The Cauchy problem for Schrodinger flows into Kahler manifolds

This paper has been withdrawn by the authors.

1. Introduction. We consider maps u : R n → N where N is a k-dimensional compact Kähler manifold with complex structure J and Kähler form ω (so that ω is a nondegenerate, skew-symmetric two-form). Thus N is a complex manifold and J is an endomorphism of the tangent bundle whose square, at each point, is minus the identity. N has a Riemannian metric g defined by g(·, ·) = ω(·, J·).
The condition that N is Kähler is equivalent to assuming that ∇J = 0 where ∇ is the Levi-Civita covariant derivative with respect to g. The energy of a map u is defined by where the energy density |du| 2 is simply the trace with respect to the Euclidean metric of the pullback of the metric g under u, |du| 2 = Tr u * (g). In local coordinates we have (We use the Einstein summation convention and sum over repeated indices.) The L 2 -gradient of E(u) is given by minus the tension of the map, −τ (u), τ (u) is a vector field on N which can be expressed in local coordinates as τ (u) = (τ (u) 1 , . . . , τ (u) k ) with where Γ i jk (u) are the Christoffel symbols of the metric g at u(x). Critical points of the energy are Harmonic maps and are characterized by the equation τ (u) = 0. The foundational result on the existence of harmonic maps is due to Eells and Sampson [14] and is achieved by studying the harmonic map flow which is simply the gradient flow for the energy functional on the space of maps. Eells and Sampson proved the existence of harmonic maps as stationary points of this flow when the domain is a compact manifold and the target is a compact manifold of non-positive curvature. In our setting, the symplectic structure on N induces a symplectic structure on the space of maps. Let X s = H s (R n , N ) be the Sobolev space of maps between R n and N as defined below. For s ≥ n 2 + 1, X s is a Banach manifold with a symplectic structure Ω induced from that of (N, ω) as follows. The tangent space to X s at a map u is identified with sections of the pull-back tangent bundle over R n . We let Γ(V ) denote the space of sections of the bundle V , for example du ∈ Γ(T s (R n )⊗u −1 (T N )). For σ, µ ∈ Γ(u −1 (T N )) = T u X s we define Ω(σ, µ) = R n ω(σ, µ)dx.
In this setting we are interested in the Hamiltonian flow for the energy functional E(·) on (X s , Ω). This is the Schrödinger flow which takes the form ∂u ∂t = J(u)τ (u). (1.2) This natural geometric motivation for the flow (1.2) was elucidated in [12].
A key aspect of our approach to understanding the flow (1.2) is to isometrically embed N in some Euclidean space R p and study "ambient" flows of maps from R n to R p which are related to (1.2). This is also central to the Eells-Sampson treatment of the harmonic map flow. Toward this end we use the Nash embedding theorem to assume that we have an isometric embedding w : (N, g) → (R p , δ).
Using this we can now define H s (R n ; N ), the L 2 -based Sobolev spaces of maps from R n to N as follows. Note that since the domain is noncompact some care must be taken even when s = 0. Definition 1.1. For s ≥ 1 let H s (R n ; N ) = {u : R n → R p : u(x) ∈ N a.e. and ∃ y u ∈ N such that (1.3) v − w(y u ) ∈ L 2 (R n ; R p ), ∂v ∈ H s−1 (R n ; R p ), With this definition in mind we can state our main result. for the generalized Schrödinger flow has a solution whenever the initial data u 0 ∈ H s+1 (R n , N ) for s ≥ n 2 + 4. Moreover (1.4) is locally well posed in H s+1 (R n , N ) for s ≥ n 2 + 4. The question of the local and global well-posedness of equation (1.4) with data in Sobolev spaces has been previously studied by many authors (see [12,13,11,35,36,42,10,33,34,43,44,31,32,20,23,21]). A common feature in most existence results for smooth solutions of Schrödinger maps is that they are obtained by using the energy method. This method consists in finding an appropriate regularizing equation which approximates the Schrödinger flow, and for which smooth solutions exist. One then proves that the regularizing equations satisfy a priori bounds in certain Sobolev norms, independent of the approximation, and that they converge to a solution of the original equation. The differences in the distinct results and proofs lie in the type of regularization used.
