Scattering and Soliton Dynamics
Tai-Peng Tsai
Draft: 2009-01-08
Not For Circulation
Contents
Preface
4
1 Linear Schrödinger equations
1.1 Dispersion and dispersive equations . . . . . .
1.2 Solutions of free Schrödinger equation . . . .
1.3 Decay estimates . . . . . . . . . . . . . . . . .
1.4 Strichartz estimates . . . . . . . . . . . . . .
1.5 Linear Schrödinger equations with potentials
1.6 Problems and notes . . . . . . . . . . . . . . .
2 Local wellposedness
2.1 Definitions . . . . . . . . .
2.2 Uniqueness and existence
2.3 Continuous dependence on
2.4 Problems and notes . . . .
. . .
. . .
data
. . .
3 Global phenomena
3.1 Global existence . . . . . .
3.2 Finite time blowup . . . . .
3.3 Solitary waves . . . . . . . .
3.4 Solitary waves and blow-up
3.5 Problems and notes . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . . . .
. . . . . . . . . .
and conservation
. . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
5
7
8
9
12
14
. . .
. . .
laws
. . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
16
16
18
21
24
.
.
.
.
.
25
25
27
28
32
33
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4 Scattering theory
4.1 Introduction . . . . . . . . . . . . . . . . . .
4.2 Scattering theory in the weighted L2 space
4.3 Scattering theory in the energy space . . . .
4.4 Problems and notes . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
34
34
36
40
44
5 Orbital stability of solitary waves
5.1 Introduction . . . . . . . . . . . .
5.2 More on solitary waves . . . . . .
5.3 Orbital stability . . . . . . . . . .
5.4 Orbital instability . . . . . . . .
5.5 Problems and notes . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
45
45
45
46
48
51
.
.
.
.
.
.
.
.
.
.
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
CONTENTS
3
6 Asymptotic stability of small solitary waves
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Asymptotic stability of small solitary waves . . . . . . . . . . . . . . . . . .
6.3 Problems and notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
52
53
58
Appendices
A.1 Bound states for general nonlinearity . . . . . . . . . . . . . . . . . . . . . .
A.2 Solutions to problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
59
60
Bibliography
63
Preface
The goal of these lectures is to introduce the basic phenomena of dispersive equations. In
order to get to more involved results in the limited time frame, we will focus on Schrödinger
equations, which arise from quantum mechanics and nonlinear optics.
The defining property of dispersion is that waves of different frequencies propagate at
different velocities. For a linear equation, if we start with a “wave packet”, the principle
of superposition says that these different frequency components will evolve independent of
each other and eventually separate, i.e., disperse, in the physical space.
For a nonlinear equation, these frequency components interact with each other. Where
the wave is small, the nonlinearity is smaller and can be neglected. Where the wave is large,
there is a strong local interaction for some time. It is local since the wave function vanishes
at spatial infinity. Experience suggests that, as time goes to infinity, the local interaction
diminishes and we have two possibilities:
1. Nothing remains in finite region and everything goes to spatial infinity. The solution
u(t) of the nonlinear equation should converge to a solution u+ (t) of a linear equation.
[Insert picture from classical mechanics.] This is called nonlinear scattering. (Linear
scattering refers to the comparison of solutions of two linear equations.)
2. If there are nondispersing local parts, they should converge to the sum of (possibly infinitely P
many) nonlinear bound states (“solitons”), denoted as Qj (t), and u(t)
converges to j Qj (t) + u+ (t). This is the subject of “soliton dynamics.”
Both area are under active research. In particular, the phenomenon in item 2 is confirmed only for completely integrable equations such as KdV, and our knowledge for other
equations is quite limited. These phenomena are the main subject of these lectures.
There lectures were delivered at the National Taiwan University in fall 2008 with supports from National Center for Theoretical Sciences (Taipei office) and Taida Institute for
Mathematical Sciences. The hospitality of these institutes are gratefully acknowledged. I
would also like to thank all the audience for the warm atmosphere and fruitful comments.
My research was supported in part by the Natural Sciences and Engineering Research Council of Canada, grant number 261356-08.
Suggestions to these notes are very welcome.
4
Chapter 1
Linear Schrödinger equations
In this chapter we describe the basic properties of the linear equations. It is the foundation
for later chapters in which we study nonlinear equations.
We will recall the following without proof: Semigroup theory, Fourier transform, RieszThorin interpolation theorem, and spectral theory for −∆ + V .
Notation: For x, y ∈ Rd , we denote x2 = |x|2 and xy = x · y. We denote Lp = Lp (Rd ) if
the domain is not specified. We denote A . B if A ≤ CB for some constant C which may
change from line to line.
1.1
Dispersion and dispersive equations
In optics, dispersion is the phenomenon in which the (phase) velocity of a wave depends
on its frequency. Media having such a property are termed dispersive media. As a result, dispersive waves of different frequency propagate at different velocities and eventually
separate, i.e., disperse, in the physical space. It should be compared with
1. Convection. Example: transport equation ∂t u + ~v · ∇u = 0. Every component propagates in the same speed;
2. Dissipation. Example: Heat equation ∂t u − ∆u = 0 with zero boundary condition.
The solution vanishes and does not propagate.
Examples of dispersive equations:
1. Free Schrödinger equation: i∂t u + ∆u = 0 where u : R × Rd → C.
2. Airy equation (linear part of KdV): ut + uxxx = 0 where uR × R → R.
3. Wave (m = 0) and Klein-Gordon (m > 0) equations: ∂tt u − ∆u + m2 u = 0.
A general form for constant coefficient dispersive PDE is
∂t u = Lu,
u(0, x) = u0 (x),
(1.1)
where L is an anti-symmetric differential (or pseudo-differential) operator,
(f, Lg) = −(Lf, g),
5
∀f, g.
(1.2)
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
6
P
If L = |α|≤k cα ∇α , where we use the multi-index convention, (1.2) implies cα ∈ R for odd
|α| and cα ∈ iR for even |α|. The equation preserves all H k -norms since
d
(u, (−∆)k u) = (Lu, (−∆)k u) + (u, (−∆)k Lu) = 0.
dt
When the domain is the entire space Rd , we can use Fourier transform to solve the
equation. Our convention for the Fourier transform and the inverse Fourier transform are
Z
1
û(ξ) = (Fu)(ξ) =
u(x)e−ix·ξ dx,
(2π)d/2 Rd
Z
1
û(ξ)eix·ξ dξ.
u(x) = (F −1 û)(x) =
(2π)d/2 Rd
Under this convention one has F∇u = iξ û and F(f ∗ g) = (2π)d/2 fˆĝ.
Applying Fourier transform to (1.1), we get
∂t û(t, ξ) = ih(ξ)û(t, ξ),
h(ξ) = −
X
i|α|+1 cα ξ α .
Note h(ξ) is a polynomial of real coefficients. It is called the dispersion relation of the
equation. One can solve
û(t, ξ) = eih(ξ)t û0 (ξ).
(1.3)
Applying inverse Fourier transform and using the formula for û0 , one gets
ZZ
1
eih(ξ)t+i(x−y)·ξ u0 (y) dy dξ.
(1.4)
u(t, x) =
(2π)d
P
More generally, L could be a pseudo-differentialPoperator, e.g., L = cαk |∇|k ∇α where
|∇| = (−∆)1/2 and F|∇|u = |ξ|û. Then h(ξ) = − i|α|+1 cαk |ξ|k ξ α .
If û0 (ξ) is concentrated at frequency ξ = ξ0 , we get a plane wave
u(t, x) = eih(ξ0 )t+ix·ξ0 .
Since a plane wave does not belong to L2 where we study the solutions, we consider a spatial
cut-off
u(t, x) = eih(ξ0 )t+ix·ξ0 φε (x + ∇h(ξ0 )t)
(1.5)
where φε (x) = φ(ε|x|) and φ(t) is a smooth real-valued function with compact support and
φ(t) = 1 for t near 0. Thus φε converges to 1 as ε → 0+ . One can show
∂t u − Lu = O(ε2 ).
The first factor in the right side of (1.5) gives the phase ξ0 ·(x−ωt) where ω = −h(ξ0 )ξ0 /|ξ0 |2 ,
while the second factor is the envelope (amplitude). We say ω is the phase velocity and
−∇h(ξ0 ) is the group velocity. Note that the phase velocity measures how fast the peaks of
the wave propagate, while the group velocity measures how fast the envelope propagates.
Equation
L
h(ξ)
ω(ξ)
−∇h(ξ)
Schrödinger
i∆
−|ξ|2
ξ
2ξ
−∂x3
ξ3
−ξ 2
−3ξ 2
±|ξ|
ξ
∓ |ξ|
ξ
∓ |ξ|
Airy
Wave
√
Klein-Gordon
p
± |ξ|2 + m2
∓
|ξ|2 +m2 ξ
|ξ|
|ξ|
∓√
ξ
|ξ|2 +m2
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
A few words about wave and Klein-Gordon:
time) PDE as
u
0
∂t U = LU, U =
, L=
ut
∆ − m2
7
We can rewrite them as a first order (in
1
,
0
0
1
h(ξ) = −i
.
−|ξ|2 − m2 0
√
1
1
iA
0
2
,
We can diagonalize L =
where L̃ =
, A = −∆ + m , P =
iA −iA
0 −iA
and rewrite ∂t U = LU as ∂t Ũ = L̃Ũ where Ũ = P −1 U . The values of h(ξ) in the above
table correspond to the diagonal entries of L̃.
Note that the wave equation is not dispersive in the strict sense since the frequency ξ
only determines the direction of the velocity, but not the speed.
P L̃P −1
1.2
Solutions of free Schrödinger equation
Abstract linear homogeneous Schrödinger equations are of the form
i∂t u = Hu,
u(0) = u0
(1.6)
where H is a self-adjoint operator acting on a Hilbert space X with dense domain. By
semigroup theory, see [46, §IX.9] or [22], −iH is the infinitesimal generator of a group of
unitary operators {S(t)}t∈R in X. We often denote S(t) = e−iHt . For the inhomogeneous
problem
i∂t u − Hu = f, u(0) = u0 ,
(1.7)
with f in a suitable space, the solution is given by the Duhamel’s formula
Z t
u(t) = S(t)u0 − i
S(t − s)f (s) ds.
(1.8)
0
In these lectures we always let X = L2 (Rd )1 and H = −∆ + V where V is a real-valued
potential with suitable decay and regularity. The semigroup theory guarantees that: For
any u0 ∈ L2 (Rd ), u(t) = S(t)u0 is the unique solution of (1.6) in H −2 for every t ∈ R in
the class
u ∈ C(R, L2 ) ∩ C 1 (R, H −2 ).
(1.9)
Moreover, ku(t)kL2 = ku0 kL2 for every t ∈ R.
We first consider the free case H = −∆. In this case we claim without proof that the
solution of (1.7) is given by
Z
(Kt ∗ u0 )(x) =
Kt (x − y)u0 (y) dy.
(1.10)
Rd
where the fundamental solution is2
x2
Kt (x) = (4πit)−d/2 ei 4t .
1
(1.11)
For any pair of functions f, g on Rd with f ḡ ∈ L1 (Rd ), we define (f, g) = Rd f (x)g(x) dx.
If we denote the heat kernel by Γ(x, t), then Kt (x) = Γ(x, it), although we do not have a physical
meaning for imaginary time it.
2
R
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
8
2
As a tempered distribution, the Fourier transform of Kt is K̂t (ξ) = (2π)−d/2 e−iξ t . Thus
2
F(Kt ∗ u0 ) = (2π)d/2 K̂t û0 = e−iξ t û0 (ξ).
This is exactly the solution formula (1.3) with h(ξ) = −|ξ|2 . Thus, if we are convinced that
the Fourier transform of S(t)u0 is given by (1.3), then we have
S(t)u0 = Kt ∗ u0 .
(1.12)
The formula (1.10) can be written as
2
−d/2 i x4t
(S(t)u0 )(x) = (4πit)
e
Z
e−
ixy
2t
y2
ei 4t u0 (y) dy
(1.13)
x2
If we define a multiplier Mt by Mt (x) = ei 4t and a dilation operator Dt by Dt w(x) =
(2it)−d/2 w(x/2t), we have
S(t)u0 = Mt Dt FMt u0 .
(1.14)
Note that all Mt , Dt and F preserves L2 -norm. Thus the operator Kt ∗ has the same
properties of Fourier transform. In particular, it maps L1 to L∞ , L2 to L2 (with same
norm), and the Schwartz class S to itself.
1.3
Decay estimates
In this section we give an estimate of S(t)u0 in Lp (Rd ). Note that the lower p means the
better spatial decay, but less local integrability. Also note that L2 plays a more important
role than other Lp for Schrödinger equations.
Lemma 1.1 (Decay estimates) If 2 ≤ p ≤ ∞, then
−d( 12 − p1 )
kS(t)ϕkLp ≤ (4π|t|)
kϕkLp0 ,
0
∀ϕ ∈ Lp .
(1.15)
Proof. For p = 2, it is because S(t) conserves L2 -norm. For p = 1, it follows from the
solution formula (1.10). For p ∈ (1, 2), it follows from the following (see, e.g.,[3, p.2]). Theorem 1.2 (Riesz-Thorin interpolation theorem) Let r1 , r2 , p1 , p2 ∈ [1, ∞]. If T
is a linear operator defined on Lr1 + Lr2 such that
kT kLr1 →Lp1 ≤ M1 ,
kT kLr2 →Lp2 ≤ M2 ,
then T can be extended to a bounded operator from Lr to Lp with
0 < θ < 1, and
kT kLr →Lp ≤ M1θ M21−θ .
1
r
=
θ
1−θ 1
r1 + r2 , p
=
θ
1−θ
p1 + p2 ,
Remarks.
1. This estimate shows S(t) does not preserve Lr unless r = 2. Also see problem 3.
2. S(t) does not improve regularity (differentiability) since (1.6) is time reversible. The
lemma says it improves the local integrability.
[Insert the figure of ( r1j , p1j ).]
[Insert the figure of the evolution of a localized u0 ]
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
1.4
9
Strichartz estimates
Lp -spaces are not sufficient for our study since they are not preserved by S(t) unless p =
2. We will need estimates involving space-time norms. For a given time interval I and
exponents (q, r), denote
Lqt Lrx = Lqt (I → Lr (Rd )), kukLqt Lrx = kukLrx (Rd ) q .
Lt (I)
Also denote CLr = C(I → Lr ).
A pair of exponents (q, r) is said to be admissible if
2 d
d
+ = ,
q
r
2
q, r ∈ [2, ∞],
(q, r, d) 6= (2, ∞, 2).
(1.16)
The figures for ( 1r , 1q ):
(insert figures for d ≥ 3, d = 2, and d = 1.)
Note that q ≥ 2 for d ≥ 3, q > 2 for d = 2 and q ≥ 4 for d = 1. Note that the space-time
norms with admissible exponents are invariant under the natural scaling: If u(t, x) is a
solution of (1.6), then so is
uα (t, x) = λd/2 u(λ2 t, λx)
for any λ > 0. We have kuλ kLqt Lrx = kukLqt Lrx for all α > 0 in R × Rd .
Theorem 1.3 (Strichartz estimates) Fix d ≥ 1. For any admissible exponents (q, r)
and (q̃, r̃), we have the homogeneous Strichartz estimate
kS(t)u0 kLqt Lrx . ku0 kL2x
(1.17)
and the inhomogeneous Strichartz estimate
Z
k
S(t − s)f (s) dskLqt Lrx . kf kLq̃0 Lr̃0 .
s<t
t
(1.18)
x
Remarks.
1. Locally in time, it means better spatial regularity (in time average sense). Globally
in time, it means decay of Lr -norm (in time average sense). It is consistent with the
picture.
2d
2. Exponents (q, r) = (2, d−2
) for d ≥ 2 are called the endpoint exponents. Endpoint
Strichartz estimates are correct (with more involved proof) for d ≥ 3 but false for
d = 2.
3. The endpoint cases are of interest to nonlinear problems: For a nonlinear equation
i∂t u = Hu + f (u), we often need f (u(t)) ∈ L1t X for some space X to ensure the
convergence of some parameters. If f (u) contains a quadratic term, we need u ∈ L2t X̃
for some X̃.
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
10
We will postpone the proof of endpoint cases. In the proof we will use the following
lemma which implies that, to prove (1.18), it suffices to show
Z
k S(t − s)f (s) dskLqt Lrx . kf kLq̃0 Lr̃0 .
(1.19)
t
R
x
Note that (1.18) is the time localization of (1.19) to s < t.
Lemma 1.4 (Christ-Kiselev lemma) Let I be a time interval, let X, Y be Banach
spaces, let X0 be a dense subset of X and let K(t, s) be a kernel defined from X0 into
Y for every t, s ∈ I. Suppose that 1 ≤ p < q ≤ ∞ is such that
Z
k K(t, s)f (s) dskLq (I→Y ) ≤ Akf kLpt (I→X)
I
for all f ∈ Lpt (I → X) and some A > 0. Then one also has
Z
k
K(t, s)f (s) dskLq (I→Y ) ≤ Cp,q Akf kLpt (I→X) .
s<t
0
For our purpose, I = R, K(t, s) = S(t − s), X = Lr̃ , Y = Lr , and X0 = Cc∞ (Rd ).
This lemma is false for 1 < p = q < ∞, because ...
Remarks on the proof. (i) In our proof of Theorem 1.3, we will use T T ∗ method for
Fourier restriction theorem for nonendpoint cases. It only needs the decay estimates Lemma
1.1 and is thus true for more H, see next section.
(ii) Strichartz estimates usually require less assumption than the decay estimates. For
example, for the Schrödinger operator H = −∆ + A · i∇ + i∇ · A with a vector magnetic
potential A in R3 , Strichartz estimates are known while decay estimates are open [8]. In
this case the Christ-Kiselev lemma is still applicable.
Proof of Theorem 1.3 for non-endpoint cases.
We may assume I = R, f ∈ S and apply (1 − ε∆)−1 , ε → 0− , to make the following
rigorous.
R
Step 1. Let g(t) = R S(t − s)f (s) ds. By Minkowski inequality and Lemma 1.1,
Z
Z
kg(t)kLrx ≤
kS(t − s)f (s)kLrx ds ≤ C
|t − s|−σ kf (s)kLr0 ds
R
R
where σ = d( 12 − 1r ). By Riesz potential estimate ([31, p.119]) or by generalized Young’s
inequality for convolution,
kg(t)kLqt Lrx ≤ Ckf kLq0 Lr0 ,
t
1
q
x
1
q0
if 0 ≤ σ < 1 and = σ + − 1 (i.e. σ = 2/q), which is equivalent to (q, r) being admissible
and q > 2.
R
The above argument is true for g(t) = s<t S(t − s)f (s) ds, which gives (1.18) when
q = q̃.
Step 2. Since (S(−s)f (s), S(−t)f (t)) = (S(t − s)f (s), f (t)), by Hölder inequality and
Step 1,
Z Z
2
kg(0)kL2x = | ( S(t − s)f (s) ds, f (t)) dt| ≤ kg(t)kLqt Lrx kf kLq0 Lr0 . kf k2 q0 r0 .
R
R
t
x
Lt Lx
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
0
11
0
Step 3. For any f ∈ Lqt Lrx , by Step 2 we have
Z
Z
(S(t)u0 , f (t)) dt = (u0 , S(t − s)f (s) ds) ≤ ku0 kL2 kg(0)kL2 . ku0 kL2 kf kLq0 Lr0 .
R
t
R
x
Thus we get (1.17) from Rduality.
Step 4. Since g(t) = R S(t − s)f (s) ds = S(t)g(0), by (1.17) and Step 2,
kg(t)kLqt Lrx . kg(0)kL2x . kf kLq̃0 Lr̃0 .
t
x
Step 5. Estimate (1.18) now follows from Step 4 and the Christ-Kiselev lemma.
Proof of Lemma 1.4.
We may assume I = R. We will assume 1 ≤ p < q < ∞ and f (t) 6= 0 for all t, and leave
the general case to problem 5.
R
Normalize kf kLp (I,X) = 1 and define φ(t) = s<t kf (s)kpX ds. It maps R to (0, 1), is
order preserving, 1-1, and onto.
We are going to decompose (0, 1) × (0, 1) into the union of products of dyadic intervals.
An interval I is called dyadic if I = 2−j [k, k + 1) for some integers k, j. Its parent is the
unique dyadic interval I ] such that I ⊂ I ] and |I ] | = 2|I|. Here |I| denotes the length of I.
Given two dyadic intervals I and J, we say I C J if
|I| = |J|,
and sup I ] = inf J ] .
sup I < inf J,
[Insert figure]
For every I, there are at most two J so that I C J, and vice versa.
For almost every φ0 < φ in (0, 1), there is a unique pair I, J such that φ0 ∈ I, φ ∈ J,
and I C J. (It is easierR to first identify I ] and J ] .)
R
Denote T f (t) = R K(t, s)f (s) ds and T̃ f (t) = s<t K(t, s)f (s) ds. Since s < t iff
φ(s) < φ(t), we can rewrite
Z
Z
X
K(t, s)f (s) ds.
