Equivariant Schr\" odinger maps from two dimensional hyperbolic space

arXiv:1705.00260v1 [math.AP] 30 Apr 2017

¨ EQUIVARIANT SCHRODINGER MAPS FROM TWO DIMENSIONAL HYPERBOLIC SPACE JIAXI HUANG, YOUDE WANG, LIFENG ZHAO A BSTRACT. In this article, we consider the equivariant Schr¨odinger map from H2 to S2 which converges to the north pole of S2 at the origin and spatial infinity of the hyperbolic space. If the energy of the data is less than 4π, 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.

C ONTENTS 1. Introduction 2. Preliminaries 2.1. Hyperbolic spaces 2.2. Function spaces and basic inequalities 2.3. Fourier transformation 3. Local well-posedness for Schr¨odinger maps 4. The Coulomb gauge representation of the equation 4.1. The Schr¨odinger maps system in the Coulomb gauge: dynamic equations for ψk 4.2. Recovering the map from ψ + 5. The Cauchy problem 5.1. Srichartz estimates 5.2. The Cauchy theory Acknowledgments References

i

1 4 4 6 8 9 15 17 22 30 30 38 47 48

Huang, Wang, Zhao 1 1. I NTRODUCTION In this article, we consider the Schr¨odinger map equation ∂u = Jτ (u), (1.1) ∂t where u(x, t) ∶ [0, T ] × H2 → S2 , τ (u) is the tension field of u and J is complex structure on S2 . The equation admits the conserved energy 1 E(u) = ∫ ∣du∣2dvolg , 2 H2 where dvolg is the volume form of (H2 , g). The Schr¨odinger maps from Euclidean spaces have been intensely studied in the last decades. The local well-posedness of Schr¨odinger maps was established by Sulem, Sulem and Bardos [29] for S 2 target, Ding and Wang [9, 10] and McGahagan [25] for general K¨ahler manifolds. Ionescu and Kenig [14] obtained the d global well-posedness of maps into S 2 with small data in the critical Besov spaces B˙ 2 (Rd , S 2 ), Q ∈ S 2 for Q

d ≥ 3. The global well-posedness for maps Rd → S 2 , d ≥ 2 with small critical Sobolev norms was obtained by Bejenaru, Ionescu, Kenig and Tataru [3]. However, the Schr¨odinger map equation with large data is a much more dufficult problem. When the target is S 2 , there exists a collection of families Qm (see [6]) of finite energy stationary solutions for integer m ≥ 1; When the target is H2 , 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 Schr¨odinger maps R2 → S2 with energy blow the ground state and equivariant Schr¨odinger maps R2 → H2 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 Schr¨odinger maps have been considered by many authors for instance [6, 11–13, 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 Schr¨odinger maps. D’Ancona and Qidi Zhang [8] showed the global existence of equivariant wave maps from hyperbolic d d spaces Hd for d ≥ 3 to general targets for small initial data in H 2 × H 2 −1 . The problem was also intensely studied by Lawrie, Oh, Shahshahani [17–20] and Li, Ma, Zhao [23]. Since the wave maps H2 → H2 or S2 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 H2 → H2 with initial data (ψ0 , ψ1 ) ∈ Eλ for 0 ≤ λ ≤ Λ by profile decomposition. For initial data without any symmetric assumption, Li, Ma and Zhao [23] proved that the small energy harmonic maps from H2 to H2 are asymptotically stable under the wave map recently. [20] established global well-posedness and scattering for wave maps from Hd for d ≥ 4 into Riemannian manifolds of bounded geometry for small initial data in the critical Sobolev space. As a geometric flow, Schr¨odinger map is a special case of Landau-Lifshitz flow. Li and Zhao [24] proved that the solution of Landau-Lifshitz u(t, x) from H2 to H2 converges to some harmonic map as t → ∞ when the Gilbert coefficient is positive.

Schr¨odinger map from H2 to S2

2

The Schr¨odinger maps on H2 exhibits markedly different phenomena from its Euclidean counterpart. First, the most interesting feature is that there is an abundance of equivariant harmonic maps introduced λ2 by [17]. Precisely, when the target is S2 , there is a family of equivariant harmonic maps with energy 4π 1+λ 2 2 for λ ∈ [0, +∞); When the target is H , we also have a family of equivariant harmonic maps with energy 1+λ2 4π 1−λ 2 for λ ∈ [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 Schr¨odinger map can be reduced to two coupled Schr¨odinger equations. If the support of initial data is contained in a open ball Bǫ (0) for ǫ > 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 Schr¨odinger equations R2 → S2 . Third, the notable feature of the problem is the better dispersive estimates of the operator eit∆H2 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 ○ ρ = ρ ○ u for all rotations ρ ∈ SO(2). Since u is a map H2 → S2 here, in the polar coordinates, u is m-equivariant if and only if u can be written as u(r, θ) = emθR u¯(r). Here R is the generator of horizontal rotations, which is defined as ⎛0 −1 0⎞ R ∶= ⎜1 0 0⎟ , Ru = k⃗ × u. ⎝0 0 0 ⎠

where k⃗ = (0, 0, 1)T . We denote ⃗i ∶= (1, 0, 0)T and ⃗j ∶= (0, 1, 0)T . The energy of m-equivariant maps can be expressed as ∞ m2 E(u) = π ∫ (∣∂r u¯∣2 + (¯ u21 + u¯22)) sinh rdr. 0 sinh2 r If m ≠ 0, then E(u) < ∞ implies that lim u1 = lim u2 = 0. Due to the exponential decay of sinh−1 r, r→0

r→0

1−λ 2 2 2 we assume that lim u1 , lim u2 ≥ 0, which gives lim u3 = 1+λ 2 ≥ 1 for λ ∈ [0, 1) by u1 + u2 + u3 = 1. r→∞ r→∞ r→∞ The equivariant Schr¨odinger map (1.1) admits solitons, which are equivariant harmonic maps u such that u × ∆u = 0. In contrast to the Sch¨odinger maps from Euclidean spaces, the Schr¨odinger maps on H2 admit harmonic maps with any energy E(u) < 4π for S2 target and E(u) < ∞ for H2 target. In fact, for 1−λ2 u ∶ H2 → S2 with endpoint u3 (∞) = 1+λ 2 for λ ∈ [0, ∞), there exists equivariant stationary solution to (1.1) 1 − (λ tanh 2r )2 2λ tanh 2r , 0, ), Qλ = ( 1 + (λ tanh r2 )2 1 + (λ tanh 2r )2 2

1+λ λ 2 2 with energy E(Qλ ) = 4π 1+λ 2 . For u ∶ H → H with endpoint u3 (∞) = 1−λ2 for λ ∈ [0, 1), there exists equivariant stationary solution to (1.1) 2λ tanh 2r 1 + (λ tanh 2r )2 Pλ = ( , 0, ), 1 − (λ tanh 2r )2 1 − (λ tanh 2r )2 2

2

Huang, Wang, Zhao 3 λ with energy E(Pλ ) = 4π 1−λ 2. This leads us to consider the equivariant Schr¨odinger maps in the classes 2

Eλ = {u ∶ H2 → S2 ∣E(u) < ∞, lim u3 = 1, lim u3 = r→0

r→∞

1 − λ2 } , λ ∈ [0, 1), 1 + λ2

but the case λ > 0 is difficult, we will not consider here. Let u ∶ H2 → S2 ⊂ R3 be a smooth map. The Sobolev norm Hk (H2 ; S2 ) are defined by ∥u∥Hk ∶= ∑ ∫ 2

k

i=1

The main results are the following.

H2

∣∇i−1 du∣2g dvolg .

3 Theorem 1.1. If u0 ∈ H3 , then there exists T > 0, such that (1.1) has a unique solution in L∞ t ([0, T ]; H ).

Theorem 1.2. If u0 ∈ H1 is a 1-equivariant map satisfying u0 ∈ E0 and E(u0 ) < 4π, then there exists 1 T > 0, such that (1.1) has a unique solution u ∈ L∞ t ([0, T ]; H ) in the class E0 defined as the unique 3 limit of smooth solution in H . In particular, there exists ǫ > 0 such that E(u0 ) < ǫ, then for any compact 1 interval J ⊂ R, there exists a unique solution u ∈ L∞ t (J; H ).

Remark 1.2. In Theorem 1.2, we restrict the map u in the class E0 for initial data u0 ∈ H1 . To obtain the existence of solutions in H1 , we need to prove the Lipschitz continuity of u with respect to u0 . If the map u ∈ Eλ for λ > 0, then the third component u3 of u does not convergence to 1 as r → ∞, thus the argument of Lipschitz continuity fails. Therefore, we need to restrict u in E0 . 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 Schr¨odinger equations with potentials, i.e (ψ + , ψ − )-system. However, the Schr¨odinger operator admits discrete spectrum in one equation of the system which is in sharp contrast with the Schr¨odinger map from R2 . In fact, we show that the L4 L4 1 bound for ψ ± depends on the compact interval J ⊂ R, which leads to the L∞ t (J; H )-bound for u depends on interval J. Theorem 1.1 and 1.2 is of similar flavor to the result of [4, 25] in the flat domain R2 . The first step is to prove the local existence for Schr¨odinger map with data u0 ∈ H3 by approximation of wave maps (see [25]). The second step is to show the existence for equivariant Schr¨odinger map with data u0 ∈ H1 . Since we restrict ourselves to the class of equivariant Schr¨odinger 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 A1 = 0. Using the Coulomb gauge as our choice of frame on Tu H2 , we can rewrite the equations for ∂r u and ∂θ u which lead to a (ψ + , ψ − )-system of mass-critical Schr¨odinger equations with potentials. Then it suffices to consider the Cauchy problem of the (ψ + , ψ − )-system. In order to establish the well-posedness for data in the space L2 (H2 ), we prove the Strichartz estimates for Schr¨odinger operator with such potentials. In fact, we can get the dispersive estimates for 0 < t < 1 with more general potentials V ≥ 0 and V ∈ e−αr L∞ (H2 ) for α ≥ 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 Schr¨odinger map u from ψ ± , the key observation is that ψ + or ψ − contain all the information of the map as in [4]. Hence, we can recover the map u(t) from ψ ± (t) for initial data R± ψ ± (0) ∈ H 2 . Furthermore by the result in Theorem 1.1, we show that the map u(t) is a Schr¨odinger map for data u0 in H3 . At the same time, we obtain the Lipschitz continuity of u(t) with respect to u0 in H1 , which gives Theorem 1.2.

Schr¨odinger map from H2 to S2

4

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 Schr¨odinger system with potentials. Indeed, the system is composed of two coupled mass-critical Schr¨odinger equations with potentials. One of the equations admits Schr¨odinger operator with positive potential, which has only purely absolutely continuous spectrum [ 14 , ∞). 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 ∥eit(∆H2 −V ) ∥L1 →L∞ ≲ t−1 ,

(1.2)

for nonnegative potential V ∈ e−αr L∞ (H2 ), α ≥ 1. We make use of the kernel of resolvent introduced by [2] frequently. By Birman-Schwinger type resolvent expansion, the resolvent RV can be expressed as a series with respect to R0 , RV and V , then the Schr¨odinger propagator in (1.2) can be written as a series. Since the dominant terms only depend on R0 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 RV in Lemma 5.5. The other equation admits Schr¨odinger operator with negative potential which has at least a discrete spectrum 0 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 Schr¨odinger map (1.1) in H3 , i.e Theorem 1.1. In Sections 4 we introduce the Coulomb gauge, in which the Schr¨odinger map can be written as two coupled Schr¨odinger equations, i.e (ψ + , ψ − )-system. Conversely, if we have ψ + ∈ L2 , we can reconstruct the Schr¨odinger map u. In Sections 5 we provide the Strichartz estimates for operator −∆H2 + V , then we get the well-posedness of (ψ + , ψ − )-system for data ψ0± ∈ L2 . Finally,we finish the proof of Theorem 1.2. 2. P RELIMINARIES

In this section we review the geometry of hyperbolic space and the Fourier transformation. 2.1. Hyperbolic spaces. We consider the Minkowski space Rd+1 for d ≥ 2 with the Minkowski metric (dx1 )2 + ⋯ + (dxd )2 − (dxd+1 )2 , and we can define the bilinear form on Rd+1 × Rd+1 , [x, y] = −x1 y 1 − ⋯ − xd y d + xd+1 y d+1 .

Then hyperbolic space Hd is defined as

Hd = {x ∈ Rd+1 ∶ [x, x] = 1, xd+1 > 0},

and the Riemannian metric g on Hd is induced by the Minkowski metric on Rd+1 . We take the point 0 ∶= (0, ⋯, 0, 1) ∈ Rd+1 as the origin in Hd . We define (G, ○); = (SO(d, 1), ○) as the connected Lie group of (d + 1) × (d + 1) matrices that leave the bilinear form [⋅, ⋅] invariant. We have X ∈ SO(d, 1) if and only if X t ⋅ Id,1 ⋅ X = Id,1 , det X = 1, Xd+1,d+1 > 0,

where Id,1 is the diagonal matrix diag[1, ⋯, 1, −1]. Let K = SO(d) denote the subgroup of SO(d, 1) that fix the origin 0. Indeed, K is a compact subgroup of rotations acting on the variables (x1 , ⋯, xd ). We can

Huang, Wang, Zhao 5 thus identify Hd with the symmetric space G/K. For every h ∈ G we can define the map Lh ∶ Hd Ð→ Hd , Lh (x) = h ⋅ x.

A function f ∶ Hd → R is called K-invariant or radial, if for all k ∈ K and for all x ∈ Hd we have f (k ⋅ x) = f (x).

Then we have the Cartan decomposition of h ∈ G, namely

˜ ar ∈ A+ , k, k˜ ∈ K, h = k ○ ar ○ k, where 0 0 ⎞ ⎛Id−1×d−1 cosh r sinh r ⎟ , A+ ∶= {ar ∶ r ∈ [0, ∞)}. ar ∶= ⎜ 0 ⎝ 0 sinh r cosh r ⎠

We introduce two convenient global systems of coordinates on Hd . One of the systems is geodesic polar coordinates: (2.1)

φ ∶ R+ × Sd−1 → Hd ⊂ Rd+1 , φ(r, ω) = (sinh r ⋅ ω, cosh r).

