On the Global Existence of Rough Solutions of the Cubic Defocusing Schrӧdinge Equation in R^{2+1}

Vol. 4, No. 2 (2007) 1–25 c

World Scientific Publishing Company 3

ON THE GLOBAL EXISTENCE OF ROUGH SOLUTIONS OF THE

5

CUBIC DEFOCUSING SCHR ¨ODINGER EQUATION IN R2+1

Y. F. FANG

7

Department of Mathematics, Cheng Kung University Tainan, 701 Taiwan

MANOUSSOS G. GRILLAKIS

9

Department of Mathematics, University of Maryland College Park, Md 20742, USA mng@math.umd.edu 11 Received 23 May 2006 Accepted 10 Oct. 2006 13 Communicated by P. G. LeFloch

Abstract. We consider the cubic defocusing Schr¨odinger equation in two space

dimen-15

sions and prove that if the initial data are inH1/2, then there exists a global solution in time. The proof combines the argument from [5] with some new correlation estimates

17

for the Schr¨odinger equation.

Keywords: I-method; Strichartz estimates; correlation estimates; conservation laws. 19

Mathematics Subject Classification 2000: 35L40, 76N10

1. Introduction

21

The purpose of the present paper is to demonstrate global existence of rough solutions for the cubic defocusing Schr¨odinger equation in 2 + 1 dimensions. To make the statement more precise, we would like to consider the following evolution equation

i∂_tψ− ∆ψ + |ψ|2ψ = 0, (1.1a)

ψ(0, x) = ψ₀(x); (t, x)∈ R × R2, (1.1b) i.e. x∈ R2_{are the space variables and t denotes the time variable. For the evolution}

Eq. (1.1), we would like to assume that ψ₀∈ H1/2(R2) and show that there exists

23

a global in time solution with ψ(t) ∈ H1/2_(R2_{). Certain conserved quantities are}

crucial to the behaviour of the solutions. The conservation of energy means that

25 the integral E(t)def=
R2  ∇ψ2 +1 2|ψ| 4  dx (1.2a) 27 1

is independent of time. On the other hand, the evolution equation preserves the

1

total mass which means that the integral

M (t)def=

R2

ψ2dx (1.2b)

3

is also independent of time. It is rather easy to demonstrate, using the two conserved quantities above, that a solution of (1.1) exists for all time provided that the initial data are sufficiently smooth, namely in H1_{. The reason for assuming ψ}

0 ∈ H1 is

that one would like to take advantage of the conservation of energy, see the intergal in (1.2a), which is positive definite and which implies that the H1 norm of the solution is bounded as long as the solution exists. In order to explain the type of result that we would like to demonstrate in the present work, we have to explain first why the Eq. (1.1) admit local weak solutions if we only assume that the initial data are in L2_{. To see why this is so, we have to use the Strichartz estimates, more}

precisely consider the linear Schr¨odinger equation with a forcing term f ,

iψ_t− ∆ψ = f, (1.3a)

ψ(0, x) = ψ₀(x). (1.3b)

The solution of (1.3) can be written in integral form as a combination of two terms, namely ψ(t) = ψ_L(t) + φ(t) where

5

ψ_L(t)def= e−i∆tψ₀; φ(t)def=
_t

0

e−i∆(t−s)f (s)ds. (1.4) The local in time construction of solutions is based on the well known Strichartz

7

estimates

ψLL4_(R2+1₎≤ C₁ψ₀_L2_(R2₎; φ_L4_(R2+1₎≤ C₂f_L4/3_(R2+1₎. (1.5)

9

Pick a time interval, say [0, T ], with the final time T to be chosen and set up an iteration scheme as follows

11

iψ_t(k+1)− ∆ψ(k+1)=−χ_{[0,T ]}(t)|ψ(k)|2ψ(k); ψ(k+1)(0, x) = ψ₀(x) (1.6) with ψ(0) _{satisfying i∂tψ}(0)_{− ∆ψ}(0)_{= 0, ψ}(0)_{(0) = ψ}

0and χ[0,T ] the characteristic 13

function of the interval [0, T ]. The Strichartz estimate in (1.5) ψ_L_L4_(R2+1₎ ≤

C₁ψ₀_L2_(R2₎ implies that there exists some time interval, say [0, T ], such that

15

ψLL4_(R2_{×[0,2T ])}≤ , (1.7)

where  is a fixed small number which will depend only on C₂, see (1.5). Let us define a local in time norm,

X_k=def_ψ(k)_

L4_(R2_{×[0,T ])}.

Applying the estimates (1.5), we obtain an inequality of the form X_k+1≤  + CX3

k,

1

from which it follows that X_k is a bounded sequence. By considering the evolution equation for differences ψ(k+1)− ψ(k), one can show that the iteration is in fact a

3

contraction with respect to the L4_{norm, thus constructing a local solution of (1.1)}

in the time interval [0, T ]. A crucial observation is the fact that T is defined via (1.7)

5

and we do not have any control of its size. Repeating the construction starting with data at time T , we can construct a solution on successive time intervals, but there is

7

no guarantee that the intervals will not shrink rapidly thus constructing a solution only on a finite time interval. On the other hand, if we use the conservation of energy

9

this cannot happen, since the energy estimate controls the H1norm of the solution, but we need to assume that ψ₀∈ H1_{. It is an interesting question to ask whether} 11

equation (1.1) has global in time solutions if one assumes that ψ₀∈ Hsfor 0≤ s < 1. This question was first posed by Bourgain, see [1, 2], in which he demonstrated

13

global existence for s > 3/5. The range of s was subsequently improved in [5, 6] to s > 4/7 using the crucial idea of an almost conservation law for the energy. In

15

the present work we would like to demonstrate global existence for s = 1/2. There are two key ingredients needed in the proof, the first is the almost conservation law

17

and the second is a correlation estimate. The paper is divided as follows, in Sec. 2, we present the main argument leading to global existence, we essentially follow the

19

argument in [6]. In Secs. 3 and 4, we derive a bilinear estimate due to Bourgain and the almost conservation law developed in [5, 6]. Section 5 presents a method

21

for deriving correlation estimates. Recently Coliander et al. showed existence for

s > 1/2 by improving the almost conservation law. 23

2. Main Argument

Presently we would like to give the main argument of the proof of global existence for the following problem

i∂_tψ− ∆ψ + |ψ|2ψ = 0, (2.1a)

ψ(0, x) := ψ₀(x)∈ H1/2(R2). (2.1b) We will follow the technique used in [5, 6] which is based on the multiplier I and

25

the idea of an almost conservation law. Before we start let us specify some notation conventions. We would like to define the Fourier multiplier m_s,N(ξ) to be a smooth

27

function in frequency variables which is less or equal than one and has the following properties 29 m_s,N(ξ)def=  1, if |ξ| ≤ N, (N/|ξ|)1−s, if |ξ| ≥ 2N, (2.2)

for 0 < s < 1, however in the present work we will choose s = 1/2. We will use the multiplier above to define Iψ, which whenever convenient we will also denote by

ψ_I, as follows

ψ_I(t, x) = Iψ(t, x) =def_F−1_m

We will also use the following spaces introduced by Bourgain, see [1, 2]. 1 ψXs,α def₌ ξs_{ S}α_ψ L2, (2.4a)

where we adopt the notation convention

3

ξdef= |ξ| + 1; S def= τ− |ξ|2;  Sdef= | S| + 1. (2.4b) When we consider localization in time, we will use the notation X_s,αI or simply write

