Maximizers for the Strichartz norm for small solutions of mass-critical NLS

Maximizers for the Strichartz norm for small solutions of mass-critical NLS

THOMASDUYCKAERTS, FRANKMERLE ANDSVETLANAROUDENKO

Abstract. Consider the mass-critical nonlinear Schr¨odinger equations in both focusing and defocusing cases for initial data inL²in space dimensionN. By Strichartz inequality, solutions to the corresponding linear problem belong to a globalL^pspace in the time and space variables, wherep=2+ _N⁴. In 1Dand 2D, the best constant for the Strichartz inequality was computed by D. Foschi who has also shown that the maximizers are the solutions with Gaussian initial data.

Solutions to the nonlinear problem with small initial data inL²are globally defined and belong to the same globalL^pspace. In this work we show that the maximum of theL^pnorm is attained for a given small mass. In addition, in 1D and 2D, we show that the maximizer is unique and obtain a precise estimate of the maximum. In order to prove this we show that the maximum for the linear problem in 1Dand 2Dis nondegenerated.

Mathematics Subject Classification (2010):35Q55 (primary); 35P25, 35B50, 35B45 (secondary).

1.

Introduction

We study theL²-critical nonlinear Schr¨odinger (NLS) equation in space dimension

N ≥1:

i∂tu+ ¹₂u+γ|u|^N4u=0, (t,x)∈

R

R

u_t=0= f ∈L²(

R

We will consider both focusing (γ = +1) and defocusing (γ = −1) equations.

Let us first recall some properties of the linear problem:

i∂tu+1

2u=0, u_t=0= f. (1.2)

This project was partially supported by the French ANR grant ONDNONLIN. S.R. was partially supported by the NSF grant DMS-0808081.

Received December 18, 2009; accepted April 20, 2010.

Denote by u = e^it2f the solution to (1.2). The mass u(t)²_L2 of the solution is conserved. Solutions to the linear problem satisfy the Strichartz inequality (see [24]):

∀f ∈L², e^it2f

L N4+2 t,x

≤Cf_L², (1.3)

where

u

L 4 N+2 t,x

=

R×R^N|u(t,x)|^N4+2dt d x ¹

N4+2

.

By standard profile decomposition arguments, one can easily show that the maxi- mum for the Strichartz inequality is attained. The best constant and maximizers for the Strichartz estimates were computed by D. Foschi [11] (see also [13] for another proof) for N =1,2. Before stating this result, we first recall some symmetries of the equations (1.1) and (1.2).

The following group of transformations leaves the solutions invariant under the nonlinear and linear Schr¨odinger evolution. If{θ0, ρ0,t₀, ξ0,x₀} ∈

R

R

R

R

e^iθ0ρ₀^N2e^{i x·ξ0}e^−i2t|ξ0|2u

ρ₀²t +t₀, ρ0

x− t

2ξ0

+x₀

. (1.4)

This includes phase invariance, scaling, time-translation, Galilean transformation and space-translation. Another transformation of (1.1) and (1.2) is the pseudo- conformal inversion (see [25]):

1 t^N/2 exp

i|x|² 2t

u

−1 t,x

t

. (1.5)

Note that all the preceding transformations leave the mass and the L

4 N+2

t,x norm of the solutions invariant. The linear equation is of course also invariant under the multiplication by a scalar: ifu(t,x)is a solution, so isc₀u(t,x),c₀∈

R

Consider the following normalized Gaussian:

G₀(x)= 1 π^N/4e^−|x|

2

2 , thus,

R^N|G₀|²d x =1, and its linear evolution:

G(t,x)=e^{12i t}G₀= 1 π^N/4

1

(1+i t)^N/2 e^{− |x|}

2

2(1+i t). (1.6)

Theorem A (Foschi). For all f ∈ L²(

R

4 N+2 L

N4+2 t,x

≤C_Sf_L^N42⁺(R^2N), C_S=









√1

3, N =1 1

2, N =2.

Furthermore, the equality holds if and only if e^i2tf is, up to the symmetries(1.4) of the equation, one of the solutions c₀G, c₀∈

C

Let us mention that the effect of the pseudo-conformal transformation (1.5) on G may be expressed only with the invariances (1.4) and we can omit it from consideration in Theorem A.

