Van Duong Dinh
email: dinhvan.duong@math.univ-toulouse.fr

GLOBAL EXISTENCE AND SCATTERING FOR A CLASS OF NONLINEAR FOURTH-ORDER SCHR ÖDINGER EQUATION BELOW THE ENERGY SPACE

Keywords: 2010 Mathematics Subject Classification. 35G20, 35G25, 35Q55 Nonlinear fourth-order Schrödinger, Global well-posedness, Scattering, Almost conservation law, Morawetz inequality

In this paper, we consider a class of nonlinear fourth-order Schrödinger equation,

Using the I-method combined with the interaction Morawetz inequality, we establish the global well-posedness and scattering in H γ (R d ) with γ(d, ν) < γ < 2 for some value γ(d, ν) > 0.

Introduction

Consider the following nonlinear fourth-order Schrödinger equation

i∂ t u(t, x) + ∆ 2 u(t, x) = -(|u| ν-1 u)(t, x), t ∈ R, x ∈ R d , u(0, x) = u 0 (x) ∈ H γ (R d ), (NL4S)
where u(t, x) is a complex valued function in R × R d , d ≥ 5. The nonlinear exponent ν is assumed to be mass-supercritical, i.e ν > 1 + 8 d and energy-subcritical, i.e. ν < 1 + 8 d-4 . The regularity exponent γ is assumed to satisfy 0 < γ < 2.

The fourth-order Schrödinger equation was introduced by Karpman [Kar96] and Karpman-Shagalov [KS00] to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. Such a fourth-order Schrödinger equation is of the form i∂ t u + ∆ 2 u + ε∆u + µ|u| ν-1 u = 0, u(0) = u 0 ,

(1.1)

where ε ∈ {0, ±1}, µ ∈ {±1} and ν > 1. We note that (NL4S) is a special case of (1.1) by taking ε = 0 and µ = 1. The nonlinear fourth-order Schrödinger equation (1.1) has attracted a lot of interest in a past decay. The sharp dispersive estimates for the linear part of (1.1) were established in [START_REF] Ben-Artzi | Saut Disperion estimates for fourth-order Schrödinger equations[END_REF]. The local well-posedness and the global well-posedness for (1.1) has been widely studied in [Din1, Din2, Din3, Guo10, GC06, HHW06, HHW07, HJ05, MXZ09, MXZ11, MWZ15, MZ07, Pau1, Pau2, PS10] and references therein. The (NL4S) enjoys a natural scaling invariance, that is if we set for λ > 0 u λ (t, x) := λ -4 ν-1 u(λ -4 t, λ -1 x), (1.2) then for T ∈ (0, +∞],

u solves (NL4S) on (-T, T ) ⇐⇒ u λ solves (NL4S) on (-λ 4 T, λ 4 T ).

We define the critical regularity exponent for (NL4S) by

γ c := d 2 - 4 ν -1 . (1.3)
The (NL4S) is known (see [START_REF] Dinh | Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces[END_REF] or [START_REF] Dinh | On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation[END_REF]) to be locally well-posed in H γ (R d ) with γ ≥ max{0, γ c } satisfying for ν is not an odd integer, γ ≤ ν.

(1.4)

Here γ is the smallest integer greater than or equal to γ. This condition ensures the nonlinearity to have enough regularity. In the sub-critical regime, i.e. γ > γ c , the time of existence depends only on the H γ -norm of initial data. Moreover, the local solution enjoys mass conservation, i.e.

M (u(t)) := u(t) 2 L 2 (R d ) = u 0 2 L 2 (R d ) ,
and H 2 -solution has conserved energy, i.e.

E(u(t))

:= R d 1 2 |∆u(t, x)| 2 + 1 ν + 1 |u(t, x)| ν+1 dx = E(u 0 ).
The persistence of regularity (see [START_REF] Dinh | On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation[END_REF]) combined with the conservations of mass and energy yield the global well-posedness for (NL4S) in H γ (R d ) with γ ≥ 2 satisfying for ν is not an odd integer, (1.4). In the critical regime, i.e. γ = γ c , one also has (see [START_REF] Dinh | Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces[END_REF] or [START_REF] Dinh | On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation[END_REF]) the local well-posedness for (NL4S) but the time of existence depends not only on the H γ -norm of initial data but also on its profile. Moreover, for small initial data, the (NL4S) is globally well-posed, and the solution is scattering.

The main goal of this paper is to show the global well-posedness and scattering for the nonlinear fourth-order Schrödinger equation (NL4S) below the energy space. Our arguments are based on the combination of the I-method and the interaction Morawetz inequality which are similar to those of [VZ09]. However, there are some difficulties due to the high-order dispersion term ∆ 2 u. Moreover, in order to successfully establish the almost conservation law, we need the nonlinearity to have at least two orders of derivatives. This leads to the restriction in spatial space of dimensions 5 ≤ d ≤ 11.

Before stating our main result, let us recall some known results concerning the global existence below the energy space for the nonlinear fourth-order Schrödinger equation. To our knowledge, Guo in [START_REF] Guo | Global existence of solutions for a fourth-order nonlinear Schrödinger equation in n + 1 dimensions[END_REF] gave a first answer to this problem. In [START_REF] Guo | Global existence of solutions for a fourth-order nonlinear Schrödinger equation in n + 1 dimensions[END_REF], the author considered (1.1) with ν -1 = 2m, m ∈ N satisfying 4 < md < 4m+2, and established the global existence in H γ (R d ) with

1+ md-9+ √ (4m-md+7) 2 +16 4m
< γ < 2. The proof is based on the I-method which is a modification of the one invented by I-Team [CKSTT02] in the context of nonlinear Schrödinger equation. Later, Miao-Wu-Zhang in [MWZ15] studied the defocusing cubic fourth-order Schrödinger equation, i.e. ν = 3 in (NL4S), and proved the global well-posedness and scattering in H γ (R d ) with γ(d) < γ < 2 where γ(5) = 16 11 , γ(6) = 16 9 and γ(7) = 45 23 . The proof relies on the combination of the I-method and a new interaction Morawetz inequality. Recently, in [START_REF] Dinh | Global existence for the defocusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space[END_REF] the author considered the defocusing cubic higher-order Schrödinger equation including the cubic fourth-order Schrödinger equation, and showed that the (NL4S) with ν = 3 is globally well-posed in H γ (R 4 ) with 60 53 < γ < 2. The argument makes use of the I-method and the bilinear Strichartz estimate. The analysis is carried out in Bourgain spaces X γ,b which is similar to those in [CKSTT02]. In the above considerations, the nonlinearity is algebraic, i.e. ν is an odd integer. This allows to write the commutator between the I-operator and the nonlinearity explicitly by means of the Fourier transform, and then carefully control the frequency interactions using multi-linear analysis. When one considers the nonlinear fourth-order Schrödinger equation (NL4S) with ν > 1 is not an odd integer, this method does not work. We thus rely purely on Strichartz and interaction Morawetz estimates.

Let us now introduce some notations.

γ(d, ν) := max{γ 1 (d, ν), γ 2 (d, ν), γ 3 (d, ν), γ 4 (d, ν)}, (1.5)
where

γ 1 (d, ν) := 3 2 + γ c 4 , γ 2 (d, ν) := 4 -ν, γ 3 (d, ν) := 2 ν -1 + (ν -2)γ c ν -1 , γ 4 (d, ν) := min σ∈(0,σ0] γ(d, ν, σ).
Here

σ 0 satisfies    2σ 0 (16 -(ν -1)(d + 4)) < (d -5)(d(ν -1) -8), 2σ 0 (ν -3) ≤ d -5, σ 0 ≤ γ, (1.6)
and γ(d, ν, σ) is the (large if there are two) root of the equation

γ c (2 -γ)(d -5 + (8 -d)σ) = min γ -1 - γ c 2 , ν -2, (ν -2)(γ -γ c ) (γ -γ c )σ.
The main result of this paper is the following:

Theorem 1.1. Let 5 ≤ d ≤ 11. The initial value problem (NL4S) is globally well-posed in H γ (R d )
for any γ(d, ν) < γ < 2, and the global solution u enjoys the following uniform bound

u L ∞ (R,H γ (R d )) ≤ C( u 0 H γ (R d ) ).
Moreover, the solution is scattering, i.e. there exist unique

u ± 0 ∈ H γ (R d ) such that lim t→±∞ u(t) -e it∆ 2 u ± 0 H γ (R d ) = 0.
We record in the table below some best known results, and compare them with our ones. As in the table, our results are not as good as the best known results when ν is an odd integer. But our method allows to treat the non-algebraic nonlinearity. The proof of the above result is based on two main ingredients: the I-method and the interaction Morawetz inequality, which are similar to those given in [VZ09]. The I-method for the fourthorder Schrödinger equation is a modification of the one introduced by I-Team in [CKSTT02]. This method is very useful for treating the nonlinear dispersive equation at low regularity, i.e. below energy space. The idea is to replace the non-conserved energy E(u) when γ < 2 by an "almost conserved" variance E(Iu) with I a smoothing operator which is the identity at low frequency and behaves like a fractional integral operator of order 2 -γ at high frequency. Since Iu is not a solution of (NL4S), we may expect an energy increment. The key is to show that the modified energy E(Iu) is an "almost conserved" quantity in the sense that the time derivative of E(Iu) decays with respect to a large parameter N (see Section 2 for the definition of I and N ). To do so, we need delicate estimates on the commutator between the I-operator and the nonlinearity. When the nonlinearity is algebraic, we can use the Fourier transform to write this commutator explicitly, and then carefully control the frequency interactions. Once the nonlinearity is no longer algebraic, this method fails. In order to treat this case, we take the advantage of Strichartz estimate with a gain of derivatives (2.5). Thanks to this Strichartz estimate, we are able to apply the technique given in [VZ09] to control the commutator. Of course, this technique is not as good as the Fourier transform technique when the nonlinearity is algebraic, but it is more robust and allows us to treat the non-algebraic nonlinearity. The interaction Morawetz inequality for the nonlinear fourth-order Schrödinger equation was first introduced in [START_REF] Pausader | The cubic fourth-order Schrödinger equation[END_REF] for d ≥ 7. Then, it was extended for d ≥ 5 in [MWZ15]. Using this interaction Morawetz inequality and the interpolation argument together with the Sobolev embedding, we have for any compact interval J and 0

