Nonuniform Dependence on Initial Data of a Periodic Camassa-Holm System

and Applied Analysis 3 the time interval [0, T 0 ] theH-norm of the solution z(t, x) is dominated by the H-norm of the initial data z 0 (x). In order to do this, we need the following lemmas. Lemma 2 (see [29]). If r > 0, then Hr ∩ L∞ is an algebra. Moreover, 󵄩󵄩󵄩󵄩fg 󵄩󵄩󵄩󵄩Hr ≤ C ( 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Hr + 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩Hr 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩∞ ) , (8) where C is a positive constant depending only on r. Lemma 3 (see [29]). If r > 0, then 󵄩󵄩󵄩󵄩[Λ r , f] g 󵄩󵄩󵄩󵄩2 ≤ C ( 󵄩󵄩󵄩󵄩fx 󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩 Λ r−1 g 󵄩󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩Λ r f 󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩∞ ) , (9) where C is a positive constant depending only on r. Theorem4. Let s > 5/2. If z = (u, ρ) is a solution of system (5) with initial data z 0 described in Theorem 1, then the maximal existence time T satisfies

Recently, some properties of solutions to the Camassa-Holm equation have been studied by many authors.Himonas et al. [15] studied the persistence properties and unique continuation of solutions of the Camassa-Holm equation.They showed that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spatial derivative, must be identically equal to zero if it also decays exponentially at a later time; see [11,22] for the same properties of solutions to other shallow water equations.Just recently, Himonas and Kenig [16] and Himonas et al. [14,17] considered the nonuniform dependence on initial data for the Camassa-Holm equation on the line and on the circle, respectively.Lv et al. [25] obtained the nonuniform dependence on initial data for - equation.Lv and Wang [26] considered the system (1) with  =  −   and obtained the nonuniform dependence on initial data.
In this paper, we will consider the nonuniform dependence on initial data to system (1).We remark that there is significant difference between system (1) and system (1) with  =  −   .It is easy to see that when  =  −   , there are some similar properties between the two equations in system (1).Thus the proof of nonuniform dependence on initial data to system (1) with  =  −   is similar to the single equation, for example, Camassa-Holm equation.But in system (1),  and  have different properties; see Theorem 1.This needs constructing different asymptotic solution; see Section 3.
This paper is organized as follows.In Section 2, we recall the well-posedness result of Hu and Yin [27] and 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   (S) ×  −1 (S) norm of the solution ((, ), (, )) in terms of   (S) ×  −1 (S) norm of the initial data ( 0 ,  0 ).In Section 3, we construct approximate solutions, compute the error, and estimate the  1 -norm of this error.In Section 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 nonuniform dependence on initial data for system (1) is established in Section 5 by constructing two sequences of solutions to system (1) in a bounded subset of the Sobolev space   (S), whose distance at the initial time is converging to zero while at any later time it is bounded below by a positive constant.During preparing our paper, we find another paper [28] where the same problem has been considered, but our method is different from theirs.
Notation.In the following, we denote by * the spatial convolution.Given a Banach space , we denote its norm by ‖ ⋅ ‖  .Since all space of functions are over S, for simplicity, we drop S in our notations of function spaces if there is no ambiguity.Let [, ] =  −  denote the commutator of linear operator  and ; see [29,30]
Next, we will give an explicit estimate for the maximal existence time .Also, we will show that at any time  in the time interval [0,  0 ] the   -norm of the solution (, ) is dominated by the   -norm of the initial data  0 ().In order to do this, we need the following lemmas.
) is a solution of system (5) with initial data  0 described in Theorem 1, then the maximal existence time  satisfies where   is a constant depending only on .Also, we have Proof.The derivation of the lower bound for the maximal existence time (10) and the solution size estimate ( 11) is based on the following differential inequality for the solution : Suppose that (12) holds.Then, integrating (12) from 0 to , we have It follows from the above inequality that Now we prove inequality (12).Note that the products   and   are only in  −1 if , ∈   .To deal with this problem, we will consider the following modified system: where for each  ∈ (0, 1] the operator   is the Friedrichs mollifier defined by Here   () = (1/)(/), and Applying the operator Λ  and Λ −1 to the first and second equations of (15), respectively, then multiplying the resulting equations by Λ     and Λ −1   , respectively, and integrating them with respect to  ∈ S, we obtain We estimate the right-hand sides of ( 17) and ( 18), and we will use the fact that Λ  and   are commutative and To estimate the first integrals in the right-hand sides of ( 17) and (18) we write them as follows: Combining ( 21)-( 24), we have For the second integral in the right-hand side of (17), we have where we have used Lemma 2 with  = −1.Similarly, for the second and third integrals in the right-hand side of (18), we get Submitting ( 25), (27), and ( 26), ( 28) into ( 17) and ( 18), respectively, we obtain

Approximate Solutions
In this section we first construct a two-parameter family of approximate solutions by using a similar method to [17] and then compute the error and last estimate the  1 ×  2 -norm of the error.Following [17], our approximate solutions  , =  , (, ) and  , =  , (, ) to (5) will consist of a low frequency and a high frequency part, that is, where  is in a bounded set of S and  is in the set of positive integers Z + .Now we compute the error.Substituting the approximate solution ( , ,  , ) into the first and second equation of ( 5), we get the following error: Direct calculation shows that Similarly, we have Let  be a generic positive constant.For any positive quantities  and , we write  ≲  ( ≳ ) meaning that  ≤  ( ≥ ) in the following.
Next, we estimate the error.We remark that the error of the periodic Camassa-Holm equation contains   ( = 1, 2, 3) and the estimate of   was contained in paper [17].In [17], they obtained that Now, we estimate  5 and   ( = 1, 2).We need the following lemma.
Estimating the  1 -Norm of  5 .By using the definition of Λ, we have where we used Lemma 5.
Estimating the  2 -Norms of  1 and  2 .Also, we have Collecting all error estimates together, we have the following Theorem.