Random Data Cauchy Problem for Supercritical Schrödinger Equations

Random data Cauchy problem for supercritical Schrödinger equations

Laurent Thomann

Université de Nantes, Laboratoire de Mathématiques J. Leray, UMR CNRS 6629, 2, rue de la Houssinière, F-44322 Nantes Cedex 03, France Received 5 February 2009; received in revised form 2 June 2009; accepted 4 June 2009

Available online 9 June 2009

Abstract

In this paper we consider the Schrödinger equation with power-like nonlinearity and confining potential or without potential.

This equation is known to be well-posed with data in a Sobolev spaceH^s ifsis large enough and strongly ill-posed issis below some critical thresholdsc. Here we use the randomisation method of the inital conditions, introduced by N. Burq and N. Tzvetkov, and we are able to show that the equation admits strong solutions for data inH^sfor somes < s_c.

©2009 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article on s’intéresse à l’équation de Schrödinger avec non-linéarité polynômiale et potentiel confinant ou sans potentiel.

Cette équation est bien posée pour des données dans un espace de SobolevH^s sisest assez grand, et fortement instable sisest sous un certain seuil critiques_c. Grâce à une randomisation des conditions initiales, comme l’ont fait N. Burq et N. Tzvetkov, on est capable de construire des solutions fortes pour des données dansH^savec dess < sc.

©2009 Elsevier Masson SAS. All rights reserved.

MSC:35A07; 35B35; 35B05; 37L50; 35Q55

Keywords:Nonlinear Schrödinger equation; Potential; Random data; Supercritical equation Mots-clés :Équation de Schrödinger non linéaire ; Potentiel ; Équation surcritique

1. Introduction

In this paper we are concerned with the following nonlinear Schrödinger equations i∂_tu+u= ±|u|^r−1u, (t, x)∈R×R^d,

u(0, x)=f (x), (1.1)

E-mail address:[email protected].

URL:http://www.math.sciences.univ-nantes.fr/~thomann/.

1 The author was supported in part by the grant ANR-07-BLAN-0250.

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.06.001

and

i∂tu+u−V (x)u= ±|u|^r−1u, (t, x)∈R×R^d,

u(0, x)=f (x), (1.2)

whereris an odd integer, and whereV is a confining potential which satisfies the following assumption.

Assumption 1.We suppose thatV ∈C^∞(R^d,R+), and that there existsk2 so that:

(i) There existsC >1 so that for|x|1, _C¹x^kV (x)Cx^k. (ii) For anyj∈N^d, there existsCj>0 so that|∂_x^jV (x)|Cjx^k−|j|.

In the following,Hwill stand for the operator, H= −+V (x).

It is well known that under Assumption 1, the operatorHhas a self-adjoint extension onL²(R^d)(still denoted byH) and has eigenfunctions(e_n)_n1which form a hilbertian basis ofL²(R^d)and satisfy

H e_n=λ²_ne_n, n1, (1.3)

withλ_n→ +∞, whenn→ +∞.

Fors∈Randp1, we define the Sobolev spaces based on the operatorH W^s,p=W^s,p

R^d

= u∈S

R^d

: H^s2u∈L^p R^d

, and the Hilbert spaces

H^s=H^s R^d

=W^s,2 R^d

= u∈S

R^d

: H^s2u∈L² R^d

, (1.4)

whereH =(1+H²)¹².

In our paper we either consider the casek=2 in all dimension or the cased=1 and anyk2. As we will see, we crucially use theL^pbounds for the eigenfunctionse_nwhich are only known in these cases.

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.

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 numerology for (1.1) is the same as (1.2) with a quadratic potential (k=2). See [14] for more references for the problem (1.1).

Assume here thatd1 andk2.

The linear Schrödinger flow enjoys Strichartz estimates, with loss of derivatives in general and without loss in the special casek=2.

