Global Well-Posedness for Coupled System of mKdV Equations in Analytic Spaces

Zennir, Khaled; Boukarou, Aissa; Alkhudhayr, Rehab Nasser

doi:https://doi.org/10.1155/2021/6614375

Journal of Function Spaces

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Approximation Methods: Theory and Applications

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Research Article | Open Access

Volume 2021 | Article ID 6614375 | https://doi.org/10.1155/2021/6614375

Global Well-Posedness for Coupled System of mKdV Equations in Analytic Spaces

Khaled Zennir,^1,2Aissa Boukarou,³and Rehab Nasser Alkhudhayr⁴

Academic Editor: Ioan Rasa

Received18 Dec 2020

Revised22 Jan 2021

Accepted11 Feb 2021

Published05 Mar 2021

Abstract

The main result in this paper is to prove, in Bourgain type spaces, the existence of unique local solution to system of initial value problem described by integrable equations of modified Korteweg-de Vries (mKdV) by using linear and trilinear estimates, together with contraction mapping principle. Moreover, owing to the approximate conservation law, we prove the existence of global solution.

1. Introduction and Main Results

For an effective approach to solving problems arising in modern science and technology, one cannot do without researching nonlinear problems of mathematical physics. The rapid development of new technology and the emergence of its high speed allow researchers to build and consider increasingly complex multidimensional models describing various phenomena, which are modeled, as a rule, using nonlinear partial differential equations (systems). However, now it has become clear that without the development of analytical methods, it is impossible to get a complete idea of the essence of the phenomenon. Analytical methods provide not only a reliable tool for debugging and comparing various numerical methods but also sometimes anticipate some scientific discoveries, make it possible to study the properties of models, to detect the presence of certain effects as a result of the existence or nonexistence of objects (solutions) with the required properties. Therefore, at present, fundamental research is being intensively carried out aimed at proving theorems of existence, uniqueness, and regularity of solutions of nonlinear partial differential equations.

In the present paper, a coupled system of modified Korteweg-de Vries equations is considered as follows:

The dynamics of solutions in the Korteweg-de Vries equations (KdV) and the modified Korteweg-de Vries equations (mKdV) are well studied due to the complete integrability of these equations (see [1–6]). For KdV equations, the studies date back to the 1970s, although some results have been obtained very recently (please see [7]). We extend the results in [7] and consider a coupled system of mKdV-type equations on the line in Equation (1).

For mKdV equations, many problems have been studied. It is proved that the mKdV equation is locally [8] and globally [9] well-posed in for . Global well-posedness in is shown in [10].

For , the author in [7] proved that the IVP (Equation (1)) is locally well-posed for the given data , . Oh in [11] used the Fourier transform restriction norm method and proved that the next IVP is locally well-posed for data with regularity .

For , the system (Equation (1)) reduces to a special case of a broad class of nonlinear evolution equations considered by Ablowitz et al. [12] in the inverse scattering context. In this case, the well-posedness issues along with existence and stability of solitary waves for this system are widely studied in the literature, using the technique developed by Kenig et al. in [13, 14].

Well-posedness for the nonperiodic gKdV equation in spaces of analytic functions has been proved by Grujic and Kalisch [15].

A class of suitable analytic functions for our analysis is the analytic Gevrey class introduced by Foias and Temam [16], defined as follows: for and with . For , the space coincides with the standard Sobolev space . For all and , we have which is the embedding property of the Gevrey spaces.

New minimal conditions are used to show the local well-posedness of solution by using linear and trilinear estimates, together with contraction mapping principle. By imposing a more appropriate conditions with the help of the approximate conservation law, we obtain an unusual global existence result in Gevery spaces.

Proposition 1 (Paley-Wiener Theorem) [17]. Let , . Then, if and only if it is the restriction to the real line of a function which is holomorphic in the strip and satisfies

Remark 2. In the view of the Paley-Wiener Theorem, it is natural to take initial data in , to obtain the best behavior of solution and may be extended to be globally in time. It means that given for some initial radius , we then estimate the behavior of the radius of analyticity over time.

The first main result on local well-posedness of Equation (1) in analytic spaces reads as follows.

