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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 410981, 17 pages
The Cauchy Problem for a Dissipative Periodic 2-Component Degasperis-Procesi System
1School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
2Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
Received 28 March 2014; Accepted 16 June 2014; Published 23 July 2014
Academic Editor: Sazzad Hossien Chowdhury
Copyright © 2014 Sen Ming 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.
The dissipative periodic 2-component Degasperis-Procesi system is investigated. A local well-posedness for the system in Besov space is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking criterions for strong solutions to the system with certain initial data are derived.
We consider the following dissipative periodic 2-component Degasperis-Procesi system: where and are nonnegative constants, , with , and denotes the unit circle.
In system (1), if , we get the classical Degasperis-Procesi equation  where represents the fluid velocity at time in direction (or equivalently the height of water’s free surface above a flat bottom). The nonlinear convection term causes the steepening of the wave form. The nonlinear dispersion effect term makes the wave form spread.
Equation (2) has attracted many researchers to discover its dynamics properties [2–15]. For example, Degasperis et al.  proved the formal integrability by constructing a Lax pair. They showed that (2) has bi-Hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions which are analogous to the Camassa-Holm peakons. The asymptotic accuracy of (2) is the same as that of Camassa-Holm equation. Dullin et al.  showed that the Degasperis-Procesi equation can be derived from the shallow water elevation equation by an appropriate Kodama transformation. Lin and Liu  proved the stability of peakons for (2) under certain assumptions. In , Yin proved the local well-posedness for (2) with initial data () and also derived the precise blow-up scenarios for the solutions. The global existence of strong solutions and global weak solutions to (2) are studied in . Escher and Kolev  and Escher and Seiler  showed that the Degasperis-Procesi equation can be reformulated as a nonmetric Euler equation on the diffeomorphism group of the circle. Vakhnenko and Parkes  derived periodic and solitary wave solutions to (2). Lundmark and Szmigielski  investigated multipeakon solutions to (2). The shock wave solutions to (2) were obtained in . Although the Degasperis-Procesi equation is similar to the Camassa-Holm equation in many aspects, especially in the structure of equation, there are some differences between the two equations. One of the famous features of Degasperis-Procesi equation is that it not only has peakon solutions with  and periodic peakon solutions  but also has shock peakons  and periodic shock waves .
In general, it is difficult to avoid the energy dissipation mechanisms in a real world. Thus different types of solutions for the dissipative Degasperis-Procesi equation have been investigated. For example, Guo et al.  studied the dissipative Degasperis-Procesi equation where () is the dissipative term. They obtained the global existence of weak solutions. Wu and Yin  established blow-up solutions and analyzed the decay of solutions to (3). In , the authors studied the long time behavior of solutions to (3). Guo  established the local well-posedness, blow-up scenario, global existence of solutions, and persistence properties for strong solutions to (3).
On the other hand, many researchers have studied the integrable multicomponent generalizations of the Degasperis-Procesi equation [24–29]. For example, Yan and Yin  investigated the 2-component Degasperis-Procesi system where . They established the local well-posedness for system (4) in Besov space with and also derived the precise blow-up scenarios for strong solutions in Sobolev space with . Zhou et al.  investigated the traveling wave solutions of the 2-component Degasperis-Procesi system. Jin and Guo  established the local well-posedness, blow-up criterions and the persistence properties of strong solutions to the system in with .
Recently, a large amount of literature was devoted to the 2-component Camassa-Holm system [30–39]. For example, Hu  studied the dissipative periodic 2-component Camassa-Holm system where . The author not only established the local well-posedness for system (5) in Besov space with but also presented global existence of solutions and the exact blow-up scenarios of strong solutions in Sobolev space with . It was shown in  that the dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton, and Novikov equations can be reduced to their nondissipative versions by means of an exponentially time dependent scaling.
Motivated by the work in [20, 28, 32, 40–43], we study the dissipative periodic 2-component Degasperis-Procesi system (1). We note that the Cauchy problem of system (1) in Besov space has not been discussed yet. One of the difficulties is that we can not obtain the estimates for , which is a conserved quantity playing a key role in studying the blow-up phenomenon of the 2-component Camassa-Holm system [32, 33]. However, this difficulty has been dealt with by establishing the estimates for , where is the first component of solution to system (1). We state our main task with two aspects. Firstly, we establish the local well-posedness for system (1) in Besov space. Secondly, we present the precise blow-up criterions for strong solutions.
We rewrite system (1) as where the operator . We write the space with , , , .
The main results of this paper are presented as follows.
Theorem 1. Let , , , and . Then there exists a time such that the Cauchy problem (1) has a unique solution . The map is continuous from a neighborhood of in into for every when and whereas .
Theorem 2. Let with and is the maximal existence time of corresponding solution to system (1). Then
Theorem 3. Let with and is the maximal existence time of corresponding solution to system (1). Then the solution blows up in finite time if and only if
Theorem 4. Let in system (1) and with . Assume that is odd, is even, , and . Then the corresponding solution to system (1) blows up in finite time. More precisely, there exists such that In addition, if with some satisfying , then there exists such that(i) if ;(ii) if ,where such that .