We say that the pair(p, q)is admissible, if 2

p+d q =d

2, 2p, q∞, (d, p, q)=(2,2,∞). (1.5)

Let 0< T 1 and assume that the pair(p, q)is admissible, then the solutionuof the equation i∂_tu−H u=0, u(0, x)=f (x), (t, x)∈R×R^d,

satisfies

u _Lp(0,T;L^q(R^d)) f _Hρ(R^d), (1.6)

with loss

ρ=ρ(p, k)=

0, ifk=2,

2

p(¹₂−¹_k)+η, for anyη >0, ifk >2. (1.7)

In the casek=2, these estimates follow from the dispersion properties of the Schrödinger–Hermite group, ob- tained thanks to an explicit integral formula. Then (1.6) follows from the standardTT^∗argument of J. Ginibre and G. Velo [10], and the endpoint is obtained with the result of M. Keel and T. Tao [11].

In the casek >2, the result is due to K. Yajima and G. Zhang [18].

Thanks to the estimates (1.6), K. Yajima and G. Zhang [18] 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) inH^sfors0 so that

s >d 2 − 2

r−1 1

2+1 k .

The next statement shows that the problem (1.2) is ill-posed below the thresholds=^d₂−_r−²₁. In particular when k=2, the well-posedness result is sharp.

Theorem 1.1(Ill-posedness). (See [7,14].) Assume that ^d₂−_r−²₁ >0and let0< σ < ^d₂−_r−²₁. Then there exist a sequencef_n∈C^∞(R^d)of Cauchy data and a sequence of timest_n→0such that

f_{n H}σ→0, whenn→ +∞,

and such that the solutionunof (1.1)or(1.2)satisfies u_n(t_n)

H^ρ→ +∞, whenn→ +∞,for allρ∈

σ

r−1

2 (^d₂−σ ), σ

.

Remark 1.2.Indeed we proved this result in [14] for the laplacian without potential. But the counterexamples con- structed 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 atu=0, and that there is even a loss of regularity in the Sobolev scale. For this range of σ, we cannot 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 indexs_c:=^d₂−_r−²₁ can be understood in the following way. Assume thatuis solution of the equation

i∂_tu+u= |u|^r−1u, (t, x)∈R×R^d, (1.8)

then for allλ >0,u_λ:(t, x)→u_λ(t, x)=λ^r−12 u(λ²t, λx)is also solution of (1.8). The homogenous Sobolev space which is invariant with respect to this scaling isH˙^sc(R^d).

Hence, fors < s_c, we say that the problems (1.1) and (1.2) aresupercritical.

Now we show that we can break this threshold in some probabilistic sense.

1.2. Randomisation of the initial condition

Let(Ω,F,p)be a probability space. In the sequel we consider a sequence of random variables(g_n(ω))_n1which satisfy

Assumption 2.The random variables are independent and identically distributed and are either (i) Bernoulli random variables:p(gn=1)=p(gn= −1)=¹₂, or

(ii) complex Gaussian random variablesgn∈NC(0,1).

A complex GaussianX∈N_C(0,1)can be understood as X(ω)=

√2 2

X₁(ω)+iX₂(ω) , whereX1, X2∈N_R(0,1)are independent.

Eachf ∈H^s can be written in the hilbertian basis(en)n1defined in (1.3) f (x)=

n1

α_ne_n(x),

and we can consider the map ω→f^ω(x)=

n1

αngn(ω)en(x), (1.9)

from(Ω,F)toH^s equipped with the Borel sigma algebra. The map (1.9) is measurable andf^ω∈L²(Ω;H^s). The random variablef^ωis called the randomisation off.

The map (1.9) was introduced by N. Burq and N. Tzvetkov [5,6] in the context of the wave equation. More precisely the authors study the problem

∂_t²u−

u+u³=0, (t, x)∈R×M, u(0, x), ∂tu(0, x)

=

f₁(x), f₂(x)

∈H^s(M)×H^s−1(M), (1.10)

whereMis a three-dimensional compact manifold.

This equation isH¹² ×H⁻¹² critical, and known to be well-posed fors¹₂ and ill-posed fors < ¹₂. Using that the randomised initial condition (f₁^ω, f₂^ω)is almost surely more regular than (f1, f2) inL^p spaces, N. Burq and N. Tzvetkov are able to show that the problem (1.10) admits a.s. strong solutions fors¹₄ (resp.s₂₁⁸) if∂M= ∅ (resp.∂M= ∅).