5

X_s,αloc, where I stands for some time interval [T₁, T₂]. This means that we consider

ψXI s,α def = inf e ψ   ψ_Xs,α : ψ(t) = ψ(t), t∈ I. (2.5) 7

The s-derivative of a function is defined in the standard way

Dsψ =F−1

|ξ|s_{f. (2.6)}

9

The energy of the function ψ_I is defined as follows, see the definition of energy in (1.2a), 11 E(Iψ)(t)def=
R2 t _∇ψ I2+1 2ψI 4 dx. (2.7)

Finally, we will employ the Strichartz estimates
t 0 e−i(t−s)∆f (s)ds Lp x,Lrt ≤ C1 f _Lp
x,Lr
t , (2.8a) e−it∆ψ_{0 L}p x,Lrt ≤ C0 ψ_{0 L2} (R2), (2.8b)

where the exponents p and r satisfy the admissibility condition, see [5, 6],

13 1 p+ 1 r = 1 2; r > 2. (2.8c)

Let us list here the ingredients that are essential for the demonstration of global

15

existence. The first observation is the fact that if ψ(t, x) is a solution of (2.1), we can scale it and obtain a new solution, namely the scaled function,

17 ψ(λ)(t, x)def= 1 λψ  t λ2, x λ  (2.9)

satisfies the same equation with initial data ψ(λ)₀ = (1/λ)ψ₀(x/λ). This scaling

19

preserves the L2 _{norm of ψ(t) as well as the L}4 _{space-time norm. Now let us scale}

the solution ψ→ ψ(λ) so that E(Iψ₀(λ))≤ 1/4, more precisely under the scaling in

21 (2.9), we have ∇Iψ(λ) 0 2L2≈ N λψ0 2 H1/2 and Iψ₀(λ)4_L4≈ 1 λ2ψ0 4 L4. (2.10) 23

Notice that from the Sobolev inequality we haveψ_L4 ≤ Cψ_H1/2. With all the

1

above in mind, we can choose λ in the following manner

λ = CNψ₀2_H1/2, (2.11)

3

where C some fixed large positive number so that the scaled function ψ₀(λ)satisfies,

E(Iψ₀(λ)) < 1/4. (2.12)

5

We will choose N at the end of the argument to be sufficiently large.

The almost conservation law is the next essential ingredient, see [5, 6] where this

7

idea was introduced. The idea is that the energy of ψ_I is almost conserved in the following sense 9 E(Iψ)(T )−E(Iψ)(0) ≤ 1 N(3−δ)/2  C₁Iψ4 X_1,(1+δ)/2[0,T ] + C2Iψ6X_1,(1+δ)/2[0,T ]  . (2.13) The third crucial ingredient is an a priori estimate which reads as follows. A solution

11

of (2.1) satisfies the a priori estimate

ψ4 L4_(R2_×[T1,T2])≤ ψ02L2  sup [T1,T2] ψ(t)2 H1/2   T₂− T₁. (2.14) 13

This type of estimate for three space dimensions was derived first in [6, 7]. A general method for developing this type of apriori estimates will be given in Sec. 5. In order

15

to present the proof of global existence, we will need a series of technical lemmata. The first lemma that we will need is the following, see [3].

17

Lemma 2.1. Assume that q(ξ₁, ξ₂), with ξ_1,2∈ R2_{, is a smooth function satisfying} |∂α_{q| ≤ C(α)(1 + |ξ|)}−|α|_{; ξ = (ξ}

1, ξ2) (2.15)

19

for every multiindex α and define a quadratic expression as follows Q[f, g](x) =F−1 
q(ξ− η, η) f (ξ− η)g(η)dη  . (2.16) 21

The following estimate holds

Q[f, g]_Lp≤ CfLp1gLp2, where 1 p₁ + 1 p₂ = 1 p. (2.17) 23

This lemma appears in [3] and we will omit its proof. We will use Lemma 2.1 above to claim that as far as the Lpnorm is concerned we can think that DI(ψ3_)∼ 25

(ψ2)(DIψ). In order to apply Lemma 2.1 we need to assume that the multiplier I has smooth symbol which can be achieved by considering a smooth approximation

27

Theorem 2.2. A pair of exponents is called admissible if it satisfies 1 1 p+ 1 r = 1 2; r > 2 (2.18a)

and with this choice of exponents the following Strichartz type estimates holds true 3

ψLp

x,Lrt ≤ C(δ)ψX0,(1+δ)/2. (2.18b)

These estimates are well known, see Bourgain [1, 2], also see next section. Next

5

we will need a lemma that says that if the space-time L4 _{norm of ψ is small on}

some interval [0, T ], we can control all higher order norms.

7

Lemma 2.3 (Local existence withHs(R2) data, wheres ≥ 0). Let us define

the quantity, 9 µ([0, t])def=
D[0,t] ψ4 dxdt. (2.19a)

If µ([0, T ])≤ µ₀, where µ₀ is some universal constant then the following estimates are true, Dsψ _Lp xLrt(D[0,T ])≤ C Dsψ_{0 L2}, (2.19b) DIψ L4_{(D[0,T ])}≤ C DIψ0 _L2. (2.19c)

Proof. The proof imitates the argument for local existence given in the

introduc-11

tion, the function Dsψ satisfies the equation,

i∂_tDsψ− ∆Dsψ=−Dsχ_{[0,T ]}(t)|ψ|2ψ (2.20)

13

in the interval [0, T ], where χ_{[0,T ]} denotes the characteristic function of the time interval [0, T ]. Let us take p = r = 4 for simplicity. Let us write D := R2× [0, T ] for

15

simplicity. Applying Strichartz estimates as stated in (2.8a) and (2.8b), we obtain Dsψ _L4 (D) ≤ C1 D s_ψ 0 _L2+ C2 Ds  |ψ|2_ψ L4/3_(D). (2.21) 17

Because of Lemma 2.1, we can write Ds(|ψ|2_{ψ)∼ ψ}2_(Ds_{ψ) and using Holder on}

the right-hand side of (2.21) with exponents 4 and 4/3, we obtain

19 Dsψ L4_(D)≤ C1 Dsψ0 _L2+ C2  µ1/2([0, T ]) Dsψ L4_(D). (2.22)

We need to assume C₂2µ([0, T ])≤ 1/4 in order to complete the proof of the lemma. 21

The proof of (2.19b) is similar so we will omit it. Lemma 2.4. Assume D1/2_ψ

0 _L2 ≤ B where B is some fixed constant, and

23

µ([0, T ]) ≤ µ₁(B) where µ₁(B) is a constant that depends only on B, then we

have the estimate 25

Iψ _X[0,T ]

1,(1+δ)/2 ≤ C

DIψ_{0 L2}

Proof. Let us consider the evolution equation below

1

i∂_tDIψ₁− ∆DIψ₁=−DIχ_{[0,T ]}(t)|ψ|2ψ, (2.24) so that DIψ(t) = DIψ₁(t) in the interval [0, T ]. Again we will denote D := R2_×

[0, T ]. For Eq. (2.24) the following estimate holds true for any δ > 0, see [1, 2], S(1+δ)/2DIψ_{1 L2}

(D)≤ C1 DIψ0 L2+ C(δ) S

−(1−δ)/2_DI_χ

[0,T ]|ψ|2ψ _L2_(D),

(2.25) where C(δ)∼ δ−1. Employing Lemma 2.1 again, we have that DI(ψ3)∼ ψ2(DIψ).

