Global existence for the defocusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space

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Global existence for the defocusing mass-critical nonlinear fourth-order Schrödinger equation below the

energy space

Van Duong Dinh

To cite this version:

Van Duong Dinh. Global existence for the defocusing mass-critical nonlinear fourth-order Schrödinger

equation below the energy space. 2017. �hal-01544689�

GLOBAL EXISTENCE FOR THE DEFOCUSING MASS-CRITICAL NONLINEAR FOURTH-ORDER SCHR ¨ ODINGER EQUATION BELOW THE

ENERGY SPACE

VAN DUONG DINH

Abstract. In this paper, we consider the defocusing mass-critical nonlinear fourth-order Schr¨ odinger equation. Using the I-method combined with the interaction Morawetz estimate, we prove that the problem is globally well-posed in H

^γ

( R

^d

),5 ≤ d ≤ 7 with γ(d) < γ < 2, where γ(5) =

⁸₅

, γ(6) =

⁵₃

and γ(7) =

¹³₇

.

1. Introduction

Consider the defocusing mass-critical nonlinear fourth-order Schr¨ odinger equation, namely i∂ t u(t, x) + ∆ ² u(t, x) = −(|u|

^d8

u)(t, x), t ≥ 0, 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 .

The fourth-order Schr¨ odinger equation was introduced by Karpman [Kar96] and Karpman- Shagalov [KS00] taking into account the role of small fourth-order dispersion terms in the prop- agation of intense laser beams in a bulk medium with Kerr nonlinearity. The study of nonlinear fourth-order Schr¨ odinger equation has attracted a lot of interest in the past several years (see [Pau1], [Pau2], [HHW06], [HHW07], [HJ05], [MXZ09], [MXZ11], [MWZ15] and references therein).

It is known (see [Din1] or [Din2]) that (NL4S) is locally well-posed in H ^γ ( R ^d ) for γ > 0 satisfying for d 6= 1, 2, 4,

dγe ≤ 1 + 8

d . (1.1)

Here dγe is the smallest integer greater than or equal to γ. This condition ensures the nonlinearity to have enough regularity. 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)) := ku(t)k ² _L

2

( R

^d

) = ku 0 k ² _L

2

( R

^d

) , and H ² -solution has conserved energy, i.e.

E(u(t)) :=

Z

R

^d

1

2 |∆u(t, x)| ² + d

2d + 8 |u(t, x)|

^2d+8d

dx = E(u 0 ).

The conservations of mass and energy together with the persistence of regularity (see [Din2]) yield the global well-posedness for (NL4S) in H ^γ ( R ^d ) with γ ≥ 2 satisfying for d 6= 1, 2, 4, (1.1). We also have (see [Din1] or [Din2]) the local well-posedness for (NL4S) with initial data u 0 ∈ L ² ( R ^d ) but the time of existence depends on the profile of u 0 instead of its H ^γ -norm. The global existence holds for small L ² -norm initial data. For large L ² -norm initial data, the conservation of mass

2010 Mathematics Subject Classification. 35G20, 35G25, 35Q55.

Key words and phrases. Nonlinear fourth-order Schr¨ odinger; Global well-posedness; Almost conservation law;

Morawetz inequality.

1

does not immediately give the global well-posedness in L ² ( R ^d ). For the global well-posedness with large L ² -norm initial data, we refer the reader to [PS10] where the authors established the global well-posedness and scattering for (NL4S) in L ² ( R ^d ), d ≥ 5.

The main goal of this paper is to prove the global well-posedness for (NL4S) in a low regularity space H ^γ ( R ^d ), d ≥ 5 with γ < 2. Since we are working with low regularity data, the conservation of energy does not hold. In order to overcome this problem, we make use of the I-method and the interaction Morawetz inequality. Due to the high-order term ∆ ² u, we requires the nonlinearity to have at least two orders of derivatives in order to successfully establish the almost conservation law. We thus restrict ourself in spatial space of dimensions d = 5, 6, 7.

Let us recall some known results about the global existence below the energy space for the nonlinear fourth-order Schr¨ odinger equation. To our knowledge, the first result to address this problem belongs to Guo in [Guo10], where the author considered a more general fourth-order Schr¨ odinger equation, namely

i∂ t u + λ∆u + µ∆ ² u + ν|u| ^2m u = 0, and established the global existence in H ^γ ( R ^d ) for 1 + ^md−9+

√

(4m−md+7)

²

+16

4m < γ < 2 where m is an integer satisfying 4 < md < 4m + 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 Schr¨ odinger equation. Later, Miao-Wu-Zhang studied the defocusing cubic fourth-order Schr¨ odinger equation, namely

i∂ t u + ∆ ² u + |u| ² u = 0,

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 I-method and a new interaction Morawetz inequality. Recently, the author in [Din3] showed that the defocusing cubic fourth-order Schr¨ odinger equation is globally well-posed in H ^γ ( R ⁴ ) with ⁶⁰ ₅₃ < γ < 2. The analysis is carried out in Bourgain spaces X ^γ,b which is similar to those in [CKSTT02]. Note that in the above considerations, the nonlinearity is algebraic. This allows to write explicitly the commutator between the I-operator and the nonlinearity by means of the Fourier transform, and then control it by multi-linear analysis. When one considers the mass-critical nonlinear fourth-order Schr¨ odinger equation in dimensions d ≥ 5, this method does not work. We thus rely purely on Strichartz and interaction Morawetz estimates. The main result of this paper is the following:

Theorem 1.1. Let d = 5, 6, 7. The initial value problem (NL4S) is globally well-posed in H ^γ ( R ^d ), for any γ(d) < γ < 2, where γ(5) = ⁸ ₅ , γ(6) = ⁵ ₃ and γ(7) = ¹³ ₇ .

