On the high–low method for NLS on the hyperbolic space

Staffilani, Gigliola; Yu, Xueying

doi:10.1063/5.0012061

In this paper, we first prove that the cubic, defocusing nonlinear Schrödinger equation on the two dimensional hyperbolic space with radial initial data in $H^{s} (H^{2})$ is globally well-posed and scatters when s > 3/4. Then, we extend the result to nonlinearities of order p > 3. The result is proved by extending the high–low method of Bourgain in the hyperbolic setting and by using a Morawetz type estimate proved by Staffilani and Ionescu.

I. INTRODUCTION

In this paper, we first consider the cubic nonlinear Schrödinger (NLS) initial value problem on the hyperbolic plane $H^{2}$ ⁠,

\{\begin{cases} i \partial_{t} u + Δ_{H^{2}} u = {| u |}^{2} u, & t \in R, x \in H^{2}, \\ u (0, x) = ϕ (x), \end{cases})

(1.1)

where u = u(t, x) is a complex-value function in spacetime $R \times H^{2}$ and ϕ is a radial initial datum.

The solution of (1.1) conserves both the mass

M (u (t)) : = \int_{H^{2}} {| u (t, x) |}^{2} d x = M (u_{0})

(1.2)

and the energy

E (u (t)) : = \int_{H^{2}} \frac{1}{2} {| \nabla_{H^{2}} u (t, x) |}^{2} + \frac{1}{4} {| u (t, x) |}^{4} d x = E (u_{0}) .

(1.3)

Conservation laws of mass and energy give the control of the L² and Ḣ¹ norms of the solutions, respectively.

Our goal in this paper is to prove the global well-posedness and scattering of (1.1) with the regularity of the initial data below H¹.

In order to best frame the problem and to emphasize its challenges, we start by recalling the results in $R^{d}$ ⁠, a setting that has been extensively considered in recent years. Consider the evolution equation in (1.4) with general non-linearities,

i \partial_{t} u + Δ u = {| u |}^{p - 1} u, p > 1,

(1.4)

in $R^{d}$ ⁠. Let us first recall that the critical scaling exponent in $R^{d}$ is

s_{c} ≔ \frac{d}{2} - \frac{2}{p - 1} .

(1.5)

It is well-known that in the sub-critical and critical regimes (s > s_c and s = s_c, respectively), the initial value problem (1.4) is locally well-posed;^1–4 we recall here that with local well-posedness, we mean local in time existence, uniqueness, and continuous dependence of the data to the solution map. Thanks to the conservation laws of energy and mass, the H¹-subcritical initial value problem and the L²-subcritical initial value problem are globally well-posed in the energy space H¹ and mass space L², respectively. The questions about scattering are much more delicate. What we mean with scattering here will be made more precise later (see, for example, Theorem 1.1), but in general terms, with scattering, we intend that the nonlinear solution as time goes to infinity approaches a linear one.

Before we talk about the global results in the more general subcritical case with data with regularity between L² (mass) and H¹ (energy), that is, in H^s, 0 < s < 1, let us denote with $g_{M}^{p}$ the regularity index above which one obtains the global well-posedness for the NLS problem on the manifold $M$ with power nonlinearity p and with $s_{M}^{p}$ the index above which one obtains scattering (with the global well-posedness) again on the manifold $M$ with power nonlinearity p.

The very first global well-posedness result in the subcritical case between the two (mass and energy) conservation laws (0 < s < 1) was given by Bourgain,⁵ where he developed the high–low method to prove global well-posedness for the cubic (p = 3) NLS in two dimensions for initial data in H^s, s > 3/5. According to the above notation, the regularity index in Ref. 5 is $g_{R^{2}}^{3} = 3 / 5$ ⁠.

We now describe the high–low method of Bourgain because it is the inspiration for part of our current work. To start, the initial datum is decomposed into a (smoother) low frequency part and a (rougher) high frequency part. The first step is to solve the NLS globally for the smoother part, for which the energy is finite, and then solve a difference equation for the rougher part. The miracle in this argument, which allows one to continue with an iteration, is that, in fact, the Duhamel term in the solution to the difference equation is small and smoother in an interval of time that is inversely proportional to the size of the low frequency part of the initial datum. At the next iteration, one merges this smoother part with the evolution of the low frequency part of the datum and repeats. It is worth mentioning that in order to obtain the miracle step, Bourgain⁵ used a Fourier transform based space X^s,b that captures particularly well the behavior of solutions with low regularity initial datum. Let us remark that there is no scattering result in the high–low method proposed by Bourgain.

Colliander–Keel–Staffilani–Takaoka–Tao⁶ improved the global well-posedness index $g_{R^{2}}^{3}$ of the initial data to 4/7 by introducing a different method, now known as the I-method. This is also based on an iterative argument. One first defines a Fourier multiplier that smooths out the initial data into the energy space and proves that the energy of the smoothed solution is almost conserved, that is, at each iteration, the growth of such modified energy is uniformly small. The index $g_{R^{2}}^{3}$ is derived by keeping the accumulation of energy controlled. As a result, Colliander et al.⁶ obtained a polynomial growth of the sub-energy Sobolev norm of the global solution. The cubic NLS in $R^{3}$ was also considered in Ref. 6, and the index $g_{R^{3}}^{3} = 5 / 6$ ⁠.

Later, in Ref. 7, by combining the Morawetz estimate with the I-method and a bootstrapping argument, the same authors were able to lower the index $g_{R^{3}}^{3}$ to 4/5 and proved for the first time that the global solution also scatters, and hence, $g_{R^{3}}^{3} = s_{R^{3}}^{3} = 4 / 5$ ⁠. It is important to recall though that Bourgain⁸ had already proved global well-posedness for general data with index $g_{R^{3}}^{3} = 11 / 13$ and scattering for radial data with index $s_{R^{3}}^{3} = 5 / 7$ ⁠.

To prove scattering, one needs to show that a spacetime norm of the solution is uniformly bounded. To this end, an iteration of local well-posedness would not suffice. Instead, one uses a Morawetz estimate that gives a uniform bound of the L⁴ spacetime norm of the solution, combined with the I-method. More in detail, one splits the timeline into a finite number of intervals I_j, of possible infinite length, on which the $L_{I_{j}}^{4}$ of the solution is small. The smallness allows for a better spacetime bound of the global solution on each interval I_j, and then, one uses an iteration on the finite number of these intervals, which finally gives the desired spacetime uniform bound for the solution and hence scattering.

More results on the high–low method and the I-method both in $R^{d}$ and compact manifolds can be found in Refs. 7, 9–21, and 22.

We now consider the initial value problem,

\{\begin{cases} i \partial_{t} u + Δ_{H^{d}} u = {| u |}^{p - 1} u, & t \in R, x \in H^{d}, \\ u (0, x) = ϕ (x), \end{cases})

(1.6)

with p > 1. Compared to what we recalled above, we expect even better results in $H^{d}$ ⁠. In fact, the negative curvature of the ambient manifold allows for more dispersion in $H^{d}$ than in the Euclidean spaces. Mathematically, we can see this in the Strichartz estimates on $H^{d}$ ⁠, a family of estimates that is broader than the one obtained for the Euclidean space (see Refs. 36 and 37). The fact that the family of Strichartz estimates is larger in $H^{d}$ reminds us of another case in which this is true. In fact also for the wave equation, the Strichartz estimates form a larger family. In this case though, it is not the curvature of the ambient manifold that generates a larger number of estimates, but instead it is the fact that the wave operator has a strong smoothing effect pointwise in time, a property that is not enjoyed by the Schrödinger operator. As a consequence, when one considers a nonlinear wave equation, the smoother and more plentiful estimates provide more suitable control of the nonlinear terms, and this is the reason why in the nonlinear wave setting, the miracle step in the high–low method in Ref. 23 does not need the Fourier type spaces X^s,b mentioned above. However, in contrast, the larger range of the Strichartz estimates for the Schrödinger operator in the hyperbolic space still is not readily enough to handle the miracle step since although one obtains better spacetime estimates, there is no pointwise smoothing effect; hence, the context we work in is more challenging than the one in Ref. 23. At this point, one may guess that using some hyperbolic version of the space X^s,b may do the trick. While this is indeed the case when the problem is posed in $T^{d}$ ⁠, see, for example, Ref. 11, in $H^{d}$ the space, it is not clear how to define the Fourier transform based X^s,b type spaces in a way that is useful to handle nonlinearities. A naive definition using the Helgason–Fourier transform in Ref. 24 is deficient because of the following two reasons: first, the eigenfunctions of the Laplace–Beltrami operator on $H^{d}$ lead to a very different Fourier inversion formula and Plancherel theorem. In particular, we cannot claim that the Fourier transform of a product is the convolution of Fourier transforms, which is a fundamental fact used in the estimates of nonlinear terms via the space X^s,b. Second, the frequency localization based on the Helgason–Fourier transform does not behave well in $L^{p} (H^{d})$ ⁠, which causes difficulties in defining an effective Littlewood–Paley decomposition. We anticipate here that our approach to recover the miracle step, where one has to prove a gain of smoothness for the solution to the nonlinear difference equation, takes advantage of a Kato type smoothing effect. This smoothing is not pointwise in time, like for the wave operator, but average in time, hence much weaker. In order to make up for this weakness, we need to use a maximal function estimate combined with a better Sobolev embedding, which, in turn, forces us to assume radial symmetry for our initial data. We expect though that our global well-posedness and scattering results are true in general, and we believe that the more sophisticated smoothing effect in Ref. 25 may play an important role.

We now move to a summary of results that have been proved in the context of well-posedness and scattering for NLS in $H^{d}$ ⁠. Although the initial value problem (1.6) cannot be properly scaled, we still use the same index s_c defined in (1.5) to guide us in gauging the difficulty of proving the global well-posedness and scattering for (1.6). The subcritical initial value problem in the hyperbolic setting was first considered in Ref. 26, where the authors proved scattering for a family of power-type nonlinearity NLS with radial H¹ data. Banica and Duyckaerts²⁷ showed global well-posedness, scattering, and blow-up results for energy-subcritical focusing NLS also on the hyperbolic space. Recall that an NLS equation is called focusing when the nonlinearity in (1.4) has a negative sign, that is, $i \partial_{t} u + Δ_{H^{2}} u = - {| u |}^{p - 1} u$ ⁠.

In the critical setting Ionescu et al.²⁸ proved global well-posedness and scattering of the energy-critical NLS in $H^{3}$ ⁠. This result uses an ad hoc profile decomposition technique to transfer the already available result in $R^{3}$ (Ref. 29) into the $H^{3}$ setting. A similar technique was also used in $T^{3}$ for the same energy critical problem.³⁰ We do not think that this method, which is well suited for critical settings, could work in our subcritical setting, when the initial data are in $H^{s} (H^{2}), 0 < s < 1$ ⁠, but it may work to transfer in $H^{2}$ the result that Dodson proved for mass critical in $R^{2}$ ⁠.³¹ To the best of the authors’ knowledge, there are no known subcritical global well-posedness and scattering results with initial data not at the conservation law level in hyperbolic spaces.

We now state the main result of this work for the initial value problem (1.1). Later, in Sec. V, we state a similar result for the more general version (1.6) with p > 3.

Theorem 1.1.

The initial value problem () with radial initial data

ϕ \in H^{s} (H^{2})

with s > 3/4 is globally well-posed and scattering holds, that is, there exists

u_{\pm} \in H^{s} (H^{2})

such that

\lim_{t \to \pm \infty} ‖ u (t) - e^{i t Δ_{H^{2}}} u_{\pm} ‖_{H_{x}^{s} (H^{d})} = 0 .

(1.7)

Remark 1.2.

Here, we conduct a discussion on the indices of regularity for the global well-posedness, and we make a comparison with other results.

As discussed above, the equivalent case we consider here but in $R^{2}$ was treated by Bourgain without redial symmetry using the X^s,b space. Since we cannot use the same approach in $H^{2}$ ⁠, we decided first to rework this case using different tools such as Kato smoothing effect, maximal function estimates, and better Sobolev embedding. We did this because in $R^{2}$ ⁠, we have a Littlewood–Paley decomposition that works very well. Using these tools in the implementation of the high–low method, we obtained that the cubic radial NLS is globally well posed when s > 4/5, that is, $g_{R^{2}}^{3} = 4 / 5$ ⁠. Recall that Bourgain’s result gives $g_{R^{2}}^{3} = 3 / 5$ ⁠, which is better than what we can do in $R^{2}$ ⁠, and it is for general data. However, what we achieved in this first step is a blueprint that is generalizable to the $H^{2}$ space. We did not report our calculations for $R^{2}$ here, but the curious reader can check a longer version of this work in Ref. 32.

One notes that the index $s_{H^{2}}^{3} = 3 / 4$ that we attained in Theorem 1.1 is smaller than the one we obtained in $R^{2}$ ⁠, where we worked out only the global well-posedness, not the scattering. This is because of the better radial Sobolev embedding in $H^{2}$ and of the help coming from the strong Morawetz estimate used in the local theory.

Now, let us discuss a little bit of history concerning the indices of regularity for the global well-posedness. In Bourgain’s paper, where the high–low method was introduced,⁵ the global existence index is $g_{R^{2}}^{3} = 3 / 5$ ⁠. Later in Ref. 6, where the I-method was used, the index $g_{R^{2}}^{3}$ was improved to 4/7, and later in Ref. 33, thanks to a sophisticated treatment of the Fourier multiplier involved in the I-method, the global existence index $g_{R^{2}}^{3}$ was lowered further to 1/2. The global well-posedness of cubic NLS in two dimensions with H^1/2 data was proved in Ref. 17. In addition, $g_{R^{2}}^{3}$ was improved to 1/3 in Ref. 9 and to 1/4 in Ref. 13. In Refs. 34, 35, and 31 Dodson proved global well-posedness and scattering for the mass-critical NLS in any dimension. As a consequence, via the persistence of regularity property, mass-critical NLS equations with any subcritical initial data are globally well-posed as well.

