Journal of Applied Mathematics

Volume 2014 (2014), Article ID 410981, 17 pages

http://dx.doi.org/10.1155/2014/410981

Research Article

The Cauchy Problem for a Dissipative Periodic 2-Component Degasperis-Procesi System

Sen Ming,¹ Han Yang,¹ and Ls Yong²

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.

Abstract.

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.

1. Introduction

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 [1]

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. [2] 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. [3] showed that the Degasperis-Procesi equation can be derived from the shallow water elevation equation by an appropriate Kodama transformation. Lin and Liu [16] proved the stability of peakons for (2) under certain assumptions. In [17], 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 [18]. Escher and Kolev [4] and Escher and Seiler [5] showed that the Degasperis-Procesi equation can be reformulated as a nonmetric Euler equation on the diffeomorphism group of the circle. Vakhnenko and Parkes [7] derived periodic and solitary wave solutions to (2). Lundmark and Szmigielski [8] investigated multipeakon solutions to (2). The shock wave solutions to (2) were obtained in [9]. 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 [2] and periodic peakon solutions [18] but also has shock peakons [9] and periodic shock waves [19].

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. [20] studied the dissipative Degasperis-Procesi equation

where () is the dissipative term. They obtained the global existence of weak solutions. Wu and Yin [21] established blow-up solutions and analyzed the decay of solutions to (3). In [22], the authors studied the long time behavior of solutions to (3). Guo [23] 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 [28] 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. [27] investigated the traveling wave solutions of the 2-component Degasperis-Procesi system. Jin and Guo [25] 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 [40] 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 [41] 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.

2. Preliminary

This section is concerned with some basic facts in periodic Besov space and the theory of transport equation. One may check [33, 44–49] for more details.

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 7 (see [44, 46]). Let , , . The nonhomogeneous periodic Besov space is defined by , where

Moreover, the low frequency cut-off is defined as for all .

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 [46]). 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 .

Lemma 10 (see [46]). Let be defined as in Lemma 9. Assume for some ,  . if or ; and if . Then, (21) has a unique solution and (23) holds true. If , then .

Lemma 11 (see [32]). Let . Assume ,  , and . If satisfies (21), then , and there exists a constant depending only on such that the statements hold:

or

where .

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 .

Since all the data , Lemma 10 enables us to show that, for all , system (28) has a global solution which belongs to .

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 .

Finally, from (6), one has if , and in otherwise. Thus . A standard use of a sequence of viscosity approximate solutions for (6) which converges uniformly in gives the continuity of solution .

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 [50]. 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 [51]). 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

where denotes the first component of solution to system (1).

Lemma 16 (see [52]). Let with . Then (59) has a unique solution . Moreover, the map is an increasing diffeomorphism of for all and

Lemma 17. Let with and is the maximal existence time of corresponding solution to (6). Then for all

Moreover, if there exists such that for all , then for all ,

Proof of Lemma 17. Differentiating the left side of (61) with respect to , using (59) and the second equation in (1), we obtain

Applying Gronwall’s inequality and (59) yields (61).

From Lemma 16, (61) and the assumption in Lemma 17, one deduces

This completes the proof of Lemma 17.

In what follows we derive the estimates for .

Lemma 18. Let with and is the maximal existence time of corresponding solution to system (1). Assume that there exists such that , for all . Then for all , we have

where .

Proof of Lemma 18. As mentioned before, here we assume to prove Lemma 18. Let . Then we rewrite the first equation in (1) as

Noting or , one has

Combining the above three equalities, one derives

Using Gronwall’s inequality, we have

Thus

Noting

and using the assumption in Lemma 18, we obtain

Hence

Applying Gronwall’s inequality yields (65).

Now we present the proof of (66). Note that, for all , if , where denotes the integer part of , then for all . It follows from some calculations that is continuous and decreasing on interval and increasing on interval :  , , ≤ , and . Applying Young’s inequality, one has

Using (59), we obtain

For the first equation in system (6), using (65) and the facts above, one derives

It follows from Gronwall’s inequality that

From Lemma 16, we obtain (66).

Lemma 19. Let with and is the maximal existence time of corresponding solution to system (6). If , then for all , we have

where

Proof of Lemma 19. It follows from the proof of Lemma 17 that

Using similar arguments as in the proof of Lemma 18, one completes the proof of Lemma 19.

4.1. The Proof of Theorem 2

We present the proof of Theorem 2 by inductive arguments with respect to the index ().