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 [1,2], P. Zhidkov [20], N. Tzvetkov [17,16,15], N. Burq and N. Tzvetkov [4]. See also our forthcoming paper [3]. However, to the best of the author’s knowledge, [5,6] 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 dispersive equation.

In this paper, we adapt these ideas to the study of the problem (1.1).

1.3. The main results

1.3.1. The cubic Schrödinger equation with quadratic potential

Our first result deals with the caseV (x)∼ x²in all dimension, for the cubic equation i∂tu+u−V (x)u= ±|u|²u, (t, x)∈R×R^d,

u(0, x)=f (x). (1.11)

Theorem 1.3.LetV satisfy Assumption1withk=2.

Assume that d2. Letσ > ^d₂ −1−_d+¹₃ andf ∈H^σ(R). Consider the functionf^ω∈L²(Ω;H^σ(R))given by the randomisation(1.9). Then there existss >^d₂−1such that:for almost allω∈Ω there existT_ω>0and a unique solution to(1.11)with initial conditionf^ω of the form

u(t,·)=e^{−it H}f^ω+C

[0, Tω];H^s

R^d

(p,q)admissible

L^p

[0, Tω];W^s,q R^d

. (1.12)

More precisely:For every0< T 1there exists an eventΩ_T so that p(ΩT)1−Ce^−c/Tδ, C, c, δ >0,

and so that for allω∈Ω_T, there exists a unique solution to(2.12)in the class(1.12).

In the cased=1, the same conclusion holds forσ >−¹₄ with ans0.

Remark 1.4.Our method allows to treat every power-like nonlinearity. The gauge invariance structure of the nonlin- earity plays no role, as we only work in Strichartz spaces.

Remark 1.5.As is [5], we can replace Assumption 2 made on (g_n)_n1 by any sequence of independent, centred random variables which satisfy some integrability conditions. However the eventΩ_T in Theorem 1.3 will generally be of the form

p(ΩT)1−CT^δ.

Remark 1.6.Let ε >0 and s∈R. If f ∈H^s is such that f /∈H^s+ε, then for almost all ω∈Ω,f^ω ∈H^s and f^ω∈/H^s+ε, hence the randomisation has no regularising effect in theL²scale. See [5, Lemma B.1] for a proof of this fact.

1.3.2. The cubic Schrödinger equation

We are also able to consider the case of the cubic Schrödinger equation without potential i∂_tu+u= ±|u|²u, (t, x)∈R×R^d,

u(0, x)=f (x). (1.13)

Denote byH^s(R^d)the space defined in (1.4) whenV (x)= |x|²(the harmonic oscillator). Then we have

Theorem 1.7.Letd 2. Let σ >^d₂−1−_d+¹₃ andf ∈H^σ(R). Consider the functionf^ω∈L²(Ω;H^σ(R))given by the randomisation(1.9). Then there exists s > ^d₂ −1 such that:for almost allω∈Ω there existT_ω>0,u₀∈ C([0, Tω];H^σ(R^d))and a unique solution to(1.13)with initial conditionf^ωin a space continuously embedded in

Y_ω=u₀+C

[0, Tω];H^s R^d

.

In the cased=1, the same conclusion holds forσ >−¹₄with ans0.

Remark 1.8.In factu₀can be writtenu₀(t,·)=Le^{−it H2}f^ω, whereLis a linear operator defined in (6.1) and (6.4), andH₂= −+ |x|²is the harmonic oscillator.

Remark 1.9.Denote byH^s(R^d)the usual Sobolev space onR^d. Fors >0, we haveH^s(R^d)⊂H^s(R^d), hence our result cannot be compared to the classical deterministic well-posedness results. However, in the case of the dimensions d=1,2, we obtain a result for initial conditions with negative regularity. Moreover we can observe thatH^s(R^d)⊂ H^s(R^d)fors0, therefore we can read the previous result in the usual setting.