The proof of the above theorem is based on the combination of the I-method and the inter-

action Morawetz inequality which is similar to those given in [DPST07]. The I-method was first

introduced by I-Team in [CKSTT02] in order to treat the nonlinear Schr¨ odinger equation at low

regularity. The idea is to replace the non-conserved energy E(u) when γ < 2 by an “almost con-

served” 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. Note

that in our setting, the nonlinearity is not algebraic. Thus we can not apply the Fourier transform

technique. Fortunately, thanks to a special Strichartz estimate (2.4), we are able to apply the

technique given in [VZ09] to control the commutator. The interaction Morawetz inequality for

the nonlinear fourth-order Schr¨ odinger equation was first introduced in [Pau2] for d ≥ 7, and was

extended for d ≥ 5 in [MWZ15]. With this estimate, the interpolation argument and Sobolev embedding give for any compact interval J,

kuk M (J) := kuk

L

8(d−3) d

t

L

2(d−3) d−4 x

. |J |

^{8(d−3)d−4}

ku 0 k

1 d−3

L

²_x

kuk

d−4 d−3

L

^∞_t

H ˙

1 x2

. (1.2)

As a byproduct of the Strichartz estimates and I-method, we show the almost conservation law for the modified energy of (NL4S), that is if u ∈ L ^∞ (J, S ( R ^d )) is a solution to (NL4S) on a time interval J = [0, T ], and satisfies kIu 0 k H

_x²

≤ 1 and if u satisfies in addition the a priori bound kuk _{M (J)} ≤ µ for some small constant µ > 0, then

sup

t∈[0,T]

|E(Iu(t)) − E(Iu ₀ )| . N ^{−(2−γ+δ)} . for max

3 − ⁸ _d , ⁸ _d < γ < 2 and 0 < δ < γ + _d ⁸ − 3.

We now briefly outline the idea of the proof. Let u be a global in time solution to (NL4S).

Observe that for any λ > 0,

u λ (t, x) := λ ⁻

^d2

u(λ ⁻⁴ t, λ ⁻¹ x) (1.3) is also a solution to (NL4S). By choosing

λ ∼ N

^2−γγ

, (1.4)

and using some harmonic analysis, we can make E(Iu λ (0)) ≤ ¹ ₄ by taking λ sufficiently large depending on ku 0 k _H

_x^γ

and N . Fix an arbitrary large time T. The main goal is to show

E(Iu λ (λ ⁴ T )) ≤ 1. (1.5)

With this bound, we can easily obtain the growth of ku(T )k _H

_x^γ

, and the global well-posedness in H ^γ ( R ^d ) follows immediately. In order to get (1.5), we claim that

ku _λ k _{M ([0,t])} ≤ Kt

^{8(d−3)d−4}

, ∀t ∈ [0, λ ⁴ T],

for some constant K. If it is not so, then there exists T 0 ∈ [0, λ ⁴ T] such that ku λ k _{M ([0,T}

₀

_]) > KT

d−4 8(d−3)

0 , (1.6)

ku λ k M ([0,T

₀

]) ≤ 2KT

d−4 8(d−3)

0 . (1.7)

Using (1.7), we can split [0, T ₀ ] into L subintervals J _k , k = 1, ..., L so that ku λ k _M(J

_k

₎ ≤ µ.

The number L must satisfy

L ∼ T

d−4 d

0 . (1.8)

Thus we can apply the almost conservation law to get sup

[0,T

₀

]

E(Iu λ (t)) ≤ E(Iu λ (0)) + N ^{−(2−γ+δ)} L.

Since E(Iu _λ (0)) ≤ ¹ ₄ , in order to have E(Iu _λ (t)) ≤ 1 for all t ∈ [0, T ₀ ], we need N ^{−(2−γ+δ)} L 1

4 . (1.9)

Combining (1.4), (1.8) and (1.9), we obtain the condition on γ. Next, using (1.2) together with some harmonic analysis, we estimate

ku λ k _{M ([0,T}

₀

_]) . T

d−4 8(d−3)

0 ku 0 k

1 d−3

L

²_x

sup

[0,T

₀

]

ku 0 k

³⁴

kIu λ (t)k _H

¹⁴

_˙

₂

x

+ N ⁻

³⁴

kIu λ (t)k H ˙

²_x

^d−4_d−3

.

Since kIu λ (t)k H ˙

_x²

. E(Iu λ (t)) ≤ 1 for all t ∈ [0, T ₀ ], we get ku λ k _M([0,T

₀

_]) ≤ CT

d−4 8(d−3)

0 ,

for some constant C > 0. This leads to a contradiction to (1.6) for an appropriate choice of K.

Thus we have the claim and also

E(Iu λ (t)) ≤ 1, ∀t ∈ [0, λ ⁴ T].

For more details, we refer the reader to Section 4.

This paper is organized as follows. In Section 2, we introduce some notations and recall some results related to our problem. In Section 3, we show the almost conservation law for the modified energy. Finally, the proof of our main result is given in Section 4.

2. 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 hai := 1 + |a|.

