Global bounds for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension

Abstract

This article is concerned with the small data problem for the cubic nonlinear
Schr\"odinger equation (NLS) in one space dimension, and short range
modifications of it. We provide a new, simpler approach in order to prove that
global solutions exist for data which is small in $H^{0,1}$. In the same
setting we also discuss the related problems of obtaining a modified scattering
expansion for the solution, as well as asymptotic completeness.

Full text

arXiv:1404.7581v2 [math.AP] 12 Oct 2014
GLOBAL BOUNDS FOR THE CUBIC NONLINEAR SCHR ¨
ODINGER
EQUATION (NLS) IN ONE SPACE DIMENSION
MIHAELA IFRIM AND DANIEL TATARU
Abstract. This article is concerned with the small data problem for the cubic nonlinear
Schr¨odinger equation (NLS) in one space dimension, and short range modifications of it.
We provide a new, simpler approach in order to prove that global solutions exist for data
which is small in H0,1. In the same setting we also discuss the related problems of obtaining
a modified scattering expansion for the solution, as well as asymptotic completeness.
1. Introduction
We consider the cubic nonlinear Schr¨odinger equation (NLS) problem in one space dimen-
sion
(1.1) iut+1
2uxx =λu|u|2, u(0) = u0,
where uis a complex valued function, u:R×R→C, and λ= 1 or −1 corresponding to the
defocusing, respectively the focusing case.
Our results and proofs apply equally to short range modifications of it
(1.2) iut+1
2uxx =λu|u|2+uF (|u|2), u(0) = u0,
where Fsatisfies
|F(r)|.|r|1+δ,|F′(r)|.|r|δ, δ > 0.
A common feature of these two equations is that they exhibit Galilean invariance as well as
the phase rotation symmetry, both of which are used in our arguments.
The question at hand is that of establishing global existence and asymptotics for solu-
tions to (1.1) and then (1.2), provided that the initial data is small and spatially localized.
Traditionally this is done in Sobolev spaces of the form Hm,k, whose norms are defined by
kuk2
Hm,k := k(1 −∂2
x)m
2uk2
L2+k(1 + |x|2)k
2uk2
L2, m, k ≥0.
The problem (1.1) is completely integrable, which allows one to use very precise techniques,
i.e., the inverse scattering method, to obtain accurate long range asymptotics, even for large
data in the defocusing case. These have the form
u(t, x)≈t−1
2eix2
2t+λ|W(x/t)|2log tW(x/t).
One notes that this is not linear scattering, but rather a modified linear scattering. Indeed,
in work of Deift and Zhou [2], the inverse scattering method is used to show that the above
The first author was supported by the Simons Foundation.
The second author was partially supported by the NSF grant DMS-1266182 as well as by the Simons
Foundation.
1

asymptotics hold for data in H1,1, not only for (1.1), but also for (1.2) with a restricted
range of powers.
In the meantime, two alternate approaches have emerged, which do not depend on the
complete integrability of the problem. The first, initiated by Hayashi and Naumkin [3], and
refined by Kato-Pusateri [7], is based on deriving an asymptotic equation for the Fourier
transform of the solutions,
d
dt ˆu(t, ξ) = λit−1ˆu(t, ξ)|ˆu(t, ξ)|2+OL∞(t−1−ǫ).
This gives a result for data in H1,1.
A second approach, introduced by Lindblad-Soffer [9], is based on deriving an asymptotic
equation in the physical space along rays
(t∂t+x∂x)u(t, x) = λit−1u(t, x)|u(t, x)|2+OL∞(t−1−ǫ).
This argument yields a similar result, though the precise regularity at which this works is
not computed, and is likely higher.
The method in the present paper in some sense interpolates between the two ideas above.
Instead of localizing sharply on either the Fourier or the physical side, we use a mixed wave
packet style phase space localization, loosely inspired from the analysis in [10]. The idea
is that using wave packets one can produce a more accurate approximate solution to the
linear Schr¨odinger equation, and use that to test for the long range behavior in the nonlinear
equation.
Our interest in this problem arose from working on two dimensional water waves, where
a similar situation occurs. There, a global result was independently proved by Ionescu-
Pusateri [5] and Alazard-Delort [1] using methods based on the two ideas above. However,
implementing either of these strategies brings on considerable difficulties. Many of these
difficulties are bypassed by the authors in [4], where a simpler proof of the global result is
given.
The present paper contains the implementation of the ideas in [4] for the simpler problems
(1.1), (1.2). Our goal is two-fold, namely to provide a simpler proof of the global result with
fewer assumptions, and also to give a more transparent introduction to the work in [4]. Our
main result is Galilean invariant:
Theorem 1. a) (Global well-posedness and decay) Consider either the equation (1.1) or
(1.2), with initial data u0which is small in H0,1,
(1.3) ku0kH0,1≤ǫ≪1.
Then there exists a unique global solution uwith regularity e−it
2∂2
xu∈C(R;H0,1(R)) which
satisfies the pointwise estimate
(1.4) kukL∞.ǫ|t|−1
2,
as well as the energy bound
(1.5) ke−it
2∂2
xukH0,1.ǫ(1 + t)Cǫ2.
2