ν d γ c γ(d, ν) (best known results) γ(d, ν) (our results)
< σ ≤ γ, u M σ (J) := u L d-5+4σ σ t (J,L 2(d-5+4σ) d-5+2σ x ) u 0 L 2 x u L ∞ t (J, Ḣ 1 2 x ) 2σ d-5+4σ u d-5 d-5+4σ L ∞ t (J, Ḣσ x ) .
(1.7)

As a byproduct of the Strichartz estimates and I-method, we show the "almost conservation law" for (NL4S), that is if u ∈ L ∞ (J, S (R d )) is a solution to (NL4S) on a time interval J = [0, T ], and satisfies Iu 0 H 2 x ≤ 1 and if u satisfies in addition the a priori bound u M σ (J) ≤ µ for some small constant µ > 0, then

sup t∈[0,T ] |E(Iu(t)) -E(Iu 0 )| N -(2-γ+δ) ,
for some δ > 0.

We now give an outline of the proof. Let u be a global in time solution to (NL4S) with initial data u 0 ∈ C ∞ 0 (R d ). Our goal to to show the uniform bounds

u M σ (R) ≤ C( u 0 H γ x ), (1.8) u L ∞ t (R,H γ x ) ≤ C( u 0 H γ x ), (1.9)
Thanks to (1.9), the global existence follows immediately by a standard density argument. Since E(Iu 0 ) is not necessarily small, we will use the scaling (1.2) to make E(Iu λ (0)) small in order to apply the "almost conservation law". By choosing

λ ∼ N 2-γ γ-γc , (1.10)
and using some harmonic analysis, we can make E(Iu λ (0)) ≤ 1 4 . We will show that there exists an absolute constant C such that

u λ M σ (R) ≤ Cλ γc+ σ(4-d)γc 2(d-5+4σ) .
(1.11)

We then obtain (1.8) by undoing the scaling. In order to prove (1.11), we perform a bootstrap argument. Note that (1.11) is equivalent to

u λ M σ ([0,t]) ≤ Cλ γc+ σ(4-d)γc 2(d-5+4σ) , ∀t ∈ R.
Assume by contraction, it is not so. Since u λ M σ ([0,t]) is a continuous function in t, there exists T > 0 so that

u λ M σ ([0,T ]) > Cλ γc+ σ(4-d)γc 2(d-5+4σ) ,
(1.12)

u λ M σ ([0,T ]) ≤ 2Cλ γc+ σ(4-d)γc 2(d-5+4σ) .
(1.13)

Using (1.13), we can split [0, T ] into L subintervals J k , k = 1, ..., L so that

u λ M σ (J k ) ≤ µ.
The number L must satisfy

L ∼ λ γc(d-5+(8-d)σ) σ
.

(1.14)

We thus can apply the "almost conservation law" to get sup

[0,T ] E(Iu λ (t)) ≤ E(Iu λ (0)) + N -(2-γ+δ) L. Since E(Iu λ (0)) ≤ 1 4 , we need N -(2-γ+δ) L 1 4 (1.15)
in order to guarantee E(Iu λ (t)) ≤ 1 for all t ∈ [0, T ]. Combining (1.10), (1.14) and (1.15), we get a condition on γ. Next, by (1.7) and some harmonic analysis, we have

u λ M σ ([0,T ]) ≤ C( u 0 L 2 x )λ γc+ σ(4-d)γc 2(d-5+4σ) sup [0,T ]
Iu λ (t)

1 4 Ḣ2 x + Iu λ (t) 1 2γ Ḣ2 x 2σ d-5+4σ × sup [0,T ] Iu λ (t) σ 2 Ḣ2 x + Iu λ (t) σ γ Ḣ2 x d-5 d-5+4σ . Since Iu λ (t) 2 Ḣ2 x E(Iu λ (t)) ≤ 1 for all t ∈ [0, T ], we get u λ M σ ([0,T ]) ≤ Kλ γc+ σ(4-d)γc 2(d-5+4σ) ,
for some constant K > 0. This contradicts with (1.12) by taking C larger than 2K. We thus obtain (1.8) and also E(Iu λ (t)) ≤ 1, ∀t ∈ [0, ∞). This also gives the uniform bound (1.9). In order to prove the scattering property, we will upgrade the uniform Morawetz bound (1.8) to the uniform Strichartz bound, namely

u S γ (R) := sup (p,q)∈B ∇ γ u L p t (R,L q x ) ≤ C( u 0 H γ x ).
Here (p, q) ∈ B means that (p, q) is biharmonic admissible (see again Section 2 for the definition). With this uniform Strichartz bound, the scattering property follows by a standard argument. We refer the reader to Section 4 for more details. This paper is organized as follows. We firstly introduce some notations and recall some results related to our problem in Section 2. In Section 3, we show the almost conservation law for the modified energy. Finally, we give the proof of our main result in Section 4.

Preliminaries

In the sequel, the notation A B denotes an estimate of the form A ≤ CB for some constant C > 0. The notation A ∼ B means that A B and B A. We write A B if A ≤ cB for some small constant c > 0. We also use a := 1 + |a|.

2.1. Nonlinearity. Let F (z) := |z| ν-1 z be the function which defines the nonlinearity in (NL4S). The derivative of F (z) is defined by

F (z) := (∂ z F (z), ∂ z F (z)),
where

∂ z F (z) = ν + 1 2 |z| ν-1 , ∂ z F (z) = ν -1 2 |z| ν-1 z z .
We also define its norm as

|F (z)| := |∂ z F (z)| + |∂ z F (z)|. It is clear that |F (z)| = O(|z| ν-1
). For a complex-valued function u, we have the following chain rule

∂ k F (u) = F (u)∂ k u, for k ∈ {1, • • • , d}. In particular, ∇F (u) = F (u)∇u. (2.1)
In order to estimate the nonlinearity, we need to recall the following fractional chain rules.

Lemma 2.1 ([CW91], [KPV93]

). Suppose that G ∈ C 1 (C, C), and α ∈ (0, 1). Then for

1 < q ≤ q 2 < ∞ and 1 < q 1 ≤ ∞ satisfying 1 q = 1 q1 + 1 q2 , |∇| α G(u) L q x G (u) L q 1 x |∇| α u L q 2 x . Lemma 2.2 ([Vis06]). Suppose that G ∈ C 0,β (C, C), β ∈ (0, 1). Then for every 0 < α < β, 1 < q < ∞, and α β < ρ < 1, |∇| α G(u) L q x |u| β-α ρ L q 1 x |∇| ρ u α ρ L α ρ q 2 x , provided 1 q = 1 q1 + 1 q2 and 1 -α βρ q 1 > 1.
The reader can find the proof of Lemma 2.1 in the case 1 < q 1 < ∞ in [CW91, Proposition 3.1] and [KPV93, Theorem A.6] when q 1 = ∞. For the proof of Lemma 2.2, we refer to [Vis06, Proposition A.1].

Strichartz estimates. Let

I ⊂ R and p, q ∈ [1, ∞]. The Strichartz norm is defined as u L p t (I,L q x ) := I R d |u(t, x)| q dx 1 q 1 p
with a usual modification when either p or q are infinity. When there is no risk of confusion, we write L p t L q x instead of L p t (I, L q x ). When p = q, we also use L p t,x . Definition 2.3. A pair (p, q) is said to be Schrödinger admissible, for short (p, q) ∈ S, if

(p, q) ∈ [2, ∞] 2 , (p, q, d) = (2, ∞, 2), 2 p + d q ≤ d 2 .
We denote for (p,

q) ∈ [1, ∞] 2 , γ p,q = d 2 - d q - 4 p . (2.2) Definition 2.4. A pair (p, q) is called biharmonic admissible, for short (p, q) ∈ B, if (p, q) ∈ S, γ p,q = 0.
Proposition 2.5 (Strichartz estimates for the fourth-order Schrödinger equation [START_REF] Dinh | Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces[END_REF]). Let γ ∈ R and u be a (weak) solution to the linear fourth-order Schrödinger equation, namely

u(t) = e it∆ 2 u 0 + t 0 e i(t-s)∆ 2 F (s)ds,
for some data u 0 , F . Then for all (p, q) and (a, b) Schrödinger admissible with q < ∞ and b < ∞,

|∇| γ u L p t (R,L q x ) |∇| γ+γp,q u 0 L 2 x + |∇| γ+γp,q-γ a ,b -4 F L a t (R,L b x ) . (2.3)
Here (a, a ) and (b, b ) are conjugate pairs, and γ p,q , γ a ,b are defined as in (2.2).

