Nlsnotes (Final2)
Nlsnotes (Final2)
Nlsnotes (Final2)
LUCA FANELLI
Contents
1. Introduction 1
2. Dispersive properties of the free Schrödinger group 2
2.1. Exercises 11
3. The local Cauchy problem for NLS 11
3.1. A scaling argument 12
3.2. Local existence: fixed point arguments 12
3.3. Exercises 16
4. The focusing NLS with data in H 1 17
4.1. Energy conservation 17
4.2. Exercises 19
5. Morawetz identities and local smoothing 20
5.1. Exercises 24
6. Interaction Morawetz identities 24
6.1. Applications: a scattering result 26
6.2. Exercises 27
6.3. Some obviously incomplete bibliographical notes 28
References 28
Antongiulio y Giacomo,
bienvenidos a este mundo.
1. Introduction
These are the notes of a short Ph.D. course of five lectures, given in
May 2010 at the Universidád Autonoma de Madrid. Our aim is to give an
introduction about the mathematical techniques and tool which have been
developed in the last years to treat some problem arising in the study of the
nonlinear Schrödinger equation (NLS). In what follows, we will refer to NLS
each time we consider an equation of the form
i∂t u(t, x) + ∆x u(t, x) ± |u|p−1 u(t, x) = 0, (1.1)
We can recollect all the previous information, to get now the following fun-
damental Lemma.
Lemma 2.1. Let u0 ∈ L2 (Rn ) and denote by
1 i|x|2
Kt (x) = n e 4t , t 6= 0, x ∈ Rn . (2.4)
(4πit) 2
Then, the unique solution u ∈ C(R \ {0}; L2 (Rn )) of (2.1) is given by
Z
1 i|x|2 ix·y i|y|2
u(t, x) = Kt ∗ u0 = n e 4t e− 2t e 4t u0 (y) dy. (2.5)
(4πit) 2
Proof. A direct computation shows that
b t (ξ) = e−it|ξ|2 ;
K (2.6)
as a consequence, denoting by v(t, ξ) the space Fourier transform of the
function u = Kt ∗ u0 , we have
b t (ξ) · û0 (ξ) = û0 (ξ)e−it|ξ|2 ,
v(t, ξ) = K
which proves the claim in view of the previous considerations. ¤
Remark 2.1. Observe again that the initial condition is attained in the fol-
lowing sense:
L2 − lim u(t, ·) = u0 (·).
t→0
These considerations show that the space L2 and, more in general, any
Hilbert space H s , is the natural setting for the Schrödinger equation. In
fact, due to the imaginary unit in the Schrödinger operator, the group eit∆
is unitary on these spaces, in which the solution, as a consequence, well
propagates (in both the directions of time).
SPANISH LECTURES ON NLS 4
\0 (ξ) =
Proof. First observe that the case q = 2 is trivial; indeed, since S(t)u
−it|ξ| 2
e û0 (ξ), by Plancherel identity we have
kS(t)u0 kL2 = ku0 kL2 , (2.8)
for all times t ∈ R. Moreover, by (2.4) we see that Kt ∈ L∞ , for all times
n
t 6= 0, and kKt kL∞ . |t|− 2 ; hence, by Young’s Theorem we can estimate
n
kS(t)u0 kL∞ = kKt ∗ u0 kL∞ . |t|− 2 ku0 kL1 . (2.9)
Estimate (2.7) now follows by interpolation (Riesz-Thorin Theorem) be-
tween (2.8) and (2.9). ¤
Remark 2.3. Estimates (2.7) affirm that the spacial Lp -norms of a free
Schrödinger solution, for q ≥ 2, decay in time, with a precise decay-rate
depending on n and q. We remark that this is a property which does not
characterize the class of dispersive equations. For example, also some para-
bolic equations satisfy this kind of a priori estimates (see e.g. the so called
strongly contractive estimates for the Heat Equation). We will spend some
more precise words about the nomenclature dispersive equations in the fol-
lowing lectures.
Although estimates (2.7) represent a precise information about solutions,
they are not ready to be used to treat nonlinear perturbations of (2.1). On
the other hand, as we see in a while, they have some strong consequences
about the boundedness of some global norms (in space and time) of solutions.
We are now ready to introduce the so called Strichartz estimates. Let us
first give a preliminary definition.
Definition 2.3. Let n ≥ 2; we say that a couple of numbers (p, q) is
Schrödinger-admissible if the following relations hold:
2 n n
= − , p ≥ 2, (p, q, n) 6= (2, ∞, 2). (2.10)
p 2 q
SPANISH LECTURES ON NLS 5
Remark 2.5. At the left-hand side of estimate (2.13) the acting operator is
the dual of the Schrödinger group S(t). Indeed, by estimate (2.12) we have
that
S : L2 3 f 7→ S(t)f ∈ Lp Lq ;
by duality, (2.12) and (2.13) are equivalent to each other, since the adjoint S ∗
with respect to the inner product in L2 is given by S ? (t) = e−it∆ . Observe
also that Z Z
i(t−s)∆
e F (s, ·) ds = S(t)S ∗ (s)F ds.
