Modified scattering for the cubic Schrödinger equation on product spaces and applications
We consider the cubic nonlinear Schrödinger equation posed on the spatial domain . We prove modified scattering and construct modified wave operators for small initial and final data respectively (. The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when . As a consequence, we obtain global strong solutions (for ) with infinitely growing high Sobolev norms .
Key words and phrases:Modified Scattering, Nonlinear Schrödinger equation, wave guide manifolds, energy cascade, weak turbulence
The purpose of this work is to study the asymptotic behavior of the cubic defocusing nonlinear Schrödinger equation posed on the wave-guide manifolds :
where is a complex-valued function on the spatial domain . In particular, we want to understand how this asymptotic behavior is related to a resonant dynamic, in a case when scattering does not occur. Our results can be directly extended to the case of a focusing nonlinearity ( in the left hand-side of (1.1)) but we will however be concerned with small data, so this distinction on the nonlinearity will not be relevant. On the other hand the result of Corollary 1.4 providing solutions blowing-up at infinite time is more striking in the defocusing case because in the focusing case one may have blow-up in finite time (via the quite different mechanism of self-focusing).
1.1. Motivation and background
The question of the influence of the geometry on the global behavior of solutions to the nonlinear Schrödinger equation
dates back at least to . The first natural question is the issue of global existence of solutions, and many works have investigated this problem in different geometric settings [4, 5, 6, 14, 19, 23, 24, 25, 30, 37, 50, 52, 57, 58, 59, 60, 61, 62, 69, 71, 72, 77, 84, 89]. The conclusion that could be derived from these works is that the geometry of the spatial domain turned out to be of importance in the context of the best possible Strichartz inequalities or the sharp local in time well-posedness results (see e.g. [6, 22, 23]). However, the analysis in [58, 59, 60, 61, 62, 77] seems to indicate that, at least in the defocusing case111In the focusing case and for large data, it is likely that existence or nonexistence questions to elliptic problems also plays an important role., the only relevant geometric information for the global existence in the energy space is the “local dimension”, i.e. the dimension of the tangent plane.
The next natural question concerns the asymptotic behavior. There the geometry must play a more important role. This is the question in which we are interested in this paper, focusing on the simpler case of noncompact quotients of .
When the domain is the Euclidean space, , this question is reasonably well-understood at least when the nonlinearity is defocusing and analytic ( odd integer). In this case, global smooth solutions disperse and in many cases even scatter to a linear state (possibly after modulation by a real phase when , ) [28, 30, 33, 35, 36, 55, 69, 70, 76, 81, 89, 91].
In contrast, much less is known for compact domains. The most studied example is that of the torus . In this case, many different long-time behaviors can be sustained even on arbitrarily small open sets around zero, ranging from KAM tori [12, 38, 74, 80] to heteroclinic orbits [31, 47] and coherent out-of-equilibrium frequency dynamics222Interestingly, all these long-time results derive from an analysis of resonant interactions that will play a central role in this work as well. . One may also mention [7, 10, 20, 86], where invariant measures for (1.1) are constructed, when the problem is posed on , or the dimensional ball for (with radial data). These works establish the existence of a large set of (not necessarily small) recurrent dynamics of (1.1).
In light of the above sharp contrast in behavior between and its compact quotient , considerable interest has emerged in the past few years to study questions of long-time behavior on “in between” manifolds, like the ones presented by the non-compact quotients of Euclidean space [52, 59, 60, 84, 85, 87].
In the generality of non-compact Riemannian manifolds , it seems plausible that a key role is played by the parameter for which solutions to the linear NLS equation ((1.2) with ) with smooth compactly supported initial data decay like . In light of the Euclidean theory on , one can draw the following hypothetical heuristics: H1) when , global solutions (sufficiently small in the focusing case) scatter and no further information is needed about the geometry “at infinity”; H2) if , global solutions scatter, but the geometry “at infinity” plays an important role in the analysis of certain sets of solutions (e.g. in the profile decomposition); H3) if , no nontrivial solution can scatter and H4) if , global solutions exhibit some “modified scattering” characterized by a correction to scattering on a larger time-scale. We are interested in this latter regime to which (1.1) belongs.
