Equivariant Schrödinger map flow on two dimensional hyperbolic space

In this article, we consider the Schrodinger flow of maps from two dimensional hyperbolic space \begin{document}$ {{\mathbb{H}}}^2 $\end{document} to sphere \begin{document}$ {{\mathbb{S}}}^2 $\end{document} . First, we prove the local existence and uniqueness of Schrodinger flow for initial data \begin{document}$ u_0\in\mathbf{H}^3 $\end{document} using an approximation scheme and parallel transport introduced by McGahagan [ 32 ]. Second, using the Coulomb gauge, we reduce the study of the equivariant Schrodinger flow to that of a system of coupled Schrodinger equations with potentials. Then we prove the global existence of equivariant Schrodinger flow for small initial data \begin{document}$ u_0\in\mathbf{H}^1 $\end{document} by Strichartz estimates and perturbation method.

1. Introduction. Let (H 2 , g) be the two dimensional hyperbolic space, the Schödinger map flow is defined by the initial value problem    ∂u ∂t = J(u)τ (u), where u(x, t) : (H 2 , g) × [0, ∞) → S 2 , J is the complex structure on S 2 , and τ (u) is the tension field of u. In the local coordinates (x 1 , x 2 ) on H 2 and (y 1 , y 2 ) on S 2 , τ (u) is given by [32] for general Kähler manifolds. The first global well-posedness of Schrödinger flow for maps R d → S 2 , d ≥ 3 with small data in the critical Besov spaces was proved by Ionescu and Kenig [16], and independently by Bejenaru [3]. This was later improved to global regularity for small data in the critical Sobolev spaces by Bejenaru, Ionescu and Kenig [4] (d ≥ 4) and Bejenaru, Ionescu, Kenig and Tataru [5] (d ≥ 2) for target S 2 , and by Li [27,28] for general compact Kähler manifolds. However, the Schrödinger map flow with large data is a much more difficult problem. When the target is S 2 , there exists a collection of families Q m (see [8]) of finite energy stationary solutions for integer m ≥ 1; When the target is H 2 , there is no nontrival equivariant stationary solutions with finite energy. Hence, Bejenaru, Ionescu, Kenig and Tataru [6,7] proved the global well-posedness and scattering for equivariant Schrödinger flow of maps R 2 → S 2 with energy below the ground state and maps R 2 → H 2 with finite energy. When the energy of maps is larger than that of ground state, the dynamic behaviors are complicated. The asymptotic stability and blow-up for Schrödinger flow have been considered by many authors for instance [13,14,15,8,33,34]. We refer to [18] for more open problems in this field. It's natural to consider geometric flows starting from curved manifolds. Because the hyperbolic spaces are symmetric and noncompact, the geometric flows from hyperbolic spaces are natural starting points. An interesting model is the heat flow from hyperbolic spaces, which is related to the Schoen-Li-Wang conjecture (see Lemm,Markovic [24]). By solving the heat flow, Li and Tam [30] gave the sufficient conditions to ensure the existence of the harmonic map between hyperbolic spaces. In recent years, there are many works concerning wave maps on hyperbolic spaces which are expected to share many similar phenomena with Schrödinger flow. D'Ancona and Zhang [10] showed the global existence of equivariant wave maps from hyperbolic spaces H d for d ≥ 3 to general targets with small data in H d 2 ×H d 2 −1 . The problem was also intensely studied by Lawrie, Oh and Shahshahani [20,21,22,23], Li, Ma and Zhao [29] and Li [25,26]. Since the wave maps H 2 → H 2 or S 2 admit a family of equivariant harmonic maps, the stability of stationary k-equivariant wave maps was proved by analyzing spectral properties of the linearized operator in [20] and [21], and the soliton resolution for equivariant wave maps H 2 → H 2 with finite energy data was proved by profile decomposition in [22]. For data without any symmetric assumption, Li, Ma and Zhao [29] proved that the small energy harmonic maps from H 2 to H 2 are asymptotically stable under the wave map, then Li [25,26] further showed the asymptotic stability of large energy harmonic maps. Lawrie, Oh and Shashahani [23] established the global well-posedness and scattering for wave maps from H d for d ≥ 4 into Riemannian manifolds of bounded geometry for small data in the critical Sobolev space. As a geometric flow, Schrödinger flow is a special case of Landau-Lifshitz flows. Li and Zhao [31] proved that the solution of Landau-Lifshitz flow u(t, x) from H 2 to H 2 converges to some harmonic map P (x) as t → ∞, i.e lim t→∞ sup x∈H 2 d H 2 (u(t, x), P (x)) = 0, when the Gilbert coefficient is positive.