T̃ f (t) =
K(t, s)f (s) ds =
φ(s)<φ(t)
ICJ: φ(t)∈J
φ(s)∈I
Denote 1E as the characteristic function of E, we have
Z
X
X
1φ(t)∈J
Tj (t),
T̃ f (t) =
K(t, s)f (s)1φ(s)∈I ds =
ICJ
R
P
where Tj (t) = ICJ, |I|=2−j 1φ−1 (J) (t) · T (f 1φ−1 (I) )(t). For a fixed j, since for each I there
are at most two J, and the functions 1φ−1 (J) have essentially disjoint supports, we have
X
kTj (t)kqq .
kT (f 1φ−1 (I) )kqq .
I: |I|=2−j
By the assumption on T , this is bounded by
X
.
kf 1φ−1 (I) kqp .
I: |I|=2−j
By construction kf 1φ−1 (I) kp = 2−j/p , and we have 2j such I. Thus
kTj (t)kq ≤ 2
−j( p1 − 1q )
,
and the claim follows from summing in j and the hypothesis p < q.
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
1.5
12
Linear Schrödinger equations with potentials
In this section we consider the linear Schrödinger equation
i∂t u = Hu,
u(0) = u0 ,
H = −∆ + V,
where V is a real-valued potential. Formally it has two constants of time: the L2 -norm and
the Hamiltonian,
Z
Z
1
|u(t, x)|2 dx,
|∇u|2 + V |u|2 dx.
2
R
This should be compared to the free case H = −∆, for which |∇k u|2 dx are constant for
all integers k ≥ 0.
There are two basic questions:
Q1. Under what conditions on V can H be identified as a self-adjoint operator and give
a unitary group S(t) = e−itH ?
Q2. Under what conditions on V does the unitary group S(t) = e−itH satisfy decay and
Strichartz estimates?
The required conditions for Q1 are pretty mild. For example, for d = 3, one only needs
V ∈ L2 + L∞ , i.e., V = V2 + V∞ with V2 ∈ L2 and V∞ ∈ L∞ . This is a consequence of the
following theorem, which applies to more general operators.
Theorem 1.5 (Kato-Rellich Theorem) Suppose that A is self-adjoint, B is symmetric,
D(A) ⊂ D(B), and for some a ∈ (0, 1), b > 0,
kBf k ≤ akAf k + bkf k,
∀f ∈ D(A).
Then A + B is self-adjoint with domain D(A).
For −∆ + V on R3 , we have A = −∆, D(A) = H 2 (R3 ), B = V , and we only to check
that, if V ∈ L2 + L∞ , then for some a ∈ (0, 1) and b > 0,
kV f kL2 ≤ ak∆f kL2 + bkf kL2 ,
∀f ∈ H 2 (R3 ).
The answer for Q2 is more involved. First of all, one cannot allow eigenfunctions in the
estimates. A function ϕ ∈ H 2 (Rd ) is an eigenfunction of H if
Hϕ = λϕ
for some λ ∈ C called its eigenvalue. Since (e−itH ϕ)(x) = e−itλ ϕ(x) is a nondispersing
solution, it has to be excluded from the estimates.
We now recall a bit of the spectral theory. The spectrum σ(H) of a self-adjoint operator
is always a subset of R. If H = −∆ + V on Rd where V (x) is real-valued with sufficient
spatial decay and regularity, then there are finitely many eigenvalues, all of which are nonpositive and with finite multiplicity. Denote these eigenvalues by λj and the corresponding
orthogonal projections onto their eigenspaces by Pj . The collection of these eigenvalues is
the discrete spectrum of H and denoted as σd (H).
The continuous spectrum of H is σc (H) = [0, ∞). Its corresponding projection is Pc =
P
1− j Pj . For each ξ ∈ Rd \ {0}, there is a solution of the differential equation −∆φ+V φ =
|ξ|2 φ of the form
φ(x, ξ) = eix·ξ + φ0 (x, ξ)
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
13
where φ0 is a “nice” function with some regularity and spatial decay. It is called a distorted
plain wave. For any f ∈ L2 (Rd ), there is some f˜ ∈ L2 so that
Z
(Pc f )(x) =
f˜(ξ)φ(x, ξ) dξ.
Rd
This is called the eigenfunction expansion, or the generalized Fourier transform. The projection Pc f is the dispersive (radiation) part of f .
When V = 0, these reduce to the free case and the usual Fourier transform.
Now back to the decay and Strichartz estimates. It should be clear now we need to
add Pc in the left sides of (1.15), (1.17), and (1.18). For example, for p ∈ [2, ∞] and (q, r)
admissible,
−d( 21 − p1 )
ke−itH Pc ϕkLpx ≤ C|t|
kϕkLp0 ,
ke−itH Pc ϕkLq Lr ≤ CkϕkL2 .
(1.20)
However, we still need to assume that 0 is a regular point of σ(H), that is, 0 is neither an
eigenvalue nor a resonance of H.
One would ask: What is a resonance? More importantly, since Pc removes all eigenfunctions, why do we need 0 not to be an eigenvalue?
A resonance is a function ψ 6∈ L2 (Rd ) which satisfies the differential equation −∆ψ +
V ψ = 0 and belongs to some function space with less spatial decay than L2 (Rd ). For d = 1
1
we require ψ ∈ L∞ \ L2 . For d = 3 we require ψ ∈ L2,− 2 −ε \ L2 for some ε > 0. Here L2,s
for s ∈ R is the Banach space equipped with the norm
kf kL2,s = k hxis f kL2 (Rd ) ,
hxi = 1 + |x|.
(1.21)
It is known that there is no resonance for d ≥ 5 and at most one resonance for d = 3. For
d = 2, the span of all resonance functions may have dimension up to 3.
To answer the second question, we will explain how (1.20) are proved. In the end it is
because of the properties of the resolvent R(z) = (H − z)−1 .
The starting point to proving (1.20) is the following weighted estimate
ke−itH Pc ϕkL2,−s ≤ C hti−d/2 kϕkL2,s ,
s > d/2.
(1.22)
It assumes strong localization of ϕ and controls only the local part of e−itH Pc ϕ. It is implied
by the L1 → L∞ decay. For example, for d = 3, H = −∆, and t > 1, we have
keit∆ ϕkL2,−3/2−ε . keit∆ ϕkL∞ . hti−3/2 kϕkL1 . hti−3/2 kϕkL2,3/2+ε .
Although weaker, estimate (1.22) controls the low frequency part of ϕ, which is usually the
most troublesome part. One first proves (1.22) since it is in terms of weighted L2 -spaces
and is understood better.
To prove (1.22), one uses the formula
Z ∞
X
−itH
−itλj
e−itλ E 0 (λ)dλ.
(1.23)
e
−
e
Pj − P0 =
j
0
The summation on the left side runs through all negative eigenvalues λj and its corresponding projections Pj , and P0 is the orthogonal projection onto the 0-eigenspace if it is
nontrivial. On the right side, E(λ) is the spectral measure associated to H and E 0 (λ) is the
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
14
density. From the functional calculus form of the spectral theory for unbounded self-adjoint
operators, we have
Z
Z
H=
λ dE(λ),
g(H) =
R
g(λ) dE(λ),
R
for suitable functions g. Equation (1.23) is the special case with g(λ) = e−itλ . It is usually
understood in L2 and does not provide any decay in t. To get t-decay, we use weighted
spaces and
πE 0 (λ) = Im R+ (λ), R+ (λ) =
lim
R(z) (λ > 0).
z→λ,Im z>0
Since kR(z)kL2 →L2 ∼ dist(z, σ(H))−1 , the above limit is not in B(L2 → L2 ). Instead, it is
in B(L2,s → L2,−s ), s > 1/2. The above limit does not exist for λ = 0, for which we use
the expansion in B(H −1,s → H 1,s ) for small z, for d = 3,
R(z) = −z −1 P0 − iz −1/2 B−1 + B0 + iz 1/2 B1 + · · ·
where Im z ≥ 0, Im z 1/2 ≥ 0,
B−1 = P0 V G3 V P0 − h·, ψi ψ,
2
G3 is the integral operator with kernel |x−y|
24π , and ψ is the resonance. From this formula
one can get
X
e−itH −
e−itλj Pj − P0 = Ct−1/2 B−1 + Ct−3/2 B1 + · · ·
j
in weighted spaces. Thus in general one can only assert t−1/2 decay. Suppose 0 is an
eigenvalue but not a resonance. If ϕ ∈ L2,r and ϕ = Pc ϕ, the above formula say e−itH ϕ
has t−3/2 decay. However, if ϕ ∈ L2,r and ϕ 6= Pc ϕ, we do not have e−itH Pc ϕ = 0 +
Ct−3/2 B1 Pc ϕ + · · · unless we know Pc ϕ ∈ L2,s (to validate the above equality), which
depends on the decay property of the 0-eigenfunctions.
We now summarize some properties of V which ensures the decay estimates (and thus
the nonendpoint Strichartz estimates): V (x) should be a real-valued potential so that 0 is
neither an eigenvalue nor a resonance, and
1. For d = 1, hxi V ∈ L1 .
2. For d = 2, |V (x)| . hxi−3−ε .
3. For d = 3, V (x) ∈ L3/2−ε ∩ L3/2+ε .
4. For d ≥ 3, V̂ ∈ L1 and hxis V : H ν → H ν is bounded for some s > d + 4 and ν > 0.
Note that, for d = 3 the condition |V (x)| . hxi−2−ε ensures the above condition. For
d ≥ 4, there are examples that V (x) being continuous with compact support is insufficient.
Thus some regularity assumption like V̂ ∈ L1 is necessary.
1.6
Problems and notes
Problems
CHAPTER 1. LINEAR SCHRÖDINGER EQUATIONS
15
1. Lemma 1.1 fails for p < 2.
0
Remark. If p < 2, the target Lp is more localized than the domain Lp , which is
inconsistent with the fact that the equation is dispersive. This can be also seen from
the relation (1.12) between S(t) and Fourier transform.
2. For any ϕ ∈ L2 , S(t)ϕ converges to 0 weakly in L2 , i.e.,
(S(t)ϕ, ζ) → 0
as t → ∞,
∀ϕ, ζ ∈ L2 .
Remark. This problem shows that S(t)ϕ must spread out in space, which is another
evidence of the dispersive nature of S(t). Note that there is no convergence rate:
For any rate one can construct ϕ and ζ with that rate as a lower bound. (Another
problem!)
3. For any fixed T > 0, there does not exist a constant C so that
kS(T )ϕkL1 ≤ CkϕkL1 ,
∀ϕ ∈ L1 .
Remark. This shows that S(t) does not preserve L1 .
R
4. We have shown F (t) = s<t S(t − s)f (s) ds ∈ L∞ L2 . Show that F (t) ∈ CL2 .
5. In the proof of the Christ-Kiselev lemma, we assumed f (t) 6= 0 for all t. How do we
deal with the general case? Also consider the case q = ∞.
Notes
In preparing these notes, I have used [37] for section 1.1, [6] for sections 1.2 and 1.3,
[36, 37] for section 1.4, and [1, 15, 25] for section 1.5.
Chapter 2
Local wellposedness
In this chapter we start our study on nonlinear Schrödinger equations by studying the local
in time behavior of their solutions. All the Sobolev spaces below are over Rd .
2.1
Definitions
A typical nonlinear Schrödinger equation is for an unknown function u(t, x) : I × Rd → C
where I is an interval containing 0. The equation reads
i∂t u + ∆u = f (u),
u(0) = u0 .
(2.1)
Usually we want to work with solutions in L∞ H 1 . Then ∂t u, ∆u ∈ H −1 in the free case
f = 0. Thus it is reasonable to assume f : H 1 → H −1 so that the equation makes sense in
H −1 .
Example (i) f (u) = λ|u|p−1 u with λ ∈ C and p ∈ (1, ∞).
The condition f ∈ C(H 1 , H −1 ) means
|(v, f (u))| ≤ CkvkH 1
∀v ∈ H 1 .
Since vf (u) ∈ (H 1 )1+p , it is equivalent to 2 ≤ p + 1 ≤ 2∗ where
1 < p < ∞, we have f ∈ C(H 1 , H −1 ) iff p ≤ pmax , where
pmax = ∞
for d = 1, 2;
pmax =
4
+1
d−2
1
2∗
=
1
2
− d1 . Thus, for
for d ≥ 3.
0
Note that H 1 ⊂ Lr and Lr ⊂ H −1 if 2 ≤ r ≤ 2∗ .
Example (ii) f (u) = V u where V is a complex-valued background potential.
Example (iii) f (u) = (W ∗ |u|2 )u (Hartree type nonlinearity). Here W is a convolution
kernel. It is called “non-local” since f (u) at a point x0 does not depend on u(x0 ) alone.
Definition 2.1 Let I be an interval containing 0, let f ∈ C(H 1 , H −1 ), and let u0 ∈ H 1 .
i) A function u is called a weak H 1 solution of (2.1) if
u ∈ L∞ (I, H 1 ) ∩ W 1,∞ (I, H −1 ),
(2.2)
u satisfies (2.1) in H −1 for almost all t ∈ I, and u(0) = u0 .
ii) A function u is called a strong H 1 solution of (2.1) if
u ∈ C(I, H 1 ) ∩ C 1 (I, H −1 ),
u satisfies (2.1) in H −1 for all t ∈ I, and u(0) = u0 .
16
(2.3)
CHAPTER 2. LOCAL WELLPOSEDNESS
17
Since we will work solely in H 1 for the rest of these notes, we will drop H 1 and simply
say weak and strong solutions.
The condition u(0) = u0 is meaningful for a weak solution because of the following.
¯ L2 ).
Lemma 2.2 If u satisfies (2.2), then u ∈ C(I,
¯ L2 ) and u(t) = ũ(t) for a.e. t.
Strictly speaking, it means there is ũ ∈ C(I,
The proof is left as Problem 6.
Lemma 2.3 (Duhamel’s formula) A function u ∈ L∞ (I, H 1 ) is a weak solution of (2.1)
on I if and only if
Z
t
u(t) = eit∆ u0 − i
ei(t−s)∆ f (u(s)) ds
(2.4)
0
for almost all t ∈ I. It is a strong solution if and only if (2.4) holds for all t ∈ I.
Proof. We will prove the case of strong solutions. We always have f (u) ∈ L∞ (I, H −1 ).
If u(t) satisfies (2.4) for all t, then u(0) = u0 , u ∈ C(I, H 1 ), f (u) ∈ C(I, H −1 ), and
1
1
1
[u(t + ε) − u(t)] = [eiε∆ − 1]u(t) − i
ε
ε
ε
Z
t+ε
ei(t+ε−s)∆ f (u(s)) ds.
t
As ε → 0, the right side converges in H −1 to i∆u − if (u), showing (2.1) for all t. We thus
have (2.3).
Suppose (2.3) holds and (2.1) holds for all t. We have f (u) ∈ C(I, H −1 ). Denote the
right side of (2.4) as v(t). Then v ∈ C(I, H 1 ), v(0) = u0 and vt = i∆v − if (u) for all t.
The difference w satisfies w ∈ C(I, H 1 ) ∩ C 1 (I, H −1 ), w(0) = 0, and i∂t w + ∆w = 0. Thus
w(t) = 0 for all t, showing (2.4).
The case of weak solutions is left as Problem 7.
Definition 2.4 Problem (2.1) is said to be locally wellposed in H 1 if it satisfies the following:
1. (Uniqueness) Any two weak solutions on any interval I 3 0 with same initial value
are the same.
2. (Existence) For any initial value u0 ∈ H 1 there is a strong solution on some time
interval.
3. (Continuous dependence on data) Suppose u(t) is a strong solution on a compact
interval I 3 0 and uk (t) is a sequence of solutions with uk (0) → u(0) in H 1 . Then for
k large enough uk (t) exist on I and uk (t) → u(t) in C(I, H 1 ).
Note we require uniqueness in a wider class than the rest.
When uniqueness holds, one can talk about maximal interval of existence: If u1 (t)
defined on I1 and u2 (t) defined on I2 are both solutions, then u1 = u2 on I1 ∩ I2 . One can
then define u = u1 or u = u2 on I1 ∪ I2 . The union of all such solutions gives a solution on
a maximal interval (−Tmin , Tmax ) where Tmin , Tmax ∈ (0, ∞].
We say the problem satisfies the blow-up alternative if whenever Tmax (u0 ) < ∞, one has
limt↑Tmax ku(t)kH 1 = +∞.
When f (u) possesses some symmetries, we have conserved quantities. For example, if
f (u) = λ|u|p−1 u + V u + (W ∗ |u|2 )u and λ, V, W are real-valued, then formally ku(t)kL2
CHAPTER 2. LOCAL WELLPOSEDNESS
18
d
(the charge) is conserved since dt
ku(t)k2L2 = 0. This condition holds for more general f if f
u
is gauge-invariant in the sense that f (u) = f (|u|) |u|
. Further, f is the Gâteaux derivative
1
1
1
of some G R∈ C (H → R) with H considered
a real Hilbert space with inner product
R
(f, g) = Re Rd f ḡ dx, i.e., G0 (u)w = Re Rd f (u)w̄ dx, then the energy
Z
1
|∇u|2 dx + G(u)
E(u) =
2
d
R
is also conserved. For f (u) = λ|u|p−1 u + V u + (W ∗ |u|2 )u with λ, V, W real-valued, if W is
even, then
Z λ
1
1
|∇u|2 +
|u|p+1 + V |u|2 + (W ∗ |u|2 )|u|2 dx.
E(u) =
2
p+1
2
Rd
If W is not even, we cannot define G.
Depending on f , there may also be conservation of momentum and angular momentum.
2.2
Uniqueness and existence
Assumption (F1): Assume f = f1 +· · ·+fk with fj ∈ C(H 1 , H −1 ) and for some exponents
rj , bj ∈ [2, 2∗ ) for d ≥ 2 or rj , bj ∈ [2, ∞] for d = 1,
kfj (u) − fj (v)k
b0
j
L
≤ C(M )(ku − vkLrj ) ∀j
if kukH 1 + kvkH 1 ≤ M .
The restrictions on the exponents rj , bj ensure that their corresponding time exponents
in the Strichartz estimates are strictly larger than 2. This will be seen essential in the
proofs.
In the case f (u) = λ|u|p−1 u, we can take k = 1 and b = r. Since kf (u)kLr0 = kukpLpr0 ,
we can take pr0 = r, which gives r = p + 1.
In general, e.g., f (u) = λ|u|p−1 u + |u|q + V u + (W ∗ |u|2 )u, we need to take k > 1.
In the following we will use Strichartz estimates in Sobolev norms. If u is a solution to
i∂t u + ∆u = 0, then so is ∂x∂ j u for any j. Thus we can replace Lr by W 1,r in the Strichartz
estimates, e.g., kS(t)u0 kLq W 1,r ≤ Cku0 kH 1 for any admissible pair (q, r).
Theorem 2.5 (Uniquesness) Suppose f satisfies assumption (F1). Any two weak solutions on [0, T ) with the same initial value must be the same.
Proof. Choose qj , aj ∈ (2, ∞] so that (qj , rj ) and (aj , bj ) are admissible. Denote these two
solutions by u and v. Let M = max(kukL∞ H 1 , kvkL∞ H 1 ) and w = u − v. By Duhamel’s
formula, for a.e. t,
Z t
Pk
w(t) = j=1 wj (t), wj (t) = −i
S(t − s)[fj (u(s)) − fj (v(s))] ds.
0
For any fixed j and J = [0, t) with 0 < t < T , by Strichartz estimates and assumption
(F1),
k
X
kwj kLq` Lr` ≤ Ckfj (u) − fj (v)k a0j b0 ≤ CC(M )ku − vk a0j r
J
`=1
LJ L
j
LJ L
j
CHAPTER 2. LOCAL WELLPOSEDNESS
19
Summing in j, we get
Pk
j=1 kwkL j Lrj
J
q
P
≤ CC(M ) kj=1 kwk
a0
LJj Lrj
Denoting φj (t) = kw(t)kLrj , we get
P
P
L(t) := kj=1 kφj kLqj (0,t) ≤ CC(M ) kj=1 kφj k
a0
j (0,t)
L
Let
τ = sup {t ∈ [0, T ) : L(t) = 0} .
Then φj (t) = 0 for a.e. t ∈ [0, τ ). We claim τ = T . If τ < T , for small ε ∈ (0, T − τ ) to be
chosen, by the above and Hölder inequality,
P
P
L(τ + ε) ≤ CC(M ) kj=1 kφj k a0j
≤ CC(M ) kj=1 εσj kφj kLqj (τ,τ +ε)
L
where σj =
1
a0j
−
1
qj
(τ,τ +ε)
> 0 since a0j < 2 < qj . Let σ = min σj > 0. We get L(τ + ε) =
CC(M )εσ L(τ + ε). Thus L(τ + ε) = 0 for ε sufficiently small, which contradicts the
definition of τ . Thus L(T ) = 0 and u = v on [0, T ).