The estimate (2.3) is exactly the one given in [START_REF] Miao | Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations[END_REF], [START_REF] Pausader | Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case[END_REF] or [START_REF] Pausader | The cubic fourth-order Schrödinger equation[END_REF] where the author considered (p, q) and (a, b) are either sharp Schrödinger admissible, i.e.

p, q ∈ [2, ∞] 2 , (p, q, d) = (2, ∞, 2), 2 p + d q = d 2 ,
or biharmonic admissible. We refer the reader to [Din1, Proposition 2.1] for the proof of Proposition 2.5. The proof is based on the scaling technique instead of using a dedicate dispersive estimate of [START_REF] Ben-Artzi | Saut Disperion estimates for fourth-order Schrödinger equations[END_REF] for the fundamental solution of the homogeneous fourth-order Schrödinger equation.

The following result is a direct consequence of (2.3).

Corollary 2.6. Let γ ∈ R and u be a (weak) solution to the linear fourth-order Schrödinger equation for some data u 0 , F . Then for all (p, q) and (a, b) biharmonic admissible satisfying q < ∞ and b < ∞,

u L p t (R,L q x ) u 0 L 2 x + F L a t (R,L b x ) , (2.4)
and

|∇| γ u L p t (R,L q x ) |∇| γ u 0 L 2 x + |∇| γ-1 F L 2 t (R,L 2d d+2 x
) .

(2.5) 2.3. Littlewood-Paley decomposition. Let ϕ be a radial smooth bump function supported in the ball |ξ| ≤ 2 and equal to 1 on the ball |ξ| ≤ 1. For M = 2 k , k ∈ Z, we define the Littlewood-Paley operators

P ≤M f (ξ) := ϕ(M -1 ξ) f (ξ), P >M f (ξ) := (1 -ϕ(M -1 ξ)) f (ξ), P M f (ξ) := (ϕ(M -1 ξ) -ϕ(2M -1 ξ)) f (ξ),
where • is the spatial Fourier transform. Similarly, we can define for M, M 1 ≤ M 2 ∈ 2 Z ,

P <M := P ≤M -P M , P ≥M := P >M + P M , P M1<•≤M2 := P ≤M2 -P ≤M1 = M1<M ≤M2 P M .
We recall the following standard Bernstein inequalities (see e.g. [BCD11, Chapter 2] or [Tao06, Appendix]).

Lemma 2.7 (Bernstein inequalities). Let γ ≥ 0 and 1 ≤ p ≤ q ≤ ∞. We have

P ≥M f L p x M -γ |∇| γ P ≥M f L p x , P ≤M |∇| γ f L p x M γ P ≤M f L p x , P M |∇| ±γ f L p x ∼ M ±γ P M f L p x , P ≤M f L q x M d p -d q P ≤M f L p x , P M f L q x M d p -d q P M f L p x .
2.4. I-operator. Let 0 ≤ γ < 2 and N 1. We define the Fourier multiplier I N by

I N f (ξ) := m N (ξ) f (ξ),
where m N is a smooth, radially symmetric, non-increasing function such that

m N (ξ) := 1 if |ξ| ≤ N, (N -1 |ξ|) γ-2 if |ξ| ≥ 2N.
We shall drop the N from the notation and write I and m instead of I N and m N . We collect some basic properties of the I-operator in the following lemma.

Lemma 2.8 ([Din3]). Let 0 ≤ σ ≤ γ < 2 and 1 < q < ∞. Then If L q x f L q x ,
(2.6)

|∇| σ P >N f L q x N σ-2 ∆If L q x ,
(2.7)

∇ σ f L q x ∆ If L q x , (2.8) f H γ x If H 2 x N 2-γ f H γ x ,
(2.9)

If Ḣ2 x N 2-γ f Ḣγ x .
(2.10)

We refer to [Din3, Lemma 2.7] for the proof of these estimates. We also recall the following product rule which is a modified version of the one given in [VZ09, Lemma 2.5] in the context of nonlinear Schrödinger equation.

Lemma 2.9 ([Din3]). Let γ > 1, 0 < δ < γ -1 and 1 < q, q 1 , q 2 < ∞ be such that 1 q = 1 q1 + 1 q2 . Then I(f g) -(If )g L q x N -(2-γ+δ) If L q 1 x ∇ 2-γ+δ g L q 2
x .

(2.11)

We again refer the reader to [Din3, Lemma 2.8] for the proof of this lemma. A direct consequence of Lemma 2.9 and (2.1) is the following corollary.

Corollary 2.10. Let γ > 1, 0 < δ < γ -1 and 1 < q, q 1 , q 2 < ∞ be such that

1 q = 1 q1 + 1 q2 . Then ∇IF (u) -(I∇u)F (u) L q x N -(2-γ+δ) ∇Iu L q 1 x ∇ 2-γ+δ F (u) L q 2 x .
(2.12) 2.5. Interaction Morawetz inequality. We now recall the interaction Morawetz inequality for the nonlinear fourth-order Schrödinger equation.

Proposition 2.11 (Interaction Morawetz inequality [Pau2], [MWZ15]

). Let d ≥ 5, J be a compact time interval and u a solution to (NL4S) on the spacetime slab J × R d . Then we have the following a priori estimate:

|∇| -d-5 4 u L 4 t (J,L 4 x ) u 0 1 2 L 2 x u 1 2 L ∞ t (J, Ḣ 1 2 x )
.

(2.13)

This estimate was first established by Pausader in [START_REF] Pausader | The cubic fourth-order Schrödinger equation[END_REF] for d ≥ 7. Later, Miao-Wu-Zhang in [MWZ15] extended this interaction Morawetz estimate to d ≥ 5. By interpolating (2.13) and the trivial estimate

u L ∞ t (J, Ḣσ x ) ≤ u L ∞ t (J, Ḣσ x ) , we obtain u M σ (J) u 0 L 2 x u L ∞ t (J, Ḣ 1 2 x ) 2σ d-5+4σ u d-5 d-5+4σ L ∞ t (J, Ḣσ x ) , (2.14)
where

u M σ (J) := u L d-5+4σ σ t L 2(d-5+4σ) d-5+2σ x .
(2.15)

Almost conservation law

For any spacetime slab J × R d , we define

Z I (J) := sup (p,q)∈B ∆ Iu L p t (J,L q x ) .
Note that in our considerations, the biharmonic admissible condition (p, q) ∈ B ensures q < ∞.

Let us start with the following commutator estimates.

Lemma 3.1. Let 5 ≤ d ≤ 11, 2+γc 2 < γ < 2, 0 < δ < min{2γ -γ c -2, γ -1}, 0 < σ ≤ γ and max 8(d -5 + 4σ) d(d -5 + 2σ) + 8σ , 1 < ν -1 < min d -5 + 4σ 2σ , 8 d -2γ . Assume that u M σ (J) ≤ µ,
for some small constant µ > 0. Then

∇IF (u) -(I∇u)F (u) L 2 t (J,L 2d d+2 x ) N -(2-γ+δ) Z I (J) µ θ Z 1-θ I (J) + Z I (J) ν-1 (3.1) ∇IF (u) L 2 t (J,L 2d d+2 x ) N -(2-γ+δ) Z ν I (J) + µ (ν-1)θ Z 1+(ν-1)(1-θ) I (J), (3.2)
where

θ := (d -5 + 4σ)(8 -(d -4)(ν -1)) 2(ν -1)(2(d -5) + (12 -d)σ) ∈ (0, 1). (3.3)
Proof. For simplifying the presentation, we shall drop the dependence on the time interval J. Denote

ε := 4(ν -1)σ d -5 + 4σ -2(ν -1)σ .
It is easy to see from our assumptions that ε > 0. We next apply (2.12) with q = 2d d+2 , q 1 = 2d(2+ε) (d-2)(2+ε)-8 and q 2 = d(2+ε) 2ε+8 to get

∇IF (u) -(I∇u)F (u) L 2d d+2 x N -α ∇Iu L 2d(2+ε) (d-2)(2+ε)-8 x ∇ α F (u) L d(2+ε) 2ε+8 x
, where α = 2 -γ + δ. Note that q 1 is well-defined since (d -2)(2 + ε) -8 > 0. We then apply Hölder's inequality in time to have

∇IF (u) -(I∇u)F (u) L 2 t L 2d d+2 x N -α ∇Iu L 2+ε t L 2d(2+ε) (d-2)(2+ε)-8 x ∇ α F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x . (3.4)
For the first factor in the right hand side of (3.4), we use the Sobolev embedding to obtain ∇Iu

L 2+ε t L 2d(2+ε) (d-2)(2+ε)-8 x ∆Iu L 2+ε t L 2d(2+ε) d(2+ε)-8 x Z I , (3.5)
where 2 + ε, 2d(2+ε)

d(2+ε)-8
is a biharmonic admissible pair. To treat the second factor in the right hand side of (3.4), we note that α < γ -γ c by our assumption on δ. Thus

∇ α F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x ∇ γ-γc F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x + |∇| γ-γc F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x . (3.6) Since F (u) = O(|u| ν-1
), we bound the first term in (3.6) as

F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x u ν-1 L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x .
By the choice of ε, we have

2(ν -1)(2 + ε) ε = d -5 + 4σ σ , d(ν -1)(2 + ε) 2ε + 8 = d(ν -1)(d -5 + 4σ) 4(d -5 + 4σ -(ν -1)σ)
.