b) (Asymptotic behavior) Let ube a solution to either (1.1) or (1.2) as in part (a). Then
there exists a function W∈H1−Cǫ2(R)such that
u(x, t) = 1
√teix2
2tW(x/t)eilog t|W(x/t)|2+errx,
(1.6)
ˆu(ξ, t) = e−itξ2
2W(ξ)eilog t|W(ξ)|2+errξ,
(1.7)
where
errx∈ǫOL∞((1 + t)−3
4+Cǫ2)∩OL2
x((1 + t)−1+Cǫ2),
errξ∈ǫOL∞((1 + t)−1
4+Cǫ2)∩OL2
ξ((1 + t)−1
2+Cǫ2).
c) (Asymptotic completeness for small data) Let Cbe a large universal constant. For each
Wsatisfying
kWkH1+Cǫ2(R)≪ǫ≪1
there exists u0∈H0,1satisfying (1.3) so that (1.6) and (1.7) hold for the corresponding
solution uto (1.1) or (1.2).
The next section contains the proof of the theorem. We begin with the proof of part (a),
which is a self contained argument. The argument for part (b) is based on a more careful
analysis of the outcome of (1a). Finally, the proof of the asymptotic completeness is again
a self contained argument, which is a simpler lower regularity version of the original result
in [6]. Several remarks may be of interest:
Remark 1.1. Since one goal of this article is to present a clear and simple statement, the
result and the proofs are done in the setting of H0,1data. However, with some extra work,
the same method will also work for data in H0,s with 1
2< s ≤1.
Remark 1.2. One may ask whether one does not have W∈H1, with a smooth one to
one correspondence between u0∈H0,1and W. The work [2] of Deift and Zhou shows that
this is not the case, and that there is necessarily some logarithmic type correction to such a
property. We leave open the question of providing a direct proof of such a correspondence in
a suitable functional setting.
2. Proof of the Theorem 1.3
2.1. Local well-posedness. While the equation (1.1) is locally well-posed for data in L2,
working with data in H0,1requires a brief discussion. The initial data space has norm
ku0k2
H0,1=ku0k2
L2+kxu0k2
L2.
However, we cannot use this same space at later times since the weight xdoes not commute
with the linear Schr¨odinger flow. Instead, we introduce the vector field L=x+it∂x, which
is the conjugate of xwith respect to the linear flow, eit
2∂2
xx=Leit
2∂2
x, as well as the generator
for the Galilean group of symmetries. Naturally we have
i∂t+1
2∂2
x, L= 0, L(λu|u|2) = 2λ|u|2Lu −λu2Lu.
Next, we state and prove a preliminary global result:
3

Proposition 2.1. The equation (1.1) is (globally) well-posed for initial data in H0,1, in the
sense that it admits a unique solution u∈C(R, L2)such that Lu ∈C(R, L2). Further, such
a solution is continuous away from t= 0, and satisfies u∈C(R\ {0}, L∞). Furthermore,
near t= 0 we have
(2.1) |u(t, x)|.t−1
2ku0kH0,1.
Proof. We start with the L2well-posedness, which is based on the Strichartz estimate for
the linear inhomogeneous problem
(i∂t+1
2∂2
x)u=f, u(0) = u0,
which has the form
(2.2) kukL∞
tL2
x+kukL4
tL∞
x.ku0kL2+kfkL1
tL2
x.
This allows us to treat the nonlinearity perturbatively and obtain the unique local solution
via the contraction principle in the space L∞
t(0, T ;L2
x)∩L4
t(0, T ;L∞
x) provided that Tis
small enough1,T≪ ku0k4
L2. The local well-posedness in L2implies global well-posedness
due to the conservation of the mass kuk2
L2.
To switch to the H0,1data we need to write the equation for Lu, which has the form
(2.3) (i∂t+1
2∂2
x)Lu = 2λ|u|2Lu −λu2Lu.
We remark that this is exactly the linearization of the equation (1.1). The L2well-posedness
of this problem also follows from the Strichartz estimate (2.2).
Finally, we consider pointwise bounds. Denoting w=ue−ix2
2t, we have ie−ix2
2tLu =it∂xw.
Hence, away from t= 0 we have w∈C(R\{0};H1), and the continuity property of w, namely
w∈Cloc(R\ {0};C0(R)), follows from the Sobolev embedding H1(R)⊂C0(R). Since whas
limit zero at infinity, the similar property for ualso follows. Finally, the pointwise bound
(2.1) is a consequence of the Gagliardo-Nirenberg type estimate
kwkL∞.kwk1
2
L2k∂xwk1
2
L2.