Step  1. For , using Lemma 11 and the second equation in (6), one obtains

From (17), we have

Thus

On the other hand, using (3) of Lemma 9 and the first equation in (6) derives

Applying (7) of Proposition 8 yields

Hence

Combining (84) and (87), one deduces

Applying Gronwall’s inequality yields

Therefore, if satisfies , from (89), Lemma 19, and the fact that , we have

Thus

which contradicts the assumption that is the maximal existence time. This completes the proof for .

Step  2. For , applying (1) of Lemma 9 to the second equation in (6) derives

Thus

which together with (87) and the fact that , makes one deduce

It follows from Gronwall’s inequality that

Therefore, if satisfies , from (95) and , we obtain

which contradicts with the assumption that is the maximal existence time. This completes the proof for .

Step  3. For , differentiating the second equation in (6) with respect to , we obtain

Using Lemma 11 derives

Thanks to (6) of Proposition 8, one obtains

Thus

which together with (87) and (84) with instead of derives

Similar to the arguments in Step  1, one completes the proof for .

Step  4. For and , differentiating the second equation in (6) times with respect to derives

From Lemma 9, we have

Using the algebraic properties of derives

Thus

which together with (87), (84) with instead of derives

Using Gronwall’s inequality, we obtain

If satisfies , using the uniqueness of solutions in Theorem 1, one obtains that is uniformly bounded. Then

which contradicts the assumption that the maximal existence time . This completes the proof for and .

Step  5. For and , differentiating the second equation in system (6) times with respect to , we obtain

Using Lemma 11 with , one derives

For sufficiently small , using (19) and the fact that , one has

Making use of (110) and (111) yields

which together with (87) and (84) with instead of derives

Thanks to Gronwall’s inequality, one has

Using the uniqueness of solutions in Theorem 1, we obtain that

is uniformly bounded by the induction assumption. Then

which leads to a contradiction.

Thus from Step  1 to Step  5, one completes the proof of Theorem 2.

4.2. The Proof of Theorem 3

Using simple density arguments, here we only need to prove the theorem for . Assume that there exists and such that

Applying Lemma 17 yields

Differentiating the first equation in (6) with respect to and using yield

Noting

and combining (119), (120), and , , , (62), (66), one deduces

Using Lemmas 17 and 18, one deduces that there exists such that

For , integrating the above inequality with respect to on interval , we have

Thus

which together with (117) and derives

This contradicts with the results in Theorem 2.

On the other hand, applying Sobolev’s embedding theorem, one deduces

and then the solution blows up in finite time. This completes the proof.

Remark 20. Theorem 3 implies that the blow-up phenomenon of solution to system (6) only depends on the slope of the first component . In other words, the first component blows up before the second component in finite time.

4.3. The Proof of Theorem 4

We use Lemmas 17 and 18 to prove Theorem 4. For simplicity, we assume here. Noting the assumption is odd, is even, and the structure of system (6), one deduces that is odd and is even with respect to for all . Thus and . Thanks to the second equation in system (6) at the point , we have

which derives .

Differentiating the first equation in system (6) with respect to variable yields

Noting the assumption in Theorem 4, one obtains . Setting and combining with (128) yield

By the assumption , we have . We claim that is true for all . In fact, if the claim is not true for all , then from the continuity of , we deduce that there exists such that for , and . Combining this with (129) derives a.e. on . Since is absolutely continuous on , one gets the contradiction . This completes the proof of the claim.

Thus we obtain that is strictly decreasing on . Let such that . From (129), we have

Since is locally Lipschitz on and strictly negative, thus is also locally Lipschitz on . It follows that

Integrating (131) with respect to over yields

Since on , one obtains that the maximal existence time . Moreover, using the assumption derives

which completes the first part proof of Theorem 4.

On the other hand, differentiating the second equation in (6) with respect to yields

Taking and noting , together with the definition of in Lemma 15, one deduces . Thus

Using the assumption in Theorem 4, (57) and letting yield . From (135), one deduces

Thanks to (133), for all , we have

Note as . Thus, if , then from (136), for , one has

On the other hand, if , it follows from (136) that, for , one deduces

This completes the proof of Theorem 4.

4.4. The Proof of Theorem 5

Let be the corresponding solution to system (1) with initial data . Differentiating the first equation in (1) with respect to , one has

Differentiating the second equation in system (6) with respect to yields

We obtain that is also a solution to system (1) provided that is a solution to system (1). Note the initial data and are odd; one derives . Using the uniqueness of solutions yields , and is odd for all . Thus for all . From the above analysis and (140), we have

Similar to the proof of Theorem 4, we complete the first part proof of Theorem 5. From (141), one deduces

From (143), we get

As before, we also obtain that is decreasing with . Thus , which combined with (144) completes the proof of Theorem 5.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (71003082) and Fundamental Research Funds for the Central Universities (SWJTU12CX061 and SWJTU09ZT36).

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