A. Blueprint of the proof

In this subsection, we summarize the three main parts of the proof of Theorem 1.1. In general terms, we combine the high–low method with a Morawetz type estimate that gives a bound for the spacetime L⁴ norm.

The first part of the proof deals with the analysis of the energy increment. Following Bourgain’s high–low method, we first decompose the initial datum into a high and low frequency part. Then, we write the solution u as the sum of the linear evolution of the high frequency part and a remainder ζ that solve a difference equation that evolves from the low frequency part of the original initial datum. In this first step, we assume that in an interval [0, τ], where τ could be infinity, the L⁴ spacetime norm of the solution is small. We then prove an estimate for the energy increment of ζ. This is the content of Proposition 3.1. To prove this energy increment estimate, we further decompose ζ = ζ₁ + ζ₂, where ζ₁ is the nonlinear solution starting from the low frequency part of the datum and ζ₂ solves a difference equation with zero datum. This part is similar to the high–low method of Bourgain, but here the interval of time is not small and the smallness comes from the L⁴ norm. The miracle step is then to be able to show that ζ₂ is smoother and small in the appropriate norms. In the second part of the proof, we assume that the total L⁴ spacetime norm of the solution is bounded and we subdivide the timeline into finitely many intervals in which this norm is small. Here, we apply the first part described above, and we prove a global energy increment for ζ, this is Proposition 3.2. In the last part, we use a bootstrapping argument to show that indeed, the L⁴ norms of the solution u are bounded. This part requires a modification of the Morawetz estimate in Ref. 36 (see Proposition 4.1), and it uses the global energy increment proved in Proposition 3.2.

To summarize, the rest of this paper is organized as follows: In Sec. II, we discuss the geometry of the domain $H^{2}$ and collect the useful analysis tools in $H^{2}$ ⁠. Next, in Sec. III, we present the calculation of the energy increment of the smoother part of the solution. In Sec. IV, we run a bootstrapping argument based on the estimates derived from Sec. III and modified Morawetz estimates and complete the Proof of Theorem 1.1. Finally, in Sec. V, we generalize this theorem for p > 3.

II. PRELIMINARIES

A. Notations

We define

‖ f ‖_{L_{t}^{q} L_{x}^{r} (I \times H^{2})} : = {[\int_{I} {(\int_{H^{2}} {| f (t, x) |}^{r} d x)}^{\frac{q}{r}} d t]}^{\frac{1}{q}},

where I is a time interval.

We use the Japanese bracket notation in the following sense:

‖ ⟨Ω⟩ f ‖_{X} = ‖ f ‖_{X} + ‖ Ω f ‖_{X},

where X is one of the normed spaces we use below.

We adopt the usual notation that A ≲ B or B ≳ A to denote an estimate of the form A ≤ CB for some constant 0 < C < ∞ depending only on the a priori fixed constants of the problem. If A ≲ B ≲ A, we write A ∼ B or A ≃ B.

B. Geometry of the domain $H^{2}$

We consider the Minkowski space $R^{2 + 1}$ with the standard Minkowski metric,

- {(d x^{0})}^{2} + {(d x^{1})}^{2} + {(d x^{2})}^{2},

and we define the bilinear form on $R^{2 + 1} \times R^{2 + 1}$ ⁠,

[x, y] = x^{0} y^{0} - x^{1} y^{1} - x^{2} y^{2} .

The hyperbolic space $H^{2}$ is defined as

H^{2} = {x \in R^{2 + 1} : [x, x] = 1 and x^{0} > 0} .

An alternative definition for the hyperbolic space is

H^{2} = {x = (t, s) \in R^{2 + 1}, (t, s) = (\cosh r, \sinh r ω), r \geq 0, ω \in S^{1}} .

One has

d t = \sinh r d r, d s = \cosh r ω d r + \sinh r d ω

and the metric induced on $H^{2}$ is dr² + sinh² r dω², where dω² is the metric on the sphere $S^{1}$ ⁠.

Then, one can rewrite integrals as

\int_{H^{2}} f (x) d x = \int_{0}^{\infty} \int_{S^{1}} f (r, ω) \sinh r d r d ω .

Let 0 = {(1, 0, 0)} denote the origin of $H^{2}$ ⁠. The distance of a point to 0 is

d ((\cosh r, \sinh r ω), 0) = r .

More generally, the distance between two arbitrary points is

d (x, x^{'}) = \cosh^{- 1} ([x, x^{'}]) .

The general definition of the Laplace–Beltrami operator is given by

Δ_{H^{2}} = \partial_{r}^{2} + \frac{\cosh r}{\sinh r} \partial_{r} + \frac{1}{\sinh^{2} r} Δ_{S^{1}} .

Remark 2.1.

The form of the Laplace–Beltrami operator implies that there will be no scaling symmetry in $H^{2}$ as we usually have in the $R^{d}$ setting.

C. Tools on $H^{2}$

In this subsection, we recall some important and classical analysis developed for the hyperbolic spaces.

1. Fourier transform on $H^{d}$

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

h_{λ, θ} (x) = {[x, Λ (θ)]}^{i λ - \frac{d - 1}{2}},

where Λ(θ) denotes the point of $R^{d + 1}$ given by (1, θ), are generalized eigenfunctions of the Laplace–Beltrami operator. Indeed, we have

- Δ_{H^{d}} h_{λ, θ} = (λ^{2} + \frac{{(d - 1)}^{2}}{4}) h_{λ, θ} .

The Fourier transform on $H^{d}$ is defined as

\hat{f} (λ, θ) ≔ \int_{H^{d}} h_{λ, θ} (x) f (x) d x,

and the Fourier inversion formula on $H^{d}$ takes the form of

f (x) = \int_{- \infty}^{\infty} \int_{S^{d - 1}} {\bar{h}}_{λ, θ} (x) \hat{f} (λ, θ) \frac{d θ d λ}{| c (λ) |^{2}},

where c(λ) is the Harish–Chandra coefficient

\frac{1}{{|c (λ)|}^{2}} = \frac{1}{2 {(2 π)}^{d}} \frac{{|Γ (i λ + \frac{d - 1}{2})|}^{2}}{{|Γ (i λ)|}^{2}} .

2. Strichartz estimates

In this subsection, we recall the Strichartz estimates proved in the hyperbolic space. We say that a couple (q, r) is admissible if (1/q, 1/r) belong to the triangle

T_{d} = \{(\frac{1}{q}, \frac{1}{r}) \in (0, \frac{1}{2}] \times (0, \frac{1}{2}) |) \frac{2}{q} + \frac{d}{r} \geq \frac{d}{2}\} \cup \{(0, \frac{1}{2})\} .

We have the following theorem by Anker–Piefelice.³⁶

Theorem 2.2

(Strichartz estimates in Ref. 37). Assume that u is the solution to the inhomogeneous initial value problem,

\{\begin{cases} i \partial_{t} u + Δ_{H^{d}} u = F, & t \in R, x \in H^{d}, \\ u (0, x) = ϕ (x) . \end{cases})

Then, for any admissible exponents (q, r) and

(\tilde{q}, \tilde{r})

, we have the Strichartz estimates,

‖ u ‖_{L_{t}^{q} L_{x}^{r} (R \times H^{d})} ≲ ‖ ϕ ‖_{L_{x}^{2} (H^{d})} + ‖ F ‖_{L_{t}^{{\tilde{q}}^{'}} L_{x}^{{\tilde{r}}^{'}} (R \times H^{d})} .

Remark 2.3.

Strichartz estimates are better in $H^{d}$ in the sense that the set T_d of admissible pairs for $H^{d}$ is much wider than the corresponding set I_d for $R^{d}$ ⁠, which is just the lower edge of the triangle (see also Fig. 1).

FIG. 1.

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Strichartz admissible pair regions (d = 2 and d ≥ 3) for the hyperbolic space.

Definition 2.4

(Strichartz spaces). We define the Banach space

S^{0} (I) = \{f \in C (I : L^{2} (H^{2})) : ‖ f ‖_{S^{0} (I)} = \sup_{(q, r) admissible} ‖ f ‖_{L_{t}^{q} L_{x}^{r} (I \times H^{2})} < \infty\} .

In addition, we define the Banach space S^σ(I), where σ > 0,

S^{σ} (I) = \{f \in C (I : H^{σ} (H^{2})) : ‖ f ‖_{S^{σ} (I)} = ‖ {(- Δ)}^{\frac{σ}{2}} f ‖_{S^{0} (I)} < \infty\} .

3. Local smoothing estimates in the hyperbolic space

Theorem 2.5

(Theorem 1.2 in Ref. 38: Local smoothing estimates in

H^{2}

⁠). For any ɛ > 0,

\begin{array}{l} ‖ {⟨x⟩}^{- \frac{1}{2} - ε} {|\nabla|}^{\frac{1}{2}} e^{i t Δ} f ‖_{L_{t, x}^{2} (R \times H^{2})} ≲ ‖ f ‖_{L_{x}^{2} (H^{2})}, \\ ‖ {⟨x⟩}^{- \frac{1}{2} - ε} \nabla \int_{0}^{t} e^{i (t - s) Δ} F (s, x) d s ‖_{L_{t, x}^{2} (R \times H^{2})} ≲ ‖ {⟨x⟩}^{\frac{1}{2} + ε} F ‖_{L_{t, x}^{2} (R \times H^{2})} . \end{array}

Remark 2.6.

Kaizuka³⁸ considered more general manifolds that they are denoted with X. To obtain the above theorem, one needs to take $p (λ) = {|λ|}^{2}$ ⁠, $p (D) = - Δ_{X} - {|ρ|}^{2}$ ⁠, and m = 2.

4. Heat-flow-based Littlewood–Paley projections and functional inequalities on $H^{2}$

The Littlewood–Paley projections on $H^{2}$ that we use in this paper are based on the linear heat equation e^sΔ. It turned out, in fact, that for us, this is a great substitute for the standard Littlewood–Paley decomposition used in $R^{d}$ ⁠, since in $H^{d}$ ⁠, one cannot localize in frequencies efficiently. We report below several results that first appeared in Ref. 25.

Definition 2.7

(Section 2.7.1 in Ref. 25: Heat-flow-based Littlewood–Paley projections). For any s > 0, we define

P_{\geq s} f = e^{s Δ} f, P_{s} f = s (- Δ) e^{s Δ} f .

By the fundamental theorem of calculus, it is straightforward to verify that

P_{\geq s} f = \int_{s}^{\infty} P_{s^{'}} f \frac{d s^{'}}{s^{'}} for s > 0 .

In particular, we have

f = \int_{0}^{\infty} P_{s^{'}} f \frac{d s^{'}}{s^{'}},

which is the basic identity that relates f to its Littlewood–Paley resolution

{P_{s} f}_{s \in (0, \infty)}

⁠. We also have

P_{\leq s} f = \int_{0}^{s} P_{s^{'}} f \frac{d s^{'}}{s^{'}} .

Remark 2.8.

Intuitively, P_sf may be interpreted as a projection of f to frequencies comparable to s^−1/2. P_≥s and P_≤s can be viewed as the projections into low and high frequencies, respectively.

Lemma 2.9

(Lemma 2.5 in Ref. 25). Let 1 < p < ∞ and p ≤ q ≤ ∞. Let ρ₀ satisfy

0 < ρ_{0}^{2} < \frac{1}{2} \min {\frac{1}{p}, 1 - \frac{1}{p}} .

For

f \in L_{x}^{p} (H^{2})

and s > 0, we have

‖ e^{s Δ} f ‖_{L_{x}^{q} (H^{2})} + ‖ s Δ e^{s Δ} f ‖_{L_{x}^{q} (H^{2})} ≲ s^{- (\frac{1}{p} - \frac{1}{q})} e^{- ρ_{0}^{2} s} ‖ f ‖_{L_{x}^{p} (H^{2})} .

Remark 2.10.

In particular, if p = q in Lemma 2.9, using Definition 2.7, we have

‖ P_{\geq s} f ‖_{L_{x}^{p} (H^{2})} + ‖ P_{s} f ‖_{L_{x}^{p} (H^{2})} ≲ ‖ f ‖_{L_{x}^{p} (H^{2})} .

Lemma 2.11

(Corollary 2.7 in Ref. 25). Let 0 < α < 1 and 1 < p < ∞. For

f \in L^{p} (H^{2})

, we have

‖ s^{α} {(- Δ)}^{α} e^{s Δ} f ‖_{L_{x}^{p} (H^{2})} ≲ ‖ f ‖_{L_{x}^{p} (H^{2})} .

Lemma 2.12

(Lemma 2.9 in Ref. 25: Boundedness of Riesz transform). Let

f \in C_{0}^{\infty} (H^{2})

. Then, for 1 < p < ∞, we have

‖ \nabla f ‖_{L_{x}^{p} (H^{2})} ≃ ‖ {(- Δ)}^{\frac{1}{2}} f ‖_{L_{x}^{p} (H^{2})} .

Lemma 2.13

(Lemma 2.10 in Ref. 25: L^p interpolation inequalities). Let

f \in C_{0}^{\infty} (H^{2})

. Then, for any 0 ≤ β ≤ α and 1 < p < ∞, we have

‖ {(- Δ)}^{β} f ‖_{L_{x}^{p} (H^{2})} ≲ ‖ f ‖_{L_{x}^{p} (H^{2})}^{1 - \frac{β}{α}} ‖ {(- Δ)}^{α} f ‖_{L_{x}^{p} (H^{2})}^{\frac{β}{α}} .

Moreover, for 1 < p < ∞, p ≤ q ≤ ∞, we have

‖ f ‖_{L_{x}^{q} (H^{2})} ≲ ‖ f ‖_{L_{x}^{p} (H^{2})}^{1 - θ} ‖ {(- Δ)}^{α} f ‖_{L_{x}^{p} (H^{2})}^{θ}, where 0 < θ = \frac{1}{α} (\frac{1}{p} - \frac{1}{q}) < 1 .

