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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 810725, 11 pages
On the Cauchy Problem for the Two-Component Novikov Equation
1College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China
2College of Mathematics and Computer Sciences, Yangtze Normal University, Chongqing, Fuling 408100, China
Received 9 January 2013; Accepted 27 March 2013
Academic Editor: M. Lakshmanan
Copyright © 2013 Yongsheng Mi 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.
We are concerned with the Cauchy problem of two-component Novikov equation, which was proposed by Geng and Xue (2009). We establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory which is motivated by that in Danchin's cerebrated paper (2001). Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend some results of Himonas (2003) to more general equations.
In this paper, we consider the following Cauchy problem of the two-component Novikov equation The two-component system in (1) was found by Geng and Xue . It was shown in  that the system (1) is exactly a negative flow in the hierarchy and admits exact solutions with -peakons and an infinite sequence of conserved quantities. Moreover, a reduction of this hierarchy and its Hamiltonian structures are discussed.
For , (1) becomes the Degasperis-Procesi equation Degasperis et al.  proved the formal integrability of (2) by constructing a Lax pair. They also showed that it has bi-Hamiltonian structure and an infinite sequence of conserved quantities and admits exact peakon solutions. The direct and inverse scattering approach to pursue it can be seen in . Moreover, in , they also presented that the Degasperis-Procesi equation has a bi-Hamiltonian structure and an infinite number of conservation laws and admits exact peakon solutions which are analogous to the Camassa-Holm peakons. It is worth pointing out that solutions of this type are not mere abstractizations: the peakons replicate a feature that is characteristic for the waves of great height-waves of the largest amplitude that are exact solutions of the governing equations for irrotational water waves (cf. the papers [5–7]). The Degasperis-Procesi equation is a model for nonlinear shallow water dynamics (cf. the discussion in ). The numerical stability of solitons and peakons, the multisoliton solutions, and their peakon limits, together with an inverse scattering method to compute -peakon solutions to Degasperis-Procesi equation, have been investigated, respectively, in [9–11]. Furthermore, the traveling wave solutions and the classification of all weak traveling wave solutions to Degasperis-Procesi equation were presented in [12, 13].
For , (1) becomes the Novikov equation which has been recently discovered by Vladimir Novikov in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity . The perturbative symmetry approach yields necessary conditions for a PDE to admit infinitely many symmetries. Using this approach, Novikov was able to isolate (3) and find its first few symmetries, and he subsequently found a scalar Lax pair for it, then proved that the equation is integrable, which can be thought as a generalization of the Camassa-Holm equation. In , it is shown that the Novikov equation admits peakon solutions like the Camassa-Holm. Also, it has a Lax pair in matrix form and a bi-Hamiltonian structure. Furthermore, it has infinitely many conserved quantities, like Camassa-Holm. The most important quantity conserved by a solution to Novikov equation is its -norm , which plays an important role in the study of (1). In [16–19], the authors study well-posedness and dependence on initial data for the Cauchy problem for Novikov equation. Recently, in , a global existence result and conditions on the initial data were considered. Existence and uniqueness of global weak solution to Novikov equation with initial data under some conditions was proved in . The Novikov equation with dissipative term was considered in . Multipeakon solutions were studied in [15, 23]. The Cauchy problem of the Novikov equation on the circle was investigated in . An alternative modified Camassa-Holm equation was introduced in .
Motivated by the references cited above, the goal of the present paper is to establish the local well-posedness for the strong solutions to the Cauchy problem (1). The proof of the local well-posedness is inspired by the argument of approximate solutions by Danchin  in the study of the local wellposedness to the Camassa-Holm equation. However, one problematic issue is that we here deal with two-component system with a higher order nonlinearity in the Besov spaces, making the proof of several required nonlinear estimates somewhat delicate. These difficulties are nevertheless overcome by careful estimates for each iterative approximation of solutions to (1). Moreover, we also prove the analyticity of its solutions in both variables, with in and in an interval around zero, provided that the initial profile is an analytic function on the real line. Hence, this analytic result can be viewed as a Cauchy-Kowalevski theorem for (1).
