On Initial Data Problem for a Periodic Two-Component b-Family System

This paper is concerned with the initial date problem for a periodic two-component b-family system. We prove that the solution map of the Cauchy problem of the b-family system is not uniformly continuous in Hs(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s({\mathbb {S}})$$\end{document}, s>5/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>5/2$$\end{document}.


Introduction
In this paper, we study the periodic case of following nonlinear dispersive system [1] (1) where m = u − u xx . Lin-Yin [2] obtained the local well-posedness of system (1). Zou [3] considered the system (1) and obtained some properties of the solutions. (1) is also considered by Zhu and Xu [4] in Besov space. If ≡ 0 , system (1) will become a number of the b-family equation where c 0 , b and are constants. For any b ≠ −1 , (2) can be derived as the family of asymptotically equivalent shallow water wave equation that emerge at quadratic order accuracy by an appropriate Kodama transformation, see [5,6] for the details. For b = −1 , the corresponding Kodama transformation is singular and the asymptotic ordering is violated, see [5,6]. If b = 2 and = 0 , (2) will become the Camassa-Holm equation [7]. The authors in [8][9][10][11] studied the Cauchy problem of the Camassa-Holm equation. If b = 3 and c 0 = = 0 , similarly, (2) will become the Degasperis-Procesi equation [12], which has been studied by [2,13].
As for Camassa-Holm equation, it is more important to study the properties of solutions, such as the persistence properties and unique continuation of solutions, non-uniform dependence on initial data, see [17][18][19][20][21][22][23] for details. In this paper, we will consider the non-uniform dependence on initial data to periodic system (1), that is , and ̄ follows from the pointwise density by convolution against the Green's function of the Helmholtz operator 1 − 2 x (cf. [14]). If =̄−̄0 , then system (1) is called as modified Camassa-Holm system. We remark that there is significant difference between system (1) and (3) with = − xx . It is easy to see that when = − xx , there are some similar properties between the two equations in system (3). Thus the proof of non-uniform dependence on initial data to system (3) with = − xx is similar to the signal equation, for example, Camassa-Holm equation. But in system (1), and u have different properties, see Theorem 2.1. This needs construct different asymptotic solution, see Sect. 3. On the other hand, in order to obtain the non-uniform dependence on initial data to system (1), we must modify the earlier method used in [19,22]. In papers [19,22], they estimated the H 1 -norm of the difference between the approximate and actual solutions, but it does not work for system (1). It is easy to see that the method used in [19,22] is available only when k 1 = 2 and k 2 = k 3 . In this paper, there is not any restriction about the parameters k 1 , k 2 and k 3 . Comparing with [24], there are some difference. Firstly, in [24], we considered the whole space, which is different from the periodic case. And so the approximate solutions constructed in this paper is different from [24]. Secondly, we take different norm from [24], see [24,Theorems 3.1 and 4.1]. Lastly, the model considered in [24] is a special case of (1). This paper is organized as follows. In Sect. 2, we obtain the well-posedness of system (1) by using Kato's theory and then use it to prove the basic energy estimate from which we derive a lower bound for the lifespan of the solution as well as an estimate of the H s ( ) × H s−1 ( ) norm of the solution (u(t, x), (t, x)) in terms of H s ( ) × H s−1 ( ) norm of the initial data (u 0 , 0 ) , where = (0, 1) . In Sect. 3, we construct approximate solutions, compute the error and estimate the H -norm of this error. In Sect. 4, we estimate the difference between approximate and actual solutions, where the exact solution is a solution to system (1) with initial data given by the approximate solutions evaluated at time zero. The non-uniform dependence on initial data for system (1) is established in Sect. 5 by construct two sequences of solutions to (1) in a bounded subset of the Sobolev space H s ( ) × H s−1 ( ) , whose distance at the initial time is converging to zero while at any later time it is bounded below by a positive constant.