Hence, a part of the time truncation in the time interval, at the left-hand
side of (2.14) the operator SS ∗ appears. Estimate (2.14) is usually called
T T ∗ -estimate (change the nomenclature S for T in your mind). Finally, we
strongly remark that the pairs (p, q), (p̃, q̃) are not related to each other in
the T T ∗ -estimate, and this turns out to be a crucial fact for the nonlinear
applications. In fact, one should see that the solution of the inhomogeneous
problem (
i∂ut (t, x) + ∆u(t, x) = F (t, x)
(2.15)
u(0, x) = u0 (x)
SPANISH LECTURES ON NLS 6
For the proof, see [6]. Now we recall a well known result about fractional
integrals.
Lemma 2.6 (Hardy-Littlewood-Sobolev inequality). Let f : Rn → C be a
measurable function and 0 < γ < n; then
° °
°f ∗ |y|−γ ° p ≤ Cp,γ kf kLr , (2.17)
L
1 1 γ
with p = r + n − 1.
Indeed, the constant Cp,γ in (2.17) is bounded by the weak Lp -norm of
the function |y|−γ . For a simple proof of (2.17), see e.g. (CITARE STEIN).
Finally, we need an abstract functional analysis result.
Lemma 2.7. Let H be a Hilbert space and X be a Banach space, and X ∗ be
its dual space. Let D ⊂ X be a dense vector space in X and T ∈ L(D; H) be
a densely defined linear operator, with its adjoint operator T ∗ ∈ L(H; D∗ ).
Then the following three properties are equivalent:
SPANISH LECTURES ON NLS 7
for any g ∈ L2 ∩ L1 . Then, for any (p, q), (p̃, q̃) satisfying
1 σ σ
+ = , p≥2 (p, q, σ) 6= (2, ∞, 1), (2.21)
p q 2
the following estimates hold:
kU (t)f kLpt Lqx ≤ C2 kf kH (2.22)
°Z °
° °
° U ∗ (s)F (s) ds° ≤ C2 kF k p0 q0 (2.23)
° ° Lt Lx
H
°Z °
° t °
° U (t)U ∗ (s)F (s) ds° ≤ C22 kF kLp̃0 Lq̃0 . (2.24)
° ° x
0 Lpt Lqx t
Before proving the previous theorem we need to recall the following in-
terpolation lemma.
Lemma 2.9 ([2], 3.13.5 (b)). Let A0 , A1 , B0 , B1 , C0 , C1 be Banach spaces
and let T be a linear operator such that
A0 × B0 → C0
T : A0 × B1 → C1
A1 × B1 → C1 .
Then
T : (A0 , A1 )θ0 ,pr × (B0 , B1 )θ1 ,qr → (C0 , C1 )θ,r ,
for all p, q, r ∈ [1, ∞] such that
1 1
+ ≥ 1, θ0 , θ1 , θ ∈ (0, 1), θ = θ0 + θ1 .
p q
Proof of Theorem 2.8. By the arguments we introduced for the proof of The-
orem 2.4, it is sufficient to prove 2.23. Indeed, as it will be clearRduring the
t
proof, we will prove it also in the case of the³truncated
´ integral 0 dt. Here
2σ
we will just prove the endpoint case (p, q) = 2, σ−1 ; we leave to the reader
the proof in the non-endpoint case, which is completely analogous to the one
for Theorem 2.4.
Let us introduce the quadratic form
Z Z
T (F, G) = (U ∗ (s)F (s), U ∗ (t)G(t))H ds dt.
One could easily check that the endpoint version of (2.23) is equivalent to
the following estimate:
|T (F, G)| ≤ C22 kF k 2σ kGk 2σ .
L2t Lxσ+1 L2t Lxσ+1
By symmetry, we just need to consider the double integral in the zone where
s < t. For any j ∈ Z, let us define
Z Z
Tj (F, G) = (U ∗ (s)F (s), U ∗ (t)G(t))H ds dt;
2j ≤t−s<2j+1
Hence we reduced to prove the following claim:
X
|Tj (F, G)| ≤ C22 kF k 2σ kGk
σ+1
2σ . (2.25)
L2t Lx L2t Lxσ+1
j∈Z
SPANISH LECTURES ON NLS 9
for any non-endpoint couple (p, q), and by Hölder in θ (notice that p0 < 2)
we get
³ ´Z
j σq − σ−1
|Tj (F, G)| ≤ 2 2
kF (s)kL2 · kG(s + θ)kL2 ([2j ,2j+1 ];Lq0 ) .