3

Interpolating between the Strichartz type estimate ψ L4 t,x ≤ C(δ) ψ X0,(1+δ)/2 (2.26) 5

and the Plancherel identity,ψ_L2= ψ_L2, we obtain the estimate

ψ _Lp t,x ≤ C(δ) ψ _X 0,(1−δ)/2, where p = 4 1 + θ; θ = 2δ 1 + δ. (2.27) 7

The dual to the estimate above reads

S−(1−δ)/2ψ _L2 ≤ C(δ)ψ_Lp
; p
=

4

3− θ (2.28)

9

and applying (2.28) on the last term of (2.25), we obtain

S−(1−δ)/2DIχ_{[0,T ]}|ψ|2ψ _L2 ≤ C(δ) χ_{[0,T ]}ψ2(DIψ) _Lp
. (2.29)

11

Now an application of Holder’s inequality with exponents, 1 3− θ+ 2(1− θ) 3− θ + θ 3− θ = 1, (2.30) 13

where the first exponents is applied to DIψ, the second to|ψ|8(1−θ)/(3−θ) _{and the}

third to|ψ|8θ/(3−θ) gives

15

ψ2(DIψ) _Lp
≤ C(δ)ψ(1−θ)/2_L4_(D) ψ2θ_L8_(D)DIψ_L4_(D). (2.31)

The Strichartz estimate, see Lemma 2.3, gives

17

D1/2ψ _L8/3

x L8t(D)≤ C

D1/2ψ_{0 L2} (2.32)

provided that µ[0, T ]≤ µ₀ while the Sobolev embedding W1/2,8/3_{⊂ L}8 _combined

with (2.32) in (2.25) gives Iψ _X[0,T ] 1,(1+δ)/2 ≤ C1 DIψ_{0 L2}+ C(δ)D1/2ψ₀2θ_L2  µ(1−θ)/2([0, T ])Iψ_X[0,T ] 1,(1+δ)/2. (2.33) Let us choose δ small but fixed, for example δ = 2−10. From the inequality (2.33),

19

we conclude that

Iψ_X[0,T ]

1,(1+δ)/2 ≤ CDIψ0L2 (2.34a)

provided that

1

C(δ)D1/2ψ₀2θ_L2



µ(1−θ)/2([0, T ])< 1/2, (2.34b) which means that we would like to have µ([0, T ]) smaller than some fixed number,

3

say µ₁(B) since we assumed that D1/2ψ₀_L2 < B. This concludes the proof of

Lemma 2.4.

5

Finally the almost conservation law, see (2.13) combined with Lemma 2.4 and the scaling gives an estimate for the energy of Iψ(λ)_{. First recall that with the} 7

scaling in (2.11), we have

D1/2ψ(λ)_{0 L2} = (1/√λ) D1/2ψ_{0 L2} ∼ (1/CN1/2) < 1 (2.35a)

9

and λ was chosen so that E(Iψ(λ)₀ ) < 1/4, so we have E(Iψ(λ))(T )− E(Iψ(λ))(0) ≤ C N(3−δ)/2DIψ (λ) 0 4L2 ≤ C N(3−δ)/2, (2.35b) 11

provided that µ[0, T ] < µ₁(B). We are now in a position to demonstrate the global in time existence of (2.1), again we will follow the argument in [6].

13

Proof of global existence. Start with data ψ₀ ∈ C_c∞ so that ψ(t, x) is a global solution of our problem. Now scale the solution ψ(λ) _{so that E(Iψ}(λ)

0 ) ≤ 1/4, i.e. 15

the scaling parameter λ is chosen according to (2.11). The multiplier I depends on

N , see (2.2), and we will choose N at the very end of the argument to be sufficiently 17

large. For the time being let us pick T₀arbitrarily large denote D[0, t] := R2_{× [0, t]}

and define the following set

19

Sdef= t : 0 < t≤ T₀ andψ(λ)_L4_(D[0,t])≤ At1/8 (2.36)

with A a constant to be chosen as follows

21

Adef= Kψ₀3/4_L2 + 1



, (2.37)

where K is some large number to be chosen later.

23

Claim 2.5. We claim that S = [0, T₀], i.e. the L4 space-time norm on the slice R2_{× [0, t] is bounded by some constant times t}1/8_.

25

Proof of Claim 2.5: Assume not, sinceψ(λ)_

L4_(D[0,t]) is a continuous function,

there exist some T ∈ [0, T₀] with the properties, D[0, T ] := R2_{× [0, T ],} 27

ψ(λ)_

From the correlation estimate (2.14), we know that 1 ψ(λ)_ L4_{(D[0,T )])}≤ ψ₀1/2_L2 (R2)  sup [0,T ] ψ(λ)_(t)1/2 H1/2_(R2₎  T1/8, (2.39)

and our next goal is to estimate, using the almost conservation law, the quantity

3 below sup [0,T ] ψ(λ)_(t)1/2 H1/2_(R2₎. (2.40) 5

Let us decompose ψ(λ) _{in high and low frequencies, i.e. we write}

ψ(λ)= P_≤Nψ(λ)+ P_≥Nψ(λ). (2.41)

7

This is a decomposition of ψ(λ) _{on frequencies |ξ| ≤ N and |ξ| ≥ N respectively.}

Recall that s = 1/2 in (2.2) and (2.3). For the low frequencies, we have, interpolating between L2 _{and H}1

P_≤N[ψ(λ)(t) _H1/2 ≤ P_≤N[ψ(λ)(t) 1/2_L2 P_≤N[ψ(λ)(t) 1/2_H1 (2.42a)

≤ ψ01/2_L2 ∇Iψ(λ)(t) 1/2_L2. (2.42b)

For the high frequencies, again with s = 1/2, we can write P_≥N[ψ(λ)(t) _H1/2≤ 1

N1/2 ∇Iψ (λ)(t)

L2. (2.42c)

9

Combining the above inequalities (2.42a) and (2.42b) in (2.39), we have the overall estimate ψ(λ)_ L4_{(D[0,T ])} ≤ T1/8ψ₀3/4_L2 sup [0,T ] ∇Iψ(λ)(t) 1/4 L2 + T1/8ψ0 1/2 L2 N1/2 sup [0,T ] ∇ Iψ(λ)(t) 1/2_L2 . (2.43) Since we know from (2.38) that, ψ(λ)_4

L4_{(D[0,T ])} ≤ (2A)4T1/2, we can split any

interval [0, T ], where T ≤ T₀, in subintervals, say J_k, with k = 1, 2, . . . , L, write

D_k:= R2_{× J}

k and for each slice Dk, we have

ψ(λ)_4

L4_(Dk)≤ µ1(B),

(2.44) where µ₁ := min{µ₀, µ₁}, see Lemmata 2.3 and 2.4. The number of slices, which

we will call L, is at most like

11

L∼(2A) 4_T1/2

Because of (2.44) we can apply the almost conservation law (2.35) for each of the