Lemma 2.14

(Sobolev embedding).

W^{s, p} (H^{d}) ↪ L^{p} (H^{d}), if 1 < p \leq q < \infty and \frac{1}{p} - \frac{1}{q} = \frac{s}{d} .

Lemma 2.15

(Lemma 2.12 in Ref. 25: Gagliardo–Nirenberg inequality). Let

f \in C_{0}^{\infty} (H^{2})

. Then, for any 1 < p < ∞, p ≤ q ≤ ∞, and 0 < θ < 1, we have

‖ f ‖_{L_{x}^{q} (H^{2})} ≲ ‖ f ‖_{L_{x}^{p} (H^{2})}^{1 - θ} ‖ \nabla f ‖_{L_{x}^{p} (H^{2})}^{θ}, where \frac{1}{q} = \frac{1}{p} - \frac{θ}{2} .

In particular, for any s > 0,

‖ f ‖_{L_{x}^{\infty} (H^{2})} ≲ ‖ f ‖_{L_{x}^{4} (H^{2})}^{\frac{1}{2}} ‖ \nabla f ‖_{L_{x}^{4} (H^{2})}^{\frac{1}{2}} ≲ ‖ f ‖_{L_{x}^{2} (H^{2})}^{\frac{1}{4}} ‖ \nabla f ‖_{L_{x}^{2} (H^{2})}^{\frac{1}{2}} ‖ Δ f ‖_{L_{x}^{2} (H^{2})}^{\frac{1}{4}} ≲ s^{- \frac{1}{2}} ‖ ⟨s Δ⟩ f ‖_{L_{x}^{2} (H^{2})} .

Lemma 2.16

(Proposition 2.14 in Ref. 25: Sobolev product rule). For σ > 0, we have

‖ f g ‖_{H_{x}^{σ} (H^{2})} ≲ ‖ f ‖_{L_{x}^{\infty} (H^{2})} ‖ g ‖_{H_{x}^{σ} (H^{2})} + ‖ f ‖_{H_{x}^{σ} (H^{2})} ‖ g ‖_{L_{x}^{\infty} (H^{2})} .

Lemma 2.17

(General Sobolev product rule). For σ > 0, we have

‖ f g ‖_{W_{x}^{σ, r} (H^{2})} ≲ ‖ f ‖_{W_{x}^{σ, p_{1}} (H^{2})} ‖ g ‖_{L_{x}^{p_{2}} (H^{2})} + ‖ f ‖_{L_{x}^{q_{1}} (H^{2})} ‖ g ‖_{W_{x}^{σ, q_{2}} (H^{2})} .

Remark 2.18.

Lemma 2.17 allows more possible L^p norms than Lemma 2.16 in the product rule. The Proof of Lemma 2.17 is using Triebel’s argument in Ref. 39 (see, for example, Secs. 7.2.2 and 7.2.4) and can also be found in Proposition 2.14 of Ref. 25. This proof relies on a localization lemma (see, for example, Lemma 2.16 in Ref. 25) to reduce to the standard Sobolev product rule.

Lemma 2.19

(Bernstein inequalities). For 0 ≤ β < α < β + 1,

\begin{array}{l} ‖ {(- Δ)}^{β} P_{\leq s} f ‖_{L_{x}^{2} (H^{2})} & ≲ s^{α - β} ‖ {(- Δ)}^{α} f ‖_{L_{x}^{2} (H^{2})}, \\ ‖ {(- Δ)}^{α} P_{\geq s} f ‖_{L_{x}^{2} (H^{2})} & ≲ s^{β - α} ‖ {(- Δ)}^{β} f ‖_{L_{x}^{2} (H^{2})} . \end{array}

Proof.

Using Definition 2.7, Lemma 2.11, and the formal property (−Δ)^a(−Δ)^b = (−Δ)^a+b,

\begin{array}{l} ‖ {(- Δ)}^{β} P_{\leq s} f ‖_{L_{x}^{2} (H^{2})} & = ‖) \int_{0}^{s} {(- Δ)}^{β} t (- Δ) e^{t Δ} f \frac{d t}{t} {(‖}_{L_{x}^{2} (H^{2})} ≲ \int_{0}^{s} t^{α - β - 1} ‖ t^{1 + β - α} {(- Δ)}^{1 + β - α} e^{t Δ} {(- Δ)}^{α} f ‖_{L_{x}^{2} (H^{2})} d t \\ ≲ \int_{0}^{s} t^{α - β - 1} ‖ {(- Δ)}^{α} f ‖_{L_{x}^{2} (H^{2})} d t = s^{α - β} ‖ {(- Δ)}^{α} f ‖_{L_{x}^{2} (H^{2})} . \end{array}

Similarly,

‖ {(- Δ)}^{α} P_{\geq s} f ‖_{L_{x}^{2} (H^{2})} = s^{β - α} ‖ s^{α - β} {(- Δ)}^{α - β} e^{s Δ} {(- Δ)}^{β} f ‖_{L_{x}^{2} (H^{2})} ≲ s^{β - α} ‖ {(- Δ)}^{β} f ‖_{L_{x}^{2} (H^{2})} .

□

5. Radial Sobolev embeddings

Lemma 2.20

(Lemma 2.13 in Ref. 25: Radial Sobolev embeddings in

H^{2}

⁠). For any s > 0 and any function f radial,

(‖ \sinh^{\frac{1}{2}} (r) f {(‖}_{L_{x}^{\infty} (H^{2})} ≲ ‖ f ‖_{L_{x}^{2} (H^{2})}^{\frac{1}{2}} ‖ \nabla f ‖_{L_{x}^{2} (H^{2})}^{\frac{1}{2}} ≲ s^{- \frac{1}{4}} ‖ {⟨s Δ⟩}^{\frac{1}{2}} f ‖_{L_{x}^{2} (H^{2})} .

Corollary 2.21

(Frequency localized radial Sobolev embeddings in

H^{2}

⁠). For any s > 0 and f radial,

(‖ \sinh^{\frac{1}{2}} (r) P_{s} f {‖)}_{L_{x}^{\infty} (H^{2})} ≲ s^{- \frac{1}{4}} ‖ P_{\frac{s}{2}} f ‖_{L_{x}^{2} (H^{2})} .

Proof.

Taking f = P_sf in Lemma 2.20, we have

(‖ \sinh^{\frac{1}{2}} (r) P_{s} f {‖)}_{L_{x}^{\infty} (H^{2})} ≲ s^{- \frac{1}{4}} (‖ {⟨s Δ⟩}^{\frac{1}{2}} P_{s} f {‖)}_{L_{x}^{2} (H^{2})} = s^{- \frac{1}{4}} (‖ P_{s} f ‖_{L_{x}^{2} (H^{2})} + ‖ {(s Δ)}^{\frac{1}{2}} P_{s} f ‖_{L_{x}^{2} (H^{2})}) .

Both terms in parentheses are bounded by

‖ P_{\frac{s}{2}} f ‖_{L_{x}^{2} (H^{2})}

⁠. In fact, by Definition 2.7, Remark 2.10, and Lemma 2.11,

\begin{array}{l} ‖ P_{s} f ‖_{L_{x}^{2} (H^{2})} = ‖ (s Δ) e^{s Δ} f ‖_{L_{x}^{2} (H^{2})} = ‖ e^{\frac{s}{2} Δ} (s Δ e^{\frac{s}{2} Δ} f) ‖_{L_{x}^{2} (H^{2})} ≲ ‖ P_{\frac{s}{2}} f ‖_{L_{x}^{2} (H^{2})}, \\ ‖ {(s Δ)}^{\frac{1}{2}} P_{s} f ‖_{L_{x}^{2} (H^{2})} = ‖ {(s Δ)}^{\frac{1}{2}} (s Δ) e^{s Δ} f ‖_{L_{x}^{2} (H^{2})} = ‖ {(s Δ)}^{\frac{1}{2}} e^{\frac{s}{2} Δ} (s Δ e^{\frac{s}{2} Δ} f) ‖_{L_{x}^{2} (H^{2})} ≲ ‖ P_{\frac{s}{2}} f ‖_{L_{x}^{2} (H^{2})} . \end{array}

□

Corollary 2.22.

For any s > 0, 1/4 < α < 1 and f radial,

‖ \sinh^{\frac{1}{2}} (r) f ‖_{L_{x}^{\infty} (H^{2})} ≲ ‖ f ‖_{L_{x}^{2} (H^{2})}^{1 - \frac{1}{4 α}} ‖ {(- Δ)}^{α} f ‖_{L_{x}^{2} (H^{2})}^{\frac{1}{4 α}} .

Proof.

We write f into its Littlewood–Paley decomposition, and using Corollary 2.21, we have

\begin{array}{l} |\sinh^{\frac{1}{2}} (r) f| & = |\int_{0}^{\infty} \sinh^{\frac{1}{2}} (r) P_{s} f \frac{d s}{s}| ≲ \int_{0}^{\infty} s^{- \frac{1}{4}} ‖ P_{\frac{s}{2}} f ‖_{L_{x}^{2} (H^{2})} \frac{d s}{s} \\ = \int_{0}^{T} s^{- \frac{1}{4}} ‖ P_{\frac{s}{2}} f ‖_{L_{x}^{2} (H^{2})} \frac{d s}{s} + \int_{T}^{\infty} s^{- \frac{1}{4}} ‖ P_{\frac{s}{2}} f ‖_{L_{x}^{2} (H^{2})} \frac{d s}{s} : = I + I I . \end{array}

Here, T is a constant that will be chosen later. Using Remark 2.10 and Lemma 2.11, we have the following two estimates for

‖ P_{s} f ‖_{L_{x}^{p} (H^{2})}

⁠. For 0 < θ < 1,

‖ P_{s} f ‖_{L_{x}^{2} (H^{2})} = ‖ s Δ e^{s Δ} f ‖_{L_{x}^{2} (H^{2})} ≲ ‖ f ‖_{L_{x}^{2} (H^{2})},

(2.1)

‖ P_{s} f ‖_{L_{x}^{2} (H^{2})} = ‖ s^{θ} s^{1 - θ} {(- Δ)}^{1 - θ} e^{s Δ} {(- Δ)}^{θ} f ‖_{L_{x}^{2} (H^{2})} ≲ s^{θ} ‖ {(- Δ)}^{θ} f ‖_{L_{x}^{2} (H^{2})} .

(2.2)

Now, by () and (), for 1/4 < α < 1,

\begin{array}{l} I & = \int_{0}^{T} s^{- \frac{1}{4}} ‖ P_{\frac{s}{2}} f ‖_{L_{x}^{2} (H^{2})} \frac{d s}{s} ≲ \int_{0}^{T} s^{- \frac{1}{4}} s^{α} ‖ {(- Δ)}^{α} f ‖_{L_{x}^{2} (H^{2})} \frac{d s}{s} = T^{α - \frac{1}{4}} ‖ {(- Δ)}^{α} f ‖_{L_{x}^{2} (H^{2})}, \\ I I & = \int_{T}^{\infty} s^{- \frac{1}{4}} ‖ P_{\frac{s}{2}} f ‖_{L_{x}^{2} (H^{2})} \frac{d s}{s} ≲ (\int_{T}^{\infty} s^{- \frac{1}{4}} \frac{d s}{s}) ‖ f ‖_{L_{x}^{2} (H^{2})} = T^{- \frac{1}{4}} ‖ f ‖_{L_{x}^{2} (H^{2})} . \end{array}

Therefore, for any T > 0,

‖ \sinh^{\frac{1}{2}} (r) f ‖_{L_{x}^{\infty} (H^{2})} ≲ T^{α - \frac{1}{4}} ‖ {(- Δ)}^{α} f ‖_{L_{x}^{2} (H^{2})} + T^{- \frac{1}{4}} ‖ f ‖_{L_{x}^{2} (H^{2})} .

Optimizing the choice of T, we obtain for 1/4 < α < 1,

‖ \sinh^{\frac{1}{2}} (r) f ‖_{L_{x}^{\infty} (H^{2})} ≲ ‖ {(- Δ)}^{α} f ‖_{L_{x}^{2} (H^{2})}^{\frac{1}{4 α}} ‖ f ‖_{L_{x}^{2} (H^{2})}^{1 - \frac{1}{4 α}} .

□

III. ENERGY INCREMENT ON $H^{2}$

In this section, we analyze a certain energy increment. As mentioned in the Introduction, we present a modified Morawetz type estimate in Subsection IV A and in Subsection IV B, we conclude the global well-posedness and scattering proof by showing that the spacetime L⁴ norm of the solution is uniformly bounded.

Let us recall schematically below the heat-flow-based Littlewood–Paley projections,

\begin{array}{l} P_{s} f = s (- Δ) e^{s Δ} f & ⇝ a projection to frequencies comparable to s^{- \frac{1}{2}}, \\ P_{\geq s} f = e^{s Δ} f = \int_{s}^{\infty} P_{s^{'}} f \frac{d s^{'}}{s^{'}} & ⇝ a projection to frequencies lower than \sim s^{- \frac{1}{2}}, \\ P_{\leq s} f = \int_{0}^{s} P_{s^{'}} f \frac{d s^{'}}{s^{'}} & ⇝ a projection to frequencies higher than \sim s^{- \frac{1}{2}} . \end{array}

Now, we decompose the initial data ϕ into a low frequency component $η_{0} = P_{> s_{0}} ϕ$ and a high frequency component $ψ_{0} = P_{\leq s_{0}} ϕ$ ⁠, where $s_{0}^{- 1}$ is a fixed large frequency and will be determined later in the proof. Note that $s_{0}^{- 1 / 2}$ plays the same role as N₀ in Ref. 5.