θ x
Now we resume the previous estimates. Define lps = Lp (Z; 2js dj); estimates
(2.27), (2.28), (2.29) can be recollected as follows:
0 0
Tj : L2t Lax × L2t Lbx → l∞
β(a,b)
, (2.30)
provided
σ σ
β(a, b) = σ − 1 − − , (2.31)
a b
(
a=b=∞ hest. (2.27),
´ h ´
2σ 2σ
(a, b) ∈ {2} × 2, σ−1 ∪ 2, σ−1 × {2}, est. (2.28), (2.29).
(2.32)
SPANISH LECTURES ON NLS 10
2.1. Exercises.
Exercise 2.12. Prove identity (2.6).
Exercise 2.13. Formally prove that formula (2.16) defines a solution of the
inhomogeneous problem (2.15).
Exercise 2.14. Prove Lemma 2.7.
Exercise 2.15. Define Lorentz spaces as real interpolation of Lp -spaces,
namely
Lp,q = (Lp0 , Lp1 )θ,q , p−1 = (1 − θ)p−1 −1
0 + θp1 ,
provided
p0 < p 1 , p0 < q ≤ ∞, 0 < θ < 1.
Let (p, q), (p̃, q̃) be two non-endpoint Schrödinger admissible couples and r, s
satisfying
p̃0 ≤ s0 < r ≤ p, r ≥ 2.
Prove that
keit∆ f kLp,r Lq,r . kf kL2 ,
°Z °
° °
° e−it∆ F (t, ·) dt° . kF k p̃0 ,s0 q̃0 ,s0 ,
° ° 2 L L
L
°Z t °
° °
° e i(t−s)∆
F (s, ·) ds° . kF kLp̃0 ,s0 Lq̃0 ,s0 .
° °
0 Lp,r Lq,r
Be careful, because
(Lp0 Lq0 , Lp1 , Lq1 )θ,r 6= Lp,r Lq,r ,
except for a few cases (see [9]). Hence the previous estimates do not follow
by interpolation of standard Strichartz estimates.
Hint: Interpolate between time-dispersive estimates and then estimate
convolutions in Lorentz spaces (see [18]). For further details about Lorentz
spaces, see [2].
Before stating the first theorems about local well-posedness, let us state
the following Lemma.
Lemma 3.3. Let I ⊂ R be a closed time interval containing 0, and u ∈
C(I; L2 (Rn )) be a local solution of (3.1). Then
ku(t, ·)kL2x = ku0 kL2 , (3.5)
for all times t ∈ I.
Proof. Assuming first that the solution u ∈ C(I; H 1 ), then (3.5) follows by
integration by parts. Then, the general statement can be proved by density
arguments. ¤
Let us start with the L2 -theory for subcritical nonlinearities. In this case,
we can state a global existence result.
Theorem 3.4 (L2 -subcritical NLS). Let u0 ∈ L2 (Rn ), n ≥ 1 and 1 < γ <
1 + n4 . Then there exists a unique global solution u ∈ C(R; L2 (Rn )).
Proof. We search for a fixed point of the Duhamel operator T defined in
(3.4). Let I ⊂ R be an open interval containing 0, and (p, q), (p̃, q̃) be two
Schrödinger 0-admissible pairs to be chosen later; moreover, define
p q
X = L∞ 2
I L ∩ LI L , k · kX = k · kL∞
I L
2 + k · kLp Lq .
I
Now, since the mass (i.e. the L2 -norm) is conserved, as stated in Lemma
3.3, as an immediate consequence 3.4 we have that the maximal existence
interval is in fact I = R, in other words the solution is global in time. In
fact, denote for example by T+ > 0 the maximal positive existence time
and assume by contradiction that T+ < ∞. By the mass conservation, we
have limt→T+ ku(t, ·)kL2 = ku0 kL2 < ∞; hence we can start again with the
fixed point argument and find a bigger positive lifespan T = T+ + T1 , which
contradicts the fact that T+ is maximal. ¤
Remark 3.1. In fact, the proof of Theorem 3.4 shows that the local solution
which has been constructed has got an additional regularity information;
indeed, we proved that u ∈ L∞ (R; L2 ) ∩ Lp (R; Lq ), with the best possible
choice of p, q.
Remark 3.2. Notice that the continuity in time of the solution is immediate
a posteriori, once it has been obtained as a fixed point of the map (3.4).
We now pass to the study of the L2 -critical case. We first state a global
result with small data.
Theorem 3.5 (L2 -critical NLS with small data). Let u0 ∈ L2 (Rn ), n ≥ 1
and γ = 1 + n4 . There exists a small ²0 > 0 depending on n and u0 , for
which, if
ku0 kL2 < ²0 , (3.11)
then the Cauchy problem (3.1) has a unique global solution u ∈ C(R; L (Rn )).