We next split u := P ≤N u + P >N u. For the low frequency part, we estimate

P ≤N u L d-5+4σ σ t L d(ν-1)(d-5+4σ) 4(d-5+4σ-(ν-1)σ) x P ≤N u θ M σ P ≤N u 1-θ L d-5+4σ σ t L 2d(d-5+4σ) (d-4)(d-5+4σ)-8σ x P ≤N u θ M σ ∆P ≤N u 1-θ L d-5+4σ σ t L 2d(d-5+4σ) d(d-5+4σ)-8σ x µ θ Z 1-θ I , (3.7)
where θ is given in (3.3). Here the first line follows from Hölder's inequality, and the second line makes use of the Sobolev embedding. The last inequality uses the fact that d-5+4σ σ , 2d(d-5+4σ)

d(d-5+4σ)-8σ
is biharmonic admissible. Note that our assumptions ensure θ ∈ (0, 1). For the high frequency part, the Sobolev embedding gives

P >N u L d-5+4σ σ t L d(ν-1)(d-5+4σ) 4(d-5+4σ-(ν-1)σ) x |∇| γc P >N u L d-5+4σ σ t L 2d(d-5+4σ) d(d-5+4σ)-8σ x N γc-2 Z I .
(3.8)

Here d-5+4σ σ , 2d(d-5+4σ) d(d-5+4σ)-8σ
is biharmonic admissible. Thus, we obtain

u L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x µ θ Z 1-θ I + N γc-2 Z I .
(3.9)

In particular,

F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x (µ θ Z 1-θ I + Z I ) ν-1 .
(3.10)

We next treat the second term in (3.6). Since ν -1 > 1, we are able to apply Lemma 2.1 to get

|∇| γ-γc F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x F (u) L 2(ν-1)(2+ε) ε(ν-2) t L d(ν-1)(2+ε) (2ε+8)(ν-2) x |∇| γ-γc u L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x u ν-2 L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x |∇| γ-γc u L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x , (3.11)
where F (u) = O(|u| ν-2 ). The first factor in the right hand side of (3.11) is treated in (3.9). For the second factor, we split u := P ≤1 u + P 1<•≤N u + P >N u. We use Bernstein inequality and estimate as in (3.7),

|∇| γ-γc P ≤1 u L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x P ≤1 u L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x µ θ Z 1-θ I .
The intermediate term is bounded by

|∇| γ-γc P 1<•≤N u L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x |∇| γ P 1<•≤N u L 2(ν-1)(2+ε) ε t L 2d(ν-1)(2+ε) d(ν-1)(2+ε)-4ε x ∆IP 1<•≤N u L 2(ν-1)(2+ε) ε t L 2d(ν-1)(2+ε) d(ν-1)(2+ε)-4ε x Z I .
Here we use

|∇| γ (∆I) -1 L 2d(ν-1)(2+ε) d(ν-1)(2+ε)-4ε x →L 2d(ν-1)(2+ε) d(ν-1)(2+ε)-4ε x 1,
and the fact 2(ν-1)(2+ε) ε , 2d(ν-1)(2+ε)

d(ν-1)(2+ε)-4ε
is biharmonic admissible. Finally, we use (2.7) to estimate

|∇| γ-γc P >N u L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x |∇| γ P >N u L 2(ν-1)(2+ε) ε t L 2d(ν-1)(2+ε) d(ν-1)(2+ε)-4ε x N γ-2 ∆P >N u L 2(ν-1)(2+ε) ε t L 2d(ν-1)(2+ε) d(ν-1)(2+ε)-4ε x N γ-2 Z I .
Combining three terms yields

|∇| γ-γc u L 2(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x µ θ Z 1-θ I + Z I .
(3.12)

Collecting (3.4), (3.5), (3.10), (3.11) and (3.12), we show the first estimate (3.1). We now prove (3.2). By triangle inequality,

I∇F (u) L 2 t L 2d d+2 x ≤ (I∇u)F (u) L 2 t L 2d d+2 x + I∇F (u) -(I∇u)F (u) L 2 t L 2d d+2 x
.

We have from Hölder's inequality, (3.5) and (3.9) that (I∇u)F (u) . Indeed, the proof of Lemma 3.1 is valid for ε = ∞.

L 2 t L 2d d+2 x I∇u L 2+ε t L 2d(2+ε) (d-2)(2+ε)-8 x F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x ∆Iu L 2+ε t L 2d(2+ε) d(2+ε)-8 x u ν-1 L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x Z I (µ θ Z 1-θ I + N γc-2 Z I ) ν-1 µ θ(ν-1) Z 1+(1-θ)(ν-1) I + N (γc-2)(ν-1) Z ν I . ( 3
We are now able to prove the almost conservation law for the modified energy functional E(Iu), where

E(Iu(t)) = 1 2 Iu(t) 2 Ḣ2 x + 1 ν + 1 Iu(t) ν+1 L ν+1 x . Proposition 3.3. Let 5 ≤ d ≤ 11, max 3 2 + γ c 4 , 4 -ν, 2 ν -1 + (ν -2)γ c ν -1 < γ < 2, 0 < δ < min{2γ -3 -γc 2 , γ + ν -4, (ν -1)γ -2 -(ν -2)γ c }, 0 < σ ≤ γ and max 8(d -5 + 4σ) d(d -5 + 2σ) + 8σ , 1 < ν -1 < min d -5 + 4σ 2σ , 8 d -2γ . Assume that u ∈ L ∞ ([0, T ], S (R d )
) is a solution to (NL4S) on a time interval [0, T ], and satisfies Iu 0 H 2 x ≤ 1. Assume in addition that u satisfies the a priori bound

u M σ ([0,T ]) ≤ µ,
for some small constant µ > 0. Then, for N sufficiently large,

sup t∈[0,T ] |E(Iu(t)) -E(Iu 0 )| N -(2-γ+δ) . (3.14)
Here the implicit constant depends only on the size of E(Iu 0 ).

Remark 3.4. As in Remark 3.2, the estimate (3.14) is still valid for ν -1 = d-5+4σ 2σ .

Proof of Proposition 3.3. We firstly note that our assumptions on γ and δ satisfy the assumptions given in Lemma 3.1. It allows us to use the estimates given in Lemma 3.1. We begin by controlling the size of Z I . By applying I, ∆I to (NL4S), and using Strichartz estimates (2.4), (2.5), we get

Z I Iu 0 H 2 x + IF (u) L 2 t L 2d d+4 x + ∇IF (u) L 2 t L 2d d+2 x . (3.15)
Using (3.2), we have

∇IF (u) L 2 t L 2d d+2 x N -(2-γ+δ) Z ν I + µ (ν-1)θ Z 1+(ν-1)(1-θ) I . (3.16)
Next, we drop the I-operator and use Hölder's inequality together with (3.9) to estimate

IF (u) L 2 t L 2d d+4 x |u| ν-1 L 2(2+ε) ε t L d(2+ε) 2ε+8 x u L 2+ε t L 2d(2+ε) d(2+ε)-8 x u ν-1 L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x u L 2+ε t L 2d(2+ε) d(2+ε)-8 x Z I (µ θ Z 1-θ I + N γc-2 Z I ) ν-1 µ (ν-1)θ Z 1+(ν-1)(1-θ) I + N (γc-2)(ν-1) Z ν I .
Here 2 + ε, 2d(2+ε)

d(2+ε)-8
is biharmonic admissible. We thus get

Z I Iu 0 H 2 x + N -(2-γ+δ) Z ν I + µ (ν-1)θ Z 1+(ν-1)(1-θ) I
.