Remarks. i) For a fixed j, the proof allows one of qj , aj to be 2. This seems to be the
only reason why one does not show directly kwkStr(0,τ +ε) ≤ Cεσ kwkStr(0,τ +ε) as in the proof
of Theorem 2.5, which is easier!
1
ii) The above proof works if one assumes a weaker condition u, v ∈ L∞
loc ([0, T ); H )
instead of u, v ∈ L∞ H 1 .
iii) This uniqueness Theorem is in a wider class of functions space than the rest of this
section (weak vs. strong).
For the existence, we need more assumption on f .
Assumption (F2): Assume f = f1 + · · · + fk as in (F1) and, for all j,
kfj (u)k
W
1,b0
j
≤ C(M )(1 + kukW 1,rj )
for all u ∈ H 1 ∩ W 1,rj with kukH 1 ≤ M .
Theorem 2.6 (Existence) Suppose f satisfies assumptions (F1) and (F2). For each
u0 ∈ H 1 , there is a unique strong solution u of (2.1) defined on a maximal time interval (−Tmin , Tmax ), with Tmin , Tmax ≥ T1 (ku0 kH 1 ) for some positive function T1 . Moreover,
for every admissible pair (a, b), we have
u ∈ Laloc ((−Tmin , Tmax ), W 1,b ).
Proof. Let r = max(rj , bj ). Choose q, qj , aj ∈ (2, ∞] so that (q, r), (qj , rj ) and (aj , bj ) are
admissible. For M, T > 0 to be chosen, let I = (−T, T ) and
n
o
q
1
1,r
q
∞
E = u ∈ L∞
H
∩
L
W
:
kuk
:=
kuk
+
kuk
≤
M
1
1,r
E
LI H
I
L W
I
I
For u ∈ E, define
u] (t) = S(t)u0 − i
Z
t
S(t − s)f (u(s)) ds.
0
CHAPTER 2. LOCAL WELLPOSEDNESS
20
Our proof is based on a fixed point argument: For suitable chosen M and T , for all u, v ∈ E,
we will show (i) u] ∈ E and (ii)
1
ku] − v ] kStr ≤ ku − vkStr
2
where kukStr := kukL∞
2 + kukLq Lr . It is weaker than k · kE .
I L
I
We first show (i). By Strichartz estimates
P
ku] kE ≤ C1 ku0 kH 1 + C j kfj (u)k a0j
W
which is bounded by Hölder inequality in time by
P
≤ C1 ku0 kH 1 + C j T σj kfj (u)k
1,b0
j
L
Lqj W
where σj =
1
qj0
−
1
aj
1,b0
j
,
> 0. By assumption (F2) and assuming T < 1,
ku] kE ≤ C1 ku0 kH 1 + C
P
jT
σj
C(M )(1 + kukLqj W 1,rj ),
By Hölder inequality first in space and then in time, one has
1−θ
kukLqj W 1,rj ≤ CkukθL∞ H 1 kukL
q W 1,r ≤ CkukE
for some θ ∈ [0, 1]. Thus, for σ = min σj > 0, T < 1, and u ∈ E,
ku] kE ≤ C1 ku0 kH 1 + CC(M )(1 + M )T σ .
Taking M = 2C1 ku0 kH 1 and T < 1 with CC(M )(1 + M )T σ < M/2, we get ku] kE ≤ M .
To show (ii), by Strichartz estimates again,
P
ku] − v ] kStr ≤ C j kfj (u) − fj (v)k a0j 1,b0j .
L
W
By similar estimates as in the proof of Theorem 2.5, we get
1
ku] − v ] kStr ≤ CC(M )T σ ku − vkStr ≤ ku − vkStr
2
for T sufficiently small. By Banach fixed point theorem, there is a unique u ∈ E with
u] = u. By problem 7, u ∈ C(I, H 1 ).
For any compact subinterval J of (−Tmin , Tmax ), for any admissible (a, b), and any
t ∈ J, by applying Strichartz estimates to the Duhamel’s formula, one gets u ∈ La (It , W 1,b )
for some open interval It 3 t. One can take a finite subcover of {It }t∈J and show u ∈
La (J, W 1,b ). This gives the last statement of the theorem.
q
1
1,r ,
Remark. The fixed point argument gives a unique solution in E ⊂ L∞
I H ∩ LI W
q
∞
1
1,r
but not in LI H . Uniqueness under extra conditions like u ∈ LI W is called conditional
uniqueness. In contrast, Theorem 2.5 is an unconditional uniqueness result.
Corollary 2.7 (Blow-up alternative) Suppose f satisfies assumptions (F1) and (F2)
and let u be the solution of (2.1) on a maximal interval (−Tmin , Tmax ). If Tmax < ∞, then
ku(t)kH 1 → ∞ as t ↑ Tmax .
Proof. Suppose M := 2 lim inf t↑Tmax ku(t)kH 1 < ∞. Let T = T1 (M ) be the lower bound
in Theorem 2.6. Choose t0 ∈ (Tmax − T, Tmax ) with ku(t0 )kH 1 ≤ M . By existence and
uniqueness, u can be extended to t ∈ [0, t0 + T ], contradicting the definition of Tmax .
CHAPTER 2. LOCAL WELLPOSEDNESS
2.3
21
Continuous dependence on data and conservation laws
Theorem 2.8 (Continuous dependence on data) Suppose f satisfies assumptions (F1)
and (F2). Suppose u(t) is a strong solution of (2.1) on a compact interval I 3 0 and uk (t)
is a sequence of strong solutions with uk (0) → u(0) in H 1 .
(i) For some T > 0 and k large enough, uk exist on J = [−T, T ] ⊂ I and uk (t) → u(t)
in C(J, Lr ) for any b ∈ [2, 2∗ ) (or b ∈ [2, ∞] if d = 1).
(ii) If f further satisfies assumption (F3), then uk (t) → u(t) in C(I, H 1 ).
In general J can be much shorter than I. Assumption (F3) will be given later. Part (ii)
will be proved after the proof of Theorem 2.9, where the consequence of assumption (F3)
is given.
Proof. (i) Let M = kukL∞ (I,H 1 ) , which is finite since I is compact. Let T = T1 (2M ) > 0
where T1 is given in Theorem 2.6. We may choose a smaller T so that J = [−T, T ] ⊂ I.
For k large kuk (0)kH 1 ≤ 2ku0 kH 1 . By Theorem 2.6, uk exist on J and kuk kL∞
1 ≤ 2CM .
J H
The same estimate in the proof for uniqueness shows
kuk − ukStr(J) ≤ C1 kuk (0) − u(0)kL2 + C(M )T σ kuk − ukStr(J)
For
The only new term is C1 kuk (0) − u(0)kL2 . Here kf kStr(J) = kf kL∞
2 + kf kLq Lr .
J L
J
T < T (M ), we have
kuk − ukStr(T ) ≤ 2C1 kuk (0) − u(0)kL2 .
By Sobolev imbedding and b < 2∗ for d ≥ 2,
kuk − ukL∞ (J,Lb ) ≤ kuk − ukθL∞ (J,L2 ) kuk − uk1−θ
L∞ (J,H 1 )
for some θ ∈ (0, 1]. This shows uk → u in L∞ (J, Lb ).
For conservation laws, we need another assumption.
Assumption (F3): Assume Im f (u)ū = 0 a.e. for all u ∈ H 1 , and f = f1 + · · · + fk as in
(F1) with fj = G0j as Gâteaux derivative for some Gj ∈ C 1 (H 1 → R) for all j, with H 1
regarded as a real Hilbert space.
R
P
When this assumption holds, define E(u) = 12 |∇u|2 dx + kj=1 Gj (u).
When L2 is considered as a real Hilbert space, it is equipped with the inner product
Z
hf, gi = Re
f ḡ dx.
Rd
R
Correspondingly, we have hf, giH k = |α|≤k Re Rd Dα f Dα ḡ dx. Recall Gâteaux derivative
is directional derivative of maps between function spaces. Thus, f = G0 means,
P
G0 (u)v = lim ε−1 [G(u + εv) − G(u)] = hf (u), viH −1 ,H 1 ,
ε→0
∀u, v ∈ H 1 .
For example, consider f (u) = λ|u|p−1 u with λ ∈ R. It is the Gâteaux derivative of
Z
λ
G(u) =
|u|p+1 dx.
p+1
CHAPTER 2. LOCAL WELLPOSEDNESS
22
To see this, denoting a = (p + 1)/2 > 1,
|u + εv|p+1 = [|u|2 + ε(uv̄ + ūv) + ε2 |v|2 ]a = |u|p+1 [1 + ε|u|−2 (uv̄ + ūv) + O(ε2 )]a
= |u|p+1 [1 + εa|u|−2 (uv̄ + ūv) + o(ε)]
Thus, for u, v ∈ Cc∞ ,
1
G (u)v = lim [G(u + εv) − G(u)] = Re
ε→0 ε
0
Z
f (u)v̄ dx = hf (u), vi .
It is then extended to u, v ∈ H 1 .
Theorem 2.9 (Conservation of charge and energy) Suppose f satisfies (F1), (F2) and
(F3). Let u be the strong solution of (2.1) on the maximal interval I = (−Tmin , Tmax ) with
u(0) = u0 ∈ H 1 . Then
ku(t)kL2 = ku0 kL2 ,
E(u(t)) = E(u0 ),
∀t ∈ I.
Proof. For the charge, since (2.1) holds in CI H −1 and u ∈ CI H 1 , by (F3) we have
Re(∂t u, u)H −1 ,H 1 = Re(−∆u + f (u), iu)H −1 ,H 1 = 0.
Since u ∈ CI H 1 and ∂t u ∈ CI H −1 , we have
d
ku(t)k2L2 = 2 Re(∂t u, u)
dt
in the weak sense. Together with u(t) ∈ CI L2 , we get ku(t)kL2 is constant.
For the energy, formally
d
E(u(t)) = (−∆u + f (u), ∂t u) = (i∂t u, ∂t u) = 0.
dt
This is not rigorous since ∂t u 6∈ L2 . We will prove the conservation of energy by approximation. There are many ways to approximate the problem. To be suitable for a wide class
of nonlinearities, we use
Iε = (1 − ε∆)−2 , 0 < ε 1.
They are bounded on various Sobolev spaces and converge to identity as ε → 0. For Rd
case they are Bessel potentials, see [3, 31].
Let fε (w) := Iε f (Iε w) and Gε (w) = G(Iε w). We have fε0 = Gε . Let uε be the solutions
of
i∂t uε + ∆uε = fε (uε ), uε (0) = Iε u0 .
Since uε (0) → u0 in H 1 , uε are uniformly bounded in C(J, H 1 ) for ε sufficiently small
and J = [−T, T ] for some T > 0. Since uε ∈ C(J, H 1 ), fε (uε ) ∈ C(J, H 2 ), and thus
uεR ∈ C(J, H 2 ). Thus ∆u, ∂t u ∈ C(J, L2 ), the above formal computation for Eε (uε ) =
1
|∇uε |2 dx + Gε (uε ) is rigourous, and we have Eε (uε (t)) = Eε (uε (0)) for t ∈ J.
2
By difference estimate similar to that in the proof for uniqueness,
kuε − ukStr(J) ≤ aε + C(M )T σ kuε − ukStr(J)
P
where aε = kIε u0 − u0 kL2 + kIε u − ukStr(J) + j kIε fj (u) − fj (u)k
a0
b0
j (J,L j )
L
as ε → 0, we conclude uε → u in Str(J) for T sufficiently small.
. Since aε → 0
CHAPTER 2. LOCAL WELLPOSEDNESS
23
For any r ∈ [2, 2∗ ) for d ≥ 2 and r ∈ [2, ∞] for d = 1, we have
kuε − ukL∞ Lr ≤ kuε − ukθL∞ L2 kuε − uk1−θ
L∞ H 1
r
for some θ ∈ (0, 1]. Thus uε → u in L∞
J L .
R1 d
P R1
Since G(u)−G(v) = 0 ds G(sv +(1−s)u) ds = j 0 (fj (sv +(1−s)u), v −u)
b0
L j ×Lbj
ds,
by (F1) and (F2),
|G(u) − G(v)| ≤ C(M )ku − vkL2 ∩Lr ,
∀kukH 1 + kvkH 1 ≤ M.
(2.5)
Since Iε uε → u in L∞ L2 ∩ L∞ Lr , we conclude Gε (uε ) = G(Iε uε ) → G(u) in L∞ (I).
Since uε and u are uniformly bounded in CJ H 1 and uε → u in CJ L2 , we have ∇uε (t) *
∇u(t) weakly in L2 and thus ku(t)kL2 ≤ lim inf ε→0 kuε (t)kL2 for all t ∈ J. Together with
Gε (uε ) → G(u),
E(u(t)) ≤ lim inf Eε (uε (t)) = lim inf Eε (uε (0)) = E(u(0)).
ε→0
ε→0
Switching the roles of t and 0, we also have E(u(0)) ≤ E(u(t)). Thus E(u(t)) = E(u(0))
for all t ∈ J. By iteration this is true for all t ∈ (−Tmin , Tmax ).
Proof of Theorem 2.8 (ii):
Since uk and u are uniformly bounded in CJ H 1 and uk → u in CJ L2 , ∇uε (t) * ∇u(t)
weakly in L2 . Since
E(uk (t)) = E(uk (0)) → E(u(0)) = E(u(t))
and G(uk (t)) → G(u(t)), we have k∇uk (t)kL2 → k∇u(t)kL2 . Since weak convergence plus
norm convergence is equivalent to strong convergence, we get uk → u in C(J, H 1 ).
By induction, we get convergence in C(Jk , H 1 ) where Jk = [−kT, kT ] ∩ I. Note we can
take a uniform T for all k since T ≤ T1 (2M ) and M = kukL∞ (I,H 1 ) .
Remark. We have shown local wellposedness assuming that f satisfies (F1), (F2) and
(F3). In fact, it suffices to assume
1. (F1) and (F3) without (F2), see [6, Th.4.3.1], or
2. f (u)(x) = f1 (u(x)) where f1 ∈ C 1 (C → C) considered as a function R2 → R2 ,
f1 (0) = 0, and |f1 (u) − f1 (v)| ≤ C(1 + |u| + |v|)p−1 |u − v| with 1 ≤ p < pmax , see [6,
Th.4.4.1]. This includes the example f (u) = u2 which does not satisfy (F3).
In the following we summarize the local wellposedness results for common f .
Theorem 2.10 Suppose
f (u) = f1 (u) + V u + (W ∗ |u|2 )u
where
1. f1 ∈ C(C → C), f1 (0) = 0, |f1 (u) − f1 (v)| ≤ C(1 + |u| + |v|)p−1 |u − v| for some
u
p ∈ [0, pmax ) and f1 (u) = f1 (|u|) |u|
;
2. V ∈ Lp + L∞ (Rd → R), p ≥ 1, p > d/2;
3. W ∈ Lq + L∞ (Rd → R), q ≥ 1, q > d/4, W is even.
Then the problem (2.1) is locally wellposed in H 1 . Furthermore, if u0 ∈ H 2 gives a strong
H 1 solution u in a maximal time interval I, then u ∈ C(I, H 2 ).
The above f satisfies (F1) and (F3). It does not satisfy (F2) unless one further assumes
integrability of ∇V . See [6, Th.4.3.1] and [6, Th.5.3.1].
CHAPTER 2. LOCAL WELLPOSEDNESS
2.4
24
Problems and notes
Problems
¯ L2 ) such that u(t) = ũ(t) for
6. If u ∈ L∞ (I, H 1 ) ∩ W 1,∞ (I, H −1 ), there is ũ ∈ C(I,
a.e. t.
7. If the problem (2.1) is locally well posed, the continuous dependence property implies
that the function Tmax (u0 ) from H 1 to (0, ∞] is lower semi-continuous.
8. Find sufficient conditions on the potentials V and W such that f (u) = V u+(W ∗|u|2 )u
satisfies the assumptions (F1) and (F2).
9. Prove Lemma 2.3 for the case of weak solutions.
10. Suppose f = f1 + · · · + fk satisfies assumptions (F1), (F2) and
k∇fj (u) − ∇fj (v)kLrj ≤ C(M )ku − vkW 1,rj
if kukH 1 + kvkH 1 ≤ M.
(2.6)
Show that
k∇un − ∇ukStr ≤ Ckun0 − u0 kH 1 + C(M )kun − ukStr + C(M )T σ k∇un − ∇ukStr
and derive continuous dependence in H 1 . Show that f (u) = u2 for d ≤ 5 satisfies
assumptions (F1), (F2) and (2.6).
Remark. Equation (2.6) almost needs C 2 estimate of f and is too strong to assume
for general f . It is ok if f is a polynomial of u and ū.
Notes
In preparing these notes, I have used §4.2–§4.4 of [6], part of which is based on Kato
[17]. I also used [6, §5.3] for the last theorem.
For f (u) = λ|u|p−1 u one also has local wellposedness in the borderline case p = pmax for
d ≥ 3 with some refined statement, see [6, §4.5].
Chapter 3
Global phenomena
In this chapter we start our study of the global in time behaviour of solutions of a nonlinear
Schrödinger equation
i∂t u + ∆u = f (u), u(0) = u0 .
(3.1)
We will show that some solutions exist for all time, while some solutions blow up in finite
time. We will then introduce nonlinear solitary waves and their relevance to the global
existence problem.
3.1
Global existence
In this section we give two theorems which assert the global existence for some nonlinearities
and initial values.
Theorem 3.1 (Global existence from energy bound) Suppose f satisfies (F1)–(F3)
and there are some ε > 0, A > 0, and C(A) > 0 so that
−G(u) ≤
1−ε
k∇uk2L2 + C(A),
2
∀u ∈ H 1 ,
kukL2 ≤ A.
(3.2)
Then for any u0 ∈ H 1 with ku0 kL2 ≤ A, the solution u of (3.1) is global and u ∈ L∞ (R, H 1 ).
Note (3.2) only assumes a lower bound of G(u), not any upper bound.
The proof is simple: It follows from (3.2) that
ε
k∇uk2L2 ≤ E(u) + C(kukL2 ) = E(u0 ) + C(ku0 kL2 ),
2
which is a uniform-in-time upper bound of kukḢ 1 and implies the conclusion.
Example. Suppose f (u) = λ|u|p−1 u with λ = ±1 and 1 < p < pmax . We have
Z
1
λ
E(u) =
|∇u|2 +
|u|p+1 dx.
2
p+1
When λ = 1 (defocusing, repulsive case), −G(u) ≤ 0. Thus (3.2) is trivially satisfied
and all H 1 -solutions are global. We also see from the formula of E(u) that k∇u(t)k2L2 are
uniformly in t bounded.
25
CHAPTER 3. GLOBAL PHENOMENA
26
When λ = −1 (focusing, attractive case), the two terms in E(u) may both grow to
infinity while E(u) is constant. To bound −G(u) from above we use Gagliardo-Nirenberg
inequality
Z
Z
Z
|u|p+1 dx ≤ Cp (
|∇u|2 dx)a · (
|u|2 dx)b .
(3.3)
Rd
Rd
Rd
It can be derived from Sobolev inequality and optimizing the scaling parameter. By considering the dimensions of u and x, we get p + 1 = 2a + 2b and d = (d − 2)a + db. Thus
a = d4 (d − 1). If a < 1, we can bound it by Young’s inequality
Z
Z
−G(u) ≤ ε
|∇u|2 dx + C(p, ε)(
|u|2 dx)b/(1−a) ,
Rd
Rd
which gives (3.2). The condition a < 1 is equivalent to
p < pc ,
pc :=
4
+ 1.
d
Note pc ∈ (1, pmax ). Thus all solutions are global in the subcritical cases, 1 < p < pc . What
if pc ≤ p < pmax ? In that case, some solutions exist for all time while some solutions blow
up in finite time, see Theorems 3.2 and 3.4.
We usually say pc is L2 -critical exponent and pmax is H 1 -critical exponent. It is related to
the scaling. If u is a solution of i∂t u + ∆u = |u|p−1 u, then so is uλ (t, x) = λ2/(p−1) u(λ2 t, λx)
for any λ > 0. Under this scaling, kuλ kL2 is constant in λ if p = pc , and k∇uλ kL2 is constant
in λ if p = pmax .
We next show global existence of small solutions in the critical and supercritical cases
p ≥ pc .
Theorem 3.2 (Global existence of small solutions) Suppose f ∈ C(C → C), f (0) =
0, and for some pc ≤ m ≤ p < pmax ,
|f (u) − f (v)| ≤ C(|u|m−1 + |u|p−1 + |v|m−1 + |v|p−1 )|u − v|
∀u, v ∈ C.
There is a small ε0 > 0 such that, if u0 ∈ H 1 and ku0 kH 1 ≤ ε0 , then the solution u of (3.1)
is global.
The condition pc ≤ m ≤ p means f has to be sufficiently nonlinear. In particular,
f (u) = V u is not included. It also forbid a source term. It ensures that the nonlinearity
is negligible where the wave is small. In this theorem, the solution is small in H 1 to start
with. Thus its behavior is similar to that of a linear equation.
In our proof below we will use the following form of chain rule.