The Schrödinger flow on H 2 exhibits markedly different phenomena from its Euclidean counterpart. First, the most interesting feature is that there is an abundance of equivariant harmonic maps as shown in [20]. Precisely, there is a family of equivariant harmonic maps H 2 → S 2 , indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere. These maps have energies taking all values between zero and the energy of the unique co-rotational Euclidean harmonic map, Q euc , from R 2 to S 2 . Second, the notable feature of the problem is the better dispersive estimates of the operator e it∆ H 2 than the Euclidean counterpart, see [2]. The stronger dispersion is possible due to the exponential volume growth of concentric geodesic spheres on the domain. The above features make (1) an interesting model.
In this paper, we establish the local existence and uniqueness of Schrödinger flow for large data and global existence and uniqueness of equivariant Schrödinger flow for small data. To explain the main results in more detail, we need to introduce some notations.
Let u be a smooth map from H 2 to S 2 . The pullback bundle u * T S 2 is the vector bundle over H 2 whose fiber at x ∈ H 2 is the tangent space T u(x) S 2 . Smooth sections of u * T S 2 are maps V : H 2 → T S 2 so that V (x) ∈ T u(x) S 2 for each x ∈ H 2 . Let ∇ denote the connections on different vector bundles which are naturally induced by Levi-Civita connections on H 2 and S 2 . Sometimes in the context, we also use more specific notations such as ∇ H 2 and ∇ S 2 to emphasize which connection we are using. In the local coordinates (x 1 , x 2 ) on H 2 and (y 1 , y 2 ) on S 2 , if V = V α ∂ ∂y α , then the covariant derivative on u * T S 2 are given by Hence, we may define the intrinsic Sobolev spaces H k (H 2 ; S 2 ) by where dvol g is the volume form of (H 2 , g), ∇ l−1 is the (l −1)-th covariant derivative of tangent vectors ∂u, and in the local coordinates, |∇ l−1 ∂u| 2 g = g i1j1 · · · g i l j l ∇ ∂ ∂x i 1 · · · ∇ ∂ ∂x i l−1 For simplicity, denote H k := H k (H 2 ; S 2 ). It is easy to check that the flow (1) admits the conserved energy where |∇u| 2 g , in the local coordinates, is given by The first main result is the following. Remark 1. The proof of Theorem 1.1 follows closely that of [32,35]. The local existence for Schrödinger map flow with data u 0 ∈ H 3 is proved by approximation of wave maps on hyperbolic space (H 2 , g) and the uniqueness is given by parallel transport.

JIAXI HUANG, YOUDE WANG AND LIFENG ZHAO
The second result concerns the equivariant Schrödinger map flow with small data u 0 ∈ H 1 . To explain this result, we introduce the geodesic polar coordinates on H 2 . We consider the Minkowski space R 2+1 with the Minkowski metric dx 2 1 + dx 2 2 − dx 2 3 , and the bilinear form on R 2+1 × R 2+1 , [x, y] = −x 1 y 1 − x 2 y 2 + x 3 y 3 .
Then hyperbolic space H 2 is defined as x] = 1, x 3 > 0}, and the Riemannian metric g on H 2 is exactly the one induced by the Minkowski metric on R 2+1 .
We denote the origin of H 2 by 0 = (0, 0, 1). And r : H 2 → [0, ∞), r(x) = d(x, 0) denotes the Riemannian distance function to the origin. Then the geodesic polar coordinates on H 2 are defined by In these coordinates, the hyperbolic metric g is given by g = dr 2 + sinh 2 rdθ 2 , the volume element dvol g on H 2 is given by sinh rdrdθ and the Laplace-Beltrami operator is written as and only if u can be written in the polar coordinates as u(r, θ) = e mθRū (r).