By taking µ sufficiently small and N sufficiently large and using the assumption Iu 0 H 2 x ≤ 1, the continuity argument gives

Z I Iu 0 H 2 x ≤ 1. (3.17) Now, let F (u) = |u| ν-1 u. A direct computation shows ∂ t E(Iu(t)) = Re I∂ t u(∆ 2 Iu + F (Iu))dx.
By the Fundamental Theorem of Calculus,

E(Iu(t)) -E(Iu 0 ) = t 0 ∂ s E(Iu(s))ds = Re t 0 I∂ s u(∆ 2 Iu + F (Iu))dxds.
Using I∂ t u = i∆ 2 Iu + iIF (u), we see that

E(Iu(t)) -E(Iu 0 ) = Re t 0 I∂ s u(F (Iu) -IF (u))dxds = Im t 0 ∆ 2 Iu + IF (u)(F (Iu) -IF (u))dxds = Im t 0 ∆Iu∆(F (Iu) -IF (u))dxds +Im t 0 IF (u)(F (Iu) -IF (u))dxds
We next write

∆(F (Iu) -IF (u)) = (∆Iu)F (Iu) + |∇Iu| 2 F (Iu) -I(∆F (u)) -I(|∇u| 2 F (u)) = (∆Iu)(F (Iu) -F (u)) + |∇Iu| 2 (F (Iu) -F (u)) + ∇Iu • (∇Iu -∇u)F (u) +(∆Iu)F (u) -I(∆uF (u)) + (I∇u) • ∇uF (u) -I(∇u • ∇uF (u)).
Therefore, Let us consider (3.18). By Hölder's inequality, we estimate

E(Iu(t)) -E(Iu 0 ) = Im t 0 ∆Iu∆Iu(F (Iu) -F (u))dxds (3.18) +Im t 0 ∆Iu|∇Iu| 2 (F (Iu) -F (u))dxds (3.19)
|(3.18)| ∆Iu L 2 t L 2d d-4 x ∆Iu L 2+ε t L 2d(2+ε) d(2+ε)-8 x F (Iu) -F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x Z 2 I |Iu -u|(|Iu| + |u|) ν-2 L 2(2+ε) ε t L d(2+ε) 2ε+8 x Z 2 I P >N u L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x u ν-2 L 2(ν-1)(2+ε) ε t L d(ν-1)(2+ε) 2ε+8 x .
(3.24)

Combining (3.24), (3.8) and (3.9), we get

|(3.18)| N γc-2 Z 3 I (µ θ Z 1-θ I + Z I ) ν-2 . (3.25)
In order to treat (3.19), we need to separate two cases 0 < ν -2 < 1 and 1 ≤ ν -2. If 0 < ν -2 < 1, then using F (z) = O(|z| ν-2 ), we have

|F (z) -F (ζ)| |z -ζ| ν-2 , ∀z, ζ ∈ C.
Moreover, there exists k 1 so that k(ν -2) ≥ 2. By Hölder's inequality,

|(3.19)| ∆Iu L 2 t L 2d d-4 x ∇Iu 2 L 4k k-2 t L q x F (Iu) -F (u) L k t L r x ∆Iu L 2 t L 2d d-4 x |∇| 1+ γc 2 Iu 2 L 4k k-2 t L q x |Iu -u| ν-2 L k t L r x Z 3 I P >N u ν-2 L k(ν-2) t L r(ν-2) x Z 3 I |∇| γc P >N u ν-2 L k(ν-2) t L r x N (γc-2)(ν-2) Z ν+1 I ,
where

q := 4kd(ν-1) (kd-4(k-2))(ν-1)+8k , q := 2kd kd-2(k-2) , r := kd(ν-1) 4(k-1)(ν-1)-4k , r := 2kd kd(ν-2)-8 .
(3.26)

Here we drop the I-operator and use (2.8) with the fact 1 + γc 2 < γ < 2 to have the third line. Note that 4k k-2 , q and k(ν -2), r are biharmonic admissible. The last line follows from (2.7). If 1 ≤ ν -2, then

|F (z) -F (ζ)| |z -ζ|(|z| + |ζ|) ν-3 , ∀z, ζ ∈ C.
We estimate

|(3.19)| ∆Iu L 2 t L 2d d-4 x ∇Iu 2 L 4k k-2 t L q x F (Iu) -F (u) L k t L r x ∆Iu L 2 t L 2d d-4 x |∇| 1+ γc 2 Iu 2 L 4k k-2 t L q x F (Iu) -F (u) L k t L r x Z 3 I Iu -u L k(ν-2) t L r(ν-2) x |Iu| + |u| ν-3 L k(ν-2) t L r(ν-2) x Z 3 I |∇| γc P >N u L k(ν-2) t L r x |∇| γc u ν-3 L k(ν-2) t L r x N γc-2 Z ν+1 I .
Thus, collecting two cases, we obtain

|(3.19)| N min{ν-2,1}(γc-2) Z ν+1 I . (3.27)
We next estimate

|(3.20)| ∆Iu L 2 t L 2d d-4 x ∇Iu L 4k k-2 t L q x ∇Iu -∇u L 4k k-2 t L q x F (u) L k t L r x ∆Iu L 2 t L 2d d-4 x ∇Iu L 4k k-2 t L q x ∇P >N u L 4k k-2 t L q x u ν-2 L k(ν-2) t L r(ν-2) x ∆Iu L 2 t L 2d d-4 x |∇| 1+ γc 2 Iu L 4k k-2 t L q x |∇| 1+ γc 2 P >N u L 4k k-2 t L q x |∇| γc u ν-2 L k(ν-2) t L r x N γc-2 2 Z ν+1 I .
(3.28)

We next consider the term (3.21). Using the notation given in Lemma 3.1, we apply Corollary 2.10 with q = 2d d+4 , q 1 = 2d(2+ε) d(2+ε)-8 and q 2 = d(2+ε) 2ε+8 to have (∆Iu)F (u) -I(∆uF (u))

L 2d d+4 x N -α ∆Iu L 2d(2+ε) d(2+ε)-8 x ∇ α F (u) L d(2+ε) 2ε+8 x
, where α = 2 -γ + δ. By Hölder's inequality, (∆Iu)F (u) -I(∆uF (u))

L 2 t L 2d d+4 x N -α ∆Iu L 2+ε t L 2d(2+ε) d(2+ε)-8 x ∇ α F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x .
We have from (3.6), (3.10) and (3.12) that

∇ α F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x (µ θ Z 1-θ I + Z I ) ν-1 . Thus (∆Iu)F (u) -I(∆uF (u)) L 2 t L 2d d+4 x N -α Z I (µ θ Z 1-θ I + Z I ) ν-1 ,
and

|(3.21)| N -(2-γ+δ) Z 2 I (µ θ Z 1-θ I + Z I ) ν-1 . (3.29)
Similarly,

|(3.22)| ∆Iu L 2 t L 2d d-4 x (I∇u) • ∇uF (u) -I(∇u • ∇uF (u)) L 2 t L 2d d+4 x .
We next apply Lemma 2.9 with q = 2d d+4 , q 1 = 4kd(ν-1)

(kd-4(k-2))(ν-1)+8k and q 2 = 4kd(ν-1) (kd+12k-8)(ν-1)-8k to have (I∇u) • ∇uF (u) -I(∇u • ∇uF (u)) L 2d d+4 x N -α I∇u L 4kd(ν-1) (kd-4(k-2))(ν-1)+8k x × ∇ α (∇uF (u)) L 4kd(ν-1) (kd+12k-8)(ν-1)-8k x .
Using the notation (3.26), the fractional chain rule implies

∇ α (∇uF (u)) L 4kd(ν-1) (kd+12k-8)(ν-1)-8k x ∇ 1+α u L q x F (u) L r x + ∇u L q x ∇ α F (u) L r x .
The Hölder inequality then gives

(I∇u) • ∇uF (u) -I(∇u • ∇uF (u)) L 2 t L 2d d+4 x N -α I∇u L 4k k-2 t L q x × ∇ 1+α u L 4k k-2 t L q x F (u) L k t L r x + ∇u L 4k k-2 t L q x ∇ α F (u) L k t L r x .
By the Sobolev embedding (dropping the I-operator if necessary) and (2.8), we have

I∇u L 4k k-2 t L q x , ∇u L 4k k-2 t L q x |∇| 1+ γc 2 u L 4k k-2 t L q x Z I , (3.30) ∇ 1+α u L 4k k-2 t L q x ∇ 1+α+ γc 2 u L 4k k-2 t L q x Z I . (3.31)
Note that by our assumptions on δ, 1 + α + γc 2 = 3 -γ + δ + γc 2 < γ. We also have

F (u) L k t L r x u ν-2 L k(ν-2) t L r(ν-2) x |∇| γc u ν-2 L k(ν-2) t L r x Z ν-2 I . (3.32) It remains to treat ∇ α F (u) L k t L r
x . Using (3.32), we only need to bound

|∇| α F (u) L k t L r
x . To do so, we separate two cases: 1 ≤ ν -2 and 0 < ν -2 < 1.