1

slices D_k. We need to controlD1/2_ψ(λ)_(T

k)L2 by some constant. Observe that

|ξ|| ψ(ξ)|2dξ≤ψ_L2+ 1E1/2(Iψ) (2.46)

3

and since the L2norm is conserved, we can boundD1/2ψ(λ)(T_k)_L2 by a constand

B, see Lemma (2.4) as long as we know that E(Iψ(λ)_(T

k)) < 1/2. Applying the

5

estimate in (2.13) L times on the successive slices, we obtain sup

[0,T ]

EIψ(λ)(t)≤ EIψ₀(λ)+ L

N3/2−δ, (2.47)

7

where we know from scaling, see (2.12), that EIψ₀(λ) ≤ 1₄. Thus in order to guarantee that the energy of scaled function Iψ(λ)_{(t) satisfies E(Iψ}(λ)_{(t)) < 1/2 for} 9

all t∈ [0, T₀], we would like to choose

L N3/2−δ ≤ 1 4 or N 3/2−δ_{∼ 4}(2A)4T 1/2 0 µ₁(B) , (2.48) 11

so that we have the bound sup

[0,T ]

EIψ(λ)(t)≤ 1/2 (2.49)

13

and substituting (2.49) back to (2.43), we obtain

ψ(λ)_

L4_{(D[0,T ])}≤ 2ψ₀3/4_L2T1/8, (2.50)

15

which contradicts (2.38) if K 2. This concludes the proof of the claim and notice that the choice of N is given by (2.48).

17

Finally recall that we assumed that ψ₀ ∈ C_c∞ hence if ψ₀ ∈ H1/2 _{we can}

approximate it by ψ₀ ∈ C_c∞ so that ψ₀− ψ₀_H1/2 → 0. Since we considered the

19

solution in an arbitrarily large interval of time [0, T₀] and the proof of Claim 2.5 involves only the H1/2 _{norm of the solution we obtain existence of solutions for} 21

arbitrarily large time. This concludes the proof of global existence.

3. Bourgain’s Quadratic Estimate

23

Let us start by explaining a quadratic estimate derived by Bourgain [1, 2], which estimate is crucial in deriving the almost conservation law in the next section. The

25

main estimate is stated in (3.11) and rephrased in (3.14) in a manner which is convenient for our purpose. Consider two functions ψ₁(t, x) and ψ₂(t, x) and their

27

corresponding Fourier transforms, say ψ₁(τ, ξ) and ψ₂(τ, ξ) respectively. Let us write the Fourier transform of the product ψ₁ψ₂

29
ψ₁ψ₂(τ, ξ) =
R×R2  ψ₁(τ− σ, ξ − η) ψ₂(σ, η)dηdσ. (3.1)

The idea is to rewrite (3.1) in parabolic variables, let us define first the new variables,

1

udef= τ− σ − |ξ − η|2_{; v}def_{= σ− |η|}2 _(3.2)

and write ψ₂(v, η) and ψ₁(u, ξ− η) for the functions ψ₂and ψ₁respectively. At this

3

point it is convenient to define

pdef= |ξ − η|2_+|η|2_{; η}

1def= η− ξ/2, (3.3)

5

so that we can write

|η1|2=



2p− |ξ|2/4. (3.4)

7

Let us call ρ := |η₁| the length of the vector η₁ so that we can write η₁ = ρeiα, where α is the angle of the vector η₁ and express ρ as a function of p and ξ via the

9 formula ρ(p, ξ) = 1 2  2p− |ξ|2_{. (3.5)} 11

Notice that dη = dη₁= ρdρdα = (1/4)dpdα, moreover we have that p = τ− u − v, so that we can write ρ as a function of (τ, u, v, ξ)

13

ρ(τ, u, v, ξ) = 1

2 

2(τ− u − v) − |ξ|2_{. (3.6)}

Combining all these we can write

15



ψ₁ψ₂(t, x) =F−1{Φ(τ, ξ)} , (3.7a)

where Φ(τ, ξ) is given by the following integral

17

Φ(τ, ξ)def=

S1_×R×R

ψ₁(u, ξ/2− ρeiα)ψ₂(v, ξ/2 + ρeiα) dαdudv (3.7b) and ρ(u, v, ξ) is given by the formula (3.6). Now let us consider a dyadic

decompo-19

sition of ψ₁and ψ₂ as follows

u∼ 2l1_{; v∼ 2}l2_{; |ξ| ∼ 2}j1_{; |η| ∼ 2}j2 _(3.8)

21

and assume without loss of generality that 2j1  2j2_{. The crucial observation is}

that since

23

ξ/2− ρeiα∼ 2j1_{; ξ/2 + ρe}iα_{∼ 2}j2_{; 2}j1_{ 2}j2_{, (3.9)}

we have that the angle α of the vector η₁ is constrained in an interval [−α₀, α₀],

25

where the size of the interval is restricted by

α₀∼2

j2

2j1. (3.10)

27

Plancherel in (3.7a) and (3.7b) gives the estimate ψ₁ψ₂ 2

L2≤ C2j2−j12l12l2ψ12L2ψ22L2. (3.11)

The estimate above can be expressed in a form which is more convenient for our purpose. Let us define first,

sdef= _{|s| + 1; }Sdef= _{τ − |ξ|}2_{ (3.12a)} b(ξ₁, ξ₂)def= min _ξ 1 ξ2 1/2 , _ξ 2 ξ1 1/2 . (3.12b)

Using the function b we can define the quadratic expression, for δ > 0, arbitrarily

1 small b(−1+δ)ψ₁ψ₂def= F−1 
b(−1+δ)(ξ− η, η) ψ₁(τ− σ, ξ − η) ψ₂(σ, η)dσdη  , (3.13) 3

and the estimate in (3.11) implies the following estimate for the quadratic expression in (3.13)

5

b(−1+δ)(ψ₁ψ₂)

L2 ≤ C(δ) S(1+δ)/2ψ1 _L2 S(1+δ)/2ψ2 _L2; δ > 0. (3.14)

Notice that from the definition in (3.12b), we have that b−1+δ≥ 1.

7

4. Almost Conservation Law

The almost conservation law was developed in a series of papers by the authors of

9

[5, 6], for the Schrodinger equation, see [5, 6]. Presently we would like to rederive the almost conservation law. The derivation is slightly different but we follow the

11

same reasoning. Start by writing the evolution equation for ψ_I := Iψ, where Iψ :=

F−1_(mψ),

13

i∂_tψ_I− ∆ψ_I+|ψ_I|2ψ_I+I|ψ|2ψ− |ψ_I|2ψ_I= 0. (4.1) The energy of ψ_I is defined to be, see (2.7),

15 Eψ_Idef=
R2 _∇ ψ_I2+1 2ψI 4 dx, (4.2)

in imitation of the conserved energy. The energy in (4.2) is not conserved but in

17

the terminology of [5, 6] it is almost conserved. Let us define two quantities

Pdef= I|ψ|2_ψ_{; R}def_{= |ψ}

I|2ψI, (4.3)

19

so that from the evolution Eq. (4.1), we can derive an evolution equation for E(Iψ), namely 21 dE dt =  P− R ; ψ_I t  L2+  P− R ; ψ_I t  L2. (4.4)