2.2. Wave packets and the asymptotic equation. To study the global decay properties
of solutions to (1.1) and (1.2), we introduce a new idea, which is to test the solution uwith
wave packets which travel along the Hamilton flow. A wave packet, in the context here, is an
approximate solution to the linear system, with O(1/t) errors. Precisely, for each trajectory
Γv:= {x=vt}, traveling with velocity v, we establish decay along this ray by testing with
a wave packet moving along the ray.
To motivate the definition of this packet we recall some useful facts. First, this ray is
associated with waves which have spatial frequency
ξv:= v=x
t.
This is associated with the phase function
φ(t, x) := x2
2t.
1This is exactly the scaling relation.
4

Then it is natural to use as test functions wave packets of the form
Ψv(t, x) := χx−vt
√teiφ(t,x).
Here we take χto be a Schwartz function. In other related problems it might be more
convenient to take χwith compact support. For normalization purposes we assume that
Zχ(y)dy = 1.
The t1
2localization scale is exactly the scale of wave packets which are required to stay
coherent on the time scale t. To see that these are reasonable approximate solutions we
observe that we can compute
(2.4) (i∂t+1
2∂2
x)Ψv=1
2teiφ∂xt1
2χ′x−vt
√t+i(x−vt)χx−vt
√t,
and observe that the right hand side has the same localization as Ψvand size smaller by a
factor t−1. Thus one can think of Ψvas good approximate solutions for the linear Schr¨odinger
equation only on dyadic time scales ∆t≪t.
If one compares Ψvwith the fundamental solution to the linear Schr¨odinger equation,
conspicuously the t−1
2factor is missing. Adding this factor does not improve the error in
the interpretation of Ψvas a good approximate solution, so we have preferred instead a
normalization which provides simpler ode dynamics for the function γdefined below.
As a measure of the decay of ualong Γvwe use the function
γ(t, v) := Zu¯
Ψvdx.
For the purpose of proving part (a) of the theorem we only need to consider γalong a single
ray. However, in order to obtain the more precise asymptoptics in part (b) we will think of
γas a function γ(t, v).
We can also express γ(t, v) in terms of the Fourier transform of u,
γ(t, v) = Zˆu(t, ξ)¯
ˆ
Ψ(t, ξ)dξ.
Here a direct computation yields
ˆ
Ψ(t, ξ) = 1
√2πZe−ixξeix2
2tχ(t−1
2(x−vt)) dx
=1
√2πe−itξ2
2eit(ξ−v)2
2Ze−i(x−vt)(ξ−v)ei(x−tv)2
2tχ(t−1
2(x−vt)) dx
=t1
2e−itξ2
2χ1(t1
2(ξ−v)),
where χ1=eiξ2
2
[
eix2
2χis a Schwartz function with the additional property that
Zχ1(ξ)dξ =Zχ(x)dx = 1.
Then we can write
(2.5) γ(t, ξ) = eitξ 2
2ˆu(t, ξ)∗ξt1
2χ1(t1
2ξ).
5

Both the solution uof (1.1) along the ray Γvand its Fourier transform evaluated at vare
compared to γ(t, v) as follows:
Lemma 2.2. The function γsatisfies the bounds
kγkL∞.t1
2kukL∞,kγkL2
v.kukL2
x,k∂vγkL2
v.kLukL2
x.
(2.6)
We have the physical space bounds
ku(t, vt)−t−1
2eiφ(t,vt)γ(t, v)kL2
v.t−1kLukL2
x,
ku(t, vt)−t−1
2eiφ(t,vt)γ(t, v)kL∞.t−3
4kLukL2
x,
(2.7)
and the Fourier space bounds
kˆu(t, ξ)−e−itξ2
2γ(t, ξ)kL2
ξ.t−1
2kLukL2
x,
kˆu(t, ξ)−e−itξ2
2γ(t, ξ)kL∞.t−1
4kLukL2
x.
(2.8)
Proof. Denote w:= e−iφu. Then ∂vw=it∂xw=it∂x(e−iφu) = ie−iφLu, and we can express
γin terms of was a convolution with respect to the vvariable,
(2.9) t−1
2γ(t, v) = w(t, vt)∗vt1
2χ(t1
2v),
where the kernel on the right has unit integral. In other words, t−1
2γ(t, v) is a regularization
of w(t, vt) on the t−1
2scale in v, or equivalently, a localization of w(t, vt) to frequencies less
than t−1
2. Hence, via Young’s inequality, we have the straightforward convolution bounds
kγ(t, v)kL∞.t1
2kw(t, vt)kL∞=t1
2kukL∞,
kγ(t, v)kL2
v.t1
2kw(t, vt)kL2
v=kukL2
x,
as well as
k∂vγ(t, v)kL2
v.t1
2k∂vw(t, vt)kL2
v=kLukL2
x.
Here we have used the fact that the L2
v,L2
xnorms are related by
kfkL2
x=t1
2kfkL2
v.
To bound the difference t−1
2γ(t, v)−w(t, vt) we use the fact that the above kernel has unit
integral to write
|t−1
2γ(t, v)−w(t, vt)|=