In support of the heuristic H1) we cite the results in [4, 62, 66, 87, 88]. The second heuristic H2) was put to test in  where the authors study the quintic NLS equation on . There, a strong relation is drawn between the large-data scattering theory for the quintic NLS equation and the system obtained from its resonant periodic frequency interactions. The relevance of the result in  to our work here lies in the following two important messages: The first is that the asymptotic behavior of (1.2) on can be understood through: i) the asymptotic dynamics of the same equation on Euclidean spaces, and ii) the asymptotic dynamics of a related resonant system corresponding to the resonant interactions between its periodic frequency modes. The second message from  is the insight that the resonant interactions in (1.1) will play a vivid and decisive role in dictating the anticipated non-scattering asymptotic dynamics of (1.1). Indeed, as [52, 87] show that quintic interactions lead to scattering behavior for small data, and since non-resonant interactions in (1.1) can be transformed, at least formally, into quintic interactions via a normal form transformation, it is up to the resonant interactions alone to drive the system away from scattering. This is the content of our main result.
The other interesting feature of the asymptotic dynamics of (1.1) as opposed to previous modified scattering results, is that the modification dictated by its resonant system is not simply a phase correction term when , but rather a much more vigorous departure from linear dynamics. As we argue below, this will pose a new set of difficulties in comparison to previous modified scattering results in the literature, but, on the plus side, will lead us to several interesting and new types of asymptotic dynamics.
1.2. Statement of the results
Consistent with the heuristics above, we show that the asymptotic dynamic of small solutions to (1.1) is related to that of solutions of the resonant system
Here is the Fourier transform of at . Noting that the dependence on is merely parametric, the above system is none other than the resonant system for the cubic NLS equation on . The equation (1.3) is globally well-posed thanks to Lemma 4.1 below.
More precisely, our main results are as follows. Below is an arbitrary integer, and and denote Banach spaces whose norms are defined in (2.8) later. They contain all the Schwartz functions.
A similar statement holds as , and a more precise one is contained in Theorem 6.1. It is worth pointing out that for , even the global existence claim in the above theorem is new, due to the energy-supercritical nature of (1.1) in this dimension. However, the main novelty is the modified scattering statement to a non-integrable asymptotic dynamics, given by (1.3).
In addition, we construct modified wave operators in the following sense:
It is worth mentioning that a slight modification of the proof of Theorems 1.1 and 1.2 shows that similar statements hold if is replaced by the sphere (with a suitably modified resonant system). Indeed, the largest part of the analysis is exploiting the dispersion. In the case of the spectrum of the Laplace-Beltrami operator satisfies the non-resonant condition needed for the normal form analysis, and the well-posedness analysis on the sphere of [23, 24] provides the needed substitute of Lemma 7.1. A similar remark applies to the case of a partial harmonic confinement (cf. ). On the other hand, the extension of our analysis to an irrational torus is less clear because of the appearance of small denominators in the normal form analysis.
As a consequence of Theorem 1.2, all the behaviors that can be isolated for solutions of the resonant system (1.3) have counterparts in the asymptotic behavior of solutions of (1.1). Most notably, given the existence of unbounded Sobolev orbits for (1.3) as proved in  for (cf. Theorem 4.8 for an explicit construction with quantitative lower bounds on the growth), we have the following.
Corollary 1.4 (Existence of infinite cascade solutions).
Let and , . Then for every there exists a global solution of (1.1) such that
More precisely, there exists a sequence for which
Corollary 1.4 gives a partial solution to a problem posed by Bourgain [17, page 43-44] concerning the possible long time growth of the , norms for the solutions of the cubic nonlinear Schrödinger equation. This growth of high Sobolev norms is regarded as a proof of the (direct)energy cascade phenomenon in which the energy of the system (here the kinetic energy) moves from low frequencies (large scales) towards arbitrarily high frequencies (small scales). Heuristically, the solution in Corollary 1.4 can be viewed as initially oscillating at scales that are , but at later times exhibits oscillations at arbitrarily smaller length-scales. This energy cascade is a main aspect of the out-of-equilibrium dynamics predicted for (1.1) by the vast literature of physics and numerics falling under the theory of weak (wave) turbulence (cf. [75, 92]).