1.3.3. The Schrödinger equation in dimension 1

Our second result concerns the caseV (x)∼ x^k, in dimension 1.

i∂_tu+u−V (x)u= ±|u|^r−1u, (t, x)∈R×R,

u(0, x)=f (x). (1.14)

Theorem 1.10.LetV satisfy Assumption1withk2. Letr9be an odd integer. Letσ > ¹₂−_r−²₁(¹₂+¹_k)−_2k¹ and f ∈H^σ(R). Consider the functionf^ω∈L²(Ω;H^σ(R)) given by the randomisation(1.9). Then there exists s >¹₂−_r−²₁(¹₂+¹_k)such that:for almost allω∈Ω there existT_ω>0and a unique solution to(1.14)with initial conditionf^ωin a space continuously embedded in

Y_ω=e^{−it H}f^ω+C

[0, Tω];H^s(R)

. (1.15)

More precisely:For every0< ε <1and0< T 1there exists an eventΩ_T so that p(ΩT)1−Ce^−c0/Tδ, C, c₀, δ >0,

and so that for allω∈Ω_T, there exists a unique solution to(1.14)in the class(1.15).

Remark 1.11.In the caser=3,r=5 orr=7, the gain of derivative is less that _2k¹. We do not write the details.

1.4. Notations and plan of the paper

Notations. In this paperc, C denote constants the value of which may change from line to line. These constants will always be universal, or uniformly bounded with respect to the parametersp, q, κ, ε, ω, . . . .We use the notations a∼b,abif _C¹baCb,aCbrespectively.

The notationL^p_T stands forL^p(0, T ), whereasL^q=L^q(R^d), andL^p_TL^q=L^p(0, T;L^q(R^d)). For 1p∞, the numberpis so that _p¹+_p¹ =1.

The abbreviation r.v. is meant for random variable.

In this paper we follow the strategy initiated by N. Burq and N. Tzvetkov [5,6].

In Section 2 we recall theL^pestimates for the Hermite functions and we show a smoothing effect inL^p spaces for the linear solution of the Schrödinger equation, yield by the randomisation. We also show how some a priori deterministic 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 casek=2, and conclude the proof of Theorem 1.3.

In Section 5 we consider the cased=1 with any potential under Assumption 1 and conclude the proof of Theo- rem 1.10.

In Section 6 we are concerned with NLS without potential.

Remark 1.12.In our forthcoming paper [3], thanks to the construction of an invariant Gibbs measure, we will show that the following Schrödinger equations

i∂_tu+u− |x|²u=κ₀|u|^r−1u, (t, x)∈R×R,

u(0, x)=f (x), (1.16)

with(κ0= −1 andr=3)or(κ0=1 andr3)admit a large set of rough (supercritical) initial conditions leading to global solutions.

2. Stochastic estimates

In the following we will take profit on theL^pbounds for the eigenfunctions ofH. This result is due to Yajima and Zhang [19] in the case(d, k)=(1, k)and to Koch and Tataru [12] when(d, k)=(d,2).

Theorem 2.1.(See [19,12].) Letk2. Then the eigenfunctionse_ndefined by(1.3)satisfy the bound

e_{n L}q(R^d)λ⁻_n^{θ (q,k,d)} e_{n L}2(R^d), (2.1)

whereθis defined by

θ (q, k,1)=

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

2

k(¹₂−_q¹) if2q <4,

1

2k −ηfor anyη >0 ifq=4,

1

2−²₃(1−_q¹)(1−¹_k) if4< q <∞,

4−k

6k ifq= ∞,

(2.2)

and

θ (q,2, d)=

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

1

2−¹_q if2q <^2(d_d+⁺₁³⁾,

1

d+3−ηfor anyη >0 ifq=^2(d_d+⁺₁³⁾,

1

3−^d₃(¹₂−¹_q) if ^2(d_d+⁺₁³⁾< q_d^2d₋₂, 1−d(¹₂−¹_q) if _d^2d₋₂q∞.

(2.3)

Notice thatθcan be negative, but its maximum is always positive, attained for q_∗(d)=q_∗=2(d+3)

d+1 . (2.4)

Letf∈H^σ and considerf^ω given by the randomisation (1.9).

Observe that the linear solution to the linear Schrödinger equation with initial conditionf^ωis u^ω_f(t, x)=e^{−it H}f^ω(x)=

n1

α_ng_n(ω)e^−iλ2nte_n(x).

Now we state the main stochastic tool of the paper. See [5] for two different proofs of this result, one based on explicit computations, and one based on large deviation estimates.