For d = 2, φ can be written explicitly as

φ ∶ R+ × [0, 2π) → Hd ⊂ Rd+1 , φ(r, θ) = (sinh r cos θ, sinh r sin θ, cosh r),

in these coordinates, the hyperbolic metric g is given by g = dr 2 + sinh2 rdθ2 , the volume element µ(dx) on H2 is given by sinh rdrdθ and the Laplace-Beltrami operator is given by ∆H2 =

∂r2

∂θ2 + coth r∂r + . sinh2 r

The other global system of coordinates is defined as follows [15]: (2.2)

ϕ ∶ Rd−1 × R → Hd , ϕ(v, s) = ( sinh s + e−s

using these coordinates we have the induced metric

∣v∣2 ∣v∣2 −s , e v1 , ⋯ , e−s vd−1 , cosh s + e−s ), 2 2

2 + e2s ds2 ]. g = e−s [dv12 + ⋯ + dvd−1

If we fix the global orthonormal frame

eα = es ∂vα for α = 1, ⋯, d − 1, and ed = ∂s , we compute the commutators [ed , eα ] = eα , [eα , eβ ] = [ed , ed ] = 0 for any α, β = 1, ⋯, d − 1,

and the covariant derivatives

∇eα eβ = δαβ ed , ∇eα ed = −eα , ∇ed eα = ∇ed ed = 0, for α, β = 1, ⋯, d − 1.

Schr¨odinger map from H2 to S2

6

2.2. Function spaces and basic inequalities. Here we define some relevant function spaces on H2 and recall some basic inequalities. For smooth function f ∶ H2 → R, the Lp (H2 )-norm for 1 ≤ p ≤ ∞ are defined by ∥f ∥Lp (H2 ) ∶= (∫

H2

Also we can define the Sobolev norm

∣f (x)∣p dvolg )1/p , ∥f ∥L∞ (H2 ) ∶= sup ∣f (x)∣.

H k (H2 ; R)

x∈H2

of f , namely

∥f ∥H k ∶= ∑ ∫ 2

where

∇l f

(2.3) and (2.4)

l≤k

H2

∣∇l f ∣2 dvolg ,

is the l-th covariant derivative of f . By [20], we have

∥f ∥H 2l ≃ ∥(−∆)l f ∥L2 , for l = 0, 1, 2, ⋯,

∥f ∥H 2l+1 ≃ ∥∇2l+1 f ∥L2 ≃ ∥∇(−∆)l f ∥L2 , for l = 0, 1, 2, ⋯.

We will often use these equivalent definitions. As a R3 -valued function, we can define the extrinsic Sobolev spaces Hk (H2 ; R3 ). We say that u has finite Hk -norm with respect to u(∞) ∶= lim u(r) if r→∞

Denote

u ∈ {u ∶ H2 → S2 ⊂ R3 ∣ui − ui (∞) ∈ H k , for i = 1, 2, 3}. ∥u∥Hk (H2 ;R3 ) ∶= ∑ ∥ui − ui (∞)∥H k 3

i=1

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 H˙ e1 by ∥f ∥H˙ e1 ∶= ∥∂r f ∥L2 + ∥

then for such f , we have Sobolev embedding (2.5)

f ∥ , sinh r L2

∥f ∥L∞ ≲ ∥f ∥H˙ e1 .

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

Lemma 2.1. Let f ∈ Cc∞ , then for 1 < p < ∞, p ≤ q ≤ ∞, 0 < θ < 1, 1 < r < 2, r ≤ l < ∞, the following inequalities hold: (2.6) (2.7) (2.8) (2.9) (2.10)

∥f ∥L2 ≲ ∥∇f ∥L2 ,

∥f ∥Lq ≲ ∥f ∥Lp ∥∇f ∥L2 , for θ

1−θ

∥f ∥Ll ≲ ∥∇f ∥Lr ,

∥f ∥L∞ ≲ ∥(−∆) 2 f ∥L2 , α

∥∇f ∥L2 ∼ ∥(−∆) 2 f ∥ 1

L2

1 1 θ = − , q p 2 1 1 1 for = − , l r 2 for α > 1,

.

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

Huang, Wang, Zhao 7 Lemma 2.2. If T is some (r, s)-type tension or tension matrix defined on H2 , then in the distribution sense, one has ∣∇∣T ∣∣ ≤ ∣∇T ∣.

(2.11)

Lemma 2.3. Let u ∶ H2 → S2 be smooth map with u(∞) = lim u(r), then r→∞

∥u∥H3 ∼ ∥u∥H3

(2.12)

in the sense that there exist polynomials P and Q such that (2.13)

∥u∥H3 ≤ P (∥u∥H3 ), ∥u∥H3 ≤ Q(∥u∥H3 ).

Proof. In order to prove (2.13), we use the polar coordinates (2.1). For k = 1, we have 3 1 2 ∣du∣ = ⟨∂r u, ∂r u⟩ + 2 ⟨∂θ u, ∂θ u⟩ = ∑ ∣dui ∣ , sinh r i=1 2

hence, ∥u∥H1 ≤ ∥u∥H1 . Conversely, by (2.6), we obtain

∥u∥H1 ≲ ∥u∥H1 + ∑ ∥ui − ui (∞)∥L2 , 3

For k = 2, we have and

≲ ∥u∥H1 .

i=1

∣∇du∣2 = g ii g jj [∣∂ij u − Γkij ∂k u∣2 − (∂ij u ⋅ u)2 ], ∑ ∣∇2 ui ∣2 = g ii g jj ∣∂ij u − Γkij ∂k u∣2 . 3

(2.14)

i=1

Therefore, ∥u∥H2 ≲ ∥u∥H2 immediately. Conversely, ∂j u ⋅ u = 0 implies

g ii g jj (∂ij u ⋅ u)2 = g ii g jj (∂i u∂j u)2 = ∣du∣4,

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

∥u∥H2 ≲ ∥u∥H2 + ∥du∥L4 , 2

2

4

≲ ∥u∥H2 + ∥∇∣du∣∥L2 ∥du∥L2 , 2

2

2

≲ ∥u∥H2 + ∥∇du∥L2 ∥du∥L2 , 2

2

≲ ∥u∥H2 + ∥u∥H2 . 2

For k = 3, we have

2

4

∣∇2 du∣2 =g ii g jj g kk ∣∂ijk u − ∂i (Γljk ∂l u) − Γlij ∂lk u + Γlij Γplk ∂p u

− Γlik ∂jl u + Γlik Γpjl ∂p u + 3(∂ij u∂k u − Γlij ∂l u∂k u)u∣ ,

and

∣∇3 uq ∣2 =g ii g jj g kk ∣∂ijk uq − ∂i (Γljk ∂l uq ) − Γlij ∂lk up + Γlij Γplk ∂p uq − Γlik ∂jl uq + Γlik Γpjl ∂p uq ∣ . 2

2

Schr¨odinger map from H2 to S2

8 By (2.14), we have

g g g ∣(∂ij u∂k u − Γlij ∂l u∂k u)u∣ ≲ ∑ ∣∇2 ui ∣2 ∣du∣2. 3

2

ii jj kk

i=1

Then

∥u∥H3 ≲ ∑ ∥∇3 ui ∥L2 + ∑ ∥∇2 ui ∥L4 ∥du∥L4 , 3

2

3

2

2

2

i=1

i=1

≲ ∥u∥H3 + ∑ ∥∇3 ui ∥L2 ∥∇2 ui ∥L2 ∥∇du∥L2 ∥du∥L2 , 3

2

2 ≲ ∥u∥H3

i=1

+ ∥u∥H3 . 4

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

∥u∥H3 ≲ ∥u∥H3 + ∑ ∥∣∇2 ui ∣∣du∣∥L2 , 2

2

2 ≲ ∥u∥H3 2 ≲ ∥u∥H3 2 ≲ ∥u∥H3 2 ≲ ∥u∥H3

Therefore, (2.13) are obtained.

3

2

i=1

+ ∥∣∇du∣∣du∣ + ∣du∣3∥L2 , 2

+ ∥∇du∥L4 ∥du∥L4 + ∥du∥L6 , 2

2

6

+ ∥∇2 du∥L2 ∥∇du∥L2 ∥du∥L2 + ∥∇du∥L2 ∥du∥L2 , 4

2

2

+ ∥u∥H3 + ∥u∥H3 . 4

6



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 ∈ Lp be radial function, we have ∥

(2.15) (2.16) (2.17) (2.18) (2.19) (2.20)

∥

r 1 ∫ sinh sf (s)ds∥ p sinh2 r 0 L r cosh r ∥ ∫ sinh sf (s)ds∥ p sinh2 r 0 L ∞ cosh s f (s)∥ ∥∫ sinh s r Lp ∞ 1 ∥∫ f (s)∥ sinh s r Lp

≲ ∥f ∥Lp , 1 < p ≤ ∞, ≲ ∥f ∥Lp , 1 < p ≤ ∞,

≲ ∥f ∥Lp , 1 ≤ p < ∞,

≲ ∥f ∥Lp , 1 ≤ p < ∞,

∞ r −1 1 e∫ρ sinh sds f (ρ)dρ∥ ≲ ∥f ∥Lp , 1 ≤ p < ∞, ∫ sinh r r Lp r 1 ≲ ∥f ∥Lp (R2 ) , 1 < p ≤ ∞, ∥ 2 ∫ f (s)sds∥ r 0 Lp (R2 )

2.3. Fourier transformation. For ω ∈ Sd−1 and λ a real number, the functions of the type hλ,ω ∶ Hd Ð→ C, x → [x, b(ω)]iλ−

n−1 2

,

Huang, Wang, Zhao 9 are generalized eigenfunctions of the Laplacian-Beltrami operator. Indeed, we have

(n − 1)2 )hλ,ω . 4 Then we can define the Fourier transformation analogous to the Euclidean case. For f ∈ Cc∞ (Hd ), −∆Hd hλ,ω = (λ2 +

f̂(λ, ω) = ∫

f (x)hλ,ω (x)dvolg ,

Hd

and one has the Fourier inversion formula for function on Hd f (x) = ∫

+∞

hλ,ω (x)f̂(λ, ω)

∫

Sd−1

−∞

where c(λ) is the Harish-Chandra coefficient, c(λ) = C

dλdω , ∣c(λ)∣2

Γ(iλ) . + iλ)

Γ( d−1 2

For the linear Schrodinger equation on Hd , {

i∂t u + ∆Hd u = 0,

u(0) = u0 ∈ L2 (Hd ).

the solution can be written explicitly see [2] as u(t, x) = C exp−it(

d−1 2 ) 2

∫

Hd

where the kernel K d is, for ρ > 0 and d ≥ 3 odd

and for d ≥ 2 even,

K d (t, ρ) = ∫

K d (t, ρ) = ∫

+∞ −∞

e−itλ ∫

+∞ −∞

+∞

2

ρ

In particular, d = 2,

K (t, ρ) = 2

3. L OCAL

c

∣t∣ 2

3

e−itλ ( 2

u0 (y)K d (t, d(x, y))dy,

∂ρ d−1 ) 2 cos(λρ)dλ. sinh ρ

sinh s ∂s d ) 2 cos(λs)dsdλ. √ ( cosh s − cosh ρ sinh s ∫

+∞ ρ

s2

ei 4t s √ ds. cosh s − cosh ρ

WELL - POSEDNESS FOR

¨ S CHR ODINGER

MAPS

In order to prove the local well-posedness in H3 , we apply the approximating Scheme introduced by McGahagan [25]. For any δ > 0, we introduce the wave map model equation: (3.1)

{

δ 2 ∇t ∂t u − J∂t u − τ (u) = 0,

u(0, x) = u0 , ∂t u(0, x) = g0δ .

where u(t, x) ∶ [0, T ] × H2 → S2 and g0δ ∈ Tu0 (x) S2 . In this section we use the global coordinates (2.2), denote ∇i = ∇ei for i = 1, 2. For simplicity, denote u ∶= uδ . Before proving the Theorem 1.1, we need the following lemma.

Schr¨odinger map from H2 to S2

10

Lemma 3.1. For T˜ > 0, there exists a constant C > 0 independent of δ, such that for any uδ ∶ [0, T˜ ]×H2 → S2 , uδ ∈ C([0, T ]; H3 ), a solution of the approximate equation, and any 1 ≤ k ≤ 2, the following estimate holds for uδ : ∥∇k−1 ∂t uδ ∥C([0,T ];L2 ) ≤ C ∥∂t uδ (0)∥Hk−1 + C ∥uδ ∥C([0,T ];Hk+1 )

for some T > 0, depending only on the size of the solution ∥uδ ∥C([0,T˜];Hk+1 ) and on the size of the initial data ∥∂t uδ (0)∥H1 .

Proof. For k = 1, we take the inner product of the above wave map equation with J(u)∇t ∂t u, the first term will disappear by orthogonality, we get ∫H2 ⟨∂t u, ∇t ∂t u⟩ = ∫H2 ⟨Jτ (u), ∇t ∂t u⟩,

(3.2)

d ⟨Jτ (u), ∂t u⟩ − ∫ ⟨J∇t τ (u), ∂t u⟩ dt ∫H2 H2 In the system of coordinate, τ (u) can be written as τ (u) = ∇i ei (u) − (∇i ei )(u), then commute ∇i and ∇t , by integration by parts, the second term of (3.3) becomes =

(3.3)

∫ 2 ⟨J∇t τ (u), ∂t u⟩ = ∫ 2 ⟨J∇t (∇i ei (u) − (∇i ei )(u)), ∂t u⟩, H

H

= ∫ ⟨J[∇t , ∇i ]ei u + J∇i ∇t ei u − J∇t e2 u, ∂t u⟩, H2 = ∫ ⟨J[∇t , ∇i ]ei u, ∂t u⟩ + ∫ 2 H

H2

ei ⟨J∇i ∂t u, ∂t u⟩

− ∫ ⟨J∇i ∂t u, ∇i ∂t u⟩ − ∫ ⟨J∇2 ∂t u, ∂t u⟩, H2

H2

= ∫ ⟨J[∇t , ∇i ]ei u, ∂t u⟩ + ∫ ⟨J∇2 ∂t u, ∂t u⟩ − ∫ ⟨J∇2 ∂t u, ∂t u⟩, 2 2 2 H

H

H

= ∫ ⟨J[∇t , ∇i ]ei u, ∂t u⟩. 2 H

If we integrate in time,by H¨older inequality we find that (3.2) becomes 1 1 2 2 ∥∂t u(t)∥L2 = ∥∂t u(0)∥L2 + ∫ ⟨Jτ (u), ∂t u⟩(t) − ∫ ⟨Jτ (u), ∂t u⟩(0) 2 2 H2 H2 −∫

t

0

∫ 2 ⟨J[∇t , ∇i ]ei u, ∂t u⟩, H

1 2 ≤ ∥∂t u(0)∥L2 + ∥τ (u)(t)∥L2 ∥∂t u(t)∥L2 + ∥τ (u)(0)∥L2 ∥∂t u(0)∥L2 2 ∫0 ∥∂t u∥L2 ∥d(u)∥L∞ ds, t