Here R is the generator of horizontal rotations, which is defined as where k = (0, 0, 1) . The energy of m-equivariant maps can be expressed as If m = 0, then E(u) < ∞ implies lim r→0 u 1 (r) = lim r→0 u 2 (r) = 0, and then lim r→0 u 3 (r) = ±1 by |u| = 1. To fix matters we agree that this limit is 1 for all t. However, due to the exponential decay of sinh −1 r as r → ∞ in the last term in the integrand of (4), we can choose the endpoints lim r→∞ū 1 (r), lim r→∞ū 2 (r) ∈ [−1, 1] arbitrarily, which is ultimately responsible for the existence of the family of harmonic maps mentioned in the above. This stands in contrast to the corresponding problem for Schödinger map flow on Euclidean spaces where the endpoints ofū 1 (r),ū 2 (r) at r → ∞ can only be zero. For any m-equivariant Schrödinger flow of maps u : and is attained by the 1-parameter family of harmonic maps where h α λ (r) := e αR h λ (r), and h λ (r) = 2λ tanh r This leads us to consider the equivariant Schrödinger flow with data in the classes Next we state our second main result.
Theorem 1.2 (Global existence for small data). There exists a sufficiently small constant 0 > 0 such that for any 1-equivariant map u 0 with u 0 H 1 ≤ 0 and for any compact interval J ⊂ R, the equivariant Schrödinger map flow (1) has a unique solution u ∈ L ∞ t (J; H 1 ) in the class E λ0 for λ 0 satisfying lim Remark 2. (i) The result in Theorem 1.2 can be extended to all m-equivariant cases for m = 0.
(ii) Under the equivariant condition, the smallness assumption on u 0 and the inequality (5) imply that λ 0 must be small.
(iii) In light of the works [20,25,26,29], which concerns the asymptotic stability of finite energy harmonic maps from H 2 to S 2 or H 2 under the wave maps evolution, one expects the asymptotic stability of harmonic maps under the Schrödinger flow. In fact, Lawrie-Lührmann-Oh-Shahshahani [19] proved the asymptotic stability of a finite energy equvariant harmonic map Q under the Schrödinger flow with respect to non-equivariant perturbations, provided Q obeys a suitable linearized stability condition. Theorem 1.2 is of similar flavor to the result of [6] in the flat domain R 2 . First, since we restrict ourselves to the class of equivariant Schrödinger flow, the Coulomb gauge suffices in our study. In fact, in this gauge, we can rewrite the equations for ∂ r u and ∂ θ u which leads to a system of coupled Schrödinger equations on H 2 with potentials, i.e Moreover, A 2 , A 0 can be expressed in terms of ψ ± , i.e and Next, we consider the Cauchy problem of the system (8). In fact, we establish the Strichartz estimates for Schrödinger operators with nonnegative potentials, and prove the global existence of (8) for small data ψ ± 0 in the space L 2 (H 2 ). Since our interest lies in the solutions which correspond to the geometric flow, we will show that the solutions of the system satisfy the compatibility condition. Finally, to construct the Schrödinger map flow u from ψ ± , the key observation is that ψ + or ψ − contain all the information of the flow as in [6]. Hence, we can recover the flow u(t) from ψ ± (t) for small initial data ψ + (0), ψ − (0) ∈ L 2 . Furthermore by Theorem 1.1, we show that the flow u(t) is a Schrödinger map flow for data u 0 in H 3 . At the same time, we obtain the Lipschitz continuity of u(t) with respect to u 0 in H 1 , which yields Theorem 1.2.