The Strichartz estimate (1.3) is the key ingredient to prove that the Cauchy problem (1.1) is locally wellposed in L² (see [8]). For small data, the solution is also globally wellposed and the globalL

4 N+2

t,x norm is finite, which implies that the solution scatters in L². This was extended to large radial data in the defocusing case γ = −1, in [30] for N ≥ 3 and in [16] for N = 2 (in this last work, the focusing case γ = 1 below the mass of the ground-state is also treated). The proofs are mainly based on technics developed for the energy-critical NLS (see e.g.[1, 2, 26, 29] and [15]).

In all these studies, a global Strichartz norm (in the mass-critical case, the L^N4+2norm) appears as the relevant norm to control. In this work we consider

I(δ)= sup

f_L2(RN)=δ

R×R^N |u(t,x)|^N4+2dt d x,

whereδ > 0 is small anduis the solution to (1.1). The results cited above imply that I(δ)is finite for small δ, and, in the defocusing case with N ≥ 2, for large δ if we restrict the maximum to radial solutions. A natural extension to Theorem A would be to show that this maximum is achieved by a unique solution (up to symmetries) of (1.1) and give a precise estimate ofI(δ).

Our main result is the following:

Theorem 1.1. Fix γ ∈ {−1,+1}. There exists a δ0 > 0 such that for all δ in (0, δ0), the maximum I(δ)is attained: there exists a solution u_δof (1.1)with initial condition f_δsuch that

f_δL² =δ, I(δ)=

R×R^N |u_δ(t,x)|^N4+2 dt d x.

If N = 1or N = 2, the maximizer u_δ is unique up to the transformations(1.4), (1.5)of the equation. Furthermore, asδ→0,

I(δ)=C_Sδ^N4+2+γD_Nδ^N8+2+

O

π

k≥1

(2k)!

k9^k(k!)² ≈0.0867and D₂= 1 2π ln4

3 ≈0.0458.

Remark 1.2. In particular, in the focusing case in 1Dand 2D, the maximum of the Strichartz norm is, for small data, higher than in the linear case. In the defocusing case, the effect of the nonlinearity is to lower this maximum.

Remark 1.3. The constantD_N may be expressed as D_N=−

2+ 4

N

Im

|G(t)|^N4G(t) _t

0

e^i(t−s)2

|G(s)|^N4G(s)

ds dt d x. (1.8) Remark 1.4. The proof also shows that in 1D and 2D, the initial condition of any maximizer with small mass δ is (after transformations) close to δG₀, where G₀is the normalized Gaussian. See Proposition 3.3 and Remark 3.4 for a precise statement.

Estimates of Strichartz norms for critical nonlinear problems are only known in a few cases. Super-exponential bounds were obtained by T. Tao for radial defo- cusing energy-critical equations: Schr¨odinger equation in space dimension higher than 3 [26], and wave equation in 3D [27]. An equivalent of the maximizum is given in [9] for the energy-critical focusing Schr¨odinger and wave equations (in space dimensions 3, 4 and 5), close to the energy threshold given by the stationary solution.

The fact that the maximum of the Strichartz norm is attained is new for a non- linear equation. The proof of this result is based on time-dependent adaptation to concentration-compactness arguments (see e.g.[18]) and on a super-additivity property of I(δ)which we show by general estimates on small solutions of (1.1).

As stated in Proposition 2.12, the proof would extend to larger data provided the scattering of all solutions and the super-additivity properties are shown for those data also. This proof is flexible and should also easily adapt to other equations,e.g.

the energy-critical NLS and wave equations for small data and (together with the methods of [9]) close to the energy threshold.

On the other hand, the proof of the uniqueness of the maximizer and of the estimate (1.7) is specific to the mass-critical problem, and strongly relies on the results of [11] and [13]. A key element is the nondegeneracy of the Gaussian for the nonlinear problem, in the orthogonal space of the null directions related to the invariances of the equation:

Theorem 1.5. Assume N = 1,2. There exists c > 0such that ifϕ ∈ L² satisfies the following orthogonality properties(x ∈

