8 pages. The published version contains extra remarks.International audienceWe prove that for the mass critical nonlinear Schrodinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the one-dimensional case, we show a precise similarity between the mass critical nonlinear Schrodinger equation and a nonlinear Schrodinger equation of derivative type

Carles, Remi

Ozawa, Tohru

Mathematical Research Letters

Carles, Rémi

HAL Descartes

On the wave operators for the critical nonlinear Schrodinger equation

Rémi Carles

Tohru Ozawa

Crossref

On the wave operators for the critical nonlinear Schrödinger equation

HAL: Hyper Article en Ligne

EPrint Series of Department of Mathematics, Hokkaido University

ON THE WAVE OPERATORS FOR THE CRITICAL
NONLINEAR SCHRO¨DINGER EQUATION
RE´MI CARLES AND TOHRU OZAWA
Abstract. We prove that for the L2-critical nonlinear Schro¨dinger equations,
the wave operators and their inverse are related explicitly in terms of the
Fourier transform. We discuss some consequences of this property. In the one-
dimensional case, we show a precise similarity between the L2-critical nonlinear
Schro¨dinger equation and a nonlinear Schro¨dinger equation of derivative type.
1. Introduction
We consider the defocusing, L2-critical, nonlinear Schro¨dinger equation
(1.1) i∂tu+
1
2
∆u = |u|4/nu, (t, x) ∈ R× Rn.
We consider two types of initial data:
Asymptotic state: U0(−t)u(t)
∣∣
t=±∞ = u±, where U0(t) = e
i t2∆.(1.2)
Cauchy data at t = 0 : u|t=0 = u0.(1.3)
It is well known that for data u±, u0 ∈ Σ = H1 ∩ F(H1), where
Ff(ξ) = f̂(ξ) = 1
(2pi)n/2
∫
Rn
f(x)e−ix·ξdx,
(1.1)–(1.2) has a unique, global, solution u ∈ C(R; Σ) ([GV79], see also [Caz03]).
Its initial value u|t=0 is the image of the asymptotic state under the action of the
wave operator:
u|t=0 =W±u±.
Similarly, (1.1)–(1.3) possesses asymptotic states:
∃u± ∈ Σ, ‖U0(−t)u(t)− u±‖Σ −→
t→±∞ 0 : u± =W
−1
± u0.
Global well-posedness properties show that the wave operators are homeomor-
phisms on Σ. Besides this point, very few properties of these operators are known.
The main result of this paper (proved in §2) shows that the wave operators and
their inverses are easily related in terms of the Fourier transform:
Theorem 1.1. Let n > 1. The following identity holds on Σ:
(1.4) F ◦W−1± =W∓ ◦ F .
In particular, if C denotes the conjugation f 7→ f , then we have:
(1.5) W−1± = (CF)−1W± (CF) .
2000 Mathematics Subject Classification. 35B33; 35B40; 35Q55.
1
2 R. CARLES AND T. OZAWA
Using continuity properties of the flow map associated to (1.1), we infer the
following result in §3:
Corollary 1.2. The result of Theorem 1.1 still holds when Σ is replaced
• Either by F(H1),
• Or by a neighborhood of the origin in L2(Rn), for (1.1) as well as for its
focusing counterpart, i∂tu+ 12∆u = −|u|4/nu,
• Or by L2r(Rn) for n > 3, the set of radial, square integrable functions.
Remark 1.3. The usual conjecture on (1.1) implies that the result of Theorem 1.1
is expected to remain valid when Σ is replaced by L2(Rn) (but not for the focusing
