Local well-posedness and persistence properties for a model containing both Camassa-Holm and Novikov equation
© Zhou et al.; licensee Springer. 2014
Received: 25 June 2013
Accepted: 10 December 2013
Published: 8 January 2014
This paper deals with the Cauchy problem for a generalized Camassa-Holm equation with high-order nonlinearities,
where and . This equation is a generalization of the famous equation of Camassa-Holm and the Novikov equation. The local well-posedness of strong solutions for this equation in Sobolev space with is obtained, and persistence properties of the strong solutions are studied. Furthermore, under appropriate hypotheses, the existence of its weak solutions in low order Sobolev space with is established.
where and .
where the variable represents the fluid velocity at time t and in the spatial direction x, and k is a nonnegative parameter related to the critical shallow water speed . The Camassa-Holm equation (1.2) is also a model for the propagation of axially symmetric waves in hyperelastic rods (cf. ). It is well known that equation (1.2) has also a bi-Hamiltonian structure [3, 4] and is completely integrable (see [5, 6] and the in-depth discussion in [7, 8]). In , Qiao has shown that the Camassa-Holm spectral problem yields two different integrable hierarchies of nonlinear evolution equations, one is of negative order CH hierachy while the other one is of positive order CH hierarchy. Its solitary waves are smooth if and peaked in the limiting case (cf. ). The orbital stability of the peaked solitons is proved in , and the stability of the smooth solitons is considered in . It is worth pointing out that solutions of this type are not mere abstractions: the peakons replicate a feature that is characteristic for the waves of great height - waves of largest amplitude that are exact solutions of the governing equations for irrotational water waves (cf. [12–14]). The explicit interaction of the peaked solitons is given in  and all possible explicit single soliton solutions are shown in . The Cauchy problem for the Camassa-Holm equation (1.2) has been studied extensively. It has been shown that this problem is locally well-posed for initial data with [17–19]. Moreover, it has global strong solutions and also admits finite time blow-up solutions [17, 18, 20, 21]. On the other hand, it also has global weak solutions in [22–25]. The advantage of the Camassa-Holm equation in comparison with the KdV equation (1.2) lies in the fact that the Camassa-Holm equation has peaked solitons and models the peculiar wave breaking phenomena [1, 21].
Wazwaz [26, 27] studied the solitary wave solutions for the generalized Camassa-Holm equation (1.3) with , , and the peakon wave solutions for this equation were studied in [28–30], and the periodic blow-up solutions and limit forms for (1.3) were obtained in . In [30, 32], the authors have given the traveling waves solution, peaked solitary wave solutions for (1.3).
The Novikov equation (1.4) possesses a matrix Lax pair, many conserved densities, a bi-Hamiltonian structure as well as peakon solutions . These apparently exotic waves replicate a feature that is characteristic of the waves of great height-waves of largest amplitude that are exact solutions of the governing equations for water waves, as far as the details are concerned [13, 35, 36]. The Novikov equation possesses the explicit formulas for multipeakon solutions . It has been shown that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces and in Sobolev spaces and possesses the persistence properties [38, 39]. In [40, 41], the authors showed that the data-to-solution map for equation (1.4) is not uniformly continuous on bounded subsets of for . Analogous to the Camassa-Holm equation, the Novikov equation shows the blow-up phenomenon  and has global weak solutions . Recently, Zhao and Zhou  discussed the symbolic analysis and exact traveling wave solutions of a modified Novikov equation, which is new in that it has a nonlinear term instead of .
where γ is a constant. equation (1.5) was independently proposed by Fokas , by Fuchssteiner , and Olver and Rosenau  as a new generalization of integrable system by using the general method of tri-Hamiltonian duality to the bi-Hamiltonian representation of the modified Korteweg-de Vries equation. Later, it was obtained by Qiao [48, 49] from the two-dimensional Euler equations, where the variables and represent, respectively, the velocity of the fluid and its potential density. Ivanov and Lyons  obtain a class of soliton solutions of the integrable hierarchy which has been put forward in a series of woks by Qiao [48, 49]. It was shown that equation (1.5) admits the Lax-pair and the Cauchy problem (1.5) may be solved by the inverse scattering transform method. The formation of singularities and the existence of peaked traveling-wave solutions for equation (1.5) was investigated in . The well-posedness, blow-up mechanism, and persistence properties are given in . It was also found that equation (1.5) is related to the short-pulse equation derived by Schäfer and Wayne .