2.1. Nonlinearity. Let F (z) := |z|

^8d

z, d = 5, 6, 7 be the function that defines the nonlinearity in (NL4S). The derivative F ⁰ (z) is defined as a real-linear operator acting on w ∈ C by

F ⁰ (z) · w := w∂ _z F(z) + w∂ _z F (z), where

∂ z F (z) = 2d + 8

2d |z|

^d8

, ∂ z F (z) = 4 d |z|

^8d

z

z .

We shall identify F ⁰ (z) with the pair (∂ z F(z), ∂ z F (z)), and define its norm by

|F ⁰ (z)| := |∂ z F (z)| + |∂ z F (z)|.

It is clear that |F ⁰ (z)| = O(|z|

^d8

). We also have the following chain rule

∂ k F (u) = F ⁰ (u)∂ k u, for k ∈ {1, · · · , d}. In particular, we have

∇F (u) = F ⁰ (u)∇u.

We next recall the fractional chain rule to estimate the nonlinearity.

Lemma 2.1. Suppose that G ∈ C ¹ ( C , C ), and α ∈ (0, 1). Then for 1 < q ≤ q ₂ < ∞ and 1 < q ₁ ≤ ∞ satisfying ¹ _q = _q ¹

1

+ _q ¹

2

, k|∇| ^α G(u)k _L

^q

x

. kG ⁰ (u)k _L

^q1

x

k|∇| ^α uk _L

^q2 x

.

We refer the reader to [CW91, Proposition 3.1] for the proof of the above estimate when 1 <

q ₁ < ∞, and to [KPV93, Theorem A.6] for the proof when q ₁ = ∞.

When G is no longer C ¹ , but H¨ older continuous, we have the following fractional chain rule.

Lemma 2.2. Suppose that G ∈ C ^0,β ( C , C ), β ∈ (0, 1). Then for every 0 < α < β, 1 < q < ∞, and

α

β < ρ < 1,

k|∇| ^α G(u)k _L

^q_x

. k|u| ^β−

^αρ

k _L

^q1

x

k|∇| ^ρ uk

α ρ

L

α ρq2 x

,

provided ¹ _q = _q ¹

1

+ _q ¹

2

and

1 − _βρ ^α q 1 > 1.

The reader can find the proof of this result in [Vis06, Proposition A.1].

2.2. Strichartz estimates. Let I ⊂ R and p, q ∈ [1, ∞]. We define the mixed norm kuk _L

^p

t

(I,L

^q_x

) := Z

I

Z

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 may write L ^p _t L ^q _x instead of L ^p _t (I, L ^q _x ). We also use L ^p _t,x when p = q.

Definition 2.3. A pair (p, q) is said to be Schr¨ odinger admissible, for short (p, q) ∈ S, if (p, q) ∈ [2, ∞] ² , (p, q, d) 6= (2, ∞, 2), 2

p + d q ≤ d

2 . We also denote for (p, q) ∈ [1, ∞] ² ,

γ p,q = d 2 − d

q − 4

p . (2.1)

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 estimate for fourth-order Schr¨ odinger equation [Din1]). Let γ ∈ R and u be a (weak) solution to the linear fourth-order Schr¨ odinger equation namely

u(t) = e ^it∆

²

u ₀ + Z t

0

e ^i(t−s)∆

²

F (s)ds,

for some data u 0 , F . Then for all (p, q) and (a, b) Schr¨ odinger admissible with q < ∞ and b < ∞, k|∇| ^γ uk _L

^p

t

( R ,L

^q_x

) . k|∇| ^γ+γ

^p,q

u ₀ k L

²_x

+ k|∇| ^γ+γ

^p,q

^−γ

^a0,b0

⁻⁴ F k _L

a0

t

( R ,L

^b_x⁰

) . (2.2) Here (a, a ⁰ ) and (b, b ⁰ ) are conjugate pairs, and γ p,q , γ a

⁰

,b

⁰

are defined as in (2.1).

Note that the estimate (2.2) is exactly the one given in [MZ07], [Pau1] or [Pau2] where the author considered (p, q) and (a, b) are either sharp Schr¨ odinger admissible, i.e.

p, q ∈ [2, ∞] ² , (p, q, d) 6= (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 [BKS00] for the fundamental solution of the homogeneous fourth-order Schr¨ odinger equation.

The following result is a direct consequence of (2.2).

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

kuk _L

^p

t

( R ,L

^q_x

) . ku 0 k L

²_x

+ kFk _L

a0

t

( R ,L

^b_x⁰

) , (2.3)

and

k∆uk _L

^p

t

( R ,L

^q_x

) . k∆u 0 k _L

2

x

+ k∇F k

L

²_t

( R ,L

2d d+2 x

)

. (2.4)

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) ˆ f (ξ), P \ _>M f (ξ) := (1 − ϕ(M ⁻¹ ξ)) ˆ f (ξ),

P [ _M f (ξ) := (ϕ(M ⁻¹ ξ) − ϕ(2M ⁻¹ ξ)) ˆ f (ξ), where ˆ · is the spatial Fourier transform. Similarly, we can define

P <M := P _≤M − P M , P _≥M := P >M + P M , and for M 1 ≤ M 2 ,

P _M

₁

_<·≤M

₂

:= P _≤M

₂

− P _≤M

₁

= X

M

₁

<M≤M

2

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 kP _≥M f k _L

^p_x

. M ^−γ k|∇| ^γ P _≥M f k _L

^p_x

,

kP _≤M |∇| ^γ f k _L

^p_x

. M ^γ kP _≤M f k _L

^p_x

, kP M |∇| ^±γ f k _L

^p_x

∼ M ^±γ kP M f k _L

^p_x

,

kP _≤M f k _L

^q_x

. M

^dp

⁻

^dq

kP _≤M f k _L

^p_x

, kP M f k _L

^q_x

. M

^dp

⁻

^dq

kP M f k _L

^p_x

.