Ding and Wang [13] established a similar result to Theorem 1.2 for s ≥ n 2 + 3. Their work proceeds by direct study of equation (1.4) with β > 0, with a passage to the limit for β = 0. Thus the regularizing equation they use is obtained by adding the second order dissipative term βτ (u). In this paper we analyze equation (1.4) by adding a fourth order dissipative term (note that we allow the case β = 0 from the start). This term arises naturally in the geometric setting as the first variation of the L 2 -norm of the tension. We believe that our regularization of (1.4) by a fourth order equation, which is geometric in nature, is of intrinsic and independent interest. H. McGahagan [31,32] in her doctoral dissertation also proved a version of Theorem 1.2. Her work proceeds by a different regularization, this time hyperbolic, implemented by adding a term of the form −ǫ ∂ 2 u ∂t 2 which transforms the equation into one whose solutions are wave maps.
We note that while our existence proof in Theorem 1.2 is different from the ones in [13] and [31,32], our proof of uniqueness is the same, using parallel transport. In fact, in Appendix A we extend the uniqueness argument in [32], carried out there in the case when β = 0 to the case β ≥ 0, which gives the uniqueness statement in Theorem 1.2.
Our proof of Theorem 1.2 actually only shows that the mapping u 0 → u ∈ C([0, T ], H s ′ +1 (R n , N )), with s ′ < s, is continuous. However, one can show, by combining the parallel transport argument with the standard Bona-Smith regularization procedure ( [8,19,22]) that the statement in Theorem 1.2 also holds.
Equations of the type (1.4), but with N being Euclidean space are generally known as derivative Schrödinger equations and have been the object of extensive study recently (see for instance [26,16,9,15,27,28,24,25]). The results in these investigations however do not apply directly to (1.4) for two reasons. The first one is the constraint imposed by the target being the manifold N . The second one is that in these works one needs to have data u 0 in a weighted Sobolev space, a condition that we would like to avoid in the study of (1.4).
It turns out that for special choices of the target N , the equations (1.4) are related to various theories in mechanics and physics. They are examples of gauge theories which are abelian in the case of Riemann surfaces (Kähler manifolds of dimension 1 such as the 2-sphere S 2 or hyperbolic 2-space H 2 ). In the case of the 2-sphere S 2 , Schrödinger maps arise naturally from the Landau-Lifschitz equations (a U (1)gauge theory) governing the static as well as dynamical properties of magnetization [30,38]. More precisely, for maps s : R × R n → S 2 ֒→ R 3 , equation (1.4) takes the form ∂ t s = s × ∆s, |s| = 1 (1.5) which is the Landau-Lifschitz equation at zero dissipation, when only the exchange field is retained [29,38]. When n = 2 this equation is also known as the twodimensional classical continuous isotropic Heisenberg spin model (2d-CCIHS); i.e. the long wave length limit of the isotropic Heisenberg ferromagnet ( [29,38,42]). It also occurs as a continuous limit of a cubic lattice of classical spins evolving in the magnetic field created by their close neighbors [42]. The paper [42] contains, in fact, for the cases n = 1, 2, N = S 2 the first local well-posedness results for equation (1.4) or (1.5) that we are aware of. In [10], Chang-Shatah-Uhlenbeck showed that, when n = 1, (1.5) is globally (in time) well-posed for data in the energy space H 1 (R 1 ; S 2 ). When n = 2, for either radially symmetric or S 1 -equivariant maps, they show that small energy implies global existence. For global existence results see also [39]. In [33,34], the authors show that, when n = 2, the problem is locally well-posed in the space H 2+ε (R 2 ; S 2 ), while the existence was extended to the space H 3/2+ε (R 2 ; S 2 ) in [20] and [23], and the uniqueness to the space H 7/4+ǫ (R 2 ; S 2 ) in [21].