Theorem 3. Let and . Then for any , there exists and unique solution of Equation (1) on such that Moreover, the solution depends on , where Furthermore, the solution satisfies the following: with constant depending only on and .

An effective method for studying lower bounds on the radius of analyticity, including this type of problem, was introduced in [18] for 1D Dirac-Klein-Gordon equations. It was applied in [19] to the modified Kawahara equation and in [20] to the nonperiodic KdV equation (for more details, please see [20–23]).

The second result for the problem (Equation(1)) is given in the next theorem.

Theorem 4. Let , and . Assume that , then the solution in Theorem 3 can be extended to be global in time and for any , we have the following: with where can be taken arbitrarily small and is a constant depending on , and .

The third result is Gevrey’s temporal regularity of the unique solution obtained in the Theorem 3. A nonperiodic function is the Gevrey class of order , i.e., , if there exists a constant such that if is analytic.

Here, we will show that for , for every and , there exist such that i.e., in spacial variable and in time variable. Also,

where Equations (13) and (14) do not hold for.

Theorem 5. Let ,, and . If , then the solution given by Theorem 4 belongs to the Gevrey class in time variable. Furthermore, it is not belong to , in .

The proof of Theorem 5 is similar to that in [1].

The paper is organized as follows. In Section 2, we define the function spaces and linear and trilinear estimates. In Section 3, we prove Theorem 3, using the linear and trilinear estimates, together with contraction mapping principle. In Section 4, we prove the existence of fundamental approximate conservation law. In the last section, Theorem 4 will be proved using the approximate conservation law.

2. Preliminary Tools and Analytic Function Spaces

2.1. Function Spaces

We define the analytic Bourgain spaces related to the modified Korteweg-de Vries type equations. The completion of the Schwartz class is given by , for , , subjected to the norm:

We often use without mention, the definition , where

For any interval , we define the localized spaces with norm:

2.2. Linear Estimates

We have the trilinear estimate (Equations (15) and (16)) defined in the analytic Bourgain spaces. Since the spaces is continuously embedded in , provided .

Lemma 6. Let , , and . Then, for all , we have the following:

Proof. First, we note that the operator defined by satisfies where is introduced in [7]. We observe that belongs to and for some , we have the following: Thus, it follows that and

Taking the Fourier transform with respect to of the Cauchy problems (Equation (1)), after an ordinary calculation, we localize in by using a cut-off function, satisfying , with in , supp, and . We consider the operator given by the following: where and are the unitary groups associated with the linear problems.

The nonlinear terms defined by and will be treated in the next lemmas.

Lemma 7. Let and . For some constant , we have the following: for all .

Proof. By definition, we have the following: It follows that Since , we have the following:

Lemma 8. Let , , , and , then for some constant , we have the following:

Proof. Define We have, by Equation (19), the following: Thus, Owing to Lemma 6 in [7], we get the following:

This completes the proof.

Lemma 9. Let be a Schwartz function in time, , and . If , then for any , we have the following: where depends only on and .

Proof. The proof of Lemma 9 for can be found in Lemma 13 of [14], for as one merely has to replace by , where the operator is defined in Equation (19).

Lemma 10 [20]. Let , , , and . Then, for any time interval , we have the following: where is the characteristic function of and depends only on .

2.3. Trilinear Estimates

We have the trilinear estimate in the following lemmas.

Lemma 11. Let , , , and be as in Lemma 8. Then,

Proof. We observe, by considering the operator in (19), that since .
Then, Thanks to Proposition 2.3 of [7], for some , we have the following:

This completes the proof.

3. Proof of Theorem 3

3.1. Existence of Solution

We estimate terms in Equation (23). For this end, we define and , with norms and similar for

Lemma 12. Let , , and . Then, for all and , with some constant , we have the following: for all .

Proof. To prove estimate of Equation (39), we have the following: Therefore, from Equations (41) and (42), we obtain the following: For the estimate of Equation (40), we observe that where

We will show that is a contraction on the ball to , where is given in Equation (46).

Lemma 13. Let , , and . Then, for all , such that the map is a contraction, where is given by the following: with .