There are two main obstacles in the above arguments. One is the a priori higher order energy estimates for approximate wave map equations, which guarantees the uniform lifespan T > 0 for approximate solutions. In order to simplify the computation, the global system of coordinates related to the Iwasawa decomposition is used. Meanwhile the uniform estimates follows from a bootstrap argument. The other obstacle lies in the establishment of the well-posedness for the coupled Schrödinger system with potentials. In order to avoid the presence of the singular potential 2 cosh r+1 sinh 2 r in the ψ + -equation in (8), we multiply e i2θ to this equation, and denote R k ψ + (r, θ) = e ikθ ψ + (r), for k = 0, 1, 2 · · · , then the ψ + -equation can be rewritten as where F + denotes the nonlinearity of ψ + -equation in (8). Note that, the potential 2 cosh r−1 sinh 2 r in (9) is regular. In order to establish the Strichartz estimates for (9), we prove the dispersive estimates for e it(∆ H 2 −V ) with nonnegative potential V ∈ e −αr L ∞ (H 2 ), α ≥ 1. Since the dispersive estimate for t > 1 has been provided by [9], we only need to establish the similar estimate for 0 < t < 1, namely By Birman-Schwinger type resolvent expansion, the resolvent R V can be expressed as a series with respect to R 0 , R V and V , then the Schrödinger propagator in (10) can be written as a series. The estimates of the dominant terms can be derived from the pointwise bounds of the free resolvent kernel in [9] and Lemma 5.4 since these terms only depend on R 0 and V . For the remainder term, we use the meromorphic continuity of resolvent R V in Lemma 5.3. For the ψ − -equation, the negative regular potential can be regarded as a perturbation term of the nonlinearity, then we prove the global existence for small data by perturbation method (see [37]). The rest of the paper is organized as follows: In Section 2 we recall the basic properties of the hyperbolic spaces and introduce some functional spaces and basic inequalities. In Section 3 we use the approximation scheme and parallel transport to prove local existence and uniqueness for Schrödinger map flow (1) in H 3 , i.e Theorem 1.1. In the rest of the article, we consider the equivariant Schrödinger map flow with small initial data. Precisely, in Section 4, we rewrite the Schrödinger map flow in the Coulomb gauge as two coupled Schrödinger equations, i.e (ψ + , ψ − )system. Moreover, we show that the Schrödinger map flow u can be recovered from ψ + in L 2 (H 2 ) with ψ + L 2 (H 2 ) sufficiently small. In Section 5 we prove the Strichartz estimates and get the well-posedness of (ψ + , ψ − )-system for data ψ ± 0 ∈ L 2 . Finally, we give the proof of Theorem 1.2.

2.
Preliminaries. In this section we review the geometry of two dimensional hyperbolic space and the function spaces, then state some basic inequalities.
Then the metric g and volume form dvol g can be written as g = e −2y (dx 2 + e 2y dy 2 ), dvol g = e −y dxdy.

Function spaces and basic inequalities.
Here we define some relevant function spaces on H 2 and recall some basic inequalities. For a smooth function f : and for smooth function f (x, t) : H 2 × I → R, the space-time norms L p (I; L q (H 2 )) are defined by For simplicity, denote f L p I L q := f L p (I;L q (H 2 )) . We can also define the Sobolev norm H k (H 2 ; R) by where ∇ l is the l-th covariant derivative of f . In the local coordinates (x 1 , x 2 ) on H 2 , the components of ∇f are given by (∇f ) i = ∂ xi f , and those of ∇ 2 f are then As were shown in [23], for l = 0, 1, 2, · · · , For a smooth map u : ∈ H k for i = 1, 2, 3, then the Sobolev spaces of the map can also be defined extrinsically, whose H k (H 2 ; R 3 )-norm is defined as Moreover, the extrinsic Sobolev norm (15) and intrinsic Sobolev norm (2) are equivalent (see [31]) in the sense that there exist polynomials P and Q such that We now recall the Sobolev inequalities and diamagnetic inequality(see [31], [23]).
In Section 4 and 5, we will consider the equivariant Schrödinger flow (1), then we will work mainly with functions of a single variable r. Hence, for radial functions, the Lebesgue integral and spaces are with respect to the sinh rdr-measure, unless otherwise specified. In fact, for a smooth radial function f , by (13)  which is a natural space as it can be seen from the expression of E(u). The functions inḢ 1 e admit the following important properties: they are continuous and have limit 0 at r = 0. These properties imply the Sobolev embedding immediately For a radial function f and for an integer k we define By direct computation, we have We also have the following bounds Proof. (i) In the geodesic polar coordinates, we have Thus the bound (23) follows.
(ii) Since Then by (22) we have cosh r sinh r Hence, these imply that for k ≥ 2 f sinh 2 r We also obtain cosh r sinh r Thus the bound (24) follows.
Finally, the following estimates are frequently used for radial functions, which can be obtained by Schur's test easily.
