# Final State Problem for the Dirac-Klein-Gordon Equations in Two Space Dimensions

Authors: Masahiro Ikeda
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study the final state problem for the Dirac-Klein-Gordon equations (DKG) in two space dimensions. We prove that if the nonresonance mass condition is satisfied, then the wave operator for DKG is well defined from a neighborhood at the origin in lower order weighted Sobolev space to some Sobolev space.


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## Body

## 1. Introduction

We study the final state problem for the Dirac-Klein-Gordon equations(DKG) in two space dimensions:
(DKG)(∂t+α·∇+iMβ)ψ=ϕβψ,(∂t2-Δ+m2)ϕ=ψ*βψ,(t,x)∈ℝ×ℝ2,
where (ψ,ϕ) is a ℂ2×ℝ-valued unknown function of (t,x),  ψ=(ψ1,ψ2)t stands a spinor field and ϕ denotes a scalar field, M,m>0 denote masses of the spinor field and the scalar field, respectively, and ψ* denotes a transposed conjugate to ψ. The operators α·∇ and Δ are defined by α·∇=∑j=12αj∂xj and Δ=∑j=12∂xj2, respectively. Here, αj  (j=1,2) and β are Dirac matrices, that is, 2 × 2 self-adjoint matrices with constant elements such that
(1)αj2=β2=I,αjβ+βαj=O,forj=1,2,αjαk+αkαj=O,forj,k=1,2,j≠k.Our aim in the present paper is to show existence of the wave operator for the DKG system(DKG) under the nonresonance mass condition m≠2M in two space dimensions.First, we recall some well-posedness results for(DKG). Many local well-posedness results in low-order Sobolev spaces have been obtained for these ten years (for recent information see, e.g., [1, 2] and references therein). Global well-posedness results in 2d case were also obtained (see, e.g., [3]). Moreover, very recently, unconditional uniqueness in 2d case was discussed in [4, 5]. On the other hand, there are few results about scattering for (DKG) in 2d case.In [6, 7], the asymptotic behavior of solutions for DKG system was studied in 3d case by reducing it to a nonlinear Klein-Gordon system (KG). Denote 𝒟±≡∂t±(α·∇+iMβ). In view of the properties (1), we have
(2)𝒟-𝒟+=∂t2-(α·∇+iMβ)(α·∇+iMβ)=∂t2+〈∇〉M2,
where 〈∇〉M≡M2-Δ. Hence, multiplying both sides of the Dirac part by 𝒟-, we obtain
(3)(∂t2+〈∇〉M2)ψ=𝒟-(ϕβψ)=(𝒟-ϕ)βψ-iMϕIψ+ϕβ𝒟+ψ=((𝒟-ϕ)β-iMϕI+λϕ2I)ψ,
where we have used the fact that ψ is the solution of the DKG system. Thus, the solution of the DKG system satisfies the following KG one:
(4)(∂t2+〈∇〉M2)ψ=((𝒟-ϕ)β-iMϕI+λϕ2I)ψ,(∂t2+〈∇〉m2)ϕ=ψ*βψ.If we want to obtain a priori estimates to the local solution for the DKG system, we can use estimates to solutions for the above KG one. Moreover, in the present two-dimensional case, the initial value problem for nonlinear KG systems including (4) was studied in [8] (see also [9]). In [8], Sunagawa proved existence of a unique global asymptotically free solution under the nonresonance mass conditions, if the initial data are sufficiently small, smooth and decay fast at infinity. However, asymptotic behavior of solutions for DKG is not clear because (DKG) is not equivalent to (4) in general. In this paper, we will consider the DKG system itself without reducing it into (4) such as in [10]. Though the initial value problem for DKG was treated in [11], the final value problem which will be discussed in this paper is more delicate because of the derivative loss difficulties.In [10], the wave operator for the DKG system has been obtained in a three-dimensional case. They dealt with the DKG system itself. Nevertheless, from a point of time decay property for the free solutions of the DKG system, two dimensional-case is critical, that is, borderline case between the long range scattering and the short range one. Therefore, their argument cannot be applicable to the two-dimensional case. To overcome the lack of time decay property, we will use the algebraic normal form transformation developed in paper [8] and the decomposition of the Klein-Gordon operator, that is,
(5)∂t2+〈∇〉M2=𝒟+𝒟-.By this combination, we will find a suitable second approximate solution toψ (given by (42)). We note that the implicit null structure for (DKG) was discovered in [12], and it was used to prove local well-posedness in low regular setting in [2]. On the other hand, in this paper, by explicit null structure, wave operator for (DKG) will be constructed.Next, we recall the problem of existence of the wave operator for(DKG). We define the free-Dirac-and Klein-Gordon evolution groups as follows:
(6)𝒱D(t)≡Icos(t〈∇〉M)-(α·∇+iMβ)〈∇〉M-1sin(t〈∇〉M),𝒱K(t)≡(cos(〈∇〉mt)sin(〈∇〉mt)-sin(〈∇〉mt)cos(〈∇〉mt)).
For given final data (ψ+,(〈∇〉mϕ1+,ϕ2+))∈(X)4 with some Banach spaces X defined explicitly later, we put
(7)ψ0(t)≡𝒱D(t)ψ+,(ϕ0(t)〈∇〉m-1∂tϕ0(t))≡𝒱K(t)(ϕ1+〈∇〉m-1ϕ2+).
We will look for a unique time local solution of (DKG) which satisfies the final state conditions as follows:
(8)limt→∞∥ψ(t)-ψ0(t)∥X~=0,(9)limt→∞∥(〈∇〉m1/2ϕ(t)〈∇〉m-1/2∂tϕ(t))-(〈∇〉m1/2ϕ0(t)〈∇〉m-1/2∂tϕ0(t))∥X~=0,
where X~ is also a suitable Banach space. If there exist T>0 and a unique solution (ψ,〈∇〉m1/2ϕ,〈∇〉m-1/2∂tϕ)∈(C([T,∞);X~))4 for (DKG) satisfying (8)-(9), then the wave operator 𝒲+ for (DKG) is defined by the mapping as follows:
(10)𝒲+:(X)2×(〈∇〉-1X×X)⟶(X~)2×(〈∇〉-1/2X~×〈∇〉1/2X~),(ψ(t),(ϕ(t),∂tϕ(t)))=𝒲+(ψ+,(ϕ1+,ϕ2+)),mmmmmmmmmmmmmmfort∈[T,∞),
where 〈∇〉-sX≡{ϕ;∥〈∇〉sϕ∥X<∞}.