counterpart of (1.1), for which finite time blow-up may occur in H1).
Remark 1.4. So far, the existence of wave operators on F(H1) is not known. Simi-
larly, asymptotic completeness in H1 remains an open problem. Theorem 1.1 shows
that the fact that these two problems are simultaneously open is not merely a tech-
nical point: they are exactly related by (1.4). This aspect is also reminiscent of the
main result in [BC06].
Using the asymptotic expansion of the wave operators near the origin, we prove
in §4 (with an extension in Appendix A):
Corollary 1.5. Let n > 1. For every φ ∈ L2(Rn), we have:∫ ±∞
0
eit
|x|2
2 F
(
|U0(t)φ|4/nU0(t)φ
)
dt =
∫ ±∞
0
U0(t)
(
|U0(−t)φ̂|4/nU0(−t)φ̂
)
dt.
Finally, in space dimension n = 1, we relate the wave operators for (1.1) with
the wave operators for the nonlinear Schro¨dinger equation of derivative type
(1.6) i∂tψ +
1
2
∂2xψ = iλ∂x
(|ψ|2)ψ, λ ∈ R.
This equation appears as a model to study the nonlinear self-modulation for the
Benjamin-Ono equation [Tan82]. For a more general nonlinear Schro¨dinger equa-
tion of derivative type (see e.g. [KT94, Tsu94] for the Cauchy problem related to
similar equations),
i∂tψ +
1
2
∂2xψ = iλ|ψ|2∂xψ + iµψ2∂xψ,
it is proved in [Oza96] that a short range scattering theory is available for λ, µ ∈ R
if and only if λ = µ: we recover (1.6). This is apparently the only cubic, gauge
invariant nonlinearity in space dimension one, for which a short range scattering
theory is available. More precisely, for (1.6)–(1.2), the wave operators Ω± : u± 7→
u(0) are well defined from Xε to H2(R), where
Xε = {φ ∈ H4 ∩ F(H4) ;
∥∥∥(1 + ξ2)φ̂∥∥∥
L∞
< ε},
and ε > 0 is sufficiently small. The following result shows that the nonlinearity in
(1.6) should be thought of as the quintic case (1.1). This result goes in the same
spirit as the approach followed in [OT98].
Theorem 1.6. Let λ ∈ R. Consider the quintic, focusing or defocusing, equation
(1.7) i∂tu+
1
2
∂2xu =
λ2
2
|u|4u, (t, x) ∈ R× R, µ ∈ R,
WAVE OPERATORS FOR CRITICAL NLS 3
with associated wave operators W±(µ) for small L2 data. For φ ∈ L2(R), define
(Nλ±φ)(x) = φ(x) exp
(
±iλ
∫ x
−∞
|φ(y)|2dy
)
.
• If ψ solves (1.6), then Nλ−(ψ) solves (1.7).
• If u solves (1.7), then Nλ+(u) solves (1.6).
• The following identity holds when all terms are well-defined:
F ◦ Ω−1± = Nλ− ◦ F ◦W−1± ◦Nλ+ =
(
Nλ+
)−1 ◦ F ◦W−1± ◦Nλ+.
Ω± ◦ F−1 =
(
Nλ+
)−1 ◦W± ◦ F−1 ◦Nλ+.
This result is checked by elementary computations, so we leave out its proof.
2. Proof of Theorem 1.1
The proof of Theorem 1.1 relies on a series of lemmas, which are stated, and
proved, in a slightly different fashion in [Tsu85]. Introduce the transform Ψ acting
on function of (t, x) as:
(2.1) (Ψu) (t, x) =
1
(it)n/2
ei
|x|2
2t u
(−1
t
,x
t
)
, for t 6= 0.
Lemma 2.1. For n > 1 and φ ∈ L2(Rn), we have:
lim
t→±∞
∥∥U0(t)F−1φ(·)− (Ψφ)(t, ·)∥∥L2 = 0.
Proof. We recall the standard decomposition of the free group, for t 6= 0:
U0(t) =MtDtFMt,
where Mt is the multiplication by ei|x|2/(2t), and Dt is the dilation operator