Local Well-Posedness
In this section we apply the Kato's theory [25] to establish the local well-posedness of , where [x] stands for the integer part of x. Then the operator Λ −2 can be expressed as where G(x) is the Green's function associated with Λ −2 . Hence (4) is equivalent to the following system In the rest of this paper, for convenience, we rewrite the above system as Let then the first two equations of system (5) can be replaced by One can mimic the method of [2] to verify that all the assumptions of the Kato's theory holds for system (5). Thus we have the following result.
Moreover, the solution depends continuously on the initial data, i.e. the mapping is continuous.
Next, we want to get the lower bounded of the maximal existence time T. Meanwhile, we will prove that there exists a positive constant T 0 such that the H s -norm of the solution will be dominated by the H s -norm of the initial data. The following lemmas are needed.
where C is a positive constant depending only on r.
Lemma 2.2 [26] If r > 0, then where C is a positive constant depending only on r.
Due to the proof of the following is similar to that in [23], we only give the outline of proof.
is a solution of system (5) with initial data z 0 described in Theorem 2.1, then the maximal existence time T satisfies where C s is a constant depending only on s. Also, we have Proof Similar to [23], it is not hard to get that Integrating (8) from 0 to t, we have The above inequality implies that . It follows from Theorem 2.2 that

Approximate Solutions
In this section we first construct a two-parameter family of approximate solutions by using a similar method to [20], then compute the error and last estimate the H -norm of the error ( 1 < < 3 2 ). Throughout this paper, we assume that s > 5∕2 and 1 < < s − 1.
Following [20], our approximate solutions u , = u , (t, x) and , = , (t, x) to (5) will be consist of a low frequency and a high frequency part, i.e.
where is in a bounded set of and is in the set of positive integers ℤ + .
Direct calculation shows that and Similarly, we have Let C be a generic positive constant. For any positive quantities P and Q, we write P ≲ Q (P ≳ Q) means that P ≤ CQ (P ≥ CQ) in the following. Next, we estimate the error. We need the following lemma.

The above relation also holds if cos( x − ) is replaced by sin( x − ).
Estimating the H -norm of E 1 . By Lemma 3.1 and ≤ s − 1 , we have Estimating the H -norms of E 2 -E 6 . Also, we have Similarly, we can estimate the H −1 -norm of F 1 and F 2 Collecting all error estimates together, we have the following theorem.

Difference Between Approximate and Actual Solutions
In this section, we will estimate the difference between the approximate and actual solutions.
Let (u , (t, x), , (t, x)) be the solution to system (5) with initial data the value of the approximate solution (u , (t, x), , (t, x)) at time zero, that is, (u , (t, x), , (t, x)) satisfies Note that (u , (0, x), , (0, x)) ∈ H s × H s−1 , s ≥ 0 . Moreover, we have Therefore, if s > 5 2 , by using Theorems 2.1 and 2.2, we have that for any in a bounded set and ≫ 1 , problem (11)   Proof Note that Applying the operators Λ and Λ −1 to both sides of the first and second equations of system (13), respectively, and integrating them, we have Applying the Cauchy-Schwarz inequality in the first integral of right side of (16) and (17), we get In order to estimate the other terms in (16) and (17), we need the following lemma. In [20], the authors obtained that where 3∕2 < < s . We remark that, in paper [20], the authors considered the periodic Camassa-Holm equation, but it is no difference between the periodic and line. Next, we estimate the third term of right side of (16 where satisfies 3∕2 < < s − 1 . We remark there exists satisfying 3∕2 < < s − 1 because the index s considered here is strictly lager than 5/2. Integrating by part and using Lemma 4.1, we have Substituting (18)- (20) into (16) and (17), and adding the resulting equations, we get Note that we have By (9), we have where z , (t) = (u , (t), , (t)) . Since z , (0) = z , (0) , we get Letting → ∞ in the above inequality, we have Summing inequalities (31) and (32) up, it yields inequality (24). This completes the proof of this theorem. ◻

Conclusion
In this paper, we considered the initial date problem of a periodic two-component b-family system. The solution map of the Cauchy problem of the b-family system is not uniformly continuous in H s ( ) , s > 5∕2 , is obtained. Comparing with the earlier results, we studied a more general shallow water equation. By constructing the approximating solutions, we prove the solution map is continuous but not uniformly continuous in H s .
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Author Contributions All authors contributed to the writing of the present article and they read and approved the final manuscript.
Funding This work was supported by NSFC of China grants 11901158, and the Startup Foundation for Introducing Talent of NUIST.

Conflict of Interest
The authors declare that they have no competing interests.
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