More recently, in [3,17], a direct method, in the case of small data, using fixed point arguments in suitable spaces was introduced. The first global well-posedness result for (1.5) in critical spaces (precisely, global well-posedness for small data in the critical Besov spaces in dimensions n ≥ 3) was proved, independently in [18] and [4]. This was later improved to global regularity for small data in the critical Sobolev spaces in dimensions n ≥ 4 in [5]. Finally, in [6], the global well-posedness of (1.5), for "small data" in the critical Sobolev space H n 2 (R n , S 2 ), n ≥ 2, was proved.

Remark 1.
A first version of this paper was posted on arxiv:0511701, in November 2005, by C.K., D.P., G.S. and T.T. The paper was withdrawn in May 2007. The reason for this was that Jesse Holzer, at the time a graduate student of Alex Ionescu at the University of Wisconsin, Madison, discovered an error in the original argument. The source of the error was in the construction of the ambient flow equations, which were introduced in the first version of the paper. The construction of the ambient flow equations resulted in equations of quasilinear type in the leading order term, which could not be solved by the Duhamel principle as was pointed out by Holzer.
C.K., D.P., G.S. and T.T. are indebted to Jesse Holzer for pointing out this error and to T.L., who constructed new ambient flow equations, whose leading order terms are ε∆ 2 , which can then be dealt with directly by Duhamel's principle. The price one pays for this change in the ambient flow equation, is that it turns out to be more difficult to show that if the initial value takes values in the manifold, so does the whole flow. All of this is carried out in Lemma 2.5 and Lemma 2.10.
Notation. We will use C, c to denote various constant, usually depending only on s, n and the manifold N . In case a constant depends upon other quantities, we will try to make that explicit. We use A B to denote an estimate of the form A ≤ CB.

2.
A fourth order parabolic regularization. The method we employ in order to establish short-time existence to (1.4) is in part inspired by the work of Ding and Wang [12]. We seek to approximate equation (1.4) by a family (parametrized by 0 < ε < 1) of parabolic equations. We establish short time existence for these systems and use energy methods to show that the time of existence is independent of ε and obtain ε independent bounds which allow us to pass to the limit as ε → 0 and thus obtain a solution to (1.4). The regularization we use differs substantially from that of Ding and Wang because we wish to view the right hand side of (1.4) as a lower order term (in the regularization) so that we can use Duhamel's principle and a contraction mapping argument to establish and study the existence of our derived parabolic system.
The energy method we employ ultimately depends on establishing ε independent L 2 -estimates for the tension, τ (u) and its derivatives. This suggests that we regularize (1.2) by ε times the gradient flow for the functional 2.1. Geometric preliminaries. We perform many of our computations in the appropriate pull-back tensor bundles over R n . We begin by recalling alternative formulations of the tension τ (u) in this setting (see [14]). First note that du is a closed 1-form with values in u −1 (T N ). The tension is simply minus the divergence of the differential of u where δ denotes the divergence operator with respect to the metric g. In particular, this shows that a map u is harmonic if and only if its differential is a harmonic 1-form. Let ∇ denote the covariant derivative on T * (R n ) ⊗ u −1 (T N ) defined with respect to the Levi-Civita connection of the Euclidean metric on R n (i.e. the ordinary directional derivative) and the Riemannian metric g on N . For α = 1, . . . , n we let ∇ α u ∈ Γ(u −1 (T N )) be the vector field given by where (u 1 , . . . , u k ) are coordinates about u(x) ∈ N . In particular The second fundamental form of the map u is defined to be the covariant derivative of du, ∇du ∈ Γ((T 2 R n ) ⊗ u −1 (T N )). In local coordinates we have for i = 1, . . . , k and α, β ∈ 1, . . . n, Note that here the subscript α actually denotes covariant differentiation with respect to the vector field ∇ α u as defined in (2.1) and we have ∇ α ∇ β u = ∇ β ∇ α u. The tension is simply the trace of ∇du with respect to the Euclidean metric, δ = δ αβ ∂u j ∂x α ∂u k ∂x α from which we recover (1.1).

2.2.
