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The Cauchy Problem for a Weakly Dissipative 2-Component Camassa-Holm System
The weakly dissipative 2-component Camassa-Holm system is considered. A local well-posedness for the system in Besov spaces is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking mechanisms and the exact blow-up rate of strong solutions to the system are presented. Moreover, a global existence result for strong solutions is derived.
We consider the following weakly dissipative 2-component Camassa-Holm system: where are constants, , are natural numbers, , and with .
In system (1), if , we get the classical Camassa-Holm equation : where is the fluid velocity in direction (or equivalently the height of the water's free surface above a flat bottom) and is a constant related to critical shallow water wave speed. The alternative derivation of (2) as a model for water waves can be found in Constantin and Lannes . Equation (2) models for the propagation of shallow water waves have attracted attention of many researchers with two remarkable features. The first one is the presence of solutions in the form of peaked solitary waves for . The peakon with , which is a feature observed for the traveling waves of largest amplitude [3–6]. The other feature is that the equation has breaking waves. In other words, the solutions remain bounded while their slope becomes unbounded in finite time. For , the solitary waves are stable solitons [3–5, 7]. Li and Olver  not only obtained the local posedness but also gave the conditions which could lead to some solutions blowing up in finite time in Sobolev space with . For other methods to establish the local well-posedness and global existence of solutions to the Camassa-Holm equation or other shallow water models, the reader is referred to [9–18] and the references therein.
In general, it is difficult to avoid the energy dissipation mechanisms in a real world. Thus, different types of solutions for dissipative Camassa-Holm equation have been investigated. For example, Wu and Yin  studied the dissipative Camassa-Holm equation: where is the dissipative term. They obtained the global existence result and blow-up results for strong solutions in Sobolev space with by Kato's theory. In , Lai and Wu also investigated the weakly dissipative Camassa-Holm equation: where are constants, are natural numbers, and is the weakly dissipative term. They obtained the local well-posedness in Sobolev space with by using the pseudoparabolic regularization technique and some estimates derived from the equation itself and also developed a sufficient condition which guaranteed the existence of weak solutions in Sobolev space with . We note that they only studied the equation in Sobolev space .
On the other hand, the Camassa-Holm equation also admits many integrable multicomponent generalizations [20–32]. For example, where . The above system was derived in  with . In , Constantin and Ivanov gave a rigorous justification of the system which is a valid approximation to the governing equation for shallow water waves for . The represents the horizontal velocity of the fluid the is related to the free surface elevation from equilibrium with boundary assumptions, and and as . They obtained the global well-posedness with small initial data and the conditions for wave-breaking mechanism to the system. For the case , it is not physical as it corresponds to gravity pointing upwards, and as . The blow-up conditions are discussed in . For , Gui and Liu  established the local well-posedness for the system in a range of Besov spaces with . They also derived wave-breaking mechanisms and the exact blow-up rate for strong solutions to the system in with . Tian et al.  obtained the local well-posedness for the system in Besov spaces with . They also derived wave-breaking mechanisms for solutions and a result of blow-up solutions with certain profile in spaces with . Yan and Yin  investigated the 2-component Degasperis-Procesi system which is similar to the 2-component Camassa-Holm system. They obtained the local well-posedness in Besov spaces and derived a precise blow-up scenario for strong solutions. Guan and Yin  presented a new global existence result and several new blow-up results of strong solutions to an integrable 2-component Camassa-Holm shallow water system.
In fact, the 2-component Camassa-Holm system also admits some generalizations due to the energy dissipation mechanisms in a real world. In , Chen et al. investigated the weakly dissipative 2-component Camassa-Holm system: where are constants and is a natural number. They investigated the local well-posedness for the system with initial data with by Kato's theory and derived a precise blow-up scenario for strong solutions to the system.
Motivated by the work in [9, 21, 23, 27, 33, 35–38], we study the weakly dissipative Camassa-Holm system (1). We note that the Cauchy problem of system (1) in Besov spaces has not been discussed yet. We state our main tasks with three aspects. Firstly, we establish the local well-posedness of solutions to the system (1). Secondly, we present the precise blow-up criterions and exact blow-up rate for strong solutions. At last, we derive a global existence result of strong solutions. Because of the presence of high order nonlinear terms and , the system (1) loses the conservation law which plays an important role in studying system (1).
Now we rewrite system (1) as where the operators and . We define the space: with , , and .
The main results of this paper are stated as follows. Firstly, we present the local well-posedness theorem.
Theorem 1. For and . Let . There exists a time such that the initial value problem (1) has a unique solution , and the map is continuous from a neighborhood of in into for every when and whereas .
We obtain the following blow-up results.
