Research Article | Open Access
Nonuniform Dependence on Initial Data of a Periodic Camassa-Holm System
This paper is concerned with some properties of a periodic two-component Camassa-Holm system. By constructing two sequences of solutions of the two-component Camassa-Holm system, we prove that the solution map of the Cauchy problem of the two-component Camassa-Holm system is not uniformly continuous in (), .
In this paper, we consider the Cauchy problem of the following two-component periodic Camassa-Holm system: where . The Camassa-Holm equation can be obtained via the obvious reduction .
The Camassa-Holm (CH) equation has been extended to a two-component integrable system (CH2) by combining its integrability property with compressibility, or free-surface elevation dynamics in its shallow-water interpretation [1, 2]; that is, where and . Local well-posedness of system (2) with was obtained by [1, 3]. The precise blow-up scenarios and blow-up phenomena of strong solution for system (2) was established by [1, 3–6]. Just recently, Gui and Liu  studied system (1) with in Besov space and they obtained the local well-posedness. In this paper, we consider the Cauchy problem of system (1) and study some properties of it.
If , then system (2) becomes the well-known Camassa-Holm equation . In the past decade, the Camassa-Holm equation has attracted much attention because of its integrability and the existence of multipeakon solution; see [4, 8–22] for the details. The Cauchy problem and initial boundary value problem of the Camassa-Holm equation have been studied extensively [10, 23]. It has been shown that the Camassa-Holm equation is locally well-posedness  for initial data , . Moreover, it has global strong solutions [10, 18] and finite time blow-up solutions . On the other hand, it has global weak solution in [8, 9, 13, 19]. The advantage of the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the Camassa-Holm equation has peaked solutions and models wave breaking (i.e., the solution remains bounded while its slope becomes unbounded in finite time [8, 10, 24]).
Recently, some properties of solutions to the Camassa-Holm equation have been studied by many authors. Himonas et al.  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  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.  obtained the nonuniform dependence on initial data for - equation. Lv and Wang  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  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 norm of the solution in terms of norm of the initial data . In Section 3, we construct approximate solutions, compute the error, and estimate the -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 , 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  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 , for simplicity, we drop in our notations of function spaces if there is no ambiguity. Let denote the commutator of linear operator and ; see [29, 30] for the details. Set , where .
2. Local Well-Posedness
Let . Then the operator acting on can be expressed by its associated Green's function , where stands for the integer part of , as Hence (1) is equivalent to the following system: In the rest of this paper, we will consider the following system:
The following result is obtained by Hu and Yin .
Theorem 1 (see ). Given , . Then there exists a maximal existence time and a unique solution to system (5) such that Moreover, the solution depends continuously on the initial data; that is, the mapping is continuous.
Next, we will give an explicit estimate for the maximal existence time . Also, we will show that at any time in the time interval the -norm of the solution is dominated by the -norm of the initial data . In order to do this, we need the following lemmas.
Lemma 2 (see ). If , then is an algebra. Moreover, where is a positive constant depending only on .
Lemma 3 (see ). If , then where is a positive constant depending only on .
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 to , we have
It follows from the above inequality that is finite if . Let , then, for , we have
Now we prove inequality (12). Note that the products and are only in if ,. To deal with this problem, we will consider the following modified system: where for each the operator is the Friedrichs mollifier defined by Here , and is a function supported in the interval such that . Applying the operator and to the first and second equations of (15), respectively, then multiplying the resulting equations by and , respectively, and integrating them with respect to , we obtain We estimate the right-hand sides of (17) and (18), and we will use the fact that and are commutative and To estimate the first integrals in the right-hand sides of (17) and (18) we write them as follows: Using Lemma 3 and (19), we can estimate the first part in the right-hand sides of (20) where we use the fact that . Noting that which is obtained by Himonas and Kenig (see [16, Lemma 2]), and integrating by parts, we obtain 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 . 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 Consequently, Then, letting go to , we have or Since , using Sobolev's inequality we have that From (32) we obtain the desired inequality (12). This completes the proof of Theorem 4.
Recall that , where . It follows from Theorem 4 that
3. Approximate Solutions
In this section we first construct a two-parameter family of approximate solutions by using a similar method to  and then compute the error and last estimate the -norm of the error.
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.
Now, we estimate and . We need the following lemma.
Lemma 5 (see ). Let . If and then The above relation also holds if is replaced with .
Estimating the -Norm of . By using the definition of , we have where we used Lemma 5.
Estimating the -Norms of and . Also, we have Collecting all error estimates together, we have the following Theorem.
Theorem 6. Let . If is bounded, then for we have
4. Difference between Approximate and Actual Solutions
In this section, we will estimate the difference between the approximate and actual solutions.
Let be the solution to system (5) with initial data of the value of the approximate solution at time zero; that is, satisfies Note that , . Moreover, we have Therefore, if , by using Theorems 1 and 4, we have that, for any in a bounded set and , problem (44) has a unique solution with
To estimate the difference between the approximate and actual solutions, we let Then satisfies where and are defined as in Section 3.
Now we prove that the -norm of difference decays.
Theorem 7. Let ; then
Proof. Note that
Applying the operator to both sides of the first equations of system (48), we have
Substituting (52) and (53) into (50) and (51), respectively, we obtain
It is direct to calculate that
Substituting the above equalities into (54) and adding the resulting equations, we get
We first look at the last term . Integrating by parts gives
Estimates of Integrals and . Integrating by parts and applying the Cauchy-Schwarz inequality, we have
Estimates of Integrals . Integrating by parts, we get and estimate its first part by Its second part can be estimated by For the last part, integrating by parts, we obtain
Estimates of Integrals and . Integrating by parts, we can deal with the integral : Similarly, we can estimate the term ,
Estimate of the Integral . Integrating by parts, we have Combining the estimations of , we have that Note that we have . Furthermore, we have Submitting (69) into (67) and using Theorem 6, we obtain which implies that Noting that combining the above inequality, we complete the proof of Theorem 7.
5. Nonuniform Dependence
Theorem 8. If , then the data-to-solution for system (5) is not uniformly continuous from any bounded subset of into , where and . More precisely, there exist two sequences of solutions and to the differential equations of (5) in such that
Proof. Let and , where and are the unique solution to Cauchy problem (44) with initial data and , respectively.
It follows from Theorem 1 that these solutions belong in . By (48) and the assumptions after Theorem 1, we see that is independent of . Letting and using estimate (34), we have where and . If is large enough, then from Lemma 5 we have which gives Combining (75) with (77), we obtain Estimate (77) together with (78) yields Theorem 7 implies that Now, applying the interpolation inequality with and and using estimates (79) and (80), we get Hence where .
Next, we prove (73) and (74). From (44), we have