2

2

≤ C ∥∂t u(0)∥L2 + C ∥τ (u)(t)∥L2 + C ∥τ (u)(0)∥L2 2

+∫

then

t

0

2

∥∂t u∥L2 ∥d(u)∥L∞ ds, 2

2

∥∂t u(t)∥L2 ≲ ∥∂t u(0)∥L2 + ∥τ (u)∥C([0,T ];L2 ) + ∫ 2

2

2

t

2

0

∥∂t u∥L2 ∥∂u∥C([0,T ];H 2 ) ds. 2

2

Huang, Wang, Zhao 11 Therefore, by Gronwall inequality, choose T such that ∥du∥C(0,T˜;H 2 ) T small, we have ∥∂t u(t)∥L2 ≲ (∥∂t u(0)∥L2 + ∥τ (u)∥C([0,T ];L2 ) )(1 + ∥∂u∥C([0,T ];H 2 ) ). 2

(3.4)

2

2

2

For k = 2, we take ∇i on the approximate equation (3.1):

δ 2 ∇i ∇t ∂t u − J∇i ∂t u − ∇i τ (u) = 0,

then we take the inner product of the above equation with J(u)∇i ∇t ∂t u and commute ∇i and ∇t , we have 0 = ∫ ⟨J∇i ∂t u, J∇i ∇t ∂t u⟩dvolg − ∫ ⟨J∇i τ, ∇i ∇t ∂t u⟩dvolg , H2

(3.5)

=

H2

1d d 2 ∥∇i ∂t u∥L2 + ∫ ⟨∇i ∂t u, [∇i , ∇t ]∂t u⟩dvolg − ∫ ⟨J∇i τ, ∇i ∂t u⟩dvolg 2 2 dt dt H2 H + ∫ ⟨J∇t ∇i τ, ∇i ∂t u⟩dvolg − ∫ ⟨J∇i τ, [∇i , ∇t ]∂t u⟩dvolg . H2

Denote

H2

I = ∫ ⟨∇i ∂t u, [∇i , ∇t ]∂t u⟩dvolg , II = ∫ ⟨J∇t ∇i τ, ∇i ∂t u⟩dvolg , H2

(3.6)

H2

III = ∫ ⟨J∇i τ, [∇i , ∇t ]∂t u⟩dvolg . H2

Then II can be rewritten as

II = ∫ ⟨J[∇t , ∇i ]τ, ∇i ∂t u⟩ + ei ⟨J∇t τ, ∇i ∂t u⟩ − ⟨J∇t τ, ∇i ∇i ∂t u⟩dvolg , ≜II1 + II2 + II3 ,

by the representation of τ (u) and (∇j ej )u = e2 u, II2 becomes II2 = − ∫

H2

e2 ⟨∇t τ, J∇i ∂t u⟩dvolg ,

= − ∫ ⟨∇t (∇ej ej u − (∇ej ej )u), J∇t e2 u⟩dvolg , H2

= − ∫ ⟨∇j ∇t ej u, J∇t e2 u⟩ + ⟨[∇t , ∇j ]ej u, J∇t e2 u⟩dvolg , H2

for II3 , by integration by parts, we have

II3 = − ∫ ⟨J[∇t , ∇j ]ej u, ∇i ∇i ∂t u⟩ + ⟨J∇j ∇j ∂t u, ∇i ∇i ∂t u⟩ − ⟨J∇t e2 u, ∇i ∇i ∂t u⟩dvolg , H2

=−∫

H2

= ∫ ⟨[∇t , ∇j ]ej u, J∇t e2 u⟩ + ⟨J∇i [∇t , ∇j ]ej u, ∇i ∂t u⟩ + ⟨J∇t e2 u, ∇i ∇i ∂t u⟩dvolg . H2

Hence,

ei ⟨J[∇t , ∇j ]ej u, ∇i ∂t u⟩ − ⟨J∇i [∇t , ∇j ]ej u, ∇i ∂t u⟩ − ⟨J∇t e2 u, ∇i ∇i ∂t u⟩dvolg , II = ∫ ⟨J[∇t , ∇i ]τ, ∇i ∂t u⟩ + ⟨J∇i [∇t , ∇j ]ej u, ∇i ∂t u⟩dvolg . H2

Then (3.5) can be written as

d 1d 2 ∥∇∂t u∥L2 = ∫ ⟨J∇i τ, ∇i ∂t u⟩ − I − II + III. 2 dt dt H2

Schr¨odinger map from H2 to S2

12

Integrating in time, by H¨older inequality, it gives 1 1 2 2 2 ∥∇∂t u(t)∥L2 ≤ ∥∇∂t u(0)∥L2 + ∥∇τ (t)∥L2 ∥∇∂t u(t)∥L2 2 2 + ∥∇τ (0)∥L2 ∥∇∂t u(0)∥L2 + ∫

t

2

then

0

−I − II + III ds,

∥∇∂t u(t)∥L2 ≲ ∥∇∂t u(0)∥L2 + ∥∇τ (t)∥L2 + ∥∇τ (0)∥L2 + ∫ ∣ − I − II + III∣ds. 2

2

2

t

2

0

From (3.6), H¨older inequality and Lemma 2.1, we have

∣ − I − II + III∣ ≤ ∥∇∂t u∥L2 ( ∥[∇t , ∇i ]τ ∥L2 + ∥∇i [∇t , ∇j ]ej u∥L2 ) + ( ∥∂u∥H 2 + ∥∇∂t u∥L2 ) ∥[∇, ∇t ]∂t u∥L2 ,

≤ ∥∇∂t u∥L2 ( ∥∂t u∥L4 ∥τ (u)∥L4 ∥du∥L∞ + ∥du∥L∞ ∥∂t u∥L2 ∥du∥L∞ ∥∇∂t u∥L2 3

2

+ ∥∂t u∥L4 ∥du∥L∞ ∥∇2 u∥L4 ) + ( ∥∂u∥H 2 + ∥∇∂t u∥L2 ) ∥∂t u∥L4 ∥∂u∥H 2 , 2

≤ ∥∇∂t u∥L2 (∥∇∂t u∥L2 + ∥∂t u∥L2 )(∥∂u∥H 2 + ∥∂u∥H 2 ) + ∥∇∂t u∥L2 ∥∂t u∥L2 ∥∂u∥H 2 , 3

2

≤ ∥∇∂t u∥L2 (∥∂t u∥L2 ∥∂u∥H 2 + ∥∂u∥H 2 + ∥∂u∥H 2 ) + ∥∂t u∥L2 (∥∂u∥H 2 + ∥∂u∥H 2 ). 2

By (3.4),

2

2

3

2

2

3

∣ − I − II + III∣ ≤ ∥∇∂t u∥2L2 (∥∂t u(0)∥L2 + ∥∂u∥C(H 2 ) )(∥∂u∥C(H 2 ) + ∥∂u∥2C(H 2 ) ) +(∥∂t u(0)∥L2 + ∥∂u∥C(H 2 ) )P5 (∥∂u∥C(H 2 ) ). 2

By Gronwall inequality, we get

2

∥∇∂t u∥L2 ≲ P (∥g0 ∥H 1 )Q(∥u∥C(H3 ) ).



Proof of Theorem 1.1. We choose data g0δ such that ∥g0δ ∥H 1 < C and δ 2 ∥g0δ ∥H 2 < C. Without any restriction we make the bootstrap assumption (3.7)

2

∥u∥C([0,T ];H2 ) + ∥∇τ (u)∥C([0,T ];L2 ) ≤ 2C(∥u(0)∥H2 , ∥∇τ (u(0))∥L2 ).

Define the energy functional by

δ2 1 2 2 ∥du∥L2 + ∥∂t u∥L2 , 2 2 d then by (3.1), we have dt E1 = 0. Define the second order energy functional by E1 (u, ∂t u) =

E2 (u, ∂t u) =

1 δ2 2 2 ∥∇du∥L2 + ∥∇∂t u∥L2 , 2 2

by (3.1) we have 1d d 2 E2 = ∥∇du∥L2 + δ 2 ∫ ⟨∇i ∇t ∂t u, ∇i ∂t u⟩ + ⟨[∇t , ∇i ]∂t u, ∇i ∂t u⟩dvolg dt 2 dt H2 1d 2 (3.8) = ∥∇du∥L2 + ∫ ⟨J∇i ∂t u + ∇i τ (u), ∇i ∂t u⟩ + δ 2 ⟨[∇t , ∇i ]∂t u, ∇i ∂t u⟩dvolg , 2 dt H2

Huang, Wang, Zhao 13 by integration by parts and ⟨JX, X⟩ = 0, the second term of (3.8) becomes ∫H2 ⟨∇i τ (u), ∇i ∂t u⟩dvolg

= ∫ ⟨∇i (∇j ej u − (∇j ej )u), ∇i ∂t u⟩dvolg , H2

= ∫ ⟨[∇i , ∇j ]ej u, ∇i ∂t u⟩ − ⟨∇i ej u, [∇j , ∇t ]ei u⟩ − ⟨∇i ej u, ∇t ∇j ei u⟩dvolg ,

(3.9)

H2

furthermore, the last term of (3.9) becomes ∫H2 −⟨∇i ej u, ∇t ∇j ei u⟩dvolg

−⟨∇i ej u, ∇t [∇j , ∇i ]u⟩ − ⟨∇i ∇j u − (∇i ej )u, ∇t (∇i ej u − (∇i ej )u)⟩

=∫

H2

− ⟨∇i ej u, ∇t (∇i ej )u⟩ − ⟨(∇i ej )u, ∇t ∇i ej u⟩ + ⟨∇i ej u, ∇t (∇i ej )u⟩dvolg 1d 2 =− ∥∇du∥L2 + ∫ −⟨∇i ej u, ∇t [∇j , ∇i ]u + ∇t (∇i ej )u⟩ − ⟨(∇i ej )u, [∇t , ∇i ]ej u⟩ 2 dt H2 − ⟨(∇2 ej )u, ∇t ej u⟩ + ⟨∇i (∇i ej )u, ∇t ej u⟩ + ⟨∇i ej u, ∇t (∇i ej )u⟩dvolg .

Hence, by (2.9) and H¨older inequality we have

d 2 E2 ≲ (∥∂t u∥L2 + ∥∇∂t u∥L2 )P (∥u∥H3 ) + δ 2 ∥∇∂t u∥L2 ∥∂t u∥L2 ∥u∥H3 . dt Define the third order energy functional by E3 = By integration by parts gives

δ2 1 2 2 ∥∇τ (u)∥L2 + ∥∇2 ∂t u∥L2 . 2 2

d E3 dt

(3.10) (3.11)

≲∫

H2

+∫

δ 2 (∣∂t u∣∣du∣∣∇∂tu∣ + ∣du∣2∣∂t u∣2 + ∣∂t u∣2 ∣∇2 u∣ + ∣du∣∣∂t u∣2)(∣∇2 ∂t u∣ + ∣∇∂t u∣)dvolg

H2

∣du∣2∣τ (u)∣∣∇∂t u∣ + ∣∇τ (u)∣(∣∂t u∣∣du∣2 + ∣∇∂t u∣)

+ ∣∇τ (u)∣(∣du∣3 ∣∂t u∣ + ∣du∣2∣∇∂t u∣ + ∣∇2 u∣∣du∣∣∂tu∣)dvolg

By (2.7) and H¨older inequality, we have

(3.10) ≲δ 2 (∥∇2 ∂t u∥L2 + ∥∇∂t u∥L2 )(∥∂t u∥L4 ∥∇∂t u∥L4 ∥∂u∥H 2 + ∥∂t u∥L4 ∥∂u∥H 2 2

+ ∥∂t u∥L8 ∥∇2 u∥L4 + ∥∂t u∥L4 ∥∂u∥H 2 ) 2

2

≲δ 2 (∥∇2 ∂t u∥L2 + ∥∇∂t u∥L2 )(∥∇2 ∂t u∥L2 ∥∇∂t u∥L2 ∥∂t u∥L2 ∥u∥H3 1/2

1/2

+ ∥∇∂t u∥L2 ∥∂t u∥L2 ∥∂u∥H 2 + ∥∇∂t u∥L2 ∥∂t u∥L2 ∥∂u∥H 2 2

+ ∥∇∂t u∥L2 ∥∂t u∥L2 ∥∂u∥H 2 ),

≲δ 2 ∥∂t u∥H 2 ∥∇∂t u∥L2 P (∥u∥H3 ). 2

3/2

1/2

2

Schr¨odinger map from H2 to S2

14 Similarly, we have

(3.11) ≲(∥∇∂t u∥L2 + ∥∂t u∥L2 )P (∥u∥H3 ).

Hence,

d (E1 + E2 + E3 ) dt 2 ≤Cδ 2 ∥∂t u∥H 2 ∥∇∂t u∥L2 P (∥u∥H3 ) + C(∥∇∂t u∥L2 + ∥∂t u∥L2 )P (∥u∥H3 ).

(3.12)

Since we have by integration by parts

∥∇2 du∥L2 ≲ ∥∇τ (u)∥2L2 + ∥du∥6L6 + ∥∇du∥2L4 ∥du∥2L4 + ∥∇2 du∥L2 , 2

2

then by (2.7) we obtain

∥∇2 du∥L2 ≲ P (∥u∥H2 ) + ∥∇τ (u)∥L2 . 2

(3.13)

2

2

Thus, integrating (3.12) in time and taking the supremum over t ∈ [0, T ] , we have δ 2 ∥∂t u∥C(H 2 ) + ∥u∥C(H 2 ) + ∥∇τ (u)∥C(L2 ) 2

2

2

≤δ 2 ∥g(0)∥H 2 + ∥u(0)∥H 2 + ∥∇τ (u(0))∥L2 2

(3.14)

2

2

+ CT δ 2 ∥∂t u∥C(H 2 ) P (∥u∥C(H2 ) + ∥∇τ (u)∥C(L2 ) ) + CT Q(∥u∥C(H2 ) + ∥∇τ (u)∥C(L2 ) ). 2

Choosing T small such that CT P (∥u∥C([0,T˜];H2 ) + ∥∇τ (u)∥C([0,T˜];L2 ) ) < 21 , from (3.14) we have δ2 2 2 ∥∂t u∥C(H 2 ) − δ 2 ∥g0 ∥H 2 2 2 2 ≤ ∥u(0)∥H2 + ∥∇τ (u(0))∥L2 + CT Q(∥u∥C(H2 ) + ∥∇τ (u)∥C(L2 ) ). ∥u∥C(H 2 ) + ∥∇τ (u)∥C(L2 ) + 2

(3.15)

2

If ∥∂t u∥C(H 2 ) ≥ 2 ∥g0 ∥H 2 , (3.15) implies 2

2

∥u∥C(H 2 ) + ∥∇τ (u)∥C(L2 ) ≤ ∥u(0)∥H2 + ∥∇τ (u(0))∥L2 + CT Q(∥u∥C(H2 ) + ∥∇τ (u)∥C(L2 ) ). 2

2

2

2

If ∥∂t u∥C(H 2 ) < 2 ∥g0 ∥H 2 , from (3.14) we obtain 2

2

∥u∥C(H 2 ) + ∥∇τ (u)∥C(L2 ) ≤ C + ∥u(0)∥H2 + ∥∇τ (u(0))∥L2 + 3CT Q(∥u∥C(H2 ) + ∥∇τ (u)∥C(L2 ) ). 2

2

2

2

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

Therefore, by (3.13) we have

3 ∥u∥C(H 2 ) + ∥∇τ (u)∥C(L2 ) < C. 2 ∥u∥C(H3 ) ≤ C(∥u(0)∥H3 )

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



Huang, Wang, Zhao 15 4. T HE C OULOMB

GAUGE REPRESENTATION OF THE EQUATION

In this section, we rewrite the equivariant Schr¨odinger map in the Coulomb gauge, then obtain the (ψ + , ψ − )-system of coupled Schr¨odinger equations. Conversely, we can recover the map u from ψ + or ψ − at fixed time. We choose v ∈ Tu H2 such that v ⋅ v = 1 and define w = u × v. Thus w ⋅ w = 1, w ⋅ u = w ⋅ v = 0, w × u = v, v × w = u.

