Equivariant Schr\"odinger maps from two dimensional hyperbolic space

TL;DR

This paper studies the existence and global behavior of equivariant Schr"odinger maps from two-dimensional hyperbolic space to the sphere, establishing local existence for energies below 4π and global solutions for sufficiently small energies.

Contribution

It proves local existence for Schr"odinger maps with energy less than 4π and global existence for small initial data in hyperbolic space, extending previous results to this setting.

Findings

Local existence for energy < 4π

Global solutions for sufficiently small energy

Convergence to the north pole at origin and infinity

Abstract

In this article, we consider the equivariant Schr\"odinger map from $\Bbb H^2$ to $\Bbb S^2$ which converges to the north pole of $\Bbb S^2$ at the origin and spatial infinity of the hyperbolic space. If the energy of the data is less than $4\pi$, we show that the local existence of Schr\"odinger map. Furthermore, if the energy of the data sufficiently small, we prove the solutions are global in time.

Equations

$$\frac{\partial u}{\partial t}=J\tau (u),$$

$$E(u)=\frac{1}{2}{\int }_{{\mathbb{H}}^2}|du{|}^2{\mathrm{dvol}}_g,$$

$$u(r,\theta )=e^{m\theta R}\bar{u}(r).$$

$$R:=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),Ru=\vec{k}\times u.$$

$$E(u)=\pi {\int }_0^{\infty }\left(|{\partial }_r\bar{u}{|}^2+\frac{m^2}{{\sinh }^{2}r}({\bar{u}}_1^2+{\bar{u}}_2^2)\right)\sinh rdr.$$

$$Q_{\lambda }=(\frac{2\lambda \tanh \frac{r}{2}}{1+(\lambda \tanh \frac{r}{2}{)}^2},0,\frac{1-(\lambda \tanh \frac{r}{2}{)}^2}{1+(\lambda \tanh \frac{r}{2}{)}^2}),$$

$$P_{\lambda }=(\frac{2\lambda \tanh \frac{r}{2}}{1-(\lambda \tanh \frac{r}{2}{)}^2},0,\frac{1+(\lambda \tanh \frac{r}{2}{)}^2}{1-(\lambda \tanh \frac{r}{2}{)}^2}),$$

$$\mathcal{E}_{\lambda}=\left\{u:{\mathbb{H}}^{2}\rightarrow{\mathbb{S}}^{2}\big{|}E(u)<\infty,\ \lim\limits_{r\rightarrow 0}u_{3}=1,\ \lim\limits_{r\rightarrow\infty}u_{3}=\frac{1-\lambda^{2}}{1+\lambda^{2}}\right\},\ \ \lambda\in[0,1),$$

$${\Vert u\Vert }_{{\mathfrak{H}}^k}^2:=\sum _{i=1}^k{\int }_{{\mathbb{H}}^2}|{\nabla }^{i-1}du{|}_g^2{\mathrm{dvol}}_g.$$

$${\Vert e^{it({\Delta }_{{\mathbb{H}}^2}-V)}\Vert }_{L^1\to L^{\infty }}\lesssim t^{-1},$$

$$[x,y]=-x^1{y}^1-\cdots -x^d{y}^d+x^{d+1}{y}^{d+1}.$$

$${\mathbb{H}}^d=\{x\in {\mathbb{R}}^{d+1}:[x,x]=1,x^{d+1}>0\},$$

$$X^t\cdot I_{d,1}\cdot X=I_{d,1},\text{mbox}\det X=1,\text{mbox}{X}_{d+1,d+1}>0,$$

$$L_h:{\mathbb{H}}^d⟶{\mathbb{H}}^d,\text{mbox}{L}_h(x)=h\cdot x.$$

$$f(k\cdot x)=f(x).$$

$$h=k\circ a_r\circ \tilde{k},a_r\in {\mathbb{A}}_{+},k,\tilde{k}\in \mathbb{K},$$

$$a_r:=\left(\begin{array}{ccc}{I}_{d-1\times d-1}& 0& 0\\ 0& \cosh r& \sinh r\\ 0& \sinh r& \cosh r\end{array}\right),{\mathbb{A}}_{+}:=\{a_r:r\in [0,\infty )\}.$$

$$\varphi :{\mathbb{R}}_{+}\times {\mathbb{S}}^{d-1}\to {\mathbb{H}}^d\subset {\mathbb{R}}^{d+1},\varphi (r,\omega )=(\sinh r\cdot \omega ,\cosh r).$$

$$\varphi :{\mathbb{R}}_{+}\times [0,2\pi )\to {\mathbb{H}}^d\subset {\mathbb{R}}^{d+1},\varphi (r,\theta )=(\sinh r\cos \theta ,\sinh r\sin \theta ,\cosh r),$$

$${\Delta }_{{\mathbb{H}}^2}={\partial }_r^2+\coth r{\partial }_r+\frac{{\partial }_{\theta }^2}{{\sinh }^{2}r}.$$

$$\phi :{\mathbb{R}}^{d-1}\times \mathbb{R}\to {\mathbb{H}}^d,\phi (v,s)=(\sinh s+e^{-s}\frac{|v{|}^2}{2},e^{-s}{v}_1,\cdots ,e^{-s}{v}_{d-1},\cosh s+e^{-s}\frac{|v{|}^2}{2}),$$

$$g=e^{-s}[d{v}_1^2+\cdots +d{v}_{d-1}^2+e^{2s}d{s}^2].$$

$$e_{\alpha }=e^s{\partial }_{v_{\alpha }}\mathrm{for}\alpha =1,\cdots ,d-1,\mathrm{and}{e}_d={\partial }_s,$$

$$[e_d,e_{\alpha }]=e_{\alpha },[e_{\alpha },e_{\beta }]=[e_d,e_d]=0\mathrm{for}\mathrm{any}\alpha ,\beta =1,\cdots ,d-1,$$

$${\nabla }_{e_{\alpha }}{e}_{\beta }={\delta }_{\alpha \beta }{e}_d,{\nabla }_{e_{\alpha }}{e}_d=-e_{\alpha },{\nabla }_{e_d}{e}_{\alpha }={\nabla }_{e_d}{e}_d=0,\mathrm{for}\alpha ,\beta =1,\cdots ,d-1.$$

$${\Vert f\Vert }_{L^p({\mathbb{H}}^2)}:=({\int }_{{\mathbb{H}}^2}|f(x){|}^p{\mathrm{dvol}}_g{)}^{1/p},{\Vert f\Vert }_{L^{\infty }({\mathbb{H}}^2)}:=\underset{x\in {\mathbb{H}}^2}{\sup }|f(x)|.$$

$${\Vert f\Vert }_{H^k}^2:=\sum _{l\le k}{\int }_{{\mathbb{H}}^2}|{\nabla }^{l}f{|}^2{\mathrm{dvol}}_g,$$

$${\Vert f\Vert }_{H^{2l}}\simeq {\Vert (-\Delta {)}^{l}f\Vert }_{L^2},\mathrm{for}l=0,1,2,\cdots ,$$

$${\Vert f\Vert }_{H^{2l+1}}\simeq {\Vert {\nabla }^{2l+1}f\Vert }_{L^2}\simeq {\Vert \nabla (-\Delta {)}^{l}f\Vert }_{L^2},\mathrm{for}l=0,1,2,\cdots .$$

$$u\in\{u:{\mathbb{H}}^{2}\rightarrow{\mathbb{S}}^{2}\subset{\mathbb{R}}^{3}\big{|}u_{i}-u_{i}(\infty)\in H^{k},\ \ {\rm for}\ i=1,\ 2,\ 3\}.$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Hyperbolic spaces

4. 2.2 Function spaces and basic inequalities

5. 2.3 Fourier transformation

6. 3 Local well-posedness for Schrödinger maps

7. 4 The Coulomb gauge representation of the equation

8. 4.1 The Schrödinger maps system in the Coulomb gauge: dynamic equations for $\psi_{k}$

9. 4.2 Recovering the map from $\psi^{+}$

10. 5 The Cauchy problem

11. 5.1 Srichartz estimates

12. 5.2 The Cauchy theory

1. Introduction

In this article, we consider the Schrödinger map equation

[TABLE]

where $u(x,t):[0,T]\times{\mathbb{H}}^{2}\rightarrow{\mathbb{S}}^{2}$, $\tau(u)$ is the tension field of $u$ and $J$ is complex structure on ${\mathbb{S}}^{2}$. The equation admits the conserved energy

[TABLE]

where ${\rm dvol}_{g}$ is the volume form of $({\mathbb{H}}^{2},\ g)$.

The Schrödinger maps from Euclidean spaces have been intensely studied in the last decades. The local well-posedness of Schrödinger maps was established by Sulem, Sulem and Bardos [29] for $S^{2}$ target, Ding and Wang [9, 10] and McGahagan [25] for general Kähler manifolds. Ionescu and Kenig [14] obtained the global well-posedness of maps into $S^{2}$ with small data in the critical Besov spaces $\dot{B}^{\frac{d}{2}}_{Q}(R^{d},S^{2})$, $Q\in S^{2}$ for $d\geq 3$. The global well-posedness for maps ${\mathbb{R}}^{d}\rightarrow S^{2}$, $d\geq 2$ with small critical Sobolev norms was obtained by Bejenaru, Ionescu, Kenig and Tataru [3]. However, the Schrödinger map equation with large data is a much more dufficult problem. When the target is $S^{2}$, there exists a collection of families $\mathcal{Q}^{m}$ (see [6]) of finite energy stationary solutions for integer $m\geq 1$; When the target is ${\mathbb{H}}^{2}$, there is not nontrival equivariant stationary solution with finite energy. Hence, Bejenaru, Ionescu, Kenig and Tataru [4, 5] proved the global well-posedness and scattering for equivariant Schrödinger maps ${\mathbb{R}}^{2}\rightarrow{\mathbb{S}}^{2}$ with energy blow the ground state and equivariant Schrödinger maps ${\mathbb{R}}^{2}\rightarrow{\mathbb{H}}^{2}$ 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 maps have been considered by many authors for instance [11, 12, 13, 6, 26, 27]. We refer to [16] for more open problems in this field.

The above results are restricted on flat domains, naturally, we can consider geomertic flow on curved manifolds. Because the hyperbolic spaces are symmetric and noncompact, geometric flows from hyperbolic spaces are natural starting points. The heat flow between hyperbolic spaces is an interesting model because it is related to the Schoen-Li-Wang conjecture (see Lemm, Markovic [21]). For such heat flow, Li and Tam [22] obtained the sufficient conditions to ensure that the harmonic map between hyperbolic spaces can be solved by solving the heat flow. In recent years, there are many works concerning wave maps on hyperbolic spaces which are expected to have many similar phenomenon to Schrödinger maps. D’Ancona and Qidi Zhang [8] showed the global existence of equivariant wave maps from hyperbolic spaces ${\mathbb{H}}^{d}$ for $d\geq 3$ to general targets for small initial data in $H^{\frac{d}{2}}\times H^{\frac{d}{2}-1}$. The problem was also intensely studied by Lawrie, Oh, Shahshahani [17, 18, 19, 20] and Li, Ma, Zhao [23]. Since the wave maps ${\mathbb{H}}^{2}\rightarrow{\mathbb{H}}^{2}$ or ${\mathbb{S}}^{2}$ have a family of equivariant harmonic maps, [17] and [18] proved the stability of stationary $k$-equivariant wave maps by analyzing spectral properties of the linearized operator. [19] continued to consider this problem and showed the soliton resolution for equivariant wave maps ${\mathbb{H}}^{2}\rightarrow{\mathbb{H}}^{2}$ with initial data $(\psi_{0},\ \psi_{1})\in\mathcal{E}_{\lambda}$ for $0\leq\lambda\leq\Lambda$ by profile decomposition. For initial data without any symmetric assumption, Li, Ma and Zhao [23] proved that the small energy harmonic maps from ${\mathbb{H}}^{2}$ to ${\mathbb{H}}^{2}$ are asymptotically stable under the wave map recently. [20] established global well-posedness and scattering for wave maps from ${\mathbb{H}}^{d}$ for $d\geq 4$ into Riemannian manifolds of bounded geometry for small initial data in the critical Sobolev space. As a geometric flow, Schrödinger map is a special case of Landau-Lifshitz flow. Li and Zhao [24] proved that the solution of Landau-Lifshitz $u(t,x)$ from ${\mathbb{H}}^{2}$ to ${\mathbb{H}}^{2}$ converges to some harmonic map as $t\rightarrow\infty$ when the Gilbert coefficient is positive.

