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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 602753, 8 pages
Global Strong Solutions to Some Nonlinear Dirac Equations in Super-Critical Space
Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea
Received 10 April 2013; Accepted 21 May 2013
Academic Editor: Leszek Gasinski
Copyright © 2013 Hyungjin Huh. 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 initial value problem of some nonlinear Dirac equations which are critical. Corresponding to the structure of nonlinear terms, global strong solutions can be obtained in different Lebesgue spaces by using solution representation formula. The uniqueness of weak solutions is proved for the solution .
In this study, we are interested in global strong solutions of nonlinear Dirac type equations: where is a nonlinear potential function which will be specified later.
The following Thirring model with the potential has been studied in [1–4]: The initial value problem of (2) was studied in [1, 3] in terms of Sobolev space . Low regularity well-posedness was discussed in  showing that there exists a time and solution () of the Cauchy problem (2). Their results were based on the observation of the null structure of Thirring equations and application of spaces which is a certain subspace of . They also proved global well-posedness for and unconditional uniqueness for . Nonlinear Dirac equations in have been studied by several authors [5–9].
In the context of Bragg grating [10, 11], the nonlinear term takes the form which gives and . We may consider the sixth and higher orders. The following potential term is introduced in the context of the Bose-Einstein condensates : which gives and .
Several authors [4, 5, 7] have studied the initial value problem of the following Dirac equations with quadratic nonlinearities: If the nonlinear term is of , type (, , resp.), then one can obtain local well-posedness for the Sobolev space  ( , resp.). Note that the scaling properties of quadratic Dirac equations give the critical Sobolev exponent .
Now it seems natural to consider the following equations: which are generalization of the basic cases of the literature and model problem to investigate regularities of solutions according to the structure of nonlinearities. Here, is positive integer and .
The system (6) is invariant under the scaling from which we deduce a scale invariant Lebesgue space . We study the initial value problem of (6) in Lebesgue space. Let us denote Note that except for and where . We define where . We call as subcritical space for , critical space for , and supercritical space for .
For the initial data , we have proved in  that there exists a global strong solution to the initial value problem (2) in the critical space Elementary and interesting approach was made. and special structure of nonlinearity was made use of in applying Fubini's theory in integration. The following is concerned with the existence of global strong solutions for the case of or .
Theorem 1. For the initial data , there exists a global strong solution to the initial value problem (6) which satisfies
Remark 2. Related with the title of this paper, we emphasize the case where global strong solution can be constructed in the super-critical space.
Taking the embedding into account, it is likely to say that the system (6) with small has better smoothing property than equations with large . We can check, in the case of (4), the existence of solution because of .
For the case of and , we can construct a global strong solution to the initial value problem (6) which satisfies for the initial data .
Our second result is concerned with the uniqueness of weak solutions. The null structure of the nonlinearity of Thirring model which is critical problem was used in  to prove the uniqueness in . To treat the general nonlinearity in (6) of which we could not find the null structure, a different approach is considered.
Theorem 3. Let and be solutions of (6) in the distribution sense with same initial data. Moreover, one assumes that Then, one has for .
2. Proof of Theorem 1
To construct global strong solution of (6), we basically follow the idea of  with modification. We will find a solution representation formula. Then, global strong solutions can be obtained by constructing an explicit approximation and using Fubini's theorem.
2.1. Representation Formula of Classical Solution
In this subsection, we consider solutions which satisfy (6) in the classical sense. An explicit representation of its solution is given in terms of initial data [2, 13]. It is interesting to express the solution of nonlinear partial differential equations in terms of initial data.
2.2. Global Strong Solutions in
Definition 4. Consider the Cauchy problem (6) with initial data . It is said that is a strong solution to the Cauchy problem on the time interval provided that satisfy (6) in the sense of distributions. That is, for any , we have
Remark 5. We say that is a weak solution if satisfies (6) in the sense of distribution.
For the initial data , we propose that the following functions are the global strong solution of (6): For the smooth function sequence which converges to in as , we consider sequence of classical solutions of (6). First of all, we estimate the difference of . Using the representation (16) and (21), we have where we used for , . The norm of the first term (A) can be treated as follows: To estimate the norm of the second term (B), we consider the following three cases. (i)For the case , we have with (ii)For the case , we have with (iii)For other cases , we have with where
Now, we are ready to prove that is a global strong solution to the Cauchy problem (6). It is easy to check (18) by considering the representation (21). Making use of the classical solution to (6), we have Considering (23)–(26) with , the first integral (1) can be estimated as follows: where The integral (2) can be bounded easily: To take care of the integral (3), we decompose the integral by using representation (16) and (21) with the notation : where we understand that the integral (I) does not exist for and the integral (II) for .
Changing variables and , the first integral (I) can be bounded as follows: where we note that . We also have where we denote where .
For the integral (II), we consider two cases. Note that the integral (II) disappears if . For the first case , we have where because and case is excluded. For the second case , we have where we have
The integral (III) can be treated in a similar way to (II). For the case , we have For the case , we have
Taking into account for , , the integral (IV) can be bounded for the case : where supremum is taken over and . Then, we obtain For the case , we have Considering , we can bound (IV) as follows. For the case , we have For the case , we have For other cases ( and ), we finally have Then, we have as which implies that (19) holds.
Proof of Theorem 3
Here, we prove Theorem 3 which shows the uniqueness of weak solutions to (6). Before proving Theorem 3, we introduce a lemma given in  for easy reference. Consider the following equations on the time interval :
Lemma 6. Suppose that and satisfy (46) in the sense of distribution. Then, one has
Now let and be two weak solutions of (6) with the same initial data. We define . Then, we have equations for : Multiplying by complex conjugates, respectively, and taking the real parts, we obtain where we note that . Applying Lemma 6 to (49) and considering , we have where we understand .
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0015866) and also partially supported by the TJ Park Junior Faculty Fellowship.
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