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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 273959, 11 pages
Final State Problem for the Dirac-Klein-Gordon Equations in Two Space Dimensions
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560 0043, Japan
Received 25 April 2013; Accepted 15 July 2013
Academic Editor: Daniel C. Biles
Copyright © 2013 Masahiro Ikeda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
We study the final state problem for the Dirac-Klein-Gordon equations in two space dimensions: where is a -valued unknown function of , stands a spinor field and denotes a scalar field, denote masses of the spinor field and the scalar field, respectively, and denotes a transposed conjugate to . The operators and are defined by and , respectively. Here, and are Dirac matrices, that is, 2 2 self-adjoint matrices with constant elements such that
First, we recall some well-posedness results for . 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., ). Moreover, very recently, unconditional uniqueness in 2d case was discussed in [4, 5]. On the other hand, there are few results about scattering for 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 . In view of the properties (1), we have where . Hence, multiplying both sides of the Dirac part by , we obtain where we have used the fact that is the solution of the DKG system. Thus, the solution of the DKG system satisfies the following KG one:
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  (see also ). In , 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 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 . Though the initial value problem for DKG was treated in , the final value problem which will be discussed in this paper is more delicate because of the derivative loss difficulties.
In , 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  and the decomposition of the Klein-Gordon operator, that is,
By this combination, we will find a suitable second approximate solution to (given by (42)). We note that the implicit null structure for was discovered in , and it was used to prove local well-posedness in low regular setting in . On the other hand, in this paper, by explicit null structure, wave operator for will be constructed.
Next, we recall the problem of existence of the wave operator for . We define the free-Dirac-and Klein-Gordon evolution groups as follows: For given final data with some Banach spaces defined explicitly later, we put We will look for a unique time local solution of which satisfies the final state conditions as follows: where is also a suitable Banach space. If there exist and a unique solution for satisfying (8)-(9), then the wave operator for is defined by the mapping as follows: where .
2. Several Notations and Main Results
We introduce several notations to state our main results. For , and , we introduce the weighted Sobolev space as follows:
where , . We also write for simplicity , , and , and so we usually omit the index and if it does not cause a confusion.
We now state our main results in this paper. We introduce the function space as follows:
Theorem 1. Let , , and . If the norm is sufficiently small, then there exist a positive constant and a unique solution
for the system . Moreover, there exists a positive constant such that the following estimate
is true for all , where and is given by (7).
By Theorem 1, we can get existence of the wave operator for as follows.
The rest of this paper is organized as follows. In Section 3, 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 through the decomposition of the Klein-Gordon operator by the Dirac one. In Section 5, following paper , 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 .
3. Elementary Estimates and Null Forms
Through the paper, we write if there exist some positive constants such that , and we also write if there exists a positive constant such that .
We introduce the free evolution groups as follows: Then, we have the following decomposition: where is th order matrix operator. We note that for any -valued function , the following equivalency is valid:
Now, we state time decay estimates through the free evolution groups obtained in paper .
Lemma 3. Let and . Then the estimate
is true for any , where is a conjugate exponent of : .
By the lemma, we can easily get time decay estimates to free solutions for the DKG system.
Corollary 4. Under the same assumption of Lemma 3 and , the following estimates are valid for any , where is a conjugate exponent of : .
Remark 5. Let , , and . Then the following estimates
hold for any .
Next, we introduce the Strichartz estimates, which enable us to treat the problem in lower order Sobolev spaces. Denote the space-time norm where is a bounded or unbounded time interval. We define the integral operator as follows: for any , where . By the duality argument of  along with Lemma 3, we have the following (see also [10, 13]).
Lemma 6. Let and . Then for any time interval , the following estimates are true:
where , and .
Next, we introduce the Leibniz rule for fractional derivatives.
Lemma 7. Let , , , and . Then the following estimate holds:
for , where .
We introduce the quadratic null forms as follows: for , where . In particular, is called a strong null form and has an additional time decay property through the operator , obtained in  (see also [8, 13, 19], etc.).