## 2. Several Notations and Main Results

We introduce several notations to state our main results. Form,k∈ℝ, and 1≤p≤∞, we introduce the weighted Sobolev space as follows:
(11)Hpm,k={ϕ;∥ϕ∥Hpm,k≡∥〈x〉k〈∇〉mϕ∥Lp<∞},where〈x〉=(1+|x|2)1/2, 〈∇〉=(1-Δ)1/2. We also write for simplicity Hm,k=H2m,k, Hm=H2m,0, and  Hpm=Hpm,0, and so we usually omit the index 0 and p=2 if it does not cause a confusion.We now state our main results in this paper. We introduce the function space as follows:(12)Dq≡Hq/(q-1)4-4/q∩H5/2,1∩H12.Theorem 1.
Letm,M>0,  m≠2M,  4<q≤∞ and (ψ+,(〈∇〉ϕ1+,ϕ2+))∈(Dq)4. If the norm ρ≡∥(ψ+,(〈∇〉ϕ1+,ϕ2+))∥H12 is sufficiently small, then there exist a positive constant T>0 and a unique solution
(13)(ψ(t),(〈∇〉m1/2ϕ(t)〈∇〉m-1/2∂tϕ(t)))∈(C([T,∞);H1/2))4,
for the system (DKG). Moreover, there exists a positive constant C>0 such that the following estimate
(14)∥ψ(t)-ψ0(t)∥H1/2+∥(ϕ(t)〈∇〉m-1∂tϕ(t))-(ϕ0(t)〈∇〉m-1∂tϕ0(t))∥H1≤Ct-μ
is true for allt≥T, where 1/2<μ<1-2/q and (ψ0,ϕ0) is given by (7).By Theorem1, we can get existence of the wave operator for (DKG) as follows.Corollary 2.
Letm,M>0,  m≠2M, and 4≤q<∞. Then the wave operator 𝒲+ for (DKG) is well defined from a neighborhood at the origin in the space (Dq)2×(〈∇〉-1Dq×Dq) to the space (H1/2)2×(H1×L2).The rest of this paper is organized as follows. In Section3, we state some basic estimates for free solutions of the DKG system and we introduce “null forms” and state their properties. In Section 4, we decompose two harmful terms by the algebraic normal form transformation and we find a second approximation for ψ through the decomposition of the Klein-Gordon operator by the Dirac one. In Section 5, following paper [10], we will also change the transformed DKG system into another form in order to apply the Strichartz type estimates to the Dirac part. In Section 6, we will prove Theorem 1 by an iteration scheme based on paper [13].