Z(w(t, (v−z)t)−w(t, vt))χ(t1
2z)t1
2dz



≤Z|w(t, (v−z)t)−w(t, vt)||χ(t1
2z)|t1
2dz.
(2.10)
To prove the pointwise bound in (2.7) we use H¨older’s inequality to obtain
|w(t, vt)−w(t, (v−z)t)|.|z|1
2k∂vwkL2
v,
which by (2.10) leads to
|e−iφu(t, vt)−t−1
2γ(t, v)|.k∂vw(t, vt)kL2
vZ|z|1
2t1
2|χ(t1
2z)|dz ≈t−1
4k∂vw(t, vt)kL2
v=t−3
4kLukL2
x.
6

To prove the L2
vbound in (2.7) we express the right hand side in the last integrand in (2.10)
in terms of the derivative of wto obtain
|t−1
2γ(t, v)−w(t, vt)|.Z1
0Z|z||∂vw(t, (v−hz)t)|t1
2χ(t1
2z)t1
2dz dh.
Hence we can evaluate the L2norm as follows:
ke−iφu(t, vt)−t−1
2γ(t, v)kL2.k∂vw(t, vt)kL2
vZ|z|t1
2|χ(t1
2z)|dz
≈t−1
2k∂vw(t, vt)kL2
v=t−1kLukL2
x.
This concludes the proof of the bound (2.7). The estimate (2.8) is obtained in a similar
manner, but using (2.5) instead of (2.9), as well as the relation
k∂ξ(eitξ2
2ˆu(t, ξ))kL2
ξ=kLukL2
x.

By the previous Lemma 2.2 we can conclude that γis indeed a good approximation of u
along a ray, but no information on the rate of decay of γwas established. Hence, the crucial
next step is to obtain an approximate ode dynamics for γ(t, v):
Lemma 2.3. If usolves (1.1) then we have
(2.11) ˙γ(t, v) = −it−1λ|γ(t, v)|2γ(t, v)−R(t, v),
where the remainder Rsatisfies
(2.12) kRkL∞
x.t−1
4kLukL2
xt−1+kuk2
L∞
x,kRkL2
v.t−1
2kLukL2
xt−1+kuk2
L∞
x.
Proof. A direct computation yields
˙γ(t) = Zut¯
Ψv+u¯
Ψvt dx =Zi(1
2uxx −λu|u|2)¯
Ψv+u¯
Ψvt dx
=Z−iu(i∂t+1
2∂2
x)Ψv−iλu|u|2¯
Ψvdx.
Using the relation (2.4) and integrating by parts we obtain
˙γ(t) = Zi1
2t∂x(t1
2χ′+i(x−vt)χ)e−iφu dx −Ziλu|u|2¯
Ψvdx
=−Z1
2t2(t1
2χ′+i(x−vt)χ)e−iφLu dx −Ziλu|u|2¯
Ψvdx.
Hence we can write an evolution equation for γ(t) of the form
˙γ(t, v) = −iλt−1|γ(t, v)|2γ(t)−R(t, v),
where R(t, v) contains error terms which are the contributions arising from using Ψvas a good
approximation of the solution of the linear Sch¨odinger equation, and also from substituting
7