Dtφ(x) =
1
(it)n/2
φ
(x
t
)
.
Noting that Ψφ =MDφ, Plancherel formula yields:∥∥U0(t)F−1φ(·)− (Ψφ)(t, ·)∥∥L2 = ∥∥(Mt − 1)F−1φ(·)∥∥L2 .
Since |Mt(x)−1| . |x|/
√
t, the lemma follows for φ ∈ H1(Rn). By density, we infer
the result for φ ∈ L2(Rn). 
Lemma 2.2. Let v = Ψu. Suppose that there exist ψ± ∈ L2(Rn) such that
‖v(t)− ψ±‖L2 −→
t→±0
0.
Then u has asymptotic states in L2:
‖U0(−t)u(t)−F−1Rψ∓‖L2 −→
t→±∞ 0,
where R stands for the symmetry with respect to the origin, (Rφ)(x) = φ(−x).
Proof. We note that Ψ is almost an involution: Ψ2 = R. Therefore, u = ΨRv:
U0(−t)u(t)−F−1Rψ∓ = U0(−t)ΨRv
(−1
t
)
−F−1Rψ∓
= U0(−t)ΨR
(
v
(−1
t
)
− ψ∓
)
+
(
U0(−t)Ψ−F−1
)
Rψ∓.
4 R. CARLES AND T. OZAWA
Taking the L2 norm, we infer:∥∥U0(−t)u(t)−F−1Rψ∓∥∥L2 6 ∥∥∥∥v(−1t
)
− ψ∓
∥∥∥∥
L2
+ ‖ΨRψ∓ − U0(t)F−1Rψ∓‖L2 .
The first term of the right-hand side goes to zero as t→ ±∞ by assumption. The
second term goes to zero by Lemma 2.1. 
Lemma 2.3. Let v = Ψu. Suppose that u ∈ C([−T, T ];L2) for some T > 0, and
u|t=0 = u0 ∈ L2(Rn). Then∥∥U0(−t)v(t)−F−1u0∥∥L2 −→t→±∞ 0.
Proof. Since U0(−t) = U0(t)−1, we have
U0(−t)v(t) =M−1t F−1D−1t M−1t v(t) =M−1t F−1u
(−1
t
)
.
Therefore,∥∥U0(−t)v(t)−F−1u0∥∥L2 6 ∥∥∥∥u(−1t
)
− u0
∥∥∥∥
L2
+
∥∥(M−t − 1)F−1u0∥∥L2 .
The first term of the right-hand side goes to zero as t → ±∞ by assumption. So
does the second, by the standard argument recalled in the proof of Lemma 2.1. 
Proof of Theorem 1.1. Let u0 ∈ Σ: there exists a unique solution u ∈ C(R; Σ) to
(1.1)–(1.3). Set v = Ψu. Because of the conformal invariance for (1.1), v solves the
same equation as u, for t 6= 0:
i∂tv +
1
2
∆v = |v|4/nv, (t, x) ∈ R \ {0} × Rn.
Lemma 2.3 shows that ∥∥U0(−t)v(t)−F−1u0∥∥L2 −→t→±∞ 0.
Let w± denote the solutions to the scattering problems:
i∂tw± +
1
2
∆w± = |w±|4/nw± ; U0(−t)w±(t)
∣∣
t=0
= F−1u0.
By uniqueness for (1.1)–(1.2), we see that
v(t, x) =
{
w−(t, x) for t < 0,
w+(t, x) for t > 0.
In particular,
‖v(t)− w±(0)‖L2 −→t→±0 0.
From Lemma 2.2, u has asymptotics states, given by:
‖U0(−t)u(t)−F−1Rw∓‖L2 −→
t→±∞ 0,
that is, u± = F−1Rw∓. We infer:
F ◦W−1± u0 = Fu± = Rw∓ = RW∓F−1u0.
Since (1.1) is invariant by R, RW∓F−1u0 = W∓RF−1u0 = W∓Fu0. This yields
(1.4). The identity (1.5) follows from (1.4) and from the identity
W± = C ◦W∓ ◦ C,
WAVE OPERATORS FOR CRITICAL NLS 5
which was noticed in [CW92] (see also [Caz03]). 
3. Proof of Corollary 1.2
The first case follows by density, sinceW± are defined and continuous on H1(Rn)
[GV85] (see also [Gin97] for a simplified presentation), and since W−1± are defined
and continuous on F(H1) [GOV94, GV79, Tsu85].
For the second case, existence of wave operators, their asymptotic completeness,
and continuity properties, were proved by T. Cazenave and F. Weissler [CW89].
We note that Corollary 1.2 can be proved in this case like Theorem 1.1, provided
that we work in a sufficiently small neighborhood of the origin in L2(Rn).
The last case follows from the recent paper by T. Tao, M. Visan and X. Zhang