R

ϕG₀=

ϕ|x|²G₀=0,

ϕx G₀=0_RN, (1.9) then

Q(ϕ)≥cϕ²_L2,

where Q is the quadratic form associated to the second derivative of the mapping f →C_S

|f|²d x 1+_N²

− e^it2f²⁺

4 N dt d x from L²to[0,∞), at the critical point f =G₀.

We refer to (3.3) for an expression of Q. This result is an analogue, for the Strichartz estimate, to the non-degeneracy of the maximizer ¹

(1+|x|²)^N−22 for the Sobolev imbedding H˙¹(

R

R

To show Theorem 1.5, we apply a lens transform ([4, 20, 22]), related to the pseudo-conformal inversion, to the solutions of (1.1), which turns the Laplace op- erator into the harmonic operator−+ |x|². The result then follows from explicit computations and a formula of Wei-Min Wang [33] on products of eigenfunctions for the harmonic oscillator.

The outline of the paper is as follows. In Section 2 we show that the maximizer is attained and in Section 3 we prove the estimate on I(δ). In Section 4 we show the uniqueness of the maximizer. Section 5 is devoted to the proof of Theorem 1.5.

ACKNOWLEDGEMENTS. The authors would like to thank Keith Rogers for pointing out the article [33]. Part of the project was done at the Institut Henri Poincar´e in Paris during the special trimesterOndes non-lin´eaires et dispersion(April-July 2009).

2.

Existence of a maximizer

In this section, where there is no restriction on the dimension N ≥1, we show the first part of Theorem 1.1:

Proposition 2.1. There existsδ0 > 0such that ifδ ∈ (0, δ0), then there exists a solution u_δof (1.1), with initial condition f_δsuch that

f_δL²_(R^N₎=δ and

R×R^N |u_δ|^N4+2 dt d x= I(δ). (2.1) After some preliminaries (Section 2.1) we show in Section 2.2 a crucial super- additivity property of I(δ), which relies on rough estimates of I(δ)and its growth rate. In Section 2.3 we use this property to prove Proposition 2.1 by concentration- compactness arguments.

2.1. Profile decomposition

We recall here from [19] a profile decomposition adapted to the Strichartz estimate for the linear equation (1.2). We start with a long time perturbation result for the equation (1.1).

Lemma 2.2 (Long time perturbation). Let A > 0. There exists C = C(A) > 0 and a smallδ0 =δ0(A) >0such that the following holds: Let u ∈C⁰(

R

i∂tu+1

2u+γ|u|^N4u=0.

Letu˜= ˜u(x,t)∈C⁰(

R

2u˜+γ| ˜u|^N4u˜. Assume ˜u

L 4 N+2 t,x

≤ A, and for someε < δ0

e

L 2(N+2) t,xN+4

≤ε and

e^i(t−t20)(u(t₀)− ˜u(t₀)) L

4 N+2 t,x

≤ε, then

u− ˜u

L 4 N+2 t,x

≤Cε.

We skip the proof of Lemma 2.2. We refer to [2, 6, 15, 29] for similar result for the energy-critical case, [12] for a subcritical case and [31, Lemma 3.1] for a statement close to Lemma 2.2 in the mass-critical case.

We next turn to the profile decomposition. If0={ρ0,t₀, ξ0,x₀} ∈(0,+∞)×

R

R

R

0(u)=ρ₀^N2e^{i x·ξ0}e^−it2|ξ0|2u

ρ₀²t+t₀, ρ0

x− t

2ξ0

+x₀

. (2.2) As we have seen in the introduction, if uis a solution to the linear equation (1.2) (respectively, to the nonlinear equation (1.1)), then0(u)is also a solution to (1.2) (respectively, to (1.1)). We say that two sequences of transformations

¹_n

n and ²_n

nare orthogonal when

nlim→∞

ρ_n¹ ρ_n² + ρ_n²

ρ_n¹ + ξ_n¹−ξ_n²

ρ_n¹ +t_n¹−t_n²+

t_n¹ 2

ξ_n¹−ξ_n²

ρ_n¹ +x_n¹−x_n²

= +∞. (2.3) We recall from [19, Theorem 2] (see [14] in space dimension 1, [3] for general space dimension) , the following profile decomposition result:

Lemma 2.3. Let{f_n}be a bounded sequence in L²(

R

n^j

nis orthogonal to _n^k

n

and for all J ,

f_n(x)=^J

j=1

n^j

U^j

(0,x)+h_n^J(x), (2.4)

where

J→+∞lim lim sup

n→∞

e^i2th_n^J

L N4+2 t,x

=0.