The corresponding result on does not directly follow from Corollary 1.4 (nor does it imply it). This is somehow surprising because one would naturally expect that adding a dispersive direction to would drive the system closer to nonlinear asymptotic stability, and further from out-of-equilibrium dynamics (this is indeed the case if we study the equation on for as was shown by the scattering result in ). Our construction draws heavily on [31, 47, 51] where unbounded Sobolev orbits are constructed for the resonant system and applied to get finite time amplifications of the Sobolev norms on . However, in the case of the torus, nonresonant interactions do not disappear and feed back into the dynamics after a long but finite time. This is precisely where the more dispersive setting of makes a difference: in this case, nonresonant terms are transformed into quintic terms which scatter, and hence, at least heuristically do not modify the long-term dynamics.
Previous results in the spirit of Corollary 1.4 may be found in [15, 16] for linear Schrödinger equations with potential, [31, 47, 73] for finite time amplifications of the initial norm, [8, 11, 51] for NLS with suitably chosen non-local nonlinearities, and [40, 41, 42, 78, 90] for the zero-dispersion Szegö and half-wave equations. Concerning the opposite question of obtaining upper bounds on the rate of possible growth of the Sobolev norms of solutions of NLS equations we refer to [9, 18, 32, 82, 83].
Corollary 1.6 (Forward compact solutions).
Let . For functions on defined for all , we define the “limit profile set” as
(no nontrivial scattering) Assume that solves (1.1) and that is a point. If is sufficiently small, then .
(scattering up to phase correction) There exists a nontrivial solution of (1.1) and a real function such that is a point.
((quasi-)periodic frequency dynamics) There exists a global solution such that is compact but .
The proof of part (3) in the above corollary is interesting in its own right. In fact, we construct global solutions to (1.1) that asymptotically bounce their energy (and mass) between two disjoint sites in frequency space periodically in time. These correspond to periodic-in-time solutions of (1.3) that exhibit the following “beating effect” (in the nomenclature of ): there exists two disjoint subsets and in , so that for any there exists a solution of (1.3) that is supported in frequency space on in such a way that the fraction of the mass carried by each of the two sets alternates between and periodically in time. We refer to Subsection 4.2 for more constructions including asymptotically quasi-periodic dynamics.
It would be interesting to understand what is the optimal topology to obtain our results. It is probably a lot larger than the one we use. Progress in this direction would impact the following:
The results are restricted to small data. In the absence of a “correct” topology, the exact meaning of “large data” is not well established.
We cannot let any norm, grow in Corollary 1.4, partly because we want to cover all the cases in a uniform manner, using simple exponents. More careful analysis might address this point (for instance, either lowering the regularity requirement in Theorem 1.2 or a more quantified version of the construction in  would resolve this). We decided not to pursue this point here because Corollary 1.3 already captures the energy cascade phenomenon.
It is possible that a more adapted topology allows to define the scattering operator in a good Banach space.
Finally, we also mention the situation in  where the partial periodicity is replaced by adding a (partially) confining potential.
1.3. Overview of proof
1.3.1. Modified scattering
In order to describe the asymptotic behavior of a nonlinear dispersive equation like (1.1), it may be relevent to study the limiting behavior of the profile obtained by conjugating out the linear flow. If converges to a fixed function , then the solution scatters. If not, the next best thing is to find the simplest possible dynamical system that describes the asymptotic dynamics of . To find this system, one has to work on proving global a priori energy and decay estimates that allow to decompose the nonlinearity in the equation in the following way:
where is integrable in time. When this is possible, one can hope to prove that the asymptotic dynamics converge to that of the effective system
Proving the global a priori energy and decay estimates can be a daunting task depending on the problem at hand. On the other hand, the process of proving the convergence to the dynamics of (1.6) depends very much on how simple or complicated is.
