The Cauchy Problem for a Weakly Dissipative 2-Component Camassa-Holm System

Ming, Sen; Yang, Han; Wu, Yonghong

doi:https://doi.org/10.1155/2014/801941

Mathematical Problems in Engineering

On this page

Abstract Introduction Acknowledgments References Copyright Related Articles

Special Issue

Fluid Dynamics and Computational Engineering

View this Special Issue

Research Article | Open Access

Volume 2014 | Article ID 801941 | https://doi.org/10.1155/2014/801941

The Cauchy Problem for a Weakly Dissipative 2-Component Camassa-Holm System

Sen Ming,¹Han Yang,¹and Yonghong Wu²

Academic Editor: Shaoyong Lai

Received18 Nov 2013

Accepted14 Dec 2013

Published03 Feb 2014

Abstract

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.

1. Introduction

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 [1]: 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 [2]. 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 [8] 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 [19] 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 [9], 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 [33] with . In [20], 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 [25]. For , Gui and Liu [27] 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. [34] 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 [21] 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 [23] 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 [35], 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).

Theorem 6. Let in system (1) and . If for all , then the corresponding strong solution to system (1) with exists globally in time.

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.

2. Preliminary

In this section, we recall some basic facts in Besov space. One may check [21, 39–42] for more details.

Proposition 7 (see [40, 42]). Let , , and . The nonhomogeneous Besov space is defined by , where Moreover, .

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 [40]). 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 [40]). 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 [40]). 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.

Lemma 12 (see [39, 40]). Let , , and . Assume that or if . Then, there exists a constant depending only on , such that the following estimate holds true: with If , then for all , , (22) holds with .

Let us state the existence result for transport equation with data in Besov space.

Lemma 13 (see [40]). 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

For convenience, we denote and rewrite (7) as We use a standard iterative process to construct the approximate solutions to (24).

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 .

Secondly, for is a Cauchy sequence in , so it converges to some limit function . An interpolation argument insures that the convergence holds in for any . It is easy to pass to the limit in (25) and to conclude that is indeed a solution to (24). Thanks to the fact that belongs to we know that the right side of the first equation in (24) belongs to , and the right side of the second equation in (24) belongs to . In the case of , applying Lemma 13 derives for any .

Finally, with (24), we obtain that if , and in otherwise. Thus, . Moreover, a standard use of a sequence of viscosity approximate solutions for (24) which converges uniformly in gives the continuity of solution .

3.2. Uniqueness and Continuity with Initial Data

Lemma 14. Assume that , , and . Let and be two given solutions to the system (24) with initial data satisfying and . Then for every , one has

Proof. Denote , ; then which implies that , and satisfies the following transport equation: with According to Lemma 12, we have Similar to the proof of the third step in Section 3.1, here we only focus our attention on the high order terms. We deduce that Hence, which together with (56) makes us obtain Using Gronwall’s inequality, we complete the proof of Lemma 14.

Remark 15. When , the Besov space coincides with the Sobolev space . Theorem 1 implies that under the assumption with , we obtain the local well-posedness for system (1), which improves the corresponding results in [20, 25, 35] by Kato's theory, where is assumed in [25, 35] and in [20].

Remark 16. The existence time for system (1) may be chosen independently of in the following sense [43]. If is the solution of system (1) with initial data for some , then with the same time . In particular, if , then .

4. Wave-Breaking Phenomena

In this section, attention is turned to investigating conditions of wave breaking and the precise blow-up rate of strong solutions to system (1). Using Theorem 1 and Remark 16, we establish the precise wave-breaking scenarios of strong solutions to system (1).

4.1. The Proof of Theorem 2

Applying Theorem 1, Remark 16, and a simple density argument, we only need to show that Theorem 2 holds with . Here we assume to prove the theorem.

Noting that , we rewrite the system (1) as Multiplying the first equation in (60) by and integrating by parts, we deduce that Since and , then Multiplying the second equation in (60) by and integrating by parts, we have which together with (62) implies On the other hand, multiplying the first equation in (60) by and integrating by parts again, we get Multiplying the second equation in (60) by and integrating by parts, we obtain which together with (65) derives For and , we have Therefore, combining (64) with (68), one deduces that Assume that and there exists such that

It follows from (69) that Applying Gronwall’s inequality to (71) yields, for all , Differentiating the first equation in (60) with respect to and multiplying the obtained equation by , then integrating by parts, we get Differentiating the second equation twice in (60) with respect to and multiplying the resulted equation by , then integrating by parts, we deduce that which together with (73), , and one has where we have used (64), (70), and (72).

Applying Sobolev’s embedding theorem and Holder's inequality yields together with (72) and (75); thus there exists and such that Using Gronwall's inequality, for all , we have which contradicts the assumption of the maximal existence time .