Lemma 3.3 (Chain Rule) Let F : C → C satisfy F (0) = 0 and, for some L > 0, α ≥ 0,
|F (u) − F (v)| ≤ L(|u|α + |v|α )|u − v| for all u, v ∈ C. Let 1 ≤ p, q, r ≤ ∞ be such that
1
α
1
p
q
α
r = p + q . Suppose u ∈ L (Ω) and ∇u ∈ L (Ω). Then |∇F (u)| ≤ 2L|u| |∇u| a.e.,
r
α
∇F (u) ∈ L (Ω), and k∇F (u)kLr ≤ 2LkukLp k∇ukLq .
We will not prove this lemma. Those unfamiliar with it may simply assume f ∈ C 1 .
Proof of Theorem 3.2 Fix a cut-off function θ ∈ Cc1 (C → R) with θ(u) = 1 for
|u| ≤ 1. Let f1 (u) = θ(u)f (u) and f2 (u) = (1 − θ(u))f (u). Let p1 = m, p2 = p and
rj = pj + 1. Then
fj (0) = 0,
|fj (u) − fj (v)| ≤ C(|u| + |v|)pj −1 |u − v|,
j = 1, 2.
CHAPTER 3. GLOBAL PHENOMENA
27
Clearly f = f1 + f2 satisfies (F1). If we furthers assumes f ∈ C 1 , then f satisfies
|fj (u)| + |∇fj (u)| ≤ C|u|pj −1 (|u| + |∇u|),
(3.4)
and hence (F2). This ensures unique local existence in H 1 . However, even without assuming
f ∈ C 1 , we still have (F2) because Lemma 3.3.
Let
P
φ(t) = kukL∞ ((0,t),H 1 ) + j kukLqj ((0,t),W 1,rj ) .
Applying Strichartz estimate to the Duhamel’s formula, we get
P
φ(t) ≤ C1 ku0 kH 1 + C j kgj k qj0
1,r 0 .
j
L ((0,t),W
By (3.4) and pj rj0 = rj , kgj k
kgj k
where
1
qj0
=
pj −1
aj
+
1
qj .
q0
1,r 0
W j
L j ((0,t),W
)
p −1
≤ CkukLjrj kukW 1,rj . Thus
1,r 0
j)
p −1
≤ CkukLjaj ((0,t),Lrj ) kukLqj ((0,t),W 1,rj ) ,
Since H 1 ⊂ Lrj and W 1,rj ⊂ Lrj , we have kukLaj ((0,t),Lrj ) ≤ Cφ(t)
if qj ≤ aj ≤ ∞. The condition aj ≤ ∞ is trivial. The condition qj ≤ aj is equivalent to
pj ≥ pc . Thus, if pc ≤ p1 ≤ p2 < pmax , we have
φ(t) ≤ C1 ku0 kH 1 + C2 φ(t)m + C2 φ(t)p .
Take ε0 > 0 so small that 2C2 (4C1 ε0 )m−1 ≤ 1/2. If ku0 kH 1 ≤ ε0 , then φ(t) ≤ 4C1 ku0 kH 1
for t ∈ [0, T ] for some T > 0 since limt→0+ φ(t) = ku0 kH 1 and C1 ≥ 1. The above inequality
shows φ(t) ≤ C1 ku0 kH 1 + 12 φ(t), and thus φ(t) ≤ 2C1 ku0 kH 1 for t ≤ T . By continuity of
φ(t), one has φ(t) ≤ 2C1 ku0 kH 1 for all t > 0.
3.2
Finite time blowup
In this section we give an example of f and initial data that lead to finite time blowup.
Define the Hilbert space
Σ = {u ∈ H 1 : xu ∈ L2 },
(3.5)
equipped with the inner product (u, v)Σ = (u, v)H 1 + (xu, xv)L2 . The following theorem
provides unique existence in Σ. (We still need continuous dependence on data to be locally
wellposed.)
Theorem 3.4 Suppose f ∈ C(C → C), f (0) = 0, |f (u)−f (v)| ≤ C(1+|u|+|v|)p−1 |u−v| for
u
. Suppose u0 ∈ Σ and let u be the strong H 1 -solution
some p ∈ [0, pmax ) and f (u) = f (|u|) |u|
R
of (3.1) on a maximal time interval I. Then u(t) ∈ C(I, Σ), φ(t) = |x|2 |u(x, t)|2 dx ∈
C 2 (I), and
Z
Z
φ0 (t) = 4 Im
ūx · ∇u dx, φ00 (t) = 16E +
[4df (u)ū − 8(d + 2)F (u)] dx,
Rd
where F (t) =
R |t|
0
f (s)ds and E = E(u) =
Rd
R
1
2
2 |∇u|
+ F (u) dx = E(u0 ).
CHAPTER 3. GLOBAL PHENOMENA
28
[Insert formal computation]
Note φ0 has the same formula for any f satisfying Im f (u)ū = 0.
2
To make it rigorous, the proof introduces a factor e−ε|x| , ε → 0+ , in the integrals and
assumes u0 ∈ H 2 to start with.
In the case f (u) = λ|u|p−1 u,
Z
4d
00
φ = 16E +
λ(p − pc ) |u|p+1 dx.
(3.6)
p+1
Corollary 3.5 Let f (u) = −|u|p−1 u with pc ≤ p < pmax . Suppose u0 ∈ H 1 ∩ L1,1 with
E(u0 ) < 0, then Tmin (u0 ) < ∞ and Tmax (u0 ) < ∞.
Proof. In view of (3.6), when pc ≤ p < pmax , we have φ00 ≤ 16E for all t. Thus
φ(t) ≤ g(t) = 8Et2 + (φ0 (0) − 16E)t + φ(0). If E < 0, then φ(t) = 0 for some t ≤ T where T
is the positive zero of g(t). This contradicts the definition of φ(t) ≥ 0 and thus Tmax ≤ T .
Similarly Tmin < ∞.
Summarizing, for f (u) = λ|u|p−1 u with 1 < p < pmax , we have
1. If 1 ≤ p < pc : GWP in H 1 and Σ (and L2 );
2. If pc ≤ p < pmax : GWP in H 1 and Σ if small or if λ > 0.
Note that GWP in Σ follows from GWP in H 1 and Theorem 3.4.
3.3
Solitary waves
For linear Schrödinger equations with potentials,
i∂t u = (−∆ + V )u,
solutions can be decomposed as the sum of two kinds of solutions: If u0 is an eigenfunction φ
with eigenvalue λ, then u(t) = e−iλt φ which is a standing wave (bound state). If u0 belongs
to the continuous spectral subspace, then u is dispersive (radiation) and satisfies the decay
and Strichartz estimates. For nonlinear equations, if the solution is global in time, then a
similar picture is conjectured to be also true.
Solitary waves are special examples of nonlinear bound states, which, roughly speaking,
are solutions that are spatially localized for all time. There are several kinds of solitary
waves, including
1. standing waves
2. traveling waves, including single-soliton and multi-soliton
3. kinks
4. cnoidal waves
5. breathers
CHAPTER 3. GLOBAL PHENOMENA
29
[Show animations]
Example 1. Consider the Hartree equation
i∂t u + ∆u = (W ∗ |u|2 )u.
(3.7)
Here W is a real, even potential, with suitable regularity and spatial decay. Its energy is
Z
1
1
E(u) =
|∇u|2 + (W ∗ |u|2 )|u|2 .
2
4
If W is negative, the following minimization problem
min
kukL2 =N
E(u)
is attained at some Q 6= 0 if N is large enough. It satisfies the Euler-Lagrange equation
−∆Q + (W ∗ |Q|2 )Q = −ωQ
where ω is the Lagrange multiplier and can be shown to be positive. It gives rises to a
standing wave solution u(t, x) = Q(x)eiωt . We will further assume W is radial, which
makes Q also radial.
The set of all solutions of (3.7) is invariant under the symmetries of translation (in x, t),
rotation, phase, Galilean transform: If u(t, x) is a solution, then so is
Rx · v tv 2
ũ(t, x) := u (t, Rx − tv − x0 ) exp i
−
+ γ0
2
4
for any constant x0 , v ∈ Rn , γ0 ∈ R and R ∈ O(n). If W is radial, then so is Q. The
solution Q(x)eiωt then gives a family of solitary waves
v2
x·v
+ (ω − )t + γ0
Q(x − tv − x0 ) exp i
2
4
with parameters x0 , v, γ0 and N (or ω).
Example 2. Consider the focusing NLS with power nonlinearity
i∂t u + ∆u = −|u|p−1 u.
(3.8)
The set of all solutions of (3.8) is invariant under the symmetries of translation, rotation,
phase, Galilean transform and scaling: if u(t, x) is a solution, then so is
λ2/(p−1) u(λ2 t, λx),
λ > 0.
Related to these symmetries, we have the following conserved quantities:
R
R
1
|ψ|p+1 dx.
1. the energy (or Hamiltonian): E[ψ] = 21 |∇ψ|2 dx − p+1
R
2. the charge: |ψ|2 dx.
R
3. the momentum: P~ [ψ] = Im ψ̄∇ψ dx.
R
4. angular momentum: − Re ψ̄ x × ∇ψ dx.
(If d 6= 3, then ...)
CHAPTER 3. GLOBAL PHENOMENA
5. center of mass:
R
x|ψ|2 dx = εx0 + 2t Im
30
R
ψ̄∇ψ dx = ~x0 + 2P~ t.
When p = pc , there is an additional symmetry called the “pseudo-conformal transform”
to be given in (3.13).
We are interested here in solutions of (3.8) of the form
ψ(t, x) = Q(x) eit
(3.9)
where Q(x) must therefore satisfy the nonlinear elliptic equation
∆Q + |Q|p−1 Q = Q.
(3.10)
Any such solution generates a family of solitary waves by the above-mentioned symmetries.
We give a list of properties of solutions of (3.10):
1. Pohozaev identity: Suppose Q ∈ H 1 ∩ Lp+1 (Rd ), 1 < p < ∞, is a Rsolution of R(3.10).
Testing (3.10) with Q̄ and x.∇Q̄ and taking real parts, one gets 12 |∇Q|2 : 21 |Q|2 :
R
1
|Q|p+1 = a : b : 1 where a = d(p−1)
and b = d+2−(d−2)p
. Since they are all
p+1
4
4
positive, it is necessary that p < pmax . Thus we may assume p ∈ (1, pmax ).
2. Regularity and decay: For d ≥ 1 and 1 < p < pmax , if Q(x) ∈ H 1 (Rd ) is a solution of
2 and |Q(x)| + |∇Q(x)| ≤ Ce−ε|x| for some C, ε > 0.
(3.10), then Q ∈ Cloc
3. For all d ≥ 1 and 1 < p < pmax , there is a unique positive radial H 1 -solution of (3.10),
denoted Q, which further satisfies Q0g (r) < 0 for r > 0. Any positive solution of (3.10)
is Qg (x − x0 ) for some x0 .
Any solution of (3.10) obtained from Q via the symmetries of translation and phase is
called a nonlinear ground state Any other solution is called a nonlinear excited state.
4. For d = 1, there is no excited state. In fact, the ODE uxx + |u|p−1 u = u can be solved
explicitly and Q(x) = a sech2/(p−1) (bx − c) for some a, b, c.
5. For d ≥ 2 there are many excited states. In fact, there are countably infinitely
many radial real solutions, each with exactly k positive zeros as a function of |x|. In
addition, there are also non-radial (and complex-valued) solutions, for example those
with non-zero angular momenta,
d = 2,
d = 3,
Q = φ(r) eimθ ,
Q = φ(r, x3 ) eimθ ,
in polar coordinates r, θ;
in cylindrical coordinates r, θ, x3 ,
and similarly defined for d ≥ 4. The uniqueness question of these solutions with
m, k ≥ 1 is open. Also, we do not know if there are “breather” solutions, analogous
to those of the generalized KdV equations.
For d = 1, the existence and uniqueness can be shown by solving the ODE directly. For
d ≥ 2, the existence is usually obtained by variational methods, for example,
1. minR u2 =N E(u): ok for p < pc since the inf is −∞ for p > pc or p = pc with N > Nc .
2. minM (u)=0 S(u), where M (u) =
R
|∇Q|2 −
d(p−1)
2(p+1)
R
|u|p+1 . This works for p > pc .
CHAPTER 3. GLOBAL PHENOMENA
31
a
b
2
( |∇u|
) ( |u|2 )
R
.
3. minu6=0 J(u), where J(u) is the Gagliardo-Nirenberg quotient J(u) =
|u|p+1
Here a, b can be solved from dimension analysis, and happen to be the same as those
in the Pohozaev identity mentioned above.
R
R
The uniqueness is obtained by first showing that any such a solution is radial with
respect to some center x0 (Gidas-Ni-Nirenberg), and then that the solution of the resulting
ODE is unique (Kwong).
In the following we explain more the Gagliardo-Nirenberg quotient approach. Let
σ≡
inf
06=u∈H 1 (Rd )
J(u).
We have σ > 0 by the Gagliardo-Nirenberg inequality. Note that J(u) is invariant under
scaling u → µu(λx). We will assume the existence of a minimizer.1 A minimizer satisfies
the Euler-Lagrange equation
−2a∆u
2bu
p+1
δJ(u)
=R
+ R 2 − R p+1 |u|p−1 u = 0.
2
J(u)
|∇u|
|u|
|u|
(3.11)
By the scaling invariance of J(u), we may choose some µ, λ > 0 so that Q(x) = µu(λx) is
still a minimizer and satisfies (3.10).
The best (smallest) constant of the Gagliardo-Nirenberg inequality is σ −1 = J(Q)−1 .
The problem of characterizing best constant has wide extension such as adding weights and
considering it on manifolds, and is under active research.
In the critical case p = 1 + 4/d, we have a = 1, b = 2/d, and
Z
Z
Z
|u|2+4/d ≤ σ −1 |∇u|2 ( |u|2 )2/d .
−4/d
By Pohozaev identity σ −1 = J(Q)−1 = (1 + d2 ) kQk2 . Thus
2/d
Z
Z
Z
1
1
2
−4/d
|∇u|2 −
|u|2
E(u) ≥
(1 + ) kQk2
|∇u|2
2
2 + 4/d
d
Z
R 2 R 2 2/d 1
=
|u| / Q
.
|∇u|2 1 −
2
(3.12)
As a corollary, let u(t) be a solution of the critical NLS i∂Rt u + ∆u +
|u|4/d u = 0 with
R
ku(0)kL2 < kQkL2 . From (3.12), we get E = E(u(t)) > 0, δ = |u(t)|2 / Q2 < 1 and
k∇u(t)k2L2 ≤ 2E/(1 − δ 2/d ).
This uniform bound shows the solution is global. This result is sharp since we have an
explicit blowup solution whose L2 -norm is the same as kQkL2 .
Example 3. Generalized KdV: u(t, x) : R1+1 → R,
ut + (uxx + |u|p−1 u)x = 0
Here 1 < p < ∞. It has a traveling wave solution u(t, x) = Q(x − t) where Q is the same
solution of (3.10) in 1D. The equation has the symmetries of translation and scaling. It has
a two-parameter family of traveling waves of the form
u(t, x) = λ2/(p−1) Q(λx − λ3 t − γ),
λ > 0,
γ ∈ R.
The speed of these traveling waves are always positive.
1
Sketch of proof: By Schwarz symmetrization, we may assume the minimizing sequence is radial. Since
1
the imbedding Hrad
⊂ Lp+1 is compact, one gets a nontrivial limit.
CHAPTER 3. GLOBAL PHENOMENA
3.4
32
Solitary waves and blow-up
In the following we explain the connection of solitary waves to blow-up. We first give a
blowup result for the special case f (u) = −|u|p−1 u.
Theorem 3.6 (Strong instability for power nonlinearity) Assume f (u) = −|u|p−1 u
and pc < p < pmax . Let Q be the unique radial ground state satisfying −∆Q + f (Q) = −Q.
There exist φn → Q in H 1 such that Tmax (φn ) < ∞ for each n.
R
Sketch of proof. Let u be the solution with u(0) = φn and φ(t) = |x|2 |u(t, x)|2 dx.
If p = pc , recall φ̈(t) = 16E. Take φn = (1 + n1 )Q. Then E(φn ) < 0.
R p+1
R
|u| . One has
If p > Pc , then φ̈(t) = 8M (u(t)) where M (u) = |∇u|2 − d(p−1)
2(p+1)
E(Q) > 0 and M (Q) = 0 as Q gives a solitary wave. It can be also verified easily.
Take φn (x) = λd/2 Q(λx), where λ = 1 + n1 . Then N (φn ) = N (Q) and E(φn ) > 0 for
n 1. Finite time blowup follows from the following claim.
Claim: M (u(t)) ≤ −δ, where δ = E(Q) − E(φn ) > 0.
For what follows we limit ourselves to the critical focusing case p = pc ,
i∂t u + ∆u + |u|4/d u = 0.
When p = pc , there is an additional symmetry called the “pseudo-conformal transform”:
t
x
ib|x|2
−d/2
u(
ub (t, x) = (1 + bt)
exp
,
), b ∈ R.
(3.13)
4(1 + bt)
1 + bt 1 + bt
The pseudo-conformal symmetry (3.13) is related to the Virial identity: By (3.6), φ00 =
16E when p = pc , and thus
Z
|x|2 |u(x, t)|2 dx = c0 + c1 t + 8Et2 ,
(3.14)
R
R
with c0 = |x|2 |u0 |2 dx and c1 = 4 Im ū0 x · ∇u0 dx.
If we apply pseudo-conformal transform (3.13) to the solitary wave Q(x)eit with parameter b = −1, we get
−i|x|2
it
x
−d/2
U (t, x) = (1 − t)
exp
+
Q(
).
(3.15)
4(1 − t) 1 − t
1−t
This solution blows up at t = 1, and k∇U (t)kL2 ∼ (1 − t)−1 . Also note U (0, x) =
2
e−i|x| /4 Q(x). Here Q be can be either the ground state or an excited state.
Consider initial data u0 = λQ, λ > 0, and Q is the ground state. We have
• λ = 1: u(t, x) = Q(x)eit .
• λ > 1: Note that E(Q) = 0 by Pohozaev identities, thus E(λQ) < 0. The solution
blows up in finite time.
• λ < 1: ku0 kL2 < kQkL2 . It can be shown that u is global (Theorem ??? later). In
fact, u scatters (Kenig-Merle?)
The above show that the solution Q(x)eit is highly unstable.
Finally we state a result by Merle-Raphael.
CHAPTER 3. GLOBAL PHENOMENA
33
Theorem 3.7 (Stable singularity formation) Let d = 2 and u be the solution of i∂t u+
∆u + |u|2 u = 0 with u(0) = u0 . If u0 ∈ H 1 , E(u0 ) < 0 and kQkL2 < ku0 kL2 < kQkL2 + α∗
where 0 < α∗ 1. Then T = Tmax (u0 ) < ∞ and
u(t, x) =
1
x − x(t) iγ(t)
Q(
)e
+ ũ(t, x)
λ(t)
λ(t)
for some continuous functions λ(t) > 0, x(t) ∈ Rd , γ(t) ∈ R with
lim λ(t)
T −t
log | log(T − t)|
−1/2
exists, > 0,
x(t) → x(T ), and ũ → u∗ in L2 , as t → T− . Here u∗ 6∈ H s for any s > 0.
The above blowup rate was conjectured for a long time based on asymptotic analysis and
numerical computation. In the above, the assumptions on u0 are stable under perturbation.
Thus there is an open set of initial data whose corresponding solutions blow-up in the same
rate. This is call generic. On the other hand, the blowup rate t−1 of (3.15) is nongeneric.
Similar analysis was done by Martel-Merle for the critical generalized KdV
ut + (uxx + u5 )x = 0.
3.5
Problems and notes
Problems
11. Solve all bounded real solutions of the ODE u00 + |u|p−1 u = u.
12. Derive the Pohozaev identity formally. Those venturous can try to prove it rigorously
for Q ∈ H 1 ∩ Lp+1 .
13. When p = pc , what can one say about u from the conservation of energy and momentum of ub ?
Notes
In preparing these notes, I have used [6, §6.1–6.2] for section 3.1, [6, §6.5] for section
3.2, and ??? for sections 3.3 and 3.4.
Chapter 4
Scattering theory
In this chapter we study the scattering properties of solutions of the nonlinear Schrödinger
equation
i∂t u + ∆u = f (u), u(0) = u0 .
(4.1)
We will focus on the model nonlinearity f (u) = λ|u|p−1 u with λ = ±1 and 1 < p < pmax .
4.1
Introduction
Let u be a global solution of (4.1) in X, where X denotes one of H 1 , L2 , or Σ. Recall
Σ = {u ∈ H 1 : xu ∈ L2 } is defined in (3.5). We say that u scatters if the limit
u± := lim S(−t)u(t)
t→±∞
(4.2)
exist in X. Note that, in the cases X = H 1 or H = L2 , Eqn. (4.2) is equivalent to
lim ku(t) − S(t)u± kX = 0,
t→±∞
(4.3)
which is closer to the physical intuition. However, this is not true when X = Σ, since S(t)
preserves the space Σ but not Σ-norm.