If 1 ≤ ν -2, then we apply Lemma 2.1 for q = r, q 1 = r(ν-2) ν-3 , q 2 = r(ν -2) and use Hölder's inequality to have

|∇| α F (u) L k t L r x O(|u| ν-3 ) L k(ν-2) ν-3 t L r(ν-2) ν-3 x |∇| α u L k(ν-2) t L r(ν-2) x u ν-3 L k(ν-2) t L r(ν-2) x |∇| α u L k(ν-2) t L r(ν-2) x |∇| γ c u ν-3 L k(ν-2) t L r x |∇| α+γc u L k(ν-2) t L r x Z ν-2 I .
Here by our assumptions, α + γ c < γ which allows us to use (2.8) to get the last estimate. If 0 < ν -2 < 1, then we use Lemma 2.2 with

β = ν -2, α = 2 -γ + δ, q = r and q 1 , q 2 satisfying ν -2 - α ρ q 1 = α ρ q 2 = r(ν -2),
and α ν-2 < ρ < 1 to be chosen later. With these choices, we have

1 - α βρ q 1 = r > 1.
Then,

|∇| α F (u) L r x |u| ν-2-α ρ L q 1 x |∇| ρ u α ρ L α ρ q 2 x u ν-2-α ρ L ( ν-2-α ρ ) q 1 x |∇| ρ u α ρ L α ρ x
. By Hölder's inequality,

|∇| α F (u) L k t L r x u ν-2-α ρ L ( ν-2-α ρ ) p 1 t L ( ν-2-α ρ ) q 1 x |∇| ρ u α ρ L α ρ p 2 t L α ρ q 2 x u ν-2-α ρ L k(ν-2) t L r(ν-2) x |∇| ρ u L k(ν-2) t L r(ν-2) x , provided that ν -2 - α ρ p 1 = α ρ p 2 = k(ν -2).
The Sobolev imbedding then gives

|∇| α F (u) L k t L r x |∇| γc u ν-2-α ρ L k(ν-2) t L r x |∇| ρ+γc u L k(ν-2) t L r x Z ν-2 I .
Here we use (2.8) together with ρ + γ c < γ to get the last estimate. Note that

α ν -2 + γ c < ρ + γ c . If we want ρ + γ c < γ for an appropriate value of ρ, we need α ν-2 + γ c < γ. This implies γ > 2 ν-1 + ν-2 ν-1 γ c and δ < (ν -1)γ -2 -(ν -2)γ c . Collecting 2 cases, we show |∇| α F (u) L k t L r x Z ν-2 I . ( 3
L 2 t L 2d d+4 x N -α Z ν I .
Thus,

|(3.22)| N -(2-γ+δ) Z ν+1 I . (3.34)
Finally, we consider (3.23). We bound

|(3.23)| |∇| -1 IF (u) L 2 t L 2d d-2 x ∇(F (Iu) -IF (u)) L 2 t L 2d d+2 x ∇IF (u) L 2 t L 2d d+2 x ∇(F (Iu) -IF (u)) L 2 t L 2d d+2 x . (3.35) By (3.2), ∇IF (u)
L 2 t L 2d d+2 x N -(2-γ+δ) Z ν I + µ (ν-1)θ Z 1+(ν-1)(1-θ) I . (3.36)
By the triangle inequality,

∇(F (Iu) -IF (u)) L 2 t L 2d d+2 x (∇Iu)(F (Iu) -F (u)) L 2 t L 2d d+2 x + (∇Iu)F (u) -∇IF (u) L 2 t L 2d d+2 x .
We firstly use Hölder's inequality and estimate as in (3.24) to get

(∇Iu)(F (Iu) -F (u)) L 2 t L 2d d+2 x ∇Iu L 2+ε t L 2d(2+ε) d(2+ε)-12 x F (Iu) -F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x ∆Iu L 2+ε t L 2d(2+ε) d(2+ε)-8 x F (Iu) -F (u) L 2(2+ε) ε t L d(2+ε) 2ε+8 x N γc-2 Z 2 I (µ θ Z 1-θ I + Z I ) ν-2 . ( 3.37) By (3.1), (∇Iu)
F (u) -∇IF (u) L 2 t L 2d d+2 x N -(2-γ+δ) Z I (µ θ Z 1-θ I + Z I ) ν-1 . ( 3.38)
Combining (3.35) -(3.38), we get

|(3.23)| N -(2-γ+δ) Z 2 I (µ θ Z 1-θ I + Z I ) 2(ν-1) . (3.39)
Collecting (3.25), (3.27), (3.28), (3.29), (3.34), (3.39) and using (3.17), we prove (3.14). Note that our assumptions on δ implies

2 -γ + δ < min γ -1 - γ c 2 , ν -2, (ν -2)(γ -γ c ) < min 2 -γ c 2 , (ν -2)(2 -γ c ) .
The proof is complete.

Global well-posedness and scattering

In this section, we shall give the proof of the global existence and scattering given in Theorem 1.1.

Global well-posedness. By the density argument, the proof of global well-posedness will be reduced to the following.

Proposition 4.1. Let 5 ≤ d ≤ 11 and γ(d, ν) < γ < 2 with γ(d, ν) be as in (1.5). Suppose that u is a global solution to (NL4S) with initial data u 0 ∈ C ∞ 0 (R d ). Then, u M σ (R) ≤ C( u 0 H γ x ), (4.1) u L ∞ t (R,H γ x ) ≤ C( u 0 H γ x ), (4.2)
where

• M σ is given in (2.15).
Proof. The proof of this result is based on the almost conservation law given in Proposition 3.3.

To do so, we need the modified energy of initial data is small. Since E(Iu 0 ) is not necessarily small, we use the scaling (1.2) to make E(Iu λ (0)) small. We have

E(Iu λ (0)) = 1 2 Iu λ (0) 2 Ḣ2 x + 1 ν + 1 Iu λ (0) ν+1 L ν+1 x . (4.3)
By (2.10),

Iu λ (0) Ḣ2 x N 2-γ u λ (0) Ḣγ x = N 2-γ λ γc-γ u 0 Ḣγ x . (4.4) By choosing λ ≈ N 2-γ γ-γc , (4.5) we have Iu λ (0) Ḣ2 x ≤ 1 8 . We next bound Iu λ (0) L ν+1
x . Note that we can easily estimate this norm by the Sobolev embedding,

Iu λ (0) L ν+1 x u λ (0) L ν+1 x = λ d ν+1 -4 ν-1 u 0 L ν+1 x λ (d-4)(ν-1)-8 (ν+1)(ν-1) u 0 H γ x , provided that γ ≥ d(ν-1)
2(ν+1) . In order to remove this unexpected condition on γ, we use the technique of [CKSTT04] (see also [MWZ15]). We firstly separate the frequency space into the domains

Ω 1 := ξ ∈ R d , |ξ| 1 λ , Ω 2 := ξ ∈ R d , 1 λ |ξ| N , Ω 3 := ξ ∈ R d , |ξ| N ,
and then write [u λ (0)] (ξ) = (χ 1 (ξ) + χ 2 (ξ) + χ 3 (ξ))[u λ (0)] (ξ),
for non-negative smooth functions χ j supported in Ω j , j = 1, 2, 3 respectively and satisfying χ j (ξ) = 1. Thus

Iu λ (0) = χ 1 (D)Iu λ (0) + χ 2 (D)Iu λ (0) + χ 3 (D)Iu λ (0).
We now use the Sobolev embedding to have

χ 1 (D)Iu λ (0) L ν+1 x |∇| d(ν-1) 2(ν+1) χ 1 (D)Iu λ (0) L 2 x |∇| d(ν-1) 2(ν+1) χ 1 (D)I L 2 x →L 2 x u λ (0) L 2 x .
Thanks to the support of χ 1 , the functional calculus gives

|∇| d(ν-1) 2(ν+1) χ 1 (D)I L 2 x →L 2 x |ξ| d(ν-1) 2(ν+1) -α |ξ| α χ 1 (ξ) L ∞ ξ λ α-d(ν-1) 2(ν+1) , (4.6) provided 0 < α < d(ν-1) 2(ν+1) . Similarly, χ 3 (D)Iu λ (0) L ν+1 x |∇| d(ν-1) 2(ν+1) χ 3 (D)Iu λ (0) L 2 x |∇| d(ν-1) 2(ν+1) -γ χ 3 (D)I L 2 x →L 2 x u λ (0) Ḣγ x . A direct computation shows u λ (0) Ḣγ x = λ -γ u 0 Ḣγ x . (4.7)
Using the support of χ 3 , the functional calculus again gives

|∇| d(ν-1) 2(ν+1) -γ χ 3 (D)I L 2 x →L 2 x |ξ| d(ν-1) 2(ν+1) -γ χ 3 (ξ)(N |ξ| -1 ) 2-γ L ∞ ξ N d(ν-1) 2(ν+1) -γ . (4.8)
To obtain this bound, we split into two cases. When d(ν-1) 2(ν+1) ≥ γ, we simply bound

|ξ| d(ν-1) 2(ν+1) -γ χ 3 (ξ)(N |ξ| -1 ) 2-γ L ∞ ξ 1 N d(ν-1) 2(ν+1) -γ .
When γ > d(ν-1) 2(ν+1) , we write

|ξ| d(ν-1) 2(ν+1) -γ χ 3 (ξ)(N |ξ| -1 ) 2-γ L ∞ ξ = N d(ν-1) 2(ν+1) -γ (N |ξ| -1 ) γ-d(ν-1) 2(ν+1) χ 3 (ξ)(N |ξ| -1 ) 2-γ L ∞ ξ N d(ν-1) 2(ν+1) -γ .
Combining (4.7) and (4.8), we get

χ 3 (D)Iu λ (0) L ν+1 x N d(ν-1) 2(ν+1) -γ λ -γ u 0 Ḣγ x .
(4.9)