Using the evolution equation, we can write

23

φdef= ∂_tψ =−i∆ψ + i|ψ|2ψ, (4.5a) so that we can split φ = φ₁+ φ₂ where we define

25

Notice that Iφ₂= iP so if we substitute back in (4.4), we obtain 1 dE dt = 2Im  P− R ; Iφ₁_L2+ 2ImP− R ; R_L2. (4.6) The almost conservation of energy means that we would like to prove the estimate

E(T )− E(0) ≤N−(3−δ)/2C₁Iψ4_X[0,T ]

1,α + C2Iψ 6 X[0,T ] 1,α  ; α = (1 + δ)/2. (4.7) Lemma 4.1. Assume 1/2≤ s < 1. For the quantity defined below

3

J (t)def= I_s|ψ|2ψ− |I_sψ|2I_sψ ; φ, (4.8a)

the following estimate holds true 5   
_T 0 J (t)dt   ≤ C N(3−δ)/2Is1ψ3_X[0,T ] 1,α Iβ φ_X[0,T ] −1,α; α = 1 + δ 2 , (4.8b)

where s₁ and β are given by 7 s₁= s−1− δ 4 ; β = 3− δ 4 ; s≥ 1 2; δ > 0. (4.8c)

At this point let us notice that applying the lemma above to Iφ₁, see (4.5) and the first term on the right-hand side of (4.6), we obtain since s₁ < s₂ imply

m_s1 ≤ m_s2,   
_T 0  P− R ; Iφ₁dt   ≤ C N(3−δ)/2Iψ 3 X[0,T ] 1,α Iφ1X−1,α[0,T ] ≤ C N(3−δ)/2Iψ 4 X[0,T ] 1,α . (4.9) Proof. We will ignore complex conjugates since they are irrelevant here and also ignore the dependence of functions on the time variable. Writing the integral above in Fourier space, we obtain

J =
 1− m(ξ2+ ξ3+ ξ4) m(ξ₂)m(ξ₃)m(ξ₄)   ψ_I(ξ₂) ψ_I(ξ₃) ψ_I(ξ₄) φ(ξ₂+ ξ₃+ ξ₄) dξ₂dξ₃dξ₄. (4.10a) Let us write for simplicity,

9

m_s,N(ξ) = 

1, if |ξ| < N,

(|ξ|/N)1−s, if |ξ| ≥ N, (4.10b)

and call ξ₁:= ξ₂+ ξ₃+ ξ₄. Rewrite the integral appearing in (4.10a) in the following manner J =
 1− m(ξ1) m(ξ₂)m(ξ₃)m(ξ₄)  _|ξ 1| |ξ2||ξ3||ξ4| × |ξ2| ψI(ξ2)|ξ3| ψI(ξ3)|ξ4| ψI(ξ4)|ξ1|−1φ(ξ 1)dξ2dξ3dξ4. (4.10c)

Employing the notation b_2,4 := b(ξ₂, ξ₄) and b_1,3 := b(ξ₁, ξ₃) for the expression b defined in (3.12a and 3.12b) and keeping in mind the estimate in (3.14), write the

integral J in the following way J =
 _|ξ 1| |ξ2||ξ3||ξ4| − a(ξ₁)

a(ξ₂)a(ξ₃)a(ξ₄)  b1−δ_2,4 (ξ₂, ξ₄)b1−δ_1,3 (ξ₁, ξ₃) ×|ξ2| ψI(ξ2)|ξ4| ψI(ξ4)b−1+δ2,4   |ξ1|−1φ(ξ 1)|ξ3| ψI(ξ3)b−1+δ1,3  dξ₂dξ₃dξ₄, (4.10d) where a(ξ) stands for the function

1

a(ξ) =|ξ|m_s,N(ξ) = 

|ξ|, if |ξ| ≤ N,

N1−s_|ξ|s_{, if |ξ| > N.}. (4.10e) Give a name to the first part of the integrand in (4.10d),

3 Qdef=  |ξ1| |ξ2||ξ3||ξ4|− a(ξ₁)

a(ξ₂)a(ξ₃)a(ξ₄) 

b1−δ_2,4 (ξ₂, ξ₄)b1−δ_1,3 (ξ₁, ξ₃). (4.11) The desired estimate in (4.8b) will follow from the quadratic estimates in (3.14) if

5

we demonstrate the bound below.

Claim 4.2. The quantity Q defined in (4.11) satisfies the bound

7

|Q| ≤ N−(3−δ)/2_(λ

1λ2λ3λ4)(1−δ)/4; δ > 0, (4.12a) where λ_j are the quantities

9

λ(ξ) =



1, if |ξ| ≤ N,

N/|ξ|, if |ξ| ≥ N; λj= λ(ξj). (4.12b)

Proof of Claim 4.2. We can assume without loss of generality that|ξ₄| ≤ |ξ₃| ≤

11

|ξ4| and write

ρ_j=|ξ_j|; j = 1, 2, 3, 4, (4.13)

13

so that ρ₄ ≤ ρ₃ ≤ ρ₂. First let us observe that if ρ₄ ≤ ρ₃ ≤ ρ₂ ≤ N and ρ₁ ≤ N then Q = 0. We will examine the size of the quantity Q in three different cases.

15

Case 1. Assume that ρ₄ ≤ ρ₃ ≤ N ≤ ρ₂. The quantity Q is largest when ρ₄ ≤

ρ₃ N hence ρ₁≈ ρ₂. In this case we have,

Q =  ρ₁ ρ₂ρ₃ρ₄ − ρs₁ ρs₂ρ₃ρ₄   ρ₄ ρ₂ (1−δ)/2_ρ 3 ρ₁ (1−δ)/2 =  1 + ρ1− ρ2 ρ₂  −  1 +ρ1− ρ2 ρ₂ _s 1 (ρ₁ρ₂)(1−δ)/2_(ρ3ρ4)(1+δ)/2 (4.14a) ≈ ρ1− ρ2 ρ₂(ρ₁ρ₂)(1−δ)/2_(ρ 3ρ4) 1+δ 2 ≤ ρ₃+ ρ₄ (ρ₃ρ₄)(1+δ)/2  N ρ₂ 3−δ 2 _N ρ₁ 1−δ 2 N−2+δ,

where we used the fact|ρ₁−ρ₂| ≤ ρ₃+ ρ₄ N and (1+x)s−1 ≤ Cx for x ∈ (0, 1).

1

The estimate in (4.14a) can be expressed in the following manner

Q ≤Cλ(3−δ)/2₂ λ(1−δ)/2₁ N−2+δ. (4.14b)

3

Case 2. Assume ρ₄ ≤ N ≤ ρ₃ ≤ ρ₂. The quantity Q is largest when ρ₃ ≈ N,

ρ₄ N, hence ρ₁≈ ρ₂. In this case we have the estimates

Q =  ρ₁ ρ₂ρ₃ρ₄ − N1−s_ρs 1 N2(1−s)_(ρ2ρ3)s_ρ4   ρ₄ρ₃ ρ₂ρ₁ (1−δ)/2 ≈  ρ₁ ρ₂ρ₃ s ₁ ρ₄     ρ₁ ρ₂ρ₃ 1−s − 1 N1−s     ρ₃ρ₄ ρ₁ρ₂ (1−δ)/2 (4.15a) ≤  ρ₁ ρ₂ρ₃ _s ρ₃ρ₄ ρ₁ρ₂ (1−δ)/2 ₁ N1−s ≈  N ρ₃ _{s−(1−δ)/2} N2 (ρ₁ρ₂)1−δ2 N−3+δ2 .