Remark 2.4. As a consequence of the orthogonality of the transformationsn^j, the following Pythagorean expansions hold for all J ≥1:

f_n²_L2− J

j=1

U^j(0)²

L²−h_n^J2

L² −→

n→+∞0, (2.5) e^i2tf_n

4 N+2 L

4 N+2 t,x

− J

j=1

U^j

4 N+2 L

4 N+2 t,x

−e^it2h_n^J

4 N+2 L

4 N+2 t,x

n→+∞−→ 0. (2.6)

Let{f_n}nbe a sequence inL²and assume that the corresponding solution to (1.1) is globally defined and satisfiesf_n

L 4 N+2 t,x

<∞. Consider the profile decomposition given by Lemma 2.3. LetV^j be the nonlinear profile associated to{U^j,t_n^j}n, that is the unique solution of (1.1) such that

nlim→∞

U^j

t_n^j −V^j

t_n^j

L²=0.

Assume also that the V^j’s are globally defined and such thatV^j

L^2+N4 is finite for all j. Combining Lemmas 2.2 and 2.3, one gets a nonlinear version of the decomposition (2.4):

Corollary 2.5. Let{f_n}nis as above and{u_n}nbe the sequence of solutions to(1.1) with initial conditions{f_n}n. Then

u_n(t,x)=^J

j=1

n^j

V^j

(t,x)+h_n^J(t,x)+r_n^J(t,x) (2.7)

with

J→+∞lim lim

n→+∞

r_n^J

L 4 N+2 t,x

+sup

t∈R

r_n^J(t)

L²

=0.

Remark 2.6. Using the orthogonality of the sequences of transformations{n^j}n, it is easy to check that

Jlim→∞lim sup

n→∞

un^N4+2

L N4+2 t,x

− J

j=1

V^j

4 N+2 L

4 N+2 t,x

=0. (2.8)

2.2. A superadditivity property of the maximum

In this paragraph we give various estimates on I(δ). The main result is the follow- ing proposition, which is one of the steps (along with a concentration-compactness argument) in showing that the maximizer is attained:

Proposition 2.7. There existsδ0>0such that if0<

α²+β²< δ0, then I(α)+I(β) < I

α²+β²

.

Remark 2.8. Superadditivity (or subadditivity for minimizers) conditions are clas- sical in this context (see [17, Subsection I.2]).

The proof of Proposition 2.7 relies on two estimates on I(δ) that we treat in Lemmas 2.9 and 2.11 below.

Lemma 2.9. There exists a constant C₀>0such that for smallδ >0,

I(δ)−C_Sδ^N4+2≤C₀δ^N8+2, (2.9) where C_S is the best constant for the Strichartz inequality

e^it2f

4 N+2

dt d x≤C_Sf_L^N42⁺². (2.10) Before proving this lemma, we start by a straightforward consequence of the small data well-posedness theory for equation (1.1) (see [8]).

Claim 2.10. There exists a constantC >0 such that iff_L²is small, then e^it2f −u

L N4+2 t,x

≤Cf_L^N42⁺¹, whereuis the solution of (1.1) with initial condition f.

Sketch of proof. The Cauchy problem theory for (1.1) implies that for small initial data

u

L N4+2 t,x

≤2fL². Since

u(t)=e^it2f +iγ _t

0

e^i2(t−s)|u(s)|^N4u(s)ds, the claim follows from Theorem A and the Strichartz estimate

_t

0

e^i2(t−s)ϕ(s)ds L

4 N+2 t,x

≤Cϕ

L 2(N+2) t,xN+4

.

Proof of Lemma2.9. Letube a solution of (1.1) with initial condition f such that fL²(R^N)=δ. Then

e^i2tf(x)

4 N+2

dt d x−

|u(t,x)|^N4+2 dt d x

≤C

e^it2f

L 4 N+2 t,x

− u

L 4 N+2 t,x

e^it2f

4 N+1 L

N4+2 t,x

+ u^N4+1

L 4 N+2 t,x

≤C e^i2tf −u

L 4 N+2 t,x

δ^N4+1≤Cδ^N8+2,

where the last line follows from the triangle inequality and then from Claim 2.10.