Lemma 2.2.(See [5].) Let(g_n(ω))_n1be a sequence of random variables which satisfies Assumption2. Then for all r2and(cn)∈l²(N^∗)we have

n1

cngn(ω) L^r(Ω)

√

r

n1

|cn|²

1 2

.

Thanks to this result we will obtain

Proposition 2.3.Letd1,2qpr <∞,σ∈Rand0< T 1. Letf∈H^σ and letf^ωbe its randomisation given by(1.9). Then

e^{−it H}f^ω

L^r(Ω)L^p(0,T )W^{θ (q)+σ,q}(R^d)√

rT^p1 f _Hσ(R^d), (2.5)

whereθ (q)=θ (q, k, d)is the function defined in(2.2).

As a consequence, if we set E_λ,f(p, q, σ )=

ω∈Ω: e^{−it H}f^ω

L^p(0,T )W^{θ (q)+σ,q} λ ,

then there existc₁, c₂>0such that for allpq2, allλ >0andf ∈H^σ p

Eλ,f(p, q, σ )

exp

c1pT^p2 − c₂λ² f ²_Hσ

. (2.6)

Remark 2.4.The previous estimate can be compared to the known deterministic estimate H^{θ (q,k,1)2} e^{−it H}f

L^p(R;L²(0,T )) f _L2(R), (2.7)

which is proved by K. Yajima and G. Zhang in [19].

Proof of Proposition 2.3. Letf =

n1α_ne_n∈H^σ. Then we have the explicit computation H^{θ (q)2+σ}e^{−it H}f^ω=

n1

α_ng_n(ω)e^{−it λ2n} λ²_n^{θ (q)+σ}

2 e_n.

Then by Lemma 2.2 we deduce H^{θ (q)2+σ}e^{−it H}f^ω

L^r(Ω)√ r

n1

|α_n|²λ^{2(θ (q)}_n ^{+σ )}|e_n|²

1 2

.

Now, for 2qrtake theL^q(R^d)norm of the previous estimate. By Minkowski and by the bounds (2.1), we obtain

e^{−it H}f^ω

L^r(Ω)W^{θ (q)+σ,q}(R^d)=H^{θ (q)2+σ}e^{−it H}f^ω

L^r(Ω)L^q(R^d)

H^{θ (q)2+σ}e^{−it H}f^ω

L^q(R^d)L^r(Ω)

√

r

n1

|αn|²λ^{2(θ (q)}_n ^{+σ )} en 2 L^q

1 2

√

r

n1

|α_n|²λ^2σ_n

1

2 =√

r f _Hσ. (2.8)

For 2qprwe now take theL^p(0, T )norm of (2.8), and by Minkowski again e^{−it H}f^ω

L^r(Ω)L^p(0,T )W^{θ (q)+σ,q}e^{−it H}f^ω

L^p(0,T )L^r(Ω)W^{θ (q)+σ,q}

√

rT^p1 f _Hσ, which is the estimate (2.5).

By the Bienaymé–Tchebychev inequality, there existsC₀>0 such that p

E_λ,f(p, q, σ )

=pH^{θ (q)+σ2} e^{−it H}f^ωr

L^p(0,T )L^q(R)λ^r

C0√

rT^p1 f _H^σ λ

r

.

Eitherλ >0 is such that λ

f _Hσ C₀√

pT^p1e, (2.9)

then inequality (2.6) holds forc₁>0 large enough.

Or we define r:=

λ C₀T^1p f _Hσe

2

p, (2.10)

then p

E_λ,f(p, q, σ )

e^−r=exp

− cλ² f ²_Hσ

,

hence the result. 2

Recall the notation (2.4) and define the event

E_λ,f =E_λ,f(M, q_∗, ε), (2.11)

whereMis a large positive number which will be fixed in Sections 4 and 5.

We now show how the proof of the local existence of the Cauchy problem (1.2) with randomised data can be reduced to a priori deterministic estimates.

In fact we want to solve the equation

i∂_tu−H u= |u|^r−1u, (t, x)∈R×R^d, u(0)=f^ω∈L²

Ω;H^σ

, (2.12)

whereσ∈Rand the operatorH satisfies Assumption 1.