2.4. I-operator. Let 0 ≤ γ < 2 and N 1. We define the Fourier multiplier I N by I d _N f (ξ) := m _N (ξ) ˆ 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. Let 0 ≤ σ ≤ γ < 2 and 1 < q < ∞. Then

kIfk _L

^q_x

. kf k _L

^q_x

, (2.5)

k|∇| ^σ P _>N f k _L

^q_x

. N ^σ−2 k∆If k _L

^q_x

, (2.6) k h∇i ^σ f k _L

^q_x

. k h∆i If k _L

^q_x

, (2.7) kf k _H

_x^γ

. kIf k H

_x²

. N ^2−γ kf k _H

^γ_x

, (2.8) kIf k H ˙

_x²

. N ^2−γ kf k H ˙

^γ_x

. (2.9) Proof. The estimate (2.5) is a direct consequence of the H¨ ormander-Mikhlin multiplier theorem.

To prove (2.6), we write

k|∇| ^σ P _>N f k _L

^q_x

= k|∇| ^σ P _>N (∆I) ⁻¹ ∆If k _L

^q_x

.

The desired estimate (2.6) follows again from the H¨ ormander-Mikhlin multiplier theorem. In order to get (2.7), we estimate

k h∇i ^σ f k _L

^q_x

≤ kP _≤N h∇i ^σ fk _L

^q_x

+ kP >N f k _L

^q_x

+ kP >N |∇| ^σ f k _L

^q_x

.

Thanks to the fact that the I-operator is the identity at low frequency |ξ| ≤ N , the multiplier theorem and (2.6) imply

k h∇i ^σ fk _L

^q_x

. k h∆i If k _L

^q_x

+ k∆If k _L

^q_x

.

This proves (2.7). Finally, by the definition of the I-operator and (2.6), we have kf k _H

^γ_x

. kP ≤N fk _H

_x^γ

+ kP >N f k L

²_x

+ k|∇| ^γ P >N f k L

²_x

. kP ≤N Ifk _H

_x^γ

+ N ⁻² k∆Ifk L

²_x

+ N ^γ−2 k∆If k L

²_x

. kIf k H

_x²

. This shows the first inequality in (2.8). For the second inequality in (2.8), we estimate

kIfk H

²_x

. kP ≤N h∇i ² Ifk L

²_x

+ kP >N h∇i ² If k L

²_x

. N ^2−γ kf k _H

_x^γ

. Here we use the definition of I-operator to get

kP _≤N I h∇i ^2−γ k _L

2

x

→L

²_x

, kP >N I h∇i ^2−γ k _L

2

x

→L

²_x

. N ^2−γ .

The estimate (2.9) is proved as for the second estimate in (2.8). The proof is complete.

When the nonlinearity F (u) is algebraic, one can use the Fourier transform to write the com- mutator like F (Iu) − IF (u) as a product of Fourier transforms of u and Iu, and then measure the frequency interactions. However, in our setting, the nonlinearity is no longer algebraic, we thus need the following rougher estimate which is a modified version of the Schr¨ odinger context (see [VZ09]).

Lemma 2.9. Let 1 < γ < 2, 0 < δ < γ − 1 and 1 < q, q ₁ , q ₂ < ∞ be such that ¹ _q = _q ¹

1

+ _q ¹

2

. Then kI(f g) − (If )gk _L

^q_x

. N ^{−(2−γ+δ)} kIf k _L

^q1

x

k h∇i ^2−γ+δ gk _L

^q2

x

. (2.10)

The proof is a slight modification of the one given in Lemma 2.5 of [VZ09]. We thus only give a sketch of the proof.

Sketch of the proof. By the Littlewood-Paley decomposition, we write I(f g) − (If)g = I(f P ≤1 g) − (If )P ≤1 g + X

M >1

[I(P _{. M} f P M g) − (IP _{. M} f)P M g]

+ X

M >1

[I(P _M f P _M g) − (IP _M f )P _M g]

= I(P _{& N} f P _≤1 g) − (IP _{& N} f )P _≤1 g + X

M & N

[I(P _{. M} f P _M g) − (IP _{. M} f )P _M g]

+ X

M >1

[I(P _M f P _M g) − (IP _M f )P _M g]

= Term 1 + Term 2 + Term 3 . Here we use the definition of the I-operator to get

I(P _N f P _≤1 g) = (IP _N f )P _≤1 g, I(P _{. M} f P _M g) = (IP _{. M} f )P _M g,

for all M N .

