Random data Cauchy problem for supercritical Schrödinger equations

0.31 MB·

by  Laurent Thomann

RANDOM DATA CAUCHY PROBLEM FOR
SUPERCRITICAL SCHRO¨DINGER EQUATIONS
by
9
0 Laurent Thomann
0
2

n
a
J

7
2
Abstract. — InthispaperweconsidertheSchr¨odingerequationwithpower-

like nonlinearity and confining potential or without potential. This equation
]
P isknowntobewell-posedwithdatainaSobolevspaceHs ifsislargeenough
A and strongly ill-posed is s is below some critical threshold sc. Here we use
. the randomisation method of the inital conditions, introduced in N. Burq-N.
h Tzvetkov [7, 8] and we are able to show that the equation admits strong
at solutions for data in Hs for some s<sc.
m In the appendix we prove the equivalence between the smoothing effect for a
Schr¨odinger operator with confining potential and the decay of the associate
[
spectral projectors.

1
v R´esum´e. — Dans cet article on s’int´eresse `a l’´equation de Schr¨odinger
8 avec non-lin´earit´e polynˆomiale et potentiel confinant ou sans potentiel. Cette
3 ´equation est bien pos´ee pour des donn´ees dans un espace de Sobolev Hs si s
2 est assez grand, et fortement instable si s est sous un certain seuil critique
4 sc. Grˆace `a une randomisation des conditions initiales, comme l’ont fait N.
.
1 Burq-N. Tzvetkov [7, 8], on est capable de construire des solutions fortes
0 pour desdonn´ees dans Hs pourdes s<sc.
9 Dansl’appendice,ondonneunecaract´erisation del’effetr´egularisant pourun
0 op´erateur de Schr¨odinger avec potentiel confinant par la d´ecroissance de ses
:
v projecteurs spectraux.
i
X
r
a
2000 Mathematics Subject Classification. — 35A07; 35B35 ; 35B05 ; 37L50 ; 35Q55.
Key words and phrases. — NonlinearSchr¨odingerequation,potential,smoothingeffect,
random data, supercritical equation.
The authorwas supported in part by thegrant ANR-07-BLAN-0250.
2 LAURENT THOMANN
1. Introduction
In this paper we are concerned with the following nonlinear Schro¨dinger
equations
i∂ u+∆u= u r−1u, (t,x) R Rd,
t
(1.1) ±| | ∈ ×
(u(0,x) = f(x),
and
i∂ u+∆u V(x)u = u r−1u, (t,x) R Rd,
t
(1.2) − ±| | ∈ ×
(u(0,x) = f(x),
where r is an odd integer, and where V is a confining potential which satisfies
the following assumption
Assumption 1. — We suppose that V ∞(Rd,R ), and that there exists
+
∈ C
k 2 so that
≥
(i) There exists C > 1 so that for x 1, 1 x k V(x) C x k.
| | ≥ Ch i ≤ ≤ h i
(ii) For any j Nd, there exists C > 0 so that ∂jV(x) C x k−|j|.
∈ j | x | ≤ jh i
In the following, H will stand for the operator,
(1.3) H = ∆+V(x).
−
It is well known that under Assumption 1, the operator H has a self-ajoint
extensiononL2(Rd)(stilldenotedbyH)andhaseigenfunctions e which
n n≥1
form an Hilbertian basis of L2(Rd) and satisfy
(cid:0) (cid:1)
(1.4) He =λ2e , n 1,
n n n ≥
with λ + , when n + .
n
−→ ∞ −→ ∞
For s R and p 1, we define the Sobolev spaces based on the operator H
∈ ≥
s,p = s,p(Rd) = u ′(Rd) : H s2u Lp(Rd) ,
W W ∈ S h i ∈
and the Hilbert spaces (cid:8) (cid:9)
s = s(Rd) = s,2(Rd) = u ′(Rd) : H s2u L2(Rd) ,
H H W ∈ S h i ∈
where H = (1+H2)12. (cid:8) (cid:9)
h i
In our paper we either consider the case k = 2 in all dimension or the case
d = 1 and any k 2. As we will see, we crucially use the Lp bounds for the
≥
eigenfunctions e which are only known in these cases.
n
Our results for the Cauchy problem (1.1) will be deduced from the study of
(1.2) with the harmonic oscillator, thanks to a suitable transformation.
SUPERCRITICAL NLS 3
Let’s recall some results about the Cauchy problems (1.1) and (1.2).
1.1. Previous deterministic results. —
Here we mainly discuss the results concerning the problem (1.2). The nu-