Conversely, using Sobolev's embedding theorem , we derive that if condition (9) in Theorem 2 holds, the corresponding solution blows up in finite time, which completes the proof of Theorem 2.

4.2. The Proof of Theorem 3

Firstly, we present three lemmas which are used to prove the blow-up mechanisms and the global existence of solutions to system (1).

Lemma 17 (see [44]). Let and . Then for all , there exists at least one point with The function is absolutely continuous on with Now consider the following trajectory equation: where denotes the first component of solution to system (1).

Lemma 18 (see [45]). Let with . Then, (81) has a unique solution: . Moreover, the map is an increasing diffeomorphism of for every and

Lemma 19 (see [23]). Let with , and let be the maximal existence time of the corresponding solution to system (1). Thus,

Proof of Theorem 3. The technique used here is inspired from [20]. Similar to the proof of Theorem 2, we only need to show that Theorem 3 holds with some .
By Lemma 17, we obtain that there exists with such that Hence, we have .
On the other hand, since is diffeomorphism for any , there exists such that , for all . Using the second equation in system (1) and the trajectory equation (81), we have Hence, Taking in (85), together with , for all , then Using the assumption with the point defined by in Theorem 3 and letting , we deduce that . From (87), we get If , then for all , so we have . For , differentiating the first equation in system (7) with respect to the variable yields Note inequality (see page 347, (5.8) in [28]). Using Sobolev’s embedding theorem and (64), we obtain Here we denote , and thus .
Denoting and combining with (64) yield that . Together with the definition of , (88), and for a.e. , from (90) we derive that Using Young's inequality, we have for all , which together with (92) makes us deduce that where By the assumption of , we have . We now claim that is true for any . In fact, assume that for some ; since is continuous, we deduce that there exists such that for , but . Combining this with (94) gives a.e. on . Since is absolutely continuous on , so we get the contradiction . Thus we prove the claim.
Now we deduce that is strictly decreasing on . Choosing that such that , then we obtain from (94) that For is locally Lipschitz on and strictly negative, we get is also locally Lipschitz on . This gives Integration of this inequality yields Since on , we obtain that the maximal existence time . Moreover, thanks to again, we get that which completes the proof of Theorem 3.

Remark 20. From (81) and (85), one deduces that if with and the component does not break in finite time , then the component is uniformly bounded for all . In fact, if there exists such that , for all , thus We find that there is a similar estimation in [25].

Remark 21. Because of the presence of high order term in system (1), it is difficult to obtain the estimate for the term without the assumption of .

4.3. The Proof of Theorem 4

Similar to the proof of Theorem 2, here we only need to show that Theorem 4 holds with . Note that system (1) with is invariant under the transformation and . Thus, if is odd and is even, then the corresponding solution satisfies that is odd and is even with respect to for all , where is the maximal existence time. Hence, .

Thanks to the transport equation at the point , we have which derives that . Noting that , we obtain For , differentiating the first equation in system (7) with respect to the variable yields Let , and we have Denoting and combining with (64) yield that . By using Young's inequality, we deduce for all that which together with (93) and (105) derives where Similarly as in the proof of Theorem 3, from (108) one gets that for all , and there exists such that . Thus, which implies the maximal existence time . Then, which completes the first part of the proof of Theorem 4.

Differentiating the transport equation with respect to the variable , together with the trajectory equation (81), we get Taking and noting that and , we have Noting the assumption, (84), and , yields that Thanks to (111), for any , we have Note that .

Therefore, if , from (115), for , we have On the other hand, if , for , it follows from (115) that Now we complete the proof of Theorem 4.

Remark 22. Under the same conditions in Theorem 3 with and , we follow a similar argument of Theorem 4 to obtain the same blow-up results (i) and (ii) in Theorem 4.

Remark 23. Thanks to the assumption of , system (1) is invariant under the transformation and .

4.4. Blow-Up Rate

Having established the wave-breaking results for system (1), we give the estimate for the blow-up rate of solutions to system (1).

Proof of Theorem 5. Noting that , we have Similar to the proof of Theorem 3, we obtain where Therefore, choosing and using (99), we find such that . Since is locally Lipschitz, it follows that is absolutely continuous. Integrating (119) on interval with , we deduce that is decreasing on and It deduces from (119) that Noting that on , we obtain Integrating the above relation on with and noting that we have .
Since is arbitrary, it deduces from (95) and (124) that which completes the proof of Theorem 5.

Remark 24. Let be the blow-up time of the corresponding solution to system (1). The initial data with satisfies the assumptions of Theorem 4. Then,

Proof. The argument is similar to the proof of Theorem 5; here we omit it.

