Global solutions for 2D quadratic Schrödinger equations
Abstract.
We prove global existence and scattering for a class of quadratic Schrödinger equations in dimension 2. The proof relies on the idea of spacetime resonance.
1. Introduction
In the present article we examine global existence and asymptotic behavior of solutions with small initial data for nonlinear Schrödinger equations with quadratic nonlinearities in dimension 2. We believe that this particular model is a good representative of a wider class of weakly dispersive nonlinear equations, i.e. nonlinear dispersive equations where the linear decay due to dispersion is not strong enough a priori to overcome the nonlinear effects over large intervals of time: estimates relying only on the power of the nonlinearity, but not on its structure, fail. As we will argue later, the key concept in this setting becomes spacetime resonances.
1.1. Known results
Consider a nonlinear Schrödinger equation
where is complexvalued, and a (nonlinear) function of which is homogeneous of order . We will review results concerning global existence and asymptotic behavior for small solutions. We refer to the textbooks by Cazenave [2] and Tao [31] for a more general discussion.
The simplest problem occurs if the decay given by the linear part is strong enough, or large enough to use dispersive or Strichartz estimates to conclude that asymptotic completeness holds for small data, i.e. the wave operators are defined, and are one to one. For smaller , more interesting effects appear, and the structure of the nonlinearity starts to play a role. Two values of are particularly important: the Strauss exponent [30], and the short range exponent , whose values are displayed below for small dimensions.
space dimension  short range exponent  Strauss exponent 
For larger than the Strauss exponent, one expect the existence of global solutions for small data, as well as some kind of asymptotic completeness. For larger than the short range exponent one expects the existence of wave operators, while for less than the short range exponent one expects small solutions not to be asymptotically free. Various global existence results for small solutions will be illustrated below.
Wave operators for small data at
Suppose first that lies above the shortrange exponent, ; an immediate computation shows that the solution becomes asymptotically free if it decays in at the rate prescribed by the linear part: . In all known cases, wave operators can be constructed for small data for this range of , but no general result seems available. For the nonlinearity , see in particular Cazenave and Weissler [3], Ginibre, Ozawa and Velo [11] and Nakanishi [23]. For small within this range, the spaces in which these wave operators exist involve weights or vector fields.
Consider now the case where lies below the shortrange exponent . For the nonlinearity , it was proved by Barab [1] that non trivial asymptotically free states cannot exist. Modified wave operators were subsequently constructed by Ozawa [24], and Ginibre and Ozawa [10] if . For the nonlinearity , in dimension 2, it was proved by Shimomura [26] and Shimomura and Tsutsumi [28] that non trivial asymptotically free states cannot exist either. However, for the nonlinearities , , in dimension 1, and , in dimension 2, wave operators were constructed by Moriyama, Tonegawa and Tsutsumi [21] and Shimomura and Tonegawa [27]; see also Hayashi, Naumkin, Shimomura and Tonegawa [18]. Finally, Gustafson, Nakanishi and Tsai [12] proved, in dimensions 2 and 3, the existence of wave operators for the nonlinearity arising from the GrossPitaevskii equation.
Global existence and asymptotic behavior for small data at
For larger than the Strauss exponent one can construct global solutions for small data using simply a fixed point theorem and dispersive estimates [30]^{2}^{2}2This can be seen as follows: let prescribed by the linear Schrödinger flow; then is larger than the Strauss exponent if and only if . Thus one can easily get the global a priori estimate for solving , and small (for the sake of simplicity, we ignore the divergence of the integral for close to 0): be the decay of
For other nonlinearities, there are few known examples of global existence for small data below the Strauss exponent. In dimension 3 however, global existence and scattering are known for and : see Hayashi and Naumkin [16], Kawahara [20] and Germain, Masmoudi and Shatah [7]. For , this is also the case (Cazenave and Weissler [3]), but for only almost global existence is known (Ginibre and Hayashi [9]). For the GrossPitaevskii equation, Gustafson, Nakanishi and Tsai [13] proved the existence of global solutions which scatter for large time.
It is interesting to notice that for the Schrödinger equation, there is to our knowledge no known example of a nonlinearity which yields blow up in finite time for small, smooth and localized data. Such a nonlinearity should, as we have seen, necessarily correspond to a power below the Strauss exponent. For the nonlinear wave equation, we know since John [19] and Schaeffer [25] that blow up occurs for nonlinearities which have the homogeneity of the Strauss exponent.
Global existence if the nonlinearity involves derivatives of
As will become clear in this article, derivatives in a nonlinearity can play the role of a null form, thus making estimates easier as far as resonances are concerned. However in the presence of derivatives in the nonlinearity one needs to recover the derivative loss in the estimates, thus making them more complicated. To shorten the discussion, we focus here on recent developments corresponding to nonlinearities of low power.