For the second term, using Lemma 2.7 and Lemma 2.8, we estimate kI(P _{. M} f P M g) − (IP _{. M} f )P M gk _L

^q_x

. kP _{. M} f k _L

^q1

x

kP M gk _L

^q2

x

, M & N . M

N 2−γ

kIf k _L

^q1

x

kP M gk _L

^q2 x

. M ^−δ N ^−(2−γ) kIf k _L

^q1

x

k|∇| ^2−γ+δ gk _L

^q2 x

. Summing over all N . M ∈ 2 ^Z , we get

kTerm ₂ k _L

^q_x

. N ^{−(2−γ+δ)} kIf k _L

^q1

x

k|∇| ^2−γ+δ gk _L

^q2 x

. For the third term, we write

I(P _M f P _M g) − (IP _M f )P _M g = X

1k∈N

[I(P ₂

k

M f P _M g) − (IP ₂

k

M f )P _M g]

= X

1k∈N N.2k M

I(P ₂

k

M f P _M g) − (IP ₂

k

M f )P _M g].

We note that

[I(P ₂

k

M f P M g) − (IP ₂

k

M f )P M g] b (ξ) = Z

ξ=ξ

₁

+ξ

₂

(m N (ξ 1 + ξ 2 ) − m N (ξ 1 )) P \ ₂

k

M f (ξ 1 )[ P M g(ξ 2 ).

For |ξ 1 | ∼ 2 ^k M & N and |ξ 2 | ∼ M , the mean value theorem implies

|m N (ξ 1 + ξ 2 ) − m N (ξ 2 )| . |∇m N (ξ 1 )||ξ 2 | . 2 ^−k 2 ^k M N

γ−2

.

The Coifman-Meyer multiplier theorem (see e.g. [CM75, CM91]) then yields kI(P ₂

k

M f P M g) − (IP ₂

k

M f )P M gk _L

^q_x

. 2 ^−k 2 ^k M

N γ−2

kP ₂

k

M f k _L

^q1

x

kP M gk _L

^q2 x

. 2 ^−k M ^{−(2−γ+δ)} kIf k _L

^q1

x

k|∇| ^2−γ+δ gk _L

^q2 x

.

By rewrite 2 ^−k M ^{−(2−γ+δ)} = 2 ^{−k(γ−1−δ)} (2 ^k M ) ^{−(2−γ+δ)} , we sum over all k 1 with γ − 1 > δ and N . 2 ^k M to get

kTerm 3 k _L

^q_x

. N ^{−(2−γ+δ)} kIf k _L

^q1

x

k|∇| ^2−γ+δ gk _L

^q2 x

.

Finally, we consider the first term. It is proved by the same argument as for the third term. We estimate

kTerm 1 k _L

^q_x

. X

k∈ N ,2

^k

& N

kI(P ₂

k

f P _≤1 g) − (IP ₂

k

f )P _≤1 gk _L

^q_x

. X

k∈ N ,2

^k

& N

2 ^−k kIf k _L

^q1 x

kgk _L

^q2

x

. N ⁻¹ kIf k _L

^q1 x

kgk _L

^q2

x

.

Note that the condition γ − 1 > δ ensures that N ⁻¹ . N ^{−(2−γ+δ)} . This completes the proof.

As a direct consequence of Lemma 2.9 with the fact that

∇F (u) = ∇uF ⁰ (u),

we have the following corollary. Note that the I-operator commutes with ∇.

Corollary 2.10. Let 1 < γ < 2, 0 < δ < γ − 1 and 1 < q, q 1 , q 2 < ∞ be such that ¹ _q = _q ¹

1

+ _q ¹

2

. Then

k∇IF (u) − (I∇u)F ⁰ (u)k _L

^q_x

. N ^{−(2−γ+δ)} k∇Iuk _L

^q1

x

k h∇i ^2−γ+δ F ⁰ (u)k _L

^q2

x

. (2.11) 2.5. Interaction Morawetz inequality. We end this section by recalling the interaction Morawetz inequality for the nonlinear fourth-order Schr¨ odinger equation. This estimate was first established by Pausader in [Pau2] for d ≥ 7. Later, Miao-Wu-Zhang in [MWZ15] extended this interaction Morawetz estimate to d ≥ 5.

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:

k|∇| ⁻

^d−54

uk _L

⁴

t

(J,L

⁴_x

) . ku 0 k _L

¹²2 x

kuk

¹²

L

^∞_t

(J, H ˙

1 x2

)

. (2.12)

By interpolating (2.12) and the trivial estimate kuk

L

^∞_t

(J, H ˙

1

x2

) ≤ kuk

L

^∞_t

(J, H ˙

1 x2

) , we obtain

kuk

L

^2(d−3)_t

(J,L

2(d−3) d−4

x

)

. ku 0 k _L

2

x

kuk

L

^∞_t

(J, H ˙

1 x2

)

_d−3¹

kuk

d−5 d−3

L

^∞_t

(J, H ˙

1 x2

)

= ku 0 k

1 d−3

L

²_x

kuk

d−4 d−3

L

^∞_t

(J, H ˙

1 x2

)

.

Using Sobolev embedding in time, we get kuk _M(J) := kuk

L

8(d−3) d

t

(J,L

2(d−3) d−4

x

)

. |J |

^{8(d−3)d−4}

ku ₀ k

1 d−3

L

²_x

kuk

d−4 d−3

L

^∞_t

(J, H ˙

1 x2

)

. (2.13)

Here _8(d−3)

d , ^2(d−3) _d−4

is a biharmonic admissible pair.

3. Almost conservation law For any spacetime slab J × R ^d , we define

Z I (J) := sup

(p,q)∈B

k h∆i Iuk _L

^p

t

(J,L

^q_x

) .