Motivated by the results mentioned above, the goal of this paper is to establish the well-posedness of strong solutions and weak solutions for problem (1.1). First, we use Kato’s theorem to obtain the existence and uniqueness of strong solutions for equation (1.1).
In [38, 56, 57], the spatial decay rates for the strong solution to the Camassa-Holm Novikov equation were established provided that the corresponding initial datum decays at infinity. This kind of property is so-called the persistence property. Similarly, for equation (1.1), we also have the following persistence properties for the strong solution.
uniformly in the time interval .
uniformly in the time interval .
Finally, we have the following theorem for the existence of a weak solution for equation (1.1).
Theorem 1.4 Suppose that with and . Then there exists a life span such that problem (1.1) has a weak solution in the sense of a distribution and .
The plan of this paper is as follows. In the next section, the local well-posedness and persistence properties of strong solutions for the problem (1.1) are established, and Theorems 1.1-1.3 are proved. The existence of weak solutions for the problem (1.1) is proved in Section 3, and this proves Theorem 1.4.
2 Well-posedness and persistence properties of strong solutions
where . Let denote the class of continuous functions from to and .
which completes the proof of Theorem 1.2. □
Next, we give a simple proof for Theorem 1.3.
Proof of Theorem 1.3 We should use Theorem 1.3 to prove this theorem.
uniformly in the time interval . This completes the proof of Theorem 1.3. □
3 Existence of solution of the regularized equation
where , , and a, k are constants. One can easily check that when , equation (3.1) is equivalent to the IVP (1.1).
Before giving the proof of Theorem 1.4, we give several lemmas.
Lemma 3.1 (See )
Lemma 3.2 Let with . Then the Cauchy problem (3.1) has a unique solution where depends on . If , the solution exists for all time. In particular, when , the corresponding solution is a classical globally defined solution of problem (3.1).
defines a bounded linear operator on the indicated Sobolev spaces.
Taking T sufficiently small so that , we deduce that maps to itself. It follows from the contraction-mapping principle that the mapping has a unique fixed point u in .
Now we study the norms of solutions of equation (3.1) using energy estimates. First, recall the following two lemmas.
Lemma 3.3 (See )
here c is a constant depending only on r.
Lemma 3.4 (See )
where denotes the commutator of the linear operators A and B, and c is a constant depending only on r.
Proof Using and (3.3) derives (3.4).
Integrating both sides of the above inequality with respect to t results in inequality (3.5).
with a constant . This completes the proof of Theorem 3.1. □
has a unique solution , in which may depend on ϵ.
For an arbitrary positive Sobolev exponent , we give the following lemma.
where c is a constant independent of ϵ.
where is independent of ϵ.
Theorem 3.2 If with such that . Let be defined as in the system (3.21). Then there exist two constants, c and , which are independent of ϵ, such that of problem (3.21) satisfies for any .
has a unique solution . From the above inequality, we know that the variable T only depends on c and . Using the theorem present on p.51 in  or Theorem II in Section 1.1 in  one derives that there are constants and independent of ϵ such that for arbitrary , which leads to the conclusion of Theorem 3.2. □
where , and . It follows from Aubin’s compactness theorem that there is a subsequence of , denoted by , such that and their temporal derivatives are weakly convergent to a function and its derivative in and , respectively. Moreover, for any real number , is convergent to the function u strongly in the space for and converges to strongly in the space for . Thus, we can prove the existence of a weak solution to equation (1.1).
with and . Since is a separable Banach space and is a bounded sequence in the dual space of X, there exists a subsequence of , still denoted by , weakly star convergent to a function v in . As weakly converges to in , as a result almost everywhere. Thus, we obtain . □
The authors are very grateful to the anonymous reviewers for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by NSFC grant 11301573 and in part by the funds of Chongqing Normal University (13XLB006 and 13XWB008) and in part by the Program of Chongqing Innovation Team Project in University under Grant No. KJTD201308.
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