Abstract
In this article, we establish some blowup results for a modified twocomponent CamassaHolm system in Sobolev spaces. We also obtain the existence of the weak solutions of this system in H^{s }× H^{s1}, s > 5/2.
Keywords:
the modified twocomponent CamassaHolm equations; blow up; weak solutions.1. Introduction
The wellknown twocomponent CamassaHolm equations [1]
where m = u  u_{xx }and σ = ± 1. Constantin and Ivanov [2] 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 singleparticle probability distribution for geodesic motion on the simplectomorphisms. While in the largedeformation diffeomorphic approach to image matching, system (1.1) is summoned in a type of matching procedure called metamorphosis (see [4] and the references therein). This system appeared originally in [5]. Based on the deformation of biHamiltonian structure of the hydrodynamic type, Chen et al. [6] 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 Laxpair. The mathematical properties of system (1.1) have been studied further in many articles, see, e.g., [715]. In [4], Holm and Ivanov generalized the Laxpair formulation of system (1.1) to produce an integrable multicomponent family, CH(n, k), of equations with n components and 1 ≤ k ≤ n velocities. They determined their LiePoisson Hamiltonian structures and gave numerical examples of their soliton solution behavior. Recently, a new global existence result and several new blowup results of strong solutions for the Cauchy problem of Equation (1.1) with σ = 1 were obtained in [8]. Gui and Liu [14] established the local well posedness for the twocomponent CamassaHolm system in a range of the Besov spaces. Chen and Liu [16] discussed the wavebreaking phenomenon of a generalized twocomponent CamassaHolm system, and determined the exact blowup rate of such solutions. The existence and uniqueness of global weak solutions to Equation (1.1) have also been discussed by Guan and Yin [17].
In this article, we consider a twocomponent generalization of Equation (1.1), that is
with initial data
where m ≥ 1. It can be reduced to (1.1) as m = 1.
The purpose of this article is to study the well posedness, local weak solution, and blowup for Cauchy problem (1.2) and (1.3). System (1.2) also conserves conservation laws. Our starting point is to obtain the local well posedness by using Kato's theory, Next, we derive some blowup results of the solutions by the following transport equation,
which is a crucial ingredient to obtain the blowup 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 [18]. 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 [18]. As for the blowup phenomenon, we get some new results of (1.2) and (1.3).
Guan and Yin [17,19] got the global weak solutions for twocomponent CamassaHolm 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 [20] also got global weak solutions for a periodic twocomponent μHunterSaxton system. However, in this article, we add highorder 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 blowup 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.
2. Blowup
Lemma 2.1: Given , s > 5/2, then there exists a maximal , and a unique solution to Cauchy problem (1.2) and (1.3) such that
Moreover, the solution depends continuously on the initial data, i.e., the mapping
is continuous.
The proof is similar with Theorem 4.1 in [21].
Let , then (1.2) is equivalent to
Consider the following initial value problem,
where u is the first component of the solution z to Equation (1.2).
To prove the blowup result, we need the following lemma.
Lemma 2.2: Let z_{0}∈ H^{s }× H^{s1}, (s > 5/2), and let T > 0 be the maximal existence time of the corresponding solution z to Equation (2.1), then we have
Proof. Differentiating the lefthand side of Equation (2.3) with respect t. It follows from (2.1) and (2.2), that
This completes the proof of this lemma.
Theorem 2.1: Let , (s > 5/2), and T be the maximal time of the solution z to Equation (1.2) with the initial data z_{0}. Assume that there exists x_{0 }∈ R such that and
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).
Proof. Let be the solution of Equation (2.1) with the initial data z_{0}, and T be the maximal time of z, and let
From (2.1) and (2.2), we have
Differentiating the first equation in (2.1) with respect x, we have
Note that , Then, by Lemma4.1, we have γ(t) = 0, ∀t ∈[0, T).
Thus
Note that . If let then we have .
With the inequality above, we get
Since , there exists , such that , i.e., .
This completes the proof of the theorem.
3. Local weak solution
Definition 3.1: ([22]) Let (u_{0}, ρ_{0}) ∈ H^{1}(R) × H^{1}(R). If (u, ρ) belongs to and satisfies the identity
for all , where , the set of all the restrictions to ([0,T) × R) × ([0,T) × R) of smooth functions on R^{2 }× R^{2 }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).