Denote R+ the set of all u0 so that the limit u+ = limt→+∞ S(−t)u(t) exists, U+ the
map from u0 to u+ , and U+ = U+ R+ . Denote R− , U− , U− similarly.
There are two basic questions in scattering theory:
1. For which u0 ∈ X does the limit u+ exists? If this holds for all u0 ∈ X, we say the
problem (4.1) is asymptotic complete.
2. For which u+ ∈ X does there exist a u0 ∈ X so that U+ u0 = u+ ? The map from u+
to u0 is called the wave operator. This is the question of existence of wave operator.
Denote the wave operator by W± . The scattering operator S : U− → U+ , defined by
S = W− U+ , is physically significant. Its basic properties can be read from those of U± and
W± and will not be discussed here.
The second question is equivalent to the existence question of the final value problem
i∂t u + ∆u = f (u),
lim S(−t)u(t) = u+ .
t→+∞
34
(4.4)
CHAPTER 4. SCATTERING THEORY
35
It is often easier, since it put asides uniqueness issue, and also avoids the blowup question
for the initial value problem (4.1).
In this chapter we will study these questions for the model nonlinearity f (u) = λ|u|p−1 u
with 1 < p < pmax . The following is a table of known results in the case d ≥ 2. Note
2/d ≤ 4/(d + 2) < p0 − 1 < 4/d if d ≥ 2.
Range of p
Defocusing λ > 0
Focusing λ < 0
No scattering
No scattering
L2 -scattering with Σ data
No Scattering
SW in Σ if small
SW in Σ if small
(p0 , pmax )
AW in Σ
SW in Σ if small
(pc , pmax )
AW in H 1
SW in H 1 if small
(1, 1 + d2 ]
(1 + d2 , 1 +
(1 +
4
d+2 )
4
d+2 , p0 ]
Above, p0 is the positive root of dp(p − 1) = 2(p + 1).
S means scattering.
A means asymptotic completeness.
W means existence of wave operator.
4
When p = 1 + d+2
, there is still L2 -scattering for Σ-data when λ > 0. I don’t know
what happens with λ < 0.
For d = 1 and λ > 0: No scattering for p ≤ 3, SW in Σ for p > 3, SW in H 1 for p > 5.
For d = 1 and λ < 0: ??
We now give some heuristics on a few exponents. Our equation can be considered as a
linear Schrödinger equation with a time dependent potential
i∂t u = −∆u + V u,
V = λ|u|p−1 .
The case V ∼ |x|−s (for |x| large) is called long range for s ≤ 1, and short range for s > 1.
Being short range means that the potential is not felt when a particle is far away from the
center of the potential, and is a necessary condition for scattering even when V is timeindependent. The borderline is V ∼ |x|−1 ∈ Ldwk (the Coulomb potential), which in the
case V = |u|p−1 means p − 1 = 2/d.
Another borderline case is V ∼ |x|−2 ∼ ∆, or V ∈ Ld/2 , which corresponds to the case
p = pc .
R ∞ − 2 (p−1)
The exponent p0 arises from an integral t s q−2
ds which converges iff p > p0 .
1
Here q is given by 2q = d( 12 − p+1
), and p = p0 iff q = 2p.
In the focusing case, since solitary waves exist for all p ∈ (1, pmax ) and they do not
scatter, the only chance to get scattering is to restrict ourselves to a class which excludes
solitary waves. One usual class is the class of small solutions, and such a result is called
low energy scattering. Denote Qλ (x) = λβ Q(λx), β = 2/(p − 1). For low energy scattering
in X, it is necessary that
inf kQλ kX > 0.
λ>0
Since
kQλ kL2 ∼ λm ,
k∇Qλ kL2 ∼ λm+1 ,
kxQλ kL2 ∼ λm−1 ,
m = β − d/2,
CHAPTER 4. SCATTERING THEORY
36
We have
1. inf kQλ kL2 > 0 iff m = 0, i.e., p = pc ,
2. inf kQλ kH 1 > 0 iff m ≤ 0 ≤ m + 1, i.e., pc ≤ p ≤ pmax ,
3. inf kQλ kΣ > 0 iff m − 1 ≤ 0 ≤ m + 1, i.e.,
4
d+2
+ 1 ≤ p ≤ pmax .
In particular, this shows no scattering for the focusing case when 1 < p < 1 +
4.2
4
d+2 .
Scattering theory in the weighted L2 space
Lemma 4.1 (Pseudoconformal conservation law) Suppose f (u) = λ|u|p−1 u. Let u0 ∈
Σ and u ∈ C(Imax , Σ) be the corresponding maximal solution. Then
k(x +
where C1 =
2it∇)u(t)k2L2
4d
p+1 λ(p
2
+ 8t G(u(t)) =
− pc ) and G(u) =
1
p+1
R
kxu0 k2L2
Z
− C1
t
Z
s
|u|p+1 dxds,
(4.5)
0
|u|p+1 dx.
Proof. Let
h(t) = k(x + 2it∇)u(t)k2L2 + 8t2 G(u(t))
Z
2
= kxu(t)kL2 − 4t4 Im
ūx · ∇u dx + 4t4 k∇u(t)k2L2 + 8t2 G(u(t)).
Rd
Recall Theorem 3.4 and Eq (3.6) that φ(t) = kxu(t)k2L2 is C 2 and
Z
Z
0
00
φ (t) = 4 Im
ūx · ∇u dx, φ = 16E + C1 |u|p+1 dx.
Rd
Thus
h(t) = φ(t) − tφ0 (t) + 8t2 E
is C 1 and
h0 (t) = −tφ00 (t) + 16tE = −C1 t
Integrating in t, we get the lemma.
Z
|u|p+1 dx.
When p = pc , C1 = 0 and this lemma reduces to the Virial identity (3.14).
It should be noted that kxS(t)u0 kL2 ∼ t for general u0 ∈ Σ. [Insert picture] One can
replace x by t in estimates (the property of finite speed propagation). This is in contrast to
the above lemma.
In the Lemma, one sees the operator
J ≡ x + 2it∇.
It has the following useful properties.
CHAPTER 4. SCATTERING THEORY
37
1. It satisfies the identity
J = SxS −1 = M (2it∇)M −1 .
(4.6)
ix2
Here S = S(t) = eit∆ and M = M (t) is the multiplication operator exp( 4t ). Note x
and 2it∇ are the two components of J. The second equality is because 2it∇(M −1 ϕ) =
M −1 Jϕ. For the first equality, note ∂t S = S(∂t + i∆) and thus, if ∂t ϕ = 0,
∂t SxS −1 ϕ = S(i∆x − xi∆)S −1 ϕ = S(2i∇)S −1 ϕ = 2i∇ϕ.
Integrating in time we get SxS −1 = x + 2it∇.
Exercise:
x
−1
2t SM
x
= SM −1 (−i∇) and −i∇M S −1 = M S −1 2t
.
2. (Linear part). Note J(i∂t + ∆) = (i∂t + ∆)J because i∂t + ∆ = Si∂t S −1 . Thus,
if u solves (i∂t + ∆)u = f and u(0) = u0 , then w = Ju solves (i∂t + ∆)w = Jf and
w(0) = J(0)u0 = xu0 . In integral form, this is
Z t
Ju(t) = S(t)x0 u − i
S(t − s)(Jf )(s) ds.
(4.7)
0
This can be also derived by applying J(t) to the integral equation of u(t),
J(t)S(t − s) = S(t)xS(−t)S(t − s) = S(t − s)S(s)xS(−s) = S(t − s)J(s).
3. Since J = M (2it∇)M −1 , we have |Ju| = |2t∇v| where v = M −1 u. Thus the left hand
side of Lemma 4.1 is equal to 8t2 E(v(t)).
4. (Nonlinear part). More generally, if F (u) satisfies F (u) = F (|u|)u/|u|, then JF (u) =
M 2it∇F (v) and |JF (u)| = |2t∇F (v)| where F is differentiable. If F : C → C satisfies
F (0) = 0 and |F (u) − F (v)| ≤ C(|u| + |v|)r−2 |u − v| for some r ∈ [2, 2∗ ) (or r ∈ [2, ∞)
if d = 1), then by Chain rule Lemma 3.3,
r−2
r
r
kJF (u)kLr0 = Ctk∇F (v)kLr0 ≤ Ctkvkr−2
Lr k∇vkL = CkukLr kJukL .
(4.8)
From now on we consider the defocusing case λ > 0.
Lemma 4.2 (Lp+1 -decay) Suppose f (u) = λ|u|p−1 u with λ = 1 (defocusing) and 1 < p <
pmax . For any solution u of (4.1) with u(0) = u0 ∈ Σ, we have
ku(t)kLp+1 ≤ C|t|−2/q ,
1
1
2
= d( −
).
q
2 p+1
1
1
Remark. One can prove ku(t)kLr ≤ C|t|−d( 2 − r )(1−ε) for any Lr ⊃ H 1 . Here ε =
(r−p−1)(4+d−dp)
(r−2)(2+d+2p−dp) ∈ (0, 1) if p ∈ (1, pc ) and r > p + 1. Otherwise ε = 0. These estimates are
the same as those for the free Schrödinger equation if ε = 0.
2
Proof. Let v(t, x) = e−ix /(4t) u(t, x). If pc ≤ p, C1 ≥ 0 and Lemma 4.1 shows
8t2 E(v(t)) ≤ kxu0 k2L2 . Thus k∇v(t)kL2 ≤ Ct−1 . For any r ∈ [2, 2∗ ], we have ku(t)kLr =
kv(t)kLr ≤ Ckv(t)k1−σ
k∇v(t)kσL2 with σ = d( 12 − 1r ).
L2
If p ∈ (1, pc ), C1 < 0 and the lemma says
Z t
8t2 E(v(t)) = kxu0 k2L2 + |C1 |
s−1 h(s) ds
0
CHAPTER 4. SCATTERING THEORY
38
R
where h(s) = t2 |u(s)|p+1 dx. Subtracting from this identity its value at t = 1, we get
Rt
8t2 E(v(t)) = 8E(v(1)) + |C1 | 1 s−1 h(s) ds. Thus
Z t
2
h(t) ≤ (p + 1)t E(v(t)) ≤ C + C2
s−1 h(s) ds, (t > 1)
1
where C2 = 18 (p + 1)|C1 | = d2 (pc − p). By Gronwall inequality h(t) ≤ CtC2 .
1
ku(t)kLp+1 ≤ Ct−σ where σ = −(C2 − 2)/(p + 1) = d( 21 − p+1
).
Thus
In the following three theorems we assume p > p0 .
Theorem 4.3 (Space-time bounds) Suppose f (u) = λ|u|p−1 u with λ = 1 (defocusing)
and p0 < p < pmax . For any solution u with u(0) = u0 ∈ Σ, for any admissible (q, r),
u, ∇u, Ju ∈ C(R, L2 ) ∩ Lq (R, Lr ).
By local wellposedness, we have
u, ∇u ∈ Cloc (R, L2 ) ∩ Lqloc (R, Lr ),
Ju ∈ Cloc (R, L2 ).
It remains to show Ju ∈ ∩Lqloc (R, Lr ) and the global estimates.
Proof. It suffices to prove the case r = p + 1. The general case (including L∞ (R, L2 ))
follows from Strichartz estimates.
Step 1. Show Ju ∈ ∩Lqloc (R, Lr ). I will only show a priori estimate. What we want can
be proved by approximation, which I skip.
Step 2. Show u ∈ Lq (R, W 1,r ).
Step 3. Show Ju ∈ ∩Lq (R, Lr ).
(more later)
Theorem 4.4 (Scattering in Σ) Suppose f (u) = λ|u|p−1 u with λ = 1 (defocusing) and
p0 < p < pmax . For any solution u with u(0) = u0 ∈ Σ, there is a unique u+ ∈ Σ so that
lim kS(−t)u(t) − u+ kΣ = 0.
t→+∞
In addition, ku+ kL2 = ku0 kL2 and
1
2
R
|∇u+ |2 = E(u0 ).
Remark.
u0 → u+ is continuous in Σ.
R The map
1
2
Note 2 |∇u+ | = E0 (u+ ), the energy of the free Schrödinger equation.
Proof. Let v(t) = S(−t)u(t). For 0 < τ < t we have
Z t
v(t) − v(τ ) = −i
S(−s)f (u(s))ds.
τ
Thus, with r = p + 1 and (q, r) admissible,
kv(t) − v(τ )kH 1 . kf (u)kLq0 ((τ,t),W 1,r0 ) . kukp−1q
L q−2
(p−1)
((τ,t),Lr )
· kukLq ((τ,t),W 1,r ) .
The first factor is finite and the second factor goes to 0 as τ → ∞. Thus v(t) converges to
some u+ in H 1 as t → ∞.
CHAPTER 4. SCATTERING THEORY
39
Since J0 S−t = S−t Jt ,
Z
x(v(t) − v(τ )) = −i
t
S(−s))(x + 2is∇)f (u(s))ds.
τ
Thus
kLHSkL2 = kS(t)LHSkL2 ≤ k(x + 2is∇)f (u(s))kLq0 ((τ,t),Lr0 ) .
. kukp−1q
L q−2
(p−1)
((τ,t),Lr )
· kJukLq ((τ,t),Lr ) .
The first factor is finite and the second factor goes to 0 as τ → ∞. Thus xv(t) converges
to xu+ in L2 as t → ∞.
R
The last statement follows from H 1 -convergence and Rd |u(x, t)|p+1 dx → 0.
Theorem 4.5 (Wave operator in Σ) Suppose f (u) = λ|u|p−1 u with λ = 1 (defocusing)
and p0 < p < pmax . For any u+ ∈ Σ, there is a unique u0 ∈ Σ such that the maximal
solution u with u(0) = u0 satisfies limt→+∞ kS(−t)u(t) − u+ kΣ = 0.
Remark. The map u+ → u0 is continuous in Σ.
Proof. Let
kukE(T ) = kukLq ((T,∞),W 1,r ) + kJukLq ((T,∞),Lr ) + sup |t|2/q ku(t)kLr ,
t≥T
and K = kS(t)u+ kE(−∞) . For T 1 to be chosen, let
E = u : kukE = kukE(T ) ≤ 2K .
Rt
Let Φ(u)(t) = −i ∞ S(t − s)f (u(s))ds and A(u)(t) = S(t)u+ + Φ(u)(t). One verifies that,
if u, v ∈ E and T is sufficiently large,
kΦ(u)kE ≤ CT −m+1 K p < K,
kA(u)kE < 2K,
1
kA(u) − A(v)kLq Lr ≤ CT −m+1 K p−1 ku − vkLq Lr ≤ ku − vkLq Lr .
2
2
2
Here m = min( q−2
(p − 1), 2p/q) > 1 since q−2
(p − 1) > 1 iff q < 2p iff p > p0 . Thus A(u)
is a contraction mapping on E and has a fixed point which solves (4.1) for t ∈ (T, ∞). We
can extend the solution to t > 0 by global wellposedness.
Suppose we have two solutions u1 and u2 both converging to u+ as t → ∞, similar
estimates show
1
ku1 − u2 kLq ((T,∞),Lr ) ≤ ku1 − u2 kLq ((T,∞),Lr )
2
if T is large enough. This shows uniqueness.
In the following we state two theorems, whose proofs use the pseudoconformal transform
and are omitted, see [6, §7.5].
Theorem 4.6 (L2 -scattering with Σ data) Suppose f (u) = λ|u|p−1 u, λ = ±1, and u ∈
C(R+ , Σ) is a global solution.
(i) If 1 < p ≤ min(2, 1 + 2/d) and limt→∞ S(−t)u(t) converges in L2 , then u0 = 0.
(ii) If d = 1, 2 < p ≤ 3 and limt→∞ S(−t)u(t) converges in Σ, then u0 = 0.
(iii) If 1 + 2/d < p < pmax and λ > 0, then limt→∞ S(−t)u(t) converges in L2 .
CHAPTER 4. SCATTERING THEORY
40
Theorem 4.7 (Continuity and asymptotic completeness) Suppose f (u) = λ|u|p−1 u,
λ = ±1, and 1 + 4/(d + 2) < p < pmax (3 < p if d = 1). The set R+ of those u0 which
scatter is open in Σ. It is Σ if λ > 0 and p ≥ p0 . The set U+ of those u+ for which the
wave operator is defined is open in Σ. It is Σ if λ > 0, or if λ < 0 and p < pc . The map
U+ : R+ → U+ is bicontinuous bijection in Σ topology.
This theorem gives, in particular, low energy scattering for λ < 0.
When λ > 0 and 1 + 4/(d + 2) < p < p0 , whether R+ = Σ is open.
4.3
Scattering theory in the energy space
In this section we consider the scattering theory in the energy space H 1 (Rd ). In contrast
to the theory in weighted space in which the data is localized and ku(t)kLp+1 has explicit
decay rate, in H 1 -theory the data is not localized and ku(t)kLp+1 will be shown to decay to
0 as t → ∞ but with no rate.
Both theories rely on showing the finiteness of kukLq (R,Lp+1 ) , which in turn depends on
the decay of ku(t)kLp+1 . In the Σ case it was derived from the pseudoconformal conservation
law. In H 1 case it will be derived from the Morawetz estimate.
To make sense of various terms in the proof of the Morawetz estimate, one uses the
following.
Lemma 4.8 (Generalized Hardy inequality) If 0 ≤ q ≤ p < ∞ and q < d, then
Z
|u|p
p q
q
dx ≤ (
) kukp−q
∀u ∈ W 1,p (Rd ).
Lp k∇ukLp ,
q
|x|
d
−
q
d
R
The usual Hardy inequality has q = p < d. The proof is the same: Assume u ∈ Cc∞ . For
any ε > 0, by divergence theorem,
Z
Z
Z
Z
x
x
|u|p
p
p x
(d − q)
=
|u|
∇
·
=
|u|
·
ndS(x)
+
− q · ∇|u|p
q
q
q
|x|
|x|
|x|
|x|
|x|>ε
|x|=ε
|x|>ε
|x|>ε
The first term is ≤ 0 since the outernormal n = −x/|x|. The second term is, using ∇|u| ≤
|∇u|,
Z
Z
p−q
p
|u|p q−1
p−1
q kuk q k∇uk p .
≤
|u|
|∇u|
≤
p(
)
L
Lp
q−1
q
|x|>ε |x|
|x|>ε |x|
Canceling the first factor and sending ε → 0, we get the lemma.
Theorem 4.9 (Morawetz’s estimate) Assume d ≥ 3. Let u0 ∈ H 1 and u ∈ C(Imax , H 1 )
be the corresponding maximal solution of (4.1) with f (u) = |u|p−1 u, 1 < p < pmax . Then
Z ∞Z
|u|p+1
dx dt < ∞.
−∞ Rd |x|
Nakanishi proved the following estimate for all d ≥ 1, which is weaker in both spatial
and temporal infinity, and obtained energy scattering for d = 1, 2:
Z ∞Z
t2
|u|p+1 dx dt < ∞.
3
Rd (|x| + t)
1
CHAPTER 4. SCATTERING THEORY
41
R
Proof. Let (f, g) = Re f ḡ and r = |x|. We may assume u0 ∈ H 2 and thus u ∈
C(Imax , H 2 ). The general case follows from approximation. We have
d
(iu, ∂r u) = (iut , ∂r u) + (iu, ∂r ut ) = (iut , ∂r u) + (i∂r∗ u, ut ) = (−∆u + |u|p−1 u, Su)
dt
d−1
where ∂r∗ = −∇( xr ·) = −∂r − d−1
and S = ∂r − ∂r∗ = 2x
r
r · ∇ + r . We first show
(−∆u, Su) ≥ 0:
Z
(−∆u, Su) = Re ∂j u∂j (2r−1 xk ∂k ū + (d − 1)r−1 ū)
Z
= Re ∂j u −2r−3 xj xk ∂k ū + 2r−1 ∂j ū + 2r−1 xk ∂k ∂j ū − (d − 1)r−3 xj ū + (d − 1)r−1 ∂j ū
Z
d − 1 −3
r xj ∂j |u|2 + (d − 1)r−1 |∇u|2 .
= −2r−1 |∂r u|2 + 2r−1 |∇u|2 + r−1 xk ∂k |∇u|2 −
2
The sum of the first and the second terms is ≥ 0. Since ∂k (r−1 xk ) = (d − 1)r−1 , the
third and the fifth terms cancel. The fourth term is C 2 |u(0)|2 if d = 3, and for d ≥ 4
∂j (r−3 xj ) = (d − 3)r−3 > 0. Thus (−∆u, Su) ≥ 0.
The other term
Z
Z
2x
d − 1 p+1
p − 1 d − 1 p+1
p−1
p+1
(|u| u, Su) =
∇|u|
+
|u|
=
|u| ,
(p + 1)r
r
p+1 r
which has the same sign as (−∆u, Su). Integrating over t ∈ [−T, T ] we get
Z T Z
p − 1 d − 1 p+1
|u|
≤ (iu, ∂r u)(T ) − (iu, ∂r u)(−T ) ≤ Ckuk2L∞ H 1 .
p+1 r
−T
Sending T → ∞ we get the claim.
