On the instability for the cubic nonlinear Schrodinger equation

0.13 MB·English

by  Rémi Carles

ON THE INSTABILITY FOR THE CUBIC NONLINEAR
SCHRO¨DINGER EQUATION
7
0
0
RE´MICARLES
2

n
a Abstract. WestudytheflowmapassociatedtothecubicSchro¨dingerequa-
J tion in space dimension at least three. We consider initial data of arbitrary
size inHs, where0<s<sc, sc the criticalindex, andperturbations in Hσ,
9
whereσ<sc isindependent ofs. Weshowaninstabilitymechanisminsome
2
Sobolev spaces ofordersmallerthan s. Theanalysis reliesontwofeatures of

super-criticalgeometricoptics: creationofoscillation,andghosteffect.
]
P
A
.
h 1. Introduction
t
a
m WeconsidertheCauchyproblemforthecubic,defocusingSchr¨odingerequation:
1
[ (1.1) i∂ ψ+ ∆ψ =|ψ|2ψ, x∈Rn ; ψ =ϕ.
t 2 |t=0
1
Formally,the massandenergyassociatedto this equationareindependentoftime:
v
8
Mass: M[ψ](t)= |ψ(t,x)|2dx≡M[ψ](0)=M[ϕ],
5
ZRn
8
1 Energy: E[ψ](t)= |∇ψ(t,x)|2dx+ |ψ(t,x)|4dx≡E[ψ](0)=E[ϕ].
0 ZRn ZRn
7 Scaling arguments yield the critical value for the Cauchy problem in Hs(Rn):
0
n
/
h sc = −1.
2
t
a Assume n > 3, so that s > 0. It was established in [3] that (1.1) is locally well-
c
m posed in Hs(Rn) if s > s . On the other hand, (1.1) is ill-posed in Hs if s < s
c c
: ([4]). Moreover, the following norm inflation phenomenon was proved in [4] (see
v
i also [1, 2]): if 0<s<sc, we can find (ϕj)j∈N in the Schwartz class S(Rn) with
X
r (1.2) kϕjkHsj→−→+∞0,
a
and a sequence of positive times τ → 0, such that the solution ψ to (1.1) with
j j
initial data ϕ satisfy:
j
kψj(τj)kHs −→ +∞.
j→+∞
In [2], this was improved to: we can find t →0 such that
j
s
kψj(tj)kHkj→−→+∞+∞, ∀k ∈(cid:21)1+sc−s,s(cid:21).
Note that (1.2) means that we consider the flow map near the origin. We show
thatinsideringsofHs,thesituationisyetmoreinvolved: fordataboundedinHs,
2000 Mathematics Subject Classification. 35B33;35B65;35Q55;81Q05;81Q20.
SupportbytheANRprojectSCASENisacknowledged.
1
2 R.CARLES
with 0<s <s , we consider perturbations which are small in Hσ for any σ < s ,
c c
and infer a similar conclusion.
Theorem 1.1. Let n>3 and 06s<s = n−1. Fix C ,δ >0. We can find two
c 2 0
sequences of initial data (ϕj)j∈N and (ϕj)j∈N in the Schwartz class S(Rn), with:
C0−δ 6kϕjkHs,kϕjkHs 6C0+δe ; kϕj −ϕjkHσ −→ 0, ∀σ <sc,
j→+∞
and a sequence of positivee times t →0, such that theesolutions ψ and ψ to (1.1),
j j j
with initial data ϕ and ϕ respectively, satisfy:
j j e
s
kψj(tj)−ψj(tj)keHkj→−→+∞+∞, ∀k ∈(cid:21)1+sc−s,s(cid:21) (if s>0),
e
lji→m+in∞fkψj(tj)−ψj(tj)kH1+ssc−s >0.
e
The main novelty in this result is the fact that the initial data are close to each
other in Hσ, for any σ <s . In particular, this range for σ is independent of s.
c
Remark 1.2. Like in [1, 2], we consider initial data of the form
ϕj(x)=jn2−sa0(jx),
for some a ∈ S(Rn) independent of j. The above result holds for all a ∈ S(Rn)
0 0
with, say1, ka0kHs =C0, and ϕj(x)=(jn2−s+j)a0(jx) (see Section 2).
Considering the case s= n, we infer from the proof of Theorem 1.1:
4 e
Corollary 1.3. Let n>5 and C ,δ >0. We can find two sequences of initial data
0
(ϕj)j∈N and (ϕj)j∈N in the Schwartz class S(Rn), with:
C −δ 6E[ϕ ],E[ϕ ]6C +δ ; M[ϕ ]+M[ϕ ]+E[ϕ −ϕ ] −→ 0,
0 e j j 0 j j j j
j→+∞
and a sequence of positieve times t →0, such that theesolutions ψ eand ψ to (1.1)
j j j
with initial data ϕ and ϕ respectively, satisfy:
j j e
liminfE[ψ −ψ ](t )>0.
e j j j
j→+∞
e
2. Reduction of the problem: super-critical geometric optics
We now proceed as in [2]. We set ε=js−sc: ε→0 as j →+∞. We change the
unknown function as follows:
uε(t,x)=js−n2ψj(cid:18)jsc+t2−s,xj(cid:19).
Note that we have the relation:
kψ (t)k =jm−s uε jsc+2−st .
j H˙m H˙m
With initial data of the form ϕj(x)=jn2(cid:13)(cid:13)−sa0(cid:0)(jx)+ja(cid:1)1(cid:13)(cid:13)(jx), (1.1) becomes:
ε2
(2.1) iε∂ uε+ ∆uε =|uε|2uε ; uε(0,x)=a (x)+εa (x).
t 0 1
2
