The investigation of local weak solutions for a generalized Novikov equation
© Lai; licensee Springer. 2014
Received: 4 February 2014
Accepted: 27 May 2014
Published: 4 June 2014
The existence of local weak solutions for a generalized Novikov equation is established in the Sobolev space with . The pseudo-parabolic regularization technique and several estimates derived from the equation itself are used to prove the existence.
which has a matrix Lax pair [1, 2] and is shown to be related to a negative flow in the Sawada-Kotera hierarchy. Several conservation quantities and a bi-Hamiltonian structure were found in . Himonas and Holliman  applied the Galerkin-type approximation method to prove the well-posedness of strong solutions for Eq. (1) in the Sobolev space with on both the line and the circle. Its Hölder continuity properties were studied in Himonas and Holmes . The abstract Kato theorem was employed in Ni and Zhou  to show the existence and uniqueness of local strong solutions in the Sobolev space with . The persistence properties of the strong solution were found. The local well-posedness for the periodic Cauchy problem of the Novikov equation in the Sobolev space with is done in Tiglay . If the initial data are analytic, the existence and uniqueness of analytic solutions for Eq. (1) are obtained in . It is worthy to mention that if the Sobolev index and sign conditions hold, the orbit invariants are applied to show the existence of periodic global strong solution. The scattering theory is used by Hone et al.  to search for non-smooth explicit soliton solutions with multiple peaks for Eq. (1). This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations (see [8–13]).
Here, we address that all the generalized versions of the Novikov equation in previous works do not involve the nonlinearly dissipative terms . This is the motivation of our work to investigate Eq. (2). We establish the existence of local weak solutions for Eq. (2) in the lower order Sobolev space with . Several estimates of solutions for the associated regularized equation for Eq. (2) are derived to prove the existence.
This paper is organized as follows. The main results are given in Section 2. In Section 3, we prove the local existence and uniqueness of solutions for the associated regularized Novikov equation (2) by using a contraction argument. In Section 4, we derive that Eq. (2) subject to the initial value has a weak solution in the sense of a distribution.
2 Main result
where . Let denote the class of continuous functions from to where . We set . For simplicity, throughout this article, we let c denote any positive constant which is independent of parameter ε.
Before giving the main result of this work, we give two lemmas which are related to the regularized problem (3).
Lemma 2.1 Assume with . Then there exists a unique solution to Cauchy problem (3) in the space where depends on . If , the solution .
has a unique solution .
Lemma 2.2 If with such that . Let . Then there exist two constants and c independent of ε such that the solution of problem (4) satisfies for any .
Now we state the main result of this work.
Theorem 2.1 Assume with and . Then there exists a such that problem (5) has at least one weak solution in the sense of distribution and .
3 Proof of Lemma 2.1
where is independent of . Choosing T sufficiently small such that , we know that Γ is a contracted mapping and . This means that Γ maps to itself. By the contraction-mapping principle, we see that the mapping Γ has a unique fixed point v in .
The global existence result follows a routine argument by using the integral (14). □
4 Proofs of Lemma 2.2 and Theorem 2.1
Lemma 4.1 (Kato and Ponce )
where c is a constant depending only on r.
Lemma 4.2 (Kato and Ponce )
Integrating Eq. (26) with respect to t results in inequality (16).
where c is a constant independent of ε.
The proof of Lemma 4.4 can be found in .
where is independent of ε.
has a unique solution . Using the theorem at p.51 in  shows that there are constants and independent of ε such that for arbitrary , which leads to the conclusion of Lemma 2.2. □
where and any . 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 v strongly in the space for and converges to strongly in the space for . Now, we can prove the existence of a weak solution to Eq. (2).
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 u in . One derives from the being weakly convergent to in that almost everywhere. Thus, we obtain . □
This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).
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