Using the above decomposition, we would like to write u into the sum of the following two solutions ψ and ζ, where $ψ = e^{i t Δ} P_{\leq s_{0}} ϕ$ solves the linear Schrödinger with high frequency data,

\{\begin{cases} i \partial_{t} ψ + Δ_{H^{2}} ψ = 0, \\ ψ (0, x) = ψ_{0} = P_{\leq s_{0}} ϕ, \end{cases})

(3.1)

and ζ solves the difference equation with low frequency data,

\{\begin{cases} i \partial_{t} ζ + Δ_{H^{2}} ζ = {|u|}^{2} u = G (ζ, ψ), \\ ζ (0, x) = η_{0} = P_{> s_{0}} ϕ, \end{cases})

(3.2)

and here, $G (ζ, ψ) = {|ζ + ψ|}^{2} (ζ + ψ) = {|ζ|}^{2} ζ + O (ζ^{2} ψ) + O (ζ ψ^{2}) + O (ψ^{3})$ ⁠.

A. Main results in this section

The main results in this section are a local energy increment (Proposition 3.1) and a conditional global energy increment (Proposition 3.2) for the solution ζ.

Proposition 3.1

(Local energy increment). Consider u as in () defined on

I \times H^{2}

, where I = [0, τ], such that

‖ u ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} = ε

(3.3)

for some universal constant ɛ. Then, for s > 3/4 and sufficiently small s₀, the solution ζ, under the decomposition u = ψ + ζ defined as in () and (), satisfies the following energy increment:

E (ζ (τ)) \leq E (ζ (0)) + C s_{0}^{\frac{3}{2} s - \frac{5}{4}} .

Proposition 3.2

(Conditional global energy increment). Consider u as in () defined on

[0, T] \times H^{2}

, where

‖ u ‖_{L_{t, x}^{4} ([0, T] \times H^{2})}^{4} \leq M

for some constant M. Then, for s > 3/4 and sufficiently small s₀, the energy of ζ satisfies the following energy increment:

E (ζ (T)) \leq E (ζ (0)) + C \frac{M}{ε} s_{0}^{\frac{3}{2} s - \frac{5}{4}},

where ɛ is the small constant in Proposition 3.1.

Remark 3.3.

T could be infinity. In fact, the ultimate goal of this paper is to show that the spacetime L⁴ norm is bounded for all time intervals, which implies scattering.

B. Proof of Proposition 3.1

To analyze the behavior of the solution ζ more carefully, we first make a further decomposition. That is, we would like to separate the differential equation (3.2) into a cubic NLS with low frequency data,

\{\begin{cases} I \partial_{t} ζ_{1} + Δ_{H^{2}} ζ_{1} = {|ζ_{1}|}^{2} ζ_{1}, \\ ζ_{1} (0, x) = η_{0} = P_{> s_{0}} ϕ, \end{cases})

(3.4)

and a difference equation with zero initial value,

\{\begin{cases} i \partial_{t} ζ_{2} + Δ_{H^{2}} ζ_{2} = {|u|}^{2} u - {|ζ_{1}|}^{2} ζ_{1}, \\ ζ_{2} (0, x) = 0 . \end{cases})

(3.5)

Hence, ζ = ζ₁ + ζ₂, and the full solution u is the sum of these three solutions,

u = ζ_{1} + ζ_{2} + ψ .

(3.6)

It is worth mentioning that the decomposition in Bourgain’s work⁵ is a cubic NLS with low frequency data,

\{\begin{cases} i \partial_{t} u_{0} + Δ u_{0} = {|u_{0}|}^{2} u_{0}, \\ u_{0} (0, x) = ϕ_{0} (x) = P_{< N_{0}} ϕ, \end{cases})

(3.7)

and a difference equation with high frequency data,

\{\begin{cases} i \partial_{t} v + Δ v = F (u_{0}, v), \\ v (0, x) = ψ_{0} (x) = P_{\geq N_{0}} ϕ, \end{cases})

(3.8)

where $F (u_{0}, v) = {|v|}^{2} v + 2 u_{0} {|v|}^{2} + ū_{0} v^{2} + u_{0}^{2} \bar{v} + 2 {|u_{0}|}^{2} v$ ⁠. Then, the full solution u is u = u₀ + v. In our work, we need to be more careful.

Note that $s_{0}^{- 1 / 2}$ plays the same role as N₀ in Ref. 5. When comparing these two decomposition, we can relate them in the following sense: ζ₁ is the same as u₀ and ψ + ζ₂ is, in fact, v, where ψ is the linear solution in (3.8) and ζ₂ is the Duhamel term w in (3.8).

ζ_{1} ↭ u_{0}, ζ_{2} ↭ w, ψ ↭ e^{i t Δ} P_{\leq s_{0}} ϕ .

Step 1: Understanding the decomposed initial data. Recall that we decomposed the initial data ϕ = η₀ + ψ₀, where $η_{0} = P_{> s_{0}} ϕ$ and $ψ_{0} = P_{\leq s_{0}} ϕ$ ⁠. Here, we list several facts of the decomposed initial data η₀ and ψ₀.

Fact 3.4.

For the low frequency data η₀,

\begin{array}{l} (1) η_{0} \in H^{1}, ‖ η_{0} ‖_{H_{x}^{1} (H^{2})} ≲ s_{0}^{\frac{1}{2} (s - 1)}, & (2) ‖ η_{0} ‖_{H_{x}^{σ} (H^{2})} ≲ s_{0}^{\frac{σ}{2} (s - 1)} for 0 < σ < 1, & (3) E (η_{0}) ≲ s_{0}^{s - 1} . \end{array}

In fact, (1) follows from Bernstein inequality (Lemma 2.19), (2) is by interpolating L² and H¹ norms, and (3) is due to Sobolev embedding (Lemma 2.14) and (2).

Fact 3.5.

For the high frequency data ψ₀,

\begin{array}{l} (1) ‖ ψ_{0} ‖_{L_{x}^{2} (H^{2})} ≲ s_{0}^{\frac{1}{2} s}, & (2) ‖ ψ_{0} ‖_{H_{x}^{s} (H^{2})} ≲ 1 . \end{array}

Here, (1) follows from Bernstein inequality (Lemma 2.19), while (2) is due to the fact that ϕ being in H^s.

Step 2: Estimation on the solution ψ of (3.1).

In fact, the solution ψ of the linear equation (3.1) is global, although it lives in a rough space H^s. Moreover, from the linear Strichartz estimates, Lemma 2.19 and (1) in Fact 3.5, one has

‖ ψ ‖_{L_{t, x}^{4} (R \times H^{2})} = ‖ e^{i t Δ} ψ_{0} ‖_{L_{t, x}^{4} (R \times H^{2})} ≲ ‖ ψ_{0} ‖_{L_{x}^{2} (H^{2})} ≲ s_{0}^{\frac{1}{2} s} .

(3.9)

More generally,

‖ ψ ‖_{S^{0} (R)} ≲ s_{0}^{\frac{1}{2} s} and ‖ ψ ‖_{S^{σ} (R)} ≲ s_{0}^{\frac{1}{2} (s - σ)} for 0 \leq σ \leq s .

(3.10)

Step 3: Estimation on the solutionζ₁of(3.4).

Lemma 3.6.

Due to the low frequency component η₀(x) of ϕ being in H¹, ζ₁(t) is a global solution and

‖ ζ_{1} (t) ‖_{H_{x}^{1} (H^{2})}

is conserved. More precisely,

\begin{array}{l} (1) ζ_{1} (t) exists globally and E (ζ_{1}) (t) = E (η_{0}) ≲ s_{0}^{s - 1}, & (2) ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} ≲ ε + s_{0}^{2 s} + ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} . \end{array}

Remark 3.7.

Ultimately, we will show $‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} ≲ ε$ in Corollary 3.9, and here, (2) is an intermediate step.

Proof of Lemma 3.6.

First, with the conservation of E(ζ₁) and (3) in Fact 3.4, it is easy to see that

E (ζ_{1} (t)) \equiv E (η_{0}) ≲ s_{0}^{s - 1} .

With η₀ being in H¹, thanks to Ref. 36, ζ₁ is globally well-posed, which proves (1).

For (2), recall the decomposition of u in (), then we simply use the triangle inequality, the assumption (), and () and obtain

‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} ≲ ‖ u ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} + ‖ ψ ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} + ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} ≲ ε + s_{0}^{2 s} + ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} .

□

Step 4: Estimation on the solution ζ₂ of (3.5) and extra estimates on ζ₁.

Recall (3.5)

\{\begin{cases} i \partial_{t} ζ_{2} + Δ_{H^{2}} ζ_{2} = F (ζ_{1}, ζ_{2}, ψ), \\ ζ_{2} (0, x) = 0, \end{cases})

where

F (ζ_{1}, ζ_{2}, ψ) = {|u|}^{2} u - {|ζ_{1}|}^{2} ζ_{1} = O (ζ_{2}^{3}) + O (ψ^{3}) + O (ζ_{2}^{2} ζ_{1}) + O (ψ^{2} ζ_{1}) + O (ζ_{2} ζ_{1}^{2}) + O (ψ ζ_{1}^{2}) .

Lemma 3.8.

The solution ζ₂ satisfies the following estimates on I:

\begin{array}{l} (1) ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})} ≲ s_{0}^{\frac{1}{2} s}, & (2) ‖ ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})} ≲ s_{0}^{\frac{1}{2} s}, & (3) ‖ ζ_{2} ‖_{L_{t}^{\infty} H_{x}^{1} (I \times H^{2})} ≲ s_{0}^{s - \frac{3}{4}} . \end{array}

Proof of Lemma 3.8.

Noting that () has zero initial value, we write out the integral equation using its Duhamel formula,

ζ_{2} (t) = i \int_{0}^{t} e^{i (t - s) Δ_{H^{2}}} F d s .

By Strichartz estimates and Hölder inequality, we have

‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})} ≲ ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})}^{3} + ‖ ψ ‖_{L_{t, x}^{4} (I \times H^{2})}^{3} + ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})} ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{2} + ‖ ψ ‖_{L_{t, x}^{4} (I \times H^{2})} ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{2} .

(3.11)

Note that here there should be two more nonlinear terms in F that contribute to

‖ F ‖_{L_{t, x}^{4 / 3}}

in (), which are

O (ζ_{2}^{2} ζ_{1})

and

O (ψ^{2} ζ_{1})

⁠. However, we dropped them, since their contributions are controlled by a multiple of those of the four nonlinear terms that are written in (). We will also drop them in the rest of this paper.

Using () and (2) in Lemma 3.6, we write () into

\begin{aligned} ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})} ≲ ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})}^{3} + s_{0}^{\frac{3}{2} s} + ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})} (ε^{\frac{1}{2}} + s_{0}^{s} + ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})}^{2}) + s_{0}^{\frac{1}{2} s} (ε^{\frac{1}{2}} + s_{0}^{s} + ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})}^{2}) . \end{aligned}

Noting that initially ζ₂(0) = 0, then by a continuity argument, we obtain

‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})} ≲ s_{0}^{\frac{1}{2} s},

which proves (1). The estimate in () also works for

‖ ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})}

⁠, and hence, (2) holds.

We postpone the Proof of (3) to step 6.□

With enough estimates on ζ₂ in hand, as we promised in Lemma 3.6, we will finish the analysis of ζ₁.

Corollary 3.9.

As a consequence of (1) in Lemma 3.8,

we improve the bound in Lemma 3.6 by $‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} ≲ ε + s_{0}^{2 s} ≲ ε$ and
obtain $‖ ζ_{1} ‖_{S^{σ} (I)} ≲ s_{0}^{\frac{σ}{2} (s - 1)}$ ⁠, where 0 ≤ σ ≤ 1.

Proof of Corollary 3.9.

Combining (2) in Lemma 3.6 and (1) in Lemma 3.8, it is easy to see that (1) holds.

For (2), we use the integral equation corresponding to the initial value problem via the Duhamel principle, Strichartz estimates, and (1), and we obtain

\begin{array}{l} ‖ ζ_{1} ‖_{S^{0} (I)} & ≲ ‖ η_{0} ‖_{L_{x}^{2} (H^{2})} + ‖ {|ζ_{1}|}^{2} ζ_{1} ‖_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} ≲ 1 + ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{3} ≲ 1 + ε^{\frac{3}{4}}, \\ ‖ ζ_{1} ‖_{S^{1} (I)} & ≲ ‖ \nabla η_{0} ‖_{L_{x}^{2} (H^{2})} + ‖ \nabla {|ζ_{1}|}^{2} ζ_{1} ‖_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} ≲ s_{0}^{\frac{1}{2} (s - 1)} + ‖ ζ_{1} ‖_{S^{1} (I)} ε^{\frac{1}{2}} . \end{array}

Therefore,

‖ ζ_{1} ‖_{S^{0} (I)} ≲ 1 and ‖ ζ_{1} ‖_{S^{1} (I)} ≲ s_{0}^{\frac{1}{2} (s - 1)} .

(3.12)

Then, interpolating (), we have that for 0 < σ < 1,

‖ ζ_{1} ‖_{S^{σ} (I)} ≲ s_{0}^{\frac{σ}{2} (s - 1)} .

□

Step 5: Local energy increment.

Now, we are ready to compute the energy increment from 0 to τ and show such an increment is as described in Proposition 3.1. That is, we will show

E (ζ (τ)) = E (ζ_{1} (τ) + ζ_{2} (τ)) = E (ζ_{1} (τ)) + (E (ζ_{1} (τ) + ζ_{2} (τ)) - E (ζ_{1} (τ))) \leq E (η_{0}) + C s_{0}^{\frac{3}{2} s - \frac{5}{4}} .

