Global wellposedness and scattering for the defocusing cubic Schrödinger equation on waveguide
Abstract
We consider the problem of large data scattering for the defocusing cubic nonlinear Schrödinger equation on . This equation is critical both at the level of energy and mass. The key ingredients contain a globalintime Stricharz estimate, resonant system approximation and profile decomposition. Assuming the large data scattering for the 2d cubic resonant system, we can prove the large data scattering for this problem.
1 Introduction
Let us consider the defocusing cubic nonlinear Schrödinger equation on ,
(1.1)  
where is the LaplaceBeltrami operator on and is a complexvalued function. A natural question is that given a finite (in the sense of norm) initial data, can we obtain a global wellposed solution that scatters?
The equation (1.1) has the following conserved quantities:

Energy : .

Mass : .

Full energy : .

Momentum: .
Actually the equation (1.1) is a special case of the general nonlinear equation on the waveguide :
(1.2)  
As for the background, we know that there are many existing results regarding NLS problems on Euclidean space. In this paper, the nonlinear Schrödinger equation is discussed on a semiperiodic space, i.e. . The motivation is to better understand the broad question of the effect of the geometry of the domain on the asymptotic behavior of large solutions to nonlinear dispersive equations.
The study of solutions of the nonlinear Schrödinger equation on compact or partially compact domains has been the subject of many works. Such equations have also been studied in applied sciences and various background. Those spaces are also called waveguide manifolds. In paper [23, 24], there is a good overview of the main results for waveguide manifolds.
The studies on global wellposedness for energy critical and subcritical equations seem to point to the absence of any geometric obstruction to global existence. Moreover, it is clear that the geometry influences the asymptotic dynamics of solutions. Thus, it is meaningful to explore when one can obtain the simplest asymptotic behavior, i.e. scattering, which means that all nonlinear solutions asymptotically resemble linear solutions. Based on the theory of NLS on Euclidean space, i.e. , the equation
(1.3)  
would scatter in the range , the equation (1.3) is mass critical; when , the equation (1.3) is energy critical; when , the equation is mass subcritical; when , the equation (1.3) is energy supercritical; when , the equation is mass supercritical and energy subcritical. . When
Naturally, we are also interested in the range of for wellposedness and scattering of the NLS on . Based on the existing results and theories, we expect that the solution of (1.2) globally exists and scatters in the range ) in equation (1.1) lies in the range (exactly at the endpoints of the interval), it is reasonable for us to consider this problem. . And fortunately the index (
Also, we are mainly inspired by a similar result [13] which studies the defocusing quintic NLS on space by Zaher Hani and Benoit Pausader. For that problem, the defocusing NLS equation is also critical both at the level of energy and mass. [15, 16, 17, 18, 19] are also some of the important related results. Also, the introduction of [8] has a summary of the main known results about NLS problems on waveguides, i.e. .
As in [13], we also need to assume the large data scattering of a cubic resonant system (1.5). The next task for us to do is to deal with the regarding resonant system, i.e. to prove this conjecture 1.2. An inspiring thing is that [33] (by K. Yang and L. Zhao) has proved the large data scattering for a similar resonant system recently. Conjecture 1.1 seems to be a reachable problem to work on, and we leave it for a later work.
The first result asserts that the small data leads to solutions that are global and scatter.
Theorem 1.1
There exists such that any initial data satisfying
leads to a unique global solution which scatters in the sense that there exists such that
(1.4) 
The uniqueness space was essentially introduced by HerrTataruTzvetkov [15]. In order to extend our analysis to large data, we use a method formalized in [20, 21]. One key ingredient is a linear and nonlinear profile decomposition for solutions with bounded energy. The socalled profiles correspond to sequences of solutions exhibiting an extreme behavior. It is there that the “energy critical” and “mass critical” nature of our equation become manifest.