Theorem 4.10 (Lr -decay without rate) Suppose d ≥ 3, f (u) = |u|p−1 u, and pc < p <
pmax . Every solution u ∈ C(R, H 1 ) of (4.1) satisfies
ku(t)kLr → 0
Proof. Step 1. To show
Z
as t → ∞,
|u(t, x)|r dx → 0
∀r ∈ (2, 2∗ ).
as t → ∞.
|x|>t log t
Let θ(x) = 1 for |x| > 1 and θ(x) = |x| for |x| ≤ 1. Let θM (x) = θ(x/M ). We have
d
1
C
(θM u, u) = (θM u, ut ) = (θM u, i(∆u − |u|p−1 u)) = −(u∇θM , i∇u) ≤
kuk2H 1 ≤
.
2dt
M
M
R
2
Thus (u, θM u) = Ct
M + θ|u0 | for all t > 0. Taking M = t log t, we get
Z
Z
C
|u(t, x)|2 ≤
+ θt log t |u0 |2 .
log t
|x|>t log t
The last term converges to 0 as t → ∞ by dominated convergence theorem. This shows the
claim for r = 2. The general case r < 2∗ follows from Hölder inequality.
CHAPTER 4. SCATTERING THEORY
42
Step 2. For any ε > 0 and t, τ > 1, there is t0 > t + 2τ so that
Z t0 Z
|u(s, x)|p+1 dxds ≤ ε.
t0 −2τ
|x|≤s log s
R 2τ (k+1) R
p+1 dxds and θ = [2τ (k + 1) log 2τ (k + 1)]−1 . We
Let ak = 2τ k
|x|≤s log s |u(s, x)|
P
P k
have
ak θk < ∞ by Morawetz estimate. Since
θk = ∞, we have lim inf k→∞ ak = 0.
Step 3. For any ε > 0 and T, τ > 1, there is t0 > max(T, τ ) so that
sup
t∈[t0 −τ,t0 ]
ku(t)kLr (Rd ) ≤ ε,
r = p + 1.
Decompose, for some τ to be chosen,
Z t−τ
Z
u(t) = S(t)u0 − i
S(t − s)f (u(s))ds − i
0
(4.9)
t
S(t − s)f (u(s))ds
t−τ
= v(t) + w(t) + z(t).
For v, by decay estimate, we can find t1 so that kv(t)kLr ≤ ε/4 for t > t1 .
For w, we choose ` = ∞ for p ≥ 2 and ` = 2/(2 − p) for 1 < p < 2. It is the largest
possible satisfying p`0 ≥ 2. Let σ = d( 12 − 1` ). We have
Z t−τ
Z t−τ
−σ
p
kw(t)kL` .
|t − s| k|u(s)| kL`0 ds .
|t − s|−σ ds = Cτ 1−σ .
−∞
0
In the above we have used p`0 ∈ [2, 2∗ ], u ∈ L∞ H 1 , and σ > 1 since p > 1 + 2/d. We also
have kw(t)kL2 ≤ 2ku0 kL2 because w(t) = S(τ )(u(t − τ ) − v(t − τ )). Since 2 < r = p + 1 < `,
1−θ
θ
(1−σ)θ
,
kw(t)kLr ≤ kw(t)kL
2 kw(t)kL` ≤ Cτ
1
θ
(1−σ)θ = ε/4.
where p+1
= 1−θ
2 + ` . We now choose τ = τε so that Cτ
For z, denoting µ = d( 12 − 1r ) and choosing 1 < m < min(1/µ, r),
Z t
Z t
Z t
0
p
−µ
−µm
1/m
1/m0
r
kz(t)kL .
|t − s| ku(s)kLr ds . (
|t − s|
ds)
(
ku(s)kpm
Lr ds)
t−τ
t−τ
Z
−µ+1/m
.τ
(
t−τ
t
ku(s)krLr ds)1/m
0
t−τ
pm0
where we have used µm < 1,
≥ r and u ∈ L∞ H 1 . By steps 1 and 2, for some t0
sufficiently large, for any t ∈ [t0 − τ, t0 ], one has [t − τ, t] ⊂ [t0 − 2τ, t0 ] and thus
0
kz(t)kLr . Cτ −µ+1/m (2ε1 )1/m ≤ ε/4.
Step 4. We revise the estimate of z(t) in step 3 by choosing m = 1 to get
ku(t)kLr (Rd ) ≤
ε
ε
+ kz(t)kLr ≤ + Cτ 1−µ sup ku(s)kpLr .
2
2
s∈[t−τ,t]
Let t0 be as in step 3 and let T = sup{t ≥ t0 : ku(s)kLr ≤ ε, ∀s ∈ [t0 − τ, t]}. We have
T ≥ t0 by step 3. Assume T < ∞. We have ku(T )kLr = ε and thus
ε ≤ ε/2 + Cτ 1−µ εp−1 ε.
CHAPTER 4. SCATTERING THEORY
43
We would get a contradiction if Cτ 1−µ εp−1 = Cτ 1−µ τ (1−σ)θ(p−1) < 1/2, or if
1 − µ + (1 − σ)θ(p − 1) < 0.
√
This is so if d ≥ 4 and p > pc , or if d = 3 and p > p1 = 12 (1 + 17).
For d = 3 and p ∈ (pc , p1 ], we need to revise the proof by choosing r sufficiently close
to 2∗ in step 3. The choice of r goes to 2∗ as p goes down to pc . See Cazenave [6, Th.7.9.2]
for details.
Remark. If p ≤ 2 (which is the case if d ≥ 4 and p ≤ pc ), we have ` = 2/(2 − p),
µ = dα/2r, σ = dα/2 with α = p − 1, θ = 1/r,
1 − µ + (1 − σ)θ(p − 1) = 1 −
dα 1
p
dα
+ (1 −
) α = (4 − dα),
2r
2 r
2r
which is nonnegative if 4 − dα ≥ 0, i.e., p ≤ pc . In particular, this shows the necessity of
p > pc when d ≥ 4.
Theorem 4.11 (Space-time bounds) Suppose d ≥ 3, f (u) = |u|p−1 u and pc < p <
pmax . Every solution u ∈ C(R, H 1 ) of (4.1) satisfies u ∈ Lq (R, W 1,r (Rd )) for any admissible
(q, r).
Proof. It suffices to prove the case r = p + 1. The general case follows from Strichartz
estimates. Let r = p + 1 and 2q = d( 12 − 1r ) < 1. For t > T > 0,
Z
t
u(t) = S(t − T )u(T ) − i
S(t − s)f (u(s))ds
T
By Strichartz estimates and Hölder inequalities,
kukLq ((T,t),W 1,r ) ≤ Cku(T )kH 1 + Ckukp−1q
L q−2
(p−1)
((T,t),Lr )
· kukLq ((T,t),W 1,r ) .
Denote g(s) = ku(s)kLr . Using u ∈ L∞ H 1 , we have
Z t
q−2
q p−1
p−1
kuk q (p−1)
= ( g(s) q−2 ds) q ≤ ε(T )kgkq−2
Lq
L q−2
if
p−1
q−2
((T,t),Lr )
T
> 1, which is exactly p > pc . Denote φ(t) = kukLq ((T,t),W 1,r ) . We have
φ(t) ≤ a + εφ(t)q−1
φ(T ) = 0,
∀t > T.
Note q − 1 > 1. The function g(φ) = a − φ + εφq−1 has a zero φ1 > 0 and is negative
in (φ1 , φ1 + δ) for some small δ > 0 if ε > 0 is sufficiently small. Since φ(T ) = 0, g(φ(t))
is continuous and ≤ 0, we conclude φ(t) ≤ φ1 for all t > T . Sending t → ∞ we get
kukLq ((T,∞),W 1,r ) ≤ φ1 .
Theorem 4.12 (Scattering in H 1 ) Suppose d ≥ 3, f (u) = |u|p−1 u and pc < p < pmax .
For every solution u ∈ C(R, H 1 ) of (4.1), there is a unique u+ ∈ H 1 so that
lim kS(−t)u(t) − u+ kH 1 = 0.
t→+∞
In addition, ku+ kL2 = ku0 kL2 and
1
2
R
|∇u+ |2 = E(u0 ).
CHAPTER 4. SCATTERING THEORY
44
Proof. Let v(t) = S(−t)u(t). For 0 < τ < t we have
Z
v(t) − v(τ ) = −i
t
S(−s)f (u(s))ds.
τ
Thus, by the proof of the previous theorem,
kv(t) − v(τ )kH 1 . kf (u)kLq0 ((τ,t),W 1,r0 ) . ε(τ )kukq−1
,
Lq ((τ,t),W 1,r )
which goes to 0 as τ → ∞.
Theorem 4.13 (Wave operator in H 1 ) Suppose d ≥ 3, f (u) = |u|p−1 u and pc < p <
pmax . For every u+ ∈ H 1 , there is a unique solution u ∈ C(R, H 1 ) of (4.1) so that
lim kS(−t)u(t) − u+ kH 1 = 0.
t→+∞
Proof. For T 1 to be chosen, let kukE = kukLq ((T,∞),W 1,r ) + supt≥T ku(t)kLr and
KT = kS(t)u+ kE . We have limT →∞ KT = 0. Let
E = {u : kukE ≤ 2KT } .
Rt
Let Φ(u)(t) = −i ∞ S(t − s)f (u(s))ds and A(u)(t) = S(t)u+ + Φ(u)(t). One verifies that,
if u, v ∈ E and KT is sufficiently small,
kΦ(u)kE ≤ CkukpE ≤ CKTp < KT ,
kA(u)kE < 2KT ,
kA(u) − A(v)kLq Lr ≤ CKTp−1 ku − vkLq Lr .
Thus A(u) is a contraction mapping on E and has a fixed point which solves (4.1) for
t ∈ (T, ∞). We can extend the solution to t > 0 by global wellposedness.
4.4
Problems and notes
Problems
14.
Notes
In preparing these notes, I have used Cazenave [6, §7.2–§7.4] for §4.2 and [6, §7.6–§7.8]
for §4.3.
Chapter 5
Orbital stability of solitary waves
5.1
Introduction
Definitions: Orbits, orbital stability, asymptotic stability.
(more later)
5.2
More on solitary waves
(Discussions on the special case f (u) = −|u|p−1 u is moved to §3.3.)
Consider the nonlinear Schrödinger equation for u(t, x) : R1+d → C,
i∂t u + ∆u = f (u),
(5.1)
where f (u) is real for real u and f (eiθ u) = eiθ f (u) for θ ∈ R. Let
Z
Z
1
1
|u|2 .
|∇u|2 + F (u), N (u) =
E(u) =
2
2
They are both conserved by (5.1). Eq. (5.1) can be written as
∂t u(t) = −iE 0 (u(t)).
Here and thereafter, we use the real inner product (f, g) = Re
Gateaux derivative of a functional.
Consider a solitary wave solution of (5.1) of the form
R
Rd
f ḡ dx when we take
u(t, x) = φω (x)eiωt .
(5.2)
−∆φω + f (φω ) = −ωφω .
(5.3)
It follows that φω satisfies
Denote the action (or Lagrangian)
S(u) = E(u) + ωN (u).
(5.4)
The equation (5.3) can be written as
E 0 (φω ) = −ωN 0 (φω ),
or S 0 (φω ) = 0.
(5.5)
Solutions of (5.5) are critical points of E(u) restricted to a manifold N (u) = constant.
45
CHAPTER 5. ORBITAL STABILITY OF SOLITARY WAVES
46
LemmaR 5.1 (Pohozaev identity) Fix d ≥ 1. Suppose f ∈ C(R, R) with f (0) = 0, and let
t
d
2
d
F (t) = 0 f (s) ds. If u satisfies ∆u = f (u)+ωu in distribution, u ∈ L∞
loc (R ), ∇u ∈ L (R ),
ω
2
1
d
and F (u) + 2 |u| ∈ L (R ), then
Z
Z
2−d
ω 2
|∇u|2 dx.
F (u) + |u| dx =
2
2d
Rd
Rd
R
It follows that S(u) = d1 Rd |∇u|2 dx > 0.
R
R
Let us first give a formal argument. Let T (u) = |∇u|2 and V (u) = Rd F (u)+ ω2 |u|2 dx.
Consider uσ (x) = u(x/σ). We have T (uσ ) = σ d−2 T (u) and V (uσ ) = σ d V (u). Hence
S(uσ ) = 12 σ d−2 T (u) + σ d V (u). Since u is a critical point of S, one has ∂σ S(uσ )|σ=1 = 0,
which gives the identity. A rigourous proof can be obtained by integrating by parts the
equation multiplied with x · ∇u over a ball BR , and showing that the boundary term
converges to zero for a sequence Rj → ∞. See [2, p.320].
See appendix for existence and uniqueness results.
5.3
Orbital stability
Assumption (A0). We assume there is an open interval I and for any ω ∈ I there is a
solitary wave solution of (5.1) of the form φω (x)eiωt , with φω > 0.
It turns out that the stability of φω depends on whether the following is true.
Assumption (A1).
d
N (φω ) > 0.
(5.6)
dω
This is related to Corollary 5.5 below.
The first stability result is by Cazenave-Lions 1982 [7] which uses the method of concentration compactness. Weinstein 1985-86 [44, 45] proves it using energy method and further
gives the continuity of x0 (t) and γ(t). Shatah-Strauss 1985 [28] proves instability when
d
dω N (φω ) < 0 and d ≥ 2.
Denote the action
S(u) = Sω (u) = E(u) + ωN (u).
Note that
d
d
S(φω ) = (E 0 (φω ) + ωN 0 (φω ),
φω ) + N (φω ) = N (φω ).
dω
dω
R
Here and thereafter (u, v) = Re uv̄. Thus, e.g., E 0 (u) = −∆u + f (u).
We expand S around φ = φω : Suppose u = φ + h where h = h1 + ih2 is small in H 1
and h1 , h2 are real. Note (S 0 (φ), h) = 0 because of (5.5). Thus
S(φ + h) − S(φ) = (h1 , L+ h1 ) + (h2 , L− h2 ) + β(khkH 1 )
(5.7)
where s−2 β(s) → 0 as s → 0+ and
L+ = −∆ + ω + f 0 (φ),
L− = −∆ + ω + f (φ)/φ.
Note that
L− φ = 0,
L+ ∇φ = 0,
L+ ∂ω φ = −φ.
The first equation is (5.3). The second equation follows from taking gradient of (5.3). The
last equation follows from differentiating (5.3) in ω.
CHAPTER 5. ORBITAL STABILITY OF SOLITARY WAVES
47
It follows from assumptions (A0), (A1), and some additional generic assumptions, the
following property. For the special cases d = 1 or f (u) = −|u|p−1 u, the additional generic
assumptions are not necessary.
Assumption (A2). There is λ > 0 so that, for real-valued v ∈ H 1 ,
(v, L− v) ≥ λkvk2H 1
(v, L+ v) ≥
λkvk2H 1
if v ⊥ Q;
(5.8)
if v ⊥ Q, ∇Q.
(5.9)
Since L± depend on ω continuously, the above estimates for one ω implies those for all
nearby ω. Explain the easy part and the power nonlinearity case:
(more later)
Theorem 5.2 Assume assumptions (A0), (A1) and (A2). Then φω (x)eiωt is orbitally
stable.
Proof. Fix ω and v0 near φω in H 1 .
Step 1. By (A1), there is a nearby ω̃ so that kφω̃ kL2 = kvkL2 . Since the two trajectories
{φω eiγ : γ ∈ R} and {φω̃ eiγ : γ ∈ R} are close to each other, if we can show the solution v(t)
remains uniformly close to the second trajectory, then it is also close to the first. Without
loss of generality we may assume ω̃ = ω.
Step 2. Decompose v(t, x) as
v(t, x + x0 )e−iγ(t) = φω (x) + h(t, x),
where the parameters x0 (t) and γ(t) are chosen to “almost” minimize khkL2 . At the minimizing parameters, we should have ∂γ (h, h) = ∂x0j (h, h) = 0, which give
(h2 , φω ) = O(h2 ),
(h1 , ∇φω ) = O(h2 ).
What we will choose is
(h2 (t), φω ) = 0,
(h1 (t), ∇φω ) = 0.
(5.10)
These parameters can be uniquely chosen for h(t) sufficiently close to φω by implicit function
theorem, and can be chosen at least for short time.
Step 3. Note kφω k2L2 = kvk2L2 = kφω + hk2L2 gives
(h1 (t), φω ) = − 12 (h, h).
Let h̃ = h + kφω where k =
(h,h)
2(φω ,φω ) .
Then
h̃1 ⊥ φω , ∇φω ,
h̃2 = h2 ⊥ φω .
By (A2),
kh̃k2H 1 . (h̃1 , L+ h̃1 ) + (h2 , L− h2 ).
For khkH 1 1, khkH 1 ∼ kh̃kH 1 . Thus
kh(t)k2H 1 . (h1 , L+ h1 ) + (h2 , L− h2 ).
Step 4. By (5.7),
kh(t)k2H 1 . S(φω + h(t)) − S(φω ) + β(t),
CHAPTER 5. ORBITAL STABILITY OF SOLITARY WAVES
48
where β(t) = β(kh(t)kH 1 ). Thus
kh(t)k2H 1 . S(v(t)) − S(φω ) + β(t) = S(v(0)) − S(φω ) + β(t).
The equality is by conservation of G. By (5.7) again at t = 0,
kh(t)k2H 1 . kh0 k2H 1 + β(t).
Since kh(t)kH 1 1, we conclude a uniform in time bound
kh(t)kH 1 . kh0 kH 1 .
This uniform bound guarantees that (5.10) can be always performed.
For the special case f (u) = −|u|p−1 u, we have φω (x) = ω 1/(p−1) Q(ω 1/2 x). Thus φ2ω =
R 2
ω
Q which is strictly increasing if and only if p < pc . This should be compared
with the strong instability result Theorem 3.6 for p ≥ pc .
R
2
− d2
p−1
5.4
Orbital instability
1 (Rd ), considered as a real Hilbert space.
We assume d ≥ 2 in this section. Let X = Hrad
Existence of φω is guaranteed by the following assumptions on f .
Assumption
(F0). f ∈ C 1 , f (eiθ u) = eiθ f (u) for θ ∈ R, f (u) ∈ R for u ∈ R,
Rt
F (t) = 0 f (s) ds satisfies F (u0 ) − ω2 u20 < 0 for some u0 > 0 and ω near ω0 . |f 0 (s)| = o(|s|` )
as |s| → ∞, where ` = 4/(d − 2) for d ≥ 3 and ` < ∞ for d = 2. lims→0 f (s)/s = 0.
Denote
S(u) = Sω (u) = E(u) + ωN (u),
1
K(u) = Sω (u) −
d
Z
|∇u|2 dx.
In view of the Pohozaev identity, a solution φ of (5.3) satisfies K(φ) = 0.
Lemma 5.3 Assume d ≥ 2 and (F0). There exists a positive minimizer φ of
R
δ(ω) = min d1 |∇u|2 dx : 0 6= u ∈ X, K(u) ≤ 0 .
(5.11)
Any such φ satisfies K(φ) = 0, Sω0 (φ) = 0, and
R
R
δ(ω) = inf {S(u) : u ∈ X, K(u) = 0} = inf S(u) : u ∈ X, |∇u|2 = |∇φ|2 .
The existence of positive minimizers of (5.11) is shown by Berestycki-Gallouet-Kavian
1983 for d = 2 and by Shatah 1985 for d ≥ 3. They have the least action among all
nontrivial solutions of Sω0 (φ) = 0. For the rest of this section we assume
Assumption (A3). Assume d ≥ 2 and (F0). For any ω in a neighborhood of ω0 there
is a minimizer φω > 0 of (5.11), and the map ω 7→ φω ∈ X is C 2 .
Denote φ0 = φω0 . By the lemma,
R
δ(ω) = d1 |∇φω |2 dx = S(φω ),
δ 0 (ω) = N (φω ).
Lemma 5.4 Assume d ≥ 2 and (A3). Let ψω (x) = φω (x/λ(ω)) where λ(ω)d = N (φ0 )/N (φω )
so that N (ψω ) = N (φ0 ). We have E(ψω ) ≤ δ(ω) − δ 0 (ω0 )ω. Equality holds if ω = ω0 .
CHAPTER 5. ORBITAL STABILITY OF SOLITARY WAVES
49
Proof.
Z
1 d−2
λ |∇φω |2 + λd (F (φω ) + ω2 φ2ω )
2
Z
1 d−2 d − 2 d
0
= −ωδ (ω0 ) +
λ
−
λ
|∇φω |2 .
2
2d
0
E(ψω ) = −ωN (ψω ) + Sω (ψω ) = −ωδ (ω0 ) +
Let g(λ) = 12 λd−2 −
lemma.
d−2 d
2d λ
− 1/d. Since g(1) = 0 and g 0 (λ)(λ − 1)(d − 2) ≥ 0, we get the
Corollary 5.5 δ 00 (ω0 ) ≥ 0 if and only if E restricted to M0 = {u ∈ X : N (u) = N (φ0 )}
has a local minimum at φ0 .