merology for (1.1) is the same as (1.2) with a quadratic potential (k = 2). See
[20] for more references for the problem (1.1).
Assume here that d 1 and k 2.
≥ ≥
The linear Schro¨dinger flow enjoys Strichartz estimates, with loss of deriva-
tives in general and without loss in the special case k = 2.
We say that the pair (p,q) is admissible, if
2 d d
(1.5) + = , 2 p,q , (d,p,q) = (2,2, ).
p q 2 ≤ ≤ ∞ 6 ∞
Let 0 < T 1 and assume that the pair (p,q) is admissible, then the solution
≤
u of the equation
i∂ u Hu = 0, u(0,x) = f(x), (t,x) R Rd,
t
− ∈ ×
satisfies
(1.6) u . f ,
k kLp(0,T;Lq(Rd)) k kHρ(Rd)
with loss
0, if k = 2,
(1.7) ρ= ρ(p,k) =
( 1(1 1)+η, for any η > 0, if k > 2.
p 2 − k
In the case k = 2, these estimates follow from the dispersion properties of the
Schro¨dinger-Hermite group, obtained thanks to an explicit integral formula.
Then (1.6) follows from the standard TT∗ argument of J. Ginibre and G. Velo
[12], and the endpoint is obtained with the result of M. Keel and T. Tao [16].
In the case k > 2, the result is due to K. Yajima and G. Zhang [24].
Thanks to the estimates (1.6), K. Yajima and G. Zhang [24] are able to use a
fixed point argument in a Strichartz space and show that the problem (1.2) is
well-posed (with uniform continuity of the flow map) in s for s 0 so that
H ≥
d 2 1 1
s > ( + ).
2 − r 1 2 k
−
Thenextstatement showsthattheproblem(1.2)isill-posedbelow thethresh-
old s= d 2 . In particular when k = 2, the well-posedness result is sharp.
2 − r−1
4 LAURENT THOMANN
Theorem 1.1 (Ill-posedness [1, 20]). — Assume that d 2 > 0 and let
2 − r−1
0 < σ < d 2 . Then there exist a sequence f ∞(Rd) of Cauchy data
2 − r−1 n ∈ C
and a sequence of times t 0 such that
n
−→
f 0, when n + ,
n Hσ
k k −→ −→ ∞
and such that the solution u of (1.1) or (1.2) satisfies
n
σ
kun(tn)kHρ −→ +∞, when n −→ +∞, for all ρ∈ r−1(d σ),σ .
i 2 2 − i
Remark 1.2. — Indeedweprovedthisresultin[20]forthelaplacianwithout
potential. But the counterexamples constructed in the proof are functions
which concentrate exponentially at the point 0, so that a regular potential
plays no role.
This result shows that the flow map (if it exists) is not continuous at u = 0,
and that there is even a loss of regularity in the Sobolev scale. For this range
of σ, we can not solve the problems (1.1) or (1.2) with a classical fixed point
argument, as the uniform continuity of the flow map is a corollary of such a
method.
The index s := d 2 can be understood in the following way. Assume that
c 2−r−1
u is solution of the equation
(1.8) i∂ u+∆u= ur−1u, (t,x) R Rd,
t
| | ∈ ×
then for all λ > 0, uλ : (t,x) uλ(t,x) = λr−21u(λ2t,λx) is also solution of
7−→
(1.8). The homogenous Sobolev space which is invariant with respect to this
scaling is H˙sc(Rd).
Hence, for s < s , we say that the problems (1.1) and (1.2) are supercritical.
c
Now we show that we can break this threshold in some probabilistic sense.
1.2. Randomisation of the initial condition. —
Let (Ω, ,p) be a probability space. In the sequel we consider a sequence of
F
random variables (g (ω)) which satisfy
n n≥1
Assumption 2. — The random variables are independent and identically
distributed and are either
(i) Bernoulli random variables : p(g = 1) = p(g = 1) = 1,
n n − 2
SUPERCRITICAL NLS 5
or
(ii) complex Gaussian random variables gn C(0,1).
∈ N
A complex Gaussian X C(0,1) can be understood as
∈ N
√2
X(ω) = X (ω)+iX (ω) ,
1 2
2
where X1,X2 R(0,1) are indepe(cid:0)ndent. (cid:1)
∈ N
Each f s can be written in the hilbertian basis (e ) defined in (1.4)
n n≥1
∈ H
f(x)= α e (x),
n n
n≥1
X
and we can consider the map
(1.9) ω fω(x) = α g (ω)e (x),