Since u is 1-equivariant it is natural to work with 1-equivariant frame, that is v = eθR v¯(r), w = eθR w(r), ¯

where v¯, w¯ are unit symmetric vectors in H2 . On one hand in such a frame we obtain the differentiated fields ψk and the connection coefficients Ak , by ψk = ∂k u ⋅ v + i∂k u ⋅ w, Ak = ∂k v ⋅ w.

On the other hand, given ψk and Ak we can return to the frame (u, v, w) via the ODE system: ⎧ ∂ u = (Rψk )v + (Iψk )w, ⎪ ⎪ ⎪ k (4.1) ⎨ ∂k v = −(Rψk )u + Ak w, ⎪ ⎪ ⎪ ⎩ ∂k w = −(Iψk )u − Ak v. If we introduce the covariant differentiation

Dk = ∂k + iAk , k ∈ {0, 1, 2},

then the compatibility conditions are imposed

Dk ψl = Dl ψk , l, k ∈ {0, 1, 2}.

(4.2)

Moreover, the curvature of this connection is given by

Dk Dl − Dl Dk = i(∂k Al − ∂l Ak ) = iI(ψk ψ¯l ).

(4.3)

An important geometric feature is that ψ2 , A2 are closely related to the original map. Precisely, for A2 we have A2 = (−v2 , v1 , 0) ⋅ (w1 , w2 , w3 ) = u3 , and ψ2 = w3 − iv3 . Hence we obtain ∣ψ2 ∣2 = u21 + u22 , and the following important conservation law A22 + ∣ψ2 ∣2 = 1.

We now turn to choose the orthonormal frame (¯ v , w) ¯ on S2 . For the equivariant Schr¨odinger map, we ∂2 use the Coulomb gauge divA = 0, namely, in the polar coordinate, ∂r A1 + sinhθ2 r A2 = 0. Since A2 = u3 is radial, we can choose A1 = 0, i.e ∂r v¯ ⋅ w¯ = 0, which can be represented as ODE (4.4)

Then for matrix U = (¯ u, v¯, w), ¯ we have

where M = −u ⋅ ∂r

uT

+ ∂r

u ⋅ uT

∂r v¯ = −(¯ v ⋅ ∂r u¯)¯ u.

∂r U = M ⋅ U, is an antisymmetric matrix.

Schr¨odinger map from H2 to S2

16

The ODE (4.4) need to be initialized at some point. To avoid introducing a constant time-dependent potential into the equation via A0 , we need to choose this initialization uniformly with respect to t. Since we restrict the data lim u¯(r, t) = k⃗ for any t, we can fix the choice of v¯ and w¯ at infinity, r→∞

lim v¯(r, t) = ⃗i,

(4.5)

r→∞

lim w(r, ¯ t) = ⃗j.

r→∞

The existence and uniqueness of (4.4) satisfying (4.5) is standard. Indeed, for u ∈ H1 , using the Picard iteration scheme v¯ = ∑ v¯i , v¯0 = v¯(∞) = ⃗i, v¯i (r) = ∫ ∞

r

M(s)¯ vi−1 ds.

∞

i=0

By H¨older inequality, we have

∥¯ vi (r)∥C([R,∞)) ≤ ∥∂r u¯∥L2 (R,∞) ∥

1 ∥ ∥¯ vi−1 (r)∥C(R,∞) , sinh r L2 (R,∞) ≲ ∥u∥H1 ∥¯ vi−1 (r)∥C(R,∞) ,

and

∥∂r v¯i ∥L2 (R,∞) ≤ ∥¯ vi−1 ∥C(R,∞) ∥u∥H1 .

vi (r)∥C([R,∞)) ≤ ǫi ∥¯ v0 ∥C([R,∞)) . Hence, we choose R large enough such that ∥¯ u∥H˙ e1 ([R,∞)) < ǫ, we have ∥¯ 1 there exists unique solution ∥¯ v∥C(R,∞) + ∥∂r v¯∥L2 (R,∞) ≲ ∥u∥H1 . Then by u ∈ H , in a similar argument, for any ǫ > 0,there exists δ > 0 sufficiently small, such that ∥u∥H1 (R−δ,R) ∥sinh−1 r∥L2 (R−δ,R) ≪ 1 for R − δ > ǫ, the solution can be extended to r = ǫ. Finally, we extend the solution to r = 0. The first two components of v¯i can be estimated immediately ∥¯ vi1,2 ∥C(0,ǫ) ≲ ∥¯ vi−1 ∥C(0,ǫ) ∥u∥H1 ,

for the third component of v¯i , by integration by parts and u¯(r) → k⃗ as r → 0, we have ∥¯ vi3 ∥C(0,ǫ) = ∥∫

and

r

ǫ

⃗ u3 ) − ∂r v¯i−1 ⋅ (¯ ⃗ u3 − v¯i−1 ⋅ (¯ ⃗ r u¯3 ds∥ ∂r (¯ vi−1 ⋅ (¯ u − k)¯ u − k)¯ u − k)∂

⃗ ≲ (∥∂r v¯i−1 ∥L2 (0,ǫ) + ∥¯ vi−1 ∥C(0,ǫ) )(∥u∥H1 (0,ǫ) + ∥¯ u − k∥ ), C(0,ǫ) ∥∂r v¯i ∥L2 (0,ǫ) ≲ ∥¯ vi−1 ∥C(0,ǫ) ∥u∥H1 (0,ǫ) .

,

C(0,ǫ)

⃗ we choose ǫ small such that ∥u∥H1 (0,ǫ) + ∥¯ u − k∥ ≪ 1, then the iteration scheme gives the unique C(0,ǫ) ⃗ ) ∥¯ v0 ∥C(0,ǫ) . Therefore, by the solution in (0, ǫ) with ∥¯ v ∥C(0,ǫ) + ∥∂r v¯∥L2 (0,ǫ) ≲ (∥u∥H1 (0,ǫ) + ∥¯ u − k∥ C(0,ǫ) above procedure, there exists a unique solution of (4.4) satisfying (4.5), moreover, we have (4.6)

∥¯ v ∥C(0,∞) + ∥∂r v¯∥L2 (0,∞) ≲ ∥u∥H1 (0,∞) .

Huang, Wang, Zhao 17 4.1. The Schr¨odinger maps system in the Coulomb gauge: dynamic equations for ψk . We derive the Schr¨odinger equations for the differentiated fields ψ1 and ψ2 . In the geodesic polar coordinate, the Schr¨odinger map flow can be written as 1 D2 ψ2 ). sinh2 r Applying the operators D1 and D2 to both sides of this equation, we obtain ⎧ ψ1 2 cosh r 1 ⎪ ⎪ D1 D2 ψ2 − − D1 ψ0 =i(D1 D1 ψ1 + coth rD1 ψ1 + ⎪ 2 2 3 D2 ψ2 ), ⎪ ⎪ sinh r sinh r sinh r (4.8) ⎨ D D ψ ⎪ 2 2 2 ⎪ D ψ =i(D D ψ + coth rD ψ + ⎪ ). ⎪ 2 0 2 1 1 2 1 ⎪ ⎩ sinh2 r By the compatibility condition (4.2), curvature of the connection (4.3) and the Coulomb gauge A1 = 0, we can derive the equations for ψ1 and ψ2 , ⎧ A22 1 2i cosh rA2 iI(ψ1 ψ¯2 )ψ2 ⎪ ⎪ + )ψ + ψ − , (i∂ + ∆)ψ =(A + ⎪ 1 2 t 1 0 ⎪ ⎪ sinh2 r sinh2 r sinh3 r sinh2 r (4.9) ⎨ ⎪ A22 ⎪ ⎪ ⎪ (i∂ + ∆)ψ =(A + )ψ1 + iI(ψ1 ψ¯2 )ψ1 . 2 0 ⎪ ⎩ t sinh2 r ψ0 = i(D1 ψ1 + coth rψ1 +

(4.7)

where ∆ ∶= ∂rr + coth r∂r . Then (4.9) can be written as ⎧ A22 − 1 2i cosh r ψ2 2 ⎪ ⎪ ⎪ )ψ − )ψ1 =(A + (i∂ 1 0 t +∆− ⎪ ⎪ ⎪ sinh2 r sinh2 r sinh r sinh2 r ⎪ ⎪ ⎪ ⎪ ψ2 2i cosh r(A2 − 1) ψ¯2 ⎪ ⎪ ⎪ )) , + ( − iI(ψ ⎪ 1 2 ⎪ ⎪ sinh r sinh r sinh r (4.10) ⎨ ⎪ A22 − 1 ψ2 2 2i cosh r ψ2 ⎪ ⎪ ⎪ (i∂ + ∆ − + ) ψ =(A + t 1 0 ⎪ 2 2 2 ) ⎪ sinh r ⎪ sinh r sinh r sinh r sinh r ⎪ ⎪ ⎪ ⎪ 2i cosh r(A2 − 1) ψ¯2 ⎪ ⎪ ⎪ −( − iI(ψ1 ))ψ1 . ⎪ 2 ⎪ sinh r ⎩ sinh r where A0 and A2 −1 can be expressed in terms of ψ1 and ψ2 . In fact, from the curvature (4.3) for k = 1, l = 2 and compatibility condition (4.2), we have (4.11) ∂r A2 = I(ψ1 ψ¯2 ), ∂r ψ2 = iA2 ψ1 . Since A2 (0) = 1, (4.11) gives

(4.12)

A2 − 1 = ∫

From (4.3) when k = 0, l = 1 and (4.7), we have

r

0

I(ψ1 ψ¯2 )(s)ds.

∂r A0 = I(ψ1 ψ¯0 ) = −

1 2 2 2 2 ∂r (sinh r∣ψ1 ∣ − ∣ψ2 ∣ ), 2 sinh r which together with initial data of (¯ v , w) ¯ frame, yields

+∞ cosh s ψ2 2 ψ2 2 1 ∣ )+∫ (∣ψ1 ∣2 − ∣ ∣ )ds. A0 (t, r) = − (∣ψ1 ∣2 − ∣ 2 sinh r sinh s sinh s r Therefore the two variables ψ1 and ψ2 are not independent.

(4.13)

Schr¨odinger map from H2 to S2

18

Since the linear part of this system is not decoupled, we introduce the two new variables ψ + and ψ − , defined as ψ2 ψ2 (4.14) ψ + = ψ1 + i , ψ − = ψ1 − i . sinh r sinh r From (4.10) and A22 + ∣ψ2 ∣2 = 1, we obtain ¯ ⎧ cosh r(A2 − 1) cosh r + 1 + ⎪ + ψ2 ⎪ )ψ = (A + 2 − I(ψ ))ψ + , (i∂ + ∆ − 2 ⎪ 0 t 2 2 ⎪ ⎪ sinh r sinh r sinh r (4.15) ⎨ ¯ ⎪ cosh r(A2 − 1) cosh r − 1 − ⎪ − ψ2 ⎪ ⎪ )ψ = (A − 2 + I(ψ ))ψ − . (i∂t + ∆ + 2 0 ⎪ ⎩ sinh r sinh2 r sinh2 r It turns out that the linear part of ψ ± -system is decoupled. The compatibility condition (4.2) is reduced to ∂r sinh r(ψ + − ψ − ) = −A2 (ψ + + ψ − ),

(4.16)

and the coefficients A0 and A2 − 1 can be expressed in terms of ψ ± ,

∣ψ + ∣2 − ∣ψ − ∣2 sinh sds, 4 0 ∞ cosh s 1 R(ψ¯+ ψ − )ds. A0 = − R(ψ¯+ ψ − ) + ∫ 2 sinh s r A2 − 1 = ∫

(4.17) (4.18) Define V ± as the vector

r

u × (k × u) ∈ Tu (H2 ), sinh r then ψ ± is the representation of V ± in the coordinate frame (v, w) and the energy of u has a new representation, i.e ∞ ∣¯ u × (k × u¯)∣2 ) sinh rdr, E(u) =π ∫ (∣∂r u¯∣2 + 0 sinh2 r V ± = ∂r u ±

(4.19)

=π ∫

0

∞

∣V ± ∣2 sinh r ∓ 2∂r u¯ ⋅ (¯ u × (k × u¯))dr,

=π ∥V ± ∥L2 ∓ 2π ∫ 2

0

∞

∂r u¯3 dr,

=π ∥ψ ± ∥L2 ∓ 2π(¯ u3 (∞) − u¯3 (0)), 2

=π ∥ψ ± ∥L2 . 2

2 Hence, ∥ψ ± ∥L2 is conserved for all time. Moreover, if we assume that ∥ψ ± ∥L2 , ∥ψ˜± ∥L2 < 2 and ∥u − u˜∥H1 ≪ 1, we obtain the Lipschitz continuity of ψ ± with u, namely ∥ψ ± − ψ˜± ∥ 2 ≲ ∥u − u˜∥ 1 . (4.20) L

H

In fact, by the above assumptions, (4.17) implies u3 , u˜3 ≳ 1 − ∥ψ − ∥L2 > 0. On interval [R, ∞), by (4.4), we have 1 4