The Schrödinger maps on ${\mathbb{H}}^{2}$ exhibits markedly different phenomena from its Euclidean counterpart. First, the most interesting feature is that there is an abundance of equivariant harmonic maps introduced by [17]. Precisely, when the target is ${\mathbb{S}}^{2}$, there is a family of equivariant harmonic maps with energy $4\pi\frac{\lambda^{2}}{1+\lambda^{2}}$ for $\lambda\in[0,+\infty)$; When the target is ${\mathbb{H}}^{2}$, we also have a family of equivariant harmonic maps with energy $4\pi\frac{1+\lambda^{2}}{1-\lambda^{2}}$ for $\lambda\in[0,1)$. Naturally, the dynamic behaviors of solutions with energy above the harmonic maps are of great interest. Second, the maps still exhibit features of mass critical equation, though it lacks scaling symmetry. Indeed, in the Coulomb gauge, the Schrödinger map can be reduced to two coupled Schrödinger equations. If the support of initial data is contained in a open ball $B_{\epsilon}(\textbf{0})$ for $\epsilon>0$ small, then the solutions will not exhibit the global geometry of the domain and thus can be approximated by solutions to the corresponding scaling invariant mass critical Schrödinger equations ${\mathbb{R}}^{2}\rightarrow{\mathbb{S}}^{2}$. Third, the notable feature of the problem is the better dispersive estimates of the operator $e^{it\Delta_{{\mathbb{H}}^{2}}}$ than the Euclidean counterpart. The stronger dispersion are possible due to the more robust geometry at infinity of noncompact symmetric spaces compared to Euclidean spaces. The above features make (1.1) an interesting model for investigating the well-posedness for large data and the stability of stationary solutions.

In this paper, we establish the local well-posedness for large data and global well-posedness for small initial data.

To explain the main results in more detail, we give a more precise account. As both the domain and the target are rotationally symmetric, the map $u$ is called $m$-equivariant, if $u$ satisfies $u\circ\rho=\rho\circ u$ for all rotations $\rho\in SO(2)$. Since $u$ is a map ${\mathbb{H}}^{2}\rightarrow{\mathbb{S}}^{2}$ here, in the polar coordinates, $u$ is $m$-equivariant if and only if $u$ can be written as

[TABLE]

Here $R$ is the generator of horizontal rotations, which is defined as

[TABLE]

where $\vec{k}=(0,\ 0,\ 1)^{T}$. We denote $\vec{i}:=(1,\ 0,\ 0)^{T}$ and $\vec{j}:=(0,\ 1,\ 0)^{T}$. The energy of $m$-equivariant maps can be expressed as

[TABLE]

If $m\neq 0$, then $E(u)<\infty$ implies that $\lim\limits_{r\rightarrow 0}u_{1}=\lim\limits_{r\rightarrow 0}u_{2}=0$. Due to the exponential decay of $\sinh^{-1}r$, we assume that $\lim\limits_{r\rightarrow\infty}u_{1},\ \lim\limits_{r\rightarrow\infty}u_{2}\geq 0$, which gives $\lim\limits_{r\rightarrow\infty}u_{3}=\frac{1-\lambda^{2}}{1+\lambda^{2}}\geq 1$ for $\lambda\in[0,1)$ by $u_{1}^{2}+u_{2}^{2}+u_{3}^{2}=1$. The equivariant Schrödinger map (1.1) admits solitons, which are equivariant harmonic maps $u$ such that $u\times\Delta u=0$. In contrast to the Schödinger maps from Euclidean spaces, the Schrödinger maps on ${\mathbb{H}}^{2}$ admit harmonic maps with any energy $E(u)<4\pi$ for ${\mathbb{S}}^{2}$ target and $E(u)<\infty$ for ${\mathbb{H}}^{2}$ target. In fact, for $u:{\mathbb{H}}^{2}\rightarrow{\mathbb{S}}^{2}$ with endpoint $u_{3}(\infty)=\frac{1-\lambda^{2}}{1+\lambda^{2}}$ for $\lambda\in[0,\infty)$, there exists equivariant stationary solution to (1.1)

[TABLE]

with energy $E(Q_{\lambda})=4\pi\frac{\lambda^{2}}{1+\lambda^{2}}$. For $u:{\mathbb{H}}^{2}\rightarrow{\mathbb{H}}^{2}$ with endpoint $u_{3}(\infty)=\frac{1+\lambda^{2}}{1-\lambda^{2}}$ for $\lambda\in[0,1)$, there exists equivariant stationary solution to (1.1)

[TABLE]

with energy $E(P_{\lambda})=4\pi\frac{\lambda^{2}}{1-\lambda^{2}}$.

This leads us to consider the equivariant Schrödinger maps in the classes

[TABLE]

but the case $\lambda>0$ is difficult, we will not consider here. Let $u:{\mathbb{H}}^{2}\rightarrow{\mathbb{S}}^{2}\subset{\mathbb{R}}^{3}$ be a smooth map. The Sobolev norm $\mathfrak{H}^{k}({\mathbb{H}}^{2};{\mathbb{S}}^{2})$ are defined by

[TABLE]

The main results are the following.

Theorem 1.1**.**

If $u_{0}\in\mathfrak{H}^{3}$, then there exists $T>0$, such that (1.1) has a unique solution in $L^{\infty}_{t}([0,T];\mathfrak{H}^{3})$.

Theorem 1.2**.**

If $u_{0}\in\mathfrak{H}^{1}$ is a 1-equivariant map satisfying $u_{0}\in\mathcal{E}_{0}$ and $E(u_{0})<4\pi$, then there exists $T>0$, such that (1.1) has a unique solution $u\in L^{\infty}_{t}([0,T];\mathfrak{H}^{1})$ in the class $\mathcal{E}_{0}$ defined as the unique limit of smooth solution in $\mathfrak{H}^{3}$. In particular, there exists $\epsilon>0$ such that $E(u_{0})<\epsilon$, then for any compact interval $J\subset{\mathbb{R}}$, there exists a unique solution $u\in L^{\infty}_{t}(J;\mathfrak{H}^{1})$.

Remark 1.2. In Theorem 1.2, we restrict the map $u$ in the class $\mathcal{E}_{0}$ for initial data $u_{0}\in\mathfrak{H}^{1}$. To obtain the existence of solutions in $\mathfrak{H}^{1}$, we need to prove the Lipschitz continuity of $u$ with respect to $u_{0}$. If the map $u\in\mathcal{E}_{\lambda}$ for $\lambda>0$, then the third component $u_{3}$ of $u$ does not convergence to $1$ as $r\rightarrow\infty$, thus the argument of Lipschitz continuity fails. Therefore, we need to restrict $u$ in $\mathcal{E}_{0}$.

Remark 1.3. In Theorem 1.2, scattering for small data is not expected generally. Represented in the Coulomb gauge, (1.1) can be reduced to the coupled mass-critical Schrödinger equations with potentials, i.e $(\psi^{+},\ \psi^{-})$-system. However, the Schrödinger operator admits discrete spectrum in one equation of the system which is in sharp contrast with the Schrödinger map from ${\mathbb{R}}^{2}$. In fact, we show that the $L^{4}L^{4}$-bound for $\psi^{\pm}$ depends on the compact interval $J\subset{\mathbb{R}}$, which leads to the $L^{\infty}_{t}(J;\mathfrak{H}^{1})$-bound for $u$ depends on interval $J$.

Theorem 1.1 and 1.2 is of similar flavor to the result of [25, 4] in the flat domain ${\mathbb{R}}^{2}$. The first step is to prove the local existence for Schrödinger map with data $u_{0}\in\mathfrak{H}^{3}$ by approximation of wave maps (see [25]). The second step is to show the existence for equivariant Schrödinger map with data $u_{0}\in\mathfrak{H}^{1}$. Since we restrict ourselves to the class of equivariant Schrödinger maps, the symmetry allow us to use Coulomb gauge. The Coulomb gauge condition impose some restriction on the connection form $A$ , which allow us to choose the particular solution $A_{1}=0$. Using the Coulomb gauge as our choice of frame on $T_{u}{\mathbb{H}}^{2}$, we can rewrite the equations for $\partial_{r}u$ and $\partial_{\theta}u$ which lead to a $(\psi^{+},\ \psi^{-})$-system of mass-critical Schrödinger equations with potentials. Then it suffices to consider the Cauchy problem of the $(\psi^{+},\ \psi^{-})$-system. In order to establish the well-posedness for data in the space $L^{2}({\mathbb{H}}^{2})$, we prove the Strichartz estimates for Schrödinger operator with such potentials. In fact, we can get the dispersive estimates for $0<t<1$ with more general potentials $V\geq 0$ and $V\in e^{-\alpha r}L^{\infty}({\mathbb{H}}^{2})$ for $\alpha\geq 1$. Since our interest lies in the solutions which correspond to the geometric flow, we show that the solutions of the system satisfy the compatibility condition. To construct the Schrödinger map $u$ from $\psi^{\pm}$, the key observation is that $\psi^{+}$ or $\psi^{-}$ contain all the information of the map as in [4]. Hence, we can recover the map $u(t)$ from $\psi^{\pm}(t)$ for initial data $R_{\pm}\psi^{\pm}(0)\in H^{2}$. Furthermore by the result in Theorem 1.1, we show that the map $u(t)$ is a Schrödinger map for data $u_{0}$ in $\mathfrak{H}^{3}$. At the same time, we obtain the Lipschitz continuity of $u(t)$ with respect to $u_{0}$ in $\mathfrak{H}^{1}$, which gives 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 uniformly estimates follows from a bootstrap argument. The other obstacle lies in the establishment of the well-posedness for the coupled Schrödinger system with potentials. Indeed, the system is composed of two coupled mass-critical Schrödinger equations with potentials. One of the equations admits Schrödinger operator with positive potential, which has only purely absolutely continuous spectrum $[\frac{1}{4},\ \infty)$. The dispersive estimate for $t>1$ has been provided by [7]. So we only need to establish the similar estimate for $0<t<1$, namely

[TABLE]

for nonnegative potential $V\in e^{-\alpha r}L^{\infty}({\mathbb{H}}^{2})$, $\alpha\geq 1$. We make use of the kernel of resolvent introduced by [2] frequently. 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 (1.2) can be written as a series. Since the dominant terms only depend on $R_{0}$ and $V$, we will use the pointwise bounds for free resolvent kernel and the Lemma 5.6. For the remainder term, we use the meromorphic continuity of resolvent $R_{V}$ in Lemma 5.5. The other equation admits Schrödinger operator with negative potential which has at least a discrete spectrum [math] even though it is extremely difficult to describe. Since we are dealing with the small data problem, the potential can be regarded as a perturbation term of the nonlinearity here.

The rest of the paper is organized as follows: In Section 2 we recall the hyperbolic spaces, function spaces, basic inequalities and the Fourier transformation. In Section 3 we use the approximating scheme to prove local well-posedness for Schrödinger map (1.1) in $\mathfrak{H}^{3}$, i.e Theorem 1.1. In Sections 4 we introduce the Coulomb gauge, in which the Schrödinger map can be written as two coupled Schrödinger equations, i.e $(\psi^{+},\ \psi^{-})$-system. Conversely, if we have $\psi^{+}\in L^{2}$, we can reconstruct the Schrödinger map $u$. In Sections 5 we provide the Strichartz estimates for operator $-\Delta_{{\mathbb{H}}^{2}}+V$, then we get the well-posedness of $(\psi^{+},\ \psi^{-})$-system for data $\psi^{\pm}_{0}\in L^{2}$. Finally,we finish the proof of Theorem 1.2.

2. Preliminaries

In this section we review the geometry of hyperbolic space and the Fourier transformation.

2.1. Hyperbolic spaces

We consider the Minkowski space $\mathbb{R}^{d+1}$ for $d\geq 2$ with the Minkowski metric $(dx^{1})^{2}+\cdots+(dx^{d})^{2}-(dx^{d+1})^{2}$, and we can define the bilinear form on $\mathbb{R}^{d+1}\times\mathbb{R}^{d+1}$,

[TABLE]

Then hyperbolic space $\mathbb{H}^{d}$ is defined as

[TABLE]

and the Riemannian metric $g$ on $\mathbb{H}^{d}$ is induced by the Minkowski metric on $\mathbb{R}^{d+1}$. We take the point $\textbf{0}:=(0,\cdots,0,1)\in{\mathbb{R}}^{d+1}$ as the origin in ${\mathbb{H}}^{d}$.