Theorem 1. Let and . Assume that . 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 .
Theorem 2. If the initial data is real analytic on the line and belongs to a space , for some , then there exist an and a unique solution to the Cauchy problem (1) that is analytic on .
The rest of this paper is organized as follows. In Section 2, we prove the local well-posedness of the initial value problem (1) in the Besov space. Section 3 is devoted to the study of the analyticity of the Cauchy problem (1) based on a contraction type argument in a suitably chosen scale of the Banach spaces.
2. Local Well-Posedness in the Besov Spaces
In this section, we will establish local well-posedness of the initial value problem (1) in the Besov spaces.
First, for the convenience of the readers, we recall some facts on the Littlewood-Paley decomposition and some useful lemmas.
Notation. stands for the Schwartz space of smooth functions over whose derivatives of all order decay at infinity. The set of temperate distributions is the dual set of for the usual pairing. We denote the norm of the Lebesgue space by with , and the norm in the Sobolev space with by .
Proposition 3 (Littlewood-Paley decomposition ). Let , and , . There exist two radial functions and such that
Furthermore, let and . Then for all , the dyadic operators and can be defined as follows: Hence, where the right-hand side is called the nonhomogeneous Littlewood-Paley decomposition of .
Lemma 4 (Bernstein's inequality ). Let be a ball with center in and a ring with center in . A constant exists so that, for any positive real number , any nonnegative integer , any smooth homogeneous function of degree , and any couple of real numbers with , there hold for any function .
Definition 5 (Besov space). Let . The inhomogeneous Besov space ( for short) is defined by where If , .
Proposition 6 (see ). Suppose that , , . One has the following.Topological properties: is a Banach space which is continuously embedded in . Density: is dense in , . Embedding: , if and . locally compact, if . Algebraic properties: for all , is an algebra. Moreover, is an algebra, provided that or and . Complex interpolation:Fatou lemma: If is bounded in and in , then andLet and f be an -multiplier (i.e., is smooth and satisfies that for all , there exists a constant , s.t. for all ). Then the operator is continuous from to .
Now we state some useful results in the transport equation theory, which are crucial to the proofs of our main theorems later.
Lemma 7 (see [26, 28]). Suppose that and . Let be a vector field such that belongs to if or to otherwise. Suppose also that , and that solves the -dimensional linear transport equations Then there exists a constant depending only on , and such that the following statements hold. If or , then or hold, where if and else. If and , and , then with . If , then for all , the estimate (14) holds with . If , then . If , then for all .
Lemma 8 (existence and uniqueness see [26, 28]). Let and with . Assume that , . Let be a time-dependent vector field such that for some , and if and if or and . Then the transport equations have a unique solution and the inequalities in Lemma 7 hold true. Moreover, , then one has .
Lemma 9 (1- Morse-type estimates [26, 28]). Assume that , , the following estimates hold.(i) For , (ii) For all ( if ) and , one has (iii) In Sobolev spaces , one has for , where is a positive constant independent of and .
Definition 10. For , and , we set
In the following, we denote a generic constant only depending on , , . Uniqueness and continuity with respect to the initial data are an immediate consequence of the following result.
Proposition 11. Let , and . Suppose that be two given solutions of the initial-value problem (1) with the initial data . Then for every , one has
Proof. Denote , , and . It is obvious that which implies that and solves the transport equations with According to Lemma 7, we have For , by Lemma 9, we have Therefore, inserting the above estimates to (24), we obtain Hence, thanks to and then applying the Gronwall's inequality, we reach (20).
Now let us start the proof of Theorem 1, which is motivated by the proof of local existence theorem about the Camassa-Holm equation in . Firstly, we will use the classical Friedrichs’ regularization method to construct the approximate solutions to the Cauchy problem (14).