For this problem, in view of the scalinginvariant of the IVP (1.1) under
There are two situations:
When , the manifolds will be similar to and we can use the 4D energy critical result [23] by (E. Ryckman and M. Visan). The appearance is a manifestation of the energycritical nature of the nonlinearity.
When , the manifolds becomes thinner and thinner and resembles . The problem will become similar to the cubic mass critical NLS problem on :
(1.5) 
The cubic resonant system:
We consider the cubic resonant system,
(1.6)  
with unknown , where .
In the special case when for , it is exactly the equation (1.5). Similar systems of nonlinear Schrödinger equations arise in the study of nonlinear optics in waveguides.
As we show in Section 8, the system (1.6) is Hamiltonian and it has a nice local theory and retains many properties of (1.5). In view of this and of the result of Dodson [9], it seems reasonable to assume the following conjecture. Another reason for us to assume this is that K. Yang and L. Zhao [33] have proved the large data scattering for a similar resonant system when index .
Conjecture 1.1
Let . For any smooth initial data satisfying:
There exists a global solution with conserved satisfying:
(1.7) 
for some finite nondecreasing function .
Remark. As for a more general case, when and , the Initial Value Problem (1.2) is also both critical at the level of mass and energy. If , would no longer be an integer, which may cause some trouble for the resonant system approximation.
We now give the main result of this paper which asserts the large data scattering for (1.1) conditioned on Conjecture 1.1.
Theorem 1.2
Assume that Conjecture 1.1 holds for all , then any initial data satisfying
leads to a solution which is global, and scatters (in the sense of (1.3)). In particular if , then all solutions of (1.1) with finite energy and mass scatter.
As a consequence of the local theory for the system (1.6), Conjecture 1.1 holds below a nonzero threshold , so Theorem 1.2 is nonempty and indeed strengthens Theorem 1.1. Another point worth mentioning is that while Theorem 1.2 is stated as an implication, it is actually an equivalence as it is easy to see that one can reverse the analysis needed to understand the behavior of largescale profile initial data for (1.1) in order to control general solutions of (1.6) and prove Conjecture 1.1 assuming that Theorem 1.2 holds (cf. Section 8). The “scattering threshold” for (1.1) and the resonant system (1.6) are same.
The proof of the Theorem 1.2 follows from a standard skeleton based on the KenigMerle machinery [20, 21] which illustrates a classical and vivid road map for global wellposedness (and scattering) problem. Mainly there are three important points: global Strichartz estimates, largescale profile and the resonant cubic system and profile decomposition.
The paper is organized as follows: in Section 2, we introduce some notations and function spaces; in Section 3, we prove the global Strichartz estimates that will be used later; in Section 4, we will prove the local wellposedness and small data scattering of (1.1); in Section 6, we obtain a good linear profile decomposition that leads us to analyze, which is what we do in Section 5 for the Euclidean and largescale profiles. In Section 7, we prove the contradiction argument leading to Theorem 1.2 (Main Theorem). In Section 8, we prove the local theory for the cubic resonant system (1.6) and also give a proof of a lemma (localintime estimate) in Section 3.
2 Notations and function spaces
About the notation, we write to say that there is a constant such that . We use when . Particularly, we write to express that for some constant depending on .
In addition to the usual isotropic Sobolev spaces , we have nonisotropic versions. For we define:
(2.1) 
Particularly is a Hilbert space with inner product:
We can also define a discrete analogue. For a sequence of realvariable functions, we let
(2.2) 
We can naturally identify and by via the Fourier transform in the periodic variable .
Function spaces. In this paper, we will use some function spaces. For example, the space was essentially introduced by HerrTataruTzvetkov [15].
For , we denote by the translate by and define the sharp projection operator as follows is the Fourier transform): and
We use the same modifications of the atomic and variation space norms that were employed in some other papers [15, 16]. Namely, for , we define:
and similarly we have,
where the and are the atomic and variation spaces respectively of functions on taking values in . There are some nice properties of those spaces. We refer to [15, 16] for the description and properties. For convenience, we also give the some definitions here.
Definition 2.1
Let , and be a complex Hilbert space. A atom is a piecewise defined function, ，
where and with . Here we let be the set of finite partitions of the real line.
The atomic space consists of all functions such that
for atoms , ,
with norm
Definition 2.2
Let , and be a complex Hilbert space. We define as the space of all functions such that
where we use the convention . Also, we denote the closed subspace of all rightcontinuous functions such that by .
Definition 2.3
For , we let resp. be the spaces of all functions such that is in resp. , with norms
For this problem, we choose to be . Norms and are both stronger than the norm and weaker than the norm . Moreover, they satisfy the following property (for ):
For an interval , we can also define the restriction norms and in the natural way: inf satisfying .
And similarly for .
A modification for to :
equipped with the norm:
(2.3) 
Our basic space to control solutions is Also we use to express the set of all solutions in whose norm is finite for any compact subset .
In order to control the nonlinearity on interval , we need to define ‘ Norm’ as follows, on an interval we have:
(2.4) 
And then we can define the following spacetime norm, i.e. ‘Znorm’ by
where . Here we decompose into . is a weaker norm than , in fact:
It follows from Strichartz estimate.
We also need the following theorem which has analogues in [13, 15, 16].
Theorem 2.1
[[13, 15, 16]] If , then
Also, we have the following estimate holds for any smooth function on an interval :
Proof: The proof follows from [15, Proposition 2.11] and [16, Proposition 2.10].
3 Global Strichartz estimate
Theorem 3.1
Then we can prove the following Strichartz Estimate:
(3.1) 
whenever
(3.2) 
Proof: The main idea of the proof is similar to [13, Theorem 3.1], i.e. use argument, a partition of unity and then estimate the diagonal part and nondiagonal part separately. One remarkable difference is that in the diagonal estimate part, we can not use Bourgain’ s estimate directly on ([2]) as in [13] since we need a Stricharz estimate with a threshold less than 4 (precisely it is ) to do the interpolation later. And we use HardyLittlewood circle method as in [19, Proposition 2.1] to obtain the localintime estimate.
First, let us prove a more precise conclusion and we can get the estimate by duality:
Lemma 3.2
For any , then there holds that
(3.3)  
for any (p,q) satisfies (3.2).
Proof: In order to distinguish between the large and small time scales, we choose a smooth partition of unity with supported in . We also denote by . Using the semigroup property and the unitarity of we can get :
Here we have,
Here ‘’ is short for ‘diagonal’ and ‘’ is short for ‘nondiagonal’.
We will estimate the diagonal part and the nondiagonal part by using different methods.
For the diagonal part: First we need a localintime estimate as follows:
Lemma 3.3
Let , then for any , ,and ,
(3.4) 
We will give the proof of Lemma 3.3 in the Appendix (Section 8).
According to the estimate (3.4) above, by duality we have
(3.5) 
where is supported in . And consequently,
(3.6)  
This finished the estimate for the diagonal part.
For the nondiagonal part: We need a lemma (Lemma 3.4) that we will introduce later and we can apply it to estimate the nondiagonal part by using Hlder’s inequality and the discrete HardySobolev inequality as below:
(3.7)  
Lemma 3.4
Suppose satisfies and that . For any function , there holds that:
Proof: The proof of this lemma is similar as in [13]. The main idea of the proof is to study the Kernel and decompose the corresponding index set into three parts and estimate over the three parts separately. Notice that a significant difference is the nonstationary phase estimate because of the dimension, we will have:
instead of
And
still holds.
Also the Kernel is defined as:
(3.8)  
And we define , and so we have .
For a dyadic number, we define which has modulus in . We define similarly for . And we have the following decomposition:
(3.9)  
where , , are three index sets. And similarly in this case we have the following decomposition:

,

,

for a large constant to be decided later. For fixed , we will decompose and estimate them as follows.
(3.10) 