Theorem 5. Let in system (1) and with . Assume that and are odd, . Then the corresponding solution to system (1) blows up in finite time. More precisely, there exists such that In addition, the inequalities hold:(i) if ;(ii) if .
The remainder of this paper is organized as follows. In Section 2, several 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 Theorems 2, 3, 4, and 5 are presented in Section 4.
Notation. We denote the norm in Lebesgue space , , by , the norm in Sobolev space , , by , and the norm in Besov space , , by . Since functions in all the spaces are over , for simplicity, we drop in our notations if there is no ambiguity. We denote , where is a sufficiently small number.
Proposition 6 (see [44, 46]). There exists a couple of smooth functions valued in , such that is supported in the ball , and is supported in the ring . Moreover, Then, for all , we define the nonhomogeneous dyadic blocks as follows: Thus , which is called the nonhomogeneous Littlewood-Paley decomposition of .
Proposition 8 (see [44, 49]). Let , ; then consider the following.(1)Density: is dense in , .(2) Embedding: , if , . is locally compact if .(3) Algebraic properties: for all , is an algebra. is an algebra or and .(4) Complex interpolation: consider (5)Fatou’s Lemma: if is bounded in and in , then and (6)1-D Morse-type estimates.(i) For , (ii)For , ( if ), and , then (iii) In Sobolev space , for , we have (7)The lifting property: let and ; then if and only if
Lemma 9 (see ). Let , , . Assume , , and if or to otherwise. If satisfies the 1-D transport equation
where stands for a given time dependent vector field, and are known data. There exists a constant depending only on , , and such that the following statements hold.
(1) If or , or where
(2) If , , , , then where .
(3) If , then for all , (23) holds true with .
(4) If , then . If , then for all .
3. The Proof of Theorem 1
We finish the proof with two subsections.
3.1. Existence of Solutions
We use a standard iterative process to construct approximate solutions to system (6).
Step 1. Starting from , we define by induction a sequence of smooth functions satisfying where .
Step 2. Now we are in the position to prove that is uniformly bounded in .
According to Lemma 9, for all , one has We know if , then is an algebra. And if , then is an algebra. Moreover, combining (7) of Proposition 8 and one deduces Using (6) of Proposition 8 yields Therefore, from (29) to (32), one gets Let us choose a such that and Pluging (34) into (33) yields Therefore, is uniformly bounded in . From Proposition 8 and the embedding properties one obtains Thus, we conclude that and are uniformly bounded in . In the same way we have that and are uniformly bounded in . Using (28), one obtains that is uniformly bounded, which yields that is uniformly bounded in .
Step 3. We demonstrate that is a Cauchy sequence in .
In fact, according to (28), for all , one has
(1) For the case , firstly, we estimate the right side of (38). From (17) and (18), we obtain Secondly, we estimate the right side of (39). Using (18), one gets For all , it is deduced from Lemma 9 that Since is uniformly bounded in and there exists a constant independent of such that for all By induction, one obtains Since , are bounded independent of , there exists a new constant such that Consequently, is a Cauchy sequence in .
(2) For the case , using (4) of Proposition 8, one has where , , and where , .
One deduces that is a Cauchy sequence in for the critical case.
Step 4. We end the proof of existence of solutions.
Firstly, since is uniformly bounded in , according to Fatou’s Lemma in Besov space, it guarantees that belongs to .
Secondly, since is a Cauchy sequence in , it converges to limit function . An interpolation argument insures that the convergence holds in for any . Taking the limit in (28) derives that is indeed a solution to (6). Thanks to the fact , we know that the right side of the first equation in (6) belongs to , and the right side of the second equation in (6) belongs to . For the case , applying Lemma 9 derives for any .
3.2. Uniqueness and Continuity with Initial Data
Lemma 12. Let , , . Assume that and are two given solutions to the Cauchy problem (6) with initial data satisfying , and . Then, for all ,
Proof. Let ; then which derives that , and satisfies the transport equation where According to Lemma 9, one deduces Similar to the arguments in Step 3 in Section 3.1, one derives Applying Gronwall’s inequality completes the proof of Lemma 12.
Remark 13. For the critical case , the proof is similar to Step 3 in Section 3.1.
Remark 14. Note that, for every . The existence time of system (1) may be chosen independently of in the following sense . If is a solution to system (1) with initial data for some , then with the same time . In particular, if , then .
4. Wave-Breaking Phenomena
This section is devoted to investigating conditions of wave breaking mechanisms of strong solutions to system (1). Using Theorem 1 and a simple density argument, we deduce that the desired results are valid for . Here we take in the proof for simplicity. We begin with three lemmas.
Lemma 15 (see ). Let and . Then for all there exists at least one point , such that The function is absolutely continuous on with We consider the trajectory equation