In dimension 3, Hayashi and Naumkin [17] were able to prove global existence and scattering for small data and for any quadratic nonlinearity involving at least one derivative: , , … In dimension 2, Cohn [5] obtained the same result for a nonlinearity of the type (his proof, relying on a normal form transform and the use of pseudoproduct operators, is actually very similar to parts of the arguments of the present article). Finally, the main result is due to Delort [6], who proved global existence for a nonlinearity of the form or . His method combines the vector fields method, a normal form transform, and microlocal analysis; it enables him to prove global existence, but not scattering. Our approach to the question of global existence is quite different from his. The Fourier analysis we develop is essentially a new point of view on the vector field and normal form methods.
1.2. The notion of spacetime resonance
The concept of spacetime resonance is a natural generalization of resonance for ODEs. If one considers a linear dispersive equation on
then the quadratic time resonances can be found by considering plane wave solutions In this case time resonance for corresponds to
However, time resonances tell only part of the story for dispersive equations when one considers spatially localized solutions. Specifically, if one considers two solutions and with data localized in space around the origin and in frequency around and , respectively, then the solutions and at large time will be spatially localized around and . Thus quadratic spatial resonance is defined as the set where
We define quadratic spacetime resonance as
The idea is that only frequencies in play a significant role in the longterm behavior of nonlinear dispersive equations. Indeed, the interaction between frequencies which are not time resonant is harmless, whereas frequencies which are not space resonant cannot interact since they have disjoint support  to be precise, this last point is valid only if the nonlinearity is local.
We believe that spacetime resonances provide a key to understand the global behavior of nonlinear dispersive equations, for small data at least. We have been using this notion, along with its natural analytical framework, to study threedimensional nonlinear Schrödinger equations [7], and more recently, water waves [8].
What heuristic understanding of quadratic nonlinear Schrödinger equations does the notion of spacetime resonance give? The three possible polynomial nonlinearities are , , and . An elementary computation (see Section 2) shows that for the two first, is reduced to a point, whereas it is a dimensional subspace for the third one. This explains why, in dimension , global existence can be proved relatively easily for and , whereas for only almost global existence is known. In dimension , the decay given by the linear Schrödinger equation is only ; in other words, quadratic nonlinearities are shortrange, making global existence results very delicate. Actually, the only known results hold for nonlinearities of the type or ; more precisely: the nonlinearities for which global existence holds exclude interactions between and , and involve derivatives. Why are derivatives in the nonlinearity helpful as far as global existence is concerned? This can be understood by going back to the spacetime resonant set, which is equal to the zero frequencies of the interacting waves; these zero frequencies are canceled by derivatives.
The above considerations lead us to the choice of a quadratic nonlinearity in the theorem below. For low frequencies, which is where resonances occur, a derivative is needed to play the role of a null form, thus will look like . Taking of the same form for high frequencies would lead to a problem distinct of resonances, which is our primary focus, namely: how to use the smoothing effect of the equation to “recover” derivatives. Since we want to avoid this technical complication, we simply define to be a standard product for high frequencies.
1.3. Main result
Consider the following equation on , a complexvalued function of ,
(NLS) 
where , are complex numbers and is defined by
denoting the Fourier transform, and where the symbol is smooth, linear for , and equal to for . Thus is like a derivative for low frequencies, and the identity for high frequencies.
Remark.
The fact that the data are given at time does not have a deep meaning: it is simply more convenient when performing estimates, since the decay of given by the linear part of the equation is not integrable at .
Before stating the theorem, let us introduce the profile given by .
Theorem 1.
There exists such that if satisfies
then there exists a global solution of such that
Furthermore, this solution scatters i.e. there exists such that
Remark.
Using the tools developed in this article, more general nonlinear Schrödinger equations can be treated, we give below a few examples.

The conclusion of the theorem still holds if any cubic terms of polynomial type are added, that is for the following equation
(notice that it is not trivial to obtain the decay proved in the theorem even if the nonlinearity consists only of cubic terms).

The theorem can be extended in a straightforward way to systems for which no quadratic or cubic spacetime resonances occur.

Finally, it is possible to handle more general pseudoproducts than . It should also be possible to extend our result to the case where by analyzing high frequencies more carefully than we have done. Finally, we remark that the fact that is linear for low frequencies simplifies some manipulations in the following, but is not essential.
1.4. Plan of the proof
The article is structured as follows:
In Section 2, we analyze the resonant structure of the different terms of the equation, and perform a normal form transform on a certain part of the nonlinearity. This yields two terms, and , which have different behaviors, and will satisfy different estimates, stated in section 3: (6) for and (7) for . The proof of these estimates, performed in sections 6 to 10 will give Theorem 1.
In Section 4 we recall or establish basic linear harmonic analysis results .
In Section 5 we turn to basic multilinear harmonic analysis, specifically pseudoproduct operators.
Finally, in the appendix A, we prove boundedness of multilinear operators with flag singularities, a fundamental result of harmonic analysis that is needed in the proof.
1.5. Notations
We denote by constants that may vary from one line to another, and use the standard notation if there exists a constant such that , and if and . The Fourier transform of is denoted by or ; the normalisation is the following
The Fourier multiplier with symbol is given by
2. Computation of the resonances and first transformation of the equation
Recall that denotes the profile of or . Then
(1) 
thus
(2) 
where
2.1. Computation of the resonances
The analysis that we will perform will rely on our understanding of resonances between two or three wave packets. In the present section, we describe the space, time, and spacetime resonant sets; then we define cutoff functions, which split the or plane into the different types of resonant sets.
