- Open Access
The Cauchy problem for the modified Novikov equation
© Hou and Zheng; licensee Springer. 2014
- Received: 18 March 2014
- Accepted: 26 June 2014
- Published: 24 September 2014
In this paper, we are concerned with the Cauchy problem for the modified Novikov equation. By using the transport equation theory and Littlewood-Paley decomposition as well as nonhomogeneous Besov spaces, we prove that the Cauchy problem for the modified Novikov equation is locally well posed in the Besov space with and and show that the Cauchy problem for the modified Novikov equation is locally well posed in the Besov space with the aid of Osgood lemma.
MSC: 35G25, 35L05, 35R25.
- Cauchy problem
- modified Novikov equation
was discovered by Vladimir Novikov  and it possesses the bi-Hamiltonian structure, infinite conservation laws. The well-posedness and blow-up of the Cauchy problem for the Novikov equation in Sobolev spaces and Besov spaces have been investigated by some authors –. The weak solution of the Cauchy problem for the Novikov equation has been investigated by some authors , , . Recently, Li and Yan  considered the Cauchy problem for the KdV equation with higher dispersion.
To the best of our knowledge, the well-posedness and blow-up of the Cauchy problem for (1.3) and (1.4) in Besov spaces are open up to now. More precisely, in this paper, motivated by , , using Littlewood-Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for (1.4) is locally well posed in the Besov space with and we give a blow-up criterion.
The main results of this paper are as follows.
Letandand. Then there exists a timesuch that problem (1.3) and (1.4) has a unique solutionin. The mapis continuous from a neighborhood ofinintofor every. When, the solution to problem (1.3) and (1.4) is continuous in.
When, (1.3) and (1.4) is locally well posed in the sense of Hadamard.
The remainder of this paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we establish local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces. In Section 4, we prove Theorem 1.2.
There exists a couple of smooth radial functionsvalued insuch thatis supported in the ballandis supported in the ring. Moreover,
with the aid of Young’s inequality, where is a positive constant independent of .
In particular, . Let , and . Define .
Topological properties: is a Banach space which is continuously embedded in .
Density: is dense in .
- (3)Embedding: if and .
Algebraic properties: , is a Banach algebra. is a Banach algebra ⇔ ⇔ or ( and ). In particular, is continuously embedded in and is a Banach algebra.
1-D Moser-type estimates:
- (i)For ,
- (ii)( if ) and , we have
- (6)Complex interpolation:
- (7)Real interpolation: , , , there exists a constant such that
- (8)Fatou lemma: if is bounded in and in , then and
Let and be an -multiplier (i.e., is smooth and satisfies that , ∋ a constant , s.t. for all ). Then the operator is continuous from to . Notice that is continuous from to and is continuous from to .
The usual product is continuous from to .
- (11)There exists a constant such that the following interpolation inequality holds:
(A priori estimates in Besov spaces)
- (1)If or , then
- (2)If , and and , then
If , then for all , (1) holds true when .
If , then . If , then for all .
(Existence and uniqueness)
Let, , , andbe as in the statement of Lemma 2.3. Assume thatfor someandandiforandandif. Then problem (2.2) and (2.3) has a unique solutionand the inequalities of Lemma 2.3can hold true. Moreover, if, then.
By using the following six steps, we will complete the proof of Theorem 1.1.
It is easily checked that , by using Lemma 2.4 and the inductive method, for all , we have that (3.1) and (3.2) has a global solution which belongs to .
for all .
Combining (3.6)-(3.7) with (3.4), we have (3.3).
which yields .
Third step: Convergence. We will derive that is a Cauchy sequence in .
From (3.31), we have that . Thus, . Consequently, is a Cauchy sequence in ; moreover, is convergent to some limit function .
Consequently, is a Cauchy sequence in and converges to some limit function .
Fourth step: Existence of solution in. Existence of solution can be proved similarly to .
Fifth step: Uniqueness of solution. We consider case and case , respectively. In fact, this can be proved similarly to .
Sixth step: Continuity with respect to the initial data. Continuity with respect to the initial data can be proved similarly to .
Thus, is uniformly bounded in . From (3.1), by using (4) in Lemma 2.2, we can prove that is uniformly bounded with respect to in .
and with the aid of (2.1), we can prove that is a Cauchy sequence in .
We would like to thank the reviewers for careful reading and valuable comments on the original draft.
- Zhao L, Zhou SG: Symbolic analysis and exact travelling wave solution to a new modified Novikov equation. Appl. Math. Comput. 2010, 217: 590-598. 10.1016/j.amc.2010.05.093MathSciNetView ArticleGoogle Scholar
- Novikov VS: Generalizations of the Camassa-Holm equation. J. Phys. A 2009., 42: 10.1088/1751-8113/42/34/342002Google Scholar
- Ni L, Zhou Y: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 2011, 250: 3002-3021. 10.1016/j.jde.2011.01.030MathSciNetView ArticleGoogle Scholar
- Wu XL, Yin ZY: Global weak solutions for the Novikov equation. J. Phys. A, Math. Theor. 2011., 44: 10.1088/1751-8113/44/5/055202Google Scholar
- Wu XL, Yin ZY: A note on the Cauchy problem for the Novikov equation. Appl. Anal. 2013, 92: 1116-1137. 10.1080/00036811.2011.649735MathSciNetView ArticleGoogle Scholar
- Yan W, Li YS, Zhang YM: The Cauchy problem for the integrable Novikov equation. J. Differ. Equ. 2012, 253: 298-318. 10.1016/j.jde.2012.03.015MathSciNetView ArticleGoogle Scholar
- Yan W, Li YS, Zhang YM: The Cauchy problem for the Novikov equation. Nonlinear Differ. Equ. Appl. 2013, 20: 1157-1169. 10.1007/s00030-012-0202-1MathSciNetView ArticleGoogle Scholar
- Lai SY: Global weak solutions to the Novikov equation. J. Funct. Anal. 2013, 265: 520-544. 10.1016/j.jfa.2013.05.022MathSciNetView ArticleGoogle Scholar
- Li Y, Yan W: Well-posedness and ill-posedness of KdV equation with higher dispersion. J. Math. Anal. Appl. 2014, 416: 647-658. 10.1016/j.jmaa.2014.01.035MathSciNetView ArticleGoogle Scholar
- Danchin R: A few remarks on the Camassa-Holm equation. Differ. Integral Equ. 2001, 14: 953-988.MathSciNetGoogle Scholar
- Danchin R: Fourier Analysis Method for PDEs. 2005.Google Scholar
- Chemin JY: Localization in Fourier space and Navier-Stokes. In Phase Space Analysis of Partial Differential Equations. Scuola Norm. Sup., Pisa; 2004:53-136.Google Scholar
- Danchin R: A note on well-posedness for Camassa-Holm equation. J. Differ. Equ. 2003, 192: 429-444. 10.1016/S0022-0396(03)00096-2MathSciNetView ArticleGoogle Scholar
- Yan W, Li YS, Zhang YM: The Cauchy problem for the generalized Camassa-Holm equation. Appl. Anal. 2014, 93: 1358-1381. 10.1080/00036811.2013.833325MathSciNetView ArticleGoogle Scholar
- Yan W, Li YS, Zhang YM: The Cauchy problem for the generalized Camassa-Holm equation in Besov space. J. Differ. Equ. 2014, 256: 2876-2901. 10.1016/j.jde.2014.01.023MathSciNetView ArticleGoogle Scholar
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