[TVZ]. The proof of Corollary 1.2 then relies on asymptotic completeness (in the
same space), along with continuous dependence upon the initial data. For n > 3,
let X = L2r(Rn); X is invariant under the action of the Fourier transform. For
φ ∈ X, let φj be a sequence in Σ, converging to φ in X. Define u±j as the solutions
to:
i∂tu
±
j +
1
2
∆u±j = |u±j |4/nu±j ; U0(−t)u±j (t)
∣∣
t=±∞ = φ̂j .
There exists u±0j = u
±
j (0) = W±φ̂j ∈ Σ. Since u±0j = FW−1∓ φj from Theorem 1.1,
the results in [CW89, TVZ] imply that there exists u±0 ∈ X such that ‖u±0j −
u±0 ‖L2 → 0 as j →∞. Let u± solve
i∂tu
± +
1
2
∆u± = |u±|4/nu± ; u±|t=0 = u±0 .
We have
‖U0(−t)u±(t)− φ̂‖L2 6
∥∥U0(−t) (u±(t)− u±j (t))∥∥L2 + ‖U0(−t)u±j (t)− φ̂j‖L2
+ ‖φj − φ‖L2 .
The global well-posedness for (1.1) in X implies
lim sup
t→±∞
‖U0(−t)u±(t)− φ̂‖L2 6F
(‖u±0 − u±0j‖L2)+ ‖φj − φ‖L2 ,
where F is a continuous function such that F (0) = 0. Finally, by letting j → ∞,
we see that u± solves
i∂tu
± +
1
2
∆u± = |u±|4/nu± ; U0(−t)u±(t)
∣∣
|t=±∞φ̂.
Let V be a neighborhood of φ in L2. From [CW89], we see by Strichartz estimates
and a bootstrap argument that the problem (1.1)–(1.2) is well-posed in L∞(] −
∞,−T ];V) (we consider only the minus sign for simplicity) for some T > 0 possibly
depending on V. By uniqueness, we infer
∃W±φ̂ = u±0 .
Since under our assumptions, W−1± are homeomorphisms on X, we also have:
u±0 = lim
j→∞
u±0j = lim
j→∞
FW−1∓ φj = FW−1∓ lim
j→∞
φj = FW−1∓ φ,
hence Corollary 1.2.
6 R. CARLES AND T. OZAWA
4. Proof of Corollary 1.5
Corollary 1.5 is a consequence of Theorem 1.1 and of the asymptotic expansion
of the wave operators near the origin in L2:
Proposition 4.1. Let n > 1 and φ ∈ L2(Rn). Then for ε > 0 sufficiently small
W±(εn/4φ) and W−1± (ε
n/4φ) are well defined in L2(Rn), and, as ε→ 0:
W±
(
ε
n
4 φ
)
= ε
n
4 φ∓ iε1+n4
∫ ±∞
0
U0(−t)
(
|U0(t)φ|4/nU0(t)φ
)
dt+O
(
ε2+
4
n
)
,
W−1±
(
ε
n
4 φ
)
= ε
n
4 φ± iε1+n4
∫ ±∞
0
U0(−t)
(
|U0(t)φ|4/nU0(t)φ
)
dt+O
(
ε2+
4
n
)
.
Proof. The proof follows from the same perturbative analysis as in [Ge´r96] (see also
[Car01] for the nonlinear Schro¨dinger equation). First, it follows from [CW89] that
W±(εn/4φ) and W−1± (ε
n/4φ) are well defined in L2(Rn) for ε > 0 sufficiently small.
We prove the asymptotic formula for the minus sign, since the proof of the
formula for the plus sign is similar. Consider uε solving:
i∂tu
ε +
1
2
∆uε = |uε|4/nuε ; U0(−t)uε(t)
∣∣
t=−∞ = ε
n/4φ.
Plugging an expansion of the form uε = εn/4(ϕ0+εϕ1+εrε) into the above equation,
and ordering in powers of ε, it is natural to impose the following conditions:
• Leading order: O(εn/4).
i∂tϕ0 +
1
2
∆ϕ0 = 0 ; U0(−t)ϕ0(t)
∣∣
t=−∞ = φ.
• First corrector: O(ε1+n/4).
i∂tϕ1 +
1
2
∆ϕ1 = |ϕ0|4/nϕ0 ; U0(−t)ϕ1(t)
∣∣
t=−∞ = 0.
The first equation yields
ϕ0(t) = U0(t)φ.
From the second equation, we have:
ϕ1(t) = −i
∫ t
−∞
U0(t− s)
(
|ϕ0(s)|4/nϕ0(s)
)
ds.
We also have:
i∂tr
ε +
1
2
∆rε = G (ϕ0 + εϕ1 + εrε)−G(ϕ0) ; U0(−t)rε(t)
∣∣
t=−∞ = 0,
where G(z) = |z|4/nz. Let γ = 2 + 4/n, and denote Lrt,x = Lr(] −∞,−t] × Rn).