uby γin the cubic nonlinearity. We write the remainder R(t, v) as a sum of three quantities
which can be easily bounded:
R(t, v) := −Z1
2t2(t1
2χ′+i(x−vt)χ)e−iφLu dx −iλ Zu¯
Ψv(|u|2− |u(t, vt)|2)dx
+iλγ(|u(t, vt)|2−t−1|γ(t, v)|2)
:=R1+R2+R3.
The integral R1is expressed as a convolution in v,
R1=−1
t(t1
2χ′(t1
2v) + itvχ(t1
2v)) ∗v(∂vw(t, vt)).
Hence, by H¨older’s inequality we obtain the pointwise bound
|R1|.t−3
4k∂vw(t, vt)kL2
v=t−5
4kLukL2
x,
while estimating the convolution kernel in L1
vyields the L2
xbound
kR1kL2
v.t−1k∂vw(t, vt)kL2
v=t−3
2kLukL2
x.
Since |u|=|w|, the second term R2:= −iλ Ru¯
Ψv(|u|2− |u(t, vt)|2)dx is bounded by
|R2(t, v)|.kuk2
L∞Z|χ(t−1
2(x−vt))|(|w(t, x)−w(t, vt)|)dx
=t1
2kuk2
L∞Z|χ(t1
2z)|(|w(t, (v−z)t)−w(t, vt)|)t1
2dz,
where the last integrand is the same as in (2.10). Then R2is estimated exactly as in the
proof of (2.7) following (2.10).
Finally, for R3it suffices to combine the estimates (2.6) and (2.7).

2.3. Proof of the global well-posedness result. From Proposition 2.1 we know that a
global solution exists, so it remains to establish the bounds (1.5) and (1.4). Proposition 2.1
also shows that ku(t)kL∞is continuous in time away from t= 0. Then a continuity argument
implies that it suffices to prove these bounds under the additional bootstrap assumption:
(2.13) kukL∞≤Dǫ|t|−1
2,
where Dis a large constant such that 1 ≪D≪ǫ−1. Then we want to prove the energy
bound (1.5), and then show that (1.4) holds with an implicit constant which does not depend
on D.
The energy estimate for Lu:To advance frome time 0 to time 1 we use the local well-
posedness result above. This gives
kLu(1)kL2.kxu(0)kL2≤ǫ.
To move forward in time past time 1 we use energy estimates in (2.3) and then (2.13) to
obtain
kLu(t)kL2≤ kLu(1)kL2+Zt
1ku(s)k2
∞kLu(s)kL2ds
≤ kLu(1)kL2+D2ǫ2Zt
1
s−1kLu(s)kL2ds.
8

Applying Gronwall’s inequality gives
(2.14) kLu(t)kL2.ǫ(1 + t)D2ǫ2,
which, combined with the conservation of mass, leads to
ke−it
2∂2
xu(t)kH0,1.ǫ(1 + t)D2ǫ2.
The pointwise decay bound: From the bound (2.7) in Lemma 2.2 and (2.14) we get
ke−iφu−t−1
2γkL∞
x.t−3
4kLukL2
x.ǫ(1 + t)−3
4+D2ǫ2,
so it remains to estimate γ. At time t= 1 we can use (2.1) and the pointwise part of (2.6)
to conclude that
kγ(1, v)kL∞.ǫ.
On the other hand, using our bootstrap assumption (2.13) and the L2bound (2.14) in
Lemma 2.3 we obtain a good bound for R(t, v), namely
kR(t, v)kL∞.ǫ(1 + D2ǫ2)t−5
4+D2ǫ2.
Then integrating in (2.11) we obtain
|γ(t, v)| ≤ |γ(1, v)|+Zt
1|R(s, v)|ds .ǫ(1 + D2ǫ2),
which leads to
|u|.(ǫ+D2ǫ3)|t|−1
2.
Under the constraint 1 ≪D≪ǫ−1we obtain (1.4), and conclude the bootstrap argument.
2.4. The asymptotic expansion of the solution. To construct the asymptotic profile
Wwe use the ode in Lemma 2.3 for γ(t, v). The inhomogeneous term R(t, v) is estimated
in L∞and L2
vby combining (2.12) with (1.4) and (2.14) to obtain
(2.15) kR(t, v)kL∞.ǫt−5
4+D2ǫ2,kR(t, v)kL2
v.ǫt−3
2+D2ǫ2.
The ODE for γ, namely
˙γ(t) = −i
t|γ(t)|2γ(t)−R(t, v)
can be explicitly solved in polar coordinates. Since R(t, v) in uniformly integrable in time,
it follows that for each v,γ(t, v) is well approximated at infinity by a solution to the unper-
turbed ODE corresponding to R1= 0, in the sense that
(2.16) γ(t, v) = W(v)ei|W(v)|2log t+OL∞
v(ǫt−1
4+D2ǫ2).
Integrating the L2
vpart of (2.15) leads to a similar L2
vbound
(2.17) γ(t, v) = W(v)ei|W(v)|2log t+OL2
v(ǫt−1
2+ǫ2D2).
Then the asymptotic expansions in (1.6), (1.7) follow directly from (2.7) and (2.8), where
kLukL2is bounded as in (2.14).
It remains to establish the regularity of W. By conservation of mass we have
ku(0)kL2
x=ku(t)kL2
x=t1
2kw(t, vt)kL2
v.
Hence by (2.7) and (2.17) we obtain
kWkL2
x=kukL2
x.
9