We define $(\mathbb{G},\circ);=(SO(d,1),\circ)$ as the connected Lie group of $(d+1)\times(d+1)$ matrices that leave the bilinear form $[\cdot,\cdot]$ invariant. We have $X\in SO(d,1)$ if and only if

[TABLE]

where $I_{d,1}$ is the diagonal matrix ${\rm diag}[1,\cdots,1,-1]$. Let $\mathbb{K}=SO(d)$ denote the subgroup of $SO(d,1)$ that fix the origin 0. Indeed, $\mathbb{K}$ is a compact subgroup of rotations acting on the variables $(x^{1},\cdots,x^{d})$. We can thus identify $\mathbb{H}^{d}$ with the symmetric space $\mathbb{G}/\mathbb{K}$. For every $h\in\mathbb{G}$ we can define the map

[TABLE]

A function $f:{\mathbb{H}}^{d}\rightarrow{\mathbb{R}}$ is called $\mathbb{K}$-invariant or radial, if for all $k\in\mathbb{K}$ and for all $x\in{\mathbb{H}}^{d}$ we have

[TABLE]

Then we have the Cartan decomposition of $h\in\mathbb{G}$, namely

[TABLE]

where

[TABLE]

We introduce two convenient global systems of coordinates on ${\mathbb{H}}^{d}$. One of the systems is geodesic polar coordinates:

[TABLE]

For $d=2$, $\phi$ can be written explicitly as

[TABLE]

in these coordinates, the hyperbolic metric $g$ is given by $g=dr^{2}+\sinh^{2}rd\theta^{2}$, the volume element $\mu(dx)$ on ${\mathbb{H}}^{2}$ is given by $\sinh rdrd\theta$ and the Laplace-Beltrami operator is given by

[TABLE]

The other global system of coordinates is defined as follows [15]:

[TABLE]

using these coordinates we have the induced metric

[TABLE]

If we fix the global orthonormal frame

[TABLE]

we compute the commutators

[TABLE]

and the covariant derivatives

[TABLE]

2.2. Function spaces and basic inequalities

Here we define some relevant function spaces on ${\mathbb{H}}^{2}$ and recall some basic inequalities. For smooth function $f:{\mathbb{H}}^{2}\rightarrow{\mathbb{R}}$, the $L^{p}({\mathbb{H}}^{2})$-norm for $1\leq p\leq\infty$ are defined by

[TABLE]

Also we can define the Sobolev norm $H^{k}({\mathbb{H}}^{2};{\mathbb{R}})$ of $f$, namely

[TABLE]

where $\nabla^{l}f$ is the $l$-th covariant derivative of $f$. By [20], we have

[TABLE]

and

[TABLE]

We will often use these equivalent definitions.

As a ${\mathbb{R}}^{3}$-valued function, we can define the extrinsic Sobolev spaces $\mathcal{H}^{k}({\mathbb{H}}^{2};{\mathbb{R}}^{3})$. We say that $u$ has finite $\mathcal{H}^{k}$-norm with respect to $u(\infty):=\lim\limits_{r\rightarrow\infty}u(r)$ if

[TABLE]

Denote

[TABLE]

In the polar coordinate (2.1), the equivariant maps are easily reduced to maps of a single variable $r$. For smooth radial function $f$, we define a natural space $\dot{H}^{1}_{e}$ by

[TABLE]

then for such $f$, we have Sobolev embedding

[TABLE]

We now recall the Sobolev inequalities (see [24], [20]).

Lemma 2.1**.**

Let $f\in C_{c}^{\infty}$, then for $1<p<\infty$, $p\leq q\leq\infty$, $0<\theta<1$, $1<r<2$, $r\leq l<\infty$, the following inequalities hold:

[TABLE]

We also recall the diamagnetic inequality (see [24], [20]).

Lemma 2.2**.**

If $T$ is some $(r,\ s)$-type tension or tension matrix defined on ${\mathbb{H}}^{2}$, then in the distribution sense, one has

[TABLE]

Lemma 2.3**.**

Let $u:{\mathbb{H}}^{2}\rightarrow{\mathbb{S}}^{2}$ be smooth map with $u(\infty)=\lim\limits_{r\rightarrow\infty}u(r)$, then

[TABLE]

in the sense that there exist polynomials $P$ and $Q$ such that

[TABLE]

Proof.

In order to prove (2.13), we use the polar coordinates (2.1). For $k=1$, we have

[TABLE]

hence, $\left\lVert u\right\rVert_{\mathfrak{H}^{1}}\leq\left\lVert u\right\rVert_{\mathcal{H}^{1}}$. Conversely, by (2.6), we obtain

[TABLE]

For $k=2$, we have

[TABLE]

and

[TABLE]

Therefore, $\left\lVert u\right\rVert_{\mathfrak{H}^{2}}\lesssim\left\lVert u\right\rVert_{\mathcal{H}^{2}}$ immediately. Conversely, $\partial_{j}u\cdot u=0$ implies

[TABLE]

then, by (2.7) and (2.11), we have

[TABLE]

For $k=3$, we have

[TABLE]

and

[TABLE]

By (2.14), we have

[TABLE]

Then

[TABLE]

Conversely, by (2.7) and (2.11), we have

[TABLE]

Therefore, (2.13) are obtained. ∎

Finally, we state the following estimates, which are often used for radial functions and obtained by Schur’s test easily.

Lemma 2.4**.**

Let $f\in L^{p}$ be radial function, we have

[TABLE]

2.3. Fourier transformation

For $\omega\in\mathbb{S}^{d-1}$ and $\lambda$ a real number, the functions of the type

[TABLE]

are generalized eigenfunctions of the Laplacian-Beltrami operator. Indeed, we have

[TABLE]

Then we can define the Fourier transformation analogous to the Euclidean case. For $f\in C_{c}^{\infty}({\mathbb{H}}^{d})$,

[TABLE]

and one has the Fourier inversion formula for function on ${\mathbb{H}}^{d}$

[TABLE]

where $c(\lambda)$ is the Harish-Chandra coefficient,

[TABLE]

For the linear Schrodinger equation on ${\mathbb{H}}^{d}$,

[TABLE]

the solution can be written explicitly see [2] as

[TABLE]

where the kernel $K^{d}$ is, for $\rho>0$ and $d\geq 3$ odd

[TABLE]

and for $d\geq 2$ even,

[TABLE]

In particular, $d=2$,

[TABLE]

3. Local well-posedness for Schrödinger maps

In order to prove the local well-posedness in $\mathfrak{H}^{3}$, we apply the approximating Scheme introduced by McGahagan [25]. For any $\delta>0$, we introduce the wave map model equation:

[TABLE]

where $u(t,x):[0,T]\times{\mathbb{H}}^{2}\rightarrow{\mathbb{S}}^{2}$ and $g_{0}^{\delta}\in T_{u_{0}(x)}{\mathbb{S}}^{2}$. In this section we use the global coordinates (2.2), denote $\nabla_{i}=\nabla_{e_{i}}$ for $i=1,\ 2$. For simplicity, denote $u:=u^{\delta}$.

Before proving the Theorem 1.1, we need the following lemma.

Lemma 3.1**.**

For $\tilde{T}>0$, there exists a constant $C>0$ independent of $\delta$, such that for any $u^{\delta}:[0,\tilde{T}]\times{\mathbb{H}}^{2}\rightarrow{\mathbb{S}}^{2}$, $u^{\delta}\in C([0,T];\mathfrak{H}^{3})$, a solution of the approximate equation, and any $1\leq k\leq 2$, the following estimate holds for $u^{\delta}$:

[TABLE]

for some $T>0$, depending only on the size of the solution $\left\lVert u^{\delta}\right\rVert_{C([0,\tilde{T}];\mathfrak{H}^{k+1})}$ and on the size of the initial data $\left\lVert\partial_{t}u^{\delta}(0)\right\rVert_{\mathfrak{H}^{1}}$.

Proof.

For $k=1$, we take the inner product of the above wave map equation with $J(u)\nabla_{t}\partial_{t}u$, the first term will disappear by orthogonality, we get

[TABLE]

In the system of coordinate, $\tau(u)$ can be written as $\tau(u)=\nabla_{i}e_{i}(u)-(\nabla_{i}e_{i})(u)$, then commute $\nabla_{i}$ and $\nabla_{t}$, by integration by parts, the second term of (3.3) becomes

[TABLE]

If we integrate in time,by Hölder inequality we find that (3.2) becomes

[TABLE]

then

[TABLE]

Therefore, by Gronwall inequality, choose $T$ such that $\left\lVert du\right\rVert_{C(0,\tilde{T};H^{2})}T$ small, we have

[TABLE]

For $k=2$, we take $\nabla_{i}$ on the approximate equation (LABEL:approximate_equation):

[TABLE]

then we take the inner product of the above equation with $J(u)\nabla_{i}\nabla_{t}\partial_{t}u$ and commute $\nabla_{i}$ and $\nabla_{t}$, we have

[TABLE]

Denote

[TABLE]

Then ${\rm II}$ can be rewritten as

[TABLE]

by the representation of $\tau(u)$ and $(\nabla_{j}e_{j})u=e_{2}u$, ${\rm II}_{2}$ becomes

[TABLE]

for ${\rm II}_{3}$, by integration by parts, we have

[TABLE]

Hence,

[TABLE]

Then (3.5) can be written as

[TABLE]

Integrating in time, by Hölder inequality, it gives

[TABLE]

then

[TABLE]

From (3.6), Hölder inequality and Lemma 2.1, we have

[TABLE]

By (3.4),

[TABLE]

By Gronwall inequality, we get

[TABLE]

∎

Proof of Theorem 1.1.

We choose data $g_{0}^{\delta}$ such that $\left\lVert g_{0}^{\delta}\right\rVert_{H^{1}}<C$ and $\delta^{2}\left\lVert g_{0}^{\delta}\right\rVert_{H^{2}}^{2}<C$. Without any restriction we make the bootstrap assumption

[TABLE]

Define the energy functional by

[TABLE]

then by (LABEL:approximate_equation), we have $\frac{d}{dt}E_{1}=0$. Define the second order energy functional by

[TABLE]

by (LABEL:approximate_equation) we have

[TABLE]

by integration by parts and $\langle JX,X\rangle=0$, the second term of (3.8) becomes

[TABLE]

furthermore, the last term of (3.9) becomes

[TABLE]

Hence, by (2.9) and Hölder inequality we have

[TABLE]

Define the third order energy functional by

[TABLE]

By integration by parts gives

[TABLE]

By (2.7) and Hölder inequality, we have

[TABLE]

Similarly, we have

[TABLE]

Hence,

[TABLE]

Since we have by integration by parts

[TABLE]

then by (2.7) we obtain

[TABLE]

Thus, integrating (3.12) in time and taking the supremum over $t\in[0,T]$ , we have

[TABLE]

Choosing $T$ small such that $CTP(\left\lVert u\right\rVert_{C([0,\tilde{T}];\mathfrak{H}^{2})}+\left\lVert\nabla\tau(u)\right\rVert_{C([0,\tilde{T}];L^{2})})<\frac{1}{2},$ from (3.14) we have

[TABLE]

If $\left\lVert\partial_{t}u\right\rVert_{C(H^{2})}^{2}\geq 2\left\lVert g_{0}\right\rVert_{H^{2}}^{2}$, (3.15) implies

[TABLE]

If $\left\lVert\partial_{t}u\right\rVert_{C(H^{2})}^{2}<2\left\lVert g_{0}\right\rVert_{H^{2}}^{2}$, from (3.14) we obtain

[TABLE]

Hence, by the bootstrap assumption (3.7), there exists $T$ small such that

[TABLE]

Therefore, by (3.13) we have

[TABLE]

for some fixed $T>0$ depending only on the size of data $u(0)$.

∎

4. The Coulomb gauge representation of the equation

In this section, we rewrite the equivariant Schrödinger map in the Coulomb gauge, then obtain the $(\psi^{+},\ \psi^{-})$-system of coupled Schrödinger equations. Conversely, we can recover the map $u$ from $\psi^{+}$ or $\psi^{-}$ at fixed time.

We choose $v\in T_{u}{\mathbb{H}}^{2}$ such that $v\cdot v=1$ and define $w=u\times v$. Thus

[TABLE]

Since $u$ is 1-equivariant it is natural to work with 1-equivariant frame, that is

[TABLE]

where $\bar{v}$, $\bar{w}$ are unit symmetric vectors in ${\mathbb{H}}^{2}$. On one hand in such a frame we obtain the differentiated fields $\psi_{k}$ and the connection coefficients $A_{k}$, by

[TABLE]

On the other hand, given $\psi_{k}$ and $A_{k}$ we can return to the frame $(u,v,w)$ via the ODE system:

[TABLE]

If we introduce the covariant differentiation

[TABLE]

then the compatibility conditions are imposed

[TABLE]

Moreover, the curvature of this connection is given by

[TABLE]

An important geometric feature is that $\psi_{2}$, $A_{2}$ are closely related to the original map. Precisely, for $A_{2}$ we have

[TABLE]

and

[TABLE]

Hence we obtain $|\psi_{2}|^{2}=u_{1}^{2}+u_{2}^{2}$, and the following important conservation law

[TABLE]

We now turn to choose the orthonormal frame $(\bar{v},\bar{w})$ on ${\mathbb{S}}^{2}$. For the equivariant Schrödinger map, we use the Coulomb gauge ${\rm div}A=0$, namely, in the polar coordinate, $\partial_{r}A_{1}+\frac{\partial_{\theta}^{2}}{\sinh^{2}r}A_{2}=0$. Since $A_{2}=u_{3}$ is radial, we can choose $A_{1}=0$, i.e

[TABLE]

which can be represented as ODE

[TABLE]

Then for matrix $U=(\bar{u},\bar{v},\bar{w})$, we have

[TABLE]

where $M=-u\cdot\partial_{r}u^{T}+\partial_{r}u\cdot u^{T}$ is an antisymmetric matrix.