2
Step 2. In order to conclude the proof, we just need to prove that the
solution we constructed in fact also in L∞ 2
R L . To do this, let I = (−T0 , T0 ),
for an arbitrary T0 > 0. By the previous step, we have that kukS < ∞,
where S = LpI Lq for a suitable choice of (p, q). Now we come back to (3.4)
and estimate by Strichartz
4
1+ n
kT ukL∞
I L
2 . ku0 kL2 + kuk
S .
Hence kukL∞
I L
2 < ∞, and the proof is complete, since T0 is arbitrary. ¤
Remark 3.3. Notice that the previous proof does not ensure kukS < ∞ in
the case I = R. In other words, we cannot prove that a Strichartz norm is
bounded in the whole lifespan of the solution with simple techniques. This
is a very difficult problem and, as we see in the following, it turns out to be
crucial to prove global existence with large data.
In view of the previous remark, we now prove the following Theorem.
Theorem 3.6 (L2 -critical NLS with large data). Let ku0 kL2 = A < ∞ and
γ = 1 + n4 . There exist a maximal interval I = (−T∗ , T ∗ ), with T∗ , T ∗ > 0,
such that problem (3.1) has a unique solution u ∈ C((−T∗ , T ∗ ); L2 (Rn )).
Proof. The proof is analogous to the previous case. With the same notations
of Theorem 3.5, by Strichartz estimate we obtain
4
1+ n
kT ukS . keit∆ u0 kS + kukS . (3.14)
Now we need to obtain the smallness ²0 by another argument. Since u0 ∈ L2 ,
we have by Strichartz that keit∆ u0 kS < ∞. Hence, by the absolute conti-
nuity of the Lebesgue integral, if the interval I is chosen to be sufficiently
small, i.e. |I| ≤ ², for a suitable ² > 0, we have that keit∆ u0 kS < ²0 , for
some small ²0 depending on u0 and the constant in the Strichartz estimate.
Consequently, we obtain
4
1+ n
kT ukS . ²0 + kukS . (3.15)
Now the proof follows exactly as in the previous case. ¤
Remark 3.4. Notice that now we cannot ensure that the maximal existence
interval is the whole line R. On the other hand, the previous proof suggests
the following sentence:
• let u be the local solution of the L2 -critical problem (3.1) with max-
imal time interval, and let S = Lp ((−T∗ , T ∗ ); Lq ) be the Strichartz
norm we used to close the fixed point. Assume that
kukS < ∞; (3.16)
then T∗ , T ∗ = +∞, i.e. the solution is global.
It should be now clear which is the simple proof of the previous sentence.
In other words, if one can ensure that a suitable Strichartz norm stays finite
also when t approaches the maximal existence time, the solution can be
extended to all times.
In the following, we will refer to this kind of statements as global exis-
tence criteria.
SPANISH LECTURES ON NLS 16
3.3. Exercises.
Exercise 3.7. Let t0 =
6 0, δ > 0 and consider the following nonlinear
equation:
i∂t u + |t − t0 |δ ∆u = ±|u|γ−1 u (3.17)
Conjecture the critical exponent γc,s,δ for the local well-posedness of the
Cauchy problem associated to (3.17) in Ḣ s , with s ≥ 0. Moreover, prove
the local theory in L2 for any subcritical γ and for the critical value γc,s,δ
in the case of small data.
Hint. For the L2 -theory, first transform (3.17) into a nonlinear Schrödinger
equation with time dependent nonlinearity, by a change o variables. Then
use the Lorentz version of Strichartz estimates (see Exercise 2.15).
Exercise 3.8. Check all the algebraic conditions on the exponents in The-
orems 3.4 and 3.5.
Exercise 3.9. Let u ∈ C((−T, T ); H 2 (Rn )) be a solution of
i∂t u + ∆u = ±|u|γ−1 u,
Define the energy E(t) as follows:
Z Z
1 2 1
E(t) = |∇u(t, x)| dx ± |u(t, x)|γ+1 dx. (3.18)
2 γ+1
Prove that E(t1 ) = E(t2 ), for any t1 , t2 ∈ (−T, T ).
Exercise 3.10. Consider the Cauchy problem (3.1) with datum u0 ∈ Ḣ 1 (Rn ).
4
Prove that, if 1 < γ < γc,1 = 1 + n−2 , then there exists a unique local solu-
1 n
tion u ∈ C((−T, T ); Ḣ (R )).
4
Moreover, prove the same when γ = 1+ n−2 and ku0 kḢ 1 < ²0 , with ²0 > 0
sufficiently small.
Exercise 3.11. In space dimension n = 3, consider the quintic Schrödinger
equation with datum u0 ∈ Ḣ 1 . Prove that the boundedness of the norm in
L10 (I; L10 ), where I is the maximal existence interval, is a criterion for the
global well-posedness in Ḣ 1 .
Exercise 3.12. In space dimension n = 3, consider the cubic Schrödinger
1
equation with datum u0 ∈ Ḣ 2 . Prove that the boundedness of the norm in
L5 (I; L5 ), where I being the maximal existence interval, is a criterion for
1
the global well-posedness in Ḣ 2 .