Lemma 2.4. Let f ∈ L p be radial function, we have cosh r sinh 2 r r 0 sinh sf (s)ds 3. Local well-posedness for Schrödinger maps. In order to prove the local well-posedness in H 3 , we apply the approximation scheme and parallel transport introduced by McGahagan [32], see also [31,35]. For any δ > 0, we introduce the wave map model equation: where u δ (x, t) : H 2 × [0, T ] → S 2 and g δ 0 ∈ T u0(x) S 2 . In this section we use the global coordinates (11) and the global orthonormal frame (12). For simplicity, denote ∇ i := ∇ ei , for i = 1, 2, denote u = u δ and C(H k ) = C([0, T ]; H k ) in the proofs of Lemma 3.1 and Theorem 1.1. The covariant derivatives on u * T S 2 do not commute, and their lack of commutation is measured by the curvature tensor R S 2 on S 2 : where V (x) ∈ T u(x) S 2 . Before proving Theorem 1.1, we need the following lemma.
Lemma 3.1. ForT > 0, there exists a constant C > 0 independent of δ, and polynomials P and Q such that for any u δ : , a solution of the approximate equation, and any 1 ≤ k ≤ 2, the following estimate holds for u δ : for some 0 < T <T , depending only on the size of the solution u δ C([0,T ];H 3 ) and on the size of the initial data g δ 0 H 1 .
Proof. For k = 1, take the inner product of the above wave map equation with J(u)∇ t ∂ t u, then the first term will disappear by orthogonality and we get In the system of coordinates, τ (u) can be written as τ (u) = ∇ i e i (u) − (∇ i e i )(u), then commute ∇ i and ∇ t . Hence by integration by parts, the second term of (30) becomes Integrating (29) in time, we have Therefore, choosing T such that T ∂u C(0,T ;H 2 ) small, we have by Gronwall inequality that For k = 2, applying ∇ i to the approximate equation (28): then we take the inner product of the above equation with J(u)∇ i ∇ t ∂ t u and commute ∇ i and ∇ t , we have which, toghether with integration in time and Hölder inequality, gives Thus, Since II can be rewritten as

JIAXI HUANG, YOUDE WANG AND LIFENG ZHAO
Due to τ (u) and that (∇ j e j )u = e 2 u, II 2 becomes For II 3 , integration by parts gives Hence, From (32), (34), Hölder inequality and (17), we have This, together with (31), implies Hence, combining this with (33) and Gronwall inequality, we get This completes the proof of the lemma. Proof of Theorem 1.1. Step 1. Prove the existence of (1). We choose data g δ 0 such that g δ 0 H 1 < C and δ 2 g δ 0 2 H 2 < C. Without any restriction we make the bootstrap assumption Define the energy functional by then by (28), we have d dt E 1 = 0. Define the second order energy functional by By integration by parts and JX, X = 0, the second term of (36) becomes (37) and the last term of (37) becomes Hence, by (18), Hölder inequality, (36), (37) and (38), we have Define the third order energy functional by Then integration by parts gives By (17) and Hölder inequality, we have Hence, Since integration by parts yields gives Thus, integrating (41) in time and taking the supremum over t ∈ [0, T ] , we have Hence, by the bootstrap assumption (35), there exists T small such that Therefore, by (42) we have for some fixed T > 0 depending only on the size of data u(0). This concludes the local existence of (1) in H 3 .
Step 2. Prove the uniqueness of (1). The proof of the uniqueness is standard.
Here, we only show that the uniqueness holds in H 3 using the ideas of McGahagan [32] and Song-Wang [35].
are two solutions to the Schrödinger flow (1) with the same initial map u 0 ∈ H 3 .
By S 2 ⊂ R 3 and (1), we have for λ = 1, 2 This, together with (20), implies From this, there exists T > 0 such that |u (1) −u (2) | < δ 0 for any (t, x) ∈ [0, T ]×H 2 , and hence, there exists a unique minimizing geodesic x) vary, the family of geodesics give rise to a map U : x) (s). Therefore, we can define a global bundle morphism P : u (2) * T S 2 → u (1) * T S 2 by the parallel transportation along each geodesic. By the similar argument to [35, Section 3.3-3.6], we have where the constant C depends on the L ∞ -norm of ∇u (1) and ∇u (2) , and

JIAXI HUANG, YOUDE WANG AND LIFENG ZHAO
where the constant C depends on the Riemannian curvature of H 2 and S 2 and the L ∞ -norm of ∇u (1) and ∇u (2) . Using the interpolation inequality (17) and Sobolev embedding (18), we obtain (2) ) be the intrinsic distance. Since S 2 has bounded geometry, by (17) and the estimate in Lemma 2.2 of [35], we have T ] by Gronwall's inequality. By repeating the above argument, we can prove u (1) = u (2) on the whole interval [0, T ] and finish the proof of the uniqueness. 4. The Coulomb gauge representation of the equation. In this section, we rewrite the equivariant Schrödinger map flow (1) in the Coulomb gauge and obtain the (ψ + , ψ − )-system (8) of coupled Schrödinger equations. Conversely, we can recover the map u from ψ + or ψ − at fixed time.