5. The Proof of Theorem 6

In this section, we first give a lemma on global strong solutions to system (1). Then, we present the proof of Theorem 6. We need to pay more attention to the assumption and for all .

Lemma 25. Assume in system (1) and with . Let be the maximal existence time of corresponding solution to system (1) in the case of with initial data . If for all , then there exists a constant such that where is a fixed constant defined in (132).

Proof. From Lemma 18, we have for all . Let , and then Noting and using (90) yield where From the proof of Theorems 3 and 5, one deduces that For convenience, here we denote From (85), we obtain that has the same sign with for all . For with , using Sobolev's embedding theorem, we have and there exists a constant such that for all . Since and for all , it follows that Taking , then for all and Now we consider the following Lyapunov function: Applying Sobolev’s embedding theorem yields Differentiating (135) with respect to and using (129) and (132), we obtain By using Gronwall’s inequality and (136), we deduce that for all . On the other hand, from (135), we have which together with (138) yields that for all , Then, by the definition of , we complete the proof of Lemma 25.

Proof of Theorem 6. Combining the results of Theorem 1, (72) and Lemma 25, we complete the proof of Theorem 6.

Remark 26. Assume in system (1) and with , and for all . Let be the maximal existence time of the corresponding solution to system (1) with , and then the corresponding solution blows up in finite time if and only if

Remark 27. For the same difficulty stated in Remark 21, here we only obtain the global existence result for solutions to system (1) with the assumption .

Conflict of Interests

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

Acknowledgments

The authors are grateful to the anonymous referees for a number of valuable comments and suggestions. This paper is supported by National Natural Science Foundation of China (71003082) and Fundamental Research Funds for the Central Universities (SWJTU12CX061 and SWJTU09ZT36).