This problem has the integral formulation u(t,·)=u^ω_f(t )−i

t 0

e^{−i(t−τ )H}|u|^r−1u(τ,·)dτ, whereu^ω_f stands for e^{−it H}f^ω.

Writeu=u^ω_f +v. Therefore,vsatisfies the integral equation

v(t,·)= −i t 0

e^{−i(t−τ )H}u^ω_f +v^r−1 u^ω_f +v

(τ,·)dτ, thus we are reduced to find a fixed point of the map

K_f^ω: v→ −i t 0

e^{−i(t−τ )H}u^ω_f +v^r−1 u^ω_f +v

(τ,·)dτ.

Indeed the next proposition shows how a priori estimates onKimply the local well-posedness results.

Proposition 2.5.(See [5].) Let0< T 1andσ∈R. Letf ∈H^σ andf^ω∈L²(Ω;H^σ)be its randomisation. Assume there existsσand a spaceX_T^s ⊂C([0, T];H^s)and constantsκ >0,C >0so that for everyv, v₁, v₂∈X^s_T,λ >0 andω∈E_λ,f^c we have

K_f^ω(v)

X^s_T CT^κ

λ^r+ v ^r_Xs T

, (2.13)

and

K_f^ω(v₁)−K_f^ω(v₂)

X_T^s CT^κ v₁−v_{2 X}^s

T

λ^r−1+ v₁ ^r−1

X^s_T + v₂ ^r−1

X_T^s

. (2.14)

Then for every0< T 1there exists an eventΩ_T so that p(ΩT)1−Ce^−c0/Tδ, C, c₀, δ >0,

and so that for allω∈ΩT, there exists a unique solution to(2.12)of the form u(t,·)=e^{−it H}f^ω+X_T^s.

Proof. Here we can follow the proof given in [5].

Let 0< μ <1 be small. Defineδ=_r^κ2, whereκis given by Proposition 2.5, and let 0< T 1 be such thatT^δμ.

Take alsoω∈E_λ,f^c .

In a first time, we will show that the applicationKis a contraction on the ballB(0,2Cλ^r)inX^s_T forλ=μT^−δ (1), ifμis chosen small enough, depending only on the absolute constantC.

By (2.13) and (2.14), to have a contraction, it suffices to findμ >0 such that the following inequalities hold CT^κ

λ^r+

2Cλ^rr

2Cλ^r and CT^κ

λ^r−1+2

2Cλ^rr−1

1

2, which is the case forμμ(C), with our choice of the parameterλ1.

Now define Ω_T =E^c

λ=μT^−δ,T ,f and Σ=

nn0

Ω1 n

,

wheren0is such thatn⁻₀^δμ. Then we deduce that p(ΩT)1−Ce^−c/T2δ and p(Σ )=1, which ends the proof of Theorem 1.3. 2

3. Deterministic estimates in the spaceW^s,p

We will need the following technical lemmas.

Lemma 3.1(Sobolev embeddings). Let1q₁q₂∞ands∈R. The following inequalities hold v _L^q2 v _W^s,q1 fors=d

1 q1− 1

q2

, whenq₂<∞, (3.1)

v _L∞ v _W^s,q1 fors > d

q₁. (3.2)

The results (3.1) and (3.2) are classical and left here.

We will also need the following lemma. See [13, Proposition 1.1, p. 105].

Lemma 3.2(Product rule). Lets0, then the following estimates hold

uv _Ws,q u _L^q1 v _Ws,q1 + v _L^q2 u _Ws,q2, (3.3)

with1< q <∞,1< q₁, q₂∞and1q₁, q₂<∞so that 1

q = 1 q₁+ 1

q₁= 1 q₂+ 1

q₂. In particular

uv _Ws,q u _L∞ v _Ws,q + v _L∞ u _Ws,q, (3.4)

for any1< q <∞. 4. Proof of Theorem 1.3

In this section, we consider the cubic Schrödinger equation with quadratic potential.

In the casek=2, there is no loss of derivative in the Strichartz estimates (1.6). Then, thanks to the Christ–Kiselev lemma, we deduce that the solution to the problem

i∂_tu−H u=F, (t, x)∈R×R^d, u(0)=f ∈H^s,

satisfies

u _Lp1(0,T;W^s,q1(R^d)) f _Hs + F

L^p2(0,T;W^s,q2(R^d)), (4.1)

where 0< T 1 and(p₁, q₁),(p₂, q₂)are any admissible pairs, in the sense of (1.5).