Since u is 1-equivariant it is natural to work with 1-equivariant frame, that is v(r, θ, t) = e θRv (r, t), w(r, θ, t) = e θRw (r, t), wherev,w ∈ u * T S 2 are unit vectors depending on r and t. Let the differentiation operators ∂ 0 , ∂ 1 , ∂ 2 stand for ∂ t , ∂ r , ∂ θ , respectively. On one hand in such a frame we obtain the differentiated fields ψ k and the connection coefficients A k , by where ψ k , A k , k = 0, 1, 2, are radial functions. On the other hand, suppose ψ k and A k are given, the frame (u, v, w) can be recovered via the system: If we introduce the covariant differentiation then it is easy to check the compatibility conditions Moreover, the curvature of this connection is given by It is important to note that ψ 2 , A 2 are closely related to the original map. Precisely, the definitions of ψ k and A k (46) imply A 2 = u 3 and ψ 2 = w 3 − iv 3 . Hence we obtain |ψ 2 | 2 = u 2 1 + u 2 2 , and the important conservation law We now turn to choose the orthonormal frame (v,w) on S 2 . For the equivariant Schrödinger map flow u, the Coulomb gauge divA = 0 can be reformulated in the polar coordinate, The ODE (52) need to be initialized at some point. To avoid introducing a constant time-dependent potential into the equation via A 0 , we need to choose this initialization uniformly with respect to t. Since we restrict the solution u ∈ E λ , by rotation we may restrict the data lim , then we can fix the choice ofv andw at infinity, The existence and uniqueness of (52) with boundary condition (53) and u H 1 1 is standard. Indeed, using the Picard iteration schemē it's easily obtained from (54), integration by parts, Hölder and (21) v L 2 ) u H 1 . Therefore, by the above Picard iteration scheme procedure, there exists a unique solution of (52) satisfying (53). Moreover, we have v C(0,∞) + ∂ rv L 2 (0,∞) |v(∞)|.
4.2. The Schrödinger maps system in the Coulomb gauge: Dynamic equations for ψ ± . In the subsection we derive the Schrödinger equations for the differentiated fields ψ ± .
In the geodesic polar coordinates, by (45), (46) and (48), the Schrödinger map flow (1) can be written as Applying the operators D 1 and D 2 to both sides of this equation, we obtain By (49), (50) and A 1 = 0, one can derive the equations for ψ 1 and ψ 2 , which can be further written as where A 0 and A 2 − 1 can be expressed in terms of ψ 1 and ψ 2 . In fact, (49) and (50) for k = 1, l = 2 imply Since A 2 (0, t) = 1 for all t, (58) gives By (50) when k = 0, l = 1 and (55), we have which, together with (53), yields Therefore the connection coefficients A 0 and A 2 depend on ψ 1 and ψ 2 .
Since the linear part of this system is not decoupled, we introduce the two new variables ψ + and ψ − , defined as By (51) and (61), the system (57) can be reduced to Thus the linear part of ψ ± -system is decoupled. By (61), the compatibility condition (49) is rewritten as and the coefficients A 0 (60) and A 2 − 1 (59) can be expressed in terms of ψ ± , If define V ± as the vector then ψ ± is the representation of V ± in the coordinate frame (v, w) and the energy of u can be reformulated as From (62), ψ ± 2 L 2 is conserved for all time, then by the above equality and lim r→0 u 3 (r) ≡ 1 imply lim r→∞ u 3 (r) is fixed.
In order to prove ψ + sinh r ∈ L 2 , we rewrite .