n n n
7−→
n≥1
X
from (Ω, ) to s equipped with the Borel sigma algebra. The map (1.9)
F H
is measurable and fω L2(Ω; s). The random variable fω is called the
∈ H
randomisation of f.
The map (1.9) was introduced by N. Burq and N. Tzvetkov [7, 8] in the
context of the wave equation. More precisely the authors study the problem
(∂2u ∆)u+u3 = 0, (t,x) R M,
(1.10) t − ∈ ×
((u(0,x),∂tu(0,x) = (f1(x),f2(x)) Hs(M) Hs−1(M),
∈ ×
where M is a three dimensional compact manifold.
This equation is H12 H−21 critical, and known to be well-posed for s 1
× ≥ 2
and ill-posed for s < 1. Using that the randomised initial condition (fω,fω)
2 1 2
is almost surely more regular than (f ,f ) in Lp spaces, N. Burq and N.
1 2
Tzvetkov are able to show that the problem (1.10) admits a.s. strong solu-
tions for s 1 (resp. s 8 ) if ∂M = (resp. ∂M = ).
≥ 4 ≥ 21 ∅ 6 ∅
Some authors have used random series to construct invariant Gibbs measures
for dispersive PDEs, in order to get long-time dynamic properties of the
flow map, see J. Bourgain [2, 3], P. Zhidkov [26], N. Tzvetkov [23, 22, 21],
N. Burq-N. Tzvetkov [6]. However, to the best of the author’s knowledge,
[7, 8] is the first work in which stochastic methods are used in the proof of
existence itself of solutions for a dispersive PDE. But above all, it is the only
well-posedness result for a supercritical equation.
6 LAURENT THOMANN
In this paper, we adapt these ideas for the study of the problem (1.1).
1.3. The main results. —
1.3.1. The cubic Schro¨dinger equation with quadratic potential. —
Our first result deals with the case V(x) x 2 in all dimension, for the cubic
∼ h i
equation
i∂ u+∆u V(x)u = u 2u, (t,x) R Rd,
t
(1.11) − ±| | ∈ ×
(u(0,x) = f(x).
Theorem 1.3. — Let V satisfy Assumption 1 with k = 2, and d 1.
≥
Let σ > d 1 1 and f σ. Consider the function fω L2(Ω; σ)
2 − − d+3 ∈ H ∈ H
given by the randomisation (1.9). Then there exists s > d 1 such that : for
2 −
almost all ω Ω there exist T > 0 and a unique solution to (1.11) with initial
ω
∈
condition fω of the form
(1.12)
u(t, ) = e−itHfω + [0,T ]; s(Rd) Lp [0,T ]; s,q(Rd) ,
ω ω
· C H W
(p,q)admissible
(cid:0) (cid:1) \ (cid:0) (cid:1)
More precisely : For every 0< T 1 there exists an event Ω so that
T
≤
p(Ω ) 1 Ce−c/Tδ, C,c,δ > 0,
T
≥ −
and so that for all ω Ω , there exists a unique solution to (2.12) in the class
T
∈
(1.12).
Remark 1.4. — Our method allows to treat every power-like nonlinearity.
The gauge invariance structure of the nonlinearity plays no role, as we only
work in Strichartz spaces.
Remark 1.5. — As is [7], we can replace the Assumption2 madeon (g )
n n≥1
by any sequence of independent, centred random variables which satisfy some
integrability conditions. However the event Ω in Theorem 1.3 will generally
T
be of the form
p(Ω ) 1 CTδ.
T
≥ −
Remark 1.6. — Let ε > 0 and s R. If f s is such that f s+ε, then
∈ ∈ H 6∈ H
for almost all ω Ω, fω s and fω s+ε, hence the randomisation has
∈ ∈ H 6∈ H
no regularising effect in the L2 scale. See Lemma B.1. in [7] for a proof of
this fact.
SUPERCRITICAL NLS 7
1.3.2. The cubic Schro¨dinger equation. —
WearealsoabletoconsiderthecaseofthecubicSchro¨dingerequationwithout
potential
i∂ u+∆u= u 2u, (t,x) R Rd,
t
(1.13) ±| | ∈ ×
(u(0,x) = f(x).
Theorem 1.7. — Let d 1. Let σ > d 1 1 and f σ. Consider
≥ 2 − − d+3 ∈ H
the function fω L2(Ω; σ) given by the randomisation (1.9). Then there
∈ H
exists s > d 1 such that : for almost all ω Ω there exist T > 0, u
2 − ∈ ω 0 ∈
[0,T ]; σ(Rd) and a unique solution to (1.13) with initial condition fω in
ω
C H
a space continuously embedded in
(cid:0) (cid:1)
Y = u + [0,T ]; s(Rd) .
ω 0 ω