On the other hand, from (2.16) and (2.17) we get
kW(v)−γ(t, v)e−i|γ(t,v)|2log tkL2
v.ǫt−1
2+D2ǫ2log t,
while by (2.6) and (2.14) we have
k∂v[γ(t, v)e−i|γ(t,v)|2log t]kL2
v.ǫtD2ǫ2log t.
It follows that for all large twe have
W(v) = OH1
v(ǫtD2ǫ2log t) + OL2
v(ǫt−1
2+D2ǫ2log t),
so by interpolation we obtain for large enough Cthe regularity
kWkH1−Cǫ2
v.ǫ.
2.5. The asymptotic completeness problem. Here we solve the problem from infinity.
For convenience, throughout this section, we set λ= 1. The naive idea would be to start
with the asymptotic profile
uasymptotic =1
√teix2
2tW(x/t)ei|W(x/t)|2log t,
and correct this to an exact solution uto the cubic NLS (1.1), by perturbatively solving the
equation for the difference from infinity. However, as defined above, the function uasymptotic
does not have enough regularity in order for it to be a good approximate solution. To remedy
this, we replace Win the above formula with a regularization of Won the time dependent
scale, namely
W(t, v) := W<t 1
2(v),
which selects the frequencies less than t1
2in W. This is the analogue of the function γ
defined for forward problem, with the same time dependent regularization scale. Then our
approximate solution is
uapp =1
√teix2
2tW(t, x/t)ei|W(t,x/t)|2log t.
To start with we make the more general assumption that
(2.18) kWkH1+2δ
v≤M, M, δ > 0, δ ≫M2.
Then by Bernstein’s inequality we have the bounds
kW(t, v)−W(v)kL2
v.Mt−1
2−δ,kW(t, v)−W(v)kL∞.M t−1
4−δ,
which imply that the functions uasymptotic and uapp are equally good as asymptotic profiles,
kuasymptotic −uappkL2
x.Mt−1
2−δ,kuasymptotic −uappkL∞.ǫt−1
4−δ.
To find the exact solution umatching uapp at infinity we denote by fthe error
(2.19) f= (i∂t+1
2∂2
x)uapp −uapp|uapp|2,
and then solve for the diffrence v=u−uapp
(i∂t+1
2∂2
x)v= (uapp +v)|uapp +v|2−uapp|uapp|2−f.
10

The uapp-cubic term cancels, and we are left with
(2.20) (i∂t+1
2∂2
x)v=N(v, uapp)−f, v(∞) = 0,
where
N(v, uapp) = v|v|2+v2¯uapp + 2|v|2uapp + 2v|uapp|2+ ¯vu2
app.
The solution operator for the inhomogeneous Schr¨odinger equation with zero Cauchy data
at infinity
(i∂t+1
2∂2
x)v=f, u(∞) = 0,
is given by
v(t) = iλ Z∞
t
e
(t−s)∂2
x
2f(s)ds := Φf.
Hence the equation (2.20) is rewritten in the form
(2.21) v= ΦN(v, uapp )−Φf.
We will solve this via the contraction principle, using the energy/Strichartz type bound (2.2)
(2.22) kΦfkL∞
t(T,∞;L2
x)+kΦfkL4
t(T,∞;L∞
x).kfkL1
t(T,∞;L2
x).
The equation for vwill be solved in a function space Xdefined using the above L∞
tL2
xand
L4
tL∞
xnorms, with appropriate time decay. Precisely, we set
kvkX:= sup
T≥1
T1
2+δ
(1 + M2log t)2kvkL∞
t(T,2T;L2
x)+kvkL4
t(T,2T;L∞
x).
We also want a bound for Lv, for which we need to use the larger space ˜
X, whose norm
carries a different time decay weight,
kwk˜
X:= sup
T≥1
Tδ
(1 + M2log t)3kwkL∞
t(T,2T;L2
x)+kwkL4
t(T,2T;L∞
x).
The first task at hand is to estimate the contribution of the inhomogeneous term f. This
is done in the following
Lemma 2.4. Assume that (2.18) holds with δ&M2. Then fdefined by (2.19) satisfies the
following estimates:
(2.23) kΦfkX+kΦLfk˜
X.M.
We postpone the proof of the lemma in order to conclude first the proof of the main result.
We succesively consider the equation for vand the equation for Lv.
(i) The equation for vin L2.In view of (2.23), in order to solve the equation (2.21) in
Xusing the contraction principle we need to show that the map v→ΦN(v, uapp) maps X
into Xwith a small Lipschitz constant for vin a ball of radius CM, where 1 ≪C≪M−1.
Then we obtain a solution vsatisfying
(2.24) kvkX.M.
Using the linear bound (2.2), it suffices to show that
(2.25) kN(v1, uapp)−N(v2, uapp )kL1
t(T,∞;L2
x).kv1−v2kX(M+kv1k2
X+kv2k2
X).
11