Applying the previous inequality to the initial data f =δF₀, whereF₀is the initial condition of a maximizer for Strichartz estimate (2.10), and then to a sequence {f_n}nsuch thatf_nL² =δand

|u_n|^N4+2→ I(δ), we obtain (2.9).

We next estimate the rate of growth ofI(δ). Lemma 2.11. Ifδis small andε≤ ¹₂δ, then

I(δ)+c₁δ^N4+1ε≤ I(δ+ε)≤ I(δ)+C₁δ^N4+1ε, (2.11) where c₁= _N⁴ C_S and C₁=2

4 N +2

C_S.

Proof. Step 1. We first show that there existC₂, 0 > 0 such that if f ∈ L² with fL²+≤0,uis the solution of (1.1) with the initial condition f, andv is the solution of (1.1) with the initial condition(1+)f, then

(1+)^N4+2 u^N4+2− v^N4+2

≤C₂f_L^N82⁺². First, observe thatu =(1+)uis a solution to the equation

i∂tu+ 1

2u + 1

(1+)^N4 |u|^N4u =0, u_t=0=(1+)f. We rewrite the above equation as

i∂tu+1

2u+ |u|^N4u =

1− 1

(1+)^N4

|u|^N4u, noting that for small, Strichartz estimate implies

1− 1

(1+)^N4

|u|^N4u L

2(N+2) t,xN+4

≤C|u|^1+N4

L 2(N+2)

N+4 t,x

=Cu^1+N4

L N4+2 t,x

≤Cf¹_L⁺2^N4.

Sincevis a solution of i∂tv+ 1

2v+ |v|^N4v =0, vt=0=(1+)f, by the long time perturbation Lemma 2.2, we get

u−v

L N4+2 t,x

≤Cf_L^N42⁺¹. Hence,

|u|^N4+2 dt d x−

|v|^N4+2 dt d x

≤Cu−v

L N4+2 t,x

f_L^N42⁺¹

≤Cf_L^N82⁺², which concludes Step 1.

Step 2. Letε,δ >0. First, we show the lower bound ofI(δ+ε). Let f ∈L²(

R

f_L² =δ and

|u(t,x)|^N4+2dt d x ≥I(δ)−δ^N8+1ε, (2.12) whereuis the corresponding solution of (1.1) and we used the supremum property of I(δ). Letu_ε be the solution of (1.1) with the initial condition

1+ ^ε_δ

f. Then u_ε(0)L²=δ+ε. By Step 1,

I(δ+ε)≥

|u_ε(t,x)|^N4+2dt d x

≥ 1+ ε

δ ⁴

N+2

|u(t,x)|^N4+2−C₂ε δδ^N8+2. By (2.12), we get

I(δ+ε)≥

1+ 4

N +2 ε

δ I(δ)−δ^N8+1ε

−C₂δ^N8+1ε.

Lemma 2.9 impliesI(δ)≥C_Sδ^N4+2−C₀δ^N8+2, hence, I(δ+ε)≥ I(δ)+C_S

4 N +2

δ^N4+1ε

− 4

N +2

C₀+

1+ 4

N +2 ε

δ

+C₂

δ^N8+1ε.

Now ifε < ¹₂δand

δ <

C_S 4+6C₀+C₂

N/4

,

the last term in the expression above will be less than 2C_Sδ^N4+1ε, and thus, the right side in (2.11) follows withc₁= _N⁴ C_S.

The upper bound on I(δ+ε)follows similarly from Step 1 and Lemma 2.9, obtaining the left side in (2.11) withC₁=2C_S

4 N +2

. We next prove Proposition 2.7.

Proof. Without loss of generality, we can assume 0< α≤β.

Step 1. We first show that there exists a large constantC₃>0 such that the conclu- sion of the proposition holds if

C₃β^N2+1≤α≤β. (2.13)

By Lemma 2.9,

I(α)+I(β)≤C_Sα^N4+2+C_Sβ^N4+2+2C₀β^N8+2, and C_S

α²+β²²

N+1

≤ I

α²+β²

+2C₀β^N8+2.