The gradient flow for G(u). For a given vector field ξ ∈ Γ(u −1 (T N )), we construct a variation of u : R n → N with initial velocity ξ as follows. Define the map where exp u(x) : T u(x) N → N denotes the exponential map. Set u s (x) = U (x, s) and now let ∇ denote the natural covariant derivative on T * (R n × R) ⊗ U −1 (T N ). Then where the inner products are taken with respect to g and we have used the metric compatibility of ∇. Let R = R(·, ·)· denote the Riemann curvature endomorphism of ∇. Using (2.3) and the definition of R we see that By the symmetries of the curvature we have and provided that τ (u) and ∇ α τ (u), for α = 1, . . . , n, are in L 2 (and likewise for v) we may integrate by parts to obtain The parabolic regularization of (1.4) which we now proceed to study is ∂u ∂t = −εF (u) + J(u)τ (u) + βτ (u) u(0) = u 0 (2.6) 2.3. The ambient flow equations. Rather than attempting to study the parabolic equations (2.6) directly we will focus on the induced "ambient flow equations" for v = w • u where w : (N, g) → R p is a fixed isometric embedding. We fix a δ > 0, chosen sufficiently small so that on the δ-tubular neighborhood w(N ) δ ⊂ R p , the nearest point projection map so that |ρ(Q)| = dist (Q, w(N )), and viewing ρ and Π as maps from w(N ) δ into itself we have Note that then the differentials of the maps satisfy as a linear map from R p to itself. For any map v : R n → w(N ) δ we set Therefore, in direct analogy with the functional G(·), we now consider Our point here (and hence the seemingly odd notation) is that we wish to consider T (v) for arbitrary maps into w(N ) δ whose image does not necessarily lie on N .
Definition 2.1. For v : R n → w(N ) δ , let F (v) denote the Euler-Lagrange operator of G(v) with respect to unconstrained variations. A simple computation shows that its components are given by α denotes the lower order terms. Note that the subscripts here refer to coordinate differentiation in R n (Greek indices) or R p (Roman indices).
For v = w • u, we wish to consider compactly supported tangential variations of G(v). Such variations correspond to (compactly supported) vector fields φ on w(N ) δ which satisfy dρ(φ) = 0.
Definition 2.2. If v = w • u, then the ambient form of the Schrödinger vector field J(u)τ (u), is given by the vector field f v with Note that f v is defined for maps v : R n → w(N ) δ whose image does not necessarily lie on N .
Next we have the following Lemma.
Proof. This follows from the facts that Finally we note that

Remark 2.
Note that H(v) only contains derivatives of v up to third order. Moreover this term is well-defined for every v : R n → w(N ) δ .
The regularized ambient equations are given by ∂v ∂t The basic relationship between the regularized geometric flows (2.6) and the regularized ambient flows (2.10) is provided by the following Lemma (cf. §7 of [14]).
Proof. First note that since w is an isometry we have and therefore G(v) = G(u). Given ξ ∈ Γ(u −1 (T N )) a smooth compactly supported vector field set φ = dw(ξ) ∈ Γ(u −1 (T R p )). As before we consider the variation of u given by u s (x) = exp u(x) sξ. We then have This together with (2.12) and (2.13) implies that the flows correspond as claimed.
We end this section by exhibiting in a more practical form the structure of the parabolic operator appearing in the regularized ambient flow equations (2.10).
and let ∂ j1 v * · · · * ∂ j l v denote terms which are a sum of products of terms of the form ∂ j1 v, . . . , ∂ j l v.
Proof. This follows from the explicit expressions for In the following Lemma (which is a suitable modification of Theorem 7C of [14]) Note that in this case by Lemma Proof. We start by calculating Here we used again that Π ab (v) = −ρ ab (v). Now if v is a solution of (2.10) we get that Combining these two calculations yields

Multiplying this equation with ρ(v) and using the facts that (note that
Integrating this equation over R n and using integration by parts, we have for all t ∈ (t 0 , t 1 ) ≤0.
Since ρ(v 0 ) = 0 this implies that ρ(v) = 0 for all t ∈ [t 0 , t 1 ] and hence finishes the proof of the Lemma.