In fact, a direct computation of the difference of the energy gives

\begin{array}{l} |E (ζ_{1} (τ) + ζ_{2} (τ)) - E (ζ_{1} (τ))| \\ ≲ (‖ ζ_{1} (τ) ‖_{H_{x}^{1} (H^{2})} + ‖ ζ_{2} (τ) ‖_{H_{x}^{1} (H^{2})}) ‖ ζ_{2} (τ) ‖_{H_{x}^{1} (H^{2})} + ‖) {(|ζ_{1} (τ)| + |ζ_{2} (τ)|)}^{3} |ζ_{2} (τ)| {(‖}_{L_{x}^{1} (H^{2})} : = I + I I . \end{array}

By the energy conservation of ζ₁, (1) in Lemma 3.6, and (3) in Lemma 3.8,

\begin{array}{l} I & = (‖ ζ_{1} (τ) ‖_{H_{x}^{1} (H^{2})} + ‖ ζ_{2} (τ) ‖_{H_{x}^{1} (H^{2})}) ‖ ζ_{2} (τ) ‖_{H_{x}^{1} (H^{2})} \leq E {(η_{0})}^{\frac{1}{2}} ‖ ζ_{2} (τ) ‖_{H_{x}^{1} (H^{2})} + ‖ ζ_{2} (τ) ‖_{H_{x}^{1} (H^{2})}^{2} \\ ≲ s_{0}^{\frac{1}{2} (s - 1)} s_{0}^{s - \frac{3}{4}} + s_{0}^{2 (s - \frac{3}{4})} ≲ s_{0}^{\frac{3}{2} s - \frac{5}{4}}, \end{array}

for s > 1/2. [ $‖ ζ_{1} (τ) ‖_{H_{x}^{1}} ‖ ζ_{2} (τ) ‖_{H_{x}^{1}}$ dominates in I.]

Using Sobolev embedding, Lemma 2.13, (2) and (3) in Lemma 3.8, and (1) in Lemma 3.6, we have the following L⁴ norm estimates for ζ₁ and ζ₂:

\begin{array}{l} ‖ ζ_{2} (τ) ‖_{L_{x}^{4} (H^{2})} & ≲ ‖ ζ_{2} (τ) ‖_{H_{x}^{\frac{1}{2}} (H^{2})} ≲ ‖ ζ_{2} (τ) ‖_{L_{x}^{2} (H^{2})}^{\frac{1}{2}} ‖ ζ_{2} (τ) ‖_{H_{x}^{1} (H^{2})}^{\frac{1}{2}} ≲ s_{0}^{\frac{3}{4} s - \frac{3}{8}}, \\ ‖ ζ_{1} (τ) ‖_{L_{x}^{4} (H^{2})} & ≲ E {(ζ_{1})}^{\frac{1}{4}} ≲ s_{0}^{\frac{1}{4} (s - 1)} . \end{array}

Combining with Hölder inequality, we compute

\begin{array}{l} I I & = ‖) {(|ζ_{1} (τ)| + |ζ_{2} (τ)|)}^{3} |ζ_{2} (τ)| {(‖}_{L_{x}^{1} (H^{2})} ≲ {(‖ ζ_{1} (τ) ‖_{L_{x}^{4} (H^{2})} + ‖ ζ_{2} (τ) ‖_{L_{x}^{4} (H^{2})})}^{3} ‖ ζ_{2} (τ) ‖_{L_{x}^{4} (H^{2})} \\ ≲ {(s_{0}^{\frac{1}{4} (s - 1)} + s_{0}^{\frac{3}{4} s - \frac{3}{8}})}^{3} s_{0}^{\frac{3}{4} s - \frac{3}{8}} ≲ s_{0}^{\frac{3}{2} s - \frac{9}{8}} \end{array}

for s > 1/4. [ $‖ ζ_{1} (τ) ‖_{L_{x}^{4} (H^{2})}^{3} ‖ ζ_{2} (τ) ‖_{L_{x}^{4} (H^{2})}$ dominates in II, and I dominates in E(ζ(τ)) − E(ζ(0)).]

Now, we finish the calculation of the analysis of the energy increment in Proposition 3.1.

Step 6: Proof of (3) in Lemma 3.8.

Before proving (3) in Lemma 3.8, we first state the following lemma:

Lemma 3.10.

For t ∈ I defined in (), we have for 0 ≤ σ ≤ s,

‖ ζ_{2} ‖_{S^{σ} (I)} ≲ s_{0}^{\frac{1}{2} (σ s + s - σ)} ≲ s_{0}^{\frac{1}{2} (s - σ)} .

(3.13)

Proof of Lemma 3.10.

For 0 < σ < s, by the integral equation, Bernstein inequality (Lemma 2.19), Lemma 2.17, Strichartz inequalities, and (),

\begin{align} ‖ ζ_{2} ‖_{S^{σ} (I)} & ≲ ‖) {⟨- Δ⟩}^{\frac{σ}{2}} O (ζ_{2}^{3}) {(‖}_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} + ‖) {⟨- Δ⟩}^{\frac{σ}{2}} O (ψ^{3}) {(‖}_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} \\ + ‖) {⟨- Δ⟩}^{\frac{σ}{2}} O (ζ_{2} ζ_{1}^{2}) {(‖}_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} + ‖) {⟨- Δ⟩}^{\frac{σ}{2}} O (ψ ζ_{1}^{2}) {(‖}_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} \\ ≲ ‖ ζ_{2} ‖_{S^{σ} (I)} ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})}^{2} + ‖ ψ ‖_{S^{σ} (I)} ‖ ψ ‖_{L_{t, x}^{4} (I \times H^{2})}^{2} + ‖ ζ_{1} ‖_{S^{σ} (I)} ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})} ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})} \\ + ‖ ζ_{2} ‖_{S^{σ} (I)} ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{2} + ‖ ζ_{2} ‖_{S^{σ} (I)} ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{2} + ‖ ζ_{1} ‖_{S^{σ} (I)} ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})} ‖ ψ ‖_{L_{t, x}^{4} (I \times H^{2})} . \end{align}

(3.14)

Using (1) in Lemma 3.8, (), (), and Corollary 3.9, we write () into

‖ ζ_{2} ‖_{S^{σ} (I)} ≲ ‖ ζ_{2} ‖_{S^{σ} (I)} s_{0}^{s} + s_{0}^{\frac{1}{2} (s - σ)} s_{0}^{s} + s_{0}^{\frac{σ}{2} (s - 1)} ε^{\frac{1}{4}} s_{0}^{\frac{1}{2} s} + ‖ ζ_{2} ‖_{S^{σ} (I)} ε^{\frac{1}{2}} + ‖ ζ_{2} ‖_{S^{σ} (I)} ε^{\frac{1}{2}} + s_{0}^{\frac{σ}{2} (s - 1)} ε^{\frac{1}{4}} s_{0}^{\frac{1}{2} s} .

Then, we have

‖ ζ_{2} ‖_{S^{σ} (I)} ≲ s_{0}^{\frac{σ}{2} (s - 1)} s_{0}^{\frac{1}{2} s} = s_{0}^{\frac{1}{2} (σ s + s - σ)} .

□

Finally, we arrive at the Proof of (3) in Lemma 3.8.

Proof of (3) in Lemma 3.8.

In this step, we prove the smoothness of the solution ζ₂ using the local smoothing estimate and the radial assumption of the initial data. In fact, this is the only place where the radial assumption is used, and all other steps work for all general data.

First, by Strichartz inequalities, we write

\begin{align} ‖ \nabla ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})} & ≲ ‖ \nabla O (ζ_{2}^{3}) ‖_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} + ‖ \nabla O (ψ^{3}) ‖_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} + ‖ \nabla O (ζ_{2} ζ_{1}^{2}) ‖_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} + ‖ \nabla O (ψ ζ_{1}^{2}) ‖_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} \\ : = I + I I + I I I + I V . \end{align}

(3.15)

Before estimating I–IV, we compute the following norms that are needed in the rest of this proof:

Claim 3.11.

For

r = |x|

⁠,

\begin{array}{l} (1) ‖) \sinh^{\frac{1}{2}} (r) ψ {(‖}_{L_{t, x}^{\infty} (R \times H^{2})} ≲ s_{0}^{\frac{1}{2} s - \frac{1}{4}}, & (2) ‖) \sinh^{\frac{1}{2}} (r) ζ_{1} {(‖}_{L_{t, x}^{\infty} (I \times H^{2})} ≲ s_{0}^{\frac{1}{4} (s - 1)}, & (3) ‖) \sinh^{\frac{1}{2}} (r) ζ_{2} {(‖}_{L_{t, x}^{\infty} (I \times H^{2})} ≲ s_{0}^{\frac{1}{2} s - \frac{1}{4}} . \end{array}

Proof of Claim 3.11.

Using the radial Sobolev embedding (Corollary 2.22), Strichartz estimates, and Fact 3.5,

\begin{array}{l} ‖) \sinh^{\frac{1}{2}} (r) ψ {(‖}_{L_{t, x}^{\infty} (R \times H^{2})} & ≲ ‖ ψ ‖_{L_{t}^{\infty} L_{x}^{2} (R \times H^{2})}^{1 - \frac{1}{4 α}} ‖ {(- Δ)}^{α} ψ ‖_{L_{t}^{\infty} L_{x}^{2} (R \times H^{2})}^{\frac{1}{4 α}} \\ ≲ ‖ ψ_{0} ‖_{L_{x}^{2} (H^{2})}^{1 - \frac{1}{4 α}} ‖ ψ_{0} ‖_{H_{x}^{2 α} (H^{2})}^{\frac{1}{4 α}} ≲ s_{0}^{\frac{1}{2} s \times (1 - \frac{1}{4 α})} s_{0}^{\frac{1}{2} (s - 2 α) \times \frac{1}{4 α}} = s_{0}^{\frac{1}{2} s - \frac{1}{4}} . \end{array}

By Corollaries 2.22 and 3.9,

‖) \sinh^{\frac{1}{2}} (r) ζ_{1} {(‖}_{L_{t, x}^{\infty} (I \times H^{2})} ≲ ‖ ζ_{1} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})}^{\frac{1}{2}} ‖ \nabla ζ_{1} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})}^{\frac{1}{2}} ≲ s_{0}^{\frac{1}{4} (s - 1)} .

By Corollary 2.22 and Lemma 3.10,

‖) \sinh^{\frac{1}{2}} (r) ζ_{2} {(‖}_{L_{t, x}^{\infty} (I \times H^{2})} ≲ ‖ ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})}^{1 - \frac{1}{4 α}} ‖ {(- Δ)}^{α} ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})}^{\frac{1}{4 α}} ≲ s_{0}^{\frac{1}{2} s \times (1 - \frac{1}{4 α})} s_{0}^{\frac{1}{2} (2 α s + s - 2 α) \times \frac{1}{4 α}} = s_{0}^{\frac{3}{4} s - \frac{1}{4}} .

□

Let us also recall some estimates from Subsection III A [(3.10), (3.9), Corollary 3.9, Lemmas 3.10, and 3.8],

\begin{align} ‖ ψ ‖_{S^{σ} (R)} ≲ s_{0}^{\frac{1}{2} (s - σ)} for 0 \leq σ \leq s, & ‖ ψ ‖_{L_{t, x}^{4} (R \times H^{2})} ≲ s_{0}^{\frac{1}{2} s}, \\ ‖ ζ_{1} ‖_{S^{σ} (I)} ≲ s_{0}^{\frac{σ}{2} (s - 1)} for 0 \leq σ \leq 1, & ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{4} ≲ ε, \\ ‖ ζ_{2} ‖_{S^{σ} (I)} ≲ s_{0}^{\frac{1}{2} (s - σ)} for 0 \leq σ \leq s, & ‖ ζ_{2} ‖_{L_{t, x}^{4} (I \times H^{2})} ≲ s_{0}^{\frac{1}{2} s} . \end{align}

(3.16)

Now, we continue working on (3.15). By Lemma 3.10,

I = ‖ \nabla O (ζ_{2}^{3}) ‖_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} ≲ ‖ \nabla ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})} ‖ ζ_{2} ‖_{L_{t}^{\frac{8}{3}} L_{x}^{8} (I \times H^{2})}^{2} ≲ s_{0}^{s} ‖ \nabla ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})} .

This term will be absorbed by the left-hand side of (3.15).

For II, we employ the local smoothing estimate. Since ψ is a linear solution, the linear version should be enough for this term. To implement the local smoothing estimate, we would like to introduce the weight ${⟨x⟩}^{- 1 / 2 - ε_{1}}$ ⁠, where ɛ₁ is a small positive number, and split out half derivative from the full gradient. Then, by the chain rule and Hölder inequality, we write

{‖\nabla O (ψ^{3})‖}_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} ≲ {‖{⟨x⟩}^{- \frac{1}{2} - ε_{1}}, {|\nabla|}^{\frac{1}{2}}, ({|\nabla|}^{\frac{1}{2}} ψ)‖}_{L_{t, x}^{2} (I_{\times} H^{2})} {‖{⟨x⟩}^{\frac{1}{2} + ε_{1}}, O, (ψ^{2})‖}_{L_{t, x}^{4} (I \times H^{2})} .

(3.17)

Now, we compute the two factors above separately. Using the linear local smoothing estimate (Lemma II C 3) and Lemma 2.19, we write the first factor into

{‖{⟨x⟩}^{- \frac{1}{2} - ε_{1}}, {|\nabla|}^{\frac{1}{2}}, ({|\nabla|}^{\frac{1}{2}} ψ)‖}_{L_{t, x}^{2} (I \times H^{2})} ≲ {‖{|\nabla|}^{\frac{1}{2}}, P_{\leq s_{0}}, ϕ‖}_{L_{x}^{2} (H^{2})} ≲ s_{0}^{\frac{1}{2} (s - \frac{1}{2})} .