## 3. Elementary Estimates and Null Forms

Through the paper, we writeA≃B if there exist some positive constants C1,C2>0 such that C1B≤A≤C2B, and we also write A≲B if there exists a positive constant C>0 such that A≤CB.We introduce the free evolution groups as follows:(15)𝒰±,m(t)≡e±it〈∇〉m=ℱ-1e±it〈ξ〉mℱ.
Then, we have the following decomposition:
(16)𝒱D(t)=∑±‍𝒰±,M(t)𝒜±D,
where
(17)𝒜±D≡12(I±i〈∇〉M-1(α·∇+iMβ))
is 0th order matrix operator. We note that for any ℂ2-valued function ψ, the following equivalency is valid:
(18)∥𝒜±Dψ∥Hpm,k≃∥ψ∥Hpm,k.Now, we stateLp-Lq time decay estimates through the free evolution groups 𝒰±,m(t) obtained in paper [14].Lemma 3.
Letm≠0 and 2≤p≤∞. Then the estimate
(19)∥𝒰±,m(t)ϕ∥Lp≲t2/p-1∥ϕ∥Hq2(1-2/p)
is true for anyt>0, where q is a conjugate exponent of p: 1/p+1/q=1.By the lemma, we can easily getLp-Lq time decay estimates to free solutions for the DKG system.Corollary 4.
Under the same assumption of Lemma3 and M>0, the following estimates
(20)∥𝒱D(t)ψ+∥Lp≲t2/p-1∥ψ+∥Hq2(1-2/p),∥𝒱K(t)(ϕ1+〈∇〉m-1ϕ2+)∥Lp≲t2/p-1∥(ϕ1+〈∇〉m-1ϕ2+)∥Hq2(1-2/p)
are valid for any t>0, where q is a conjugate exponent of p: 1/p+1/q=1.Remark 5.
Letκ∈ℝ,  M,m≠0, and 2≤p<∞. Then the following estimates
(21)∥𝒱D(t)ψ+∥Hpκ≲t2/p-1∥ψ+∥Hκ+2-4/p,1,∥𝒱K(t)(ϕ1+〈∇〉m-1ϕ2+)∥Hpκ≲t2/p-1∥(ϕ1+〈∇〉m-1ϕ2+)∥Hκ+2-4/p,1
hold for anyt>0.Next, we introduce the Strichartz estimates, which enable us to treat the problem in lower order Sobolev spaces. Denote the space-time norm(22)∥ϕ∥Ltr(I;Lxq)≡∥∥ϕ(t)∥Lxq∥Ltr(I),
where I is a bounded or unbounded time interval. We define the integral operator as follows:
(23)𝒢±,m[g](t)≡∫Tt‍𝒰±,m(t-τ)〈∇〉m-1g(τ)dτ
for any T∈I¯, where m>0. By the duality argument of [15] along with Lemma 3, we have the following (see also [10, 13]).Lemma 6.
Let2≤q<∞ and 2/r=1-(2/q). Then for any time interval I, the following estimates are true:
(24)∥𝒢±,m[g]∥Ltr(I;Lxq)≲∥g∥Ltr′(I;Hq′2γ-1),∥𝒢±,m[g]∥Lt∞(I;Lx2)≲∥g∥Ltr′(I;Hq′γ-1),∥𝒢±,m[g]∥Ltr(I;Lxq)≲∥g∥Lt1(I;Hγ-1),
wherer′=r/(r-1),  q′=q/(q-1) and γ=1-(2/q).Next, we introduce the Leibniz rule for fractional derivatives.Lemma 7.
Letκ>0,  1<p,q1,q2<∞,  1<r1,r2≤∞, and 1/p=1/q1+1/r1=1/q2+1/r2. Then the following estimate holds:
(25)∥uv∥Hpκ≲∥u∥Hq1κ∥v∥Lr1+∥v∥Hq2κ∥u∥Lr2.For the proof of (25) see, for example, [16].We introduce the operator𝒵=(𝒵1,𝒵2), where 𝒵k≡xk∂t+t∂k for k=1,2. Let 𝒵α=𝒵1α1𝒵2α2 for a multi-index α=(α1,α2)∈(ℕ∪{0})2. We can see the commutation relations (see [6, 17]) as follows:
(26)[𝒟+,𝒵k-(12)αk]=αk𝒟+,[∂t2-Δ+m2,𝒵k]=0,fork=1,2, where [A,B]≡AB-BA.We introduce the quadratic null forms as follows:(27)𝒬0(f,g)≡(∂tf)(∂tg)-(∇f)·(∇g),𝒬j,k(f,g)≡(∂jf)(∂kg)-(∂kf)(∂jg),
for 0≤j<k≤2, where ∂≡(∂0,∇)≡(i∂t,∂1,∂2). In particular, 𝒬j,k is called a strong null form and has an additional time decay property through the operator 𝒵k, obtained in [18] (see also [8, 13, 19], etc.).Lemma 8.
Letj,k=1,2. Then, for any smooth function f,g, the identities
(28)𝒬0,j(f,g)=t-1(∂0f)(𝒵jg)-t-1(𝒵jf)(∂0g),𝒬j,k(f,g)=t-2(𝒵jg)(𝒵kf)-t-2(𝒵jf)(𝒵kg)+t-1(∂jf)(𝒵kg)-t-1(∂jg)(𝒵kf)+t-1(𝒵jf)(∂kg)-t-1(𝒵jg)(∂kf)
are valid for any t∈ℝ∖{0}.