SPANISH LECTURES ON NLS 17
4.1. Energy conservation. Let γ > 1 and consider the Cauchy problem
(
i∂t u + ∆u + ²|u|γ−1 u = 0
(4.1)
u(0, x) = u0 (x),
¡ ¢
where ² = ±1, and u0 ∈ H 1 (Rn ). Let u ∈ C (−T, T ); H 1 (Rn ) be a local
solution of (4.1) and define, for any t ∈ [−T, T ], the energy E(t) as follows:
Z Z
1 2 ²
E(t) = |∇u(t, x)| dx − |u(t, x)|γ+1 dx.
2 γ+1
Notice that E(t) < ∞, for any t ∈ (−T, T ), since u ∈ H 1 . Moreover, the
computation made in exercise 3.9, which is justified for H 2 -solutions, can be
extended to H 1 -solutions by standard density arguments. Consequently, the
energy E(t) is conserved during the evolution also in this case, i.e. E(t) =
E(0), for all times t ∈ (−T, T ). Now observe that the sign of E(t) depends
on ². We give the following definition.
Definition 4.1. We say that equation (4.1) is defocusing if ² = −1; on
the other hand, if ² = 1, we say that (4.1) is focusing.
In the defocusing case, since E is positive we have the immediate bound
k∇u(t, ·)kL2 (Rn ) ≤ E(0) < ∞.
Moreover, the mass (i.e. the L2 -norm) is conserved, hence we can extend
the local solution in H 1 to be global in time (in the H 1 -subcritical case
4
γ < 1 + n−2 ).
On the other hand, in the focusing case the situation is quite different
and the global theory needs some additional arguments which we are going
to introduce in the following results.
In the following, we will concentrate on the fosusing NLS with data in
H 1 , for L2 -subcritical nonlinearities.
Theorem 4.2. Let u0 ∈ H 1 , and 1 < γ < 1 + n4 ; let u ∈ C((−T∗ , T ∗ ); H 1 )
be a local solution of (4.1), with ² = 1, and maximal time existence. Then
T∗ = T ∗ = ∞.
Moreover, assume γ = 1 + n4 ; then there exists ²0 > 0 such that, if
ku0 kL2 < ²0 , then T∗ = T ∗ = ∞.
SPANISH LECTURES ON NLS 18
Proof. First we remark that the local solution is constructed by a fixed point
argument (see the previous section and Exercise (3.10)). Now we want to
conclude that the solution is in fact global. In order to do this, since the L2 -
norm is conserved, it is sufficient to prove that the Ḣ 1 -norm does not blow
up when t approaches the maximal time. Consider the energy functional
Z Z
1 2 1
E(t) = |∇u(t, x)| dx − |u(t, x)|γ+1 dx,
2 γ+1
and recall the Gagliardo-Nirenberg inequality
2+(γ−1)· 2−n (γ−1) n
kf kγ+1
Lγ+1
≤ Cγ+1 kf kL2 2
k∇f kL2 2
, (4.2)
4
valid for 1 ≤ γ ≤ 1 + n−2 (notice that, for γ = 1 this is an identity, while for
4
γ = 1 + n this is the Sobolev embedding of Ḣ 1 in L(n+2)/(n−2) ). By (4.2)
we obtain
µ ¶
2 1 Cγ+1 2+(γ−1)· 2−n (γ−1) n −2
E(0) = E(t) ≥ k∇u(t)kL2 − ku(t)kL2 2
k∇u(t)kL2 2
.
2 γ+1
(4.3)
Let us first consider the case γ < 1 + n4 . Since ku(t)kL2 is constant in t, and
(γ − 1) n2 − 2 < 0, inequality (4.3) implies that
k∇u(t)kL2 ≤ CE(0), (4.4)
for some C > 0, which proves the claim.
In the case γ = 1 + n4 , by (4.3) we have
µ ¶
2 1 Cγ+1 4
E(0) ≥ k∇ukL2 − ku(0)kL2 .
n
2 γ+1
As a consequence, the choice
γ+1
²0 < (4.5)
2Cγ+1
gives (4.4), and the proof is complete. ¤
Remark 4.1. As it is shown in the previous proof, the best choice of ²0
is related to the best constant Cγ+1 in the Gagliardo-Nirenberg inequality
(4.2). As Weinstein proved in (CITARE WEINSTEIN BENE), the best
Cγ+1 is reached by the unique positive and radial solution with minimal
energy (ground state) of the elliptic equation
4
−∆Q + Q = |Q| n Q;
Weinstein gave an approximation of the L2 -norm of the ground state, which
determines a threshold for the initial data u0 ∈ H 1 in the focusing mass-
critical NLS, as we saw in the previous theorem.