Lemma 8. Let . Then, for any smooth function , the identities are valid for any .
4. Decomposition of Critical Terms
We study a structure of some harmful terms of . By the difference of and the free DKG system, it follows that where . The last two terms and are critical, both of which have the worst time decay property. Especially, since
(see Corollary 4), the -norm of these terms is not integrable with respect to time over . 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 , based on papers [19–21].
Let be a solution for the following homogeneous KG system with masses , By the masses , , we introduce the symmetric matrix as follows: We have the following.
Lemma 9 (see ). Let with . Then the quadratic term can be decomposed as where
Under the nonresonance mass condition , and , we can apply Lemma 9 to the critical terms and . Before doing so, we prepare for several notations. We put which is well defined if and . For a real-valued function and a -valued function , we define -valued functions of bilinear form: Moreover, for -valued functions , , we put the following bilinear forms: We have the following.
Corollary 10. Let , , and be a free solution for the Dirac-Klein-Gordon equations. Then the quadratic terms , can be expressed as
Proof. We consider the Dirac part of (38). Multiplying by both hand sides of , we get which implies that is also a solution of the free KG equation. Note that by the condition and , we can apply Lemma 9 with , , and to get, for , Thus, by a simple calculation, we obtain (38). Next, note that from equality (39), we see that 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 as follows: where is defined by (7) and
are the second approximate solution to , where we have used the identities , and 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: By combining Corollary 10 and this decomposition, we can rewrite as follows.
This lemma enables us to treat the Dirac-Klein-Gordon equations as well as the reduced KG system (4) in two space dimensions.
Proof. From (29), we see that is a solution of if and only if the new variable satisfies the following DKG equations: We consider the Dirac part of (47) only, since it is easier to handle the KG part. Note that by the assumption and , we can apply Corollary 10 to . Thus, we have Moreover, by the decomposition (44), we can transform the first term of the right hand side of (48) as follows: where we have used the definition of 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 was characterized in  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 (Lemma 6). 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 though we do not necessarily need the operator in dealing with the initial value problem for the DKG system (see ). We will construct the desired solution for the DKG system by the iteration scheme. Let be a sequence such that under the final conditions
for , where is given by (7). It suffices to prove that the sequence is a Cauchy one in the Banach space for some .
By the decomposition of the Klein-Gordon operator by the Dirac operator, we have Thus, from the Dirac part for (55), we can deduce the following: for , where . Therefore, from Dirac part of (55), we have
Remark 13. By properties (1) of the Dirac matrices, we can transform into another form without any derivatives of or the free solution (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: We can see that the sequence is a solution of the KG part for (55) if and only if the sequence satisfies
Remark 14. The identity holds, which enables us to reconstruct a solution for (45) from .
Inserting the identities
into the nonlinearities , , we can express (63) by the new variable only without .
At the end of this section, we will lead the integral equations associated with (63). We introduce a new unknown function sequence whose components are defined by a nonlinear term for , and a matrix-operator . Then by using these notations, (63) can be simplified as To lead the integral equations for (68), we need to study the asymptotic behavior of the new variable . We can obtain the following.
The proof of the lemma will be given in Appendix.
6. Proof of Theorem 1
In this section, we give a proof of Theorem 1. Note that the identities
hold; the nonlinearity can be expressed in terms of the space derivatives of (so excluding the time derivatives).
For , where is sufficiently large, we introduce the following function space: with the norm where , , and . We define
In order to obtain the theorem, we will show that the sequence is a Cauchy one in a closed ball for appropriate and , where .
Hereafter, we will use the notation , and for simplicity if it does not cause a confusion.
Proof. We will prove that for any by induction. In the case of , it is easy to see that for some and . We omit the details. For , we assume that for . We will show that for some and .