Lemma 12. Assume that . Let , , and . Then there exists a sequence of smooth functions solving the following linear transport equation by induction: Moreover, there is a positive such that the solutions satisfying the following properties:(i) is uniformly bounded in ,(ii) is a Cauchy sequence in .
Proof. Since all the data and belong to , Lemma 8 enables us to show by induction that for all , (28) has a global solution which belongs to . Thanks to Lemma 7 and the proof of Proposition 11, we have the following inequality for all :
Thanks to , we find is an algebra. From this, one obtains
which along with the above inequality leads to
Let us choose a such that , and suppose by induction that for all Indeed, since is an algebra, one obtains from (32) that for any And then inserting (33) and (32) into (31) leads to Hence, one can see that which implies that is uniformly bounded in . Using the Moser-type estimates (see Lemma 9), one finds that Hence, using (28), we have uniformly bounded, which yields that the sequence is uniformly bounded in .
Now, it suffices to show that is a Cauchy sequence in . In fact, for all , from (28), we have with Similar to the proof of Proposition 11, then for every , we obtain Since is uniformly bounded in and we get a constant independent of such that for all , Arguing by induction with respect to the index , one can easily prove that As , , and are bounded independently of , there exists constant independent of such that Thus, is a Cauchy sequence in .
Proof of Theorem 1. Thanks to Lemma 12, we obtain that is a Cauchy sequence in ; so, it converges to some function . We now have to check that belongs to and solves the Cauchy problem (1). Since is uniformly bounded in , according to Lemma 12, the Fatou property for the Besov spaces (Proposition 6) guarantees that also belongs to
On the other hand, as converges to in , an interpolation argument ensures that the convergence holds in , for any . It is then easy to pass to the limit in (28) and to conclude that is indeed a solution to the Cauchy problem (1). Thanks to the fact that belongs to , the right-hand side of the equation belongs to , and the right-hand side of the equation belongs to . In particular, for the case , Lemma 8 enables us to conclude that for any . Finally, using the equation again, we see that if , and otherwise. Therefore, belongs to . Moreover, a standard use of a sequence of viscosity approximate solutions for the Cauchy problem (1) which converges uniformly in gives the continuity of the solution in . The proof of Theorem 1 is complete.
3. Analyticity of Solutions
In this section, we will show the existence and uniqueness of analytic solutions to the system (1) on the line .
First, we will need a suitable scale of Banach spaces as follows. For any , we set where is the Sobolev space of order two on the real line and is the set of nonnegative integers. One can easily verify that equipped with the norm is a Banach space and that, for any , is continuously embedded in with Another simple consequence of the definition is that any in is a real analytic function on . Crucial for our purposes is the fact that each forms an algebra under pointwise multiplication of functions.
Lemma 13 (see ). Let . There is a constant , independent of , such that for any and in , one has
Lemma 14 (see ). There is a constant such that for any , one has , , and .
Theorem 15 (see ). Let be a scale of decreasing Banach spaces; namely, for any , one has and . Consider the Cauchy problem Let , , and be positive constants, and assume that satisfies the following conditions. If for the function is holomorphic in and continuous on with values in and then is a holomorphic function on with values in . For any and any , , that is, , , one has There exists such that for any , Then, there exists a and a unique function , which for every is holomorphic in with values in and is a solution to the Cauchy problem (51).
Next, we restate the Cauchy problem (1) in a more convenient form, and we can rewrite the Cauchy problem (1) as follows: Applying the operator to both sides of the first equation and second equation in (55), we obtain Differentiating with respect to on both sides of the previous equation and letting , , , and , then the problem (56) can be written as a system for , , , as follows:
Define Then, we have
Proof of Theorem 2. Theorem 2 is a straightforward consequence of the abstract Cauchy-Kowalevski Theorem 15. We only need to verify conditions (1)-(2) in the statement of the abstract Cauchy-Kowalevski Theorem 15 for