In this section, we discuss the existence of weak solution of Cauchy problem (1.2) and (1.3). To this purpose, we consider the following Cauchy problem:
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).
For any 0 < ε < 1/4 and s ≥ 1, the integral operators
and
define two bounded linear operator in the indicated Sobolev spaces.
To prove the existence of solutions to the problem (3.1) and (3.2), we apply the two operators above to both sides of (3.1) and then integrate the resulting equations with regard to t. This leads to the following equations.
A standard application of the contraction mapping theorem leads to the following existence result.
Theorem 3.1: For each initial data u_{0 }∈ H^{s }(s ≥ 1), ρ_{0 }∈ H^{s1}(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];H^{s1}) of system (3.1) and (3.2) in the sense of distribution. If u_{0 }∈ H^{s }(s ≥ 2), ρ_{0 }∈ H^{s1}(s ≥ 3), the solution (u, ρ) ∈ C([0,∞];H^{s}) × C([0, ∞];H^{s1}) exists for all time, in particular, when u_{0 }∈ H^{s }(s ≥ 4), ρ_{0 }∈ H^{s1}(s ≥ 5), the corresponding solution is a classical globally defined solution of (3.1) and (3.2).
The global existence result follows from the conservation law
admitted by (3.1) in its integral form.
Theorem 3.2: Suppose that for some s ≥ 4, the function pair u(x, t) and ρ(x, t) in the solution of Equation (3.1) corresponding to the initial data u_{0 }∈ H^{s }(s ≥ 4); ρ_{0 }∈ H^{s1}(s ≥ 5), then the following inequalities hold:
For any real number q ∈ (1, s] (s ≥ 5), there exists a constant c depending only on q, m, such that
For any q ∈ (1, s1] (s ≥ 4), there exists a constant c such that
And for any q ∈ (1, s2] (s ≥ 5), there exists a constant c such that
Proof. It is obvious that (3.3) holds. In order to prove (5.4), let . We rewrite Equation (3.1) in the following equivalent form.
For any q ∈ (1, s] (s ≥ 5), applying (Λ^{q }u)Λ^{q }to the both sides of the first equation of Equation (3.7), respectively, and integrating with regard to x, we obtain
By using Sobolev embedding theorems, we have
where we have used lemma in [23] with r = q 2 > 0. Also
where we have used Lemma in [24] with r = q 1 > 0.
Then, we get
For any q ∈ (1, s1] (s ≥ 5), applying (Λ^{q1 }ρ) Λ^{q1 }to the both sides of the second equation of Equation (3.7), respectively, then we obtain
Summing up (3.8) and (3.9), we get
For any q ∈ (1, s1] (s ≥ 4), applying (Λ^{q }u_{t})Λ^{q }to the both sides of the first equation of Equation (3.7), respectively, and integrating with regard to x, we obtain that
and
where we have used lemma in [24] with r = q 1 > 0. Then, we get
For any q ∈ (1, s2] (s ≥ 5), applying (Λ^{q1 }ρ_{t})Λ^{q1 }to the both sides of the second equation of Equation (5.7), respectively, then we obtain
This complete the proof of the theorem.
Suppose u_{0 }∈ H^{s }(s ≥ 1), ρ_{0 }∈ H^{s1}(s ≥ 2), and let u_{ε0}, ρ_{ε0 }be the convolution u_{ε0 }= φ_{ε}*u_{0}, ρ_{ε0 }= φ_{ε}*ρ_{0}, where such that the Fourier transform of φ satisfies , , and for any ξ ∈ (1,1). Then, it follows from Theorem 3.1 that for each ε with 0 < ε <1/4, the Cauchy problem
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 u_{ε0}, ρ_{ε0 }in the following lemma. The proof is similar to Lemma 5 in [25].
Lemma 3.1: Under the above assumptions, there hold
for any ε with , where c is a constant independent of ε. The proof is similar to Lemma 5 in [25].