(3.18)

To estimate the second factor in (3.17), by Hölder inequality, Sobolev embedding, (3.10), (3.9), and Claim 3.11, we have

\begin{align} {‖{⟨x⟩}^{\frac{1}{2} + ε_{1}}, O, (ψ^{2})‖}_{L_{t, x}^{4} (I \times H^{2})} & ≲ ‖ χ_{{|x| \leq 1}} O (ψ^{2}) ‖_{L_{t, x}^{4} (I \times H^{2})} + {‖χ_{{|x| > 1}}, {|x|}^{\frac{1}{2} + ε_{1}}, O, (ψ^{2})‖}_{L_{t, x}^{4} (I \times H^{2})} \\ ≲ ‖ ψ ‖_{L_{t, x}^{8} (I \times H^{2})}^{2} + ‖ \sinh^{\frac{1}{2}} (r) ψ ‖_{L_{t, x}^{\infty} (I \times H^{2})} ‖ ψ ‖_{L_{t, x}^{4} (I \times H^{2})} \\ ≲ s_{0}^{s - \frac{1}{2}} + s_{0}^{\frac{1}{2} s - \frac{1}{4}} s_{0}^{\frac{1}{2} s} ≲ s_{0}^{s - \frac{1}{2}} \end{align}

(3.19)

for all s. Then, combining (3.18) and (3.19), we obtain

I I ≲ (3.17) ≲ s_{0}^{\frac{1}{2} (s - \frac{1}{2})} s_{0}^{s - \frac{1}{2}} = s_{0}^{\frac{3}{2} s - \frac{3}{4}} .

For III and IV, we write them in a similar way as in (3.17),

I I I ≲ ‖ \nabla O (ζ_{2} ζ_{1}^{2}) ‖_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} ≲ {‖{⟨x⟩}^{- \frac{1}{2} - ε_{1}}, {|\nabla|}^{\frac{1}{2}}, ({|\nabla|}^{\frac{1}{2}} ζ_{2})‖}_{L_{t, x}^{2} (I \times H^{2})} {‖{⟨x⟩}^{\frac{1}{2} + ε_{1}}, O, (ζ_{1}^{2})‖}_{L_{t, x}^{4} (I \times H^{2})}

(3.20)

\begin{align} + {‖{⟨x⟩}^{- \frac{1}{2} - ε_{1}}, {|\nabla|}^{\frac{1}{2}}, ({|\nabla|}^{\frac{1}{2}} ζ_{1})‖}_{L_{t, x}^{2} (I \times H^{2})} {‖{⟨x⟩}^{\frac{1}{2} + ε_{1}}, O, (ζ_{1} ζ_{2})‖}_{L_{t, x}^{4} (I \times H^{2})}, \\ I V ≲ ‖ \nabla O (ψ ζ_{1}^{2}) ‖_{L_{t, x}^{\frac{4}{3}} (I \times H^{2})} & ≲ {‖{⟨x⟩}^{- \frac{1}{2} - ε_{1}}, {|\nabla|}^{\frac{1}{2}}, ({|\nabla|}^{\frac{1}{2}} ψ)‖}_{L_{t, x}^{2} (I \times H^{2})} {‖{⟨x⟩}^{\frac{1}{2} + ε_{1}}, O, (ζ_{1}^{2})‖}_{L_{t, x}^{4} (I \times H^{2})} \end{align}

(3.21)

+ {‖{⟨x⟩}^{- \frac{1}{2} - ε_{1}}, {|\nabla|}^{\frac{1}{2}}, ({|\nabla|}^{\frac{1}{2}} ζ_{1})‖}_{L_{t, x}^{2} (I \times H^{2})} {‖{⟨x⟩}^{\frac{1}{2} + ε_{1}}, O, (ζ_{1} ψ)‖}_{L_{t, x}^{4} (I \times H^{2})} .

Noting that the estimates for ζ₂ in Claim 3.11 and (3.16) are equally good or better than those for ψ, we will only present the estimation of IV. Our goal here is to prove that IV is bounded by $s_{0}^{s - 3 / 4}$ ⁠. In fact, III is also bounded by $s_{0}^{s - 3 / 4}$ ⁠.

We first start with the terms with no derivative and positive weights. By Hölder inequality, Sobolev embedding, Corollary 3.9, and Claim 3.11,

‖ {⟨x⟩}^{\frac{1}{2} + ε_{1}} O (ζ_{1}^{2}) ‖_{L_{t, x}^{4} (I \times H^{2})} ≲ ‖ ζ_{1} ‖_{L_{t, x}^{8} (I \times H^{2})}^{2} + ‖ \sinh^{\frac{1}{2}} (r) ζ_{1} ‖_{L_{t, x}^{\infty} (I \times H^{2})} ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})} ≲ s_{0}^{\frac{1}{2} (s - 1)} + s_{0}^{\frac{1}{4} (s - 1)} ≲ s_{0}^{\frac{1}{2} (s - 1)}

for s < 1.

By Hölder inequality, Sobolev embedding, Corollary 3.9, (3.10), and Claim 3.11,

\begin{array}{l} ‖ {⟨x⟩}^{\frac{1}{2} + ε_{1}} O (ζ_{1} ψ) ‖_{L_{t, x}^{4} (I \times H^{2})} & ≲ ‖ ζ_{1} ‖_{L_{t, x}^{12} (I \times H^{2})} ‖ ψ ‖_{L_{t, x}^{6} (I \times H^{2})} + ‖ \sinh^{\frac{1}{2}} (r) ψ ‖_{L_{t, x}^{\infty} (I \times H^{2})} ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})} \\ ≲ s_{0}^{\frac{1}{3} (s - 1)} s_{0}^{\frac{1}{2} (s - \frac{1}{3})} + s_{0}^{\frac{1}{2} s - \frac{1}{4}} ε^{\frac{1}{4}} ≲ s_{0}^{\frac{1}{2} s - \frac{1}{4}} \end{array}

for s > 3/4.

Then, we focus on the terms with derivatives and negative weights.

Note that we have treated $‖ {⟨x⟩}^{- \frac{1}{2} - ε_{1}} {|\nabla|}^{\frac{1}{2}} ({|\nabla|}^{\frac{1}{2}} ψ) ‖_{L_{t, x}^{2} (I \times H^{2})}$ in (3.18).

By Hölder inequality, Fact 3.4, Sobolev embedding, Corollary 3.9, and Claim 3.11,

\begin{array}{l} ‖ {⟨x⟩}^{- \frac{1}{2} - ε_{1}} {|\nabla|}^{\frac{1}{2}} ({|\nabla|}^{\frac{1}{2}} ζ_{1}) ‖_{L_{t, x}^{2} (I \times H^{2})} & ≲ ‖ {|\nabla|}^{\frac{1}{2}} η_{0} ‖_{L_{x}^{2} (H^{2})} + ‖ {⟨x⟩}^{\frac{1}{2} + ε_{1}} {|ζ_{1}|}^{2} ζ_{1} ‖_{L_{t, x}^{2} (I \times H^{2})} \\ ≲ s_{0}^{\frac{1}{4} (s - 1)} + ‖ ζ_{1} ‖_{L_{t, x}^{6} (I \times H^{2})}^{3} + ‖ \sinh^{\frac{1}{2}} (r) ζ_{1} ‖_{L_{t, x}^{\infty} (I \times H^{2})} ‖ ζ_{1} ‖_{L_{t, x}^{4} (I \times H^{2})}^{2} \\ ≲ s_{0}^{\frac{1}{4} (s - 1)} + s_{0}^{\frac{1}{6} (s - 1) \times 3} + s_{0}^{\frac{1}{4} (s - 1)} ε^{\frac{1}{2}} ≲ s_{0}^{\frac{1}{2} (s - 1)} \end{array}

for s < 1.

Therefore, continue from (3.20) and (3.21),

I I I \leq I V ≲ s_{0}^{\frac{1}{2} (s - \frac{1}{2})} s_{0}^{\frac{1}{2} (s - 1)} + s_{0}^{\frac{1}{2} (s - 1)} s_{0}^{\frac{1}{2} s - \frac{1}{4}} ≂ s_{0}^{s - \frac{3}{4}} for all s .

Combining all the terms I–IV, we write (3.15) into

‖ \nabla ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})} ≲ I + I I + I I I + I V ≲ s_{0}^{s} ‖ \nabla ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})} + s_{0}^{\frac{3}{2} s - \frac{3}{4}} + s_{0}^{s - \frac{3}{4}} + s_{0}^{s - \frac{3}{4}} .

Then, we have

‖ \nabla ζ_{2} ‖_{L_{t}^{\infty} L_{x}^{2} (I \times H^{2})} ≲ s_{0}^{s - \frac{3}{4}} .

This concludes the Proof of (3) in Lemma 3.8 and completes the analysis of the energy increment.□

C. Proof of Proposition 3.2

We divide the time interval [0, T] into [0, T] = ∪_iI_i = ∪_i[a_i, a_i+1] such that on each I_i,

‖ u ‖_{L_{t, x}^{4} (I_{i} \times H^{2})}^{4} = ε .

(3.22)

Hence,

# I_{i} \sim \frac{M}{ε} .

Let us remark that the length of such small intervals could be very long, and if some of them is an infinite interval, say [a_k, ∞), then we just call a_k+1 = ∞.

On the first interval I₁ = [0, a₁], we can apply Proposition 3.1 and have the local energy increment

E (ζ (a_{1})) \leq E (ζ (0)) + C s_{0}^{\frac{3}{2} s - \frac{5}{4}} .

On the second time interval I₂ = [a₁, a₂], we solve ζ by solving a cubic NLS with smoother data,

\{\begin{cases} i \partial_{t} ζ_{1}^{(1)} + Δ_{H^{2}} ζ_{1}^{(1)} = {|ζ_{1}^{(1)}|}^{2} ζ_{1}^{(1)}, \\ ζ_{1}^{(1)} (a_{1}, x) = ζ (a_{1}) = ζ_{1} (a_{1}) + ζ_{2} (a_{1}), \end{cases})

(3.23)

and a difference equation with zero initial value,

\{\begin{cases} i \partial_{t} ζ_{2}^{(1)} + Δ_{H^{2}} ζ_{2}^{(1)} = {|u|}^{2} u - {|ζ_{1}^{(1)}|}^{2} ζ_{1}^{(1)}, \\ ζ_{2}^{(1)} (a_{1}, x) = 0 . \end{cases})

(3.24)

Hence, $ζ = ζ_{1}^{(1)} + ζ_{2}^{(1)}$ ⁠, and the full solution will be $u = ζ + ψ = ζ_{1}^{(1)} + ζ_{2}^{(1)} + ψ$ ⁠. Applying Proposition 3.1 again, we see that

E (ζ (a_{2})) \leq E (ζ (a_{1})) + C s_{0}^{\frac{3}{2} s - \frac{5}{4}} .

The reason why we are safe to apply Proposition 3.1 on I₂ is that new decomposed initial data ζ(a₁) and ψ(a₁) satisfy all the facts in Facts 3.4 and 3.5, and all the calculations that we did in Proposition 3.1 will apply to the new systems (3.23) and (3.24). In particular, the size of new initial data ζ(a₁) in energy is the size of ζ(0) in (3.2) plus a small error from ζ₂(a₁), which can be seen from Proposition 3.1,

E (ζ (a_{1})) \leq E (ζ (0)) + \underset{small error}{\underset{︸}{C s_{0}^{\frac{3}{2} s - \frac{5}{4}}}} \sim E (ζ (0)) ≲ s_{0}^{s - 1} .

In addition, the H¹ norm of ζ(a₁) can be thought as the H¹ norm of ζ(0) plus a small error,

‖ ζ (a_{1}) ‖_{H_{x}^{1} (H^{2})} \leq ‖ ζ_{1} (a_{1}) ‖_{H_{x}^{1} (H^{2})} + ‖ ζ_{2} (a_{1}) ‖_{H_{x}^{1} (H^{2})} ≲ \underset{size of H^{1} norm of ζ (0)}{\underset{︸}{s_{0}^{\frac{1}{2} (s - 1)}}} + \underset{small error}{\underset{︸}{s_{0}^{s - \frac{3}{4}}}} \sim s_{0}^{\frac{1}{2} (s - 1)} .

Then, we can continue this iteration as long as the accumulated energy increment does not surpass the size of the initial energy of ζ(0), which guarantees that the setup for the smoother component remains the same size in the next iteration. That is,

C \frac{M}{ε} s_{0}^{\frac{3}{2} s - \frac{5}{4}} \leq E (ζ (0)) \sim s_{0}^{s - 1},

which gives

M \sim s_{0}^{- \frac{1}{2} s + \frac{1}{4}} .

(3.25)

The total energy increment is

E (ζ (T)) \leq E (ζ (0)) + C \frac{M}{ε} s_{0}^{\frac{3}{2} s - \frac{5}{4}} .

(3.26)

For now, we finish the Proof of Proposition 3.2 and give the choice of s₀.

Remark 3.12

(Boundedness of H^s norm of u). As a consequence of Proposition 3.2, we conclude that the H^s norm of u has the following bound:

‖ u (T) ‖_{H_{x}^{s} (H^{2})} ≲ s_{0}^{\frac{1}{2} (s - 1) s} .

In fact, () implies the boundedness of the H¹ norm of ζ(T),

‖ ζ (T) ‖_{H_{x}^{1} (H^{2})}^{2} \leq E (ζ (T)) \leq E (ζ (0)) + C \frac{M}{ε} s_{0}^{\frac{3}{2} s - \frac{5}{4}} ≲ s_{0}^{s - 1} .

(3.27)

The triangle inequality and the mass conservation laws of u and ψ with () give the boundedness of L² norm of ζ(T),

‖ ζ (T) ‖_{L_{x}^{2} (H^{2})} \leq ‖ u (T) ‖_{L_{x}^{2} (H^{2})} + ‖ ψ (T) ‖_{L_{x}^{2} (H^{2})} ≲ ‖ u (0) ‖_{L_{x}^{2} (H^{2})} + s_{0}^{\frac{1}{2} s} \leq 2 ‖ u (0) ‖_{L_{x}^{2} (H^{2})} .

(3.28)

Then, the H^s bound ζ(T) follows from the interpolation () and (),

‖ ζ (T) ‖_{H_{x}^{s} (H^{2})} ≲ ‖ ζ (T) ‖_{L_{x}^{2} (H^{2})}^{1 - s} ‖ ζ (T) ‖_{H_{x}^{1} (H^{2})}^{s} ≲ s_{0}^{\frac{1}{2} (s - 1) s} .