Note that in our consideration 5 ≤ d ≤ 7, the biharmonic admissible condition (p, q) ∈ B ensures q < ∞. Let us start with the following commutator estimates.

Lemma 3.1. Let 5 ≤ d ≤ 7, 1 < γ < 2 and 0 < δ < γ − 1. Then k∇IF (u) − (I∇u)F ⁰ (u)k

L

²_t

(J,L

2d d+2

x

) . N ^{−(2−γ+δ)} (Z I (J)) ¹⁺

^8d

, (3.1) k∇IF (u)k

L

²_t

(J,L

2d d+2

x

) . kuk _M

^8d

_(J) Z I (J ) + N ^{−(2−γ+δ)} (Z I (J )) ¹⁺

^d8

, (3.2) where kuk _{M (J)} is given in (2.13). In particular,

k∇IF (u)k

L

²_t

(J,L

2d d+2

x

) . (Z I (J )) ¹⁺

^8d

. (3.3)

Proof. For simplifying the notation, we shall drop the dependence on the time interval J . We apply (2.11) with q = _d+2 ^2d , q ₁ = _d ^2d(d−3)

₂

_−9d+22 and q ₂ = _2(2d−7) ^d(d−3) to get

k∇IF (u) − (I∇u)F ⁰ (u)k

L

2d d+2 x

. N ^{−(2−γ+δ)} k∇Iuk

L

2d(d−3) d2−9d+22 x

k h∇i ^2−γ+δ F ⁰ (u)k

L

d(d−3) 2(2d−7) x

.

We then apply H¨ older’s inequality to have k∇IF (u) − (I∇u)F ⁰ (u)k

L

²_t

L

2d d+2 x

. N ^−α k∇Iuk

L

2(d−3) d−4

t

L

2d(d−3) d2−9d+22 x

k h∇i ^α F ⁰ (u)k

L

^2(d−3)_t

L

d(d−3) 2(2d−7) x

,

where α = 2 − γ + δ ∈ (0, 1) by our assumptions. For the first factor in the right hand side, we use the Sobolev embedding to obtain

k∇Iuk

L

2(d−3) d−4

t

L

2d(d−3) d2−9d+22 x

. k∆Iuk

L

2(d−3) d−4

t

L

2d(d−3) d2−7d+16 x

. Z _I , (3.4)

where _2(d−3)

d−4 , _d ^2d(d−3)

2

−7d+16

is a biharmonic admissible pair. For the second factor, we estimate k h∇i ^α F ⁰ (u)k

L

^2(d−3)_t

L

d(d−3) 2(2d−7) x

. kF ⁰ (u)k

L

^2(d−3)_t

L

d(d−3) 2(2d−7) x

+ k|∇| ^α F ⁰ (u)k

L

^2(d−3)_t

L

d(d−3) 2(2d−7) x

. (3.5) Since F ⁰ (u) = O(|u|

^8d

), we use (2.7) to have

kF ⁰ (u)k

L

^2(d−3)_t

L

d(d−3) 2(2d−7) x

. kuk

^8d

L

16(d−3) d

t

L

4(d−3) 2d−7 x

. Z

8 d

I , (3.6)

where _16(d−3)

d , ^4(d−3) _2d−7

is biharmonic admissible. In order to treat the second term in (3.5), we apply Lemma 2.1 with q = _2(2d−7) ^d(d−3) , q ₁ = _−d ^2d(d−3)

₂

_+11d−26 and q ₂ = _d ^2d(d−3)

₂

_−3d−2 to get

k|∇| ^α F ⁰ (u)k

L

^2(d−3)_t

L

d(d−3) 2(2d−7) x

. kF ⁰⁰ (u)k

L

^4(d−3)_t

L

2d(d−3)

−d2 +11d−26 x

k|∇| ^α uk

L

^4(d−3)_t

L

2d(d−3) d2−3d−2 x

.

As F ⁰⁰ (u) = O(|u|

^8d

⁻¹ ), we have kF ⁰⁰ (u)k

L

^4(d−3)_t

L

2d(d−3)

−d2 +11d−26 x

. kuk

^8d

⁻¹

L

4(8−d)(d−3) d

t

L

2(8−d)(d−3)

−d2 +11d−26 x

. Z

8 d

−1

I . (3.7)

Here 4(8−d)(d−3)

d , 2(8−d)(d−3)

−d

²

+11d−26

is biharmonic admissible. Since

4(d − 3), _d ^2d(d−3)

2

−3d−2

is also a bi- harmonic admissible, we have from (2.7) that

k|∇| ^α uk

L

^4(d−3)_t

L

2d(d−3) d2−3d−2 x

. Z _I . (3.8)

Note that α < 1 < γ. Collecting (3.4), (3.6), (3.7) and (3.8), we prove (3.1).

We now prove (3.2). We have from (3.1) and the triangle inequality that k∇IF (u)k

L

²_t

L

2d d+2 x

. k(∇Iu)F ⁰ (u)k

L

²_t

L

2d d+2 x

+ N ^{−(2−γ+δ)} Z ¹⁺

8 d

I . (3.9)

The H¨ older inequality gives k(∇Iu)F ⁰ (u)k

L

²_t

L

2d d+2 x

. k∇Iuk

L

2(d−3) d−5

t

L

2d(d−3) d2−9d+26 x

kF ⁰ (u)k

L

^d−3_t

L

d(d−3) 4(d−4) x

.

