)and (5.8) hold even when m = 3. Especially, the same argument as in Sections4 and 5 yields the following estimates: (cid:13)(cid:13)(cid:13) ∂ h u ( h ) (cid:13)(cid:13)(cid:13) H s − ≤ k ψ k H s − P ⌊ s − ⌋ ( k ψ k H s − ) for t ∈ [0 , T ] , k ψ k H s − ≤ (cid:13)(cid:13)(cid:13) ∂ h u ( h ) (cid:13)(cid:13)(cid:13) H s − P ⌊ s − ⌋ ( (cid:13)(cid:13)(cid:13) ∇ u ( h ) (cid:13)(cid:13)(cid:13) H s − ) for t ∈ [0 , T ] , k ψ k L ∞ T H s − ≤ C (cid:13)(cid:13)(cid:13) ∂ h u ( h )0 (cid:13)(cid:13)(cid:13) H s − + CT (1 + T ) K k ψ k L ∞ T H s − . where C = C ( (cid:13)(cid:13) u max0 − ¯ k (cid:13)(cid:13) H s − ) > K > k ψ k L ∞ T H s − ≤ C ( (cid:13)(cid:13) u max0 − ¯ k (cid:13)(cid:13) H s ) (cid:13)(cid:13)(cid:13) ∂ h u ( h )0 (cid:13)(cid:13)(cid:13) H s − T = T ( (cid:13)(cid:13) u max0 − ¯ k (cid:13)(cid:13) H s ). Consequently, Proposition 6.1 yields (cid:13)(cid:13)(cid:13) u (1) − u (0) (cid:13)(cid:13)(cid:13) L ∞ T H s − ≤ Z (cid:13)(cid:13)(cid:13) ∂ h u ( h ) (cid:13)(cid:13)(cid:13) L ∞ T H s − ≤ Z k ψ k L ∞ T H s − P ( k ψ k L ∞ T H s − ) ≤ Z (cid:13)(cid:13)(cid:13) ∂ h u ( h )0 (cid:13)(cid:13)(cid:13) H s − C ( (cid:13)(cid:13) u max − ¯ k (cid:13)(cid:13) L ∞ T H s ) ≤ C ( (cid:13)(cid:13) u max0 − ¯ k (cid:13)(cid:13) H s ) (cid:13)(cid:13)(cid:13) u (1)0 − u (0)0 (cid:13)(cid:13)(cid:13) H s − , which gives (1.4).We next prove the continuity part. The proof here is based on the typicalargument invented in [5], while a detailed exposition of this method is availablein [10]. Let s > { u ( n )0 } n ∈ N is a sequence in ¯ k + H s satisfying u ( n )0 − u → H s . Let ϕ ∈ S ( R ) be a nonnegative function with k ϕ k L = 1satisfying F [ ϕ ] = 1 in a neighborhood of the origin. Then we define u ,η := Π ◦ (cid:0) ϕ η ∗ (cid:0) u − ¯ k (cid:1) + ¯ k (cid:1) , where ϕ η := η − ϕ ( η − · ), and also define u ( n )0 ,η in the same way. Let u , u η , and u ( n ) η be unique solutions to (1.1) corresponding to the initial data u , u ,η , and u ( n )0 ,η respectively. Then we have the following properties: Proposition 6.2.