(3.29)

Therefore, the H^s norm of u(T) is bounded due to () and the fact

ψ \in H^{s} (H^{2})

⁠,

‖ u (T) ‖_{H_{x}^{s} (H^{2})} \leq ‖ ζ (T) ‖_{H_{x}^{s} (H^{2})} + ‖ ψ (T) ‖_{H_{x}^{s} (H^{2})} ≲ s_{0}^{\frac{1}{2} (s - 1) s} + 1 ≲ s_{0}^{\frac{1}{2} (s - 1) s} .

Consequently, we also have the bound of the H^σ (0 < σ < s) norm by interpolating the H^s with L² norms,

‖ u (T) ‖_{H_{x}^{σ} (H^{2})} ≲ ‖ u (T) ‖_{H_{x}^{s} (H^{2})}^{\frac{σ}{s}} ‖ u (T) ‖_{L_{x}^{2} (H^{2})}^{1 - \frac{σ}{2}} ≲ s_{0}^{\frac{σ}{2} (s - 1)} .

IV. GLOBAL WELL-POSEDNESS AND SCATTERING ON $H^{2}$

In this section, we use a bootstrapping argument to finally show the global well-posedness and scattering results stated in Theorem 1.1. In fact, we will show that the Morawetz norm is uniformly bounded, and once we have it, the proof of scattering will be standard.

A. Morawetz estimates on $H^{2}$

We first recall that the Morawetz estimate of the cubic NLS on $H^{2}$ in Ref. 36, when u is the solution to the cubic NLS equation $i \partial_{t} u + Δ_{H^{2}} u = {|u|}^{2} u$ ⁠, reads as

‖ u ‖_{L_{t, x}^{4} ([t_{1}, t_{2}] \times H^{2})}^{4} ≲ ‖ u ‖_{L_{t}^{\infty} L_{x}^{2} ([t_{1}, t_{2}] \times H^{2})} ‖ u ‖_{L_{t}^{\infty} H_{x}^{1} ([t_{1}, t_{2}] \times H^{2})} .

Proposition 4.1.

If we modify the NLS equation, that is, u solves

i \partial_{t} u + Δ_{H^{2}} u = {|u|}^{2} u + N,

then the modified Morawetz estimate becomes

\begin{aligned} ‖ u ‖_{L_{t, x}^{4} ([t_{1}, t_{2}] \times H^{2})}^{4} ≲ ‖ u ‖_{L_{t}^{\infty} L_{x}^{2} ([t_{1}, t_{2}] \times H^{2})} ‖ u ‖_{L_{t}^{\infty} H_{x}^{1} ([t_{1}, t_{2}] \times H^{2})} + ‖ N ū ‖_{L_{t, x}^{1} ([t_{1}, t_{2}] \times H^{2})} + ‖ N \nabla \bar{u} ‖_{L_{t, x}^{1} ([t_{1}, t_{2}] \times H^{2})} . \end{aligned}

(4.1)

Remark 4.2.

The Proof of Proposition 4.1 is very similar as the Proof in Ref. 36. The difference is that we consider a more general nonlinear term, which mainly gives two extra terms that account for the two extra terms in (4.1).

B. Proof of Theorem 1.1

Now, we are ready for the bootstrapping argument.

Step 1: Setup of the open–close argument.Define

W : = \{T : ‖ u ‖_{L_{t, x}^{4} ([0, T] \times H^{2})}^{4} \leq M\},

where M > 0 is a constant. W is closed and non-empty. Now, we want to show that W is open. If T₁ ∈ W, then due to the local well-posedness theory and Remark 3.12, for some T₀ > T₁ and T₀ sufficiently close to T₁, we have

‖ u ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{4} \leq 2 M .

In fact, the H^s norm of u(T₁) is bounded, then using a standard local well-posedness argument, we can continue the solution u from time T₁ at least for a short time. Within such a short time period, due to the sub-criticality, the spacetime L⁴ norm of u will be bounded by twice the H^σ, for σ arbitrarily small, norm at T₁, which is of order $s_{0}^{σ / 2 (s - 1)}$ (see Remark 3.12). Hence, we want to ensure that

s_{0}^{\frac{σ}{2} (s - 1)} < M^{\frac{1}{4}} \sim s_{0}^{\frac{1}{4} (- \frac{1}{2} s + \frac{1}{4})},

which is achieved for any s > 3/4 by taking σ small enough (say, σ = 1/4). This guarantees the existence of such T₀.

Now, we show that T₀ ∈ W, that is,

‖ u ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{4} \leq M .

(4.2)

Recalling the decomposition of the solution u in (3.1) and (3.2) and using the smallness of L⁴ norm of ψ in (3.9), one can reduce (4.2) to

‖ ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{4} \leq \frac{1}{2} M .

(4.3)

Now, we will prove the improved bound of $L_{t, x}^{4}$ in (4.3) in steps 2 and 3.

Step 2: Improving the bound for $L_{t, x}^{4}$ . The modified Morawetz estimate in (4.1) that now gives

\begin{aligned} ‖ ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{4} ≲ ‖ ζ ‖_{L_{t}^{\infty} L_{x}^{2} ([0, T_{0}] \times H^{2})} ‖ ζ ‖_{L_{t}^{\infty} H_{x}^{1} ([0, T_{0}] \times H^{2})} + ‖ N \bar{ζ} ‖_{L_{t, x}^{1} ([0, T_{0}] \times H^{2})} + ‖ N \nabla \bar{ζ} ‖_{L_{t}^{1} L_{x}^{1} ([0, T_{0}] \times H^{2})} . \end{aligned}

(4.4)

Recall Eq. (3.2) that ζ satisfies, then in our case, $N$ is given by

N = {|u|}^{2} u - {|ζ|}^{2} ζ = {|ψ + ζ|}^{2} (ψ + ζ) - {|ζ|}^{2} ζ = {|ψ|}^{2} ψ + O (ψ^{2} ζ) + O (ψ ζ^{2}) .

Now, we estimate the right-hand side terms in (4.4).

For the second term in (4.4), by Hölder inequality, (3.9), and (4.3), we have

\begin{array}{l} ‖ N \bar{ζ} ‖_{L_{t, x}^{1} ([0, T_{0}] \times H^{2})} & ≲ ‖ ψ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{3} ‖ ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})} + ‖ ψ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})} ‖ ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{3} \\ ≲ s_{0}^{\frac{3}{2} s} ‖ ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})} + s_{0}^{\frac{1}{2} s} ‖ ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{3} ≲ s_{0}^{\frac{3}{2} s} M^{\frac{1}{4}} + s_{0}^{\frac{1}{2} s} M^{\frac{3}{4}} . \end{array}

We write the last term in (4.4) as

\begin{align} ‖ N \nabla \bar{ζ} ‖_{L_{t, x}^{1} ([0, T_{0}] \times H^{2})} & ≲ ‖ O (ζ^{2} ψ) \nabla \bar{ζ} ‖_{L_{t, x}^{1} ([0, T_{0}] \times H^{2})} + ‖ O (ψ^{3}) \nabla \bar{ζ} ‖_{L_{t, x}^{1} ([0, T_{0}] \times H^{2})} \\ ≲ ‖ ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{2} ‖ ψ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})} ‖ \nabla ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})} + ‖ ψ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{3} ‖ \nabla ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})} . \end{align}

(4.5)

Claim 4.3.

We claim that

\begin{array}{l} (1) ‖ \nabla ζ ‖_{L_{t}^{\infty} L_{x}^{2} ([0, T_{0}] \times H^{2})} ≲ s_{0}^{\frac{1}{2} (s - 1)}, & (2) ‖ \nabla ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{4} ≲ M s_{0}^{2 (s - 1)} . \end{array}

Assuming Claim 4.3, we continue the estimation of (4.5),

(4.5) ≲ M^{\frac{1}{2}} s_{0}^{\frac{1}{2} s} {(M s_{0}^{2 (s - 1)})}^{\frac{1}{4}} + s_{0}^{\frac{3}{2} s} {(M s_{0}^{2 (s - 1)})}^{\frac{1}{4}} = M^{\frac{3}{4}} s_{0}^{s - \frac{1}{2}} + M^{\frac{1}{4}} s_{0}^{2 s - \frac{1}{2}} .

Using the same calculation as in (3.28), we have

‖ ζ (t) ‖_{L_{x}^{2} (H^{2})} \leq ‖ u (t) ‖_{L_{x}^{2} (H^{2})} + ‖ ψ (t) ‖_{L_{x}^{2} (H^{2})} ≲ ‖ u (0) ‖_{L_{x}^{2} (H^{2})} + s_{0}^{\frac{1}{2} s} \leq 2 ‖ u (0) ‖_{L_{x}^{2} (H^{2})},

then the first term in (4.4) is bounded by

‖ ζ ‖_{L_{t}^{\infty} L_{x}^{2} ([0, T_{0}] \times H^{2})} ‖ ζ ‖_{L_{t}^{\infty} H_{x}^{1} ([0, T_{0}] \times H^{2})} ≲ ‖ u (0) ‖_{L_{x}^{2} (H^{2})} s_{0}^{\frac{1}{2} (s - 1)} .

Now, (4.4) becomes

‖ ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{4} ≲ s_{0}^{\frac{1}{2} (s - 1)} + (s_{0}^{\frac{3}{2} s} M^{\frac{1}{4}} + s_{0}^{\frac{1}{2} s} M^{\frac{3}{4}}) + (M^{\frac{3}{4}} s_{0}^{s - \frac{1}{2}} + M^{\frac{1}{4}} s_{0}^{2 s - \frac{1}{2}}) .

To close the argument, we need the following inequality holds for $M \sim s_{0}^{- 1 / 2 s + 1 / 4}$ ⁠:

s_{0}^{\frac{1}{2} (s - 1)} + (s_{0}^{\frac{3}{2} s} M^{\frac{1}{4}} + s_{0}^{\frac{1}{2} s} M^{\frac{3}{4}}) + (M^{\frac{3}{4}} s_{0}^{s - \frac{1}{2}} + M^{\frac{1}{4}} s_{0}^{2 s - \frac{1}{2}}) < \frac{1}{2} M .

(4.6)

This requirement of (4.6) can be achieved for s > 3/4. Now, we are left to prove Claim 4.3.

Step 3: Proof of Claim 4.3.

Proof of Claim 4.3.

The total energy increment from Proposition 3.2 implies (1),

‖ \nabla ζ ‖_{L_{t}^{\infty} L_{x}^{2} ([0, T_{0}] \times H^{2})}^{2} \leq \sup_{t} E (ζ (t)) \leq E (ζ (0)) + \frac{M}{ε} s_{0}^{\frac{3}{2} s - \frac{5}{4}} \sim s_{0}^{s - 1} .

To estimate

‖ \nabla ζ ‖_{L_{t}^{4} L_{x}^{4} ([0, T_{0}] \times H^{2})}

⁠, we consider the subintervals I_i’s defined in Proposition 3.2. We claim that

‖ \nabla ζ ‖_{L_{t, x}^{4} (I_{i} \times H^{2})} ≲ ‖ \nabla ζ (a_{i}) ‖_{L_{x}^{2} (H^{2})} .

(4.7)

In fact, by Strichartz estimates,

\begin{align} ‖ \nabla ζ ‖_{L_{t, x}^{4} (I_{i} \times H^{2})} & ≲ ‖ \nabla ζ (a_{i}) ‖_{L_{x}^{2} (H^{2})} + ‖ \nabla {(ζ + ψ)}^{3} ‖_{L_{t, x}^{\frac{4}{3}} (I_{i} \times H^{2})} \\ ≲ ‖ \nabla ζ (a_{i}) ‖_{L_{x}^{2} (H^{2})} + ‖ \nabla O (ζ^{3}) ‖_{L_{t, x}^{\frac{4}{3}} (I_{i} \times H^{2})} + ‖ \nabla O (ψ^{3}) ‖_{L_{t, x}^{\frac{4}{3}} (I_{i} \times H^{2})} . \end{align}

(4.8)

The second term in () will be absorbed by the left-hand side of (),

‖ \nabla O (ζ^{3}) ‖_{L_{t, x}^{\frac{4}{3}} (I_{i} \times H^{2})} ≲ ‖ \nabla ζ ‖_{L_{t, x}^{4} (I_{i} \times H^{2})} ‖ ζ ‖_{L_{t, x}^{4} (I_{i} \times H^{2})}^{2} ≲ ‖ \nabla ζ ‖_{L_{t, x}^{4} (I_{i} \times H^{2})} ε^{\frac{1}{2}} .

The last inequality in the above equation is due to () and the construction of I_j (),

‖ ζ ‖_{L_{t, x}^{4} (I_{i} \times H^{2})} \leq ‖ ψ ‖_{L_{t, x}^{4} (I_{i} \times H^{2})} + ‖ u ‖_{L_{t, x}^{4} (I_{i} \times H^{2})} ≲ s_{0}^{\frac{1}{2} s} + ε^{\frac{1}{4}} ≲ ε^{\frac{1}{4}} .

(4.9)

For the last term in (), the same calculation as in () gives

‖ \nabla O (ψ^{3}) ‖_{L_{t, x}^{\frac{4}{3}} (I_{i} \times H^{2})} ≲ s_{0}^{\frac{3}{2} s - \frac{3}{4}} .

Then, () becomes

‖ \nabla ζ ‖_{L_{t, x}^{4} (I_{i} \times H^{2})} ≲ ‖ \nabla ζ (a_{i}) ‖_{L_{x}^{2} (H^{2})} + ‖ \nabla ζ ‖_{L_{t, x}^{4} (I_{i} \times H^{2})} ε^{\frac{1}{2}} + s_{0}^{\frac{3}{2} s - \frac{3}{4}} .

Therefore, the claim () follows.