There is a constantκN >0 such that 1+x^N2+1+κNx ≤(1+x)^N2+1forx∈ [0,1]. As a consequence,α^N4+2+β^N4+2+κNβ^N4α²≤

α²+β²_N²₊1

. Combining with the previous estimates, we get

I(α)+I(β)+C_SκNβ^N4α²−4C₀β^N8+2≤I

α²+β²

, which yields the announced result ifC₃is chosen large in (2.13).

Step 2. We next show that the conclusion of the Proposition still holds if

0< α <C₃β^N2+1, (2.14) whereC₃is the constant defined in Step 1. Choosingδ0small enough,β ≤δ0and (2.14) imply

α² 4β ≤

α²+β²−β ≤ β 2. By Lemma 2.11, withδ=βandε=

α²+β²−β, I(β)≤I

α²+β²

−c₁β^N4+1

α²+β²−β

≤I

α²+β²

−c₁ 4β^N4α².

Combining with Lemma 2.9 we get, taking a smallerδ0if necessary, I(α)+I(β)≤I

α²+β²

−c₁

4β^N4α²+2C_Sα^N4+2

≤I

α²+β²

+α²β^N4

2C_SC

4 N

3 β^N82 − c₁ 4

,

which shows that the conclusion of the proposition holds also in this case, provided δ0>0 is small enough.

2.3. Proof of the existence of the maximizer

Let us show Proposition 2.1. We will prove the following more general result:

Proposition 2.12. Assume that there exists a constant A>0such that

i. Scattering: for all f ∈ L² such thatfL² ≤ A, the solution u of (1.1)with initial condition f is globally defined and

δ≤ A=⇒I(δ) <∞.

ii. Superadditivity: if0<

α²+β²= A, andα, β >0, then I(α)+I(β) < I(A) .

Then there exists a solution u_Aof (1.1)with initial condition f_A∈L²such that f_AL²= A,

|u_A|^2+N4 =I(A).

In view of the small data global well-posedness theory and Proposition 2.7, Propo- sition 2.12 implies Proposition 2.1. Let us prove Proposition 2.12.

Let{u_n}nbe a sequence of solutions to (1.1) with initial data f_n such that f_nL²_(R^N₎ = A, lim

n→∞

R^N|u_n|^N4+2= I(A).

We will show that there exist a subsequence of{u_n}nand a sequence{n}nof trans- formations such that{n(u_n)}nconverges strongly inL². Consider, after extraction, a profile decomposition of the sequence{f_n}n:

f_n = J

j=1

n^j

U^j

t=0+h_n^J. (2.15)

It is sufficient to show thatU^j =0 except for one j and that lim_n→∞h_n^JL² =0, which we will do in two steps.

Step 1: no dichotomy. First assume that there are at least two nonzero profiles, say U¹ =0 andU² = 0. LetV¹be the nonlinear profiles associated to{U¹,t_n¹}and Vnthe solution of (1.1) given by

V_n=_n¹(V¹).

LetW_nbe the sequence of solutions to (1.1) with initial condition W_n(0)= f_n−V_n(0).

Letr_n =u_n−V_n−W_n. By assumption (2.12), all the nonlinear profilesV^jscatter.

Thus, one can use Corollary 2.5, showing

nlim→∞sup

t∈Rr_n(t)_L² =0. Furthermore, (see (2.5) and Remark 2.6)

|f_n|² =

|V_n(0)|²+

|W_n(0)|²+o_n(1) (2.16)

|u_n|^N4+2 =

|V_n|^N4+2+

|W_n|^N4+2+o_n(1). (2.17) Letε= U¹(0)L². Then for alln,ε= V_n(0)L². By (2.16),

W_n(0)²_L2 = A²−ε²+o_n(1).

By our assumptions, ε > 0 (otherwise, U¹ would be zero) and A² − ε² > 0 (otherwise,U² would be zero). Using that

|u_n|^N4+2tends to I(A)asn → ∞, and that by Lemma 2.2, lim sup_n

|W_n|^N4+2≤I(√

A²−ε²), we get by (2.17) I(A)≤I(ε)+I

A²−ε²

. (2.18)

This contradicts assumption (2.12), concluding Step 1.