3. The Duhamel solution to the ambient flow equations. In this section we introduce a fixed point method that solves the initial value problem (2.10) in the Sobolev space H s+1 (R n , R p ), for s ≥ n 2 + 4. To simplify the notation, using Proposition 2.3, we rewrite (2.10) as ∂v ∂t We now state the well-posedness theorem for (3.1). For any fixed v 0 , define the spaces We then have the following theorem: To prove the theorem we rewrite (3.1) as an integral equation using the Duhamel principle: is the solution of the linear and homogeneous initial value problem associated to (3.1). The main idea is to consider the operator and prove that for a certain T ε the operator L is a contraction from a suitable ball To estimate L we need to study the smoothing properties of the linear solution S ε (t)v 0 . Because the order of derivatives that appears in N (v) is 3, in order to be able to estimate the nonlinear part of L in H s+1 , we should prove that the operator S ε (t) provides a smoothing effect also of order 3. We have in fact the following lemma: Define the operator D s , s ∈ R as the multiplier operator such that D s f (ξ) = |ξ| s f . Then for any t > 0 and i = 1, 2, 3, Proof. The proof follows from Plancherel theorem and the two estimates To state the next lemma, where we show how for small intervals of time the evolution S ε (t)(v 0 − γ) stays close to v 0 − γ, we need to introduce the spaceḢ s . This space denotes the homogeneous Sobolev space defined as the set of all functions f such that D s f ∈ L 2 .
Proof. By the mean value theorem which combined with the trivial bound We now write and this concludes the argument since s > n 2 + 4σ guarantees the summability after the application of Cauchy-Schwartz. Note also that We are now ready to prove Theorem 3.1.
Proof. For T ε , r > 0 and s ≥ n 2 + 4 consider the ball We want to prove that for the appropriate T ε and r, the operator L maps B r to itself and is a contraction. We start with the estimate of the linear part of L. By (3.6) we have To estimate the nonlinear term we use (3.6) (3.7), and interpolation : Thus We still need to check that Lv is continuous in time and that Lv ∈ L ∞,2 δ . The continuity follows directly from the continuity of the operator S ε (t). To prove the L ∞ and L 2 estimates one uses (3.8) with σ = 1/4 applied to f = v 0 − γ, the Sobolev inequality and estimates similar to the ones used to obtain (3.14). One gets We now take r = 3C 0 v 0 H s and so that (3.14) and (3.16) By shrinking T ε further by an absolute constant if necessary, from (3.19) and (3.20) we obtain (3.20) The contraction mapping theorem ensures that there exists a unique function 4. Analytic preliminaries. In this section we state and present the detailed proof of an interpolation inequality for Sobolev sections on vector bundles which appears in [13] (see Theorem 2.1). This inequality was first proved for functions on R n by Gagliardo and Nirenberg, and for functions on Riemannian manifolds by Aubin [1]. The justification for presenting a complete proof is that this estimate plays a crucial role in the energy estimates and therefore in the proof of the results this paper. The precise dependence of the constants involved in this inequality is vital to our argument and we feel compelled to emphasize it.
Let Π : E → R n be a Riemannian vector bundle over R n . We have the bundle Λ P T * R n ⊗ E → R n over R n which is a tensor product of the bundle E and the induced P -form bundle over R n , with P = 1, 2, . . . , n. We define T (Λ P T * R n ⊗ E) as the set of all smooth sections of Λ P T * R n ⊗ E → R n . There exists an induced metric on Λ P T * R n ⊗ E → R n from the metric on T * R n and E such that for any where {e i } is an orthonormal local frame for T R n . We define the inner product on with respect to the above inner product. To define the bundle-valued Sobolev space H k,r (R n , Λ P T * R n ⊗ E) consider ∇ the covariant derivative induced by the metric on E, then take the completion of smooth sections of E in the norm where If r = n/k − 1 = 1 then (4.5) does not hold for a = 1. The constant C that appears in (4.5) only depends on n, k, j, q, r and a.
Proof. If f is a real valued smooth function with compact support on E then Theorem 3.70 in [2] ensures that (4.5) holds.