(v − v˜)(r) = ∫

then (4.6) and u3 − u˜3 =

∞

r

u21 −˜ u21 +u22 −˜ u22 u3 +˜ u3

(v − v˜) ⋅ ∂r u u + v˜ ⋅ ∂r (u − u˜)u − ∂r v˜ ⋅ u(u − u˜)ds,

imply

∥v − v˜∥C(R,∞) ≲ ∥v − v˜∥C(R,∞) ∥∂r u∥L2 (R,∞) + ∥u − u˜∥H1 ,

Huang, Wang, Zhao 19 choose R large enough, we have ∥v − v˜∥C(R,∞) ≲ ∥u − u˜∥H1 (R,∞) . Then for any ǫ > 0 small, on interval [ǫ, R], there exists δ > 0 such that any interval I ⊂ [ǫ, R] with ∣I∣ < δ, we have ∥∂r u∥L2 (I) ∥sinh−1 r∥L2 (I) ≪ 1. By a similar argument to that on [R, ∞), we obtain ∥v − v˜∥C(ǫ,R) ≲ ∥u − u˜∥H1 (ǫ,R) . Finally, on interval (0, ǫ), by Sobolev embedding (2.5), we have ∥

then we get

u3 − u˜3 ∥ + ∥u3 − u˜3 ∥L∞ (0,ǫ) ≲ ∥u − u˜∥H1 (0,ǫ) , sinh r L2 (ǫ)

∥∂r (v − v˜)∥L2 (0,ǫ) ≲ ∥v − v˜∥C(0,ǫ) ∥∂r u∥L2 + ∥u − u˜∥H1 ,

which implies by integration by parts ∥v − v˜∥C(0,ǫ) ≲ ∥∫

ǫ

r

+ ∥∫

⃗ ⃗ − (v − v˜) ⋅ (u − k)∂ ⃗ r uds∥ ∂r ((v − v˜) ⋅ (u − k)u) − ∂r (v − v˜) ⋅ (u − k)u

C(0,ǫ)

ǫ

r

∂r (˜ v ⋅ (u − u˜)u) − ∂r v˜ ⋅ (u − u˜)u − v˜ ⋅ (u − u˜)∂r uds∥

+ ∥∂r v˜∥L2 (0,ǫ) ∥

u − u˜ ∥ , sinh r L2 (0,ǫ)

⃗ ≲ ∥v − v˜∥C(0,ǫ) (∥u − k∥ +∥ C(0,ǫ) + ∥∂r (v − v˜)∥L2 (0,ǫ) ∥

C(0,ǫ)

u − k⃗ ∥ ) sinh r L2 (0,ǫ)

u − k⃗ ∥ + ∥u − u˜∥H1 (0,ǫ) , sinh r L2 (0,ǫ)

≲ ∥v − v˜∥C(0,ǫ) ∥u∥H1 (0,ǫ) + ∥u − u˜∥H1 (0,ǫ) ,

which together with ∥u∥H1 (0,ǫ) ≪ 1, yields ∥v − v˜∥C(0,ǫ) ≲ ∥u − u˜∥H1 (0,ǫ) . Therefore, we obtain ∥v − v˜∥C(0,ǫ) ≲ ∥u − u˜∥H1 .

(4.21)

Then by (4.21), (4.19) and Sobolev embedding (2.5), the Lipschitz continuity (4.20) follows. In this paper we will work with the key system (4.15) to obtain the space-time estimates for ψ ± . Suppose ψ ± satisfies the compatibility condition (4.16) and ∥ψ ± ∥L2 < ∞, define A2 , ψ1 , ψ2 by (4.17) and (4.14), then they satisfy the relation (4.11). Furthermore, we claim that ψ1 ∈ L2 and ψ2 , A2 − 1 ∈ H˙ e1 . In fact, by (4.17) and (4.14), we have ψ1 ∈ L2 , A2 ∈ L∞ , from (4.14), (4.11) and (2.16), we get ψ2 A2 −1 2 sinh r , ∂r ψ2 , ∂r A2 , and sinh r ∈ L . + i2θ + Denote R+ ψ = e ψ and R− ψ − = ψ − . Then we have Proposition 4.1. ∥u∥H3 ∼ ∥R+ ψ + ∥H 2 + ∥R− ψ − ∥H 2 .

(4.22)

Proof. If u ∈ H1 , we easily obtain ∥du∥L2 = π ∥ψ ± ∥L2 . If u ∈ H2 , by the equivariance condition, we have 2

(4.23)

∂rr u, (

2

cosh r cosh r 1 ∂r − ∂r u3 ∈ L2 , 2 )(u1 , u2 ), sinh r sinh r sinh r

Schr¨odinger map from H2 to S2

20

Since A1 = 0, then ∇r (v + iw) = 0, which gives

∂r ψ ± =∂r (V ± ⋅ (v + iw)),

(4.24)

by the representation of V ± (4.19), we have

=∂r V ± ⋅ (v + iw),

−u3 u + k⃗ ), sinh r ∂r u3 ⋅ u + u3 ∂r u cosh r(k⃗ − u3 u) , =∂rr u ∓ ∓ sinh r sinh2 r ∂r u3 − 1 cosh r cosh r ⃗ k =(∂rr ∓ ± ∂r u ∓ 2 )u ∓ sinh r sinh r sinh r sinh2 r cosh r(u3 − 1) ∂r u3 ⋅ u , ± u∓ 2 sinh r sinh r ∂r cosh r cosh r ∂r ∂r (4.25) =((∂rr ∓ ( − − )u3 ) ))u1 , (∂rr ∓ ( ))u2 , (∂rr ∓ 2 2 sinh r sinh r sinh r sinh r sinh r cosh r(u3 − 1) ⃗ u3 − 1 cosh r(u3 − 1) ∂r u3 ±( k− − ∂r u) ± ( )u, 2 sinh r sinh r sinh r sinh2 r denote F ± = (4.25) ⋅ (v + iw), then ∂r V ± =∂r (∂r u ±

(4.26)

u3 −1 ∣ For ψ − , since ∣ sinh r ≲

∂r ψ ± = F ± ± (

u21 +u22 sinh r

ψ2 1 − u3 − cosh r − 1 ψ +i (u3 − 1) ). sinh r sinh r sinh r

∈ L1 (dr), applying e∫r

∞ u3 −1 ds sinh s

to both sides of (4.26), by F ± ∈ L2 , we have ∞ u3 −1 ψ − ) ∈ L2 . Since u3 and ψ − are radial, we obtain e∫r sinh s ds ψ − ∈ H˙ 1 , which gives ψ − ∈ L4 by ∂r (e u3 −1 4 − 2 + 2 ∣ ∣u1 ∣+∣u2 ∣ (2.7). Hence, by (4.26) and ∣ sinh r ≲ sinh r ∈ L , we have ∂r ψ ∈ L . It also follows that ∂r ψ ∈ L . ψ+ In order to prove sinh r ∈ L2 , we rewrite ∞ u3 −1 ∫r sinh s ds

ψ+ ∂r u −u3 u + k⃗ =( + ) ⋅ (v + iw), sinh r sinh r sinh2 r ∂r ∂r 1 1 ∂r − − u3 ) ⋅ (v + iw) =(( 2 )u1 , ( 2 )u2 , sinh r sinh r sinh r sinh r sinh r u21 + u22 1 − u3 1 − u3 +( u , u , ) ⋅ (v + iw), 1 2 sinh2 r sinh2 r sinh2 r

ψ 1−u3 1 2 2 ∈ L2 and (4.23), we have sinh by ∣ sinh 2 ∣ ≲ r ∈L . r sinh2 r u3 −1 Conversely, if R± ψ ± ∈ H 1 , (2.7) implies ψ ± ∈ L4 . Then by ∣ sinh r∣ ≲ u2 +u2

+

∣ψ2 ∣ sinh r

∈ L4 and (4.26), we have

cosh r r θ 2 F ± ∈ L2 , namely, [∂rr ∓ ( cosh sinh r ∂r + sinh r )]u ⋅ (v + iw) ∈ L . The part of [∂rr ∓ ( sinh r ∂r + sinh2 r )]u in the ψ normal space is −∣ψ1 ∣2 ± ∣ sinh2 r ∣ ∈ L2 by ψ ± ∈ L4 . Therefore, (4.23) is obtained. If u ∈ H3 , by (2.4) and Lemma 2.3, we obtain ∇(−∆)ui ∈ L2 for i = 1, 2, 3, then by equivariance condition, we get ∂θ2 2

(4.27)

∂r (∂rr +

cosh r 1 cosh r ∂r − ∂r )u3 ∈ L2 , 2 )(u1 , u2 ), ∂r (∂rr + sinh r sinh r sinh r

∂2

Huang, Wang, Zhao 21 and cosh r 1 1 )(u1 , u2 ) ∈ L2 . (∂rr + ∂r − sinh r sinh r sinh2 r

(4.28)

In order to prove R± ψ ± ∈ H 2 , it suffices to prove (∂rr +

(4.29) By (4.26), we have

cosh r cosh r 4 + ∂r − ∂r )ψ − ∈ L2 . 2 )ψ , (∂rr + sinh r sinh r sinh r

∂rr ψ ± =∂r ((∂rr ∓ ( (4.30) (4.31)

∂r ∂r ∂r cosh r cosh r ))u1 , (∂rr ∓ ( ))u2 , (∂rr ∓ − − )u3 ) ⋅ (v + iw) 2 2 sinh r sinh r sinh r sinh r sinh r ψ2 u3 − 1 − cosh r − 1 (u3 − 1) ∓ ψ ), ± ∂r (i sinh r sinh r sinh r u3 − 1 cosh r − 1 ψ2 ∂r (ψ + − ψ − ) cosh r − =∂r F ± − ( ψ ) ∂r ψ − ± (u3 − 1)(i + )± 2 sinh r 2i sinh r sinh r sinh2 r ∂r A2 cosh r − 1 ± (i ψ2 ∓ ψ − ). sinh r sinh r

Since R± ψ ± ∈ H 1 , (2.7) implies ψ ± ∈ L4 ∩ L6 , then by ∣u3 − 1∣ ≲ ∣ψ2 ∣2 and (4.12), the third term of (4.30) r u3 −1 and (4.31) are in L2 . From (4.27), we also have ∂r F ± ∈ L2 . Hence, ∂r (e∫∞ sinh s ds ∂r ψ − ) ∈ L2 , which further r u3 −1 u3 −1 4 − 2 gives e∫∞ sinh s ds ∂r ψ − ∈ L4 , this implies ∂r ψ − ∈ L4 . Since sinh r ∈ L , we obtain ∂rr ψ ∈ L , therefore, we + 2 also get ∂rr ψ ∈ L . Next, we estimate this term cosh r − cosh r − 1 ψ2 u3 − 1 − cosh r ∂r ψ − = (F − i (u3 − 1) + ψ ). sinh r sinh r sinh r sinh r sinh r

(4.32)

By (4.28), the first two components of as

cosh r − sinh r F

are in L2 . For the third component, which can be written

cosh r ∂r ⃗ + iw) = i(cosh r − 1) ψ2 (∂rr + ∂r )u3 + iψ2 (∂rr + ∂r )u3 . (∂rr + )u3 k(v sinh r sinh r sinh r sinh r sinh r sinh r By (4.23) and (4.27), we have ∆u3 ∈ L4 . Hence, the right hand side of (4.33) is in L2 . By a similar argument, the third term in (4.32) is also in L2 . Thus, R− ψ − ∈ H 2 . For R+ ψ + , we need to estimate (4.33)

cosh r 4 + ∂r − 2 )ψ sinh r sinh r cosh r − 1 2 cosh r − 4 3 3 cosh r − 1 cosh r − 4 3 ∂r + ∂r + )u2 , =((∂r3 + ∂r2 + ∂r + 2 3 )u1 , (∂r + 2 sinh r sinh r sinh r sinh r sinh r sinh3 r cosh r − 1 2 1 ∂r )u3 ) ⋅ (v + iw) (4.34) (∂r3 + ∂r − sinh r sinh2 r 4(1 − u3 ) u3 − 1 − 3i∂r u3 cosh r − 1 − (u3 − 1)(ψ + − ψ − ) − ψ )− iψ2 + ∂r ( ψ2 3 2 sinh r sinh r sinh r sinh2 r cosh r u3 − 1 − cosh r cosh r − 1 (u3 − 1)(ψ + − ψ − ) − ψ . + 2 sinh r sinh r sinh r sinh r (∂rr +

Schr¨odinger map from H2 to S2

22

By (4.23) and (4.28), the first two components of (4.34) are in L2 , namely for i = 1, 2

3 cosh r − 1 2 cosh r − 4 ∂r + )ui ∂r + 2 sinh r sinh r sinh3 r 1 cosh r − 1 cosh r − 1 2 =∂r ∆ui − ∂r )ui . ∆ui + ( ∂r + 4 sinh r sinh r sinh2 r For the third component, since ∂r ∆u3 ∈ L2 , it suffices to estimate (∂r3 +

cosh r 2 ⃗ cosh r − 1 1 ∂r u3 k ⋅ (v + iw) = i ψ2 ∂r2 u3 + (ψ + − ψ − )∂r2 u3 . sinh r sinh r 2 cosh r−1 2 2 By (4.23) and ∣ψ2 ∣ ≤ 1, we get sinh r ψ2 ∂r u3 ∈ L . By (2.5) and (2.9), we have ∥ψ + ∥L∞ ≲ ∥ψ + ∥H˙ e1 and ∥ψ − ∥L∞ ≲ ∥ψ − ∥H 2 , then 12 (ψ + − ψ − )∂r2 u3 ∈ L2 . Therefore, (4.34) are in L2 . The other terms are also easily obtained by Sobolev embedding and A22 + ∣ψ2 ∣2 = 1. Thus, R+ ψ + ∈ H 2 is obtained. 

4.2. Recovering the map from ψ + . Here we will keep track of ψ + ∈ L2 , since it contains all the information about the map. Indeed, by (4.11), we have the system of (A2 , ψ2 ) ⎧ ∣ψ2 ∣2 ⎪ +¯ ⎪ ⎪ ψ ) − ∂ A = I(ψ , 2 ⎪ ⎪ r 2 sinh r (4.35) ⎨ ⎪ ψ2 ⎪ ⎪ . ∂r ψ2 = iA2 ψ + + A2 ⎪ ⎪ ⎩ sinh r Then from the choice of (¯ v (∞), w(∞)) ¯ (4.5), it gives the data (A2 (∞), ψ2 (∞)) = (1, 0). Given ψ + ∈ L2 + with ∥ψ ∥L2 < 2, we reconstruct A2 − 1, ψ2 ∈ H˙ e1 by above system (4.35), then by the system in (4.1) with condition (4.5), we can return to the map u.