Step 2: non vanishing and the end of the proof. There must be one nonzero profile in (2.15). If not, then

nlim→∞

|u_n|^N4+2=0,

showing that I(A) = 0, a contradiction. It remains to show that the remainder h_n =h_n^J in (2.15) tends to 0 inL². Denote by

ε= lim

n→∞h_nL², then, using again Lemma 2.2, we get I(A) ≤ I(√

A²−ε²), which shows by as- sumption (2.12) thatε=0.

Denoting byU¹the only nonzero profile in (2.15), we have shown that(¹_n)⁻¹(u_n) tends toU¹inL², and therefore,

U¹_L²= A,

|U¹|^N4+2= I(A), concluding the proof of the proposition.

3.

Estimate of the maximum of the Strichartz norm

In the remainder of the paper, we restrict ourselves to 1D and 2D. In this section we prove the second part of Theorem 1.1:

Proposition 3.1. Assume that N =1or N =2. Then asδ→0, I(δ)=

|u_δ|^2+N4 =C_Sδ^2+N4 +γD_Nδ^2+N8 +

O

π

k≥1

(2k)!

k9^k(k!)² ≈0.0867and D₂= 1 2π ln4

3 ≈0.0458.

Before proving Proposition 3.1, we define the quadratic form associated to the maximum of the Strichartz estimate that appears in Theorem 1.5. By Theorem A, ifGis the Gaussian solution defined by (1.6) andϕ∈L², then

CS

|G₀+ϕ|² 1+_N²

− G+e^i2tϕ²⁺

4 N ≥0.

Expanding the above inequality and using thatGis a maximizer, we obtain that the linear part vanishes,i.e.,

∀ϕ∈L², C_SRe

G₀ϕ=Re

|G|^N4G e^i2tϕ. (3.1) The expansion at second order inϕyields

C_S

|G₀+ϕ|² ₁₊²

N − G+e^i2tϕ²⁺

4

N =Q(ϕ)+

O

, (3.2)

where Qis a (real) nonnegative symmetric quadratic form onL²defined by Q(ϕ)=C_S

N +2

N

|ϕ|²+ 4(N+2) N²

Re

G₀ϕ

2

− (N+2)² N²

|G|^N4 e^i2tϕ²

− 2(N +2) N² Re

|G|^N4−2G²

e^i2tϕ₂ .

(3.3)

By the transformations of the linear equation (respectively, multiplication by a real number, phase shift, space translation, Galilean invariance, scaling and time trans- lation), we have

Q(G₀)=Q(i G₀)=Q(x G₀)= Q(i x G₀)=Q(x²G₀)=Q(i x²G₀)=0, (3.4) ifN =1 and

Q(G₀)=Q(i G₀)=Q(x_jG₀)=Q(i x_jG₀)=Q(|x|²G₀)=Q(i|x|²G₀)=0, (3.5) (where j =1,2) ifN =2. Theorem 1.5, which will be proved in Section 5 states thatQ is positive definite in the subspace of functions inL²that are orthogonal to the directions in (3.4) or (3.5). This non-degeneracy property is crucial in the proof of Proposition 3.1, which is divided in two parts.

3.1. Choice of the maximizer

We first give a corollary to the linear profile decomposition that will be needed in the proof. Recall from (1.6) the definition of the normalized GaussianG.

Lemma 3.2. Let{f_n}nbe a sequence in L²(

R

nlim→∞f_nL² =1, (3.6)

and

nlim→∞ e^i2tfn

4 N+2

dt d x=CS. (3.7)

Then there exist a subsequence of{f_n}_n (still denoted by{f_n}_n), a phaseθ0and a sequence{n}nof transformations of the form(2.2)such that

nlim→∞

f_n−e^iθ0n(G)

L²=0, (3.8)

where G is the normalized Gaussian solution defined in(1.6).

Proof. This is an application of Lemma 2.3 and the uniqueness result of Foschi [11].

After extraction of a subsequence, the sequence{f_n}nadmits a profile decom- position of the form (2.4). At least one of the profiles is nonzero. Indeed, if it was not the case,e^i2tf_n

L^N4+2would tend to 0, a contradiction with (3.7). Reordering the profiles, we may assume thatU¹=0. By the Pythagorean expansion (2.6) and by (3.7)

C_S+o_n(1)= e^i2tf_n

4 N+2

dt d x ≤C_S U¹

4 N+2

L² +w_n¹

4 N+2 L²

+o_n(1).