Strichartz and Ho¨lder estimates yield
‖rε‖Lγt,x .
∥∥∥(|ϕ0|4/n + |εϕ1|4/n + |εrε|4/n) ε (|ϕ1|+ |rε|)∥∥∥
Lγ
′
t,x
.
(
‖ϕ0‖4/nLγt,x + ‖εϕ1‖
4/n
Lγt,x
+ ‖εrε‖4/n
Lγt,x
)(
ε‖ϕ1‖Lγt,x + ‖εrε‖Lγt,x
)
.
(
‖φ‖4/nL2 + ‖εrε‖4/nLγt,x
)(
ε‖φ‖L2 + ‖εrε‖Lγt,x
)
.
. ε‖φ‖1+4/nL2 + ‖εrε‖1+4/nLγt,x .
WAVE OPERATORS FOR CRITICAL NLS 7
A bootstrap argument shows that for 0 < ε¿ 1, rε ∈ Lγ(R× Rn), and
‖rε‖Lγ(R×Rn) . ε.
Using Strichartz estimates again, we infer:
‖rε‖L∞(R;L2(Rn)) . ε.
Considering uε at time t = 0 yields the first part of the proposition. The second
part can be proven in the same way, but can also be inferred from the first part
via Neumann series, since W± are small perturbations of the identity near the
origin. 
Now Corollary 1.5 follows from Corollary 1.2 and Proposition 4.1, where we
identify the terms of order ε1+4/n.
Remark 4.2. Considering the asymptotic expansion of the wave operators and their
inverse to higher order would yield other formulae, similar to Corollary 1.5. We
have not written them, for they are more intricate (they involve several integrations
in time), and we do not know if they can be of some interest.
Appendix A. Sub-critical case
In this appendix, we consider more generally the nonlinear Schro¨dinger equation
(A.1) i∂tu+
1
2
∆u = |u|2σu, (t, x) ∈ R× Rn,
in the sub-critical case σ < 2/n. Following the approach to prove Corollary 1.5, we
have:
Proposition A.1. Let σ < 2/n, with
• σ > 1/n if n 6 2.
• σ > 2/(n+ 2) if n > 2.
Then the following identities hold for every φ ∈ Σ:∫ ±∞
0
eit
|x|2
2 F (|U0(t)φ|2σU0(t)φ) dt = ∫ ±∞
0
|t|nσ−2U0(t)
(
|U0(−t)φ̂|2σU0(−t)φ̂
)
dt,∫ ±∞
0
|t|nσ−2eit |x|
2
2 F (|U0(t)φ|2σU0(t)φ) dt = ∫ ±∞
0
U0(t)
(
|U0(−t)φ̂|2σU0(−t)φ̂
)
dt.
Sketch of the proof. Let u solving (A.1). Then v = Ψu solves
(A.2) i∂tv +
1
2
∆v = |t|nσ−2|v|2σv, (t, x) ∈ R \ {0} × Rn.
It follows from [CW92] (see also [Caz03]) that wave operators exist, are continuous
and invertible, near the origin in Σ, both for (A.1) and (A.2). We can then mimic
the proof of Theorem 1.1, with the remark that in Theorem 1.1, the operatorsW−1±
on the left-hand side are associated to u, while the operators W∓ on the right-hand
side are associated to v.
Adapting Proposition 4.1 to the cases of (A.1) and (A.2) proceeds along the
same lines as the estimates in [CW92]. This yields the first identity in the above
proposition.
For the second, we simply notice that Ψ2 = R, so that we can exchange the roles
of u and v. 
8 R. CARLES AND T. OZAWA
References
[BC06] P. Blue and J. Colliander, Global well-posedness in Sobolev space implies global existence
for weighted L2 initial data for L2-critical NLS, Commun. Pure Appl. Anal. 5 (2006),
no. 4, 691–708.
[Car01] R. Carles, Remarques sur les mesures de Wigner, C. R. Acad. Sci. Paris, t. 332, Se´rie I
332 (2001), no. 11, 981–984.
[Caz03] T. Cazenave, Semilinear Schro¨dinger equations, Courant Lecture Notes in Mathematics,
vol. 10, New York University Courant Institute of Mathematical Sciences, New York,