We now state another famous result, due to Glassey, about a blow-up
criterion for the focusing NLS with data in H 1 . It is based on a convexity
method called virial identity. First we prove an important lemma on which
we will come back in the next sections.
SPANISH LECTURES ON NLS 19
Theorem 4.3 (Virial identity). Let u0 ∈ H 1 , and assume that |·|u0 (·) ∈ L2 .
Let u ∈ C((−T∗ , T ∗ ); H 1 (Rn )) be a local solution of (4.1) in the maximal
(−T∗ , T ∗ ), and define
Z
V (t) = |x|2 |u(t, x)|2 dx, (4.6)
for all t ∈ (−T∗ , T ∗ ). Since the energy is conserved and V (0) is assumed to
be finite, this shows that V (t) is a parabola. Now, notice that, if k is large
enough, then
Z Z
k2 k γ+1
E(t) := E(t; ku0 ) = |∇u0 (x)| dx − |u0 (x)|γ+1 dx < 0.
2 γ+1
Consequently, V (t) becomes negative after a finite time T + , which proves
that the solution stops to exist in a finite time. ¤
4.2. Exercises.
Exercise 4.5. Consider equation (3.17) and define the energy as follows:
Z Z
a(t) 1
E(t) = |∇u(t, x)|2 dx ± |u(t, x)|γ+1 dx,
2 γ+1
where a(t) = |t − t0 |δ . Prove that, if u is a sufficiently regular solution of
(3.17), then the following identity holds:
d 1
E(t) = a0 (t)k∇u(t, ·)k2L2 .
dt 2
Exercise 4.6. Use the result of the previous exercise to state when a local
H 1 -solution of (3.17) is in fact global, distinguishing between defocusing
and focusing cases.
Exercise 4.7. Prove the Gagliardo-Nirenberg inequality (4.2).
SPANISH LECTURES ON NLS 20
Exercise 4.8. Prove Lemma 4.3. (Hint. For regular solution, just compute
the derivatives and integrate by parts. Then argue by density.)
Exercise 4.9. In dimension n ≥ 3, prove that, for any f ∈ Ḣ 1 (Rn ) with
| · |f (·) ∈ L2 (Rn ), the following inequality holds:
kf kL2 ≤ 4k |x|f kL2 k∇f kL2
Exercise 4.10. Let u be an H 1 -solution of the focusing NLS with negative
initial energy, in space dimension ≥ 3; we saw that there exists a time T such
that V (t) = k|·|u(t)kL2 → 0, when t → T . Prove that limt→T k∇u(t, ·)kL2 =
∞. (Hint. Use the previous exercise.)
hence (5.4) follows by (5.7). In order to prove (5.5) consider identity (5.3);
since ∆ is a symmetric operator and [∆, φ] is antisymmetric, with respect
to the L2 -product, we notice that
(u, [∆, [∆, φ]]u) = (u, ∆[∆, φ]u) − (u, [∆, φ]∆u) (5.8)
= (∆u, [∆, φ]u) + ([∆, φ]u, ∆u)
= 2<(∆u, [∆, φ]u).
Again, now (5.5) follows by (5.3) and integration by parts in (5.8). ¤
Remark 5.1. The formal computation in the previous proof is justified for
Ḣ 2
R -solutions. Indeed, the term which requires more regularity is of the form
∇φ·∇u∆u, and φ is an abstract function, at this level. On the other hand,
the result of the previous Theorem holds also on H 1 -solutions, and the proof
can be performed by standard density arguments.
For the NLS, we have an analogous result.
Lemma 5.2 (Virial-type identities
R for NLS). Let φ : Rn → R be a positive
2
function and define Θ(t) = φ|u| dx, being u a solution of
iut + ∆u = ±|u|γ−1 u. (5.9)
The following formal identities hold:
Z
d
Θ(t) = 2= u∇φ · ∇u dx (5.10)
dt
Z Z Z
d2 2 2 2 2(γ − 1)
Θ(t) = 4 ∇uD φ∇u dx − |u| ∆ φ dx ± |u|γ+1 ∆φ dx,
dt2 γ+1
(5.11)
where D2 φ is the Hessian matrix of φ and ∆2 φ = ∆(∆φ) is the bi-laplace.
We leave the proof of the previous Lemma as an exercise for the reader.
Notice that the result generalize the virial identity, Theorem 4.3.
Remark 5.2. Notice that, for stationary solutions u = u(x) of (5.9), identity
(5.11) reads as
Z Z
2 n(γ − 1)
|∇u| dx ± |u|γ+1 dx = 0,
2(γ + 1)
in the case φ = |x|2 : this is the so called Pohozaev identity.