The ODE (4.7) 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 data $\lim\limits_{r\rightarrow\infty}\bar{u}(r,t)=\vec{k}$ for any t, we can fix the choice of $\bar{v}$ and $\bar{w}$ at infinity,

[TABLE]

The existence and uniqueness of (4.7) satisfying (4.8) is standard. Indeed, for $u\in\mathfrak{H}^{1}$, using the Picard iteration scheme

[TABLE]

By Hölder inequality, we have

[TABLE]

and

[TABLE]

we choose $R$ large enough such that $\left\lVert\bar{u}\right\rVert_{\dot{H}^{1}_{e}([R,\infty))}<\epsilon$, we have $\left\lVert\bar{v}_{i}(r)\right\rVert_{C([R,\infty))}\leq\epsilon^{i}\left\lVert\bar{v}_{0}\right\rVert_{C([R,\infty))}$. Hence, there exists unique solution $\left\lVert\bar{v}\right\rVert_{C(R,\infty)}+\left\lVert\partial_{r}\bar{v}\right\rVert_{L^{2}(R,\infty)}\lesssim\left\lVert u\right\rVert_{\mathfrak{H}^{1}}$. Then by $u\in\mathfrak{H}^{1}$, in a similar argument, for any $\epsilon>0$,there exists $\delta>0$ sufficiently small, such that $\left\lVert u\right\rVert_{\mathfrak{H}^{1}(R-\delta,R)}\left\lVert\sinh^{-1}r\right\rVert_{L^{2}(R-\delta,R)}\ll 1$ for $R-\delta>\epsilon$, the solution can be extended to $r=\epsilon$. Finally, we extend the solution to $r=0$. The first two components of $\bar{v}_{i}$ can be estimated immediately

[TABLE]

for the third component of $\bar{v}_{i}$, by integration by parts and $\bar{u}(r)\rightarrow\vec{k}$ as $r\rightarrow 0$, we have

[TABLE]

and

[TABLE]

we choose $\epsilon$ small such that $\left\lVert u\right\rVert_{\mathfrak{H}^{1}(0,\epsilon)}+\left\lVert\bar{u}-\vec{k}\right\rVert_{C(0,\epsilon)}\ll 1$, then the iteration scheme gives the unique solution in $(0,\epsilon)$ with $\left\lVert\bar{v}\right\rVert_{C(0,\epsilon)}+\left\lVert\partial_{r}\bar{v}\right\rVert_{L^{2}(0,\epsilon)}\lesssim(\left\lVert u\right\rVert_{\mathfrak{H}^{1}(0,\epsilon)}+\left\lVert\bar{u}-\vec{k}\right\rVert_{C(0,\epsilon)})\left\lVert\bar{v}_{0}\right\rVert_{C(0,\epsilon)}$. Therefore, by the above procedure, there exists a unique solution of (4.7) satisfying (4.8), moreover, we have

[TABLE]

4.1. The Schrödinger maps system in the Coulomb gauge: dynamic equations for $\psi_{k}$

We derive the Schrödinger equations for the differentiated fields $\psi_{1}$ and $\psi_{2}$.

In the geodesic polar coordinate, the Schrödinger map flow can be written as

[TABLE]

Applying the operators $D_{1}$ and $D_{2}$ to both sides of this equation, we obtain

[TABLE]

By the compatibility condition (4.5), curvature of the connection (4.6) and the Coulomb gauge $A_{1}=0$, we can derive the equations for $\psi_{1}$ and $\psi_{2}$,

[TABLE]

where $\Delta:=\partial_{rr}+\coth r\partial_{r}$. Then (4.12) can be written as

[TABLE]

where $A_{0}$ and $A_{2}-1$ can be expressed in terms of $\psi_{1}$ and $\psi_{2}$. In fact, from the curvature (4.6) for $k=1,\mbox{ }l=2$ and compatibility condition (4.5), we have

[TABLE]

Since $A_{2}(0)=1$, (4.14) gives

[TABLE]

From (4.6) when $k=0,\ l=1$ and (4.10), we have

[TABLE]

which together with initial data of $(\bar{v},\bar{w})$ frame, yields

[TABLE]

Therefore the two variables $\psi_{1}$ and $\psi_{2}$ are not independent.

Since the linear part of this system is not decoupled, we introduce the two new variables $\psi^{+}$ and $\psi^{-}$, defined as

[TABLE]

From (4.13) and $A_{2}^{2}+|\psi_{2}|^{2}=1$, we obtain

[TABLE]

It turns out that the linear part of $\psi^{\pm}$-system is decoupled. The compatibility condition (4.5) is reduced to

[TABLE]

and the coefficients $A_{0}$ and $A_{2}-1$ can be expressed in terms of $\psi^{\pm}$,

[TABLE]

Define $\mathcal{V}^{\pm}$ as the vector

[TABLE]

then $\psi^{\pm}$ is the representation of $\mathcal{V}^{\pm}$ in the coordinate frame $(v,w)$ and the energy of $u$ has a new representation, i.e

[TABLE]

Hence, $\left\lVert\psi^{\pm}\right\rVert_{L^{2}}^{2}$ is conserved for all time. Moreover, if we assume that $\left\lVert\psi^{\pm}\right\rVert_{L^{2}},\ \left\lVert\tilde{\psi}^{\pm}\right\rVert_{L^{2}}<2$ and $\left\lVert u-\tilde{u}\right\rVert_{\mathfrak{H}^{1}}\ll 1$, we obtain the Lipschitz continuity of $\psi^{\pm}$ with $u$, namely

[TABLE]

In fact, by the above assumptions, (4.20) implies $u_{3},\ \tilde{u}_{3}\gtrsim 1-\frac{1}{4}\left\lVert\psi^{-}\right\rVert_{L^{2}}>0$. On interval $[R,\infty)$, by (4.7), we have

[TABLE]

then (4.9) and $u_{3}-\tilde{u}_{3}=\frac{u_{1}^{2}-\tilde{u}_{1}^{2}+u_{2}^{2}-\tilde{u}_{2}^{2}}{u_{3}+\tilde{u}_{3}}$ imply

[TABLE]

choose $R$ large enough, we have $\left\lVert v-\tilde{v}\right\rVert_{C(R,\infty)}\lesssim\left\lVert u-\tilde{u}\right\rVert_{\mathfrak{H}^{1}(R,\infty)}$. Then for any $\epsilon>0$ small, on interval $[\epsilon,\ R]$, there exists $\delta>0$ such that any interval $I\subset[\epsilon,R]$ with $|I|<\delta$, we have $\left\lVert\partial_{r}u\right\rVert_{L^{2}(I)}\left\lVert\sinh^{-1}r\right\rVert_{L^{2}(I)}\ll 1$. By a similar argument to that on $[R,\infty)$, we obtain $\left\lVert v-\tilde{v}\right\rVert_{C(\epsilon,R)}\lesssim\left\lVert u-\tilde{u}\right\rVert_{\mathfrak{H}^{1}(\epsilon,R)}$. Finally, on interval $(0,\epsilon)$, by Sobolev embedding (2.5), we have

[TABLE]

then we get

[TABLE]

which implies by integration by parts

[TABLE]

which together with $\left\lVert u\right\rVert_{\mathfrak{H}^{1}(0,\epsilon)}\ll 1$, yields $\left\lVert v-\tilde{v}\right\rVert_{C(0,\epsilon)}\lesssim\left\lVert u-\tilde{u}\right\rVert_{\mathfrak{H}^{1}(0,\epsilon)}$. Therefore, we obtain

[TABLE]

Then by (4.24), (4.22) and Sobolev embedding (2.5), the Lipschitz continuity (4.23) follows.

In this paper we will work with the key system (4.18) to obtain the space-time estimates for $\psi^{\pm}$.

Suppose $\psi^{\pm}$ satisfies the compatibility condition (4.19) and $\left\lVert\psi^{\pm}\right\rVert_{L^{2}}<\infty$, define $A_{2}$, $\psi_{1}$, $\psi_{2}$ by (4.20) and (4.17), then they satisfy the relation (4.14). Furthermore, we claim that $\psi_{1}\in L^{2}$ and $\psi_{2},\ A_{2}-1\in\dot{H}^{1}_{e}$. In fact, by (4.20) and (4.17), we have $\psi_{1}\in L^{2}$, $A_{2}\in L^{\infty}$, from (4.17), (4.14) and (2.16), we get $\frac{\psi_{2}}{\sinh r},\ \partial_{r}\psi_{2},\ \partial_{r}A_{2}$, and $\frac{A_{2}-1}{\sinh r}\in L^{2}$.

Denote $R_{+}\psi^{+}=e^{i2\theta}\psi^{+}$ and $R_{-}\psi^{-}=\psi^{-}$. Then we have

Proposition 4.1**.**

[TABLE]

Proof.

If $u\in\mathfrak{H}^{1}$, we easily obtain $\left\lVert du\right\rVert_{L^{2}}^{2}=\pi\left\lVert\psi^{\pm}\right\rVert_{L^{2}}^{2}$. If $u\in\mathfrak{H}^{2}$, by the equivariance condition, we have

[TABLE]

Since $A_{1}=0$, then $\nabla_{r}(v+iw)=0$, which gives

[TABLE]

by the representation of $\mathcal{V}^{\pm}$ (4.22), we have

[TABLE]

denote $F^{\pm}=(\ref{F+-})\cdot(v+iw)$, then

[TABLE]

For $\psi^{-}$, since \big{|}\frac{u_{3}-1}{\sinh r}\big{|}\lesssim\frac{u_{1}^{2}+u_{2}^{2}}{\sinh r}\in L^{1}(dr), applying $e^{\int_{r}^{\infty}\frac{u_{3}-1}{\sinh s}ds}$ to both sides of (4.29), by $F^{\pm}\in L^{2}$, we have $\partial_{r}(e^{\int_{r}^{\infty}\frac{u_{3}-1}{\sinh s}ds}\psi^{-})\in L^{2}$. Since $u_{3}$ and $\psi^{-}$ are radial, we obtain $e^{\int_{r}^{\infty}\frac{u_{3}-1}{\sinh s}ds}\psi^{-}\in\dot{H}^{1}$, which gives $\psi^{-}\in L^{4}$ by (2.7). Hence, by (4.29) and \big{|}\frac{u_{3}-1}{\sinh r}\big{|}\lesssim\frac{|u_{1}|+|u_{2}|}{\sinh r}\in L^{4}, we have $\partial_{r}\psi^{-}\in L^{2}$. It also follows that $\partial_{r}\psi^{+}\in L^{2}$.

In order to prove $\frac{\psi^{+}}{\sinh r}\in L^{2}$, we rewrite

[TABLE]

by \big{|}\frac{1-u_{3}}{\sinh^{2}r}\big{|}\lesssim\frac{u_{1}^{2}+u_{2}^{2}}{\sinh^{2}r}\in L^{2} and (4.26), we have $\frac{\psi^{+}}{\sinh r}\in L^{2}$.

Conversely, if $R_{\pm}\psi^{\pm}\in H^{1}$, (2.7) implies $\psi^{\pm}\in L^{4}$. Then by \big{|}\frac{u_{3}-1}{\sinh r}\big{|}\lesssim\frac{|\psi_{2}|}{\sinh r}\in L^{4} and (4.29), we have $F^{\pm}\in L^{2}$, namely, $[\partial_{rr}\mp(\frac{\cosh r}{\sinh r}\partial_{r}+\frac{\partial_{\theta}^{2}}{\sinh^{2}r})]u\cdot(v+iw)\in L^{2}$. The part of $[\partial_{rr}\mp(\frac{\cosh r}{\sinh r}\partial_{r}+\frac{\partial_{\theta}^{2}}{\sinh^{2}r})]u$ in the normal space is -|\psi_{1}|^{2}\pm\big{|}\frac{\psi_{2}}{\sinh r}\big{|}\in L^{2} by $\psi^{\pm}\in L^{4}$. Therefore, (4.26) is obtained.

