Blow-up and local weak solution for a modified two-component Camassa-Holm equations
© Tian and Zhu; licensee Springer. 2012
Received: 14 November 2011
Accepted: 2 May 2012
Published: 2 May 2012
In this article, we establish some blow-up results for a modified two-component Camassa-Holm system in Sobolev spaces. We also obtain the existence of the weak solutions of this system in H s × Hs-1, s > 5/2.
where m = u - u xx and σ = ± 1. Constantin and Ivanov  derived this system in the context of shallow water theory. u can be interpreted as the horizontal fluid velocity and ρ is related to the water elevation in the first approximation [2, 3]. They showed that while small initial data develop into global solutions, for some initial data wave breaking occurs. They also discussed the solitary wave solutions. In Vlasov plasma models, system (1.1) describes the closure of the kinetic moments of the single-particle probability distribution for geodesic motion on the simplectomorphisms. While in the large-deformation diffeomorphic approach to image matching, system (1.1) is summoned in a type of matching procedure called metamorphosis (see  and the references therein). This system appeared originally in . Based on the deformation of bi-Hamiltonian structure of the hydrodynamic type, Chen et al.  obtained system (1.1) when σ = -1. They show that it has the peakon and multilink solitons, and is integrable in the sense that it has Lax-pair. The mathematical properties of system (1.1) have been studied further in many articles, see, e.g., [7–15]. In , Holm and Ivanov generalized the Lax-pair formulation of system (1.1) to produce an integrable multi-component family, CH(n, k), of equations with n components and 1 ≤ |k| ≤ n velocities. They determined their Lie-Poisson Hamiltonian structures and gave numerical examples of their soliton solution behavior. Recently, a new global existence result and several new blow-up results of strong solutions for the Cauchy problem of Equation (1.1) with σ = 1 were obtained in . Gui and Liu  established the local well posedness for the two-component Camassa-Holm system in a range of the Besov spaces. Chen and Liu  discussed the wave-breaking phenomenon of a generalized two-component Camassa-Holm system, and determined the exact blow-up rate of such solutions. The existence and uniqueness of global weak solutions to Equation (1.1) have also been discussed by Guan and Yin .
where m ≥ 1. It can be reduced to (1.1) as m = 1.
which is a crucial ingredient to obtain the blow-up phenomenon. Last, by using the conserves from laws and the contraction mapping theorem, we obtain the existence of weak solutions of Cauchy problem (1.2) and (1.3). These methods are similar to that was used in . However, because of the asymmetry and the high strength of the nonlinearity of Equation (1.3), it is more difficult to estimate the norm of u, ρ, u x , ρ x in Sobolev space. In addition, also we get Equation (5.10) which is different with that in . As for the blow-up phenomenon, we get some new results of (1.2) and (1.3).
Guan and Yin [17, 19] got the global weak solutions for two-component Camassa-Holm shallow water system; they first obtained approximate solutions for the system, then they prove the compactness of these solutions, and at last they got the global weak solutions. Using the same way, Liu and Yin  also got global weak solutions for a periodic two-component μ-Hunter-Saxton system. However, in this article, we add high-order perturbation terms in this system, and by using the conserves laws and the contraction mapping theorem, we obtain the existence of weak solutions.
The remainder of this article is organized as follows. Section 2 is the preliminary. In Section 3, the local well posedness for strong solution of Cauchy problem (1.2) and (1.3) is established by Kato's theory. In Section 4, by transport equation, some blow-up results of the solutions of Cauchy problem (1.2) and (1.3) are obtained. The proof of existence of local weak solution is carried out in Section 5.
The proof is similar with Theorem 4.1 in .
where u is the first component of the solution z to Equation (1.2).
To prove the blow-up result, we need the following lemma.
This completes the proof of this lemma.
Then, T is finite and the slope of u tends to negative infinity as t goes to T while u is uniformly bounded on [0, T).
Note that , Then, by Lemma4.1, we have γ(t) = 0, ∀t ∈[0, T).
Since , and ,
Note that . If let then we have .
Since , we obtain that .