Theorem 2. Let with and be the maximal existence time of the solution to system (1) with initial data . Then, the corresponding solution blows up in finite time if and only if
Theorem 3. Let in system (1). Assume with and the initial value satisfies , where is a fixed constant defined in (95) and with the point defined by . Then, the corresponding solution to system (1) with blows up in finite time. Namely, there exists a with such that where such that .
Theorem 4. Let in system (1). Assume with and the initial value satisfies that is odd, is even, and , where is a fixed constant defined in (109) and . Then the corresponding solution to system (1) with blows up in finite time. More precisely, there exists a with such that In addition, if with some satisfying , then there exists a with such that where such that .
We also obtain the exact blow-up rate of strong solutions to system (1).
Theorem 5. Let be the maximal existence time of the corresponding solution to system (1) with . The initial data with satisfies , where is a fixed constant defined in (120) and with the point defined by ; then
Now we present a global existence result of strong solutions to system (1).
The remainder of this paper is organized as follows. In Section 2, some properties of Besov space and a priori estimates for solutions of transport equation are reviewed. Section 3 is devoted to the proof of Theorem 1. The proofs of wave-breaking results and the precise blow-up rate of strong solutions to system (1) are given in Section 4, respectively. The proof of Theorem 6 is presented in Section 5.
Notation. In this paper, we denote the convolution on , by the norm of Lebesgue space by and the norm in Sobolev space , by , and the norm in Besov space , by . Here we denote , where is a sufficiently small number.
Proposition 8 (see [40, 42]). Let , , , and , then; consider the following.(1)Embedding: , if and . And locally compact if .(2)Algebraic properties: for any is an algebra. is an algebra or and .(3)Complex interpolation:(4)Fatou’s Lemma: if is bounded in and in , then and(5)Let and be an -multiplier (i.e., is smooth and satisfies that for every there exists a constant such that for all ). Then, the operator is continuous from to .(6)Density: is dense in .
Lemma 9 (see ). Let be an open interval of . Let and be the smallest integer such that . Let satisfy and . Assume that has values in . Then, and there exists a constant depending only on such that
Lemma 10 (see ). Let be an open interval of . Let and be the smallest integer such that . Let satisfy and . Assume that has values in . Then, there exists a constant depending only on such that
Lemma 11 (see ). Assume that and ; the following estimates hold:(i)for , then (ii)for , if , and , then where are constants independent of .
Then we present two related lemmas for the following transport equations: where stands for a given time dependent vector field and and are known data.
Let us state the existence result for transport equation with data in Besov space.
Lemma 13 (see ). Let be as in the statement of Lemma 12 and and is a time dependent vector field for some , , such that if then ; if or , then . Thus transport equation (21) has a unique solution , and (22) holds true. If , then one has .
3. The Proof of Theorem 1
We finish the proof of Theorem 1 by the following steps.
3.1. Existence of Solutions
Step 1. Starting from , we define by induction a sequence of smooth functions solving the following transport equation: where Since all the data , Lemma 13 enables us to show that for all , the system (25) has a global solution which belongs to .
Step 2. Next, we prove that is uniformly bounded in .
According to Lemma 12, we have the following inequalities for all : From Proposition 8, Lemma 11, and -multiplier property of , it yields that Using the -multiplier property of , we get (i)By the assumption of and Lemma 9, we have (ii)As we know, if , then is an algebra; if , then is an algebra. By the assumption of and Lemmas 10 and 11, we deduce that Combining (i) and (ii) yields Thanks to Lemma 11, we get Therefore from (27) to (34), we obtain Let us choose a such that and Inserting (36) into (35) yields Therefore, is uniformly bounded in . From Proposition 8, Lemma 11, and the following embedding properties: we have We deduce that and are uniformly bounded in , in the same way that are uniformly bounded in . Using the system (25), we have is uniformly bounded, which derives that is uniformly bounded in .
Step 3. Now we demonstrate that is a Cauchy sequence in .
In fact, according to (25), we note that for all , we have (1) We estimate the terms in the right side of (40). By Lemma 11, we have We use Lemma 10 to estimate the high order terms in (40). Firstly, we get Secondly, by and Lemma 11, we deduce that Hence, (2) Now we estimate the right side of (41): Combining (1), (2), and Lemma 12 yields for all Since is uniformly bounded in and there exists a constant independent of such that for all , we get By induction, we obtain that Since are bounded independently of , we conclude that there exists a new constant such that Consequently, is a Cauchy sequence in .
Step 4. We end the proof of existence of solutions.
Firstly, since is uniformly bounded in , according to the Fatou property for Besov space, it guarantees that also belongs to