Lemma 4.2. Let ψ + ∈ L2 , such that ∥ψ + ∥L2 < 2, the system (4.35) has a unique solution (A2 , ψ2 ) satisfying ψ2 , A2 − 1 ∈ H˙ e1 , and ∥ψ2 ∥H˙ e1 + ∥A2 − 1∥H˙ e1 + ∥

(4.36)

Moreover, we have the following properties: ψ2 (i) If ψ + ∈ Lp , with 1 ≤ p < ∞, then ψ − , sinh r,

A2 −1 sinh r

A2 − 1 ∥ ≲ ∥ψ + ∥L2 . 1 sinh r L (dr)

∈ Lp and

A2 − 1 ψ2 ∥ +∥ ∥ ≲ ∥ψ + ∥Lp . sinh r Lp sinh r Lp (ii) Given ǫ > 0, and R such that ∥ψ + ∥L2 (R/[R−1 ,R]) ≤ ǫ, then we have ∥ψ − ∥Lp + ∥

(4.37)

∥ψ2 ∥H˙ e1 (R/[ǫR−1 ,R]) + ∥A2 − 1∥H˙ e1 (R/[ǫR−1 ,R]) ≤ ǫ.

(4.38)

(iii) If (A˜2 , ψ˜2 ) is another solution to (4.35) corresponding to ψ˜+ , then

(4.39)

∥ψ − − ψ˜− ∥L2 + ∥ψ2 − ψ˜2 ∥H˙ 1 + ∥A2 − A˜2 ∥H˙ 1 ≲ ∥ψ + − ψ˜+ ∥L2 . e

e

(iv) If R+

ψ+

∈

H s,

then R−

ψ−

∈

Hs

for s ∈ {1, 2}.

Proof. We consider the ODE system (4.35) with boundary condition lim A2 (r) = 1, lim ψ2 (r) = 0.

r→∞

r→∞

Huang, Wang, Zhao 23 ψ2 ψ2 + The system and boundary condition imply A22 + ∣ψ2 ∣2 = 1. We define ψ − = ψ + − 2i sinh r , ψ1 = ψ − i sinh r , then we get ∂r A2 = 41 sinh r(∣ψ + ∣2 − ∣ψ − ∣2 ) from (4.35). Since A2 (∞) = 1 which yields by integration from infinity

A2 − 1 =

1 ∞ −2 (∣ψ ∣ − ∣ψ + ∣2 ) sinh sds. ∫ 4 r

Thus we have A2 > 1 − 14 ∥ψ + ∥L2 > 0. To prove existence, by choosing R large enough such that ∥ψ + ∥L2 (R,∞) ≤ ǫ. We want to seek ψ2 with the property that ∥ψ2 ∥H˙ e1 (R,∞) ≲ ǫ. This implies that ∣ψ2 ∣ ≲ ǫ, then we have A2 − 1 ≲ ǫ2 . By the relation √ A22 + ∣ψ2 ∣2 = 1, we get A2 = 1 − ∣ψ2 ∣2 . Now we only need to consider the ψ2 -equation in 2

X = {ψ2 ∈ H˙ e1 ∶ ∥ψ2 ∥H˙ e1 ≤ 2Cǫ}

Rewrite the ψ2 equation as

∂r ψ2 = iψ + + i(A2 − 1)ψ + + (A2 − 1)

ψ2 ψ2 + , sinh r sinh r

then ψ2 ψ2 = iψ + + i(A2 − 1)ψ + + (A2 − 1) , sinh r sinh r

∂r ψ2 − r

−1

Multiply by e− ∫∞ sinh

sds

on both sides, we have

∂r (e− ∫∞ sinh r

−1

sds

ψ2 ) = e− ∫∞ sinh r

−1

sds

(iψ + + i(A2 − 1)ψ + + (A2 − 1)

ψ2 ), sinh r

Integrating from infinity we obtain ψ2 (r) = ∫

r

r

e∫ρ

sinh−1 sds

∞

(iψ + + i(A2 − 1)ψ + + (A2 − 1)

Define the map T ∶ H˙ e1 (R, ∞) → H˙ e1 (R, ∞) by T (ψ2 )(r) = ∫

r

∞

r

e∫ρ

sinh−1 sds

ψ2 )dρ. sinh ρ

(iψ + + i(A2 − 1)ψ + + (A2 − 1)

ψ2 )dρ. sinh ρ

Schr¨odinger map from H2 to S2

24

Now it suffices to show that T is a contraction map in X. Indeed, the estimate (2.19) and Sobolev embedding lead to ∥T ψ2 ∥H˙ e1 (R,∞) ≤ ∥iψ + + i(A2 − 1)ψ + + (A2 − 1) + ∥sinh−1 r ∫

r

r

e∫ρ

sinh−1 sds

∞

ψ2 ∥ sinh r L2 (R,∞)

(iψ + + i(A2 − 1)ψ + + (A2 − 1)

∣ψ2 ∣2 ψ2 ∥ A2 + 1 sinh r L2 (R,∞) ψ2 + C ∥iψ + + i(A2 − 1)ψ + + (A2 − 1) ∥ , sinh r L2 (R,∞)

≤ ∥ψ + ∥L2 (R,∞) + C ∥ψ + ∥L2 (R,∞) + ∥

ψ2 )dρ.∥ , sinh ρ L2 (R,∞)

≤C(∥ψ + ∥L2 (R,∞) + ∥ψ2 ∥L∞ (R,∞) ∥ψ2 ∥H˙ e1 (R,∞) ), 2

≤C(∥ψ + ∥L2 (R,∞) + ∥ψ2 ∥H˙ e1 (R,∞) ), 3

≤C(ǫ + (2Cǫ)3 ), 0 such that ∥

By (2.19), we obtain that

Cǫ 1 ∫ r sinh−1 sds eR ψ2 (R)∥ < . sinh r 2 L2 (R−a,R)

∥J (ψ2 )∥H˙ e1 (R−a,R) ≤ ∥

1 ∫ r sinh−1 sds eR ψ2 (R)∥ sinh r L2 (R−a,R)

+C ∥iψ + + i(A2 − 1)ψ + + (A2 − 1)

ψ2 ∥ , sinh r L2 (R−a,R)

Cǫ 2 + C ∥ψ + ∥L2 (R−a,R) + C ∥ψ2 ∥L∞ ∥ψ2 ∥H˙e1 (R−a,R) , 2 Cǫ ≤ + Cǫ + (2Cǫ)3 , 2 ≤ 2Cǫ. ≤

Meanwhile we have

∥J (ψ2 ) − J (ψ˜2 )∥H˙ 1 (R−a,R) e

ψ2 − ψ˜2 ψ2 + (A˜2 − 1) ∥ , ≤ C(∥i(A2 − A˜2 )ψ + + (A2 − A˜2 ) sinh r sinh r L2 (R,∞)

2 ≤ C ∥ψ2 − ψ˜2 ∥H˙ 1 (R−a,R) (∥ψ + ∥L2 (R−a,R) + ∥ψ2 ∥H˙ e1 (R−a,R) + ∥ψ˜2 ∥H˙ 1 (R,∞) , e

e

≤ C(ǫ + 3Cǫ) ∥ψ2 − ψ˜2 ∥H˙ 1 (R−a,R) , e

≤ C ǫ ∥ψ2 − ψ˜2 ∥H˙ 1 (R−a,R) . ′

e

Therefore J is a contraction map in Y . Since the lifespan interval a only depends on ǫ and ∥ψ + ∥L2 , we can extend √ the solution to r = 0. Thus the existence of ψ2 in [0, ∞) follows, and the A2 is obtained by A2 (r) = 1 − ∣ψ2 ∣2 . ψ2 , then the system gives Next we obtain the bound for (4.36). Let G = 1+A 2 2∣G∣2 iψ + ¯ d G) + ∣G∣2 = 2R( , dr 1 + A2 sinh r

or equivalently

d 2 ψ+ ¯ ∣G∣2 − ∣G∣2 = −2I( G), dr sinh r 1 + A2

which implies

namely,

therefore

∣

∣

d ∣G∣ ∣ψ + ∣ , ∣G∣ − ∣≤ dr sinh r 1 + A2

r ∣ψ + ∣ d − ∫ r sinh−1 sds −1 , (e ∞ ∣G∣)∣ ≤ e− ∫∞ sinh sds dr 1 + A2

∣G∣(r) ≤ ∫

r

∞

r

e∫ρ

sinh−1 sds

∣ψ + ∣ (ρ)dρ. 1 + A2

Schr¨odinger map from H2 to S2

26 Since ∣A2 ∣ ≤ 1, we get

∣ψ2 ∣(r) ≲ ∫

(4.40)

∞

r

e∫ρ

sinh−1 sds

r

ψ2 ∥ 0, (4.40) gives ∥ sinh r L2 A2 −1 The bounds for ∥ sinh r ∥L2

∣ψ + ∣ (ρ)dρ. 1 + A2

≲ ∥ψ + ∥L2 . The bounds for ∥∂r ψ2 ∥L2 and ∥∂r A2 ∥L2 follow By (2.19) and A2 > A2 −1 and ∥ sinh directly from (4.35). r ∥L1 (dr) are obtained by the compatibility relation A22 + ∣ψ2 ∣2 = 1. Now we prove the additional properties (i)-(iv). First, we have the bound for (4.37). If ψ + ∈ Lp , by (2.19) ψ2 A2 −1 ∥Lp ≲ ∥ψ + ∥Lp , then the Lp -bound for ψ − and sinh and (4.40), we obtain ∥ sinhr r are obtained immediately 2 − 2 by the definition of ψ and A2 + ∣ψ2 ∣ = 1. Second, we obtain (4.38). By (4.35) and A22 + ∣ψ2 ∣2 = 1, we have ∥A2 − 1∥H˙ e1 (R/[ǫR−1 ,R]) + ∥∂r ψ2 ∥L2 (R/[ǫR−1 ,R]) ≲ ǫ + ∥

It suffices to get the L2 -bound for ∣ψ2 ∣ ≲ ∫

∞

r

r

e∫ρ

ψ2 sinh r .

sinh−1 sds

From (4.40), we have

1(0,R−1 ] ∣ψ +∣dρ + ∫

∞ r

r

e∫ρ

ψ2 ∥ . sinh r L2 (R/[ǫR−1 ,R])

sinh−1 sds

1(R−1 ,∞) ∣ψ + ∣dρ.

For the first term we use (2.19) and the smallness of ∥ψ + ∥L2 (0,R−1 ) . For the second term, by H¨older inequality, we have ∞ r −1 1 ∥ e∫ρ sinh sds 1(R−1 ,∞) ∣ψ + ∣dρ∥ ∫ sinh r r L2 (0,ǫR−1 ) ∞ 1 r −1 1 ( ∫ ∣e∫ρ sinh sds sinh−1 ρ∣2 sinh ρdρ) 2 ∥ψ + ∥L2 ∥ ≤∥ , sinh r R−1 L2 (0,ǫR−1 ) =( ∫ ≲ǫ.

ǫR−1

0

∞ 1 1 −1 2 ∫ρr sinh−1 sds 2 ∥ψ + ∥L2 , e sinh ρdρ dr) sinh r ∫R−1

Then by (2.19), we easily obtain ∞ r −1 ψ2 1 ∥ ∥ ≤∥ e∫ρ sinh sds 1[R,∞) ∣ψ + ∣dρ∥ ≲ ǫ. ∫ sinh r L2 (R,∞) sinh r r L2

Thus the L2 -bound follows. Third, we get the Lipschitz continuity (4.39). For notational convenience we denote δψ + = ψ + − ψ˜+ , δψ2 = ψ2 − ψ˜2 , δA2 = A2 − A˜2 .

Without any restriction in generality, we can make the assumption ∥δψ + ∥L2 ≪ 1 and the bootstrap assumption 1 ∥δψ2 ∥L∞ + ∥δA2 ∥L∞ ≲ ∥δψ + ∥L2 2 . By (4.35) and A22 + ∣ψ2 ∣2 = 1, we derive the equations ˜ ⎧ δψ2 ⎪ ˜+ δA2 + A2 − 1 δψ2 + ψ2 δA2 + iA2 δψ + , ⎪ + i ψ ⎪ ∂ δψ = r 2 ⎪ ⎪ sinh r sinh r sinh r ⎨ 2 ˜ ⎪ ⎪ ⎪ ∂ δA = 2δA2 + I(ψ + δψ ) + 2(A2 − 1) δA + I(δψ + ψ¯˜ ) + (δA2 ) . ⎪ ⎪ 2 r 2 2 2 ⎩ sinh r sinh r sinh r

Huang, Wang, Zhao 27 2 (δA2 )2 2) +¯ + ˜ Since ∥δψ + ∥L2 ≪ 1 and (δA sinh r is a high order term, I(δψ ψ2 ) + sinh r and iA2 δψ can be regarded as + + T error terms. Let X = (Rδψ , Iδψ , δA2 ) , we have 1 (4.41) ∂r X = LX + BX + E, sinh r where L = diag{1, 1, 2},

Rψ˜2 A2 −1 + 0 −Iψ˜+ + sinh ⎛ sinh ⎛ R(iA2 δψ ) ⎞ r r⎞ ˜ Iψ2 ⎟ A2 −1 B=⎜ , E = ⎜ I(iA2 δψ + ) ⎟ . Rψ˜+ + sinh ⎜ 0 sinh r r ⎟ ⎝I(δψ + ψ¯˜2 ) + (δA2 )2 ⎠ 2(A˜2 −1) ⎝ Iψ + Rψ + ⎠ sinh r sinh r

From (4.37) we obtain the L2 -norm of B is bounded. Then we decompose B = B1 + B2 ∶= B1≥ǫ (r) + B1 0, r there exists δ > 0 such that ∥ψ + ∥L2 (R−δ,R) < ǫ. Define Ui (r) = ∫R MUi−1 ds, denote I = [R − δ, R], we have and

∥Ui ∥C(I) ≤ ∥M1 ∥L1 (dr)(I) ∥Ui−1 ∥C(I) + ∥∂r M2 ∥L2 (I) (∫

R−δ

R

sh−1 sds)1/2 ∥Ui−1 ∥C(I) ,

∥∂r Ui ∥L2 (I) ≤ (∥M1 ∥L2 (I) + ∥∂r M2 ∥L2 (I) ) ∥Ui−1 ∥C(I) .

By (4.43), we have ∥M1 ∥L1 (dr)(I) + ∥M1 ∥L2 (I) + ∥∂r M2 ∥L2 (I) ≲ ∥ψ + ∥L2 (I) , therefore, ∥Ui ∥C(I) + ∥∂r Ui ∥L2 (I) ≲ ∥ψ + ∥L2 (I) (1 + (∫ ≲

i ∥ψ + ∥L2 (I) .