Using the notation from (4.12b) we can write (4.15a) as follows

Q ≤Cλ₁λ₂(1−δ)/2λs−(1−δ)/2₃ N−(3−δ)/2. (4.15b)

5

Case 3. Assume N ≤ ρ₄≤ ρ₃≤ ρ₂. The quantity Q is largest when ρ₄≤ ρ₃ ρ₂, hence ρ₁≈ ρ₂. Now we can estimate

7 Q =  ρ₁ ρ₂ρ₃ρ₄ − ρs₁ N2(1−s)_(ρ 2ρ3ρ4)s   ρ₃ρ₄ ρ₁ρ₂ (1−δ)/2 =  ρ₁ ρ₂ρ₃ρ₄ s _ρ 1 ρ₂ρ₃ρ₄ 1−s − 1 N2(1−s)   ρ₃ρ₄ ρ₁ρ₂ (1−δ)/2 ≤  ρ₁ ρ₂ρ₃ρ₄ _s 1 N2(1−s)  ρ₃ρ₄ ρ₁ρ₂ (1−δ)/2 ≤  ρ₁ ρ₂ s_N ρ₃ s−(1−δ)/2_N ρ₂ 1/2_N ρ₁ 1/2_N ρ₄ s−(1−δ)/2 N−2, (4.16a)

which can be written, using the notation in (4.12b)

9

Q ≤ λ₃λ₄s−(1−δ)/2λ₁λ₂(1−δ)/2N−2. (4.16b) The three cases demonstrate the inequality in (4.12a) i.e. we see that for s≥ 1/2,

11

we have that the quantity Q is bounded by

Q ≤CN−(3−δ)/2(λ₁λ₂λ₃λ₄)(1−δ)/4, (4.17)

13

where λ_j := λ(ξ_j) are given by (4.12b). This concludes the proof of the claim. Hence this completes the proof of Lemma 4.1.

Let us now apply the estimate in Lemma 4.1 for the expression below, see the

1

second term on the right-hand side of (4.6)   
_T 0  P− R; Rdt   ≤ C N(3−δ)/2Iψ 3 X1,α[0,T ]Iβ R_X[0,T ] −1,α, (4.18a) 3

so that we would like to estimate the expression  Sα/ξI_βR L2; R =|ψI| 2_ψ I. (4.18b) 5

For simplicity we can write 

S₁def= τ₂+ τ₃+ τ₄−ξ₂+ ξ₃+ ξ₄2; S_k def= τ_k−ξ_k2, k = 2, 3, 4

(4.19a) and, denoting α := (1 + δ)/2, we have the following inequality



S₁α≤ S₂α+ Sα₃ + S₄α+|ξ₂||ξ₃|α+|ξ₃||ξ₄|α+|ξ₂||ξ₄|α. (4.19b)

7

With these observations in mind we have that as far as the L2 _{norm is concerned,}

we can write 9 m_β(ξ) ξ  Sα|ψI|2ψI  ≈ m_ξβ(ξ)ψ2 I∗ SαψI+m_ξβ(ξ)ψI∗|ξ|αψI∗|ξ|αψI. (4.20) Combining the Strichartz type estimate in (2.18a) and (2.18b) with the Sobolev embedding, we obtain the following inequalities

DψIL2 xL∞t + ψI L∞_xLr t ≤ C ψI X1,α; 2 < r, (4.21a) Sαψ_{I L}q xL2t ≤ C ψ_{I X} 1,α; q≥ 2, (4.21b) Dαψ_{I L}p xL4t ≤ C ψ_{I X} 1,α; p≤ 4/δ. (4.21c)

Finally employing the inequalities in (4.21a)–(4.21c), we have for each of the terms appearing on the right-hand side of (4.20) and for any p > 1 and (1/r) + (1/r
) = 1

ξ−1Fψ2_ISαψ_I _L2 ≤ C ψ_I 2_L2pr
x L∞t Sαψ_{I L}pr x L2t ≤ C ψ_I 3_X 1,α, (4.22a) ξ−1Fψ_IDαψ_I2 _L2 ≤ C ψ_{I L}pr
x L∞t Dαψ_I 2_L2pr x L4t ≤ C ψ_I 3_X 1,α. (4.22b)

Combining (4.22a) and (4.22b) we obtain

11   
_T 0  P− R ; Rdt   ≤ C N(3−δ)/2Iψ 6 X1,α[0,T ] , (4.23)

and this estimate together with (4.9) imply the almost concervation law in (4.7).

5. Correlation Estimates

1

We would like to describe a general method for deriving correlation estimates for Schr¨odinger type equations. The estimate which is relevant here is stated in (5.40).

3

The idea is to view the evolution equation as describing the evolution of a compress-ible dispersive fluid whose pressure is a function of the density. For simplicity we will

5

consider a cubic equation, but the method works for more general nonlinearities. Let us start by considering the Nonlinear Schr¨odinger equation with a defocusing

7

cubic nonlinearity, i.e. the field ψ(t, x) satisfies the equation

iψ_t− ∆ψ + |ψ|2ψ = 0; ψ : R× Rn→ C, (5.1a)

9

where the space dimension is n = 2, 3 and C is the complex plane. The cubic nonlinearity serves as an example, one can consider more general nonlinear terms

11

as long as they are of defocusing type. Let us adopt the conventions

D[T₁, T₂]def= Rn× [T₁, T₂]; Rn_t def= (t, x) : x∈ Rn (5.1b)

13

to denote a space-time slab and a time slice for t fixed. A central observation is that the Eq. (5.1a) above conserves energy which means that the integral

15 Edef=
Rn t  |∇ψ|2₊1 2|ψ| 2  dx (5.1c)

is a constant independent of time. Equation (5.1a) has more conservation laws certain of which we wish to exploit in order to obtain apriori estimates. Let us define certain quantities that will be central in our investigation,

ρdef= 1 2|ψ| 2_{, (5.2a)} p_j def= 1 2i  ψ∇_jψ− ψ∇_jψ; j = 1, 2, . . . , n, (5.2b) p₀def= 1 2i  ψψ_t− ψψ_t, (5.2c) σ_jk def= 1 2  ∇jψ∇_kψ +∇_jψ∇_kψ; j, k = 1, 2, . . . , n. (5.2d) What I defined above is a density function ρ(t, x), a momentum vector p_j(t, x), with

p₀ the time component, and a stress tensor σ_jk(t, x). The Schr¨odinger evolution gives conservation laws for the density and the space components of the momenta, they are

∂_t{ρ} − ∇_j{pj} = 0, (5.3a)

∂_t{p_j} − ∇_k{µk_j} = 0, (5.3b)

where µ_kj(t, x) is the tensor

17

The tensor µ_kj consists of three parts, the term−∆ρ is dispersive while the term

1

2ρ2 describes the pressure as a function of density. Some remarks are in order, Eq. (5.1a) is the evolution equation of the Lagrangian

3
Rn_×R  − p0+ σ + 2ρ2  dxdt, (5.4a)

where σ = tr(σ_jk). The integral of the quantity p₀is not preserved by the flow, but

5 the integral
Rn t  − p0+ 2ρ2  dx 7

is, indeed, constant independent of time. This observation follows from a structure equation that (5.1a) satisfies, namely

9

−p0− ∆ρ + σ + 4ρ2= 0. (5.4b)

Notice that the system (5.3a) and (5.3b) describes the conservation laws of an

11

irrotational compressible and dispersive fluid.