Proof. From Lemma 12, for all , we have the following: We choose sufficiently small such that ; hence, Thus, maps into , which is a contraction, since for all . Hence, is a contraction.

3.2. The Uniqueness

Uniqueness of the solution in can be proved by the following standard argument.

Suppose that are solutions to Equation (1) with in . Setting and , we see that solves the Cauchy problem:

Thus, by Equation (50), we have the following: since we have the following:

Thanks to Equation (53), we have the following:

Integrating by parts of the last integral, we obtain the following: from which we deduce the inequality as follows:

Since , we have that and are continuous in on the compact set and are in . Thus, we can conclude that

Therefore, from Equations (56) and (57), we obtain the differential inequality:

Solving it gives the following:

Since , from Equation (59), we obtain that , , or .

Now by Equation (51), we have the following:

Solving it gives the following:

Since , from Equation (61), we obtain that , , or .

3.3. Continuous Dependence of the Initial Data

To prove continuous dependence of the initial data, we will prove the following.

Lemma 14. Let , , and . Then, for all , if and are two solutions to Equation (1) corresponding to initial data and , we have the following:

Proof. If and are two solutions to Equation (1), corresponding to initial data and , we have from Lemma 6 as follows: By taking and , we have the following: Thus, Then,

This completes the proof of Theorem 3.

4. Approximate Conservation Law

We have the following: which is conserved for a solution of Equation (1). We are going to show an approximate conservation law for a solution to Equation (1) based on the conservation of the norm of solution.

Theorem 15. Let and , be as in Theorem 3 with ; there exist and , such that for any and any solution to the Cauchy problem (Equation (1)) on the time interval , we have the estimate: Moreover, we have the following:

We need the following estimate.

Lemma 16. Given , there exist , , and ; we have the following: where , , and the operator is given by Equation (19).

Proof. Let . Then, We shall calculate the Fourier transform of as follows: Now using Corollary 7.3 in [21], let ; we have the following: For , one can see that Now by taking , we obtain the following: Then, Now let . Then, By Equations (76) and (77), we have the following:

Proof (Theorem 15). Let which are real-valued since the multiplier is even and are real-valued. Applying to Equation (1), we obtain the following: where and .
By multiplying both sides of Equation (79) by and Equation (80) by and integrating with respect to space variable, we get the following: Then, Noting that as (see [20]), we use integration by parts to obtain the following: Integrating the last equality with respect to , we obtain the following: Thus, By using Holder’s inequality, Lemma 10, Lemma 9, and the fact that Since , we obtain the following: where for ; we can conclude from Lemma 16: Therefore, Finally, by using Equation (8), we conclude that

5. Proof of Theorem 4

Let , , and be fixed, and . Then, we have to prove that the solution of Equation (1) satisfies the following: where and is a constant depending on , and . By Theorem 3, there is a maximal time , such that

If , it is done.

If , as we assume henceforth, it remains to prove the following:

5.1. The Case

Fixed ; we will show that, for , sufficiently small

In this case, by Theorem 3 and Theorem 15 with the smallness conditions on will be

Here, is the constant in Theorems 15.

By induction, we check that for , where is chosen so that . This does exist; by Theorem 3 and the definition of , we have the following:

In the first step, we cover the interval , and by Theorem 15, we have the following: since ; we used the following:

This satisfies Equation (98) for , and Equation (99) is following and using again as well as the following:

Suppose now that Equations (98) and (99) hold for some and we prove that it holds for , we estimate the following: satisfying Equation (98) with replaced by . To get Equation (99) with replaced by , it is then enough to have the following: but this holds by Equation (97), since .

Finally, Equation (97) is satisfied for such that

Thus, , where

5.2. The General Case

For all , by Equation (4), we have .

For case , it is proved that there is a , such that where depends on , and .

Applying again the embedding Equation (4), we now conclude that which imply Equation (94). The proof of Theorem 4 is now completed.

Data Availability

No data were used in this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

Rehab Nasser Alkhudhayr would like to thank the Deanship of Scientific Research, Qassim University, for funding the publication of this project.

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Copyright © 2021 Khaled Zennir et al. 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.

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