The followings hold true.(i) There exist η > , n ∈ N such that u ,η , u ( n )0 ,η are well-defined for all < η < η and n ≥ n .(ii) sup n ≥ n ,η ≥ η (cid:13)(cid:13)(cid:13) u ( n )0 ,η − ¯ k (cid:13)(cid:13)(cid:13) H s < ∞ . Thus the maximal existence time of u η and u ( n ) η are bounded from below uniformly in η and n .(iii) (cid:13)(cid:13) u ,η − ¯ k (cid:13)(cid:13) H s +1 + sup n ≥ n (cid:13)(cid:13)(cid:13) u ( n )0 ,η − ¯ k (cid:13)(cid:13)(cid:13) H s +1 ≤ Cη − .(iv) There exists T > , independent of η , such that k u η − u k L ∞ T H s + sup n ≥ n (cid:13)(cid:13)(cid:13) u ( n ) η − u ( n ) (cid:13)(cid:13)(cid:13) L ∞ T H s → as η → . (v) There exists T > , independent of η , such that (cid:13)(cid:13)(cid:13) u η − u ( n ) η (cid:13)(cid:13)(cid:13) L ∞ T H s ≤ C ( η ) (cid:13)(cid:13)(cid:13) u − u ( n )0 (cid:13)(cid:13)(cid:13) H s . Proof. (i) Since Π is defined in R \{ } , it suffices to find η > , n ∈ N satis-fying (cid:13)(cid:13)(cid:13) u ( n )0 − ¯ k − ϕ η ∗ ( u ( n )0 − ¯ k ) (cid:13)(cid:13)(cid:13) L ∞ < − (6.2)for all 0 < η < η and n ≥ n . We observe that for Φ ∈ S ( R ), the left handside is bounded by C (1 + k ϕ k L ) (cid:16)(cid:13)(cid:13)(cid:13) u − u ( n )0 (cid:13)(cid:13)(cid:13) H s + (cid:13)(cid:13) u − ¯ k − Φ (cid:13)(cid:13) H s (cid:17) + k Φ − ϕ η ∗ Φ k L ∞
30y the Young inequality and the Sobolev embeddings. Thus the conclusionfollows since S ( R ) is dense in H s ( R ).(ii), (iii) By (6.2), the mean value theorem implies (cid:13)(cid:13)(cid:13) u ( n )0 ,η − ¯ k (cid:13)(cid:13)(cid:13) L ≤ (cid:13)(cid:13)(cid:13) Π( ϕ η ∗ ( u ( n )0 − ¯ k ) + ¯ k ) − Π( u ( n )0 ) (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13) u ( n )0 − ¯ k (cid:13)(cid:13)(cid:13) L ≤ C (cid:13)(cid:13)(cid:13) ϕ η ∗ ( u ( n )0 − ¯ k ) − ( u ( n )0 − ¯ k ) (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13) u ( n )0 − ¯ k (cid:13)(cid:13)(cid:13) L ≤ C (1 + k ϕ k L ) (cid:13)(cid:13)(cid:13) u ( n )0 − ¯ k (cid:13)(cid:13)(cid:13) L , which is bounded uniformly in η and n . For σ ≥
1, we have (cid:13)(cid:13)(cid:13) ∇ σ ( u ( n )0 ,η − ¯ k ) (cid:13)(cid:13)(cid:13) L x ≤ C X j =1 (cid:13)(cid:13)(cid:13)(cid:16) ∇ Π ◦ ( ϕ η ∗ ( u ( n ) − ¯ k ) + ¯ k ) (cid:17) · ∂ j ( ϕ η ∗ ( u ( n ) − ¯ k )) (cid:13)(cid:13)(cid:13) H σ − ≤ (cid:13)(cid:13)(cid:13) ∇ (cid:16) ϕ η ∗ ( u ( n )0 − ¯ k ) (cid:17)(cid:13)(cid:13)(cid:13) H σ − P ( (cid:13)(cid:13)(cid:13) ϕ η ∗ ( u ( n ) − ¯ k ) (cid:13)(cid:13)(cid:13) H σ − ) , where the last inequality follows from the same argument as in the proof ofProposition 6.1, which leads to the conclusion.(iv) We first observe the convergence of the first term. The argument inStep 1 in the proof of Theorem 1.2 in Section 5 yields (cid:13)(cid:13) u η − ¯ k (cid:13)(cid:13) L ∞ T H s +1 ≤ C ( (cid:13)(cid:13) u ,η − ¯ k (cid:13)(cid:13) H s ) (cid:13)(cid:13) u ,η − ¯ k (cid:13)(cid:13) H s +1 for T = T ( (cid:13)(cid:13) u − ¯ k (cid:13)(cid:13) H s ). Especially, (iii) implies (cid:13)(cid:13) u η − ¯ k (cid:13)(cid:13) L ∞ T H s +1 ≤ Cη − . Thus by interpolation, it suffices to show k u η − u η ′ k H = o ( η s − ) for 0 < η ′ < η. We first note that the following H -difference estimate is true: k u η − u η ′ k L ∞ T H ≤ C k u ,η − u ,η ′ k H . Indeed, since the embedding ∇ − L ∞ ⊃ H s holds for s >