## 4. Decomposition of Critical Terms

We study a structure of some harmful terms of(DKG). By the difference of  (DKG) and the free DKG system, it follows that
(29)𝒟+(ψ-ψ0)=(ϕ-ϕ0)βψ+ϕ0β(ψ-ψ0)+ϕ0βψ0,(□+m2)(ϕ-ϕ0)=(ψ-ψ0)*βψ+ψ0*β(ψ-ψ0)+ψ0*βψ0,
where □=∂t2-Δ. The last two terms ϕ0βψ0 and  ψ0*βψ0 are critical, both of which have the worst time decay property. Especially, since
(30)ϕ0βψ0,ψ0*βψ0=O(t-1)inL2ast⟶+∞(see Corollary4), the L2-norm of these terms is not integrable with respect to time t over [1,∞). Therefore, it can not be expected that usual perturbation technique is applicable to (29). To overcome this lack of time decay property, we will decompose them into an image of a Klein-Gordon operator and a remainder term following paper [8], based on papers [19–21].Let(v1,v2) be a solution for the following homogeneous KG system with masses M1,M2>0,
(31)(□+Mj2)vj=0,(t,x)∈ℝ×ℝ2,forj=1,2.
By the masses M1,  M2, we introduce the symmetric matrix as follows:
(32)ℳ=ℳ(M1,M2)=(M12+M222M1M22M1M2M12+M22).
We have the following.Lemma 9 (see [8]).
Letm~>0 with det(m~2I-ℳ)≠0. Then the quadratic term v1v2 can be decomposed as
(33)v1v2=1det(m~2I-ℳ){(□+m~2)f-4ℛ},
where
(34)f=f(v1,v2)≡(-M12-M22+m~2)v1v2-2𝒬0(v1,v2),ℛ=ℛ(v1,v2)≡∑m=12‍𝒬0,m(∂tv1,∂mv2)+∑m=12‍𝒬0,m(∂tv2,∂mv1)-𝒬1,2(∂1v1,∂2v2)-𝒬2,1(∂2v1,∂1v2).Under the nonresonance mass conditionm,M>0, and m≠2M, we can apply Lemma 9 to the critical terms ϕ0βψ0 and  ψ0*βψ0. Before doing so, we prepare for several notations. We put
(35)ℳ~≡1m2(2M+m)(m-2M)
which is well defined if m,M>0 and m≠2M. For a real-valued function ϕ and a ℂ2-valued function ψ=(ψ1,ψ2)t, we define ℂ2-valued functions of bilinear form:
(36)fD=fD(ϕ,ψ)≡(f(ϕ,ψ1),f(ϕ,ψ2))t,ℛD=ℛD(ϕ,ψ)≡(ℛ(ϕ,ψ1),ℛ(ϕ,ψ2))t,𝒬0D=𝒬0D(ϕ,ψ)≡(𝒬0(ϕ,ψ1),𝒬0(ϕ,ψ2))t,
Moreover, for C2-valued functions φ=(φ1,φ2)t,  ψ=(ψ1,ψ2)t, we put the following bilinear forms:
(37)fK=fK(φt,ψ)≡∑j=12‍f(φj,ψj),ℛK=ℛK(φt,ψ)≡∑j=12‍ℛ(φj,ψj),𝒬0K=𝒬0K(φt,ψ)≡∑j=12‍𝒬0(φj,ψj).
We have the following.Corollary 10.
Letm,M>0,  m≠2M, and (ψ0,ϕ0) be a free solution for the Dirac-Klein-Gordon equations. Then the quadratic terms ϕ0βψ0,  ψ0*βψ0 can be expressed as
(38)ϕ0βψ0=ℳ~{(□+M2)fD(ϕ0,βψ0)-4ℛD(ϕ0,βψ0)},ψ0*βψ0=ℳ~{(□+m2)fK(ψ0*,βψ0)-4ℛK(ψ0*,βψ0)}.Proof.
We consider the Dirac part of (38). Multiplying by 𝒟- both hand sides of 𝒟+ψ0=0, we get
(39)𝒟-𝒟+ψ0=(□+M2)ψ0=0,
which implies that ψ0=(ψ0,1,ψ0,2)t is also a solution of the free KG equation. Note that by the condition m,M>0 and m≠2M, we can apply Lemma 9 with m~=M,  v1=ϕ0, and v2=ψ0,k to get, for k=1,2,
