Wave breaking and infinite propagation speed for a modified two-component Camassa-Holm system with
© Lv et al.; licensee Springer. 2014
Received: 21 January 2014
Accepted: 14 March 2014
Published: 28 March 2014
In this paper, we investigate the modified two-component Camassa-Holm equation with on the real line. Firstly, we establish sufficient conditions on the initial data to guarantee that the corresponding solution blows up in finite time for the modified two-component Camassa-Holm (MCH2) system. Then an infinite propagation speed for MCH2 is proved in the following sense: the corresponding solution with compactly supported initial data does not have compact x-support in its lifespan.
MSC:37L05, 35Q58, 26A12.
where , , u denotes the velocity field, and ρ is related to the free surface density with the boundary assumptions; expresses an averaged or filtered density, κ is a nonnegative dissipative parameter, g is the downward constant acceleration of gravity in applications to shallow water waves. For convenience we assume in this paper. Moreover, u and γ satisfy the boundary conditions: and as .
Obviously, under the constraints of and , system (1.1) reduces to the Camassa-Holm equation, which was derived physically by Camassa and Holm in  (found earlier by Fokas and Fuchssteniner  as a bi-Hamiltonian generalization of the KdV equation) by directly approximating the Hamiltonian for Euler’s equation in the shallow water region with representing the free surface above a flat bottom. There have been extensive studies on Camassa-Holm equation. Now, we mention some results that are related to our results. Firstly, wave breaking for a large class of initial data has been established in [3–6]. Recently, Zhou and his collaborators  give a direct proof for McKean’s theorem . In addition, the large time behavior for the support of momentum density of the Camassa-Holm equation was studied in . An interesting phenomenon of the propagation speed for the Camassa-Holm equation with was presented by Zhou and his collaborators in their work  in the sense that a strong solution of the Cauchy problem with compact initial profile cannot be compactly supported at any later time unless it is the zero solution. Meanwhile, for the same problem about the equation , we refer to  for details.
The CH2 system appeared initially in , and recently Constantin and Ivanov in  gave a demonstration about its derivation in view of the fluid shallow water theory from the hydrodynamic point of view. This generalization, similar to the Camassa-Holm equation, possessed the peakon, multi-kink solutions and the bi-Hamiltonian structure [13, 14] and is always integrable. The wave breaking mechanism was discussed in [15–17] and the existence of global solutions was analyzed in [12, 16, 18]. A geometric investigation can be found in [15, 19].
We cannot obtain the conservation of norm for the CH2 system.
In what follows, we always assume and .
This paper is organized as follows. In Section 2, we will present some results, which will be used in this paper. In Section 3, we will establish several sufficient conditions to guarantee that the corresponding strong solution brows up. In Section 4, we will investigate the infinite propagation speed of MCH2 with .
In this section, for completeness, we recall some elementary results. We list them and skip their proofs for conciseness. Local well-posedness for the MCH2 system (1.3) can be obtained by Kato’s semigroup theory .
Lemma 2.2 
Lemma 2.3 
with constants . If the initial data , then the solution to (2.2) goes to −∞ before t tends to .
which is always positive before the blow-up time. Therefore, the function is an increasing diffeomorphism of a line.
In this section, we establish sufficient conditions on the initial data to guarantee blow-up for system (1.3). We start this section with the following useful lemma.
for all , where is the initial value of . □
The next result describes the precise blow-up scenarios for sufficiently regular solutions to system (1.3).
This contradicts the assumption. Conversely, the Sobelev embedding result (with ) implies that if Theorem 3.2 holds, the solution blows up in finite time, which completes the proof of Theorem 3.2. □
We state our first criterion via the associated initial potential as follows.
for some point . Then the solution to our system (1.3) with initial value blows up in finite time.
Remark 3.1 This theorem is similar to the result proved by Zhou in .
In order to arrive at our result, we need the following three claims.
Claim 1. for all t in its lifespan.
Therefore, thanks to (2.4) we obtain , for all t in its lifespan.
Claim 2. For any fixed t, for all .
Claim 3. is decreasing, for all .
where we used the Claim 2.
for all , which implies can be extended to infinity.
where we used .
Let , then (3.22) is an equation of type (3.1) with . The proof is completed by applying Lemma 2.3. □
we find that the inequalities (3.19), (3.20) still hold. As is well known, McKean  states that only the sign of the initial potential , not the size of it, affects the wave breaking phenomenon. Similar to his theorem, we apply a similar initial potential to the two-component case, and it reveals that the sign of the initial density also plays an important role.
Then we give the second criterion in this paper.
Then the corresponding solution to system (1.3) blows up in finite time.
This completes the proof. □
Finally, we give the third criterion.
This completes the proof. □
4 Infinite propagation speed
In this section, we consider the infinite propagation speed for system (1.1). It can be shown as follows.
where and denote continuous nonvanishing functions with and for . Furthermore, is a strictly decreasing function, while is an increasing function.
Therefore, is an increasing function in the lifespan. From (4.1), it follows that for .
Similarly, it is easy to see that is decreasing with . Therefore, for .
Taking , we obtain what we want. Then the theorem is proved. □
This work is partially supported by NSFC (Grant No. 11101376), NSFC (Grant No. 11226176) and ZJNSF (Grant No. LQ13A010008).