Since R n div |∇s| p−2 ∇s, s = 0 (4.14) then (4.13) gives Given our choice of j = 1, k = 2 and a = 1 2 we have 1 q + 1 r = 2 p , i.e. 1 q + 1 r + p−2 p = 1. Thus Hölder's inequality yields thus For a ∈ 1 2 , 1 we consider two cases: q < n, and q ≥ n. For q < n using the convexity of log f p L p as a function of p we have where t < p < σ are such that 2 (4.20) Using (4.11) and (4.17) we have that and Combining (4.19), (4.21) and (4.22) we obtain which proves the case a ∈ 1 2 , 1 and q < n. For q ≥ n, t > 0 and b ∈ (0, 1) such that we have by (4.9) Choosing t > 0 so that we have by (4.17) Combining (4.26) and (4.28) we obtain by (4.25) and (4.27). This concludes the proof of Case 2.

Case 4:
We now proceed by induction on k. Assume that for k ≥ 2 and j < k we have proved (4.5). Let j < k < k + 1. By (4.9) we have By the induction hypothesis, applied to ∇ k−1 s, we also have and Letting q 4 = r 2 , q 3 = q, r 4 = r, r 2 = p we obtain By hypothesis for j < k and using (4.43) we have with a 0 a 1 ∈ j k+1 , 1 and which finishes the proof of the proposition.
If r = n k−1−j = 1 then (4.47) does not hold for a = 1. The constant C that appears in (4.48) only depends on n, k, j, q, r and a.
Proof. Apply (4.5) to s = ∇u a section of the bundle u * (T N ) ⊗ T * R n . Since ∇u is not necessarily compactly supported a standard approximation argument might be needed to complete the proof.
In the second part of this section we establish the equivalence of the Sobolev norms defined in either the intrinsic, geometric setting or in the ambient, Euclidean setting. These results hold when we are above the range in which these spaces have suitable multiplication properties. Since we are working with the gradients of the maps we must consider the H s spaces with s > n 2 + 1. We begin by assuming that we have chosen coordinate systems on (N, g) so that the eigenvalues of g are bounded above and below by a fixed constant C > 1, i.e. we assume that We denote these coordinates by either (y 1 , . . . , y k ) or (u 1 , . . . , u k ). As before (x 1 , . . . , x n ) denotes Euclidean coordinates on R n .
For v : R n → R p we let ) denotes the vector field along u defined in (2.1). We use the following notation for higher order derivatives.
Remark 1. Note that our use of the multi-index notation differs from the usual one.
Recall that for v : R n → R p u : R n → N and the sum is taken over all distinct multi-indices of length l + 1. The L 2 norm of each of these is computed with respect to the metric g as indicated. We use the obvious analogous definition for Our immediate goal is to show that for k > n 2 G (j1,...,j l ) (u)∂ j1 u * · · · * ∂ j l u (4.49) E (j1,...,j l ) (u)∇ j1 u * · · · * ∇ j l u (4.50) where each subscript j s ≥ 1 and ..,j l ) (u)e a and each G, E and F are smooth, bounded functions of u.
The notation a j1 * · · · * a j l corresponds to a product of the a j k 's.
Remark 2. Throughout this section whenever expressions similar to the right hand side of (4.49), (4.50) or (4.51) occur, a key point is to note that all the subscripts j s ≥ 1, for s ∈ {1, . . . , l}. This is always to be understood even if it is not explicitly stated.
Proof. We establish each of these by induction, beginning with (4.49). Note that for k = 0, ∇u = ∂u. For k = 1 G (j1,...,j l ) (u)∂ j1 u * · · · * ∂ j l u which completes the proof of (4.49). The proof of (4.50) proceeds in a similar fashion and is left to the reader.
We now proceed to bound the pointwise norms in terms of each other.
Lemma 4.5. Assume that k > n 2 + 1. There exists a constant C = C(N, k, n) such that for u ∈ C ∞ (R n , N ) constant outside a compact set of R n if v = w • u then Proof. By (4.55) we have Then by Hölder's inequality Since k ≥ n 2 + 1 then Note that to ensure that a i < 1 in (4.64) we either need n ≤ 3 or we must have . Thus to accommodate all values of n simultaneously, it is enough to choose k ≥ n 2 +1 and k ∈ N. Thus (4.5) in Proposition 4.1 yields Therefore combining (4.61), (4.63) and (4.66) we have A similar argument to the one above where Proposition 4.4 is now applied to ∇u rather than ∂v yields (4.60).