We treat the intermediate case as

χ 2 (D)Iu λ (0) L ν+1 x |∇| d(ν-1) 2(ν+1) -γ χ 2 (D)I L 2 x →L 2 x u λ (0) Ḣγ x . We have |∇| d(ν-1) 2(ν+1) -γ χ 2 (D)I L 2 x →L 2 x ξ| d(ν-1) 2(ν+1) -γ χ 2 (ξ) L ∞ ξ . When d(ν-1) 2(ν+1) ≥ γ, we bound ξ| d(ν-1) 2(ν+1) -γ χ 2 (ξ) L ∞ ξ N d(ν-1) 2(ν+1) -γ .
When γ > d(ν-1) 2(ν+1) , we write

ξ| d(ν-1) 2(ν+1) -γ χ 2 (ξ) L ∞ ξ = ξ| d(ν-1) 2(ν+1) -γ-β |ξ| β χ 2 (ξ) L ∞ ξ λ β+γ-d(ν-1) 2(ν+1) , provided d(ν-1) 2(ν+1) -γ < β < d(ν-1) 2(ν+1)
. These estimates together with (4.7) yield

χ 2 (D)Iu λ (0) L ν+1 x    N d(ν-1) 2(ν+1) -γ λ -γ u 0 Ḣγ x if d(ν-1) 2(ν+1) ≥ γ, λ β-d(ν-1) 2(ν+1) u 0 Ḣγ x if γ > d(ν-1) 2(ν+1) .
(4.10)

Collecting (4.6), (4.9), (4.10) and use (4.5), we obtain

Iu λ (0) L ν+1 x λ α-d(ν-1) 2(ν+1) + λ β-d(ν-1) 2(ν+1) + λ ((d-4)(ν-1)-8)γ 2(2-γ)(ν+1) u 0 H γ x , (4.11)
for some 0 < α < d(ν-1) 2(ν+1) and d(ν-1) 2(ν+1) -γ < β < d(ν-1) 2(ν+1) . Therefore, it follows from (4.4), (4.5) and (4.11) by taking λ sufficiently large depending on u 0 H γ x and N (which will be chosen later and depend only on u 0 H γ x ) that E(Iu λ (0)) ≤ 1 4 .

We now show that there exists an absolute constant C such that

u λ M σ (R) ≤ Cλ γc+ σ(4-d)γc 2(d-5+4σ
) . (4.12)

By undoing the scaling, using the fact that

u λ M σ (R) = λ γc+ σ(4-d) d-5+4σ u M σ (R)
, we get (4.1). We shall use the bootstrap argument to show (4.12). By time reversal symmetry, it suffices to treat the positive time only. To do so, we define

Ω 1 := t ∈ [0, ∞) | u λ M σ ([0,t]) ≤ Cλ γc+ σ(4-d)γc 2(d-5+4σ) . We want to show Ω 1 = [0, ∞). Let Ω 2 := t ∈ [0, ∞) | u λ M σ ([0,t]) ≤ 2Cλ γc+ σ(4-d)γc 2(d-5+4σ) .
In order to run the bootstrap argument successfully, we need to verify four things: 1) Ω 1 = ∅. This is obvious as 0 ∈ Ω 1 . 2) Ω 1 is closed. This follows from Fatou's Lemma. 3) Ω 2 ⊂ Ω 1 . 4) If T ∈ Ω 1 , then there exists δ > 0 such that [T, T + δ) ⊂ Ω 2 . This is a consequence of the local well-posedness and 3). It remains to prove 3). Fix T ∈ Ω 2 , we will show that T ∈ Ω 1 . We firstly use the interaction Morawetz inequality (2.14) and the mass conservation to have

u λ M σ ([0,T ]) u λ (0) L 2 x u λ L ∞ t ([0,T ], Ḣ 1 2 x ) 2σ d-5+4σ u λ d-5 d-5+4σ L ∞ t ([0,T ], Ḣσ x ) C( u 0 L 2 x )λ 2σγc d-5+4σ u λ 2σ d-5+4σ L ∞ t ([0,T ], Ḣ 1 2 x ) u λ d-5 d-5+4σ L ∞ t ([0,T ], Ḣσ x ) .
(4.13)

We now decompose u λ (t) := P ≤N u λ (t) + P >N u λ (t) to estimate the second and the third factor in the right hand side of (4.13). For the low frequency part, we interpolate between the L 2

x -norm and Ḣ2

x -norm to have

P ≤N u λ (t) Ḣ 1 2 x P ≤N u λ (t) 3 4 L 2 x P ≤N u λ (t) 1 4 Ḣ2 x C( u 0 L 2 x )λ 3γc 4
Iu λ (t)

1 4

Ḣ2

x , (4.14)

P ≤N u λ (t) Ḣσ x P ≤N u λ (t) 1-σ 2 L 2 x P ≤N u λ (t) σ 2 Ḣ2 x C( u 0 L 2 x )λ γc(2-σ) 2 Iu λ (t) σ 2 Ḣ2 x . (4.15)
Note that the I-operator is the identity on low frequency |ξ| ≤ N . For high frequency part, we interpolate between the L 2 x -norm and Ḣγ x -norm and use (2.7) to have

P >N u λ (t) Ḣ 1 2 x P >N u λ (t) 1-1 2γ L 2 x P >N u λ (t) 1 2γ Ḣγ x C( u 0 L 2 x )λ γc(1-1 2γ ) N γ-2 2γ
Iu λ (t)

1 2γ Ḣ2 x C( u 0 L 2 x )λ 3γc 4
Iu λ (t)

1 2γ Ḣ2
x , (4.16)

P >N u λ (t) Ḣσ x P >N u λ (t) 1-σ γ L 2 x P >N u λ (t) σ γ Ḣ2 x C( u 0 L 2 x )λ γc(1-σ γ ) N σ(γ-2) γ Iu λ (t) σ γ Ḣ2 x C( u 0 L 2 x )λ γc (2-σ) 2 Iu λ (t) σ γ Ḣ2 x .
(4.17)

Here we use the fact 0 < γ < 2 to get (4.16) and (4.17). Collecting (4.13) through (4.17), we get

u λ M σ ([0,T ]) C( u 0 L 2 x )λ γc+ σ(4-d)γc 2(d-5+4σ) sup [0,T ] Iu λ (t) 1 4 Ḣ2 x + Iu λ (t) 1 2γ Ḣ2 x 2σ d-5+4σ × sup [0,T ] Iu λ (t) σ 2 Ḣ2 x + Iu λ (t) σ γ Ḣ2 x d-5 d-5+4σ . (4.18)
Thus, by taking C sufficiently large depending on u 0 L 2 x , we get T ∈ Ω 1 , provided that sup

[0,T ] Iu λ (t) Ḣ2 x 1. (4.19)
We will prove that (4.19) holds for T ∈ Ω 2 . Indeed, let µ > 0 be a sufficiently small constant given in Proposition 3.3. We divide [0, T ] into subintervals J k , k = 1, ..., L in such a way that

u λ M σ (J k ) ≤ µ.
The number of possible subinterval must satisfy

L ∼ λ γc+ σ(4-d)γc 2(d-5+4σ) µ d-5+4σ σ ∼ λ γc(d-5+(8-d)σ) σ . (4.20)
We next apply Proposition 3.3 on each of the subintervals J k to have sup

[0,T ] Iu λ (t) 2 L 2 x sup [0,T ] E(Iu λ (t)) ≤ E(Iu λ (0)) + C(E(Iu λ (0)))N -(2-γ+δ) L. Since E(Iu λ (0)) ≤ 1 4 , we need N -(2-γ+δ) L 1 4 (4.
N γc (2-γ)(d-5+(8-d)σ) (γ-γc )σ -(2-γ+δ)
1. This is possible whenever γ is such that

γ c (2 -γ)(d -5 + (8 -d)σ) (γ -γ c )σ < 2 -γ + δ, or γ c (2 -γ)(d -5 + (8 -d)σ) < (2 -γ + δ)(γ -γ c )σ. (4.22) Since δ < min{2γ -3 -γc 2 , γ + ν -4, (ν -1)γ -2 -(ν -2)γ c }, we have γ > γ(d, ν, σ)
, where γ(d, ν, σ) is the (larger if there are two) root of the equation

γ c (2 -γ)(d -5 + (8 -d)σ) = min γ -1 - γ c 2 , ν -2, (ν -2)(γ -γ c ) (γ -γ c )σ.
This completes the bootstrap argument and (4.12) follows. Thus, (4.19) holds for all T ∈ R.

We now estimate u(T ) H γ x . To do so, we use the conservation of mass, the scaling (1.2) and (2.9) to have

u(T ) H γ x u(T ) L 2 x + u(T ) Ḣγ x u 0 L 2 x + λ γ-γc u λ (λ 4 T ) Ḣγ x u 0 L 2 x + λ γ-γc Iu λ (λ 4 T ) H 2 x u 0 L 2 x + λ γ-γc u λ (λ 4 T ) L 2 x + Iu λ (λ 4 T ) Ḣ2 x .
Using (4.19), we get for all T ∈ R,

u(T ) H γ x u 0 L 2 x + λ γ-γc (λ γ c u 0 L 2 x + 1) ≤ C( u 0 H γ x
). Here we use (4.5) with the fact that N is chosen sufficiently large depending only on u 0 H γ x . This proves (4.2) and the proof of Proposition 4.1 is complete.