References

R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Constantin and D. Lannes, “The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,” Archive for Rational Mechanics and Analysis, vol. 192, no. 1, pp. 165–186, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Constantin, “The trajectories of particles in Stokes waves,” Inventiones Mathematicae, vol. 166, no. 3, pp. 523–535, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Constantin and J. Escher, “Particle trajectories in solitary water waves,” Bulletin of the American Mathematical Society, vol. 44, no. 3, pp. 423–431, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Constantin and W. A. Strauss, “Stability of peakons,” Communications on Pure and Applied Mathematics, vol. 53, no. 5, pp. 603–610, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Constantin and W. A. Strauss, “Stability of the Camassa-Holm solitons,” Journal of Nonlinear Science, vol. 12, no. 4, pp. 415–422, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Constantin and W. A. Strauss, “Stability of a class of solitary waves in compressible elastic rods,” Physics Letters A, vol. 270, no. 3-4, pp. 140–148, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. A. Li and P. J. Olver, “Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,” Journal of Differential Equations, vol. 162, no. 1, pp. 27–63, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Lai and Y. Wu, “Local well-posedness and weak solutions for a weakly dissipative Camassa-Holm equation,” Scientia Sinica Mathematica, vol. 40, no. 9, pp. 901–920, 2010.
View at: Google Scholar
S. Lai and Y. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,” Journal of Differential Equations, vol. 248, no. 8, pp. 2038–2063, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Lai and Y. Wu, “Global solutions and blow-up phenomena to a shallow water equation,” Journal of Differential Equations, vol. 249, no. 3, pp. 693–706, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Lai and Y. Wu, “A model containing both the Camassa-Holm and Degasperis-Procesi equations,” Journal of Mathematical Analysis and Applications, vol. 374, no. 2, pp. 458–469, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Lai and Y. Wu, “The study of global weak solutions for a generalized hyperelastic-rod wave equation,” Nonlinear Analysis, vol. 80, pp. 96–108, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Mu, S. Zhou, and R. Zeng, “Well-posedness and blow-up phenomena for a higher order shallow water equation,” Journal of Differential Equations, vol. 251, no. 12, pp. 3488–3499, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. Tian, P. Zhang, and L. Xia, “Global existence for the higher-order Camassa-Holm shallow water equation,” Nonlinear Analysis, vol. 74, no. 7, pp. 2468–2474, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D.-X. Kong, “Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves,” Journal of Differential Equations, vol. 219, no. 2, pp. 421–450, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Guo, S. Lai, and Y. Wu, “On existence and uniqueness of the global weak solution for a shallow water equation,” Applied Mathematics and Computation, vol. 218, no. 23, pp. 11410–11420, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Mi and C. Mu, “Well-posedness and analyticity for the Cauchy problem for the generalized Camassa-Holm equation,” Journal of Mathematical Analysis and Applications, vol. 405, no. 1, pp. 173–182, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
S. Wu and Z. Yin, “Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation,” Journal of Differential Equations, vol. 246, no. 11, pp. 4309–4321, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Constantin and R. I. Ivanov, “On an integrable 2-component Camassa-Holm shallow water system,” Physics Letters A, vol. 372, no. 48, pp. 7129–7132, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Yan and Z. Yin, “On the Cauchy problem for a 2-component Degasperis-Procesi system,” Journal of Differential Equations, vol. 252, no. 3, pp. 2131–2159, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Yan and Z. Yin, “Analytic solutions of the Cauchy problem for 2-component shallow water systems,” Mathematische Zeitschrift, vol. 269, no. 3-4, pp. 1113–1127, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Guan and Z. Yin, “Global existence and blow-up phenomena for an integrable 2-component Camassa-Holm shallow water system,” Journal of Differential Equations, vol. 248, no. 8, pp. 2003–2014, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Guan and Z. Yin, “Global weak solutions for a 2-component Camassa-Holm shallow water system,” Journal of Functional Analysis, vol. 260, no. 4, pp. 1132–1154, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Escher, O. Lechtenfeld, and Z. Yin, “Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,” Discrete and Continuous Dynamical Systems, vol. 19, no. 3, pp. 493–513, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the 2-component Camassa-Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Gui and Y. Liu, “On the Cauchy problem for the 2-component camassa-Holm system,” Mathematische Zeitschrift, vol. 268, no. 1-2, pp. 45–46, 2011.
View at: Publisher Site | Google Scholar
M. Chen, S.-Q. Liu, and Y. Zhang, “A 2-component generalization of the Camassa-Holm equation and its solutions,” Letters in Mathematical Physics, vol. 75, no. 1, pp. 1–15, 2006.
View at: Publisher Site | Google Scholar
M. Zhu, “Blow-up, global existence and persistence properties for the coupled Camassa-Holm equations,” Mathematical Physics, Analysis and Geometry, vol. 14, no. 3, pp. 197–209, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z. Guo and L. Ni, “Persistence properties and unique continuation of solutions to a 2-component Camassa-Holm equation,” Mathematical Physics, Analysis and Geometry, vol. 14, no. 2, pp. 101–114, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Fu and C. Qu, “Unique continuation and persistence properties of solutions of the 2-component Degasperis-Procesi equations,” Acta Mathematica Scientia, vol. 32, no. 2, pp. 652–662, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Jin and Z. Jiang, “Wave breaking of an integrable Camassa-Holm system with 2-components,” Nonlinear Analysis, vol. 95, pp. 107–116, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
A. Shabat and L. M. Alonso, “On the prolongation of a hierarchy of hydrodynamic chains,” in New Trends in Integrability and Partial Solvability, vol. 132 of NATO Science Series II, pp. 263–280, Springer, Dordrecht, The Netherlands, 2004.
View at: Google Scholar
L. Tian, W. Yan, and G. Gui, “On the local well posedness and blow-up solution of a coupled Camassa-Holm equations in Besov spaces,” Journal of Mathematical Physics, vol. 53, Article ID 013701, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Chen, X. Deng, and J. Zhang, “Blow up and blow-up rate for the generalized 2-component Camassa-Holm equation,” International Journal of Nonlinear Science, vol. 12, no. 3, pp. 313–322, 2011.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
W. Chen, L. Tian, X. Deng, and J. Zhang, “Wave breaking for a generalized weakly dissipative 2-component Camassa-Holm system,” Journal of Mathematical Analysis and Applications, vol. 400, no. 2, pp. 406–417, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Qu, Y. Fu, and Y. Liu, “Well-posedness, wave breaking and peakons for a modified μ-Camassa-Holm equation,” Journal of Functional Analysis, vol. 266, no. 2, pp. 433–477, 2013.
View at: Publisher Site | Google Scholar
Y. Fu, G. Gui, Y. Liu, and C. Qu, “On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity,” Journal of Differential Equations, vol. 255, no. 7, pp. 1905–1938, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
R. Danchin, “A few remarks on the Camassa-Holm equation,” Differential and Integral Equations, vol. 14, no. 8, pp. 953–988, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 2005.
W. Yan, Y. Li, and Y. Zhang, “The Cauchy problem for the integrable Novikov equation,” Journal of Differential Equations, vol. 253, no. 1, pp. 298–318, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschafte, Springer, Berlin, Germany, 2011.
View at: Publisher Site | MathSciNet
Z. Yin, “On the Cauchy problem for an integrable equation with peakon solutions,” Illinois Journal of Mathematics, vol. 47, no. 3, pp. 649–666, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Constantin, “Existence of permanent and breaking waves for a shallow water equation: a geometric approach,” Annales de l'Institut Fourier, vol. 50, no. 2, pp. 321–362, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

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.

PDF Download Citation

Download other formats

Order printed copies

Views

789

Downloads

869

Citations