C H
Remark 1.8. — In fact u can be w(cid:0)ritten u (t, ) =(cid:1) e−itH2fω, where is a
0 0
· L L
linear operator defined in (6.1) and (6.4), andH = ∆+ x2 is the harmonic
2
− | |
oscillator.
1.3.3. The Schro¨dinger equation in dimension 1. —
Our second result concerns the case V(x) x k, in dimension 1.
∼ h i
i∂ u+∆u V(x)u = ur−1u, (t,x) R R,
t
(1.14) − ±| | ∈ ×
(u(0,x) = f(x).
Theorem 1.9. — Let V satisfy Assumption 1 with k 2. Let r 9 be
≥ ≥
an odd integer. Let σ > 1 2 (1 + 1) 1 and f σ. Consider the
2 − r−1 2 k − 2k ∈ H
function fω L2(Ω; σ) given by the randomisation (1.9). Then there exists
∈ H
s > 1 2 (1 + 1) such that : for almost all ω Ω there exist T > 0 and
2 − r−1 2 k ∈ ω
a unique solution to (1.14) with initial condition fω in a space continuously
embedded in
(1.15) Y = e−itHfω + [0,T ]; s(R) .
ω ω
C H
More precisely : For every 0 < ε < 1 an(cid:0)d 0 < T 1 th(cid:1)ere exists an event ΩT
≤
so that
p(Ω ) 1 Ce−c0/Tδ, C,c ,δ > 0,
T 0
≥ −
and so that for all ω Ω , there exists a unique solution to (1.14) in the class
T
∈
(1.15).
Remark 1.10. — In the case r = 3, r = 5 or r = 7, the gain of derivative is
less that 1 . We do not write the details.
2k
8 LAURENT THOMANN
1.4. Notations and plan of the paper. —
Notations. — In this paper c, C denote constants the value of which may
changefromlinetoline. Theseconstants willalwaysbeuniversal, oruniformly
bounded with respect to the parameters p,q,κ,ε,ω,... We use the notations
a b, a. b if 1b a Cb, a Cb respectively.
∼ C ≤ ≤ ≤
The notation Lp stands for Lp(0,T), whereas Lq = Lq(Rd), and LpLq =
T T
Lp(0,T;Lq(Rd)). For 1 p , the number p′ is so that 1 + 1 = 1.
≤ ≤ ∞ p p′
The abreviation r.v. is meant for random variable.
In this paper we follow the strategy initiated by N. Burq and N. Tzvetkov
[7, 8].
In Section 2 we recall the Lp estimates for the Hermite functions and we show
asmoothingeffect in Lp spaces for thelinear solution of theSchro¨dinger equa-
tion,yieldbytherandomisation. Wealsoshowhowsomeapriorideterministic
estimates imply the main results.
In Section 3 we recall some deterministic estimates in Sobolev spaces.
In Section 4 we prove the estimates of Section 2 in the case k = 2, and con-
clude the proof of Theorem 1.3.
In Section 5 we consider the case d= 1 with any potential under Assumption
1 and conclude the proof of Theorem 1.9.
In Section 6 we are concerned with NLS without potential.
In the Appendix A we show that the (deterministic) smoothing effect for the
free Schro¨dinger equation with confining potential is equivalent to the decay
of the spectral projectors.
Remark 1.11. — In our forthcoming paper [5], thanks to the construction
of an invariant Gibbs measure, we will show that the following Schro¨dinger
equation
i∂ u+∆u x 2u= u2u, (t,x) R R,
t
(1.16) −| | ±| | ∈ ×
(u(0,x) = f(x),
admits a large set of rough (supercritical) initial conditions leading to global
solutions.
Acknowledgements. — The author would like to thank N. Burq, N.
Tzvetkov and C. Zuily for many enriching discussions on the subject. He is
also indebted to D. Robert for many clarifications on eigenfunctions of the
Schro¨dinger operator.
SUPERCRITICAL NLS 9
2. Stochastic estimates
In the following we will take profit on the Lp bounds for the eigenfunctions
of H. This result is due to Yajima-Zhang [25] in the case (d,k) = (1,k) and
to Koch-Tat˘aru [17] when (d,k) = (d,2)
Theorem 2.1 ([25, 17]). — Let k 2. Then the eigenfunctions e defined
n
≥
by (1.4) satisfy the bound
(2.1) e . λ−θ(q,k,d) e ,
k nkLq(Rd) n k nkL2(Rd)
where θ is defined by
2(1 1) if 2 q < 4,
k 2 − q ≤
 1 η for any η > 0 if q =4,
(2.2) θ(q,k,1) =  2k −