The theorem is instructive but will not be needed below. Thus we skip its proof.1
Corollary 5.6 If δ 00 (ω0 ) < 0, then E(ψω ) < E(φ0 ) for ω near ω0 but ω 6= ω0 .
Proof. By Lemma 5.4, E(ψω ) ≤ g(ω) := δ(ω) − δ 0 (ω0 )ω and E(φ0 ) = g(ω0 ). Since
= δ 0 (ω) − δ 0 (ω0 ), we have g 0 (ω0 ) = 0 and g 00 (ω0 ) < 0. Thus g has a strict local
maximum at ω0 .
g 0 (ω)
Theorem 5.7 Assume d ≥ 2 and (A3). If δ 00 (ω0 ) < 0 then K = eiθ φω0 : θ ∈ R is
unstable under the flow (5.1). Specifically, there is a neighborhood U ⊂ X of K and a
continuous functional P : U → R such that, if u0 ∈ U , E(u0 ) < E(φω0 ), N (u0 ) = N (φω0 )
and P (u0 ) 6= 0 (so that u0 may be arbitrarily near K), the solution of (5.1) with initial
condition u0 exits U in a finite time.
We will show that, for ω 6= ω0 near ω0 , ψω satisfies the above conditions on u0 . Thus
there is an open set of u0 whose corresponding solutions exit U in finite times. In contrast,
the solution with u(0) = φω remains in U .
R
R
Proof. RSuppose δ 00 (ω0 ) ≥ 0. For any u ∈ M0 near φ0 , we can find ω ∼ ω0 so that |∇u|2 = |∇φω |2
2
0
0
because ∂ω |∇φω | = dδ (ω) > 0. By Lemma 5.3, E(u) = Sω (u) − ωN (u) ≥ δ(ω) − ωδ (ω0 ), which is
≥ δ(ω0 ) − ω0 δ 0 (ω0 ) = E(φ0 ) since δ 00 (ω0 ) ≥ 0. Conversely, assume E restricted to M0 has a local minimum
at φ0 . By Lemma 5.4, δ(ω0 ) − δ 0 (ω0 )ω0 = E(φ0 ) ≤ E(ψω ) ≤ δ(ω) − δ 0 (ω0 )ω, which implies δ 00 (ω0 ) ≥ 0. R
2 −d R
For the special case f (u) = −|u|p−1 u, we have φω (x) = ω 1/(p−1) Q(ω 1/2 x) and φ2ω = ω p−1 2 Q2 .
1
− 2 +d
Let ω0 = 1 and φ0 = Q. We have λ(ω)d = N (φ0 )/N (φω ) = ω p−1 2 and ψω (x) = φω (x/λ(ω)) =
ω 1/(p−1) Q(ω 2/d(p−1) x) has N (ψω ) = N (Q) and
Z
Z
1
m1
2
E(ψω ) = ω
|∇Q| − ωC1 = g(ω)C1 , C1 =
Qp+1 ,
2
p+1
1 m
4
where m = d(p−1)
and g(ω) = m
ω − ω by Pohozaev identity. Note g 0 (1) = 0 and g(ω) (and thus E(ψω ))
has a local
minimum
at
ω
=
1
if and only if m ≥ 1, i.e., p ≤ pc . This suggests Cor. 5.5. We have
R
2
δ(ω) = d1 |∇φω |2 = dm
ω σ C1 , σ = d+2+(2−d)p
> 0. Lemma 5.4 asserts
2(p−1)
f (ω) = ω m − mω −
2
2 σ
2σ
ω +
ω ≤ 0,
dm
dm
for ω ∼ 1.
Note f 0 (1) = 0 and f (1) = − (d+4−dp)
≤ 0. If p 6= pc , then f (ω) < 0 for ω ∼ 1. If p = pc then m = σ = 1
4d(p−1)
and f (ω) ≡ 0. Thus Lemma 5.4 and Cor. 5.5 remain correct for power nonlinearity when d = 1.
CHAPTER 5. ORBITAL STABILITY OF SOLITARY WAVES
50
We will choose a functional A(u) defined on U so that A(u(t)) has a monotonicity
property which ensures u(t) exits U in finite time if u(0) ∈ U \ K. Intuition and comparison
with ODE stability:
(more later)
For general A(u) we have
d
A(u(t)) = (A0 (u(t)), ∂t u(t)) = P (u(t)),
dt
where
P (u) := (A0 (u), −iE 0 (u)).
(5.12)
Lemma 5.8 For any u ∈ Uδ = {u ∈ X : ku − φ0 kX < δ} with δ > 0 sufficiently small,
there is a unique α = α(u) ∈ [0, 2π) which minimizes ku − φ0 eiα kL2 , and Re(u, iφ0 eiα ) = 0.
The latter follows from
0 = ∂α (u − φ0 eiα , u − φ0 eiα ) = 2 Re(u − φ0 eiα , iφ0 eiα ) = −2 Re(u, iφ0 eiα ).
Let
A(u) = (−iy0 , e−iα(u) u),
y0 = ∂ω |ω=ω0 ψω .
(5.13)
We have A(ueiθ ) = A(u), which implies
0 = ∂θ |θ=0 A(ueiθ ) = (A0 (u), iu).
We also have A0 (φ0 ) = −iy0 since
(A0 (φ0 ), h) = ∂ε0 A(φ0 + εh) = (−iy0 , h) + (−iy0 , −iφ0 )∂ε0 α(φ0 + εh)
and the last term is zero because 0 = ∂ω N (ψω )|ω=ω0 = (y0 , φ0 ).
For any u ∈ Uδ , define uλ ∈ U2δ for λ ∈ (1 − ε, 1 + ε) which solves the ODE
∂λ uλ = iA0 (uλ ),
u1 = u.
We have N (uλ ) = N (u) since ∂λ N (uλ ) = (uλ , ∂λ uλ ) = (uλ , iA0 (uλ )) = 0.
Claim: There is a unique λ ∈ (1 − ε, 1 + ε) so Rthat k∇uλ kL2 R= k∇φ0 kL2 .
d
The claim is because at u = φ0 and λ = 1, dλ
|∇uλ |2 dx = ∇φ0 · ∇y0 and
Z
∇φ0 · ∇y0 > 0.
(5.14)
Eq. (5.14) is because
Z
Z
Z
2 ∇φ0 · ∇y0 = ∂ω |ω=ω0 |∇ψω |2 = ∂ω |ω=ω0 [λ(ω)]d−2 |∇φω |2
= (d − 2)λ0 (ω0 )dδ(ω0 ) + dδ 0 (ω0 )
and, since λ(ω) = δ 0 (ω0 )/δ 0 (ω), we have λ0 (ω0 ) = −δ 0 (ω0 )−1 δ 00 (ω0 ) > 0.
Lemma 5.9 If u ∈ M0 ∩ Uδ \ K, there is λ ∈ (1 − ε, 1 + ε) so that
E(φ0 ) < E(u) + (λ − 1)P (u).
(5.15)
CHAPTER 5. ORBITAL STABILITY OF SOLITARY WAVES
51
Proof. We may assume α(u) = 0.
Consider ∂λ2 E(uλ )|λ=1 with u = φ0 . We have
∂λ E(uλ ) = ∂λ S(uλ ) = (E 0 (uλ ) + ω0 N 0 (uλ ), ∂λ uλ ),
∂λ2 E(uλ ) = ((E 00 (uλ ) + ω0 N 00 (uλ ))∂λ uλ , ∂λ uλ ) + (E 0 (uλ ) + ω0 N 0 (uλ ), ∂λ2 uλ ).
At λ = 1 with u = φ0 , the second term vanishes and the first term is close to
∂ω2 |ω=ω0 E(ψω ) = ((E 00 (φ0 ) + ω0 N 00 (φ0 ))y0 , y0 )
By Lemma 5.4, the function g(ω) = E(ψω ) − δ(ω) + δ 0 (ω0 )ω has a local maximum at ω0
and thus ∂ω2 g(ω0 ) ≤ 0, which amounts to ∂ω2 |ω=ω0 E(ψω ) ≤ δ 00 (ω0 ) < 0. Thus ∂λ2 E(uλ ) < 0
for u ∈ Uδ , α(u) = 0, λ ∈ (1 − ε, 1 + ε), and δ sufficiently small. By Taylor expansion,
E(uλ ) < E(u) + (λ − 1)P (u).
On the other hand, for λ = λ(u) and N (u) = N (φ0 ), we have
E(uλ ) = S(uλ ) − ω0 N (uλ ) ≥ S(φ0 ) − ω0 N (φ0 ) = E(φ0 ).
This shows the lemma.
RAs a corollary,
R since E(ψω ) < E(φ0 ) for ω 6= ω0 , we have (λ(ψω ) − 1)P (ψω ) > 0. Since
2
∂ω |∇ψω | = 2 ∇φ0 · ∇y0 6= 0 at ω = ω0 by (5.14), λ(ψω ) − 1 changes sign at ω0 . Thus
P (ψω ) also changes sign, and P (φ0 ) = 0.
Let U± = {u ∈ Uδ \ K : E(u) < E0 , ±P (u) > 0}. They are nonempty since they contain
ψω for ω 6= ω0 . If u(t) is a solution with u(0) ∈ U+ , by (5.15), for λ(t) = λ(u(t)) ∈
(1 − ε, 1 + ε),
(λ(t) − 1)P (u(t)) > ε0 = E0 − E(u(t)) > 0.
This shows P (u(t)) has a fixed sign (thus U+ is invariant) and |P (u(t))| >
d
A(u(t)) =
Since dt
u0 ∈ U− is similar.
5.5
ε0
|λ(t)−1|
≥ ε0 .
P (u(t)), this shows u(t) has to exit Uδ in finite time. The case with
Problems and notes
Problems
15.
Notes
In preparing these notes, I have used Berestycki-Lions [2] for §5.2, Weinstein [45] for
§5.3, and Shatah-Strauss [28] for §5.4.
Chapter 6
Asymptotic stability of small
solitary waves
6.1
Introduction
[Recall the concept of asymptotic stability.]
(more later)
Intrinsically, one difficulty of proving asymptotic stability with non-localized data is that
small waves (solitary or dispersive) may keep coming from spatial infinity. They are hard
to deal with even if they are getting smaller and smaller. To avoid this difficulty one may
assume localized data. In this case one is not worried with small dispersive wave but the
existence of small solitary waves with faster speed may still cause a problem. This is why
not much is known for NLS. The case of gKdV is better understood because smaller solitary
waves have smaller speeds and the dispersive wave always goes the opposite direction.
We now consider the technical difficulties. For NLS, one usually decomposes the error
term with respect to the linearized operator. If one decomposes
u(t) = φσ(t) + η(t)
(6.1)
where σ is a vector of parameters for solitary waves φσ , and η is the error term, then η
satisfies an equation of the form
∂t η = Lη + M (σ̇) + N (η)
(6.2)
where L is the linearized operator around φσ , M (σ̇) is the modulation term coming from
the time derivative of φσ(t) and is linear in σ̇, and N collects terms nonlinear in η. All L,
M , N depend on σ. One tries to decompose η into spectral subspaces of L,
η = η0 + ηd + ηc ,
(6.3)
where
1. ηc lies on the continuous spectral subspace of L and is dispersive. One hopes to show
its local decay, and thus its influence to the local dynamics is getting smaller, by decay
or Strichartz type estimates.
52
CHAPTER 6. ASYMPTOTIC STABILITY OF SMALL SOLITARY WAVES
53
2. η0 lies in E0 , the zero eigenspace of L. The eigenspace is usually generated by the
symmetries of the equation and one tries to control η0 using M (σ̇), resulting in equations of σ̇. However, E0 usually has irreducible Jordan blocks of rank ≥ 2 and whose
generator may not corresponds to a symmetry. For example, for the ground state of
i∂t u + ∆u + |u|p−1 u = 0, if p 6= pc , the dimension of symmetries is 2d + 2, the same
as that of E0 . If p = pc , then the dimension of symmetries is 2d + 3 (the extra one is
pseudoconformal transform), but the dimension of E0 is 2d + 4.
3. ηd lies in eigenspaces of nonzero eigenvalues. These nonzero eigenvalues are purely
imaginary if φσ is known to be orbitally stable. For some special cases they may
be shown to decay due to nonlinear resonance with the ground state through the
continuous spectral subspace, but very little is known.
To illustrate some of these (very rough) ideas, we will present in next section a result
on small solitary waves, which are easier to deal with because:
1. Linear estimates may be proved by perturbative methods;
2. The estimates of modulation and nonlinear terms may be absorbed by the left side;
3. Modulation parameters are more likely to converge.
6.2
Asymptotic stability of small solitary waves
R
Denote (a, b) = R3 āb and ha, bi = Re(a, b).
Consider a nonlinear Schrödinger equation for u : R × R3 → C
i∂t u = H0 u + f (u),
H0 = −∆ + V.
(6.4)
We assume V is a real-valued potential so that H0 has only one eigenvalue e0 < 0 with
eigenfunction φ0 , and f is a covariant nonlinearity satisfying f (ueiθ ) = f (u)eiθ and f (|u|) ∈
R. Under these assumptions, the L2 -norm ku(t)kL2 and the energy
Z
Z u
1
1
2
2
|∇u| + V |u| + F (u) dx, F (u) =
f (s)ds,
2
R3 2
0
are constant in time. Using these conserved quantities, and the smallness of ku0 kH 1 , one
can prove a uniform estimate supt ku(t)kH 1 1 and obtain global well-posedness.
If a solitary wave is of the form u(t, x) = Q(x)e−iEt , then the pair Q, E solve
(−∆ + V )Q + f (Q) = EQ.
(6.5)
There exists a family of small solitary waves Q = Q[z] with E = E[z], bifurcated from zero
along φ0 direction and parametrized by small z ∈ C, |z| ≤ δ 1, which solve (6.5) and
satisfy
Q[z] = zφ0 + q,
q ⊥ φ0 ,
0
E[z] = e0 + e ∈ R,
kqkH 2 ∩W 1,1 = o(z),
0
e = o(z).
(6.6)
One can obtain the equations of (q, e0 ) by subtracting zH0 φ0 = ze0 φ0 from (6.5) and solve
(q, e0 ) by a contraction mapping argument. Observe that
Q[zeiα ] = Q[z]eiα ,
(α ∈ R),
E[z] = E[|z|].
CHAPTER 6. ASYMPTOTIC STABILITY OF SMALL SOLITARY WAVES
54
Picture and Literature.
(more later)
Soffer-Weinstein [29, 30], Pillet-Wayne [23], Tsai-Yau [39]–[42], Tsai [38], GustafsonNakanishi-Tsai [13]. All but the last use localized data and decay estimates. The last uses
H 1 data and Strichartz estimates.
Q[z] is differentiable in z if we regard z as a real vector
z = z1 + iz2 ↔ (z1 , z2 ) ∈ R2 .
We will denote the z-derivatives by D1 = ∂z∂ 1 and D2 =
matrix, regarded as a R-linear map on C:
DQ[z] : C → C,
∂
∂z2 ,
and DQ[z] denotes the Jacobian
DQ[z]w 7→ D1 Q[z] Re w + iD2 Q[z] Im w.
(6.7)
DQ = O(1),
(6.8)
From (6.6), we have
Q = O(z),
and Dj Dk Q = o(z).
The gauge covariance of Q[z] implies that
DQ[z]iz = iQ[z].
(6.9)
Given a general solution u(t) of (6.4), it is natural to decompose it into solitary wave
and dispersive wave components:
u(t) = Q[z(t)] + η(t).
(6.10)
z
(When z 6= 0, we can decompose u(t) = {Q[|z(t)|] + ξ(t)} |z|
since the local interaction of
ξ with Q is easier to understand. The variable ξ is however not as good as η for global
behavior and is ambiguous when z = 0.)
For any such decomposition, the difference of (6.4) and (6.5), using EQ − iDQż =
−iDQ(ż + iEz) by (6.9), yields an equation for η
i∂t η = H[z]η − iDQ[z](ż + iEz) + F2 (z, η),
(6.11)
where H[z] is the linearized operator
H[z]η := (−∆ + V )η + ∂ε |ε=0 f (Q + εη),
(6.12)
and F2 collects terms which are higher-order in η:
F2 (z, η) := f (Q + η) − f (Q) − ∂ε |ε=0 f (Q + εη).
(6.13)
The operator H[z] is R-linear but not C-linear. It is, however, symmetric if we regard C as
R2 , and use the reduced inner product h·, ·i.
The decomposition (6.10) is not unique and we need to specify the space for η. There
are three typical choices: (i) η ⊥ φ0 , (ii) η has the minimal norm, (iii) η satisfies some
orthogonality condition so that the linear term of the parameter equation vanishes. Choice
(i) has been popular in the literature for its simplicity but is insufficient for our purpose.
CHAPTER 6. ASYMPTOTIC STABILITY OF SMALL SOLITARY WAVES
55
We will require η to belong to the following “continuous spectral subspace” Hc [z],
Hc [z] := η ∈ L2 : hiη, D1 Q[z]i = hiη, D2 Q[z]i = 0 ,
(6.14)
and we can uniquely decompose u(t) as
u(t) = Q[z(t)] + η(t),
η(t) ∈ Hc [z(t)].
(6.15)
The requirement η(t) ∈ Hc [z(t)] determines z(t) uniquely. An evolution equation for z(t)
is derived from differentiating the relation hiη, Dj Q[z]i = 0 with respect to t, and using
equation (6.11) (see (6.24)). Our goal is to prove the asymptotic stability of Q[z(t)] and
the asymptotic completeness of η(t).
Remarks.
1) The subspace Hc [z] is an invariant subspace of i(H[z] − E[z]), (which is the linearized
operator in the ∂t ξ equation), as follows from the relation
(H[z] − E[z])Dj Q[z] = (Dj E[z])Q[z]
(6.16)
(which is the result of differentiating (6.5)), together with (6.9).
∂
Q} in the inner
2) When z ∈ R+ , Hc [z] is just the orthogonal complement of {Q, i ∂|z|
product h·, ·i. This subspace is often used in the literature. The current definition using z,
instead of its magnitude and phase, is more natural because we allow z = 0, for which case
the phase is not well defined.
3) Note that we impose a time-dependent condition η(t) ∈ Hc [z(t)] instead of simpler
conditions such as η(t) ⊥ φ0 . The reason is the following. If we assume (η(t), φ0 ) = 0, then
the equations for ż + iEz yield
|ż + iEz| . |(φ0 , Aη)| + |(φ0 , F2 )|,
(6.17)
where A is some linear operator. The term (φ0 , Aη) is linear in η and hence is not integrable
in time, in light of the estimate η ∈ L2t W 1,6 . Thus we cannot conclude that |z| and E[z]
have limits as t → ∞. This term drops out if we require η(t) ∈ Hc [z], and the equation for
d
ż + iEz (and hence dt
|z|) becomes quadratic in η.
We now state the assumptions and the main theorem.
Assumption (V): V is a real-valued potential so that H0 is a self-adjoint operator
on L2 (R3 ) with domain H 2 . It has only one negative eigenvalue e0 < 0 with normalized
eigenfunction φ0 > 0. Denote the projections onto the discrete and continuous spectral
subspaces of H0 by Pd = φ0 (φ0 , ·) and Pc = 1 − Pd , and assume the following Strichartz
1
2
1,6 :
estimates with X = L∞
t H ∩ Lt W
Z t
−itH0
ke
Pc φkX . kφkH 1 , k
e−i(t−s)H0 Pc F (s)dskX . kF kL2 W 1,6/5 .
(6.18)
−∞
The above estimates may be proved if |V (x)| ≤ C(1 + |x|)−3−ε for some ε > 0, and
the bottom of the continuous spectrum of H0 , zero, is not an eigenvalue nor a resonance,
cf. §1.5.
Assumption (F): f : C → C satisfies the gauge covariance f (ueiθ ) = f (u)eiθ for θ ∈ R
and f (|u|) ∈ R. When restricted to R, f is C 2 with f (0) = f 0 (0) = 0 and |f 00 (s)| ≤
C(s1/3 + s3 ) for s ≥ 0.
Example:
f (u) = a|u|4/3 u + b|u|4 u, a, b ∈ R.
The signs of a, b do not matter since we consider small solutions.
CHAPTER 6. ASYMPTOTIC STABILITY OF SMALL SOLITARY WAVES
56
Theorem 6.1 (Asymptotic stability and completeness) Let d = 3 and assume (V)
and (F). Every solution u of (6.4) with data u0 sufficiently small in H 1 can be uniquely
decomposed as
u(t) = Q[z(t)] + η(t),
(6.19)
with differentiable z(t) ∈ C and η(t) ∈ Hc [z(t)] satisfying
kηkL2t W 1,6 ∩L∞
1 + kzkL∞ . ku0 kH 1 ,
t
t H
. ku0 k2H 1 .
kż + iE[z]zkL1t ∩L∞
t
(6.20)
Moreover, there exist z+ ∈ C with |z+ | − |z(0)| . ku0 k2H 1 , and η+ ∈ H 1 ∩ Ran Pc such that
Z t
z(t) exp i
E[z(s)]ds → z+ ,
kη(t) − e−itH0 η+ kH 1 → 0
(6.21)
0
as t → ∞. In particular, |z(t)| → |z+ | and, if z+ 6= 0, arg z(t) +
mod 2π.