(40)ϕ0ψ0,k=ℳ~{(□+M2)f(ϕ0,ψ0,k)-4ℛ(ϕ0,ψ0,k)}.
Thus, by a simple calculation, we obtain (38). Next, note that from equality (39), we see that ψ0¯ satisfies the free KG equation. Thus in the same manner as the proof of the Dirac part, we can prove the KG part, which completes the proof of the corollary.Next, we will change the DKG equations into another form without critical nonlinearities. We introduce a new unknown function(Ψ,Φ) as follows:
(41)Ψ≡ψ-ψ0-f~D≡ψ~-f~D,Φ≡ϕ-ϕ0-f~K≡ϕ~-f~K,
where (ψ0,ϕ0) is defined by (7) and
(42)f~D=f~D(ϕ0,ψ0)≡ℳ~𝒟-fD(ϕ0,βψ0)=ℳ~(fD(𝒟-ϕ0,βψ0)-iMfD(ϕ0,ψ0)),(43)f~K=f~K(ψ0)≡ℳ~fK(ψ0*,βψ0)are the second approximate solution to(ψ,ϕ), where we have used the identities αjβ+βαj=O,  β2=I and 𝒟+ψ0=0 to obtain the third equality in (42).Here, we remember that by the anticommutation relations (1) of the Dirac matrices, we can decompose the KG operator as follows:
(44)□+M2=𝒟+𝒟-.
By combining Corollary 10 and this decomposition, we can rewrite (DKG) as follows.Lemma 11.
Letm,M>0 and m≠2M. Then (ψ,ϕ) satisfies (DKG) if and only if the new variable (Ψ,Φ) defined by (41) is a solution of
(45)𝒟+Ψ=F,(□+m2)Φ=G,(t,x)∈ℝ×ℝ2,
where
(46)F=F(ϕ~,ψ~)≡ϕ~βψ~+ϕ~βψ0+ϕ0βψ~-4ℳ~ℛD(ϕ0,βψ0),G=G(ψ~)≡ψ~*βψ~+ψ~*βψ0+ψ0*βψ~-4ℳ~ℛK(ψ0*,βψ0),
andℳ~,  ℛD and ℛK are defined by (35), (36), and (37), respectively.This lemma enables us to treat the Dirac-Klein-Gordon equations(DKG) as well as the reduced KG system (4) in two space dimensions.Proof.
From (29), we see that (ψ,ϕ) is a solution of (DKG) if and only if the new variable (ψ~,ϕ~) satisfies the following DKG equations:
(47)𝒟+ψ~=ϕ~βψ~+ϕ~βψ0+ϕ0βψ~+ϕ0βψ0,(□+m2)ϕ~=ψ~*βψ~+ψ~*βψ0+ψ0*βψ~+ψ0*βψ0.
We consider the Dirac part of (47) only, since it is easier to handle the KG part. Note that by the assumption m,M>0 and m≠2M, we can apply Corollary 10 to ϕ0βψ0. Thus, we have
(48)ϕ0βψ0=ℳ~{(□+M2)fD(ϕ0,βψ0)-4ℛD(ϕ0,βψ0)}.
Moreover, by the decomposition (44), we can transform the first term of the right hand side of (48) as follows:
(49)λℳ~(□+M2)fD(ϕ0,βψ0)=ℳ~𝒟+𝒟-fD(ϕ0,βψ0)=𝒟+f~D,
where we have used the definition of f~D given by (42). Inserting (48) and (49) into the Dirac part of (47), we obtain the Dirac part of (45), which completes the proof of the lemma.Remark 12.
The null structure of(DKG) was characterized in [12] by using Fourier space. On the other hand, we note that in the above argument, Fourier space does not appear at all.