We emphasize twofeatures for the WKB analysisassociatedto (2.1). First,evenif
the initial datum is independent of ε, the solution instantly becomes ε-oscillatory.
This is the argument of the proof of [2, Cor. 1.7]. Second, the aspect which was
1Providedthatwechoosej sufficientlylarge.
INSTABILITY FOR CUBIC NLS 3
notusedintheproofof[2,Cor.1.7]iswhatwascalledghost effect ingasdynamics
([6]): a perturbation of order ε of the initial datum may instantly become relevant
at leading order. These two features are direct consequences of the fact that (2.1)
is super-critical as far as WKB analysis is concerned (see e.g. [2]).
Consider the two solutions uε and uε of (2.1) with a = 0 and a = a respec-
1 1 0
tively. Then Theorem 1.1 stems from the following proposition, which in turn is
essentially a reformulation of [2, Prop.e1.9 and 5.1].
Proposition 2.1. Let n>1 and a ∈S(Rn;R)\{0}. There exist T >0 indepen-
0
dent of ε∈]0,1], and a,φ,φ ∈C([0,T];Hs) for all s>0, such that:
1
kuε−aeiφ/εkL∞([0,T];Hεs)+kuε−aeiφ1eiφ/εkL∞([0,T];Hεs) =O(ε), ∀s>0,
where
e
kfk2 = 1+|εξ|2 s|f(ξ)|2dξ,
Hεs ZRn
(cid:0) (cid:1)
and f stands for the Fourier transform of f. In adbdition, we have, in Hs:
φb(t,x)=−t|a0(x)|2+O(t3) ; φ1(t,x)=−2t|a0(x)|2+O(t3) as t→0.
Therefore, there exists τ >0 independent of ε, such that:
liminfεskuε(τ)−uε(τ)k >0, ∀s>0.
H˙s
ε→0
3. Outline of the peroof of Proposition 2.1
The idea, due to E. Grenier [5], consists in writing the solution to (2.1) as
uε(t,x)=aε(t,x)eiφε(t,x)/ε, where aε is complex-valued, and φε is real-valued. We
assume that a ,a ∈ S(Rn) are independent of ε. For simplicity, we also assume
0 1
that they are real-valued. Impose:
1
∂ φε+ |∇φε|2+|aε|2 =0 ; φε(0,x)=0.
t
(3.1)  2
∂ aε+∇φε·∇aε+ 1aε∆φε =iε∆aε ; aε(0,x)=a (x)+εa (x).
t 0 1
2 2
Workingwith the unknown function uε = t(Reaε,Imaε,∂ φε,...,∂ φε), (3.1)
1 n
yields a symmetric quasi-linear hyperbolic system: for s > n/2+2, there exists
T >0 independent of ε∈]0,1] (and of s, from tame estimates), such that (3.1) has
a unique solution (φε,aε) ∈ C([0,T];Hs)2. Moreover, the bounds in Hs(Rn) are
independent of ε, and we see that (φε,aε) converges to (φ,a), solution of:
1
∂ φ+ |∇φ|2+|a|2 =0 ; φ(0,x)=0.
t
(3.2)  2
 1
∂ a+∇φ·∇a+ a∆φ=0 ; a(0,x)=a (x).
t 0
2
More precisely, energy estimates for symmetric systems yield:
kφε−φkL∞([0,T];Hs)+kaε−akL∞([0,T];Hs) =O(ε), ∀s>0.
Onecanprovethatφεandaεhaveanasymptoticexpansioninpowersofε. Consider
the next term, given by:
∂ φ(1)+∇φ·∇φ(1)+2Re aa(1) =0 ; φ(1) =0.
t t=0
 (cid:16) (cid:17) (cid:12)
∂ a(1)+∇φ·∇a(1)+∇φ(1)·∇a+ 1a(1)∆φ+(cid:12) 1a∆φ(1) = i∆a ; a(1) =a .
t 2 2 2 t=0 1
(cid:12)
 (cid:12)
4 R.CARLES
Then a(1),φ(1) ∈L∞([0,T];Hs) for every s>0, and
kaε−a−εa(1)kL∞([0,T∗];Hs)+kΦε−φ−εφ(1)kL∞([0,T∗];Hs) 6Csε2, ∀s>0.
Observe that since a is real-valued, (φ(1),Re(aa(1))) solves an homogeneous linear
system. Therefore, if Re(aa(1))=0 at time t=0, then φ(1) ≡0.
Considering the cases a =0 and a = a for uε and uε respectively, we obtain
1 1 0
the first assertion of Prop. 2.1. Note that the above O(ε2) becomes an O(ε) only,
sincewedivideφε andφbyε. ThisalsoexplainswhytheefirstestimateofProp.2.1
is stated in Hs and not in Hs. The rest of the proposition follows easily.
ε
Remark 3.1. We could use the ghost effect at higher order. For N ∈ N, assume
uε =(1+εN)a for instance. Then for some τ >0 independent of ε, we have
|t=0 0
e liminf εskuε(τ)−uε(τ)kH˙s ×ε1−N >0, ∀s>0.
ε→0
(cid:0) (cid:1)
Back to the functions ψ, the rangeefor k becomes:
s+(s −s)(N −1)
k> c ·
1+s −s
c
For this lower bound to be strictly smaller than s, we have to assume s>N −1.
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Universit´e Montpellier 2, Math´ematiques, UMR CNRS 5149, CC 051, Place Eug`ene
Bataillon,34095Montpelliercedex 5,France2
E-mail address: [email protected]
2
Presentaddress: WolfgangPauliInstitute, Universita¨tWien,Nordbergstr.15,A-1090Wien