2003.
[CW89] T. Cazenave and F. Weissler, Some remarks on the nonlinear Schro¨dinger equation in
the critical case, Lect. Notes in Math., vol. 1394, Springer-Verlag, Berlin, 1989, pp. 18–
29.
[CW92] , Rapidly decaying solutions of the nonlinear Schro¨dinger equation, Comm.
Math. Phys. 147 (1992), 75–100.
[Ge´r96] P. Ge´rard, Oscillations and concentration effects in semilinear dispersive wave equa-
tions, J. Funct. Anal. 141 (1996), no. 1, 60–98.
[Gin97] J. Ginibre, An introduction to nonlinear Schro¨dinger equations, Nonlinear waves (Sap-
poro, 1995) (R. Agemi, Y. Giga, and T. Ozawa, eds.), GAKUTO International Series,
Math. Sciences and Appl., Gakko¯tosho, Tokyo, 1997, pp. 85–133.
[GOV94] J. Ginibre, T. Ozawa, and G. Velo, On the existence of the wave operators for a class of
nonlinear Schro¨dinger equations, Ann. IHP (Physique The´orique) 60 (1994), 211–239.
[GV79] J. Ginibre and G. Velo, On a class of nonlinear Schro¨dinger equations. II Scattering
theory, general case, J. Funct. Anal. 32 (1979), 33–71.
[GV85] , Scattering theory in the energy space for a class of nonlinear Schro¨dinger
equations, J. Math. Pures Appl. (9) 64 (1985), no. 4, 363–401.
[KT94] S. Katayama and Y. Tsutsumi, Global existence of solutions for nonlinear Schro¨dinger
equations in one space dimension, Comm. Partial Differential Equations 19 (1994),
no. 11-12, 1971–1997.
[OT98] T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear
Schro¨dinger equations, Differential Integral Equations 11 (1998), no. 2, 201–222.
[Oza96] T. Ozawa, On the nonlinear Schro¨dinger equations of derivative type, Indiana Univ.
Math. J. 45 (1996), no. 1, 137–163.
[Tan82] M. Tanaka, Nonlinear self-modulation problem of the Benjamin-Ono equation, J. Phys.
Soc. Japan 51 (1982), no. 8, 2686–2692.
[Tsu85] Y. Tsutsumi, Scattering problem for nonlinear Schro¨dinger equations, Ann. Inst. H.
Poincare´ Phys. The´or. 43 (1985), no. 3, 321–347.
[Tsu94] , The null gauge condition and the one-dimensional nonlinear Schro¨dinger equa-
tion with cubic nonlinearity, Indiana Univ. Math. J. 43 (1994), no. 1, 241–254.
[TVZ] T. Tao, M. Visan, and X. Zhang, Global well-posedness and scattering for the mass-
critical nonlinear Schro¨dinger equation for radial data in high dimensions, preprint,
math.AP/0609692.
De´partement de Mathe´matiques, UMR CNRS 5149, CC 051, Universite´ Montpellier 2,
Place Euge`ne Bataillon, 34095 Montpellier cedex 5, France1
E-mail address: Remi.Carles@math.cnrs.fr
Department of Mathematics, Hokkaido University, Sapporo 060-810, Japan
E-mail address: ozawa@math.sci.hokudai.ac.jp
1Present address: Wolfgang Pauli Institute, Universita¨t Wien, Nordbergstr. 15, A-1090 Wien


On the wave operators for the critical nonlinear Schr\"odinger equation

On the wave operators for the critical nonlinear Schr\"odinger equation

Abstract

Similar works

Full text

Available Versions

HAL Descartes

Crossref

HAL: Hyper Article en Ligne

EPrint Series of Department of Mathematics, Hokkaido University