R−δ

R

sinh−1 sds)1/2 )( ∥Ui−1 ∥C(I) + ∥∂r Ui−1 ∥L2 (I) ),

By choosing δ small such that (∫R sinh−1 sds)1/2 is small, then we can still rely on iteration scheme to extend the solution to [ǫR−1 , ∞). On the interval (0, ǫR−1 ], by (4.38), we have ∥ψ2 ∥C((0,ǫR−1 ]) ≲ ∥ψ2 ∥H˙ e1 ((0,ǫR−1 ]) ≲ ǫ. By a similar argument to that on [R, ∞), we extend the solution to r = 0. As a byproduct, R−δ

∥U − U0 ∥C(0,∞) + ∥∂r U∥L2 (0,∞) ≲ ∥ψ + ∥L2 .

From the system (4.42), we know that u¯3 and q = w¯3 − i¯ v3 solve the system {

∂r q = i¯ u3 ψ1 ,

∂r u¯3 = I(ψ1 q¯).

with boundary condition (¯ u3 , q)(∞) = (1, 0). By uniqueness, A2 = u¯3 , ψ2 = q. Next, we construct the system of (u, v, w) by equivariant setup, that is, apply (¯ u, v¯, w) ¯ by eθR . From ψ2 = w¯3 − i¯ v3 and the orthonormality condition, (4.1) is satisfied for k = 2.

Schr¨odinger map from H2 to S2

30

˜ as above. From the construction it follows that Given ψ + , ψ˜+ ∈ L2 , we construct U and U ˜ 2 ∥U − U˜ ∥C(0,∞) + ∥∂r (U − U)∥ (4.44) ≲ ∥ψ + − ψ˜+ ∥L2 . L (0,∞)

which implies ∥∂r (u − u˜)∥L2 (0,∞) ≲ ∥ψ + − ψ˜+ ∥L2 . Since u1 = v2 w3 − v3 w2 , u˜1 = v˜2 w˜3 − v˜3 w˜2 , ψ2 = w3 − iv3 , ψ˜2 = w˜3 − i˜ v3 , by (4.44), we have 1 u1 − u˜1 ∥ ≤∥ [(v2 − v˜2 )w˜3 + v2 (w3 − w˜3 )]∥ ∥ sinh r L2 sinh r L2 1 +∥ [(v3 − v˜3 )w2 + v˜3 (w2 − w ˜2 )]∥ , sinh r L2 ψ˜2 ψ2 − ψ˜2 ∥ + ∥U∥L∞ ∥ ∥ , ≲ ∥U − U˜ ∥L∞ ∥ sinh r L2 sinh r L2 ≲ ∥ψ + − ψ˜+ ∥ (∥ψ + ∥ 2 + ∥ψ˜+ ∥ ). A similar argument shows that ∥u − u˜∥H˙ 1 ≲ ∥ψ + − ψ˜+ ∥L2 .

u2 2 −˜ ∥ usinh r ∥L2

+

L2

u3 3 −˜ ∥ usinh r ∥ L2

5. T HE C AUCHY

L

L2

≲ ∥ψ + − ψ˜+ ∥L2 (∥ψ + ∥L2 + ∥ψ˜+ ∥L2 ). Therefore, 

PROBLEM

In this section we concerned with the (ψ + , ψ − )-system which we recall here ¯ ⎧ cosh r(A2 − 1) cosh r − 1 i2θ + ⎪ + ψ2 ⎪ 2 −2 )e ψ = [A + 2 − I(ψ (i∂ + ∆ )]ei2θ ψ + , ⎪ 0 t H 2 2 ⎪ ⎪ sinh r sinh r sinh r (5.1) ⎨ ¯ ⎪ cosh r − 1 − ψ2 cosh r(A2 − 1) ⎪ ⎪ ⎪ )]ψ − . (i∂ + ∆H2 + 2 )ψ = [A0 − 2 + I(ψ − ⎪ 2 2 ⎩ t sinh r sinh r sinh r ± ± with initial data ψ (t0 ) = ψ0 . Where A0 , A2 , ψ2 are given by (4.18), (4.17), (4.14). Since the system (5.1) arised from the Schr¨odinger map (1.1), we will show that (ψ + , ψ − ) satisfy the compatibility condition. For simplicity of notations, we denote ∥f ± ∥ = ∥f + ∥ + ∥f − ∥. Since our analysis relies on Lpt Lqx -norm, we define the norm of f by ∥f ∥Lp Lq ∶= ∥f ∥Lp (I;Lq ) . Finally, we denote the nonlinearities by I

¯ cosh r(A2 − 1) + ψ2 − I(ψ )]ei2θ ψ + , sinh r sinh2 r ¯ cosh r(A2 − 1) − ψ2 + I(ψ )]ψ − . F − (ψ − ) = [A0 − 2 sinh r sinh2 r F + (ψ + ) = [A0 + 2

5.1. Srichartz estimates. To understand the well-posedness of (5.1), we need to obtain the Strichartz estimates. The ψ + -equation in (5.1) is a nonlinear Schr¨odinger equation with positive and exponential decay potential. More generally, we consider the Schr¨odinger equation (5.2)

{

(i∂t + ∆H2 − V )u = F, u(0) = f,

where V ∈ e−αr L∞ (H2 ; R) for α ≥ 1 is a positive potential. In this section we always denote potential V p for p ∈ [1, ∞]. (p, q) is called admissible pair, if as (5.2). For simplicity, we denote p′ = p−1 (p, q) ∈ {(p, q) ∈ (2, ∞) × (2, ∞) ∶

1 1 1 + = } ∪ {(∞, 2)}. p q 2

Huang, Wang, Zhao 31 Then we obtain the following Strichartz estimates. Theorem 5.1. Let (p, q), (˜ p′ , q˜′ ) be admissible pairs, I ⊂ R be open interval. (i) If f ∈ L2 (H2 ), then ∥eit(∆H2 −V ) f ∥Lp Lq ≲ ∥f ∥L2 , I

(ii) If F ∈

LpI˜Lq˜,

then

∥∫ ei(t−s)(∆H2 −V ) F (s)ds∥ 0 t

LpI Lq

≲ ∥F ∥Lp˜Lq˜ . I

Based on a standard theory, the above results are obtained by the following dispersive estimates immediately. Proposition 5.2. Assume V ∈ e−αr L∞ (H2 ; R), α ≥ 1, is a positive potential, then we have −1 ⎧ ⎪ ⎪ C∣t∣ , if 0 < ∣t∣ < 1, it(∆H2 −V ) (5.3) ∣∣e ∣∣L1→L∞ ≤ ⎨ − 23 ⎪ ⎪ ⎩ C∣t∣ , if ∣t∣ ≥ 1.

By standard convention the resolvent of Laplacian −∆H2 on H2 is written as R0 (s) = (−∆H2 −s(1−s))−1 with Rs > 12 corresponding to the resolvent set s(1 − s) ∈ C − [ 41 , ∞). The kernel of R0 (s) is R0 (s; z, w) = Q0s−1 (cosh r),

(5.4)

where Q0s−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 1 1 dΠ(λ) =2iλ[R0 ( + iλ) − R0 ( − iλ)]dλ, 2 2 1 = − 4λIR0 ( + iλ)dλ. 2 Then we use the spectral resolution to write i

it∆H2

e

+∞ e4 2 eitλ f (w)dΠ(λ; z, w)dwdλ, f (z) = ∫ ∫ 2πi 0 H2 i +∞ +∞ e4 sin λs 2 = eitλ λ ∫ √ f (w)dsdwdλ. ∫ ∫ 2πi 0 H2 r cosh s − cosh r

Similarly, from [7], the resolvent of −∆H2 + V for potential V defined as above is given by RV (s) = (−∆H2 + V − s(1 − s))−1 and the continuous component of spectral resolution is given by 1 dΠV (λ) = −4λIRV ( + iλ)dλ, 2 then the kernel of Schr¨odinger propagator can be written as i

+∞ 1 e4 2 eitλ λIRV ( + iλ; z, w)f (w)dwdλ. e f (z) = ∫ ∫ 2πi 0 2 H2 By Birman-Schwinger type resolvent expansion for all frequencies: it(∆H2 −V )

RV (s) = R0 (s) + R0 (s)[−V R0 (s)] + [R0 (s)V ]RV (s)[V R0 (s)],

Schr¨odinger map from H2 to S2

32 we get eit(∆H2 −V ) f (z) i

+∞ 1 e4 2 (5.5) eitλ λIR0 ( + iλ; z, w)f (w)dwdλ = ∫ ∫ 2πi 0 2 H2 i +∞ e4 1 1 2 + (5.6) eitλ λIR0 ( + iλ)[−V IR0 ( + iλ)]f (w)dwdλ ∫ ∫ 2 2πi 0 2 2 H i +∞ 4 e 1 1 1 2 + (5.7) eitλ λI[R0 ( + iλ)V ]RV ( + iλ)[V IR0 ( + iλ)]f (w)dwdλ. ∫ ∫ 2 2πi 0 2 2 2 H 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 ≤ λ ≤ 1 and r ∈ (0, ∞) C∣ log r∣, r ≤ 1, 1 ∣R0 ( + iλ; z, w)∣ ≤ { − 21 − 12 r Cλ e , r > 1, 2

where r ∶= d(z, w).

C∣ log r∣, r ≤ 1, 1 ∣∂λ R0 ( + iλ; z, w)∣ ≤ { − 21 −( 12 −ǫ)r , r > 1, Cλ e 2

Lemma 5.4. For the free resolvent kernel the pointweise bounds are valid for λ ≥ 1, and r ∈ (0, ∞) C∣ log r∣, λr ≤ 1, 1 ∣R0 ( + iλ; z, w)∣ ≤ { − 21 − 12 r Cλ e , λr > 1, 2

where r ∶= d(z, w).

C∣ log r∣, λr ≤ 1, 1 ∣∂λ R0 ( + iλ; z, w)∣ ≤ { − 21 −( 21 −ǫ)r , λr > 1, Cλ e 2

We also recall the meromorphic continuation from [7]. Lemma 5.5. For V ∈ e−αr L∞ (H2 ) with α > 0, the resolvent RV (s) admits a meromorphic continuation to the half-plane Rs > 21 − δ as a bounded operator RV (s) ∶ e−δr L2 (H2 ) → eδr L2 (H2 )

for δ < α2 . And there exists a constant MV such that for all λ ∈ R with ∣λ∣ ≥ MV , α 1 α ∣∣e− 2 r ∂λq RV ( + iλ)e− 2 r ∣∣L2 →L2 ≤ Cq,α ∣λ∣−1. 2

If the RV ( 12 + iλ) has no pole at λ = 0, we can extend the estimate through λ = 0 to give α 1 α ∣∣e− 2 r ∂λq RV ( + iλ)e− 2 r ∣∣L2 →L2 ≤ Cq,α⟨λ⟩−1 . 2 In order to prove Proposition 5.2, we also need the following lemma.

Huang, Wang, Zhao 33 √ ⎧ √ √ r+a t ⎪ ⎪ , r≥ , t ⎪ ∞ ei 4t (s + a) ⎪ ⎪ sinh r t √ ds ≲ ⎨ (5.8) √ ∫ ⎪ r t a cosh s − cosh r ⎪ √ ⎪ ⎪ . t(1 + ), r < ⎪ ⎩ r 2 Proof. The proof roughly follows the approach in [2]. Before proving the lemma, we recall two useful estimates, that is, Lemma 5.6.

(s+a)2

1 c c √ ≤ ≤ , for s > ρ ≥ 0, cosh s − cosh ρ (s − ρ) cosh ρ s − ρ

(5.9) and

c 1 , for s > ρ > 0. ≤√ cosh s − cosh ρ (s − ρ) sinh ρ

(5.10) √

Case 1: r ≥ 2t . τt + r, then Let s = r+a

i

LHS(5.8) = ∫

∞

Denote Φ(τ ) ∶=

(

0

=2tei

(5.11)

e 4t

(r+a)2 4t

√ ∫

τt cosh( r+a

τt 1 e 4(r+a)2 2 ( 2(r+a) 2 + 2) dτ. √ τt + r) − cosh r cosh( r+a i(

∞ 0

τ 2t

+τ )

+ τ2 , then (5.11) can be written as ∞ √ i (r+a)2 √ r + a eiΦ(τ ) Φ′ (τ ) √ r+a 2 te 4t dτ ∫ τt sinh r 0 t sinh r (cosh( r+a + r) − cosh r)

τ 2t 4(r+a)2

(5.12)

τt ( r+a + r + a) t dτ, r+a + r) − cosh r

τ 2 t2 +(r+a)2 +τ t) (r+a)2

Since ∫0 = ∫0 + ∫1 , (5.12) can be split into √ i (r+a)2 √ r + a (I1 + I2 ). (5.13) 2 te 4t sinh r ∞

1

∞

For I1 , by (5.10) and r ≥

√ 2

t

we have

I1 ≲ ∫

1

≲∫

1

0

≲1. For I2 , Let (5.14)

α(τ ) = [

0

√

τt 2(r+a)2

+ 21

r+a τ t t sinh r r+a

sinh r

dτ,

1 √ dτ, τ

1 τt r+a (cosh( + r) − cosh r)]− 2 . t sinh r r+a

Schr¨odinger map from H2 to S2

34 By integrating by parts in I2 , we get

I2 = ∫

∞ 1

∂τ eiΦ(τ ) α(τ )dτ,

≤∣eiΦ(τ ) α(τ )∣∞ 1 ∣ +∣∫ Notice that α(1) ≲ 1 and

α′ (τ )

≲ sup ∣α(τ )∣ + ∫ τ ≥1

∞

1

< 0 by (5.14), hence,

∞

1

eiΦ(τ ) α′ (τ )∣dτ,

∣α′ (τ )∣dτ.

∣I2 ∣ ≲ ∣α(1)∣ ≲ 1.

That is I1 and I√2 are bounded. Therefore (5.8) follows (5.13) in the region r ≥ Case 2: r < 2t . Let us split the left hand side of (5.8) into three parts: LHS(5.8) = ∫

2r+a

r+a

+∫

√

t+a

2r+a

+ ∫√

∞ t+a

√

t 2 .

=∶ J1 + J2 + J3 .

