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The investigation of local weak solutions for a generalized Novikov equation
Journal of Inequalities and Applications volume 2014, Article number: 230 (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.
Recently, many scholars have paid attention to the study of the integrable Novikov equation 
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]).
In this work, we study the following generalized dissipative Novikov equation:
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
The space of all infinitely differentiable functions with compact support in is denoted by . We let () be the space of all measurable functions such that . We define with the standard norm . For any real number s, we let denote the Sobolev space with the norm defined by
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.
For the generalized Novikov equation (2), we consider the Cauchy problem of its associated regularized equation
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 .
For a real number s with , suppose that the function , and let be the convolution of the function and such that the Fourier transform of φ satisfies , , and for any . Then we have . It follows from Lemma 2.1 that for each ε satisfying , the Cauchy problem
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 .
We write the Cauchy problem for Eq. (2)
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
Lemma 3.1 Let r and q be real numbers such that , then
Proof of Lemma 2.1 Defining the operator , we know that is a bounded linear operator. Using the operator D to both sides of the first equation of problem (3) and then integrating the resultant equation over the interval give rise to
We choose that v and g belong to the closed ball of radius about the zero function in and Γ is the operator on the right-hand side of Eq. (6). For fixed , we obtain
where may depend on ε. Applying the algebraic property of the space with , we will give estimates for the right-hand side of inequality (7). In fact, we have
The first inequality of Lemma 3.1 yields
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 .
For , using the first equation of system (3) gives rise to
which derives the conservation law
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 )
If , then is an algebra. Moreover,
where c is a constant depending only on r.
Lemma 4.2 (Kato and Ponce )
Let . If and , then
Lemma 4.3 Let and the function is a solution of problem (3) and the initial data . Then
If , there is a constant c independent of ε such that
If , there is a constant c independent of ε such that
Using and the Parseval equality, we have
If , using to multiply both sides of the first equation of system (3) and combining with Eq. (18), we get
We will estimate every term on the right-hand side of Eq. (19), separately. By using Cauchy-Schwarz inequality and Lemmas 4.1 and 4.2, we have
For the third term, using , the Cauchy-Schwarz inequality and Lemma 4.1, we obtain
For the fourth term, writing , we get
in which we have used
Using Lemma 4.1 repeatedly results in
For the last term in Eq. (19), using Lemma 4.1 yields
Integrating Eq. (26) with respect to t results in inequality (16).
Using the operator to both sides of the first equation of system (3), we obtain
Applying to both sides of Eq. (27) for gives rise to
In fact, we have
We know the identity
Using the Cauchy-Schwarz inequality, Lemma 4.1, and yields
Furthermore, we have
For the last term in Eq. (28), we get
Applying the Cauchy-Schwarz inequality and Lemmas 4.1 and 4.2 gives rise to
Lemma 4.4 The following estimates hold for any ε with and
where c is a constant independent of ε.
The proof of Lemma 4.4 can be found in .
Remark For , using , , Eqs. (15), (37), and (38), we know
where is independent of ε.
Proof of Lemma 2.2 For simplicity, we use notation and differentiate Eq. (27) with respect to x to obtain
Letting be an integer and multiplying the above equation by and then integrating the resulting equation with respect to x yield
applying the Hölder inequality yields
Since as for any , we integrate both sides of the inequality (44) with respect to t and take the limit as to obtain
Using the inequality (41) yields
Using Eq. (46), Lemma 4.1, and Lemma 4.3 gives rise to
From Eqs. (37), (38), (49), and (50), one has
For and , it follows from Eqs. (45), (48), and (51) that
It follows from the contraction-mapping principle that there is a such that the equation
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. □
Using the conclusion of Lemmas 2.2 and 4.3, Eq. (41), , the notation , and the Gronwall inequality results in the inequalities
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).
Proof of Theorem 2.1 From Lemma 2.2, we know that () is bounded in the space . Thus, the sequences and () are weakly convergent to v and in for any , respectively. Therefore, v satisfies the equation
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 . □
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This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).
The author declares that he has no competing interests.
The article is work of one author, who approves the final manuscript.
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Lai, S. The investigation of local weak solutions for a generalized Novikov equation. J Inequal Appl 2014, 230 (2014). https://doi.org/10.1186/1029-242X-2014-230
- a generalized Novikov equation
- local weak solution