If $u\in\mathfrak{H}^{3}$, by (2.4) and Lemma 2.3, we obtain $\nabla(-\Delta)u_{i}\in L^{2}$ for $i=1,\ 2,\ 3$, then by equivariance condition, we get

[TABLE]

and

[TABLE]

In order to prove $R_{\pm}\psi^{\pm}\in H^{2}$, it suffices to prove

[TABLE]

By (4.29), we have

[TABLE]

Since $R_{\pm}\psi^{\pm}\in H^{1}$, (2.7) implies $\psi^{\pm}\in L^{4}\cap L^{6}$, then by $|u_{3}-1|\lesssim|\psi_{2}|^{2}$ and (4.15), the third term of (4.33) and (4.34) are in $L^{2}$. From (4.30), we also have $\partial_{r}F^{\pm}\in L^{2}$. Hence, $\partial_{r}(e^{\int_{\infty}^{r}\frac{u_{3}-1}{\sinh s}ds}\partial_{r}\psi^{-})\in L^{2}$, which further gives $e^{\int_{\infty}^{r}\frac{u_{3}-1}{\sinh s}ds}\partial_{r}\psi^{-}\in L^{4}$, this implies $\partial_{r}\psi^{-}\in L^{4}$. Since $\frac{u_{3}-1}{\sinh r}\in L^{4}$, we obtain $\partial_{rr}\psi^{-}\in L^{2}$, therefore, we also get $\partial_{rr}\psi^{+}\in L^{2}$.

Next, we estimate this term

[TABLE]

By (4.31), the first two components of $\frac{\cosh r}{\sinh r}F^{-}$ are in $L^{2}$. For the third component, which can be written as

[TABLE]

By (4.26) and (4.30), we have $\Delta u_{3}\in L^{4}$. Hence, the right hand side of (4.36) is in $L^{2}$. By a similar argument, the third term in (4.35) is also in $L^{2}$. Thus, $R_{-}\psi^{-}\in H^{2}$.

For $R_{+}\psi^{+}$, we need to estimate

[TABLE]

By (4.26) and (4.31), the first two components of (4.37) are in $L^{2}$, namely for $i=1,\ 2$

[TABLE]

For the third component, since $\partial_{r}\Delta u_{3}\in L^{2}$, it suffices to estimate

[TABLE]

By (4.26) and $|\psi_{2}|\leq 1$, we get $\frac{\cosh r-1}{\sinh r}\psi_{2}\partial_{r}^{2}u_{3}\in L^{2}$. By (2.5) and (2.9), we have $\left\lVert\psi^{+}\right\rVert_{L^{\infty}}\lesssim\left\lVert\psi^{+}\right\rVert_{\dot{H}^{1}_{e}}$ and $\left\lVert\psi^{-}\right\rVert_{L^{\infty}}\lesssim\left\lVert\psi^{-}\right\rVert_{H^{2}}$, then $\frac{1}{2}(\psi^{+}-\psi^{-})\partial_{r}^{2}u_{3}\in L^{2}$. Therefore, (4.37) are in $L^{2}$. The other terms are also easily obtained by Sobolev embedding and $A_{2}^{2}+|\psi_{2}|^{2}=1$. Thus, $R_{+}\psi^{+}\in H^{2}$ is obtained.

∎

4.2. Recovering the map from $\psi^{+}$

Here we will keep track of $\psi^{+}\in L^{2}$, since it contains all the information about the map. Indeed, by (4.14), we have the system of $(A_{2},\psi_{2})$

[TABLE]

Then from the choice of $(\bar{v}(\infty),\ \bar{w}(\infty))$ (4.8), it gives the data $(A_{2}(\infty),\ \psi_{2}(\infty))=(1,\ 0)$. Given $\psi^{+}\in L^{2}$ with $\left\lVert\psi^{+}\right\rVert_{L^{2}}<2$, we reconstruct $A_{2}-1,\ \psi_{2}\in\dot{H}^{1}_{e}$ by above system (4.38), then by the system in (4.4) with condition (4.8), we can return to the map $u$.

Lemma 4.2**.**

Let $\psi^{+}\in L^{2}$, such that $\left\lVert\psi^{+}\right\rVert_{L^{2}}<2$, the system (4.38) has a unique solution $(A_{2},\psi_{2})$ satisfying $\psi_{2},\mbox{ }A_{2}-1\in\dot{H}^{1}_{e}$, and

[TABLE]

*Moreover, we have the following properties:

(i)  If $\psi^{+}\in L^{p}$, with $1\leq p<\infty$, then $\psi^{-},\mbox{ }\frac{\psi_{2}}{\sinh r},\mbox{ }\frac{A_{2}-1}{\sinh r}\in L^{p}$ and*

[TABLE]

(ii)  Given $\epsilon>0$, and $R$ such that $\left\lVert\psi^{+}\right\rVert_{L^{2}({\mathbb{R}}\backslash[R^{-1},R])}\leq\epsilon$, then we have

[TABLE]

(iii)  If $(\tilde{A}_{2},\ \tilde{\psi}_{2})$ is another solution to (4.38) corresponding to $\tilde{\psi}^{+}$, then

[TABLE]

(iv)  If $R_{+}\psi^{+}\in H^{s}$, then $R_{-}\psi^{-}\in H^{s}$ for $s\in\{1,\ 2\}$.

Proof.

We consider the ODE system (4.38) with boundary condition

[TABLE]

The system and boundary condition imply $A_{2}^{2}+|\psi_{2}|^{2}=1$. We define $\psi^{-}=\psi^{+}-2i\frac{\psi_{2}}{\sinh r}$, $\psi_{1}=\psi^{+}-i\frac{\psi_{2}}{\sinh r}$, then we get $\partial_{r}A_{2}=\frac{1}{4}\sinh r(|\psi^{+}|^{2}-|\psi^{-}|^{2})$ from (4.38). Since $A_{2}(\infty)=1$ which yields by integration from infinity

[TABLE]

Thus we have $A_{2}>1-\frac{1}{4}\left\lVert\psi^{+}\right\rVert_{L^{2}}^{2}>0$.

To prove existence, by choosing $R$ large enough such that $\left\lVert\psi^{+}\right\rVert_{L^{2}(R,\infty)}\leq\epsilon$. We want to seek $\psi_{2}$ with the property that $\left\lVert\psi_{2}\right\rVert_{\dot{H}^{1}_{e}(R,\infty)}\lesssim\epsilon$. This implies that $|\psi_{2}|\lesssim\epsilon$, then we have $A_{2}-1\lesssim\epsilon^{2}$. By the relation $A_{2}^{2}+|\psi_{2}|^{2}=1$, we get $A_{2}=\sqrt{1-|\psi_{2}|^{2}}$. Now we only need to consider the $\psi_{2}$-equation in

[TABLE]

Rewrite the $\psi_{2}$ equation as

[TABLE]

then

[TABLE]

Multiply by $e^{-\int^{r}_{\infty}\sinh^{-1}sds}$ on both sides, we have

[TABLE]

Integrating from infinity we obtain

[TABLE]

Define the map $\mathcal{T}:\mbox{ }\dot{H}^{1}_{e}(R,\infty)\rightarrow\dot{H}^{1}_{e}(R,\infty)$ by

[TABLE]

Now it suffices to show that $\mathcal{T}$ is a contraction map in $X$. Indeed, the estimate (2.19) and Sobolev embedding lead to

[TABLE]

And the map $\mathcal{T}$ is Lipschitz with a small Lipschitz constant,

[TABLE]

Therefore there exists a unique solution $\psi_{2}\in\dot{H}^{1}_{e}(R,\infty)$.

Next we extend the solution to $r=0$. Consider the equation $\partial_{r}\psi_{2}=iA_{2}\psi^{+}+A_{2}\frac{\psi_{2}}{\sinh r}$ with data $\psi_{2}(R)$. By Duhamel formula, it suffices to consider the map

[TABLE]

and the space

[TABLE]

Since $\frac{1}{\sinh r}e^{2\int_{R}^{r}\sinh^{-1}sds}$ is bounded, there exists $a=a(\epsilon,\mbox{ }\left\lVert\psi^{+}\right\rVert_{L^{2}})>0$ such that

[TABLE]

By (2.19), we obtain that

[TABLE]

Meanwhile we have

[TABLE]

Therefore $\mathcal{J}$ is a contraction map in $Y$. Since the lifespan interval $a$ only depends on $\epsilon$ and $\left\lVert\psi^{+}\right\rVert_{L^{2}}$, we can extend the solution to $r=0$. Thus the existence of $\psi_{2}$ in $[0,\infty)$ follows, and the $A_{2}$ is obtained by $A_{2}(r)=\sqrt{1-|\psi_{2}|^{2}}$.

Next we obtain the bound for (4.39). Let $G=\frac{\psi_{2}}{1+A_{2}}$, then the system gives

[TABLE]

or equivalently

[TABLE]

which implies

[TABLE]

namely,

[TABLE]

therefore

[TABLE]

Since $|A_{2}|\leq 1$, we get

[TABLE]

By (2.19) and $A_{2}>0$, (4.43) gives $\left\lVert\frac{\psi_{2}}{\sinh r}\right\rVert_{L^{2}}\lesssim\left\lVert\psi^{+}\right\rVert_{L^{2}}$. The bounds for $\left\lVert\partial_{r}\psi_{2}\right\rVert_{L^{2}}$ and $\left\lVert\partial_{r}A_{2}\right\rVert_{L^{2}}$ follow directly from (4.38). The bounds for $\left\lVert\frac{A_{2}-1}{\sinh r}\right\rVert_{L^{2}}$ and $\left\lVert\frac{A_{2}-1}{\sinh r}\right\rVert_{L^{1}(dr)}$ are obtained by the compatibility relation $A_{2}^{2}+|\psi_{2}|^{2}=1$.

Now we prove the additional properties (i)-(iv). First, we have the bound for (4.40). If $\psi^{+}\in L^{p}$, by (2.19) and (4.43), we obtain $\left\lVert\frac{\psi_{2}}{sinhr}\right\rVert_{L^{p}}\lesssim\left\lVert\psi^{+}\right\rVert_{L^{p}}$, then the $L^{p}$-bound for $\psi^{-}$ and $\frac{A_{2}-1}{\sinh r}$ are obtained immediately by the definition of $\psi^{-}$ and $A_{2}^{2}+|\psi_{2}|^{2}=1$.

Second, we obtain (4.41). By (4.38) and $A_{2}^{2}+|\psi_{2}|^{2}=1$, we have

[TABLE]

It suffices to get the $L^{2}$-bound for $\frac{\psi_{2}}{\sinh r}$. From (4.43), we have

[TABLE]

For the first term we use (2.19) and the smallness of $\left\lVert\psi^{+}\right\rVert_{L^{2}(0,R^{-1})}$. For the second term, by Hölder inequality, we have

[TABLE]

Then by (2.19), we easily obtain

[TABLE]

Thus the $L^{2}$-bound follows.

Third, we get the Lipschitz continuity (4.42). For notational convenience we denote

[TABLE]

Without any restriction in generality, we can make the assumption $\left\lVert\delta\psi^{+}\right\rVert_{L^{2}}\ll 1$ and the bootstrap assumption

[TABLE]

By (4.38) and $A_{2}^{2}+|\psi_{2}|^{2}=1$, we derive the equations

[TABLE]

Since $\left\lVert\delta\psi^{+}\right\rVert_{L^{2}}\ll 1$ and $\frac{(\delta A_{2})^{2}}{\sinh r}$ is a high order term, $\Im(\delta\psi^{+}\bar{\tilde{\psi}}_{2})+\frac{(\delta A_{2})^{2}}{\sinh r}$ and $iA_{2}\delta\psi^{+}$ can be regarded as error terms. Let $X=(\Re\delta\psi^{+},\ \Im\delta\psi^{+},\ \delta A_{2})^{T}$, we have

[TABLE]

where $L={\rm diag}\{1,\ 1,\ 2\}$,

[TABLE]

From (4.40) we obtain the $L^{2}$-norm of $B$ is bounded. Then we decompose $B=B_{1}+B_{2}:=B\textbf{1}_{\geq\epsilon}(r)+B\textbf{1}_{<\epsilon}(r)$ for small $\epsilon$. By the $L^{2}$-bound for $B$, we have $|rB|\rightarrow 0,\ as\ r\rightarrow 0$, which gives $|B_{2}|\ll\frac{1}{r}$ in $(0,\ \epsilon)$. We also easily obtain $\left\lVert B_{1}\right\rVert_{L^{1}(dr)}\lesssim|\log\epsilon|^{1/2}\left\lVert B\right\rVert_{L^{2}}$ by Hölder inequality. Then we can construct the bounded matrix $e^{-\int_{\infty}^{r}B_{1}ds}$ such that $\partial_{r}e^{-\int_{\infty}^{r}B_{1}ds}=-e^{-\int_{\infty}^{r}B_{1}ds}B_{1}$. Hence (4.44) can be written as

[TABLE]

then

[TABLE]

By the above expression of $X$ and (2.19), we have

[TABLE]

and

[TABLE]

where $c\ll 1$. Hence, $\left\lVert\frac{X}{\sinh r}\right\rVert_{L^{2}}\lesssim\left\lVert\delta\psi^{+}\right\rVert_{L^{2}}$ and $\left\lVert X\right\rVert_{L^{\infty}}\lesssim\left\lVert\delta\psi^{+}\right\rVert_{L^{2}}$. Furthermore, by (4.44) we have $\left\lVert\partial_{r}X\right\rVert_{L^{2}}\lesssim\left\lVert\delta\psi^{+}\right\rVert_{L^{2}}$.