## 5. Reduction to Some First Order System

To construct a solution for the final value problem of the DKG system, we will use the Strichartz type estimates (Lemma6). However, it seems difficult to apply these estimates to the Dirac part for (45) due to a derivative loss difficulty. To gain first order differentiability properties of nonlinear term, we use the matrix operators
(50)ℬ±D≡12I(1∓i〈∇〉M-1∂t)=∓i2〈∇〉M-1ℒ∓D,ℒ±D≡(∂t∓i〈∇〉m)I,
though we do not necessarily need the operator ℬ in dealing with the initial value problem for the DKG system (see [11]). We will construct the desired solution (ψ,ϕ) for the DKG system by the iteration scheme. Let {(ψl,ϕl)}l≥0 be a sequence such that
(51)𝒟+ψl+1=ϕlβψl,(□+m2)ϕl+1=(ψl)*βψl,l≥0,(ψ0,ϕ0)=(ψ0,ϕ0),
under the final conditions
(52)limt→∞∥ψl(t)-ψ0(t)∥H1/2=0,(53)limt→∞∥(ϕl(t)〈∇〉m-1∂tϕl(t))-(ϕ0(t)〈∇〉m-1∂tϕ0(t))∥H1=0,forl≥0, where (ψ0,ϕ0) is given by (7). It suffices to prove that the sequence {ψl,(〈∇〉m1/2ϕl,〈∇〉m-1/2∂tϕl)}l≥0 is a Cauchy one in the Banach space (C([T,∞);H1/2))4 for some T>0.As the previous section, we introduce the new sequence{(Ψl,Φl)} as follows:
(54)Ψl≡ψl-ψ0-f~D≡ψ~l-f~D,Φl≡ϕl-ϕ0-f~K≡ϕ~l-f~K.
By Lemma 11, the sequence {(ψl,ϕl)} is a solution of (51) if and only if the new one {(Ψl,Φl)} satisfies the transformed DKG equations as follows:
(55)𝒟+Ψl+1=Fl,(□+m2)Φl+1=Gl,l≥1,(56)(Ψ0,Φ0)=-(f~D,f~K),
where
(57)Fl≡F(ϕ~l,ψ~l),Gl≡G(ψ~l),forl≥0 (f~D,  f~K, and  F and G are defined by (42)-(43) and (46), resp.).By the decomposition of the Klein-Gordon operator by the Dirac operator, we have(58)ℒ±Dℬ±D=∓i2〈∇〉M-1I(∂t2+〈∇〉M2)=∓i2〈∇〉M-1𝒟-𝒟+.
Thus, from the Dirac part for (55), we can deduce the following:
(59)ℒ±Dℬ±DΨl+1=∓i2〈∇〉M-1𝒟-𝒟+Ψl+1=〈∇〉M-1F±l,
for l≥0, where F±l≡∓(i/2)𝒟-Fl. Therefore, from Dirac part of (55), we have
(60)ℒ±Dℬ±DΨl+1=〈∇〉M-1F±l,l≥0,ℬ±DΨ0=-ℬ±Df~D.Remark 13.
By properties (1) of the Dirac matrices, we can transform F±l into another form without any derivatives of ψ~ or the free solution ψ0 (see (78)-(79), precisely). This fact enables us to use the Strichartz estimates for (60).Next we will also transform the KG part of (55) as in [10, 13]. We also use the operator (1-component version of the Dirac part) as follows:
(61)ℬ±K≡12(1∓i〈∇〉m-1∂t),ℒ±K≡∂t∓i〈∇〉m.
We can see that the sequence {Φl} is a solution of the KG part for (55) if and only if the sequence {ℬ±KΦl} satisfies
(62)ℒ±Kℬ±KΦl+1=〈∇〉m-1G±l,forl≥0,ℬ±KΦ0=-ℬ±Kf~K,whereG±l≡G±l(ψ~l)≡∓(i/2)Gl.Therefore, by (60) and (62), we get
(63)ℒ±Dℬ±DΨl+1=〈∇〉M-1F±l,ℒ±Kℬ±KΦl+1=〈∇〉m-1G±l,forl≥0,(64)(ℬ±DΨ0,ℬ±KΦ0)=-(ℬ±Df~D,ℬ±Kf~K).Remark 14.
The identity∑±‍ℬ±*=I holds, which enables us to reconstruct a solution (Ψ,Φ) for (45) from (ℬ±DΨ,ℬ±KΦ).Inserting the identities(65)ψ~l=∑±‍ℬ±DΨl+f~D,ϕ~l=∑±‍ℬ±KΦl+f~K,into the nonlinearitiesF±l,  G±l, we can express (63) by the new variable (ℬ±DΨl,ℬ±KΦl) only without (ϕ~l,ψ~l).At the end of this section, we will lead the integral equations associated with (63). We introduce a new unknown function sequence {vl} whose components are defined by
(66)vl≡(ℬ+DΨl,ℬ-DΨl,〈∇〉m1/2ℬ+KΦl,〈∇〉m1/2ℬ-KΦl)t,
a nonlinear term
(67)𝒩=𝒩(vl)≡(〈∇〉M-1F+l,〈∇〉M-1F-l,〈∇〉m-1/2G+l,〈∇〉m-1/2G-l)t
for l≥0, and a matrix-operator ℒ≡diag(ℒ+D,ℒ-D,ℒ+K,ℒ-K). Then by using these notations, (63) can be simplified as
(68)ℒvl+1=𝒩(vl)forl≥0.
To lead the integral equations for (68), we need to study the asymptotic behavior of the new variable vl. We can obtain the following.Lemma 15.
Let(ψ+,(〈∇〉ϕ1+,ϕ2+))∈(H5/2,1)4. The function (ψl,ϕl) defined by (51) satisfies (52)-(53) for any l≥0 if and only if the new function vl satisfies (68) and
(69)limt→∞∥vl∥H1/2=0,forl≥0.The proof of the lemma will be given in Appendix.We introduce a matrix evolution operator as follows:(70)𝒰(t)≡diag(𝒰+,M(t),𝒰-,M(t),𝒰+,m(t),𝒰-,m(t)).
From Lemma 15, we can lead the integral equations associated with (68) as follows:
(71)vl+1(t)=-∫t∞‍𝒰(t-s)𝒩(vl)ds.