Previous modified scattering results that we are aware of, only concerned equations (or systems) posed on , quasilinear or semilinear [1, 26, 33, 34, 54, 55, 56, 64, 65, 68, 76, 91] and had an integrable asymptotic system for (1.6). This often allowed for a simple phase conjugation (in physical or Fourier space) to give the modification. In contrast, our limiting system is given by (1.3) which is not only a non-integrable system, but also allows for the growth of norms of its solutions as we saw in Corollary 1.4. This requires a robust approach to modified scattering, that tolerates the growth of the limiting system as long as the decay of in (1.5) is sufficiently fast to trump the divergence effects of the effective part .
1.3.2. Isolating the resonant system: heuristics
To isolate the effective interactions ( above), we can argue formally by looking at (1.1) in Fourier space:
where and where denotes the Fourier transform of at . Roughly speaking, a stationary phase argument in the integral implies333see  for a previous use of this remark. that for very large times the equation for can be written as
This is essentially an ODE system for each . As is well-known, resonant interactions corresponding to for which play a particularly important role in the dynamics of such an ODE, especially given the decay of . This suggests that the expression above can be simplified to
As a result, one should expect the asymptotic dynamics of to be dictated by the ODE system given by the first term on the right-hand side above. The latter system can be seen to be autonomous when written in terms of the slow time scale in which it has the form (1.3). Note that this system was previously studied and shown to have interesting dynamics [27, 31, 39, 51].
1.3.3. Norms and the control of the solution
As mentioned above, establishing a priori energy and decay estimates is a precursor to isolating the leading order dynamics. In the scalar case [55, 68], the needed energy estimates follow easily once we guarantee the decay for the norm. Indeed, schematically speaking, if is an appropriate energy of the system that controls its strong norms, then one has the relation
which barely allows to close any polynomial-growth bootstrap for . The decay can be bootstrapped by relying on the boundedness of the Fourier transform, which follows from the equation satisfied by . An almost identical energy method argument works in the case , but reaches its limit there. Indeed, for , we do not have access to the sharp linear decay which was crucial to closing the energy bootstrap above. To overcome this difficulty, we need additional estimates coming from the low-regularity theory. We use a hierarchy of three norms.
The -norm is bounded and essentially corresponds to the strongest information that remains a priori bounded uniformly in time.
The -norm controls the number of periodic derivatives we want to consider. It grows slowly with time, but the difference with the asymptotic dynamics decays in this norm.
The -norm is slightly stronger than the -norm. It is allowed to grow slowly, but still yields better control on objects in the -norm. In particular, it controls the same number of derivatives in the periodic directions as the -norm.
While the choice of the norm is dictated by the resonant system, there is considerable flexibility in the choices of the two other norms. Another possible choice might be a variation444But for the moment, it seems difficult in the proof of the modified wave operator to work with an intermediate norm controlling no weight in . of . One of the main problems complicating the situation here is the need for a bounded linearized operator around a solution of the asymptotic system, which is not trivial in view of the missing decay of .
The significance of the norm stems from the following two key facts: 1) it is conserved for the resonant system555Ultimately, this leads to the key non-perturbative ingredient, see (4.4) and (6.5)., and 2) it is a controlling norm for the existence and growth of its solution666In particular, this forces the restriction in Theorems 1.1 and 1.2. in view of Lemma 4.3. This, combined with Lemma 7.3, provides the extent to which we can get decay for solutions of (1.1). Interestingly, all this global analysis of the resonant system (1.3) relies heavily on using local in time Strichartz estimates on the torus in order to get global-in-time bounds for the the norm of the nonlinearity (see Lemma 7.1). At this place our view point is quite different form a naïve 1d vector valued analysis (as is the case in ).
We also note that although our approach is close in spirit to recent developments in global existence for quasilinear equations [43, 44, 63, 64, 65], some of the key estimates really pertain to the low-regularity theory (see Lemma 7.1 and Lemma 7.2777This is somewhat parallel to the energy method in the quasilinear results.).