Lemma 4.6. There exists a constant C = C(N, n) such that if u ∈ C ∞ (R n , N ) is constant outside a compact set of R n and v = w • u then for 1 ≤ k ≤ n 2 + 1 (4.69) Proof. The proof is very similar to that of Lemma 4.5, where the a i 's and p i 's in the interpolation are taken as follows where s 0 = n 2 + 2, and Remark 3. Proposition 4.4 holds for s ∈ C m c (E) where E is a finite dimensional C m vector bundle over R n provided k < m. Similarly Lemma 4.5 holds for u ∈ C m (R n , N ) and u constant outside a compact set of R n , provided once again that m > k. A simple approximation theorem ensures that Lemma 4.5 holds for u ∈ C m (R n , N ) ∩ H k (R n , N ) with m > k. Corollary 4.2. Assume that k ≥ n 2 + 4. There exists a constant C = C(N, k, n) such that for u ∈ C k+1 (R n , N ) ∩ H k (R n , N ) (4.74) Proof. Recall that ∇u = |∂v| if v = w • u, and by Sobolev embedding theorem and Therefore combining (4.60), (4.69) and (4.75) we have Note that (4.55) ensures that Note that (4.55) also ensures that To prove that (1.4) has a solution we need to show that (2.10) has a solution for ε = 0. To do this we need to show that each v ε extends to a solution in C([0, T ], H s+1 ∩ L 2,∞ δ ) where T > 0 is independent of ε. This is accomplished by proving ε-independent energy estimates for the function v ε . It turns out that thanks to the geometric nature of this flow, if one assumes enough regularity (i.e. s > [ n 2 ] + 4), it is easier to prove ε-independent energy estimates for the corresponding u ε . Lemma 4.5 and Lemma 4.6 then allows us to translate these into estimates for v ε .
Let l ∈ N. We denote by α the multi-index of length l, α = (α 1 · · · α l ), and ∇ α u = ∇ α1··· ∇ α l u. The following lemma and corollaries establish some computational identities which are very useful. N )), s ∈ N, s > n 2 + 2. Let X ∈ T N for 1 ≤ l ≤ s and |α| = l. We have Proof. The proof is done by induction on the length of the multi-index α, i.e., on l. We prove (5.4) and leave (5.2) and (5.3) to the reader, as the proofs are very similar. If l = 1
Remark 4. Note that in particular (5.10) applied to X = τ (u) yields N )) with s ∈ N and s ≥ n 2 + 4 be a solution of (5.1). Then for n 2 + 4 ≤ l ≤ s and l ∈ N we have d dt For l = n 2 + 2 and l = n 2 + 3 we have d dt Proof. We first compute the evolution where we have used the fact that for f ∈ L 1 (R n , R n ) R n div f = 0 as well as integration by parts. Note that using integration by parts and Cauchy-Schwarz we have Combining (5.15) and (5.16) we have For 1 ≤ l ≤ s applying (5.3) we have Consider each term separately Using (5.12) and (5.7) and integrating by parts we have that We now look at the third term in (5.19) and recall that ∇J = 0 and JX, X = 0 for X ∈ T N . Integrating by parts and applying (5.7) we obtain for γ = (α 2 · · · α l ) Thus (5.23) yields |∇ l−2 τ (u)| |∇ j1 u| · · · |∇ jm u|.