## 6. Proof of Theorem1

In this section, we give a proof of Theorem1. Note that the identities
(72)∂tΨl=i〈∇〉M(v1l-v2l),∂tΦl=i〈∇〉m1/2(v3l-v4l)hold; the nonlinearity𝒩(vl) can be expressed in terms of the space derivatives of vl (so excluding the time derivatives).ForT>1, where T is sufficiently large, we introduce the following function space:
(73)XT={v∈(C([T,∞);H1/2))6;∥v∥XT<∞},
with the norm
(74)∥v∥XT≡supt∈[T,∞)tμ(∥v∥Lt4(I;L4)+∥v∥Lt∞(I;H1/2)),
where 1/2<μ<1-2/q,  4<q≤∞, and I=[t,∞). We define
(75)A≡C∥(ψ+,(〈∇〉ϕ1+,ϕ2+))∥Hq/(q-1)4-4/q∩H5/2,1.In order to obtain the theorem, we will show that the sequence{vl} is a Cauchy one in a closed ball XT,A for appropriate T and ρ, where XT,A≡{v∈XT;∥v∥XT≤A}.Hereafter, we will use the notationLtrX=Ltr(I;X),  𝒟=𝒟- and
(76)ℬΨ=ℬ±DΨ,ℬΦ=ℬ±KΦ,
for simplicity if it does not cause a confusion.Proof.
We will prove thatvl∈XT,A for any l≥0 by induction. In the case of  l=0, it is easy to see that v0∈XT,A for some T and ρ. We omit the details. For l≥1, we assume that vk∈XT,A for 0≤k≤l. We will show that vl+1∈XT,A for some T and ρ.
First, by the identities𝒟+ψ0=0 and 𝒟+ψl=λϕl-1βψl-1 for l≥1, we get, for l≥1,
(77)𝒟-(ϕ~lβψ~l)=(𝒟-ϕ~l)βψ~l-iMϕ~lIψ~l+λϕ~lϕ~l-1Iψ~l-1+λϕ~lϕ~l-1Iψ0+λϕ~lϕ0Iψ~l-1+λϕ~lϕ0Iψ0,𝒟-ℛD(ϕ0,βψ0)=ℛD(𝒟-ϕ0,βψ0)-iMℛD(ϕ0,βψ0).
From these identities, we can express F±l as follows:
(78)F±l=∓i2∑j=13‍Fjl+“remainder”forl≥1,
where
(79)F1l≡(𝒟-ϕ~l)βψ~l,F2l≡(𝒟-ϕ0)βψ~l+(𝒟-ϕ~l)βψ0,(80)F3l≡4iℳ~ℛD(𝒟-ϕ0,βψ0).
Here, we note that “remainder” (given by (78)) can be handled in the same manner as Fjl(j=1,2,or3). Thus, we will omit the estimate of them. We also decompose G±l as G±l=∓(i/2)∑j=13Gjl, where
(81)G1l=(ψ~l)*βψ~l,G2l=(ψ~l)*βψ0+ψ0*βψ~l,G3l=4iℳ~ℛK(ψ0*,βψ0).
Taking Lt4Lx4-norm and Lt∞H1/2-norm of (71) and applying Lemma 6 with (q,r,γ)=(4,4,1/2) and (2,∞,0), we have
(82)∥vl+1∥Lt4Lx4+∥vl+1∥Lt∞H1/2≲∥F1l∥Lt4/3Lx4/3+∥G1l∥Lt4/3H4/31/2+∑j=2,3‍(∥Fjl∥Lt1H-1/2+∥Gjl∥Lt1Lx2).
Moreover, we remember that(ϕ~l,ψ~l) is expressed as (65).
Now, we will estimateF1l. By the Hölder inequality, we have
(83)∥(𝒟ℬΦl)ℬΨl∥Lt4/3Lx4/3≲∥∥ℬΦl(s)∥H1∥ℬΨl(s)∥Lx4∥Lt4/3≲A∥s-μ∥ℬΨl(s)∥Lx4∥Lt4/3≤A∥ℬΨl∥Lt4Lx4∥s-μ∥Lt2(I)≲A2t1/2-2μ,
for any t≥T since vk∈XT,A for 0≤k≤l. By the Hölder inequality and Remark 5 with p=8, we obtain
(84)∥f~D(s)∥L4≲∥ϕ0(s)∥H82∥ψ0(s)∥H81≲A2s-3/2,
for any s≥t. In the same manner as the proof of the estimate (83), we also obtain
(85)∥(𝒟ℬΦl)f~D∥Lt4/3Lx4/3≲A3t-3/4-μ,
for all t≥T, due to vk∈XT,A for 0≤k≤l and (84). By the Hölder inequality and Remark 5 with p=8/3,8, we obtain
(86)∥𝒟f~K(s)∥L2≲∥ψ0(s)∥H8/32∥ψ0(s)∥H81≲s-1∥ψ+∥H8/55/2∥ψ+∥H8/75/2≲A2s-1
for any s≥t, where we have used properties (1) of α,β, and 𝒟+ψ0=0. Thus, in the same manner as the proof of the estimate (83), we obtain
(87)∥(𝒟f~K)ℬΨl∥Lt4/3Lx4/3≲A3t-1/2-μ,
for all t≥T due to vl∈XT,A and (86). By the Hölder inequality and estimates (84) and (86), we get
(88)∥(𝒟f~K)f~D∥Lt4/3Lx4/3≲∥∥𝒟f~K(s)∥Lx2∥f~D(s)∥Lx4∥Lt4/3(I)≲A4t-7/4,
for all t≥T. Thus by combining (83), (85), and (87)-(88), we obtain
(89)∥F1l∥Lt4/3Lx4/3≲A2t1/2-2μ,
for t≥T≥1 since μ<1. Next, we consider F2l. We have
(90)∥F2l∥Lt1H-1/2≤∥(𝒟ϕ~l)ψ0∥Lt1Lx2+∥(𝒟ϕ0)ψ~l∥Lt1Lx2.
By Corollary 4 with p=∞, we have
(91)∥(𝒟ℬΦl)ψ0∥Lt1Lx2≲∥∥ℬΦl(s)∥H1∥ψ0(s)∥Lx∞∥Lt1≲ρAt-μ,
for all t≥T since vk∈XT,A for 0≤k≤l. In the same manner as the estimate (91), we get
(92)∥(𝒟f~K)ψ0∥Lt1Lx2≲∥∥𝒟f~K(s)∥Lx2∥ψ0(s)∥Lx∞∥Lt1≤ρA2t-1,