We now introduce two important radial multipliers:
• the virial multiplier φ(x) = |x|2
• the Morawetz multiplier φ(x) = |x|
For the virial we have
D2 |x|2 = 2Id, ∆2 |x|2 = 0,
SPANISH LECTURES ON NLS 22
hence ∆2 |x| < 0. Moreover, we use the following formula, which is true for
radial functions,
φ0
∇uD2 φ∇u = φ00 |∂r u|2 + |∂τ u|2 , (5.14)
r
where ∂r , ∂τ denote, respectively, the radial and tangential components of
the gradient. Then we obtained
Z Z
|∇τ u(t, x)|2 d
0≤ dx . = u∂r u dx,
|x| dt
for all times t ∈ R. Integrating in time yields
Z T Z ¯Z ¯ ¯Z ¯
|∇τ u(t, x)|2 ¯ ¯ ¯ ¯
dx dt . ¯ u(−T )∂r u(−T ) dx¯ + ¯ u(T )∂r u(T ) dx¯¯
¯ ¯ ¯
−T |x|
. ku(−T )k 12 + ku(T )k 12 ,
Ḣ Ḣ
1
for any T ∈ R, and the proof follows now by the conservation of the Ḣ 2 -
norm. ¤
SPANISH LECTURES ON NLS 23
Notice that estimate (5.13) is a global (in space and time) tangential
smoothing for solutions of the free Schrödinger equation. This kind of result
is in general false for the complete gradient. A famous local result in this
direction is the following.
Theorem 5.4 (Local smoothing). Let n ≥ 1, and u be a solution of (5.1),
1
with u0 ∈ Ḣ 2 . The following a priori estimate hold:
Z Z
1
sup |∇(t, x)|2 dx dt . ku0 k2 1 . (5.15)
R>0 R |x|≤R Ḣ 2
¤
Remark 5.4. Notice that the local smoothing multiplier used in the proof
of Theorem 5.4 permits to obtain further informations on the solutions.
Indeed, one can easily check that the following trace estimate was proved:
Z Z
1
sup 2 |u|2 dσ(x) dt . ku0 k2 1 ,
R>0 R |x|=R Ḣ 2
Hence it is clear that we can reduce to the computations made in the previous
section, to prove the following result.
Theorem 6.1. Let u be a solution of the NLS (5.9), φ = φ(|x|) : Rn →
[0, ∞) be a positive radial multiplier and consider the quantity H(t) in (6.1).
Then, the following formal identities hold:
Z Z
Ḣ(t) = 4= j(x) · ∇φ(x − y)m(y) dx dy (6.2)
Z Z
Ḧ(t) = 4 G(x, y)D2 φ(x − y)G(y, x) dx dy (6.3)
Z Z
− m(x)m(y)∆2 φ(x − y) dx dy
Z Z
4(γ − 1)
± |u(x)|γ+1 |u(y)|2 ∆φ(x − y) dx dy
γ+1
Z Z
=4 G(x, y)D2 φ(x − y)G(y, x) dx dy (6.4)
Z Z
+ ∇(m(x)) · ∇(m(y))∆φ(x − y) dx dy
Z Z
2(γ − 1)
± |u(x)|γ+1 |u(y)|2 ∆φ(x − y) dx dy,
γ+1
where m(x) = |u(x)|2 , j(x) = =u(x)∇u(x), and
G(x, y) = u(y)∇u(x) − u(x)∇u(y).
We leave the proof as a relevant exercise for the reader.
Remark 6.1. It is also possible to define an interactive quantity H(t) con-
taining two different solutions of (5.9), see the bibliographical notes in the
following.
Now we concentrate on identity (6.3). Notice that, when we use the virial
multiplier φ = |x|2 we obtain by (6.3) that
Z Z Z Z
2 2 4n(γ − 1)
Ḧ(t) = 16 |u(x)| |∇u(x)| dx ± |u(x)| dx |u(x)|γ+1 dx.
2
γ+1
Now we state the analogous result which can obtained by using the Morawetz
multiplier φ = |x|.
Theorem 6.2 (Colliander-Keel-Staffilani-Takaoka -Tao [7]). In space di-
mension n = 3, consider the defocusing Schrödinger equation (5.9) with
1 < γ ≤ 1 + 4/(n − 2) and initial datum u0 ∈ H 1 (R3 ); let u be a local H 1 -
solution in a interval I. Prove that the following a priori estimate holds:
Z Z
3 1
|u(t, x)|4 dx dt . M (0) 2 E(0) 2 , (6.5)
I
Remark 6.4. The property (6.10) can be rephrased, using the terminology
of Scattering Theory, in terms of asymptotic completeness of the wave
operators associated to free hamiltonian H0 = −∂x2 and the perturbed one
via the nonlinear term in (6.9).