Organization of the paper
Section 2 introduces the notations used in this paper. Section 3 provides a decomposition of the nonlinearity as in (1.5). Section 4 introduces the resonant system (1.3) and gives some properties of its solutions. Section 5 shows existence of the modified wave operators and proves Theorem 1.2. Section 6 shows the modified scattering statement and proves Theorem 1.1. Finally, Section 7 collects various additional estimates needed in the proofs.
2.1. Standard notations
In this paper . We will often consider functions and functions . To distinguish between them, we use the convention that lower case letters denote functions defined on , capitalized letters denote functions defined on , and calligraphic letters denote operators, except for the Littlewood-Paley operators and dyadic numbers which are capitalized most of the time.
We define the Fourier transform on by
Similarly, if depends on , denotes the partial Fourier transform in . We also consider the Fourier transform of ,
and this extends to . Finally, we also have the full (spatial) Fourier transform
We will often use Littlewood-Paley projections. For the full frequency space, these are defined as follows:
where , when and when . Next, we define
Many times we will concentrate on the frequency in only, and we therefore define
and define similarly to . By a slight abuse of notation, we will consider indifferently as an operator on functions defined on and on . We shall use the following commutator estimate
Below, we will need a few parameters. We fix and888The exact value of can be significantly lowered e.g. by allowing more weights in the norm in (2.8). . For a positive number, we let be an arbitrary function satisfying
Particular examples are the characteristic functions , with the natural interpretation of the integral on of .
We will use the following sets corresponding to momentum and resonance level sets:
In particular, note that if and only if are the vertices of a rectangle.
2.2. Duhamel formula
We will prove all our statements for . By time-reversal symmetry, one obtains the analogous claims for . In studying solutions to (1.1), it will be convenient to factor out the linear flow and write a solution of (1.1) as
We then see that solves (1.1) if and only if solves
We will denote the nonlinearity in (2.3) by , where the trilinear form is defined by
Now, we can compute the Fourier transform of the last expression which leads to the identity
One verifies that
Thus one may also write
According to our previous discussion, we now define the resonant part of the nonlinearity999 corresponds to in (1.5). as
We have a remarkable Leibniz rule for , namely
A similar property holds for the whole nonlinearity , where can also be a derivative in the transverse direction, . Property (2.7) will be of importance in order to ensure the hypothesis of the transfer principle displayed by Lemma 7.4.
We will often consider sequences and we define the following norm on these:
For functions, we will often omit the domain of integration from the description of the norms. However, we will indicate it by a subscript (for ), (for ) or (for ). We will use mainly three different norms: a weak norm
and two strong norms
We have the following hierarchy
To verify the middle inequality, using (2.1) and the elementary inequality
one might observe that
squaring and multiplying by , we find that (using interpolation too)
We also remark that the operators , and the multiplication by are bounded in , , , uniformly in .
The space-time norms we will use are
In most of the cases, in order to sum-up the estimates we make use of the following elementary bound
As a warm up, we can prove the following simple estimates which are sufficient for the local theory.
The following estimates hold:
However, these estimates fall short of giving a satisfactory global theory.
Coming back to (2.5), we readily obtain
Assume . We use the basic dispersive bound for the Schrödinger equation and (2.10) to get
Estimate (2.16) allows us to write for any
If , we use Sobolev estimates instead of (2.16) and get
3. Structure of the nonlinearity
The purpose of this section is to extract the key effective interactions from the full nonlinearity in (1.1). We first decompose the nonlinearity as
where is given in (2.6). Our main result is the following
Assume that for , , , : satisfy
Then for , we can write
where the following bounds hold uniformly in ,
where . Assuming in addition
we also have that
We will give a proof of Proposition 3.1 at the end of this section. It depends on various lemmas that we prove first. Among these lemmas, Lemma 3.2, Lemma 3.3 and the first part of Lemma 3.7 are essentially based on arguments, while Lemma 3.6 and the second part of Lemma 3.7 are based on regularity in Fourier space.
3.1. The high frequency estimates
Assume that . The following estimates hold uniformly in :