Putting all the small intervals together and using (2), we get

‖ \nabla ζ ‖_{L_{t, x}^{4} ([0, T_{0}] \times H^{2})}^{4} ≲ \frac{M}{ε} \sup_{I_{i}} ‖ \nabla ζ (a_{i}) ‖_{L_{x}^{2} (H^{2})}^{4} \leq \frac{M}{ε} ‖ \nabla ζ ‖_{L_{t}^{\infty} L_{x}^{2} ([0, T_{0}] \times H^{2})}^{4} ≲ \frac{M}{ε} s_{0}^{2 (s - 1)} .

Now, we finish the Proof of Claim 4.3.□

V. GENERAL NONLINEARITIES

Our result for the cubic NLS in Theorem 1.1 can be generalized to a larger class of nonlinearities. In fact, we have the following result.

Theorem 5.1.

The initial value problem (1.6) with radial initial data $ϕ \in H^{s} (H^{2})$ is globally well-posed and scatters in $H^{s} (H^{2})$ when $s > \frac{3 p - 6}{3 p - 5}$ .

Note that the scaling of (1.6) is $s_{c} = 1 - \frac{2}{p - 1}$ ⁠.

A. Sketch of the proof

We will briefly present how the method presented above for the cubic NLS can be generalized to nonlinearities of order p.

As in Sec. III, we decompose u in ψ and ζ, where $ψ = e^{i t Δ} P_{\leq s_{0}} ϕ$ solves the linear Schrödinger with high frequency data and ζ solves the difference equation with low frequency data,
$\{\begin{cases} i \partial_{t} ψ + Δ_{H^{2}} ψ = 0, \\ ψ (0, x) = ψ_{0} = P_{\leq s_{0}} ϕ, \end{cases}) \{\begin{cases} i \partial_{t} ζ + Δ_{H^{2}} ζ = {|u|}^{p - 1} u, \\ ζ (0, x) = η_{0} = P_{> s_{0}} ϕ . \end{cases})$
(5.1)
Then, using similar analysis, we obtain the global energy increment given the boundedness of u in the critical spacetime $L_{t, x}^{2 (p - 1)}$ (see Propositions 5.2 and 5.3). The analogs of all the estimates that we used in the cubic case can be found in (5.3).
Similarly, a bootstrapping argument on the $L_{t, x}^{2 (p - 1)}$ norm gives both the global existence and scattering.

Note that the only difference in the general case is that the spacetime $L_{t, x}^{2 (p - 1)}$ in the local theory (Proposition 5.2) is different from the Morawetz norm. Hence, in the bootstrapping argument, an intermediate step is needed. In fact, in this step, we first obtain and improve the estimates on the Morawetz norm, then we bootstrap the $L_{t, x}^{2 (p - 1)}$ norm with the better Morawetz bound. Note that $L_{t, x}^{2 (p - 1)}$ agrees with the Morawetz norm when p = 3, and hence, such a step is not needed in Sec. IV.

B. Analogs of the main propositions

We now present the analogs of Propositions 3.1 and 3.2 on the energy increment.

Proposition 5.2

(Local energy increment). Consider u as in () defined on

I \times H^{2}

⁠, where I = [0, τ], such that

‖ u ‖_{L_{t, x}^{2 (p - 1)} (I \times H^{2})}^{2 (p - 1)} = ε

for some universal constant ɛ. Then, for

s > \frac{p}{p + 1}

and sufficiently small s₀, the solution ζ, under the decomposition u = ψ + ζ defined as in (), satisfies the following energy increment:

E (ζ (τ)) \leq E (ζ (0)) + C s_{0}^{\frac{p + 3}{4} s - \frac{p + 2}{4}} .

Proposition 5.3

(Conditional global energy increment). Consider u as in () defined on

[0, T] \times H^{2}

, where

‖ u ‖_{L_{t, x}^{2 (p - 1)} ([0, T] \times H^{2})}^{2 (p - 1)} \leq M

for some constant M. Then, for

s > \frac{p}{p + 1}

and sufficiently small s₀, the energy of ζ satisfies the following energy increment:

E (ζ (T)) \leq E (ζ (0)) + C \frac{M}{ε} s_{0}^{\frac{p + 3}{4} s - \frac{p + 2}{4}},

where ɛ is the small constant in Proposition 5.2.

C. Analogs of the main estimates

Within the proofs of above two propositions, we need a further decomposition for ζ as is (3.4)–(3.6),

\{\begin{cases} i \partial_{t} ζ_{1} + Δ_{H^{2}} ζ_{1} = {|ζ_{1}|}^{p - 1} ζ_{1}, \\ ζ_{1} (0, x) = η_{0} = P_{> s_{0}} ϕ, \end{cases}) \{\begin{cases} i \partial_{t} ζ_{2} + Δ_{H^{2}} ζ_{2} = {|u|}^{p - 1} u - {|ζ_{1}|}^{p - 1} ζ_{1}, \\ ζ_{2} (0, x) = 0 . \end{cases})

(5.2)

Hence, the full solution u is the sum of these three solutions u = ζ₁ + ζ₂ + ψ.

The analog of (3.16) can be computed similarly as follows:

\begin{align} ‖ ψ ‖_{S^{σ} (R)} ≲ s_{0}^{\frac{1}{2} (s - σ)} for 0 \leq σ \leq s, & ‖ ψ ‖_{L_{t, x}^{2 (p - 1)} (R \times H^{2})} ≲ s_{0}^{\frac{1}{2} (s - s_{c})}, \\ ‖ ζ_{1} ‖_{S^{σ} (I)} ≲_{‖ ϕ ‖_{H_{x}^{s_{c}}}} \{\begin{cases} 1 & for 0 \leq σ \leq s_{c} \\ s_{0}^{\frac{σ - s_{c}}{2 (1 - s_{c})} (s - 1)} & for s_{c} \leq σ \leq 1, \end{cases}) & ‖ ζ_{1} ‖_{L_{t, x}^{2 (p - 1)} (I \times H^{2})}^{2 (p - 1)} ≲_{‖ ϕ ‖_{H_{x}^{s_{c}}}} ε, \end{align}

(5.3)

‖ ζ_{2} ‖_{S^{σ} (I)} ≲_{‖ ϕ ‖_{H_{x}^{s_{c}}}} s_{0}^{\frac{1}{2} (s - σ)} for 0 \leq σ \leq s, ‖ ζ_{2} ‖_{L_{t, x}^{2 (p - 1)} (I \times H^{2})} ≲_{‖ ϕ ‖_{H_{x}^{s_{c}}}} s_{0}^{\frac{1}{2} (s - s_{c})} .

Claim 3.11 will be the same in the general setting. Most importantly, the analog of (3) in Lemma 3.8 is

‖ ζ_{2} ‖_{L_{t}^{\infty} H_{x}^{1} (I \times H^{2})} ≲ s_{0}^{\frac{p + 1}{4} s - \frac{p}{4}} .

The choice of s₀ in Proposition 5.3 is given by

M \sim s_{0}^{\frac{1}{2} (\frac{1 - s}{1 - s_{c}} - \frac{1}{2})} .

(5.4)

It is also worth mentioning that the hidden constant in (5.3) at the second iteration is bounded by the $H^{s_{c}}$ norm of ϕ plus a small error,

‖ (ζ_{1} + ζ_{2}) (a_{1}) ‖_{H_{x}^{s_{c}} (H^{2})} \leq ‖ ζ_{1} (a_{1}) ‖_{H_{x}^{s_{c}} (H^{2})} + ‖ ζ_{2} (a_{1}) ‖_{H_{x}^{s_{c}} (H^{2})} ≲ ‖ ϕ ‖_{H_{x}^{s_{c}} (H^{2})} + s_{0}^{\frac{1}{2} s} s_{0}^{- \frac{1}{2} s_{c} (\frac{1}{2} + \frac{1 - s}{1 - s_{c}})}

for $s > \frac{p}{p + 1}$ ⁠. Then, the accumulated gain of this hidden constant will be dominated by the size of the $H^{s_{c}}$ norm of ϕ, hence not growing.

D. A different bootstrapping argument

We consider

W : = \{T : ‖ u ‖_{L_{t, x}^{2 (p - 1)} ([0, T] \times H^{2})}^{2 (p - 1)} \leq M\} .

We are then reduced to showing that for T₀ chosen in the same manner as in step 1 in Sec. IV,

‖ ζ ‖_{L_{t, x}^{2 (p - 1)} ([0, T_{0}] \times H^{2})}^{2 (p - 1)} \leq \frac{1}{2} M .

Things are different here. First, interpolating $‖ ζ ‖_{L_{t, x}^{2 +} ([0, T_{0}] \times H^{2})}^{2 +} ≲ M$ with the bound of the $L_{t, x}^{2 (p - 1)}$ norm in the assumption gives an estimate on the Morawetz norm,

‖ ζ ‖_{L_{t, x}^{p + 1} ([0, T_{0}] \times H^{2})}^{p + 1} ≲ M .

(5.5)

Using (5.5) and the modified Morawetz estimate (4.1), we obtain as before

‖ ζ ‖_{L_{t, x}^{p + 1} ([0, T_{0}] \times H^{2})}^{p + 1} ≲ s_{0}^{\frac{1}{2} (s - 1)} .

If we simply require $s_{0}^{1 / 2 (s - 1)} < M$ ⁠, here there will be no room to improve the $L_{t, x}^{2 (p - 1)}$ norm at all. Hence, to this end, we demand it to be much smaller than M, that is, for α ∈ (0, 1),

s_{0}^{\frac{1}{2} (s - 1)} \leq M^{α} ≪ M,

(5.6)

hence recalling (5.4), we get the first restriction on s

s > 1 - \frac{α}{2 + α (p - 1)} .

With this better Morawetz bound, we can improve the $L_{t, x}^{2 (p - 1)}$ norm by making it smaller than M by Hölder inequality, (5.6), and Proposition 5.3 with the choice of M as in (5.4),

‖ ζ ‖_{L_{t, x}^{2 (p - 1)} ([0, T_{0}] \times H^{2})}^{2 (p - 1)} ≲ ‖ ζ ‖_{L_{t, x}^{p + 1} ([0, T_{0}] \times H^{2})}^{p + 1 -} ‖ ζ ‖_{L_{t, x}^{\infty -} ([0, T_{0}] \times H^{2})}^{p - 3 +} ≲ M^{α -} ‖ {⟨\nabla⟩}^{1 -} ζ ‖_{L_{t}^{\infty -} L_{x}^{2 +} ([0, T_{0}] \times H^{2})}^{p - 3 +} ≲ M^{α} s_{0}^{\frac{1}{2} (s - 1) (p - 3)} ≪ M .

This with (5.4) implies the second restriction on s,

s > 1 - \frac{1 - α}{2 (p - 3) + (1 - α) (p - 1)} .

Therefore, combining both conditions on s, we choose $s > s_{H^{2}}^{p}$ to be the best possible scattering index, where

s_{H^{2}}^{p} = \min_{α \in (0, 1)} \max \{1 - \frac{α}{2 + α (p - 1)}, 1 - \frac{1 - α}{2 (p - 3) + (1 - α) (p - 1)}\} = 1 - \frac{1}{3 p - 5} = \frac{3 p - 6}{3 p - 5} .

ACKNOWLEDGMENTS

G.S. and X.Y. graciously acknowledge the support from the Jarve Seed Fund. G.S. was also supported, in part, by the grant NSF (Grant No. DMS-1764403) and X.Y. by an AMS-Simons Travel Grants. G.S. and X.Y. would also like to thank A. Lawrie and S. Shahshahani for very insightful conversations.

VI. DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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2020

Author(s)

On the high–low method for NLS on the hyperbolic space

I. INTRODUCTION

A. Blueprint of the proof

II. PRELIMINARIES

A. Notations

B. Geometry of the domain $H^{2}$

C. Tools on $H^{2}$

1. Fourier transform on $H^{d}$

2. Strichartz estimates

3. Local smoothing estimates in the hyperbolic space

4. Heat-flow-based Littlewood–Paley projections and functional inequalities on $H^{2}$

5. Radial Sobolev embeddings

III. ENERGY INCREMENT ON $H^{2}$

A. Main results in this section

B. Proof of Proposition 3.1

C. Proof of Proposition 3.2

IV. GLOBAL WELL-POSEDNESS AND SCATTERING ON $H^{2}$

A. Morawetz estimates on $H^{2}$

B. Proof of Theorem 1.1

V. GENERAL NONLINEARITIES

A. Sketch of the proof

B. Analogs of the main propositions

C. Analogs of the main estimates

D. A different bootstrapping argument

ACKNOWLEDGMENTS

VI. DATA AVAILABILITY

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

On the high–low method for NLS on the hyperbolic space

I. INTRODUCTION

A. Blueprint of the proof

II. PRELIMINARIES

A. Notations

B. Geometry of the domain H2

C. Tools on H2

1. Fourier transform on Hd

2. Strichartz estimates

3. Local smoothing estimates in the hyperbolic space

4. Heat-flow-based Littlewood–Paley projections and functional inequalities on H2

5. Radial Sobolev embeddings

III. ENERGY INCREMENT ON H2

A. Main results in this section

B. Proof of Proposition 3.1

C. Proof of Proposition 3.2

IV. GLOBAL WELL-POSEDNESS AND SCATTERING ON H2

A. Morawetz estimates on H2

B. Proof of Theorem 1.1

V. GENERAL NONLINEARITIES

A. Sketch of the proof

B. Analogs of the main propositions

C. Analogs of the main estimates

D. A different bootstrapping argument

ACKNOWLEDGMENTS

VI. DATA AVAILABILITY

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only

B. Geometry of the domain $H^{2}$

C. Tools on $H^{2}$

1. Fourier transform on $H^{d}$

4. Heat-flow-based Littlewood–Paley projections and functional inequalities on $H^{2}$

III. ENERGY INCREMENT ON $H^{2}$

IV. GLOBAL WELL-POSEDNESS AND SCATTERING ON $H^{2}$

A. Morawetz estimates on $H^{2}$