Let us assume that we have two different solutions of (5.1a) for which we have

13

the corresponding densities ρa and momenta pa where a = 1, 2. Now write the conservation law for the momenta again emphasizing the space and time dependence

15 ∂_tp1_j(t, x)− ∇_k µ1,k_j (t, x)  = 0. (5.5)

Multiplying the equation above with ρ2_{(t, y) and using the conservation of mass}

equation we obtain an equation, of the form

∂_tp1_j(t, x)ρ2(t, y)− ∇_y,kp1_j(t, x)p2,k(t, y)− ∇_x,k µ1,k_j (t, x)ρ2(t, y)  = 0. (5.6) In Eq. (5.6) above, we used the convention∇_x,∇_yto keep track of which variable

17

is differentiated. Change variables as follows x→ x + y

2 ; y→ x− y

2 (5.7)

19

and symmetrize in the a = 1, 2 indices, i.e. switch the roles of 1 and 2 and add them so that Eq. (5.6) can be written as

21 ∂_t Q(12)_j (t, x; y)  − ∇y,k M(12),k_j (t, x; y)  − ∇x,k N(12),k_j (t, x; y)  = 0, (5.8) where we use the convention (12) to denote the symmetrized quantities and the quantities Q(12)_j , M_kj(12) and N_kj(12) are defined below,

Q(12)_j (t, x; y)def= p(1_j (t, x + y)ρ2)(t, x− y), (5.9a)

M_kj(12)(t, x; y)def= µ(1_jk(t, x + y)ρ2)(t, x− y) − p(1_j (t, x + y)p2)_k(t, x− y), (5.9b)

N_kj(12)(t, x; y)def= µ(1_jk(t, x + y)ρ2)(t, x− y) + p(1_j (t, x + y)p2)_k(t, x− y). (5.9c) In what follows we will occasionally skip the indices a = 1, 2 for the two solutions

23

confusion. If we switch y→ −y, we obtain another equation like (5.8) with the sign

1

of the second term switched. Subtracting the two equations we obtain a symmetrized version of (5.8), which reads

3

∂_t{Q_A,j} − ∇_y,kM_S,jk − ∇_x,kN_A,jk = 0, (5.10) where the relevant symmetrized or anti-symmetrized quantities are defined below

Q_A,j(t, x; y)def= Q_j(t, x; y)− Q_j(t, x;−y), (5.11a)

M_S,kj(t, x; y)def= M_kj(t, x; y) + M_kj(t, x;−y), (5.11b)

N_A,kj(t, x; y)def= N_kj(t, x; y)− N_kj(t, x;−y). (5.11c) Notice that a similar argument, starting from the conservation of mass Eq. (5.3a)

5

gives the equation

∂_t{D(t, x; y} − ∇_y,kQk_A(t, x; y)− ∇_x,kQk_S(t, x; y)= 0, (5.12a)

7

where D(t, x; y) and Q_S,k(t, x; y) are defined as follows

D(12)(t, x; y) = ρ(1(t, x + y)ρ2)(t, x− y), (5.12b)

Q(12)_S,k(t, x; y) = p(1_k(t, x + y)ρ2)(t, x− y) + p(1_k(t, x− y)ρ2)(t, x + y). (5.12c) Symmetrizing Eq. (5.8) gives an equation similar to (5.10)

∂_t{Q_S,j} − ∇_y,kM_A,jk − ∇_x,kN_S,jk = 0, (5.13)

9

where M_A,jk and N_S,jk stand for the anti-symmetrized and symmetrized versions of M_jk and N_jkrespectively but we are not going to make use of (5.13) here.

11

A crucial observation is the fact that we can express the stress tensor σ_kjin terms of the density function ρ and the momenta p_j, namely an elementary calculation,

13

using the fact that

∇jρ = 1

2 

ψ∇_jψ + ψ∇_jψ 15

shows that we can write

σ_kj = 1

2ρ[∇kρ∇jρ + pkpj] . (5.14)

17

The equation above suggests that we split σ_kj in two parts, a potential part say π_kj and a kinetic part, say κ_kj, in the following manner

19

π_kj def= 1

2ρ∇kρ∇jρ; κkj

def₌ 1

2ρpkpj. (5.15)

The trace of the potential part of the stress tensor tr(π_kj) contains part of the potential energy of the field ψ due to compressibility, while the trace of the kinetic part tr(κ_kj) contains the kinetic energy of the field. Our next step is to examine carefully the tensor M_S,kj. We would like to suppress the dependence on t since it

is not relevant in what follows. If we substitute (5.15) in the expression of µ_kj, we can write a long formula for M_S,kj as follows

M_S,kj(12)=−p(1_j (x + y)p_k2)(x− y) − p(1_j (x− y)p2)_k (x + y) + 2κ(1_kj(x + y)ρ2)(x− y) + 2κ(1_kj(x− y)ρ2)(x + y) + 2π(1_kj(x + y)ρ2)(x− y) + 2π_kj(1(x− y)ρ2)(x + y)

− δkj[∆ρ(1(x + y)ρ2)(x− y) + ∆ρ(1(x− y)ρ2)(x + y)]

+ δ_kj[2(ρ(1)2(x + y)ρ2)(x− y) + 2(ρ(1)2(x− y)ρ2)(x + y)]. (5.16) Let us define two vectors, using the momenta and density

1 J_j1,2(t, x; y)def= p 1 j(x + y)  ρ1_{(x + y)}  ρ2_{(x− y) −} p 2 j(x− y)  ρ2_(x− y)  ρ1_{(x + y) (5.17)}

and similarly J_j2,1by switching the roles of 1, 2 so that the first four terms in (5.14) can be expressed as a tensor product of the vector J_ja,b, i.e. we can write

− p(1

j (x + y)p2)k(x− y) − p(1j (x− y)p2)k (x + y) + 2κ(1kj(x + y)ρ2)(x− y) + 2κ_kj(1(x− y)ρ2)(x + y) = J_k1,2J_j1,2+ J_k2,1J_j2,1. (5.18) It is worthwhile to notice here that Q_A,j can be expressed using D and J_j , namely

3 Q(12)_A,j =√D1,2_J1,2 j + √ D2,1_J2,1 j , (5.19)

where D1,2 _{:= ρ}1_{(x + y)ρ}2_{(x− y) and similarly for D}2,1_{. Taking into account the} 5

observation in (5.18), we can write the tensors M_S,kj as a sum of four terms,

M_S,kj(12)= J_k1,2J_j1,2+ J_k2,1J_j2,1+ Φ_kj+ δ_kj[W + P ] , (5.20)

7

where the tensor Φ_kj and the scalar potentials W and P are

Φ_kj def= 2π(1_kj(x + y)ρ2)(x− y) + 2π_kj(1(x− y)ρ2)(x + y), (5.21a)