For J1 , we assume r > 0, otherwise J1 = 0 immediately, then 2r+a u √ J1 ≲ ∫ du, r+a cosh(u − a) − cosh r 2r+a u du, ≲∫ √ r+a (u − a − r) sinh r √ r . ≲(2r + a) sinh r Since we are in the case r
0, by (5.20) we have

√ u

√ du ≲ ∫ 0 cosh u − 1

1

√ √ ∞ u u du + ∫ √ du < ∞. u 1 cosh u

√ √ s − r0 + r0 √ (5.21) ≲ ∫ ds, r0 cosh s − cosh r0 √ √ ∞ ∞ r0 u du + ∫ √ √ ≲∫ ds, r0 0 cosh s − cosh r0 cosh(u + r0 ) − cosh r0 √ ∞ 1 u du + √ √ , ≲∫ 0 sinh r0 (cosh u − 1) cosh r0 √ r0 1 +√ . ≲√ cosh r0 sinh r0 The third integral can be estimated similar to the second one. If r0 = 0, ∞ ∞ s s √ √ ds < ∞. ds = ∫ ∫ r0 0 cosh s − cosh r0 cosh s − 1 If r0 > 0, ∞ ∞ ∞ s − r0 r0 s ds = ∫ ds + ∫ ds, √ √ √ ∫ r0 r0 r0 cosh s − cosh r0 cosh s − cosh r0 cosh s − cosh r0 1 r0 ≲√ +√ . cosh r0 sinh r0 In conclusion, we obtained √ r1 1 1/2 ( 1 √t (r1 ) + 1< √t (r ) ) (5.16) ≲t [ √ 1 sinh r1 ≥ 2 2 sinh r0 1 1 √ 1 1 √ 1≥ √t (r1 ) + √ 1 t (r1 )], +√ cosh r0 sinh r1 2 cosh r0 r1 < 2 √ 1 1 1 r1 1/2 )1 √t (r1 ) ≲t [( √ +√ √ sinh r0 sinh r1 cosh r0 sinh r1 ≥ 2 1 1 1 + (√ )1 √t (r1 )]. +√ sinh r0 cosh r0 r1 < 2 ∞

Huang, Wang, Zhao 37 Therefore, √ 1 1 r1 1 √ (5.15) ≲t−1 ∫ V (z1 )[( √ )1 √t (r1 ) +√ z0 ,z1 sinh r0 sinh r1 cosh r0 sinh r1 ≥ 2 1 1 1 +√ )1 √t (r1 )]g(z0 )dz0 dz1 , + (√ sinh r0 cosh r0 r1 < 2 ≲∣t∣−1 ∥g∥L1 .

Finally, we estimate the (5.7). By duality, it suffices to prove for ∥h∥L1 = 1, (5.22)

⟨∫

∞

eitλ λ ∫ 2

0

H2

I[R0 V ]RV [V R0 ]g(w)dwdλ, h⟩ ≲ ∣t∣−1 ∥g∥L1 .

write 1 A(g)(z) = ∫ V (z)R0 (− + iλ; z, z0 )g(z0 )dz0 , 2 1 B(g)(z) = ∫ V (z)∂λ R0 (− + iλ; z, z0 )g(z0 )dz0 . 2 Then by integration by parts and Lemma 5.6, we have (5.22) = ∫

∞

0

1 1 2 eitλ λ ∫ IR0 (− + iλ; y0 , y1 )V (y1 )RV (− + iλ; y1 , z1 ) 2 2

1 V (z1 )R0 (− + iλ; z1 , z0 )g(z0 )dz0 dz1 dy1 h(y0 )dy0 dλ, 2 ∞ 1 1 2 = ∫ eitλ λ ∫ RV (− + iλ; y, z)V (z)R0 (− + iλ); z, z0 )g(z0 )dz0 dz 2 2 0 ∫ V (y)R0 (y, y0 )h(y0 )dy0 dydλ,

=∣t∣−1 ∫

∞

0

eitλ [∫ ∂λ RV (y, z)A(g)(z)dzA(h)(y)dy 2

+ ∫ RV (y, z)B(g)(z)dzA(h)(y)dy

+ ∫ RV (y, z)A(g)(z)dzB(h)(y)dy]dλ,

≲∣t∣−1 ∫

∞

0

⟨λ⟩−1 ( ∥e 4 ∣z∣ A(g)(z)∥ 1

+ ∥e 4 ∣z∣ B(g)(z)∥ 1

+ ∥e

1 ∣z∣ 4

A(g)(z)∥

∥e 4 ∣y∣ A(h)(y)∥ 1

L2

L2

∥e 4 ∣y∣ A(h)(y)∥ 1

L2

∥e

1 ∣y∣ 4

L2

B(h)(y)∥

L2

)dλ.

L2

Schr¨odinger map from H2 to S2

38

By Lemma 5.3 and Lemma 5.4, Young’s inequality and Holder’s inequality, we have ∥e 4 ∣z∣ A(g)(z)∥

≲ ∥∫ e 4 ∣z∣ V (z)(∣ log r∣1≤1(r) + e− 2 r 1>1 (r))1≤1 (λ)g(z0 )dz0 ∥ 1

1

1

L2

+ ∥∫ e

1 ∣z∣ 4

V (z)(∣ log r∣1≤1(λr) + λ e − 21

− 12 r

1>1 (λr))1>1 (λ)g(z0 )dz0 ∥

≲1≤1 (λ) ∥g∥L1 (∥∣ log r∣1≤1 (r)∥L2 + ∥e− 2 r 1>1 (r)∥ 1

L∞

)

+ 1>1 (λ) ∥g∥L1 (∥∣ log r∣1≤1 (λr)∥L2 + ∥λ− 2 e− 2 r 1>1 (λr)∥ 1

1

≲(1≤1 (λ) + 1>1 (λ)λ− 2 ) ∥g∥L1 , 1

≲⟨λ⟩− 2 ∥g∥L1 .

L2

L∞

, L2

),

1

Similarly, we have

∥e 4 ∣z∣ B(g)(z)∥ 1

Therefore,

≲ ⟨λ⟩− 2 ∥g∥L1 . 1

L2

(5.7) ≲ sup ∣t∣−1 ∫ ⟨λ⟩− 2 ∥g∥L1 ∥h∥L1 dλ ≲ ∣t∣−1 ∥g∥L1 . 0 ∞

3

∥h∥L1 =1

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 operar−1 tor −∆H2 − 2 cosh has discrete spectrum, we use perturbation method (see [30]) to prove global wellsinh2 r posedness. Theorem 5.7. Consider the problem (5.1) with data ∥ψ0± ∥L2 < 2, where A0 , A2 , ψ2 are given by (4.18), (4.17), (4.14). Then there exists a unique maximal-lifespan solution pair (ψ + , ψ − ) ∶ I × R2 Ð→ C × C with t0 ∈ I and ψ ± (t0 ) = ψ0± with the following additional properties: (i) If ∥ψ0± ∥L2 < 2, then there exists T = T (ψ0± ) > 0, and a unique solution (ψ + , ψ − ) of the system in the time interval [−T, T ] with (ψ + , ψ − ) ∈ L4 ([−T, T ]; L4 ) ∩ C([−T, T ]; L2 ). (ii) If (ψ + , ψ − ) ∶ (T0 , T1 ) × R+ Ð→ C × C, ∣T1 − T0 ∣ < ∞, is a solution to (5.1) with ∥ψ ± ∥L4 L4 (T0 ,T1 ) < ∞, then ψ ± can be extended to a solution on a larger time interval. (iii) There exists ǫ > 0 such that ∥ψ0± ∥L2 ≤ ǫ, then for any compact interval J ⊂ R, (5.1) has a unique global solution ψ ± (t) ∈ L4 (J; L4 ) ∩ C(J; L2 ), moreover, ∥ψ ± ∥L4 L4 ≲ C(J, ∥ψ0± ∥L2 ). J (iv) For every A > 0, and ǫ > 0, there is δ > 0 such that if ψ ± is a solution satisfying ∥ψ ± ∥L4 L4 ≤ A and I M(ψ ± − ψ˜± ) ≤ δ, then there exists a solution such that ∥ψ ± − ψ˜± ∥ 4 4 ≤ ǫ, and M(ψ ± − ψ˜± ) ≤ ǫ, ∀ t ∈ I. 0

0

(v) Assume that

(5.23)

R± ψ0±

∈

H s,

for s ∈ 1, 2. If

∥ψ ± ∥

LI L

L4I L4

≤ M, then the solution ψ ± satisfies

∥R± ψ ± ∥H s ≲M ∥R± ψ0± ∥H s + 1, ∀ t ∈ I,

and it has Lipschitz dependence with respect to the initial data. Proof. (i) Consider the system (4.15) in the space

X = {(ψ + , ψ − ) ∈ C([0, T ]; L2 ) ∩ L4T L4 ∶ ∥ψ ± ∥C([0,T ];L2 ) ≤ 2 ∥ψ0± ∥L2 , ∥ψ ± ∥L4 L4 ≤ ǫ}. T

Huang, Wang, Zhao 39 Given the formulas for A0 , A2 and ψ2 by (4.18), (4.17), (4.14), using Lemma 2.4, we obtain ∥A0 ∥L2 + ∥

(5.24)

cosh r(A2 − 1) ψ2 ∥ ≲ ∥ψ ± ∥L4 , ∥ ∥ ≲ ∥ψ ± ∥L4 . 2 2 sinh r sinh r L L4

In a similar argument, we also obtain that

2 ∥F ±(ψ ± ) − F ± (ψ̃± )∥L 43 ≲ ∥ψ ± − ψ̃± ∥L4 (∥ψ ± ∥2L4 + ∥ψ̃± ∥L4 ).

(5.25)

r−1 Denote V = 2 cosh , then by Duhamel formula, define the maps sinh2 r

T + (ψ + ) = eit(∆−V ) ei2θ ψ0+ − i ∫

(5.26)

T − (ψ − ) = eit∆ ψ0− − i ∫

t

0

t 0

ei(t−s)(∆−V ) F + (ψ + )ds,

ei(t−s)∆ (F − (ψ − ) − V ψ − )ds.

For any ǫ > 0, there exists φ±0 ∈ C0∞ , such that ∥ψ0± − φ±0 ∥L2 < ǫ 4C , then dispersive estimates and (5.24) imply ∥T + (ψ + )∥L4

T1

L4

≤ ∥eit(∆−V ) ei2θ (ψ0+ − φ+0 )∥L4 + ∥∫

t

0

T1

L4

and there exists T1 > 0, s.t T14 ∥φ±0 ∥H˙ 12
0 such that if E(ψ0± ) = ∥ψ0± ∥L2 ≤ ǫ, then (5.27) has a unique global solution u± ∈ C(R; L2 ) ∩ L4 L4 , moreover, ∥u± ∥L∞ L2 ∩L4 L4 (R×H2 ) ≤ Cǫ. Now we show using a perturbative argument that (5.1) is global well-posed for E(ψ0± ) < ǫ. First we show that for T sufficiently small depending only on E(ψ0± ), and V , the solution (ψ + , ψ − ) to (5.1) on [0, T ] satisfies an a priori estimate (5.28)

Fix a small parameter η > 0, since V =

∥ψ ± ∥L∞ L2 ∩L4 L4 (0,T ) ≤ 8Cǫ.

cosh r−1 , sinh2 r

there exists T > 0 for I = [0, T ], such that

∥V ∥L2 L2 < η. I

Further, by Duhamel formula, Strichartz estimates and ∥u± ∥L∞ L2 ∩L4 L4 ≤ Cǫ, we have ∥eit(∆−V ) u+(0)∥L4 L4 ≤ ∥u+∥L4 L4 + ∥∫ ei(t−s)(∆−V ) F + (u+ )ds∥ t

I

I

(5.29)

0

L4I L4

≤Cǫ + C(Cǫ)3 , ≤2Cǫ.

and ∥eit∆ u−(0)∥L4 L4 ≤ ∥u−∥L4 L4 + ∥∫ ei(t−s)∆ F − (u− )ds∥ t

(5.30)

I

I

≤2Cǫ.

0

, L4I L4

,

Huang, Wang, Zhao 41 Since (ψ + , ψ − ) satisfies (5.1) and u+(0) = ei2θ ψ0+ , u− (0) = ψ0− , apply the Duhamel formula, (5.29) and (5.30) to obtain ∥ψ + ∥L4 L4 ≤ ∥eit(∆−V ) u+ (0)∥L4 L4 + C ∥ψ ± ∥L4 L4 , 3

I

(5.31) and

≤2Cǫ + C

I

I

3 ∥ψ ± ∥L4 L4 I

.

∥ψ − ∥L4 L4 ≤ ∥eit∆ u− (0)∥L4 L4 + C ∥ψ ± ∥L4 L4 + C ∥V ∥L2 L2 ∥ψ − ∥L4 L4 , 3

I

≤2Cǫ + C

I

3 ∥ψ ± ∥L4 L4 I 1 3,

Choose η sufficiently small such that Cη
0. In particular, if in addition E(u0 ) < ǫ2 for sufficiently small ǫ, we can construct the fields ψ ± on interval [0, T ] satisfying (4.15) and ∥ψ + ∥L2 = ∥ψ − ∥L2 < ǫ as in Section 4.1. By Theorem 5.7 (iii), the solution ψ ± is defined on J ⊂ R for any compact interval J and with ∥ψ ± ∥L4 L4 ≤ C(J, ∥ψ0± ∥L2 ). Then by Theorem 5.7 (v) and Proposition 4.3, we construct a map u(t) ∈ H3 J coincide with the Schr¨odinger map on [0, T ] from ψ ± (t), moreover, E(u(T )) < ǫ2 . Then repeat the procedure the map reconstructed from ψ ± (t) is in fact a Schr¨odinger map. For initial data u0 ∈ H1 , there exists u0,n ∈ H3 such that ∥u0 − u0,n ∥H1 < n1 . By (4.20), we obtain the ± ± Lipschitz continuity of ψ0,n , i.e ∥ψ0,n − ψ0± ∥L2 ≲ ∥u0,n − u0 ∥H1 . Then from Theorem 5.7 (iv), the solution ± of (4.15) is Lipschitz continuous with respect to initial data, we have ∥ψn± − ψ ± ∥L2 ≲ ∥ψ0,n − ψ0± ∥L2 ≲ ∥u0,n − u0 ∥H1 for any t ∈ I. From Proposition 4.3, we get ∥u(t) − un (t)∥H1 ≲ ∥ψn± − ψ ± ∥L2 ≲ ∥u0,n − u0 ∥H1 for any t ∈ I. Hence, we obtain the desired result.  ACKNOWLEDGMENTS The first author thanks Dr. Ze Li for helpful discussions.

48

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Jiaxi Huang, Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, China. E-mail: [email protected]; Youde Wang, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, P.R. China.

E-mail: [email protected] Lifeng Zhao, Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, China. E-mail: [email protected].