Rt
0
E[z(s)]ds − arg z+ → 0
Sketch of proof. We will show that
kukL∞
<δ
1
t H [0,T ]
implies that kukL∞
< δ/2,
1
t H [0,T ]
(6.22)
for any T > 0, provided δ and δ 0 were chosen sufficiently small. Then, by continuity in
time, this bound and the solution together extend globally in time. In the argument below
we will not explicitly specify the time interval.
Step 1. z estimates. Recall we impose hiη, Dj Q[z]i = 0, j = 1, 2. Differentiating this
relation with respect to t and plugging equation (6.11) into that, we obtain
0 = hHη − iDQ(ż + iEz) + F2 , Dj Qi + hiη, Dj DQżi ,
where H, E and Q all depend on z. By the symmetry of H and (6.16),
hHη, Dj Qi = hη, HDj Qi = hη, Dj (EQ)i = hη, EDj Qi = hiη, EDj DQizi ,
(6.23)
where we have used hiη, DQi = 0 and (6.9) for the last two equalities. Thus we obtain
X
(hiDj Q, Dk Qi + hiη, Dj Dk Qi)(ż + iEz)k = − hF2 , Dj Qi .
(6.24)
k=1,2
In view of (6.8), the matrix on the left hand side is estimated as
hiDj Q, Dk Qi + hiη, Dj Dk Qi = j − k + o(δ).
(6.25)
Inverting this matrix, we obtain
|ż + iEz| . | hF2 , DQ[z]i | . kF2 kL1 +L∞ .
(6.26)
Note |F2 | . (1 + |Q| + |η|)4 |η|2 and kF2 kL1 +L∞ . (1 + kQkL6 + kηkL6 )4 kηk2L6 . Therefore
kż + iEzkL1 ∩L∞ . kηk2X .
(6.27)
CHAPTER 6. ASYMPTOTIC STABILITY OF SMALL SOLITARY WAVES
57
Step 2. η-estimate. We estimate η by writing the equation (6.11) in the form
i∂t η = (−∆ + V )η + F1
(6.28)
F1 := f (Q + η) − f (Q) − iDQ(ż + iEz).
(6.29)
with
Denote ηc := Pc η. We have kηc k ∼ kηk in any reasonable norm and the map ηc → η − ηc is
compact. Applying the Strichartz estimates to (6.28), we get
kηkX . kηc kX . kη(0)kH 1 + kPc F1 kL2t W 1,6/5 . ku0 kH 1 + kF1 kL2t W 1,6/5 .
(6.30)
We claim
kf (Q + η) − f (Q)kL2t W 1,6/5 . α(kQkH 2 + kηkH 1 )kηkW 1,6
(6.31)
P
where α(s) = s4/3 + s4 . To get an idea, note that |f (Q + η) − f (Q)| . p |Q|p−1 |η| + |η|p
with p = 7/3, 5, and kη p kL6/5 is controlled by kηkp−1
kηkL6 if p = 7/3 and by kηkp−1
kηkL6
L2
L6
if p = 5. We get
kF1 kL2t W 1,6/5 . kż + iEzkL2 + δkηkL2t W 1,6 .
(6.32)
Step 3. From (6.27), (6.30) and (6.32), we deduce that
1/2
kηkX + kż + iEzkL2 + kF1 kL2t W 1,6/5 . ku0 kH 1 < δ 0 ,
(6.33)
if we take δ sufficiently small. Choosing δ 0 even smaller, we obtain the desired bootstrapping
estimate (6.22), and so the solution, as well as all the estimates, extends globally.
Moreover, we have
k∂t |z|kL1 ≤ kż + iEzkL1 . kηk2L2 L6,2 . ku0 k2H 1 ,
t
(6.34)
so |z(t)| and E[z(t)] = E[|z(t)|] converge as t → ∞.
Step 4. Finally, we prove that η is asymptotically free. We have the integral equation
Z t
ηc (t) = eit(∆−V ) ηc (0) − i
e−is(∆−V ) Pc F1 (s)ds .
(6.35)
0
By the Strichartz estimate, for any T > S > 0 we have
Z T
k
e−is(∆−V ) Pc F1 (s)dskH 1 . kF1 kL2t W 1,6/5 [S,T ] → 0,
S
as T > S → ∞, by the Lebesgue dominated convergence theorem, and the finiteness of
kF1 kL2t W 1,6/5 (0,∞) . Thus the integral in (6.35) converges in H 1 as t → ∞, and we obtain
Z ∞
−it(∆−V )
lim e
ηc (t) = ηc (0) − i
e−is(∆−V ) Pc F1 (s)ds =: η+ .
t→∞
0
In particular, ηc (t) converges to 0 weakly in H 1 . Then η(t) − ηc (t) converges to 0 strongly
in H 1 . Therefore we conclude that
kη(t) − eit(∆−V ) η+ kH 1 → 0.
CHAPTER 6. ASYMPTOTIC STABILITY OF SMALL SOLITARY WAVES
6.3
Problems and notes
Problems
16.
Notes
In preparing these notes, I have used [13] for section 6.2.
58
Appendix
A.1
Bound states for general nonlinearity
We can rewrite (5.3), with g(u) = −ωu − f (u), as
−∆u = g(u).
(A.1)
Lemma A.2
with g(0) = 0, and
R t (Pohozaev identity) Fix d ≥ 1. Suppose g ∈ C(R, R)
∞
let G(t) = 0 g(s) ds. If u satisfies −∆u = g(u) in distribution, u ∈ Lloc (Rd ), ∇u ∈ L2 (Rd ),
and G(u) ∈ L1 (Rd ), then
Z
Z
d−2
|∇u|2 dx.
G(u) dx =
2d
d
d
R
R
Denote the action (or Lagrangian)
Z
S(u) =
Rd
It follows that S(u) =
1
d
R
Rd
1
|∇w|2 − G(u) dx.
2
(A.2)
|∇u|2 dx > 0 if u solves (A.1).
Theorem A.3R(Unique existence for d = 1 [2]) Suppose g ∈ C(R, R) with g(0) = 0,
t
and let G(t) = 0 g(s) ds. The following problem
−u00 = g(u),
u ∈ C 2 (R),
u(x0 ) > 0 for some x0 ,
lim u(x) = 0
x→±∞
has a solution if and only if
u0 = inf{s > 0 : G(s) = 0} exists,
u0 > 0,
g(u0 ) > 0.
In this case, the solution is unique up to translation. After a suitable translation, u is
positive, even, u(0) = u0 , u0 (x) < 0 for x > 0.
If one assumes in addition that lims→0 g(s)/s = −m < 0, then u, u0 , u00 have exponential
decay at infinity.
In the following we consider the cases d ≥ 2.
Theorem A.4 (Radial symmetry [10]) Let u be a C 2 positive solution vanishing at infinity of −∆u = g(u) in Rd , d ≥ 2, with g ∈ C 1+µ (Rd ), µ > 0, g(0) = 0 and g 0 (0) < 0.
Then u is radial about some point x0 ∈ Rd and ∂u
∂r < 0 for 0 < r = |x − x0 |.
59
APPENDICES
60
Theorem A.5 (Existence
of ground state for d ≥ 2) Let d ≥ 2. Suppose g ∈ C(R, R)
Rt
is odd, and let G(t) = 0 g(s) ds. If g satisfies
lim g(s)s−1 = −m < 0,
s→0
lim sup g(s)s−` ≤ 0,
s→∞
G(s0 ) > 0 for some s0 > 0,
where ` = d+2
d−2 for d ≥ 3 and can be any positive number for d = 2, then the equation
−∆u = g(u) possesses a nonzero solution u ∈ H 1 (Rd ) which is positive, radial, C 2 , and
decreases with respect to r = |x|. Itself and its derivatives up to order 2 have exponential
decay at infinity.
The above theorem
is shown in [2] for d ≥ 3 and ??? for d = 2. The last condition
R
is necessary since G(u) > 0. The middle condition is necessary in view of the example
g(u) = −u + up . For the first condition, there is no solution if m < 0. The case m = 0 is
delicate.
It is obtained as a (rescaled) constrained minimizer of
Z
Z
2
1
d
|∇w| : w ∈ H (R ),
G(w) dx = d − 2
(A.3)
min
Rd
as in Berestycki-Lions [2] for d ≥ 3 and ??? for d = 2, or
Z
1
2
1
d
|∇w| dx : w ∈ H (R ), K(w) ≤ 0
(A.4)
min
n
R
where K(w) = S(w)− n1 |∇w|2 dx, as in Shatah-Strauss 1985 [28] for d ≥ 3 and Brezis-Lieb
for d = 2.
It is shown in [2, Th.3] that, if u is a solution obtained from minimizing (A.3) and v is
another solution, then
0 < S(u) ≤ S(v).
That is, a ground state has the minimal action among all bound states.
In general, the uniqueness of φω is not known for d ≥ 2. Kwong [19] proves the case for
g(u) = −ωu + up , P
1 < p < pmax
. A generalization by Chen-Lin [5] shows uniqueness for
P
d ≥ 3 and f (u) = Ii=1 ai upi − Jj=1 bj uqj − ωu, where ai , bj , ω > 0, 1 < qj < p < d+2
d−2 for
d
I = 1 and 1 < qj < pi < d−2 for I ≥ 2.
A.2
Solutions to problems
1. For p < 2, Lemma 1.1 is a growth estimate since the exponent in the estimate becomes
positive. Let us only consider the case p = 1. If it is true for all t > 0, then for any
ϕ ∈ L∞ ∩ L2 , by sending t → 0 in
|(S(t)ϕ, ϕ)| ≤ kS(t)ϕkL1 kϕkL∞ ≤ Ctd/2 kϕk2L∞
we derive ϕ = 0 in L2 . We further claim that, for any fixed t > 0, there does not exist
a constant C1 so that
kS(t)ϕkL1 ≤ C1 kϕkL∞ ,
∀ϕ ∈ L∞ .
Otherwise, for any f ∈ L1 ,
kS(2t)f kL1 = kS(t)S(t)f kL1 ≤ C1 kS(t)f kL∞ ≤ C2 kf kL1
where C2 = C1 (4πt)−d/2 . This contradicts problem 3.
APPENDICES
61
2. One method from your fellow classmate is to consider the Fourier transform
2
(S(t)ϕ, ζ) = (e−itξ ϕ̂, ζ̂).
2
Since the factor e−itξ is highly oscillatory as t → ∞, it should converge to 0, in
the same spirit of the Riemann-Lebesgue lemma. The solution I prepared is the
following: Normalize kϕkL2 = kζkL2 = 1. For any small ε > 0 choose R(ε) ≥ 1 so
that kϕ(1 − 1R )kL2 + kζ(1 − 1R )kL2 < ε/4. Here 1R is the characteristic function of
|x| < R. Then
|(S(t)ϕ, ζ) − (S(t)1R ϕ, 1R ζ)| < ε/2.
Since 1R ϕ, 1R ζ ∈ L1 , |(S(t)1R ϕ, 1R ζ)| < ε/2 for t > T = T (R(ε)).
3. Assume there exist such T and C. Then for any ϕ ∈ L1 ∩ L2 ,
kϕk2L2 = (S(T )ϕ, S(T )ϕ) ≤ kS(T )ϕkL1 kS(T )ϕkL∞ ≤ C3 kϕk2L1
where C3 = C(4πT )−d/2 . This cannot be true for all ϕ ∈ L1 ∩ L2 .
4. For small ε > 0, (ε < 0 is similar), we have
F (t + ε) − F (t) = I + II
where
Z
t+ε
Z
S(t + ε − s)f (s) ds = S(t + ε)
I=
t
and
S(−s)f (s)1[t,t+ε] (s) ds
R
Z
(S(t + ε − s) − S(t − s))f (s)ds = S(ε)F (t) − F (t).
II =
s<t
Since
kIkL2 ≤ kf (s)1[t,t+ε] (s)kLq0 Lr0 ,
it goes to 0 as ε → 0. Since F (t) ∈ L2 and S(ε) is continuous in ε in L2 , II → 0 as
ε → 0.
5. Extend f (t) by zero to t ∈ R\ I, and let E = {t ∈ R : f (t) = 0}. Define
fε (t, x) = f (t, x) + ε
1E (t)
ζ(x)
1 + t2
where ζ is a fixed function in X0 . Then our proof applies to fε for all ε > 0. Taking
ε → 0 we get the desired estimate for f .
The case q = ∞ is postponed.
6. (later)
7. (later)
8. (later)
9. (later)
10. (later)
APPENDICES
62
11. From the assumption u ∈ L∞ (R), the ODE shows uxx ∈ L∞ (R). Thus u is continuous,
2 . We may assume u(0) > 0, which implies u(x) ∈ R. Multiplying
and then u ∈ Cloc
the equation by 2ux and integrating, we get
u2x +
p+1
2
p+1 |u|
− u2 = const.
Since u ∈ H 1 (R), the constant is 0. If u(x0 ) = 0 for some x0 , then ux (x0 ) = 0 and
thus u ≡ 0 by uniqueness of ODE. Thus u(x) > 0 for all x ∈ R.
Let u(x) = av β (y), y = cx, for some a, b, c ∈ R. We get
a2 β 2 c2 v 2β−2 vy2 +
Setting c = 1/β, β =
2
p−1 ,
p+1 β(p+1)
2
v
p+1 a
− a2 v 2β = 0.
2 1/(p−1)
and a = ( p+1
)
, we get
vy2 + v 4 − v 2 = 0.
When v(y0 ) < 1, locally v(y) < 1 and vy 6= 0 has the same sign. Thus
p
vy = ε v 2 − v 4 , ε = sgn vy (y0 ).
This is separable and can be integrated, v(y) = sech(y + y0 ).
Thus u(x) = a sechβ (x/β) up to translation and phase.
12. (later)
13. (later)
Bibliography
[1] Agmon, Shmuel: Spectral properties of Schrdinger operators and scattering theory.
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218.
[2] Berestycki, H.; Lions, P.-L. Nonlinear scalar field equations. I. Existence of a ground
state. Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345.
[3] J. Bergh and J. Löfström, Interpolation spaces, An introduction. Grundlehren Math.
Wiss. 223, Springer, Berlin–Heiderberg–New York, 1976.
[4] T. Cazenave, Semilinear Schrödinger equations, Amer. Math. Soc., 2003.
[5] Chen, Chiun Chuan; Lin, Chang Shou Uniqueness of the ground state solutions of
∆u + f (u) = 0 in Rn , n ≥ 3. Comm. Partial Differential Equations 16 (1991), no. 8-9,
1549–1572.
[6] Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New
York; American Mathematical Society, Providence, RI, 2003.
[7] Cazenave, T.; Lions, P.-L. Orbital stability of standing waves for some nonlinear
Schrdinger equations. Comm. Math. Phys. 85 (1982), no. 4, 549–561.
[8] Erdogan, M. Burak; Goldberg, Michael; Schlag, Wilhelm: Strichartz and smoothing
estimates for Schrdinger operators with large magnetic potentials in R3 . J. Eur. Math.
Soc. (JEMS) 10 (2008), no. 2, 507–531.
[9] J. Fröhlich, S. Gustafson, L. Jonsson, and I.M. Sigal, Solitary wave dynamics in an
external potential. Comm. Math. Phys. 250 (2004) 613-642.
[10] Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry of positive solutions of nonlinear
elliptic equations in Rn . Mathematical analysis and applications, Part A, pp. 369–402,
Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.
[11] M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations. Comm. Pure Appl. Math. 41 (1988), no. 6, 747–774.
[12] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal. 74 (1987), no. 1, 160–197.
[13] S. Gustafson, K. Nakanishi and T.-P. Tsai, Asymptotic stability and completeness in
the energy space for nonlinear Schrödinger equations with small solitary waves, Int.
Math. Res. Not. 2004 (2004) no. 66, 3559–3584.
63
BIBLIOGRAPHY
64
[14] J. Iaia and H. Warchall, Nonradial solutions of a semilinear elliptic equation in two
dimensions. J. Differential Equations 119 (1995), no. 2, 533–558.
[15] Jensen, Arne; Kato, Tosio Spectral properties of Schrödinger operators and time-decay
of the wave functions. Duke Math. J. 46 (1979), no. 3, 583–611.
[16] Journé, J.-L.; Soffer, A.; Sogge, C. D. Decay estimates for Schrdinger operators. Comm.
Pure Appl. Math. 44 (1991), no. 5, 573–604.
[17] Kato, Tosio: On nonlinear Schrdinger equations. II. H s -solutions and unconditional
well-posedness. J. Anal. Math. 67 (1995), 281–306.
[18] Keel, Markus; Tao, Terence Endpoint Strichartz estimates. Amer. J. Math. 120 (1998),
no. 5, 955–980.
[19] M. K. Kwong, Uniqueness of positive solutions of ∆u − u + up = 0 in Rn , Arch. Rat.
Mech. Anal. 105 (1989), 243–266.
[20] P. L. Lions, Solutions complexes d’équations elliptiques semilinéaires dans RN . C. R.
Acad. Sci. Paris Sér. I Math. 302 (1986), no. 19, 673–676.
[21] T. Mizumachi, Vortex solitons for 2D focusing nonlinear Schrödinger equation. Differential Integral Equations 18 (2005), no. 4, 431–450.
[22] Pazy, A. Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279
pp. QA 377 P299
[23] C. A. Pillet and C. E. Wayne: Invariant manifolds for a class of dispersive, Hamiltonian,
partial differential equations. J. Differential Equations 141, no. 2, (1997), 310–326.
[24] S.I. Pohozaev, Eigenfunctions of the equation ∆u + λf (u) = 0. Sov. Math. Doklady 5
(1965) 1408-1411.
[25] Reed, Michael; Simon, Barry Methods of modern mathematical physics. II. Fourier
analysis, self-adjointness. Academic Press, 1975.
[26] I. Rodnianski, W. Schlag, A. Soffer, Dispersive analysis of charge transfer models,
Comm. Pure Appl. Math. 58 (2005), no. 2, 149–216.
[27] I. Rodnianski, W. Schlag, A. Soffer, Asymptotic stability of N-soliton states of NLS,
preprint, http://arxiv.org/abs/math.AP/0309114
[28] Shatah, Jalal; Strauss, Walter, Instability of nonlinear bound states. Comm. Math.
Phys. 100 (1985), no. 2, 173–190.
[29] A. Soffer and M.I. Weinstein, Multichannel nonlinear scattering theory for nonintegrable equations I, Comm. Math. Phys. 133 (1990), 119–146.
[30] A. Soffer and M.I. Weinstein, Multichannel nonlinear scattering theory for nonintegrable equations II, J. Diff. Eqns. 98, (1992), 376–390.
[31] Stein, Elias M. Singular integrals and differentiability properties of functions. Princeton
Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.
BIBLIOGRAPHY
65
[32] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory
integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series,
43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ,
1993.
[33] Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math.
Phys. 55 (1977), 149–162.
[34] Walter A. Strauss, Nonlinear wave equations, CBMS Regional Conference Series in
Mathematics, 73. American Mathematical Society, Providence, RI, 1989.
[35] C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equations: self-focusing and
wave collapse, Springer, 1999.
[36] T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional
Schrödinger equation. Comm. PDE 25 (2000) no. 7-8 1471-1485.
[37] T. Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, 106. American Mathematical Society, Providence, RI, 2006.
[38] T.-P. Tsai, Asymptotic dynamics of nonlinear Schrödinger equations with many bound
states, J. Diff. Equations 192 (2003), no.1, 225–282.
[39] T.-P. Tsai and H.-T. Yau, Asymptotic dynamics of nonlinear Schroedinger equations:
resonance dominated and dispersion dominated solutions, Comm. Pure Appl. Math.
55 (2002) 0153–0216.
[40] ——, Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res.
Not. 2002 (2002), no.31, 1629–1673.
[41] ——, Stable directions for excited states of nonlinear Schrödinger equations, Comm.
Partial Diff. Equ. 27 (2002), no. 11&12, 2363–2402.
[42] ——, Classification of asymptotic profiles for nonlinear Schrödinger equations with
small initial data, Adv. Theor. Math. Phys. 6 (2002), no.1, 107–139.
[43] Weinstein, Michael I. Nonlinear Schrdinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1982/83), no. 4, 567–576.
[44] ——, Modulational stability of ground states of nonlinear Schrdinger equations. SIAM
J. Math. Anal. 16 (1985), no. 3, 472–491.
[45] ——, Lyapunov stability of ground states of nonlinear dispersive evolution equations,
Comm. Pure Appl. Math. 39, (1986), 51–68.
[46] Kôsaku Yosida, Functional analysis. Reprint of the sixth (1980) edition. Classics in
Mathematics. Springer-Verlag, Berlin, 1995. xii+501 pp.