W def= −[∆ρ(1_{(x + y)ρ}2)_{(x− y) + ∆ρ}(1_{(x− y)ρ}2)_{(x + y)], (5.21b)} P def= 2ρ(12(x + y)ρ2)(x− y) + 2ρ(1)2(x− y)ρ2)(x + y). (5.21c) At this point we are ready to derive apriori estimates. For simplicity we will assume that ψ1_{= ψ}2_{, thus ρ}1_{= ρ}2_{etc. in what follows, the assumption makes the} 9

will be identity (5.10), which if we contract with a vector field Xj(y) that depends

1

only on the y variables, we obtain an equation of the form

∂_tQ_A,jXj− ∇_y,kM_S,jk Xj− ∇_x,kN_A,jk Xj+ R = 0, (5.22)

3

where the remainder R is a sum of four terms

R = R₀+ R₁+ R₂+ R₃ (5.23)

5

given by the following expressions

R₀def= (∇kX_j) JkJj; R₁def= (∇kX_j) Φ_kj, (5.24a)

R₂def= (divX) W ; R₃def= (divX) P. (5.24b) Let us choose the vector field Xj(y) to be of the particular form, we would like to choose

7

Xj(y)def= b(|y|/N)uj; uj def= yj/|y|, (5.25) where b(s) is an increasing bounded function and N is a large parameter to be chosen appropriately later. Adopt the convention r =|y| and compute

∇kX_j = b(r/N ) r [δkj− ukuj] + 1 Nb
_{(r/N )u} ku_j, (5.26a) divX = 1 r(b(r/N ) + (r/N )b
_{(r/N )) . (5.26b)}

Observe that because b(s) is increasing the terms R₀, R₁ and R₃ are positive. We

9

will look more carefully at the R₂term. First let us examine the three dimensional, case, i.e. n = 3 with the simple choice b = 1. Let us observe that in three space

11

dimensions we have, see (5.21) for the expression of W

R₂=
R3_×R3_×R W |y| dxdydt = C
R3_×R ρ1(t, x)ρ2(t, x) dxdt. (5.27) 13

Recall the notation in (5.1b) and let us define a correlation function R[T₁, T₂] for the density and another correlation, say L(t), between the momentum and the density in the following manner

R[T₁, T₂]def=
D[T1,T2] ρ1(t, x)ρ2(t, x) dxdt, (5.28a) L(t)def=
Rn t×Rnt  Q_A,jXj dxdy. (5.28b)

With these conventions, after we integrate (5.22), we obtain the estimate

R[0, T ]≤ C [L(0) − L(T )] , (5.29)

15

and one can show, as in [6], that

|L(t)| ≤ Cψ1_(t)2

L2ψ2(t)2_H1/2+ψ2(t)2_L2ψ1(t)2_H1/2. (5.30)

It is interesting to recast L(t) in a different light, define a potential function Ua(t, x)

1

and a vector field P_ja(t, x) by solving Laplace’s equation

−∆Ua _{= ρ}a_{; −∆P}a

j = paj; a = 1, 2, (5.31)

3

now we can write, omiting the indices a = 1, 2 for simplicity,

L(t) =
R3 t  ∆U P_j− U∆P_jxjdx. (5.32) 5

Consider the two dimensional case, i.e. n = 2. Assume for simplicity that the two states are equal and make the choice, see (5.25)

7

b(s)def= s log(√e/s), if s≤ 1/√e, (5.33) otherwise we require that b(s) is smooth increasing and bounded by one for s >

9

1/√e. With this choice, we have, with r =|y|

divX = 2

N log(N/r), if r < N/ √

e, (5.34)

11

so that the Laplacian of divX can be written

−∆ (divX) = 2 Nδ(y) +

c(r/N )

N3 , (5.35)

13

where c(r/N ) is a bounded function supported in the region{r > N/√e}.

Integrat-ing (5.22) we obtain the identity below

15 L(T₂)− L(T₁) =
_T2 T1
R2 t×R2t R dxdydt = 0. (5.36)

Observe that because Xj is bounded, we can obtain a bound,

17
R2 T×R2T  Q_A,jXj dxdy≤ ψ(T )2_L2ψ(T )2_H1/2, (5.37)

which estimate follows from the observation

19   
R2 t Xj(y)p_j(t, x + y)dx   ≤ Xj∇jψ H−1/2ψH1/2. (5.38)

Let us make the choice N = (T₂− T₁)12 so that we have a bound

21
_T2 T1
R2 t×R2t c(|y|/N) N3 ρ(t, x + y)ρ(t, x− y) dxdydt ≤ ψ 2 L2/(T2− T1)1/2, (5.39)

which follows from the fact that _R2

tρ(t, x) dx =ψ0

2

L2. Finally from the integral

23

of the term we called R₂, see (5.24b), we obtain a decay estimate
D[T1,T2] ρ2(t, x) dxdt≤ C sup t  ψ(t)2 L2ψ(t)2_H1/2  (T₁− T₂)12. (5.40) 25

A remark is in order concerning the name correlation for this type of estimate.

1

Let d(y) be the function with the property∇_jd(y) = X_j(y). Multiply (5.12a) with

d and differentiate with respect to time, and use (5.10) to obtain the equation 3

∂_t2{dD} + ∇_y,kM_S,jk Xj− d∂_tQk_A+∇_x,kP_A,jk Xj− d∂_tQk_S= R. (5.41) Integrating with respect to the space variables, we obtain

5 d2 dt2 
R2 t×R2t {dD} dxdy  =
R2 t×R2t {R} dxdy. (5.42)

Notice that the integral

7 C[t]def=
R2 t×R2t {dD} dxdy =
R2 t×R2t

d(y)ρ(1(t, x + y)ρ2)(t, x− y) dxdy, (5.43) expresses a correlation between the density functions ρa(t, x). Since C[t] is positive

9

and its derivative is integrable in the case where the space dimension is three we obtain that C[t]∼ t for large time. The integral C[t] can be written, after a change of

11 variables, C[t] = 2
R2 t×R2t |x − y|ρ1_{(t, x)ρ}2_{(t, y)dxdy.} 13

We can derive a true space-time estimate for the two dimensional case. Let d(t, y) and Xj(t, y) to be chosen. The conservation laws that we would like to use are

∂_t{D} − ∇_y,j Qj_A  − ∇x,j Qj_S  = 0, (5.44a)

∂_t{Q_A,j} − ∇_y,kM_S,jk − ∇_x,kN_A,jk = 0. (5.44b) Multiply the first equation with d(t, y) and contract the second equation with

Xj(t, y) and add them to obtain,

∂_tdD + XjQ_A,j− ∇_y,kdQk_A+ M_S,jk Xj− ∇_x,kdQk_S+ N_A,jk Xj

+

−∂_tdD +∇_jd− ∂_tX_jQj_A+∇_kXjM_S,jk

= 0. (5.45)

Try the vector field of the form

Xj= b  |y| tα  uj (5.46) 15

with α to be chosen, so that we have

∇kXj= b(|y|/t α₎ |y| [δkj − ukuj] + 1 tαb
_(|y|/tα_)u kuj (5.47a) divX = 1 |y|b(|y|/tα) + 1 tαb
_(|y|/tα_{). (5.47b)}

We assumed that b(s) = s log√e/sfor s≤ 1/√e and using this we can compute

divX = 2

tαlog



tα/|y| for|y| ≤ tα/√e, (5.48)