We use the Sobolev embedding to estimate k∇Iuk

L

2(d−3) d−5

t

L

2d(d−3) d2−9d+26 x

. k∆Iuk

L

2(d−3) d−5

t

L

2d(d−3) d2−7d+20 x

. Z I . (3.10)

Here _2(d−3)

d−5 , _d ^2d(d−3)

₂

_−7d+20

is biharmonic admissible. Since F ⁰ (u) = O(|u|

^8d

), we have kF ⁰ (u)k

L

^d−3_t

L

d(d−3) 4(d−4) x

. kuk

^8d

L

8(d−3) d

t

L

2(d−3) d−4 x

= kuk _M

^8d

. (3.11)

Combining (3.9), (3.10) and (3.11), we obtain (3.2). The estimate (3.3) follows directly from (3.2) and (2.7). Note that _8(d−3)

d , ^2(d−3) _d−4

is biharmonic admissible. The proof is complete.

We are now able to prove the almost conservation law for the modified energy functional E(Iu), where

E(Iu(t)) = 1

2 kIu(t)k ² _{H ˙}

₂

x

+ d

2d + 8 kIu(t)k

^2d+8d

L

2d+8 xd

.

Proposition 3.2. Let 5 ≤ d ≤ 7, max

3 − ⁸ _d , ⁸ _d < γ < 2 and 0 < δ < γ + ⁸ _d − 3. Assume that u ∈ L ^∞ ([0, T ], S ( R ^d )) is a solution to (NL4S) on a time interval J = [0, T ], and satisfies kIu 0 k _H

2

x

≤ 1. Assume in addition that u satisfies the a priori bound kuk M (J) ≤ 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)} . (3.12) Here the implicit constant depends only on the size fo E(Iu 0 ).

Proof. We again drop the notation J for simplicity. Our first step is to control the size of Z I . Applying I, ∆I to (NL4S), and then using Strichartz estimates (2.3), (2.4), we have

Z I . kIu 0 k H

_x²

+ kIF (u)k

L

²_t

L

2d d+4 x

+ k∇IF (u)k

L

²_t

L

2d d+2 x

. (3.13)

Using (3.2), we have k∇IF (u)k

L

²_t

L

2d d+2 x

. kuk _M

^8d

Z I + N ^{−(2−γ+δ)} Z _I ¹⁺

^d8

. µ

^8d

Z I + N ^{−(2−γ+δ)} Z _I ¹⁺

^8d

. (3.14) We next drop the I-operator and use H¨ older’s inequality to estimate

kIF (u)k

L

²_t

L

2d d+4 x

. k|u|

^8d

k

L

^d−3_t

L

d(d−3) 4(d−4) x

kuk

L

2(d−3) d−5

t

L

2d(d−3) d2−7d+20 x

. kuk

^8d

L

8(d−3) d

t

L

2(d−3) d−4 x

kuk

L

2(d−3) d−5

t

L

2d(d−3) d2−7d+20 x

. kuk _M

^8d

Z I . µ

^8d

Z I . (3.15)

The last inequality follows from (2.7) and the fact _2(d−3)

d−5 , _d ^2d(d−3)

₂

_−7d+20

is biharmonic admissible.

Collecting from (3.13) to (3.15), we obtain

Z I . kIu 0 k H

_x²

+ µ

^8d

Z I + N ^{−(2−γ+δ)} Z ¹⁺

8 d

I .

By taking µ sufficiently small and N sufficiently large, the continuity argument gives Z I . kIu 0 k _H

2

x

≤ 1. (3.16)

Next, we have from a direct computation that

∂ t E(Iu(t)) = Re Z

I∂ t u(∆ ² Iu + F (Iu))dx.

By the Fundamental Theorem of Calculus, E(Iu(t)) − E(Iu 0 ) =

Z t

0

∂ s E(Iu(s))ds = Re Z t

0

Z

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

Z t

0

Z

I∂ _s u(F (Iu) − IF (u))dxds

= Im Z t

0

Z

∆ ² Iu + IF (u)(F (Iu) − IF (u))dxds

= Im Z t

0

Z

∆Iu∆(F (Iu) − IF (u))dxds +Im

Z t

0

Z

IF (u)(F (Iu) − IF (u))dxds.

We next write

∆(F(Iu) − IF (u)) = (∆Iu)F ⁰ (Iu) + |∇Iu| ² F ⁰⁰ (Iu) − I(∆F ⁰ (u)) − I(|∇u| ² F ⁰⁰ (u))

= (∆Iu)(F ⁰ (Iu) − F ⁰ (u)) + |∇Iu| ² (F ⁰⁰ (Iu) − F ⁰⁰ (u)) + ∇Iu · (∇Iu − ∇u)F ⁰⁰ (u) +(∆Iu)F ⁰ (u) − I(∆uF ⁰ (u)) + (I∇u) · ∇uF ⁰⁰ (u) − I(∇u · ∇uF ⁰⁰ (u)).