2, we can use inSection 3 the energy method instead of Yudovich argument, which yields theestimate of the form G ( T ) ≤ CG (0) (see also [17]). Hence we have η − s k u η − u η ′ k L ∞ T H ≤ Cη − s k u ,η − u ,η ′ k H ≤ Cη − s (cid:13)(cid:13) ( ϕ η − ϕ η ′ ) ∗ ( u − ¯ k ) (cid:13)(cid:13) H ≤ Cη − s Z R |F [ ϕ η ]( ξ ) − F [ ϕ η ′ ]( ξ ) | |F [ u − ¯ k ]( ξ ) | h ξ i dξ. (6.3)31y the definition of ϕ , the Taylor theorem and interpolation yield |F [ ϕ η ]( ξ ) − | ≤ Cη s − | ξ | s − sup | ξ ′ |≤ η | ξ | | ∂ s F ϕ ( ξ ′ ) | + sup | ξ ′ |≤ η | ξ | | ∂ s +1 F ϕ ( ξ ′ ) | ! . Thus (6.3) is bounded by C Z R sup | ξ ′ |≤ η | ξ | | ∂ s F ϕ ( ξ ′ ) | + sup | ξ ′ |≤ η | ξ | | ∂ s +1 F ϕ ( ξ ′ ) | ! |F [ u − ¯ k ]( ξ ) | h ξ i s dξ → η → Z R sup | ξ ′ |≤ η | ξ | | ∂ s F ϕ ( ξ ′ ) | + sup | ξ ′ |≤ η | ξ | | ∂ s +1 F ϕ ( ξ ′ ) ||F [ u ( n )0 − u ]( ξ ) | h ξ i s dξ ≤ C (cid:13)(cid:13)(cid:13) u ( n )0 − u (cid:13)(cid:13)(cid:13) H s → n → ∞ uniformly in η .(v) Applying (1.4), we have (cid:13)(cid:13)(cid:13) u η − u ( n ) η (cid:13)(cid:13)(cid:13) L ∞ T H s ≤ C ( (cid:13)(cid:13) u ,η − ¯ k (cid:13)(cid:13) H s +1 , (cid:13)(cid:13)(cid:13) u ( n )0 ,η − ¯ k (cid:13)(cid:13)(cid:13) H s +1 ) (cid:13)(cid:13)(cid:13) u ,η − u ( n )0 ,η (cid:13)(cid:13)(cid:13) H s ≤ C ( η ) (cid:13)(cid:13)(cid:13) u − u ( n )0 (cid:13)(cid:13)(cid:13) H s , which completes the proof.We finally show that Proposition 6.2 implies the continuity. Let ǫ > T > η ≥ η and n > n ,we have (cid:13)(cid:13)(cid:13) u − u ( n ) (cid:13)(cid:13)(cid:13) L ∞ T H s ≤ k u − u η k L ∞ T H s + (cid:13)(cid:13)(cid:13) u η − u ( n ) η (cid:13)(cid:13)(cid:13) L ∞ T H s + (cid:13)(cid:13)(cid:13) u ( n ) η − u ( n ) (cid:13)(cid:13)(cid:13) L ∞ T H s ≤ k u − u η k L ∞ T H s + C ( η ) (cid:13)(cid:13)(cid:13) u − u ( n )0 (cid:13)(cid:13)(cid:13) H s + sup n ≥ n (cid:13)(cid:13)(cid:13) u ( n ) − u ( n ) η (cid:13)(cid:13)(cid:13) L ∞ T H s . We fix sufficiently small η , then (iv) gives (cid:13)(cid:13)(cid:13) u − u ( n ) (cid:13)(cid:13)(cid:13) L ∞ T H s ≤ ǫ + C ( η ) (cid:13)(cid:13)(cid:13) u − u ( n )0 (cid:13)(cid:13)(cid:13) H s . Therefore, taking n → ∞ , we have lim n →∞ (cid:13)(cid:13) u − u ( n ) (cid:13)(cid:13) L ∞ T H s = 0 since ǫ > Acknowledgments . The author would like to thank Yoshio Tsutsumi for thesupervision and giving him a lot of useful suggestions. The author also wishesto thank Stephen Gustafson for encouraging him to study the problem in thepresent paper and giving him helpful comments. The author was supported byGrand-in-Aid for JSPS Fellows 18J21037.32 eferences [1] Bejenaru, I., Ionescu, A.D., Kenig, C.E.: Global existence and uniquenessof Schr¨odinger maps in dimensions d ≥
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