First, by the identities and for , we get, for , From these identities, we can express as follows: where Here, we note that “remainder” (given by (78)) can be handled in the same manner as . Thus, we will omit the estimate of them. We also decompose as , where Taking -norm and -norm of (71) and applying Lemma 6 with and , we have
Moreover, we remember that is expressed as (65).
Now, we will estimate . By the Hölder inequality, we have for any since for . By the Hölder inequality and Remark 5 with , we obtain for any . In the same manner as the proof of the estimate (83), we also obtain for all , due to for and (84). By the Hölder inequality and Remark 5 with , we obtain for any , where we have used properties (1) of , and . Thus, in the same manner as the proof of the estimate (83), we obtain for all due to and (86). By the Hölder inequality and estimates (84) and (86), we get for all . Thus by combining (83), (85), and (87)-(88), we obtain for since . Next, we consider . We have By Corollary 4 with , we have for all since for . In the same manner as the estimate (91), we get for any , where we have used the estimate (86). Moreover, we also have for all since . In the same proof as the estimate (84), by the Hölder inequality and Remark 5 with , we get for any . By estimate (94) and Corollary 4 with , we have for all . Therefore, by combining estimates (90)–(93) and (95), we obtain for any since . Next, we consider . By the definition of , we have where we put . By Lemma 8, we can express as for , where By applying the Hölder inequality, we have By Corollary 4 with , we get for any . 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 since . Thus, by combining (100)–(102), we get for any . By the Hölder inequality, we have since in the same manner as the proof of estimates (102), we obtain for any . Therefore, combining (97)-(98), (103), and (104), we have
for all since .
Next, we will estimate . By the Leibniz formula (25) with , , , and and the Hölder inequality, we obtain for any since . By the fractional Leibniz rule (25) again and Remark 5 with , we have for any . In the same manner as the proof of the estimate (107), we obtain for any due to and (108). In the same manner as the proof of the estimate (109), we get for all . Thus, by combining the estimates (107) and (109)-(110), we obtain for since . In the same manner as the proof of the estimates (96) and (106), we obtain for any . Finally, by combining (82), (89), (96), (106), and (111)-(112), we obtain for . By the estimate (113) and , there exist a large and a small such that . In the same manner as the proof of (113), we can prove the estimate for if is sufficiently large and is sufficiently small, which implies that is a Cauchy sequencein . Theorem 1 is proved.
In this section, we give a proof of Lemma 15. First, we prepare the following.
Lemma 16 (see ). Let and let be a -valued given function. Then, for any -valued function , the equivalency
holds for all .
For the proof of the lemma, see .
By the lemma and a decay property of given by (42), we also have the following.
Before proving the corollary, we remember some properties of the operators given by (17) (see  in detail). We note that the identity holds due to properties (1) of Dirac matrices. Hence, by a direct calculation, we get the following identities: We put .
Proof. By Lemma 16, we see that (8) with is equivalent to
By decomposition (16) and identities (A.4), we have
By estimate (18), the fractional Leibniz rule (25) with and , and Remark 5 with , we get
for all , which completes the proof of the corollary.
Next we will prove Lemma 15.
Proof of Lemma 15. First we prove the Dirac part. By Corollary 17, we see that (52) is equivalent to
Note that the identity
holds. From the Dirac part of (55), we have
Thus, it is sufficient to show that
By the Sobolev inequality and the Hölder inequality, we have, for ,
By Remark 5 with , we get
Thus, by assumptions and estimates (A.12)-(A.13), we obtain (A.11) for . In the case of , it is easy to see (69). We omit the details. Conversely, assume (69) and will prove (52). By the decomposition , we have only to show that
By the Hölder inequality and Remark 5 with , we obtain
Since the remainder terms in (A.15) can be estimated in the same manner as the proof of (A.17), we obtain
from which (A.14) follows.
Next, we consider the KG part. By the identity we can see that (53) is equivalent to In the same manner as the proof of estimate (A.18), we can obtain which completes the proof of the lemma.
The author would like to express deep gratitude to an anonymous referee for the useful suggestions and comments.
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