Thus we have
To prove the existence, we consider the ψ 2 -equation in Multiply by e −A2(∞) r ∞ sinh −1 sds on both sides and integrating from infinity we obtain Define the map T :Ḣ 1 e (0, ∞) →Ḣ 1 e (0, ∞) by Now it suffices to show that T is a contraction map in X. Indeed, A 2 (∞) > 1 2 , (26) and (21) lead to And the map T is Lipschitz with a small Lipschitz constant, Therefore there exists a unique solution ψ 2 ∈Ḣ 1 e and the A 2 is obtained by A 2 (r) = 1 − |ψ 2 | 2 .
By the above expression of X, (26) and (84), we have Therefore, we have Using the method of continuity, we get Furthermore, by (89) we have ∂ r X L 2 δψ + L 2 . Hence, the Lipschitz continuity (86) is obtained.

Strichartz estimates.
To understand the well-posedness of (95), we need to obtain the Strichartz estimates. The R + ψ + -equation in (95) is a nonlinear Schrödinger equation with nonnegative and exponential decay potential. More generally, we consider the Schrödinger equation where V ∈ e −αr L ∞ (H 2 ; R) for α ≥ 1 is a nonnegative potential. In this section we always denote potential V as (96). For simplicity, we denote p = p p−1 for p ∈ [1, ∞].
Then we obtain the following Strichartz estimates.
Based on a standard theory, the above results are obtained by the following dispersive estimates immediately.
Proposition 4 (Dispersive estimates). Assume V ∈ e −αr L ∞ (H 2 ; R), α ≥ 1, is a nonnegative potential, then By standard convention the resolvent of Laplacian −∆ H 2 on H 2 is written as where Q 0 s−1 is Legendre function, r := d(z, w). The continuous part of the spectral resolution is given by (see [9])

JIAXI HUANG, YOUDE WANG AND LIFENG ZHAO
Then one can use the spectral resolution to write (see [2]) Similarly, from [9], the resolvent of −∆ H 2 + V for potential V defined as above is given by R V (s) = (−∆ H 2 + V − s(1 − s)) −1 and the continuous component of spectral resolution is given by then the kernel of Schrödinger propagator can be written as By Birman-Schwinger type resolvent expansion for all frequencies: we get Before proving Proposition 4, we recall the pointwise bounds on the resolvent kernel from [9]. These bounds will be crucial for the dispersive estimates.

Lemma 5.3 (Meromorphic continuation).
For V ∈ e −αr L ∞ (H 2 ) with α > 0, the resolvent R V (s) admits a meromorphic continuation to the half-plane s > 1 2 − δ as a bounded operator And there exists a constant M V such that for all λ ∈ R with |λ| ≥ M V , If the R V ( 1 2 + iλ) has no pole at λ = 0, one can extend the estimate through λ = 0 to give In order to prove Proposition 4, we also need the following lemma. (102) Proof. The proof roughly follows the approach in [2]. Before proving the lemma, we recall two useful estimates, that is, Denote Φ(τ ) := τ 2 t 4(r+a) 2 + τ 2 and α(τ ) := [ r+a t sinh r (cosh( τ t r+a + r) − cosh r)] − 1 2 , (105) can be written as Now it suffices to prove the boundedness of I 1 and I 2 . For I 1 , by (104) and r ≥

JIAXI HUANG, YOUDE WANG AND LIFENG ZHAO
For I 2 , integration by parts and α(1) 1, α (τ ) < 0 give Therefore (102) follows from (106) in the region r ≥ Let us split the left hand side of (102) into three parts: For J 1 , we assume r > 0, otherwise J 1 = 0 immediately, then (104) and r < t 2 give For J 2 , by (103) we have For J 3 , let s = √ tτ , we get Denote ψ(τ ) := τ 2 4 + aτ Therefore, (102) follows in the region r < √ t 2 . Proof of Proposition 4. The estimate for |t| ≥ 1 in (97) has been proved in [9], we only prove the case 0 < |t| < 1 here. In order to estimate e it(∆ H 2 −V ) f for f ∈ L 1 , it suffices to bound (99)-(101) respectively. (99) is indeed e it∆ H 2 f , which can be estimated in [1]. To estimate (100), we rewrite it by (98) as It suffices to estimate the integrals in the right hand side. For r 0 > 0, we have from which implies the following estimate immediately In conclusion, we obtained Therefore, Finally, we estimate the (101). By duality, it suffices to prove for any h L 1 = 1, Denote Then by integration by parts and Lemma 5.4, we have Similarly, we have e Therefore, Thus Proposition 4 follows.