2.1.1. Quadratic resonances
Due to our choice of nonlinearity, the only type of quadratic interactions occuring are two waves giving a wave or two waves giving a wave, or for short: “ gives ” and “ gives ”. The corresponding phase functions are
A simple computation gives that the space, time, and spacetime resonant sets are: for
and for
In both cases the spacetime resonant set is reduced to a point! This is to a large extent the key of the above theorem.
Further notice that as far as is concerned, ; thus for this type of interaction, we shall not have to take space resonances into account for the analysis.
We take this opportunity to analyze the interaction ( gives ) and explain why this interaction is out of the scope of our theorem. For gives one easily sees that
Thus, the spacetime resonant set is too large; this explains why global existence should not be expected, or at least why our method does not apply.
2.1.2. Cubic resonances
All the possible cubic interactions, namely “ gives ”, “ gives ”, “ gives ”“   gives ”, occur for as will become clear in the next section. They correspond respectively to the phase functions
(3) 
A small computation shows that the spacetime resonant sets are:
Note that the spacetime resonant sets , which seems (and will be) favorable to obtain estimates. The set , which looks very problematic is actually benign since by the following identity
(4) 
it will generate “null terms”. That is when trying to establish the weighted estimate, one differentiates a certain trilinear expression in , which corresponds to adding an weight in physical space. The worst term arises when the derivative hits an oscillating term with phase , which introduces a factor of . Due to the above identity, one can substitute to this factor , which is harmless since an integration by parts in or makes it disappear. See Section 10 for the details.
2.1.3. Partition of the frequency space
The proof will rely on a decomposition of the multilinear expressions, which will be achieved by splitting the , or space; this manipulation will enable us to treat separately the different types of resonnances.
Let us first explain the procedure in the case of quadratic interactions: consider either the or the case, and define 3 smooth functions , and of such that
Of course, the splitting in the case is easier since time resonances are trivial then and one takes
The case of cubic resonances is handled similarly in the cases where the spacetime resonant set is trivial, ie , and . This gives cutoff functions
All the cutoff functions which have been defined will be dilated as time goes by, in the following way
2.2. Normal form transform and decomposition of
Split the integral occuring in (2) using the quadratic cutoff functions, and integrate by parts in the term with , using the identity
(this manipulation is nothing but a normal form transform). A small computation shows that the equation (2) can then be rewritten as
(5) 
with
and all the remaining terms are denoted by where
Thus consists of the boundary terms arising from integration by parts in , consists of terms that are strongly localized in frequency, consist of quadratic terms, and consists of cubic terms. The point here is that and satisfy different types of estimates since is less localized in space than , but is pointwise smaller.
3. A priori estimates and outline of the proof
The proof of the theorem will consist in the following a priori estimates: for ,
(6) 
and for ,
(7) 
Since , this implies
(8) 
The above estimates will be established separately for and the three components of , i.e., , and . Furthermore, it will be necessary to decompose further, by observing that can be seen as a bilinear operator and that
(9) 
Terms involving are the simplest to estimate and we shall skip them. Terms of the form and terms of the form or will be estimated in different ways.
In order to simplify the notations, we will set in the following and equal to 1, and we will denote indifferently for or its complex conjugate .
4. Linear harmonic analysis: basic results
The following are standard inequalities and notations that we include for the convenience of the reader.
4.1. A GagliardoNirenberg type inequality
For Schrödinger equation the generator of the pseudo conformal transformation plays the role of partial differentiation. Thus we have
Lemma 4.1.
The following inequality holds
Proof.
The proof relies on the observation that , with . Thus we get
where we used the standard GagliardoNirenberg inequality for the first inequality. ∎
4.2. LittlewoodPaley theory
Consider a function supported in the annulus such that
Define first
and then the Fourier multipliers
This gives a homogeneous and an inhomogeneous decomposition of the identity (for instance, in )
All these operators are bounded on spaces:
Also recall Bernstein’s lemma: if ,
(10) 
Finally, we will need the LittlewoodPaley square and maximal function estimates
Theorem 2.
(i) If , with (the latter denoting the annulus of center , inner radius , outer radius ), and ,
Furthermore, denoting ,
(ii) If , denoting , .
4.3. Fractional integration and dispersion
To some extent, the approach that we follow transforms the question “ how does the linear Schrödinger flow and resonances interact?” into “how can one combine fractional integration and the dispersive estimates for the Schrödinger group?”
The following lemma will thus be very useful. Define a smooth function such that for and for . Then set for
thus is like fractional integration of order for frequencies , and like for frequencies .
Lemma 4.2.

(i) If , and either , and
(ii) If , there holds
(iii) If , and
(iv) If , , , and
Proof.
The points and are standard. In order to prove , observe that it follows from interpolation between the estimate, which is clear, and the estimate, which reads
By scaling, it suffices to prove this estimate if and . This is done as follows
As for , it follows from , , and
(11) 