 1 2(1 1)(1 1) if 4 < q < ,
 2 − 3 − q − k ∞
4−k if q = ,
 6k ∞


and 
1 1 if 2 q < 2(d+3),
2 − q ≤ d+1
 1 η for any η > 0 if q = 2(d+3),
(2.3) θ(q,2,d) =  d+3 − d+1

 1 d(1 1) if 2(d+3) < q 2d ,
 3 − 3 2 − q d+1 ≤ d−2
1 d(1 1) if 2d q .
 − 2 − q d−2 ≤ ≤ ∞




Notice that θ can benegative, but its maximum is always positive, attained
for
2(d+3)
(2.4) q (d) = q = .
∗ ∗
d+1
Let f σ and consider fω given by the randomisation (1.9).
∈ H
Observethat thelinear solution to thelinear Schro¨dingerequation withinitial
condition fω is
uωf(t,x) = e−itHfω(x)= αngn(ω)e−iλ2nten(x).
n≥1
X
Now we state the main stochastic tool of the paper. See [7] for two different
proofs of this result, one based on explicit computations, and one based on
large deviation estimates.
10 LAURENT THOMANN
Lemma 2.2 ([7]). — Let(g (ω)) beasequenceofrandomvariableswhich
n n≥1
satisfies Assumption 2. Then for all r 2 and (c ) l2(N∗) we have
n
≥ ∈
1
c g (ω) . √r c 2 2.
n n Lr(Ω) | n|
(cid:13)nX≥1 (cid:13) (cid:16)nX≥1 (cid:17)
(cid:13) (cid:13)
Thanks to this result we will obtain
Proposition 2.3. — Let d 1, 2 q p r < , σ R and 0 < T 1.
≥ ≤ ≤ ≤ ∞ ∈ ≤
Let f σ and let fω be its randomisation given by (1.9). Then
∈ H
(2.5) ke−itHfωkLr(Ω)Lp(0,T)Wθ(q)+σ,q(Rd) . √rTp1 kfkHσ(Rd),
where θ(q)= θ(q,k,d) is the function defined in (2.2).
As a consequence, if we set
E (p,q,σ) = ω Ω : e−itHfω λ ,
λ,f { ∈ k kLp(0,T)Wθ(q)+σ,q ≥ }
then there exist c ,c > 0 such that for all p q 2, all λ > 0 and f σ
1 2
≥ ≥ ∈ H
2 c2λ2
(2.6) p(Eλ,f(p,q,σ)) ≤ exp c1pTp − f 2 .
k kHσ
(cid:0) (cid:1)
Remark 2.4. — The previous estimate can be compared to the known de-
terministic estimate
(2.7) H θ(q,2k,1)e−itHf Lp(R;L2(0,T)) . f L2(R),
kh i k k k
which is proved by K. Yajima and G. Zhang in [25].
See also Appendix A for an idea of the proof.
Proof of Proposition 2.3. — Let f = α e σ. Then we have the
n≥1 n n ∈ H
explicit computation
P
hHiθ(q2)+σe−itHfω = αngn(ω)e−itλ2nhλ2niθ(q2)+σen.
n≥1
X
Then by Lemma 2.2 we deduce
1
khHiθ(q2)+σe−itHfωkLr(Ω) . √r |αn|2λn2(θ(q)+σ)|en|2 2.
(cid:16)nX≥1 (cid:17)