Theorem 3.3: Suppose that u_{0}(x) ∈ H^{s}(R), s ∈ [1, 3/2]; ρ_{0}(x) ∈ H^{s1}(R),
s1 ∈ [1, 3/2] such that , . Let u_{ε0 }= φ_{ε}*u_{0}, ρ_{ε0 }= φ_{ε}*ρ_{0}, 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).
Proof. Use Equation (3.7) with u = u_{ε}, ρ = ρ_{ε}. Differentiating with respect to x on both sides of the first equation in Equation (3.7). Note that , we obtain
Let n > 0 be an integer. Then, multiplying the above equation by (u_{x})^{2n+1 }to integrate with respect to x, we get
It follows from Hölder inequality that
Note that as p→∞ for any f ∈ L^{∞}∩L^{2}. Integrating the above inequality over R with respect to t, and taking the limitation as n→∞, we have
It follows from (3.3) that
For any given r ∈ (1/2,1), we have
Then from (3.4), we have
and
Thus,
and
Then, we get
It follows that
where c is a constant depends on Λ^{2 }and m.
Also, we can obtain
It follows from the contraction mapping theorem that there exists a constant T > 0 such that the equation
has a unique solution f(t) ∈ C [0,T]. Theorem II in Section I.1 in [26] shows that for any t ∈ [0,T] which leads to the conclusion of this theorem.
Let u = u_{ε}, ρ = ρ_{ε}, with (3.4) used
where q ∈ (0, s], r ∈ (0, s1], t ∈ [0,T].
Then, it follows from Aubin's compactness theorem [27] that there exist subsequences of {u_{ε}}, {ρ_{ε}} denoted by , such that , are weakly convergent to u(t, x)∈ L^{2}([0,T]; H^{s}), ρ(t, x)∈ L^{2}([0,T]; H^{s1}), respectively, and , are weakly convergent to u_{t}(t, x)∈ L^{2}([0,T]; H^{s1}), ρ_{t}(t, x)∈ L^{2}([0,T]; H^{s2}), respectively. Because are weakly convergent to u(t, x)∈ L^{2}([0,T]; H^{s}), for any f∈ (L^{2}([0,T]; H^{s}))* = L^{2}([0,T]; H^{s}) when n → ∞. Applying Riesz lemma, we conclude that there exists such that
Since as n → ∞, we have . Then for any real R > 0, converges strongly to u ∈ L^{2}([0,T]; H^{q}(R, R)) for any q ∈ [0, s1); and converges to u_{t }strongly in L^{2}([0,T]; H^{r}(R, R)) for any r ∈ [0, s1]. Similarly, for any g∈ (L^{2}([0,T]; H^{s}))* = L^{2}([0,T]; H^{s}) as n → ∞. By Riesz lemma, we conclude that there exists such that
Since as n → ∞, we have . Then for any real R > 0, converges strongly to ρ ∈ L^{2 }([0,T]; H^{q1}(R, R)) for any q ∈ [0, s1), and converge to u_{t}, ρ_{t }strongly in L^{2}([0,T]; H^{r1}(R, R)) for any r ∈ [0, s1]. Hence, the existence of a weak solution to the Cauchy problem (1.2) and (1.3) is established.
Theorem 3.4: Let u_{0}(x) ∈H^{s}(R) and ρ_{0}(x) ∈H^{s1}(R) , which satisfy , . Then there exists a constant T > 0 such that the Cauchy problem (1.2) and (1.3) with the initial data has a solution
in the sense of distribution. And u_{x}, ρ_{x }∈ L^{∞}([0,T] × R).
Proof. It follows from Theorem 3.3 that , are bounded in the space L^{∞}. Hence, the sequences , , , , , , , are also weakly convergent to u^{2}, ρ^{2}, , , uρ, u_{x}ρ_{x}, u_{x}ρ, uρ_{x}∈L^{2}([0,T]; H^{r}(R, R)) for any r ∈ [0, s1] and R > 0, respectively. Therefore, u, ρ satisfy
and
with u(0,x) = u_{0}(x), ρ(0,x) = ρ_{0}(x), and any . Moreover, since X = L^{1}([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).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
LT raised the modified twocomponent CamassaHolm equations and conceived the local weak solution of the equations. MZ carried out the solution of the weak solution and its blowup phenomenon. All authors read and approved the final manuscript.
Acknowledgements
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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