Finally we prove (iv). If $s=1$, by (4.38), we have

[TABLE]

then by (4.39), (4.40) and Gagliardo-Nirenberg inequality, we obtain

[TABLE]

If $s=2$, by (4.38) and $A_{2}^{2}+|\psi_{2}|^{2}=1$, we have

[TABLE]

by Sobolev embedding $H^{2}\hookrightarrow L^{6}$, we get $\partial_{rr}\psi^{-}\in L^{2}$. Similarly, $\frac{\cosh r}{\sinh r}\partial_{r}\psi^{-}\in L^{2}$. Hence, $R_{-}\psi^{-}\in H^{2}$. ∎

Proposition 4.3**.**

Given $\psi^{+}\in L^{2}$ with $\left\lVert\psi^{+}\right\rVert_{L^{2}}<2$, then there is a unique map $u:{\mathbb{H}}^{2}\longrightarrow{\mathbb{S}}^{2}$ with the property that $\psi^{+}$ is the representation of $\mathcal{V}^{+}$ relative to a Coulomb gauge satisfying (4.22) with $E(u)=\pi\left\lVert\psi^{+}\right\rVert_{L^{2}}^{2}$. Moreover, the map $\psi^{+}\rightarrow u$ is Lipschitz continuous in the following sense:

[TABLE]

Proof.

Given $\left\lVert\psi^{+}\right\rVert_{L^{2}}<2$, by Lemma 4.2, there is a unique solution $(A_{2},\ \psi_{2})$. Let $\psi_{1}=\psi^{+}-i\frac{\psi_{2}}{\sinh r}$. Now we solve the system of $U=(\bar{u},\bar{v},\bar{w})^{T}$, that is

[TABLE]

Since $\psi_{1}=-(A_{2}-1)\psi_{1}-i\partial_{r}\psi_{2}$, $M$ can be rewritten as $M=M_{1}+\partial_{r}M_{2}$, where

[TABLE]

and by (4.39), $\left\lVert M_{1}\right\rVert_{L^{1}(dr)}+\left\lVert M_{2}\right\rVert_{\dot{H}^{1}_{e}}\lesssim\left\lVert\psi^{+}\right\rVert_{L^{2}}$. If we restrict $M_{1}$ and $M_{2}$ on $[R,\infty)$ for sufficiently large $R$, we can assume $\left\lVert M_{1}\right\rVert_{L^{1}(dr)(R,\infty)}+\left\lVert M_{2}\right\rVert_{\dot{H}^{1}_{e}(R,\infty)}\lesssim\epsilon$. This allow us to construct solutions with data at $r=\infty$ by using the iteration scheme

[TABLE]

Let $X=\{U\in C([R,\infty)):\partial_{r}U\in L^{2}([R,\infty)),\lim\limits_{r\rightarrow\infty}U(r)\ \ {\rm exists}\}$. We run the iteration scheme in $X$. For $U_{i-1}\in X$, we have

[TABLE]

from which we obtain

[TABLE]

Therefore,

[TABLE]

Then by choosing $R$ large enough, we can use the iteration scheme to construct a solution $U$ on $[R,\infty)$.

The uniqueness of (4.45) is obtained by conservation law, that is, apply $(\bar{u},\bar{v},\bar{w})$ to both side of (4.45), we have $\frac{1}{2}\partial_{r}(|\bar{u}|^{2}+|\bar{v}|^{2}+|\bar{w}|^{2})=0$.

Since $\partial_{r}\bar{u}\cdot\bar{v}=-\partial_{r}\bar{v}\cdot\bar{u}$ by (4.45), we have $\bar{u}\cdot\bar{v}(r)={\rm const}$, which together with $\lim\limits_{r\rightarrow\infty}U(r)=U(\infty)=(\vec{k},\vec{i},\vec{j})^{T}$ yields $\bar{u}\cdot\bar{v}(r)=\bar{u}\cdot\bar{v}(\infty)=0$. Similarly, we also have $\bar{u}\cdot\bar{w}=\bar{v}\cdot\bar{w}=0$ and $|\bar{v}|=|\bar{w}|=1$. Thus $U$ satisfies the orthonormality condition.

Next, the solution constructed above can be extended to $(0,\infty)$. Since $\left\lVert\psi^{+}\right\rVert_{L^{2}}<2$, then for any $\epsilon>0$, there exists $\delta>0$ such that $\left\lVert\psi^{+}\right\rVert_{L^{2}(R-\delta,R)}<\epsilon$. Define $U_{i}(r)=\int_{R}^{r}MU_{i-1}ds$, denote $I=[R-\delta,\ R]$, we have

[TABLE]

and

[TABLE]

By (4.46), we have $\left\lVert M_{1}\right\rVert_{L^{1}(dr)(I)}+\left\lVert M_{1}\right\rVert_{L^{2}(I)}+\left\lVert\partial_{r}M_{2}\right\rVert_{L^{2}(I)}\lesssim\left\lVert\psi^{+}\right\rVert_{L^{2}(I)}$, therefore,

[TABLE]

By choosing $\delta$ small such that $(\int_{R}^{R-\delta}\sinh^{-1}sds)^{1/2}$ is small, then we can still rely on iteration scheme to extend the solution to $[\epsilon R^{-1},\infty)$.

On the interval $(0,\epsilon R^{-1}]$, by (4.41), we have $\left\lVert\psi_{2}\right\rVert_{C((0,\epsilon R^{-1}])}\lesssim\left\lVert\psi_{2}\right\rVert_{\dot{H}^{1}_{e}((0,\epsilon R^{-1}])}\lesssim\epsilon$. By a similar argument to that on $[R,\infty)$, we extend the solution to $r=0$. As a byproduct,

[TABLE]

From the system (4.45), we know that $\bar{u}_{3}$ and $q=\bar{w}_{3}-i\bar{v}_{3}$ solve the system

[TABLE]

with boundary condition $(\bar{u}_{3},q)(\infty)=(1,0)$. By uniqueness, $A_{2}=\bar{u}_{3}$, $\psi_{2}=q$.

Next, we construct the system of $(u,v,w)$ by equivariant setup, that is, apply $(\bar{u},\bar{v},\bar{w})$ by $e^{\theta R}$. From $\psi_{2}=\bar{w}_{3}-i\bar{v}_{3}$ and the orthonormality condition, (4.4) is satisfied for $k=2$.

Given $\psi^{+},\tilde{\psi}^{+}\in L^{2}$, we construct $U$ and $\tilde{U}$ as above. From the construction it follows that

[TABLE]

which implies $\left\lVert\partial_{r}(u-\tilde{u})\right\rVert_{L^{2}(0,\infty)}\lesssim\left\lVert\psi^{+}-\tilde{\psi}^{+}\right\rVert_{L^{2}}$. Since $u_{1}=v_{2}w_{3}-v_{3}w_{2}$, $\tilde{u}_{1}=\tilde{v}_{2}\tilde{w}_{3}-\tilde{v}_{3}\tilde{w}_{2}$, $\psi_{2}=w_{3}-iv_{3}$, $\tilde{\psi}_{2}=\tilde{w}_{3}-i\tilde{v}_{3}$, by (4.47), we have

[TABLE]

A similar argument shows that $\left\lVert\frac{u_{2}-\tilde{u}_{2}}{\sinh r}\right\rVert_{L^{2}}+\left\lVert\frac{u_{3}-\tilde{u}_{3}}{\sinh r}\right\rVert_{L^{2}}\lesssim\left\lVert\psi^{+}-\tilde{\psi}^{+}\right\rVert_{L^{2}}(\left\lVert\psi^{+}\right\rVert_{L^{2}}+\left\lVert\tilde{\psi}^{+}\right\rVert_{L^{2}})$. Therefore, $\left\lVert u-\tilde{u}\right\rVert_{\dot{H}^{1}}\lesssim\left\lVert\psi^{+}-\tilde{\psi}^{+}\right\rVert_{L^{2}}$. ∎

5. The Cauchy problem

In this section we concerned with the $(\psi^{+},\ \psi^{-})$-system which we recall here

[TABLE]

with initial data $\psi^{\pm}(t_{0})=\psi^{\pm}_{0}$. Where $A_{0}$, $A_{2}$, $\psi_{2}$ are given by (4.21), (4.20), (4.17). Since the system (5.1) arised from the Schrödinger map (1.1), we will show that $(\psi^{+},\ \psi^{-})$ satisfy the compatibility condition.

For simplicity of notations, we denote $\left\lVert f^{\pm}\right\rVert=\left\lVert f^{+}\right\rVert+\left\lVert f^{-}\right\rVert$. Since our analysis relies on $L_{t}^{p}L_{x}^{q}$-norm, we define the norm of $f$ by $\left\lVert f\right\rVert_{L_{I}^{p}L^{q}}:=\left\lVert f\right\rVert_{L^{p}(I;L^{q})}$. Finally, we denote the nonlinearities by

[TABLE]

5.1. Srichartz estimates

To understand the well-posedness of (5.1), we need to obtain the Strichartz estimates. The $\psi^{+}$-equation in (5.1) is a nonlinear Schrödinger equation with positive and exponential decay potential. More generally, we consider the Schrödinger equation

[TABLE]

where $V\in e^{-\alpha r}L^{\infty}({\mathbb{H}}^{2};{\mathbb{R}})$ for $\alpha\geq 1$ is a positive potential. In this section we always denote potential $V$ as (5.2). For simplicity, we denote $p^{\prime}=\frac{p}{p-1}$ for $p\in[1,\ \infty]$. $(p,\ q)$ is called admissible pair, if

[TABLE]

Then we obtain the following Strichartz estimates.

Theorem 5.1**.**

*Let $(p,\ q),\ (\tilde{p}^{\prime},\ \tilde{q}^{\prime})$ be admissible pairs, $I\subset{\mathbb{R}}$ be open interval.

(i)  If $f\in L^{2}({\mathbb{H}}^{2})$, then*

[TABLE]

(ii)  If $F\in L^{\tilde{p}}_{I}L^{\tilde{q}}$, then

[TABLE]

Based on a standard theory, the above results are obtained by the following dispersive estimates immediately.

Proposition 5.2**.**

Assume $V\in e^{-\alpha r}L^{\infty}({\mathbb{H}}^{2};{\mathbb{R}}),\ \alpha\geq 1$, is a positive potential, then we have

[TABLE]

By standard convention the resolvent of Laplacian $-\Delta_{{\mathbb{H}}^{2}}$ on ${\mathbb{H}}^{2}$ is written as $R_{0}(s)=(-\Delta_{{\mathbb{H}}^{2}}-s(1-s))^{-1}$ with $\Re s>\frac{1}{2}$ corresponding to the resolvent set $s(1-s)\in{\mathbb{C}}-[\frac{1}{4},\ \infty)$. The kernel of $R_{0}(s)$ is

[TABLE]

where $Q^{0}_{s-1}$ is Legendre function, $r:=d(z,w)$. With the hyperbolic convention for spectral parameter, Stone’s formula gives the continuous part of the spectral resolution as

[TABLE]

Then we use the spectral resolution to write

[TABLE]

Similarly, from [7], the resolvent of $-\Delta_{{\mathbb{H}}^{2}}+V$ for potential $V$ defined as above is given by $R_{V}(s)=(-\Delta_{{\mathbb{H}}^{2}}+V-s(1-s))^{-1}$ and the continuous component of spectral resolution is given by

[TABLE]

then the kernel of Schrödinger propagator can be written as

[TABLE]

By Birman-Schwinger type resolvent expansion for all frequencies:

[TABLE]

we get

[TABLE]

Before proving Proposition 5.2, we recall the pointwise bounds on the resolvent kernel from [7]. This bounds will be crucial for the dispersive estimates.