To finish the estimate we need to use the interpolation result that appears in Proposition 4.4. Consider 3 ≤ m ≤ l + 2, 1 ≤ j s ≤ l − 1 and j 1 + · · · + j m = l + 2 then by
To prove the well-posedness of the Schrödinger flow (i.e. when β = 0 in (1.4)) we refer to work of Ding and Wang [12] and McGahagan [32]. By adapting the argument of Ding and Wang [12] one can show that if a solution, u ∈ C([0, T 0 ], H s+1 (R n , N )) with s ≥ n 2 + 4, to the initial value problem (1.4) (with β = 0) exists then it is unique. This argument makes explicit use of the fact that the target is compact and isometrically embedded into some Euclidean space. We present and extend here part of an argument that appears in the proof of Theorem 4.1 in [32]. These inequalities yield uniqueness and continuous dependence on the initial data for general β ≥ 0. Let u 1 , u 2 ∈ C([0, T 0 ], H s+1 (R n , N )) be solutions of (1.4) with initial data u 0 1 , u 0 2 ∈ H s+1 (R n , N ) with s ≥ n 2 + 4. Following the notation in [32] let V = ∇u 1 and W = ∇u 2 . Let V (x) represent the parallel transport of V to the point u 2 (x) along the unique geodesic joining the points. McGahagan proves (see end of the proof of Theorem 4.1 in [32]) that whenever u 0 1 − u 0 2 H [ n 2 ] +4 is small enough (depending only on the geometry of N ) and β = 0, then where C depends on the H [ n 2 ]+4 norms of u 1 and u 2 . In the case that u 0 1 = u 0 2 McGahagan concludes (using Gronwall's) that W − V 2 L 2 = u 1 − u 2 2 L 2 = 0, and that therefore u 1 = u 2 a.e.. In appendix A we show that the inequality (5.108) (and therefore also the uniqueness result) remains true for all β ≥ 0.
Since the unique solution is constructed as a limit of solutions of equation (5.2) letting ε → 0, the estimate in Theorem 5.5 yields that sup 0≤t≤T0 ∇u(t) H s ≤ 3 ∇u 0 H s . (5.109) To prove the continuous dependence on the initial data note that, in general, (5.108) yields where W 0 = ∇u 0 2 and V 0 (x) is the parallel transport of V 0 = ∇u 0 1 to u 0 2 (x). Since and ] +4 is small enough then the functions v 1 = ω • u 1 and v 2 = ω • u 2 , which are solutions to the ambient equation, depend continuously in the H s ′ +1 (R n , R p )-norm on the initial data for s ′ < s. As mentioned in the introduction by means of the standard Bona-Smith regularization procedure ( [8,19,22]) one can prove that the dependence on the initial data is continuous in H s+1 (R n , R p ). It is in this sense that we express the well-posedness of (1.4). This concludes the proof Theorem 1.
Appendix A. Proof of (5.108) for β ≥ 0. Since the proof follows closely the one of Theorem 4.1 in [32] we only sketch the main ideas here.
We let u 1 , u 2 ∈ C([0, T 0 ], H s+1 (R n , N ) be two solutions of (1.4) with initial data u 0 1 respectively u 0 2 and we assume that ||u 0 1 − u 0 2 || H [ n 2 ] +4 is small. As in [32] we let γ(x; x, t) be the unique length minimizing geodesic (parametrized by arclength s ∈ [0, l(x, t)]) between u 1 (x, t) and u 2 (x, t), where γ(0; x, t) = u 1 (x, t) and γ(l(x, t); x, t) = u 2 (x, t) (the existence of the geodesic follows from the argument on page 392 in [32]; note that this argument is also applicable if u 0 1 and u 0 2 are only close to each other in L n ). Moreover we define V k = ∂ k u 1 , W k = ∂ k u 2 and V k = X(l, 0)V k as the parallel transport of V k to the point u 2 . In the following we let X(l, 0) =: X.
Using the previous definitions and the commutator formula (A.1) we can rewrite this equation as follows Multiplying this equation with (D k W k − XD k V k ∈ T u2 N ) and integrating we get Combining this with the above equality we conclude Next we estimate the three terms on the right hand side. We start with where we used (A.2) in the last line. Using the same arguments we also get |III| ≤c||W k − XV k || L 2 ||[X, ∂ t ]V k || L 2 ≤c(||W −V || 2 L 2 + ||u 1 − u 2 || 2 L 2 ).
Altogether this implies Next we need to estimate 1 2 ∂ t R n |u 1 − u 2 | 2 . In order to do this we argue as in [32] and we consider N to be isometrically embedded into R p and we extend J as a continuous linear operator on R p . In the following we denote the second fundamental form of the embedding N ֒→ R p by A. With the help of these conventions we calculate Arguing as in [32], page 296, we get the estimate