Proof of Theorem 6.4. Let us write the global solution we already constructed
as Z t
2 2
u(t) = eit∂x u0 + i ei(t−s)∂x |u|8 u(s) ds (6.11)
0
Notice that a weaker version of estimate (6.6) is
Z Z Z Z
2 2
|∂x (|u(x)| )| dx dt + |u(x)|12 dx dt . M (0)3 E(0), (6.12)
SPANISH LECTURES ON NLS 27
where M and E are as usual the mass and the energy, which are two invari-
ants of the motion. Moreover, by the Gagliardo-Nirenberg inequality
3 1
kf kL∞ . kf kL4 6 k∂x f kL4 2 , (6.13)
as t tends to infinity. When time goes to −∞, the conclusions are analogous,
by the same computations. Now we define
Z ±∞
2
u± = u0 + i e−is∂x |u|8 u(s) ds, (6.16)
0
6.2. Exercises.
Exercise 6.5. Prove Theorem 6.1. Hint. For (6.2), (6.3) just use the
Morawetz identities of the previous section; for (6.4) integrate by parts one
term in (6.3).
Exercise 6.6. Prove Theorem 6.2. Hint. Use the multiplier φ = |x| in
Theorem 6.1.
Exercise 6.7. Prove the 1D Gagliardo-Nirenberg inequality (6.13).
Exercise 6.8. Consider the operator
Z Z
Tu = u(x)∇u(x) · ∇φ(x − y)|u(y)|2 dx dy,
R3 R3
with φ(|x|) = |x|α , x, y ∈ R3 . Find the unique α ∈ R such that the following
estimate holds:
|T u| . k∇uk2L2 .
SPANISH LECTURES ON NLS 28
References
[1] J.A. Barceló, A. Ruiz, and L. Vega, Some dispersive estimates for
Schrödinger equations with repulsive potentials J. Funct. Anal. 236 (2006), 1–24.
[2] Bergh, J., and Löfström, J., Interpolation spaces. Springer Verlag, Berlin,
1976.
[3] Bourgain, J., Global wellposedness of defocusing critical nonlinear Schrödinger
equation in the radial case, J. Amer. Math. Soc. 12, no. 1, (1999), 145–171.
[4] Cazenave T., Semilinear Schrödinger equations, Courant Lecture Notes in
Mathematics, vol. 10, American Mathematical Society, Providence, 2003.
[5] Cazenave, T.m and Weissler, F. B., The Cauchy problem for the nonlinear
Schrödinger equation in H 1 , Manuscripta Math. 61 (1988), no. 4, 477–494.
[6] M. Chirst, and A. Kiselev, Maximal functions associated to filtrations, J.
Funct. Anal. 179(2) (2001), 409–425.
[7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global
well-posedness and scattering for the energy-critical nonlinear Schrd̈inger equa-
tion in R3 , Annals of Math. 167 No. 3 (2008), 767–865.
[8] P. Constantin, and J.-C. Saut, Local smoothing properties of dispersive equa-
tions, Journ. AMS (1988), 413–439.
[9] Cwikel, M., On (LP0 (A0 ), LP1 (A1 ))θ,q , Proc. Amer. Math. Soc. 44 (1974), 286-
292.
[10] Fanelli, L., Semilinear Schrödinger equation with time dependent coefficients,
Math. Nach. 282 (2009), 976–994.
[11] Fanelli L., Montefusco E., On the blow-up threshold for weakly coupled
nonlinear Schrödinger equations, J. Phys. A 40 (2007), 14139–14150.
[12] Fanelli, L., and Vega, L., Magnetic virial identities, weak dispersion and
Strichartz inequalities, Math. Ann. 344 (2009), 249–278.
[13] Ginibre, J., and Velo, G., The global Cauchy problem for the nonlinear
Schrödinger equation revisited, Ann. I.H.P., Analyse Non Lynéare 2 (1985),
309–327.
[14] Glassey, R.T., On the blowing up of solutions to the Cauchy problem for
nonlinear Schrödinger equations, J. Math. Physics 18 (1977), 1794–1797.
[15] Keel, M., and Tao, T., Endpoint Strichartz estimates, Amer. J. Math. 120
(1998) no. 5, 955–980.
[16] Kenig, C., and Merle, F., Global well-posedness, scattering and blow-up for
the energy-critical, focusing, non-linear Schrödinger equation in the radial case,
Invent. Math. 166 (2006), 645-675.
SPANISH LECTURES ON NLS 29
[17] Morawetz, C., Time decay for the nonlinear Klein-Gordon equations, Proc.
Roy. Soc. Ser. A 306 (1968), 291–296.
[18] O’Neil, R., Convolution operators and L(p, q) spaces, Duke Math. J. 30 (1963),
129–142.
[19] Ozawa T., Remarks on proofs of conservation laws for nonlinear Schrodinger
equations. Calc. Var. PDE 25 (2006) 403408.
[20] Planchon, F., and Vega, L. Bilinear virial identities and Applications, Ann.
Scient. Ec. Norm. Sup. 42 (2009), 263–292.
[21] P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J.
55 (1987), 699–715.
[22] L. Vega, The Schrödinger equation: pointwise convergence to the initial date,
Proc. AMS 102 (1988), 874–878.