The Cauchy theory.
Here we consider the Cauchy problem for (95). The local well-posedness of (95) is implied by Strichartz estimates in Theorem 5.1. Then for small initial data, since the operator −∆ H 2 − 2 cosh r−1 sinh 2 r has discrete spectrum (see Proposition 3), we use perturbation method (see [37]) to prove global wellposedness.
For simplicity, we denote the potential in (95) as V = 2 cosh r − 1 sinh 2 r .

JIAXI HUANG, YOUDE WANG AND LIFENG ZHAO
In a similar argument, we also have Since V ∈ L 2 (H 2 ) is independent of t, the V ψ − in ψ − -equation can be regarded as a nonlinear term. Based on a standard argument, by Strichartz estimate in Theorem 5.1, (112) and (113), the system has a unique solution with ψ ± (t 0 ) = ψ ± 0 on some maximal life-time open interval I with t 0 ∈ I. Hence (i) follows. (ii) is standard in light of (113).
Since the system (95) admits energy conservation ψ ± L 2 = ψ ± 0 L 2 and T depends only on ψ ± 0 L 2 and V = 2 cosh r−1 sinh 2 r , we obtain from (i) that (ψ + , ψ − ) is a global solution with ψ ± L 4 J L 4 ≤ C|J| 0 , and ψ ± C(J;L 2 ) ≤ C 0 , for any compact interval J ⊂ R, where C depends on ψ ± 0 L 2 and V L 2 . Finally, we prove the additional regularity (iv). Applying (−∆) s 2 for s = 1, 2 to both sides of the system (95), we obtain The nonlinearities F ± can be written as Let ϕ(r) ∈ C ∞ c be a bump function with 0 ≤ ϕ ≤ 1, ϕ B1(0) = 1 and ϕ B c 2 (0) = 0, thenF ± 2 can be rewritten as Since ψ ± L 4 I L 4 ≤ M , Strichartz estimates imply ψ ± L 3 I L 6 1. Then we split the interval into I = I j such that ψ ± To obtain (111), it suffices to estimate the second term of the right hand side of (120). Define For s = 1, from (19) we easily get Since the operator 1 r 2 r 0 ·sds keeps the two dimensional frequency localization, one could use Littlewood-Paley decomposition to deal with I ± . To estimate I ± , we claim that for f radial, the following estimate holds Then we have Hence, (120), (121) and (123) imply We repeat the above procedure for I j+1 to obtain the similar estimate in I j+1 . Thus, (111) valid for s = 1.
The above theorem is only concerned with the general solutions of (95). Since the system of (ψ + , ψ − ) is derived from the Schrödinger map flow (1), if we want to reconstruct the map u by ψ ± , the solution ψ ± of (95) must satisfy the compatibility condition (63).
Proof of Theorem 1.2. Given initial data u 0 ∈ H 3 with u 0 H 1 < 0 , by Theorem 1.1 there exists a unique solution on [0, T ] for some T > 0. On this interval we use Section 4.1 to construct fields ψ ± obeying the system (95) and whose mass satisfies  Thus the bound (7) follows.
Next, we continue to prove the global existence of the equivariant Schrödinger flow (1) with small initial data u 0 ∈ H 1 . For such initial data, there exist u 0,n ∈ H 3 such that u 0 −u 0,n H 1 < 1 n . Then by the above proof, there exists a unique solution u n ∈ L ∞ (J; H 3 ) with u n (0) = u 0,n satisfying u n L ∞ (J;H 3 ) ≤ C(|J|, u 0,n H 3 ), and u n L ∞ (J;H 1 ) ≤ C u 0,n H 1 .
By (67), Theorem 5.5(ii), and Proposition 2, we obtain the Lipschitz continuity of u(t) with respect to u 0 in H 1 , i.e u − u n H 1 u 0 − u 0,n H 1 .
Thus we get the solution u ∈ L ∞ (J; H 1 ) defined as the unique limit of solutions u n satisfying u L ∞ (J;H 1 ) ≤ C u 0 H 1 .
Hence, the bound (6) follows. This completes the proof of Theorem 1.2.