For simplicity we consider the case v2= 0 and show that
(2.26) kN(v, uapp )kL1
t(T,∞;L2
x).MkvkX+kvk3
X.
The general case is identical. To bound kN(v, uapp)kL1
t(T,∞;L2
x)we we divide [T, ∞) into
dyadic subintervals, estimate N(uapp, v;f) in each such interval, and then sum up. For the
terms in Nwe succesively compute
(2.27)
kv|uapp|2kL1
t(T,2T;L2
x).Tkuappk2
L∞([T,2T]×R)kvkL∞
t(T,2T;L2
x).M2T−1
2−δ(1 + M2log t)2kvkX,
(2.28) k|v|2uappkL1
t(T,2T;L2
x).T3
4kuappkL∞([T ,2T]×R)kvkL∞
t(T,2T;L2
x)kvkL4
t(T,2T;L∞
x)
.M T −3
4+2δ(1 + M2log t)4kvk2
X,
respectively
(2.29) kv|v|2kL1
t(T,2T;L2
x).T1
2kvkL∞
t(T,2T;L2
x)kvk2
L4
t(T,2T;L∞
x).T−1−3δ(1 + M2log t)6kvk3
X.
Thus, (2.26) follows.
(ii) The equation for Lv in L2.Applying Lto (2.20) we obtain
(i∂t+1
2∂2
x)Lv =LN(v, uapp)−Lf.
Then for Lv we seek to solve the linear problem
Lv = Φ(LN(v, uapp)) −ΦLf
in the space ˜
X. The bound for ΦLf is provided by (2.23). We expand LN(v, uapp) as
LN(v, uapp ;f) : = L(v|v|2) + L(v2¯uapp) + 2L(|v|2uapp ) + 2L(v|uapp|2) + L(¯vu2
app)−Lf
=Q(Lv) + g−Lf,
where the linear part Q(Lv), respectively the inhomogeneous term gare given by
Q(Lv) := 2|v|2Lv −v2Lv + 2¯uappvLv −u2
appLv + ¯vuappLv −vuappLv +|uapp|2Lv,
g:= 2uapp¯vLuapp −v2Luapp +|v|2Luapp +v¯uappLuapp −vuappLuapp .
We can use again (2.2), so it remains to estimate Q(Lv) and gin L1
t(T, ∞;L2
x). For uapp we
make use only of the pointwise bound kuappkL∞.M t−1
2and the L2
xbound for Luapp
kLuappkL2
x.M(1 + M2log t),
while for vwe use the Xnorm bound (2.24). The same type of analysis as in the proof of
(2.27)-(2.29) leads to the estimate
kQ(Lv)kL1
t(T,2T;L2
x).M2T−δ(1 + M2log T)3kLvk˜
X,
where the worst term in Q(Lu) is the last one. After dyadic summation this yields
kQ(Lv)kL1
t(T,∞;L2
x).δ−1M2T−δ(1 + M2log T)3kLvk˜
X.
12