Denote by X^s_T =C

[0, T];H^s

∩L^p

[0, T];W^s,q

, (4.2)

where the intersection is meant over all admissible pairs(p, q).

Recall thatEλ,f =Eλ,f(M, q_∗, σ )which is defined in (2.11). Then forMlarge enough, independent ofλandT we have the following proposition.

Proposition 4.1.Let0< T 1,d1 andσ > ^d₂−1−_d+¹₃. Letf ∈H^σ. Then there exists > ^d₂ −1,κ >0and C >0so that for everyv, v1, v2∈X_T^s,λ >0andω∈E_λ,f^c we have

K_f^ω(v)

X_T^s CT^κ

λ³+ v ³_Xs T

, (4.3)

and K_f^ω(v₁)−K_f^ω(v₂)

X^s_T CT^κ v₁−v_{2 X}^s

T

λ²+ v₁ ²_Xs

T + v₂ ²_Xs T

. (4.4)

For the proof of Proposition 4.1, we distinguish the casesd=1,d=2 andd3.

4.1. Cased3

Denote byq_d=_d^2d₋₂, so that(2, qd)is the end point in the Strichartz estimates (4.1). Then the resolution spaceX_T^s defined in (4.2) reads

X_T^s =C

[0, T];H^s

∩L²

[0, T];W^s,qd .

Letq_∗be defined by (2.4), then as 2q_∗q_d, the following inclusion holds X_T^s ⊂L^p∗

[0, T];W^s,q∗ ,

withp_∗2 so that(p_∗, q_∗)is an admissible pair, i.e.p_∗=^2(d_d⁺³⁾. Proof of Proposition 4.1, cased3. In this proof, we will writeu=u^ω_f.

The term|u+v|²(u+v)is a homogenous polynomial of degree 3. We expand it, and for sake of simplicity in the notations, we forget the complex conjugates. Hence

|u+v|²(u+v)=O

0j3

u^jv^3−j . (4.5)

By (4.1), we only have to estimate each term of the right-hand side inL¹_TH^s+L²_TW^s,qd, withq_d =_d^2d₊₂. Letε >0 so that

σ=d

2 −1− 1 d+3+ε.

Recall thatθ (q_∗)=_d+¹₃−η, for anyη >0. In the following we chooseη=ε/2 and we set s=θ (q_∗)+σ=d

2 −1+ε

2. (4.6)

With this choice ofs, by (3.2), the following embedding holds u _L^q0 u _W^s,q∗, withq₀=1

3d(d+3). (4.7)

Moreover, ass >_q^d

d, by (3.2), it is straightforward to check that there existsκ >0 so that v _L2

TL^∞T^κ v _L2W^s,qd T^κ v _X^s

T. (4.8)

Now assume thatω∈E_λ,f^c and turn to the estimation of each term in the r.h.s. of (4.5).

•We estimate the termv³inL¹_TH^s=L¹_TW^s,2. Use the inequality (3.4) withq=2 v³

H^s v ²_L∞ v _Hs, and thus by (4.8)

v³

L¹_TH^s v ²

L²_TL^∞ v _L∞

TH^s T^κ v ³_Xs

T. (4.9)

•We estimate the termuv²inL¹_TH^s. By (3.3) uv²

W^s,2 u _L^q1v²

W^s,q1+v²

L^q2 u _Ws,q2

u _L^q1 v _L∞ v _Ws,q1 + v ²

L^2q2 u _Ws,q2. DefineA₁= u _L^q1 v _L∞ v _Ws,q1.

We chooseq₁=2d, thenq₁=_d^2d₋₁. Observe that 2< q₁< q_dand ford3,q₁q₀, whereq₀in given by (4.7).

Therefore by (4.8) we infer A_{1 L}1

T T^κλ v _L2

TL^∞ v

L^p_T¹W^s,q1T^κλ v ²_Xs

T, (4.10)

wherep₁is such that(p₁, q₁)is admissible.