for any t≥T, where we have used the estimate (86). Moreover, we also have
(93)∥(𝒟ϕ0)ℬΨl∥Lt1Lx2≲∥∥ϕ0(s)∥H∞1∥ℬΨl(s)∥Lx2∥Lt1≤Aρt-μ,
for all t≥T since vl∈XT,A. In the same proof as the estimate (84), by the Hölder inequality and Remark 5 with p=4, we get
(94)∥f~D(s)∥L2≲∥ϕ0(s)∥H42∥ψ0(s)∥H41≲A2s-1,
for any s≥t. By estimate (94) and Corollary 4 with p=∞, we have
(95)∥(𝒟ϕ0)f~D∥Lt1Lx2≲∥∥ϕ0(s)∥H∞1∥f~D(s)∥Lx2∥Lt1≲ρA2t-1,
for all t≥T. Therefore, by combining estimates (90)–(93) and (95), we obtain
(96)∥F2l∥Lt1H-1/2≲ρAt-μ,
for any t≥T≥1 since μ<1. Next, we consider F3l. By the definition of ℛD, we have
(97)∥ℛD(𝒟ϕ0,ψ0)∥Lt1H-1/2≲∑j=1,2‍∥ℛ(𝒟ϕ0,ψ0,j)∥Lt1Lx2,
where we put ψ0=(ψ0,1,ψ0,2)t. By Lemma 8, we can express ℛ as
(98)ℛ(𝒟ϕ0,ψ0,j)≡s-1Z1+s-2Z2,
for s∈ℝ∖{0}, where
(99)Z1≡(∂0∂t𝒟ϕ0)(𝒵1∂1ψ0,j)-(𝒵1∂t𝒟ϕ0)(∂0∂1ψ0,j)+similar,Z2≡-(𝒵1∂2ψ0,j)(𝒵2∂1𝒟ϕ0)+(𝒵1∂1𝒟ϕ0)(𝒵2∂2ψ0,j)+similar.
By applying the Hölder inequality, we have
(100)∥s-1Z1∥Lt1Lx2≲∫t∞‍s-1(∥ϕ0∥Hq3∥𝒵ψ0∥H2q/(q-2)1nnnnnnnnnnnnnnnnn+∥ψ0∥Hq2∥𝒵ϕ0∥H2q/(q-2)2)ds.
By Corollary 4 with p=q, we get
(101)∥ϕ0(s)∥Hq3≲s-1+2/q∥(〈∇〉ϕ1+,ϕ2+)∥Hq/(q-1)4-4/q≲As-1+2/q,∥ψ0(s)∥Hq2≲s-1+2/q∥ψ+∥Hq/(q-1)4-4/q≲As-1+2/q,
for any s≥t. On the other hand, note that the commutation relations (26) hold. By applying the Sobolev inequality and the charge and energy conservation laws, we obtain
(102)∥𝒵ψ0∥H2q/(q-2)1≲∥𝒵ψ0∥H1+2/q≲∥𝒵ψ0∥H3/2≲∥(𝒵ψ0)(0)∥H3/2≲A,∥𝒵ϕ0∥H2q/(q-2)2≲∥𝒵ϕ0∥H2+2/q≲∥𝒵ϕ0∥H5/2≲∥(𝒵ϕ0)(0)∥H5/2≲A,
since q>4. Thus, by combining (100)–(102), we get
(103)∥s-1Z1∥Lt1Lx2≲A2t-1+2/q,
for any t≥T. By the Hölder inequality, we have
(104)∥s-2Z2∥Lt1Lx2≲∫t∞‍s-2∥𝒵ψ0(s)∥H41∥𝒵ϕ0(s)∥H42ds≲A2t-1,
since in the same manner as the proof of estimates (102), we obtain
(105)∥𝒵ψ0(s)∥H41+∥𝒵ϕ0(s)∥H42≲A,
for any s≥t. Therefore, combining (97)-(98), (103), and (104), we have
(106)∥F3l∥Lt1H-1/2≲A2t-1+2/q,
for allt≥T≥1 since q>4.
Next, we will estimateG1l. By the Leibniz formula (25) with κ=1/2,  p=4/3,  q1=q2=2, and r1=r2=4 and the Hölder inequality, we obtain
(107)∥(ℬΨl)*ℬΨl∥Lt4/3H4/31/2≲∥∥ℬΨl(s)∥H1/2∥ℬΨl(s)∥Lx4∥Lt4/3≲A∥s-μ∥ℬΨl(s)∥Lx4∥Lt4/3≲A∥s-μ∥Lt2(I)∥ℬΨl∥Lt4Lx4≲A2t1/2-2μ,
for any t≥T since vl∈XT,A. By the fractional Leibniz rule (25) again and Remark 5 with p=4, we have
(108)∥f~D(s)∥H1/2≲∥ϕ0(s)∥H45/2∥ψ0(s)∥H43/2≲A2s-3/2,
for any s≥t. In the same manner as the proof of the estimate (107), we obtain
(109)∥(ℬΨl)*f~D∥Lt4/3H4/31/2≲∥∥ℬΨl(s)∥H1/2∥f~D(s)∥H1/2∥Lt4/3≲A3t-3/4-μ,
for any t≥T due to vl∈XT,A and (108). In the same manner as the proof of the estimate (109), we get
(110)∥(f~D)*f~D∥Lt4/3H4/31/2≲A4t-7/4,
for all t≥T. Thus, by combining the estimates (107) and (109)-(110), we obtain
(111)∥G1l∥Lt4/3H4/31/2≲A2t1/2-2μ,
for t≥T≥1 since μ<1. In the same manner as the proof of the estimates (96) and (106), we obtain
(112)∥G2l∥Lt1Lx2≲ρAt-μ,∥G3l∥Lt1Lx2≲A2t-1+2/q,
for any t≥T. Finally, by combining (82), (89), (96), (106), and (111)-(112), we obtain
(113)∥vl+1∥XT≲A(AT1/2-μ+ρ+AT-1+μ+2/q),
for T≥1. By the estimate (113) and 1/2<μ<1-2/q, there exist a large T>0 and a small ρ>0 such that vl+1∈XT,A. In the same manner as the proof of (113), we can prove the estimate
(114)∥vl+1-vl∥XT≤12∥vl-vl-1∥XT,
for l≥1 if T>1 is sufficiently large and ρ>0 is sufficiently small, which implies that {vl}l≥0 is a Cauchy sequencein XT,A. Theorem 1 is proved.



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*Source: 273959-2013-08-19.xml*