Scattering. We firstly show that the global Morawetz estimate 4.1 can be upgraded to the global Strichartz estimate

u S γ (R) := sup (p,q)∈B ∇ γ u L p t (R,L q x ) ≤ C( u 0 H γ x ). (4.23)
Here we refer to Section 2 for the definition of (p, q) ∈ B. Let u be a global solution to (NL4S) with initial data u 0 ∈ H γ (R d ) for 5 ≤ d ≤ 11 and γ(d, ν) < γ < 2. Using the uniform bound (4.1), we can decompose R into a finite number of disjoint intervals

J k = [t k , t k+1 ], k = 1, ..., L so that u M (J k ) ≤ δ, (4.24)
for a small constant δ > 0 to be chosen later. By Strichartz estimates (2.4) and (2.5), we have

u S γ (J k ) ∇ γ u(t k ) L 2 x + F (u) L 2 t (J k ,L 2d d+4 x ) + |∇| γ-1 F (u) L 2 t (J k ,L 2d d+2 x
) .

(4.25)

We estimate for some ε > 0,

|∇| γ-1 F (u) L 2 t (J k ,L 2d d+2 x ) |∇| γ-1 u L 2+ε t (J k ,L 2d(2+ε) (d-2)(2+ε)-8 x ) F (u) L 2(2+ε) ε t (J k ,L d(2+ε) 2ε+8 x ) |∇| γ u L 2+ε t (J k ,L 2d(2+ε) d(2+ε)-8 x ) u ν-1 L 2(ν-1)(2+ε) ε t (J k ,L d(ν-1)(2+ε) 2ε+8 x ) u S γ (J k ) u ν-1 L 2(ν-1)(2+ε) ε t (J k ,L d(ν-1)(2+ε) 2ε+8 x ) .
(4.26)

Similarly,

|F (u) L 2 t (J k ,L 2d d+4 x ) u L 2+ε t (J k ,L 2d(2+ε) d(2+ε)-8 x ) F (u) L 2(2+ε) ε t (J k ,L d(2+ε) 2ε+8 x ) u L 2+ε t (J k ,L 2d(2+ε) d(2+ε)-8 x ) u ν-1 L 2(ν-1)(2+ε) ε t (J k ,L d(ν-1)(2+ε) 2ε+8 x ) u S γ (J k ) u ν-1 L 2(ν-1)(2+ε) ε t (J k ,L d(ν-1)(2+ε) 2ε+8 x ) .
(4.27)

We now need the following result. Proof. We firstly use Hölder's inequality to have

u L 2(ν-1)(2+ε) ε t (J,L d(ν-1)(2+ε) 2ε+8 x ) ≤ u θ1 M σ (J) u 1-θ1
L ∞ t (J,L q x ) , (4.29)

provided that 1 -θ 1 q = 4(2ε + 8)σ -dε(d -5 + 2σ) 4dσ(ν -1)(2 + ε) and θ 1 := ε(d -5 + 4σ) 2(ν -1)(2 + ε)σ .

Similarly,

u L ∞ t (J,L q x ) u θ2 L ∞ t (J,L 2 x ) u 1-θ2 L ∞ t (J,L 2d d-2γ x ) u θ2 L ∞ t (J,L 2
x ) u 1-θ2 L ∞ t (J, Ḣγ In order to perform the above estimates, we need α(ε) > 0 and β(ε) > 0. We note that ε → α(ε) and ε → β(ε) are decreasing functions provided that γ > (d-4)σ d-5+4σ . Moreover, since

α(ε) → 1 - d 2γ (ν -1) + 4 γ , β(ε) → d γ ν -1 2 - 4 d as ε → 0.
As 8 d < γ < 8 d-2γ , the two limits are positive. Thus by taking ε > 0 small enough, we have α(ε) > 0 and β(ε) > 0. The proof is complete.

Remark 4.3. It is easy to see that the function σ ∈ (0, γ] → (d-4)σ d-5+4σ is increasing and attains its maximal value at σ = γ. In this case, the condition (d-4)σ d-5+4σ < γ becomes γ > 1 4 which is always satisfied in our consideration.

We now continue the proof of scattering property. By (4.25), (4.26), (4.27) and Lemma 4.2, we have

u S γ (J k ) ∇ γ u(t k ) L 2 x + u S γ (J k ) u ε(d-5+4σ) 2(2+ε)σ M σ (J k ) u α(ε) L ∞ t (J k ,L 2 x ) u β(ε) L ∞ t (J k , Ḣγ x ) ∇ γ u(t k ) L 2 x + u S γ (J k ) u ε(d-

  .13) The estimate (3.2) follows easily from (3.1) and (3.13). Note that by our assumptions, α = 2 -γ + δ < γ -γ c < 2 -γ c . The proof is complete. Remark 3.2. The estimates (3.1) and (3.2) still hold for ν -1 = d-5+4σ 2σ

0 IF

 0 I∇u) • ∇uF (u) -I(∇u • ∇uF (u))]dxds (3.22) +Im t (u)(F (Iu) -IF (u))dxds.(3.23)

  .33) By (3.30), (3.31), (3.32) and (3.33), (I∇u) • ∇uF (u) -I(∇u • ∇uF (u))

Lemma 4. 2 .

 2 Let d ≥ 5, 0 < σ ≤ γ < 2 be such that (d-4)σ d-5+4σ < γ and 8 d < ν -1 < 8 d-2γ. Then there exists ε > 0 small such that for any time intervalJ, 16σ + ε((d + 4)σ -γ(d -5 + 4σ)) 2γσ(2 + ε) , β(ε) := d γ ν -1 2 -16 + ε(d + 4) 2d(2 + ε) .

  θ 2 )(d -2γ) 2d .Thus, by (4.29) and (4.30), a direct consequence givesu ) := θ 2 (1 -θ 1 )(ν -1) = 1 -d 2γ (ν -1) + 16σ + ε((d + 4)σ -γ(d -5 + 4σ)) 2γσ(2 + ε) , β(ε) := (1 -θ 2 )(1 -θ 1 )(ν -

Table 1 .

 1 Our results compare with best known results.

		5 1 2	16 11 ≈ 1.4545 (see [MWZ15])	1.6711
	3	6 1 16 9 ≈ 1.7777 (see [MWZ15])	1.8719
		7 3 2	45 23 ≈ 1.9565 (see [MWZ15])	1.9665
	4	5 7 6 6 5 3	--	1.9257 1.9922

  By the time reversal symmetry, it is enough to treat the positive time only. We will show that e -it∆ 2 u(t) has limits in H γ x as t → +∞. By Duhamel formula,M σ ([t1,t2]) . This implies that e -it2∆ 2 u(t 2 ) -e -it1∆ 2 u(t 1 ) H γ x → 0 as t 1 , t 2 → +∞. Hence the limit +∞. The proof is now complete.

	By Strichartz estimates (2.4), (2.5) and estimating as in (4.31),
	e -it2∆ 2	u(t 2 ) -e -it1∆ 2	u(t 1 ) H γ x		i	t2 t1	e -is∆ 2	F (u(s))ds	H γ x
							F (u)	L 2 t ([t1,t2],L	2d d+4 x	)	+ |∇| γ-1 F (u)	L 2 t ([t1,t2],L	2d d+2 x	)
							u S γ ([t1,t2]) u	ε(d-5+4σ) 2(2+ε)σ M σ ([t1,t2]) u	α(ε)+β(ε) L ∞ t ([t1,t2],H γ x )
							u	1+α(ε)+β(ε) S γ ([t1,t2])	u	ε(d-5+4σ) 2(2+ε)σ
					u + 0 := lim t→+∞	e -it∆ 2	u(t)
	exists in H γ x . Moreover,
		u(t) -e it∆ 2	u + 0 = -i	+∞	e i(t-s)∆ 2	F (u(s))ds.
									t
	A same argument as above shows that
						u(t) -e it∆ 2	u + 0 H γ x → 0
	as t →
									5+4σ) 2(2+ε)σ M σ (J k )	u	α(ε)+β(ε) L ∞ t (J k ,H γ x ) .	(4.31)
	This shows that
									ε(d-5+4σ)
		u S γ (J k )		∇					2(2+ε)σ C( u 0 H γ x ).
	By taking δ > 0 small enough, we get
			u S γ (J k )		∇	γ u(t k ) L 2 x ≤ C( u 0 H γ x ).
	This proves (4.23).
	We next use the global Strichartz bound (4.23) to prove the scattering property, i.e. there exist
	unique u ± 0 ∈ H γ x such that
				lim t→±∞	u(t) -e it∆ 2	u ± 0 H γ x = 0.
			e -it∆ 2	u(t) = u 0 + i	t	e -is∆ 2	F (u(s))ds.
									0
	For 0 < t 1 < t 2 , we have
		e -it2∆ 2	u(t 2 ) -e -it1∆ 2	u(t 1 ) = i	t2	e -is∆ 2	F (u(s))ds.
									t1

γ u(t k ) L 2 x + u S γ (J k ) δ

Acknowledgments

The author would like to express his deep gratitude to Prof. Jean-Marc BOUCLET for the kind guidance and encouragement.