This is where we need the condition δ≫M2both in order to have a good dyadic summation,
and in order to gain a small Lipschitz constant. Next we bound gin L1
t(T, ∞;L2
x); this is
better since we use at least one vnorm, and we obtain
kgkL1
t(T,∞;L2
x).M3T−1
4−δ(1 + M2log T)3.
The proof of the theorem is concluded, modulo the proof of Lemma 2.4, which follows.
Proof of Lemma 2.4. We first compute f. For that we need the time derivative of W,
∂tW(t, v) = t−1Wt1
2(v),
where Wt1
2is obtained from Wvia a zero order multiplier which is localized exactly at dyadic
frequency t1
2. Then we can write
f=1
t1
2
eix2
4teilog t|W|21
thWt1
2+ 2iWlog tℜ(Wt1
2
¯
W)i
+1
t2hW′′ + 2iWlog tℜ(W′′ ¯
W)−4Wlog tℜ(W′¯
W)2i
+1
t22iW′log tℜ(W′¯
W) + 2iWlog t|W′|2,
where W′and W′′ denote the first and the second derivative with respect to v. The expression
for Lf is computed from this using the observation that L(eix2
2tg(x/t)) = ieix2
2t∂vg(x/t). From
(2.18) we have the L∞and L2
vand bounds
(2.30)
kWkL∞.M, kW′kL∞.Mt 1
4−δ,
kW′kL2
v.M, kW′′ kL2
v.Mt 1
2−δ,kW′′′kL2
v.Mt1−δ,
kWt1
2kL2
v.Mt−1
2−δ,kW′
t1
2kL2
v.Mt−δ.
Using these bounds it is easy to see that the following estimates hold
(2.31) kfkL2
x.Mt−3
2−δ(1 + M2log t)2,kLfkL2
x.Mt−1−δ(1 + M2log t)3.
Then the bound for Φfin (2.23) follows easily by time integration and (2.2). Unfortunately,
a direct integration in the bound for Lf in (2.31) yields an extra δ−1factor,
kLfkL1
t(T,∞;L2
x).δ−1Mt−δ(1 + M2log t)3,
so the bound for ΦLf in (2.23) cannot be obtained directly.
To improve on this, we first peel off the better part of Lf, which includes all terms which
do not contain either of the factors Wt1
2,W′′. Precisely, we set
h:= 1
t3
2
eix2
2t∂vZ,
where the expression of Zis given by
Z(t, v) = eilog t|W |2Wt1
2+ 2iWlog tℜ(Wt1
2
¯
W) + 1
tW′′ + 2i1
tWlog tℜ(W′′ ¯
W).
13

The function Zis essentially localized around frequency t1
2; this is seen in the estimates
below for Z, which are computed in terms of the L2
v-norm of W≤t1
2:
(2.32) k∂j
vZkL2
v.t−1+ j
2(1 + M2log t)j+1kW′′
≤t1
2kL2
v, j = 0,1,2.
Since the regularity of Wis H1+δ, this shows that the map from Wto Zis mostly diagonal
with respect to frequencies, with rapidly decaying off-diagonal tails.
The difference Lf −hcan be shown to have better time decay,
kLf −hkL2
x.M3t−5
4−δ(1 + M2log t)3,
which is stronger than needed. It remains to consider the output of h, for which it is no
longer enough to obtain a fixed time L2bound and then integrate it in time. Instead, we
consider Φhdirectly.
To estimate Φhwe first compute the Fourier transform of h,
ˆ
h(ξ) = 1
t3
2Ze−ixξeix2
2t∂vZ(t, x/t)dx =1
t1
2
eitξ2
2Zeit(ξ−v)2
2∂vZ(t, v)dv.
Interpreting the last integral as a convolution, we compute its pullback to time zero,
(eit∂2
x
2h)(t, x) = t−1eix2
2t\
(∂vZ)(t, x) = t−1eix2
2tix ˆ
Z(t, x),
which, in view of (2.32), is mainly concentrated in the dyadic region x≈t1
2. Then the
solution to the backward Schr¨odinger equation is
Φh(t) = e−it∂2
x
2z(t), z(t, x) = ix Z∞
t
s−1eix2
2sˆ
Z(s, x)ds
Now we take advantage of the fact that, in the above integral, dyadic regions in tessentially
contribute to different dyadic regions in x. This shows that
kt−1eix2
2sxˆ
Z(s, x)kl2L1
t(T,∞;L2
x).kW′
≥T1
2kL2
v+T−1
2kW′′
≤T1
2kL2
v.M T −δ(1 + M2log T)3,
where the l2norm is taken with respect to dyadic regions in frequency. After time integration
this implies that
kz(t)kl2˙
W1,1(T,∞;L2
x).M T −δ(1 + M2log T).
Here we cannot interchange the l2and the ˙
W1,1norm. However, we can do it if we relax
˙
W1,1to the space V2of functions with bounded 2 variation,
l2˙
W1,1(T, ∞;L2
x)⊂l2V2(T, ∞;L2
x)⊂V2(T, ∞;l2L2
x) = V2(T, ∞;L2
x).
Thus we obtain
kz(t)kV2(T,∞;L2
x).M T −δ(1 + M2log T)3.
Then the desired conclusion
kΦh(t)kL∞(0,T ;L2)+kΦh(t)kL4(0,T ;L∞).Mt−δ(1 + M2log t)3
follows in view of the Strichartz embeddings for V2spaces,
ke−it∂2
x
2z(t)kL∞L2+kΦh(t)kL4L∞.kzkV2L2,
see Section 4 in [8].

14

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Department of Mathematics, University of California at Berkeley
E-mail address:ifrim@math.berkeley.edu
Department of Mathematics, University of California at Berkeley
E-mail address:tataru@math.berkeley.edu
15