Therefore,

E(Iu(t)) − E(Iu 0 ) = Im Z t

0

Z

∆Iu∆Iu(F ⁰ (Iu) − F ⁰ (u))dxds (3.17)

+Im Z t

0

Z

∆Iu|∇Iu| ² (F ⁰⁰ (Iu) − F ⁰⁰ (u))dxds (3.18) +Im

Z t

0

Z

∆Iu∇Iu · (∇Iu − ∇u)F ⁰⁰ (u)dxds (3.19) +Im

Z t

0

Z

∆Iu[(∆Iu)F ⁰ (u) − I(∆uF ⁰ (u))]dxds (3.20) +Im

Z t

0

Z

∆Iu[(I∇u) · ∇uF ⁰⁰ (u) − I(∇u · ∇uF ⁰⁰ (u))]dxds (3.21) +Im

Z t

0

Z

IF (u)(F (Iu) − IF (u))dxds. (3.22)

Let us consider (3.17). By H¨ older’s inequality, we estimate

|(3.17)| . k∆Iuk ²

L

⁴_t

L

2d d−2 x

kF ⁰ (Iu) − F ⁰ (u)k

L

²_t

L

d x2

. Z _I ² k|Iu − u|(|Iu| + |u|)

^8d

⁻¹ k

L

²_t

L

d x2

. Z _I ² kP >N uk

L

16 d

t

L

⁴_x

kuk

^8d

⁻¹

L

16 d t

L

⁴_x

. (3.23)

By (2.6), we bound

kP >N uk

L

16 d

t

L

⁴_x

. N ⁻² k∆Iuk

L

16 d

t

L

⁴_x

. N ⁻² Z I , (3.24) where

16 d , 4

is biharmonic admissible. Similarly, we have from (2.7) that kuk

L

16 d

t

L

⁴_x

. Z I . (3.25)

Combining (3.23) − (3.25), we get

|(3.17)| . N ⁻² Z ²⁺

8 d

I . (3.26)

We next bound

|(3.18)| . k∆Iuk

L

⁴_t

L

2d d−2 x

k|∇Iu| ² k

L

16 11 t

L

4d 4d−11 x

kF ⁰⁰ (Iu) − F ⁰⁰ (u)k

L

¹⁶_t

L

4d 15−2d x

. k∆Iuk

L

⁴_t

L

2d d−2 x

k∇Iuk ²

L

32 11 t

L

4d−118d x

kF ⁰⁰ (Iu) − F ⁰⁰ (u)k

L

¹⁶_t

L

4d 15−2d x

. Z _I ³ k|Iu − u|

^8d

⁻¹ k

L

¹⁶_t

L

4d 15−2d x

. Z _I ³ kP >N uk

^8d

⁻¹

L

16(8−d) d

t

L

4(8−d) 15−2d x

. N ⁻² (

⁸d

−1 ) Z ²⁺

8 d

I . (3.27)

Here we drop the I-operator and apply (2.7) with the fact γ > 1 to get the third line. We also use the fact that for 5 ≤ d ≤ 7,

|F ⁰⁰ (z) − F ⁰⁰ (ζ)| . |z − ζ|

^8d

⁻¹ , ∀z, ζ ∈ C . The last estimate uses (3.24). Note that

32

11 , _4d−11 ^8d

and _16(8−d)

d , ^4(8−d) _15−2d

are biharmonic admis- sible. Similarly, we estimate

|(3.19)| . k∆Iuk

L

⁴_t

L

2d d−2 x

k∇Iuk

L

32 11 t

L

8d 4d−11 x

k∇Iu − ∇uk

L

32 11 t

L

8d 4d−11 x

kF ⁰⁰ (u)k

L

¹⁶_t

L

4d 15−2d x

. Z _I ² k∇P >N uk

L

32 11 t

L

8d 4d−11 x

kF ⁰⁰ (u)k

L

¹⁶_t

L

4d 15−2d x

.

We next use (2.6) to have k∇P _>N uk

L

32 11 t

L

8d 4d−11 x

. N ⁻¹ k∆Iuk

L

32 11 t

L

8d 4d−11 x

. N ⁻¹ Z _I . As F ⁰⁰ (u) = O(|u|

^8d

⁻¹ ), we use (2.7) to get

kF ⁰⁰ (u)k

L

¹⁶_t

L

4d 15−2d x

. kuk

^8d

⁻¹

L

16(8−d) d

t

L

4(8−d) 15−2d x

. Z

8 d

−1

I . (3.28)

We thus obtain

|(3.19)| . N ⁻¹ Z ²⁺

8 d

I . (3.29)

By H¨ older’s inequality,

|(3.20)| . k∆Iuk

L

²_t

L

d−42d x

k(∆Iu)F ⁰ (u) − I(∆uF ⁰ (u))k

L

²_t

L

2d d+4 x

.

We then apply Lemma 2.9 with q = _d+4 ^2d , q 1 = _d ^2d(d−3)

2

−7d+16 and q 2 = _2(2d−7) ^d(d−3) to get k(∆Iu)F ⁰ (u) − I(∆uF ⁰ (u))k

L

2d d+4 x

. N ^−α k∆Iuk

L

2d(d−3) d2−7d+16 x

k h∇i ^α F ⁰ (u)k

L

d(d−3) 2(2d−7) x

,

where α = 2 − γ + δ. The H¨ older inequality then implies k(∆Iu)F ⁰ (u) − I(∆uF ⁰ (u))k

L

²_t

L

2d d+4 x

. N ^−α k∆Iuk

L

2(d−3) d−4

t

L

2d(d−3) d2−7d+16 x

k h∇i ^α F ⁰ (u)k

L

^2(d−3)_t

L

d(d−3) 2(2d−7) x

.

We have from (3.5), (3.6), (3.7) and (3.8) that k h∇i ^α F ⁰ (u)k

L

^2(d−3)_t

L

d(d−3) 2(2d−7) x

. Z

8 d

I .