Lemma 5.3**.**

For the free resolvent kernel the pointwise bounds are valid for $0\leq\lambda\leq 1$ and $r\in(0,\ \infty)$

[TABLE]

[TABLE]

where $r:=d(z,w)$.

Lemma 5.4**.**

For the free resolvent kernel the pointweise bounds are valid for $\lambda\geq 1$, and $r\in(0,\infty)$

[TABLE]

[TABLE]

where $r:=d(z,w)$.

We also recall the meromorphic continuation from [7].

Lemma 5.5**.**

For $V\in e^{-\alpha r}L^{\infty}({\mathbb{H}}^{2})$ with $\alpha>0$, the resolvent $R_{V}(s)$ admits a meromorphic continuation to the half-plane $\Re s>\frac{1}{2}-\delta$ as a bounded operator

[TABLE]

for $\delta<\frac{\alpha}{2}$. And there exists a constant $M_{V}$ such that for all $\lambda\in{\mathbb{R}}$ with $|\lambda|\geq M_{V}$,

[TABLE]

If the $R_{V}(\frac{1}{2}+i\lambda)$ has no pole at $\lambda=0$, we can extend the estimate through $\lambda=0$ to give

[TABLE]

In order to prove Proposition 5.2, we also need the following lemma.

Lemma 5.6**.**

[TABLE]

Proof.

The proof roughly follows the approach in [2]. Before proving the lemma, we recall two useful estimates, that is,

[TABLE]

and

[TABLE]

Case 1: $r\geq\frac{\sqrt{t}}{2}$.

Let $s=\frac{\tau t}{r+a}+r$, then

[TABLE]

Denote $\Phi(\tau):=\frac{\tau^{2}t}{4(r+a)^{2}}+\frac{\tau}{2}$, then (5.15) can be written as

[TABLE]

Since $\int_{0}^{\infty}=\int_{0}^{1}+\int_{1}^{\infty}$, (5.16) can be split into

[TABLE]

For $I_{1}$, by (5.14) and $r\geq\frac{\sqrt{t}}{2}$ we have

[TABLE]

For $I_{2}$, Let

[TABLE]

By integrating by parts in $I_{2}$, we get

[TABLE]

Notice that $\alpha(1)\lesssim 1$ and $\alpha^{\prime}(\tau)<0$ by (5.18), hence,

[TABLE]

That is $I_{1}$ and $I_{2}$ are bounded. Therefore (5.12) follows (5.17) in the region $r\geq\frac{\sqrt{t}}{2}$.

Case 2: $r<\frac{\sqrt{t}}{2}$.

Let us split the left hand side of (5.12) into three parts:

[TABLE]

For $J_{1}$, we assume $r>0$, otherwise $J_{1}=0$ immediately, then

[TABLE]

Since we are in the case $r<\frac{\sqrt{t}}{2}$, we get that

[TABLE]

For $J_{2}$, by (5.13) we have

[TABLE]

For $J_{3}$, let $u=\sqrt{t}\tau+a$, we get that

[TABLE]

Then $J_{3}$ can be written as

[TABLE]

where $\psi(\tau)=\frac{\tau^{2}}{4}+\frac{a\tau}{2\sqrt{t}}$ and $\beta(\tau)=[\cosh(\sqrt{t}\tau)-\cosh r]^{-\frac{1}{2}}$. By integration by parts, we get

[TABLE]

Since the derivative of $\beta$ is negative, we obtain

[TABLE]

Therefore, we have

[TABLE]

in the region $r<\frac{\sqrt{t}}{2}$. ∎

Proof of Proposition 5.2.

The estimate for $|t|\geq 1$ in (5.3) has been proved in [7], we only prove the case $0<|t|<1$ here. In order to estimate $e^{it(\Delta_{{\mathbb{H}}^{2}}-V)}$, it suffices to bound (5.5)-(5.7) respectively. (5.5) is indeed $e^{it\Delta_{{\mathbb{H}}^{2}}}f$, which can be estimated in [1]. To estimate (5.6), we rewrite it by (5.4) as

[TABLE]

By Lemma 5.6, since $\sqrt{r_{1}+s}\leq\sqrt{r_{1}}+\sqrt{s}$ for $r_{1}\geq 0$ and $s\geq 0$ we get

[TABLE]

It suffices to estimate the three integrals the right hand side. By (16) of [1], for $r_{0}>0$ we have

[TABLE]

For the second integral, we make the change of variables $u=s-r_{0}$,

[TABLE]

If $r_{0}=0$,

[TABLE]

If $r_{0}>0$, by (5.24) we have

[TABLE]

The third integral can be estimated similar to the second one. If $r_{0}=0$,

[TABLE]

If $r_{0}>0$,

[TABLE]

In conclusion, we obtained

[TABLE]

Therefore,

[TABLE]

Finally, we estimate the (5.7). By duality, it suffices to prove for $\left\lVert h\right\rVert_{L^{1}}=1$,

[TABLE]

write

[TABLE]

Then by integration by parts and Lemma 5.6, we have

[TABLE]

By Lemma 5.3 and Lemma 5.4, Young’s inequality and Holder’s inequality, we have

[TABLE]

Similarly, we have

[TABLE]

Therefore,

[TABLE]

Thus Proposition 5.2 follows. ∎

5.2. The Cauchy theory

Here we consider the Cauchy problem for (5.1). The local well-posedness of (5.1) is directly by Strichartz estimates in Theorem 5.1. Then for small initial data, since the operator $-\Delta_{{\mathbb{H}}^{2}}-2\frac{\cosh r-1}{\sinh^{2}r}$ has discrete spectrum, we use perturbation method (see [30]) to prove global well-posedness.

Theorem 5.7**.**

*Consider the problem (5.1) with data $\left\lVert\psi_{0}^{\pm}\right\rVert_{L^{2}}<2$, where $A_{0}$, $A_{2}$, $\psi_{2}$ are given by (4.21), (4.20), (4.17). Then there exists a unique maximal-lifespan solution pair $(\psi^{+},\psi^{-}):I\times{\mathbb{R}}^{2}\longrightarrow{\mathbb{C}}\times{\mathbb{C}}$ with $t_{0}\in I$ and $\psi^{\pm}(t_{0})=\psi^{\pm}_{0}$ with the following additional properties:

(i) If $\left\lVert\psi_{0}^{\pm}\right\rVert_{L^{2}}<2$, then there exists $T=T(\psi^{\pm}_{0})>0$, and a unique solution $(\psi^{+},\psi^{-})$ of the system in the time interval $[-T,T]$ with $(\psi^{+},\psi^{-})\in L^{4}([-T,T];L^{4})\cap C([-T,T];L^{2})$.

(ii) If $(\psi^{+},\psi^{-}):(T_{0},T_{1})\times R^{+}\longrightarrow{\mathbb{C}}\times{\mathbb{C}}$, $|T_{1}-T_{0}|<\infty$, is a solution to (5.1) with $\left\lVert\psi^{\pm}\right\rVert_{L^{4}L^{4}(T_{0},T_{1})}<\infty$, then $\psi^{\pm}$ can be extended to a solution on a larger time interval.

(iii) There exists $\epsilon>0$ such that $\left\lVert\psi^{\pm}_{0}\right\rVert_{L^{2}}\leq\epsilon$, then for any compact interval $J\subset{\mathbb{R}}$, (5.1) has a unique global solution $\psi^{\pm}(t)\in L^{4}(J;L^{4})\cap C(J;L^{2})$, moreover, $\left\lVert\psi^{\pm}\right\rVert_{L^{4}_{J}L^{4}}\lesssim C(J,\ \left\lVert\psi^{\pm}_{0}\right\rVert_{L^{2}})$.

(iv) For every $A>0$, and $\epsilon>0$, there is $\delta>0$ such that if $\psi^{\pm}$ is a solution satisfying $\left\lVert\psi^{\pm}\right\rVert_{L^{4}_{I}L^{4}}\leq A$ and $M(\psi^{\pm}_{0}-\tilde{\psi}^{\pm}_{0})\leq\delta$, then there exists a solution such that $\left\lVert\psi^{\pm}-\tilde{\psi}^{\pm}\right\rVert_{L^{4}_{I}L^{4}}\leq\epsilon$, and $M(\psi^{\pm}-\tilde{\psi}^{\pm})\leq\epsilon$, $\forall\ t\in I$.

(v) Assume that $R_{\pm}\psi^{\pm}_{0}\in H^{s}$, for $s\in{1,\ 2}$. If $\left\lVert\psi^{\pm}\right\rVert_{L^{4}_{I}L^{4}}\leq M$, then the solution $\psi^{\pm}$ satisfies*

[TABLE]

and it has Lipschitz dependence with respect to the initial data.

Proof.

(i) Consider the system (4.18) in the space

[TABLE]

Given the formulas for $A_{0}$, $A_{2}$ and $\psi_{2}$ by (4.21), (4.20), (4.17), using Lemma 2.4, we obtain

[TABLE]

In a similar argument, we also obtain that

[TABLE]

Denote $V=2\frac{\cosh r-1}{\sinh^{2}r}$, then by Duhamel formula, define the maps

[TABLE]

For any $\epsilon>0$, there exists $\phi^{\pm}_{0}\in C_{0}^{\infty}$, such that $\left\lVert\psi_{0}^{\pm}-\phi_{0}^{\pm}\right\rVert_{L^{2}}<\frac{\epsilon}{4C}$, and there exists $T_{1}>0$, s.t $T_{1}^{\frac{1}{4}}\left\lVert\phi_{0}^{\pm}\right\rVert_{\dot{H}^{\frac{1}{2}}}<\frac{\epsilon}{4C}$, then dispersive estimates and (5.28) imply

[TABLE]

Similarly, we have

[TABLE]

Since $V\in L^{2}$ independent on t, there exists $0<T<T_{1}$, such that $C\left\lVert V\right\rVert_{L^{2}_{T}L^{2}}<\delta$, hence $\left\lVert\mathcal{T}^{-}(\psi^{-})\right\rVert_{L^{4}_{T}L^{4}}<\epsilon$. We can also show that $\mathcal{T}^{\pm}(\psi^{\pm})\in C([0,T];L^{2})$. Indeed, by Strichartz estimates, we have

[TABLE]

and

[TABLE]

Therefore, $(\mathcal{T}^{+}(\psi^{+}),\mathcal{T}^{-}(\psi^{-}))\in X$ for any $\psi^{\pm}\in X$.

Then we need to show $(\mathcal{T}^{+}(\psi^{+}),\ \mathcal{T}^{-}(\psi^{-}))$ is a contraction map. By (5.29), we get

[TABLE]

and

[TABLE]

In conclusion, $(\mathcal{T}^{+}(\psi^{+}),\mathcal{T}^{-}(\psi^{-}))$ is a contraction map in $X$, by the fixed point theorem, there exists a unique solution in $X$ for small $T$ depending only on $\psi^{\pm}_{0}$ and $\left\lVert V\right\rVert_{L^{2}}$.

(iii) Let $u^{\pm}:I\times R^{+}\longrightarrow{\mathbb{C}}$ be an approximate solution to system (5.1) in the sense that

[TABLE]

Based on standard fixed point argument, by the Strichartz estimates for Schrödinger operators $-\Delta_{{\mathbb{H}}^{2}}$ and $-\Delta_{{\mathbb{H}}^{2}}+V$, there exists $\epsilon>0$ such that if $E(\psi^{\pm}_{0})=\left\lVert\psi^{\pm}_{0}\right\rVert_{L^{2}}\leq\epsilon$, then (5.31) has a unique global solution $u^{\pm}\in C({\mathbb{R}};L^{2})\cap L^{4}L^{4}$, moreover, $\left\lVert u^{\pm}\right\rVert_{L^{\infty}L^{2}\cap L^{4}L^{4}({\mathbb{R}}\times{\mathbb{H}}^{2})}\leq C\epsilon$.

Now we show using a perturbative argument that (5.1) is global well-posed for $E(\psi^{\pm}_{0})<\epsilon$. First we show that for $T$ sufficiently small depending only on $E(\psi^{\pm}_{0})$, and $V$, the solution $(\psi^{+},\psi^{-})$ to (5.1) on $[0,T]$ satisfies an a priori estimate

[TABLE]

Fix a small parameter $\eta>0$, since $V=\frac{\cosh r-1}{\sinh^{2}r}$, there exists $T>0$ for $I=[0,T]$, such that

[TABLE]

Further, by Duhamel formula, Strichartz estimates and $\left\lVert u^{\pm}\right\rVert_{L^{\infty}L^{2}\cap L^{4}L^{4}}\leq C\epsilon$, we have

[TABLE]

and

[TABLE]

Since $(\psi^{+},\ \psi^{-})$ satisfies (5.1) and $u^{+}(0)=e^{i2\theta}\psi^{+}_{0},\ u^{-}(0)=\psi^{-}_{0}$, apply the Duhamel formula, (5.33) and (5.34) to obtain

[TABLE]

and

[TABLE]

Choose $\eta$ sufficiently small such that $C\eta<\frac{1}{3}$, which yields

[TABLE]

Combining (5.35) and (5.36), we have

[TABLE]

Then by continuity argument, we get

[TABLE]

which, together with Strichartz estimates gives

[TABLE]

Therefore (5.32) is obtained.