- Camassa R, Holm D: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 1993, 71: 1661–1664. 10.1103/PhysRevLett.71.1661MathSciNetView ArticleMATHGoogle Scholar
- Fuchssteiner B, Fokas AS: Symplectic structures, their Backlund transformations and hereditary symmetries. Physica D 1981/1982,4(1):47–66. 10.1016/0167-2789(81)90004-XMathSciNetView ArticleMATHGoogle Scholar
- Constantin A, Escher J: Well-posedness, global existence and blow-up phenomenon for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 1998, 51: 475–504. 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5MathSciNetView ArticleMATHGoogle Scholar
- Constantin A, Escher J: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998, 181: 229–243. 10.1007/BF02392586MathSciNetView ArticleMATHGoogle Scholar
- McKean HP: Breakdown of a shallow water equation. Asian J. Math. 1998, 2: 767–774.MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y: Wave breaking for a shallow water equation. Nonlinear Anal. 2004, 57: 137–152. 10.1016/j.na.2004.02.004MathSciNetView ArticleMATHGoogle Scholar
- Jiang Z, Ni L, Zhou Y: Wave breaking of the Camassa-Holm equation. J. Nonlinear Sci. 2012, 22: 235–245. 10.1007/s00332-011-9115-0MathSciNetView ArticleMATHGoogle Scholar
- Jiang Z, Zhou Y, Zhu M: Large time behavior for the support of momentum density of the Camassa-Holm equation. J. Math. Phys. 2013., 54: Article ID 081503Google Scholar
- Himonas A, Misiolek G, Ponce G, Zhou Y: Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Commun. Math. Phys. 2007, 271: 511–512. 10.1007/s00220-006-0172-4MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y, Chen H: Wave breaking and propagation speed for the Camassa-Holm equation with . Nonlinear Anal., Real World Appl. 2011,12(3):1875–1882. 10.1016/j.nonrwa.2010.12.005MathSciNetView ArticleMATHGoogle Scholar
- Olver P, Rosenau P: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 1996, 53: 1900–1906.MathSciNetView ArticleGoogle Scholar
- Constantin A, Ivanov R: On an integrable two-component Camassa-Holm shallow water system. Phys. Lett. A 2008, 372: 7129–7132. 10.1016/j.physleta.2008.10.050MathSciNetView ArticleMATHGoogle Scholar
- Chen M, Liu S, Zhang Y: A two-component generalization of the Camassa-Holm equation and its solutions. Lett. Math. Phys. 2006, 75: 1–15. 10.1007/s11005-005-0041-7MathSciNetView ArticleMATHGoogle Scholar
- Falqui G: On a Camassa-Holm type equation with two dependent variables. J. Phys. A 2006, 39: 327–342. 10.1088/0305-4470/39/2/004MathSciNetView ArticleMATHGoogle Scholar
- Escher J, Lechtenfeld O, Yin Z: Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete Contin. Dyn. Syst. 2007, 19: 493–513.MathSciNetView ArticleMATHGoogle Scholar
- Gui G, Liu Y: On the global existence and wave-breaking criteria for the two-component Camassa-Holm system. J. Funct. Anal. 2010, 258: 4251–4278. 10.1016/j.jfa.2010.02.008MathSciNetView ArticleMATHGoogle Scholar
- Guo Z, Zhu M: Wave breaking for a modified two-component Camassa-Holm system. J. Differ. Equ. 2012, 252: 2759–2770. 10.1016/j.jde.2011.09.041MathSciNetView ArticleMATHGoogle Scholar
- Guo Z: Blow-up and global solutions to a new integrable model with two components. J. Math. Anal. Appl. 2010, 372: 316–327. 10.1016/j.jmaa.2010.06.046MathSciNetView ArticleMATHGoogle Scholar
- Holm D, Ivanov R: Two component CH system: inverse scattering, peakons and geometry. Inverse Probl. 2011., 27: Article ID 045013Google Scholar
- Guan C, Karlsen KH, Yin Z: Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation. Contemp. Math. In Proceedings of the 2008–2009 Special Year in Nonlinear Partial Differential Equations. Am. Math. Soc., Providence; 2010:199–220.Google Scholar
- Guo Z, Zhu M, Ni L: Blow-up criteria of solutions to a modified two-component Camassa-Holm system. Nonlinear Anal. 2011, 12: 3531–3540. 10.1016/j.nonrwa.2011.06.013MathSciNetView ArticleMATHGoogle Scholar
- Holm D, Náraigh LÓ, Tronci C: Singular solutions of a modified two-component Camassa-Holm equation. Phys. Rev. E 2009.,3(79): Article ID 016601Google Scholar
- Jin L, Guo Z: A note on a modified two-component Camassa-Holm system. Nonlinear Anal., Real World Appl. 2012, 13: 887–892. 10.1016/j.nonrwa.2011.08.024MathSciNetView ArticleMATHGoogle Scholar
- Kato T Lecture Notes in Math. 48. In Spectral Theory and Differential Equations. Springer, Berlin; 1975:25. Dedicated to Konrad Jorgensm, Proceedings of the Symposium held Dundee 1974View ArticleGoogle Scholar
- Yan W, Tian L, Zhu M: Local well-posedness and blow-up phenomenon for a modified two-component Camassa-Holm system in Besov spaces. Int. J. Nonlinear Sci. 2012, 13: 99–104.MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y: On solutions to the Holm-Staley b -family of equations. Nonlinearity 2010, 23: 369–381. 10.1088/0951-7715/23/2/008MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y: Blow-up of solutions to a nonlinear dispersive rod equation. Calc. Var. Partial Differ. Equ. 2005, 25: 63–77.View ArticleMathSciNetMATHGoogle Scholar
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