Since , there exists , such that , i.e., .
This completes the proof of the theorem.
3. Local weak solution
for all , where , the set of all the restrictions to ([0,T) × R) × ([0,T) × R) of smooth functions on R2 × R2 with compact support contained in ((-T, T) × R) × ((-T, T) × R). Then, z is called a weak solution to Equation (1.6). If z is a weak solution on [0, T) × [0, T) for every T > 0, then it is called global weak solution to Equation (1.6).
where ε is a constant satisfying 0 < ε < 1/4. Note that when ε = 0, system (3.1) and (3.2) is just the system (1.2) and (1.3).
define two bounded linear operator in the indicated Sobolev spaces.
A standard application of the contraction mapping theorem leads to the following existence result.
Theorem 3.1: For each initial data u0 ∈ H s (s ≥ 1), ρ0 ∈ Hs-1(s ≥ 2), there exists a T > 0 depending only on the norm of , , and m such that there exists a unique solution (u, ρ) ∈ C([0,T];H s ) × C([0, T];Hs-1) of system (3.1) and (3.2) in the sense of distribution. If u0 ∈ H s (s ≥ 2), ρ0 ∈ Hs-1(s ≥ 3), the solution (u, ρ) ∈ C([0,∞];H s ) × C([0, ∞];Hs-1) exists for all time, in particular, when u0 ∈ H s (s ≥ 4), ρ0 ∈ Hs-1(s ≥ 5), the corresponding solution is a classical globally defined solution of (3.1) and (3.2).
admitted by (3.1) in its integral form.
where we have used Lemma in  with r = q -1 > 0.
This complete the proof of the theorem.
has a unique solution u ε (t, x) ∈C∞([0,∞);H∞ and ρ ε (t, x) ∈C∞([0,∞);H∞. We first demonstrate the properties of the initial data uε 0, ρε 0in the following lemma. The proof is similar to Lemma 5 in .
for any ε with , where c is a constant independent of ε. The proof is similar to Lemma 5 in .
Theorem 3.3: Suppose that u0(x) ∈ H s (R), s ∈ [1, 3/2]; ρ0(x) ∈ Hs-1(R),
s-1 ∈ [1, 3/2] such that , . Let uε 0= φ ε *u0, ρε 0= φ ε *ρ0, be defined the same as above. Then, there exist constants T > 0 and c > 0 independent of ε such that the corresponding solution u ε , ρ ε of (3.10) satisfy the inequalities , for any t ∈ [0,T).
where c is a constant depends on Λ-2 and m.
has a unique solution f(t) ∈ C [0,T]. Theorem II in Section I.1 in  shows that for any t ∈ [0,T] which leads to the conclusion of this theorem.
where q ∈ (0, s], r ∈ (0, s-1], t ∈ [0,T].
Since as n → ∞, we have . Then for any real R > 0, converges strongly to ρ ∈ L2 ([0,T]; Hq-1(-R, R)) for any q ∈ [0, s-1), and converge to u t , ρ t strongly in L2([0,T]; Hr-1(-R, R)) for any r ∈ [0, s-1]. Hence, the existence of a weak solution to the Cauchy problem (1.2) and (1.3) is established.
in the sense of distribution. And u x , ρ x ∈ L∞([0,T] × R).
with u(0,x) = u0(x), ρ(0,x) = ρ0(x), and any . Moreover, since X = L1([0,T] × R) is a separable Banach space and , are bounded sequences in the dual space X* = L∞([0,T] × R) of X, there are two subsequences of , (still denoted by , ) weak star convergent to two functions U, P ∈ L∞([0,T] × R), respectively. Because , are also weakly convergent to u x , ρ x ∈ L∞([0,T] × R), respectively. It follows that u x = U, ρ x = P hold almost everywhere. Hence, u x , ρ x ∈ L∞([0,T] × R).
The study was supported by the National Nature Science Foundation of China (No. 11171135, 71073072), the Nature Science Foundation of Jiangsu (No. BK 2010329), the Project of Excellent Discipline Construction of Jiangsu Province of China, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 09KJB110003).
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