Then from the system (5.1), we have energy conservation $E(\psi^{\pm})=E(\psi^{\pm}_{0})$. Since the $T$ depends only on $E(\psi^{\pm}_{0})$ and $V=2\frac{\operatorname{ch}r-1}{\operatorname{sh}^{2}r}$, by (ii) and energy conservation, it will follow that $(\psi^{+},\psi^{-})$ is a global solution with $\left\lVert\psi^{\pm}\right\rVert_{L^{4}_{J}L^{4}}\leq C(\left\lVert\psi^{\pm}_{0}\right\rVert_{L^{2}},|J|)$ for any compact interval $J\subset{\mathbb{R}}$.

(v)Applying $(-\Delta)^{\frac{s}{2}}$ for $s=1,\ 2$ to both sides of system (5.1), we obtain

[TABLE]

The nonlinearities $F^{\pm}$ can be written as

[TABLE]

Let $\varphi(r)\in C_{c}^{\infty}$ be a bump function with $0\leq\varphi\leq 1$, \varphi\big{|}_{B_{1}(0)}=1 and \varphi\big{|}_{B_{2}^{c}(0)}=0, $F_{2}^{\pm}$ can be rewritten as

[TABLE]

Since $\left\lVert\psi^{\pm}\right\rVert_{L^{4}_{I}L^{4}}\leq M$, Strichartz estimates imply $\left\lVert\psi^{\pm}\right\rVert_{L^{3}_{I}L^{6}}\lesssim 1$. Then we split the interval into $I=\bigcup I_{j}$ such that $\left\lVert\psi^{\pm}\right\rVert_{L^{3}_{I_{j}}L^{6}}<\epsilon$, $\left\lVert(-\Delta)^{1/2}\psi^{\pm}_{0}\right\rVert_{L^{2}}\left\lVert\psi^{\pm}\right\rVert_{L^{3}_{I_{j}}L^{6}}\lesssim 1$, $\left\lVert\partial_{r}^{k}V\right\rVert_{L^{3}_{I_{j}}L^{6}}\ll 1$ and $\left\lVert\frac{\cosh r}{\sinh r}\partial_{r}V\right\rVert_{L^{3}_{I_{j}}L^{6}}\ll 1$. By Duhamel’s formula and Strichartz estimates, we have

[TABLE]

Now we estimate the second term of the right hand side of (5.39). Define

[TABLE]

For $s=1$, from (2.10) we easily obtain

[TABLE]

Since the operator $\frac{1}{r^{2}}\int_{0}^{r}\cdot sds$ keeps the two dimensional frequency localization, one could use Littlewood-Paley decomposition to deal with ${\rm I}^{\pm}$. To estimate ${\rm I}^{\pm}$, we claim that for $f$ radial, $p\geq 2$ the following estimate holds

[TABLE]

Then we have

[TABLE]

Hence, (5.39), (LABEL:s=1_easy_term) and (5.42) imply

[TABLE]

We repeat the above procedure for $I_{j+1}$ to obtain the similar estimate in $I_{j+1}$. Thus, (5.27) valid for $s=1$.

For $s=2$, similarly, we also easily have

[TABLE]

Then for ${\rm I}^{\pm}$, which can be rewritten as

[TABLE]

By (2.20) and (5.41) we have

[TABLE]

for ${\rm I}_{2}^{\pm}$, from (5.41) we obtain

[TABLE]

Thus, by (5.39), (5.44), (5.45) and (5.46) we have

[TABLE]

Hence, (5.27) follows for $s=2$.

Finally, we prove (5.41). Denote $B_{r}=B_{r}(0)$ and $m_{k}(r)=\varphi(2^{-k}r)-\varphi(2^{-k+1}r)$. Since $f$ is radial, we have

[TABLE]

Then for $P_{k}f=\mathcal{F}^{-1}(m_{k}(\xi)\widehat{f}(\xi))$ , we have

[TABLE]

which implies

[TABLE]

Hence, by Littlewood-Paley decomposition and (5.49) we have

[TABLE]

from (2.20) we obtain

[TABLE]

Thus, (5.41) follows. ∎

The above theorem is only concerned with the general solutions of (5.1). Since the system of $(\psi^{+},\ \psi^{-})$ is derived from the Schrödinger map (1.1), if we want to reconstructed the map $u$ by $\psi^{\pm}$, the solution $\psi^{\pm}$ of (5.1) must satisfies the compatibility condition (4.19).

Theorem 5.8**.**

If $\psi^{\pm}_{0}\in L^{2}$ satisfies the compatibility condition, then $\psi^{\pm}(t)$ satisfies the compatibility condition for any $t\in I$. If, in addition, $R_{\pm}\psi^{\pm}_{0}\in H^{3}$, then (4.5) and (4.6) are satisfied.

Proof.

Given $\psi_{1}=\frac{\psi^{+}+\psi^{-}}{2}$, $\frac{\psi_{2}}{\sinh r}=\frac{\psi^{+}-\psi^{-}}{2i}$, $A_{1}=0$. To prove the compatibility condition (4.19), it suffices to show that $D_{1}\psi_{2}=D_{2}\psi_{1}$ is preserved for $t\in I$. For this we need to derive the equation for

[TABLE]

Before deriving the equation for $F$, we give some identities from (5.1). First, (4.20) gives

[TABLE]

Second, the system of $(\psi^{+},\psi^{-})$ (5.1) and (4.20) imply that

[TABLE]

where $\psi_{0}$ is given by (4.10) Third, (4.21) implies

[TABLE]

Finally, we obtain the following two equations from (5.1) by algebraic computation and $A_{2}^{2}+|\psi_{2}|^{2}=1$,

[TABLE]

Then combining the above two equations with (4.11), we have

[TABLE]

Apply the operator $D_{0}$ to $F$, by (5.51)-(5.52), we have

[TABLE]

So we derive equation for F:

[TABLE]

namely

[TABLE]

If $R^{\pm}\psi^{\pm}\in H^{1}$, we can write

[TABLE]

Due to the boundedness of $A_{2}$ and $\frac{\cosh r-1}{\sinh r}$, we get $\frac{F}{\sinh r}\in L^{2}$.

If $R^{\pm}\psi^{\pm}\in H^{2}$, we using $A_{2}^{2}+|\psi_{2}|^{2}=1$ and Sobolev embedding, yields $\frac{F}{\sinh r}\in\dot{H}^{1}$ by the representation

[TABLE]

Let $P_{\epsilon}$ for $\epsilon>0$ be the smoothing operator defined by the Fourier multiplier $\lambda\rightarrow e^{-\epsilon^{2}\lambda^{2}}$. Denote $N$ is the nonlinearity of (5.53). Applying $P_{\epsilon}$ to both sides of (5.53), we obtain

[TABLE]

Since $P_{\epsilon}(e^{i\theta}\frac{F}{\sinh r})$, $\partial_{r}P_{\epsilon}(e^{i\theta}\frac{F}{\sinh r})$ and $\frac{1}{\sinh r}P_{\epsilon}(e^{i\theta}\frac{F}{\sinh r})\in L^{2}$, which implies

[TABLE]

and

[TABLE]

Hence, by integration by parts and (2.9), we get

[TABLE]

which further gives

[TABLE]

Then let $\epsilon\rightarrow 0$, we obtain

[TABLE]

By using Gronwall inequality and $F(0)=0$, we get $F(t)=0$ for all $t\in I$.

In general, if $\psi^{\pm}_{0}\in L^{2}$ only, there exists $R_{+}\psi^{+}_{0,n}\in H^{2}$ such that $\left\lVert\psi^{+}_{0}-\psi^{+}_{0,n}\right\rVert_{L^{2}}\leq\frac{1}{n}$. By Lemma 4.2, we obtain compatible pair $R_{\pm}\psi^{\pm}_{0,n}\in H^{2}$ and $\left\lVert\psi^{-}_{0}-\psi^{-}_{0,n}\right\rVert_{L^{2}}\lesssim\frac{1}{n}$. By the above argument, the solutions $\psi^{\pm}_{n}$ with initial data $\psi^{\pm}_{0,n}$ satisfy compatibility condition. Then the compatibility condition for $\psi^{\pm}_{n}$ can be written as

[TABLE]

Hence, by Theorem 5.7 (iv), Lemma 2.4 and the expression of $A_{2}$ (4.20), we have

[TABLE]

which complete the proof of Theorem 5.8. ∎

Proof of Theorem 1.2.

First, we claim: Given $R_{\pm}\psi^{\pm}_{0}\in H^{2}$, $\psi^{\pm}(t)$ is the solution of (4.18), then the map $u(t)$ constructed in Proposition 4.3 is a Schrödinger map. Indeed, by Proposition 4.3, we construct $u_{0}\in\mathfrak{H}^{3}$. Then by Theorem 1.1, there exists a unique solution $u(t)\in L^{\infty}(I;\mathfrak{H}^{3})$ with data $u_{0}$. As in Section 4.1, we construct Coulomb gauge and its field component such that they satisfy (4.18) with initial data $\psi^{\pm}_{0}$. The uniqueness of the solution of (4.18) implies $\psi^{\pm}(t)$ are the gauge representation of $\mathcal{V}^{\pm}(t)$. Therefore the map reconstructed in Proposition 4.3 is the Schrödinger map $u(t)$.

Next we begin to prove the Theorem 1.2. Given initial data $u_{0}\in\mathfrak{H}^{3}$, by Theorem 3.2 we obtain a unique local solution on $[0,T]$ for some $T>0$. In particular, if in addition $E(u_{0})<\epsilon^{2}$ for sufficiently small $\epsilon$, we can construct the fields $\psi^{\pm}$ on interval $[0,\ T]$ satisfying (4.18) and $\left\lVert\psi^{+}\right\rVert_{L^{2}}=\left\lVert\psi^{-}\right\rVert_{L^{2}}<\epsilon$ as in Section 4.1. By Theorem 5.7 (iii), the solution $\psi^{\pm}$ is defined on $J\subset{\mathbb{R}}$ for any compact interval $J$ and with $\left\lVert\psi^{\pm}\right\rVert_{L^{4}_{J}L^{4}}\leq C(J,\ \left\lVert\psi^{\pm}_{0}\right\rVert_{L^{2}})$. Then by Theorem 5.7 (v) and Proposition 4.3, we construct a map $u(t)\in\mathfrak{H}^{3}$ coincide with the Schrödinger map on $[0,\ T]$ from $\psi^{\pm}(t)$, moreover, $E(u(T))<\epsilon^{2}$. Then repeat the procedure the map reconstructed from $\psi^{\pm}(t)$ is in fact a Schrödinger map.

For initial data $u_{0}\in\mathfrak{H}^{1}$, there exists $u_{0,n}\in\mathfrak{H}^{3}$ such that $\left\lVert u_{0}-u_{0,n}\right\rVert_{\mathfrak{H}^{1}}<\frac{1}{n}$. By (4.23), we obtain the Lipschitz continuity of $\psi^{\pm}_{0,n}$, i.e $\left\lVert\psi^{\pm}_{0,n}-\psi^{\pm}_{0}\right\rVert_{L^{2}}\lesssim\left\lVert u_{0,n}-u_{0}\right\rVert_{\mathfrak{H}^{1}}$. Then from Theorem 5.7 (iv), the solution of (4.18) is Lipschitz continuous with respect to initial data, we have $\left\lVert\psi^{\pm}_{n}-\psi^{\pm}\right\rVert_{L^{2}}\lesssim\left\lVert\psi^{\pm}_{0,n}-\psi^{\pm}_{0}\right\rVert_{L^{2}}\lesssim\left\lVert u_{0,n}-u_{0}\right\rVert_{\mathfrak{H}^{1}}$ for any $t\in I$. From Proposition 4.3, we get $\left\lVert u(t)-u_{n}(t)\right\rVert_{\mathfrak{H}^{1}}\lesssim\left\lVert\psi^{\pm}_{n}-\psi^{\pm}\right\rVert_{L^{2}}\lesssim\left\lVert u_{0,n}-u_{0}\right\rVert_{\mathfrak{H}^{1}}$ for any $t\in I$. Hence, we obtain the desired result. ∎

Acknowledgments

The first author thanks Dr. Ze Li for helpful discussions.