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The Cauchy problem for the seventh-order dispersive equation in Sobolev space
Boundary Value Problems volume 2014, Article number: 122 (2014)
Abstract
This paper is devoted to the Cauchy problem for the higher-order dispersive equation , . The local well-posedness of the associated Cauchy problem is established in Sobolev space with with the aid of the Fourier restriction norm method.
MSC:35K30.
1 Introduction
In this paper, we are concerned with the Cauchy problem for the following seventh-order dispersive equation:
Kenig et al. [1] established that
is locally well-posed in some weighted Sobolev spaces for small initial data and for arbitrary initial data. Recently, Pilod [2] studied the following higher-order nonlinear dispersive equation:
where and u is a real- (or complex-) valued function and proved it is locally well-posed in weighted Besov and Sobolev spaces for small initial data and proved ill-posedness results when for some in the sense that (1.5) cannot have its flow map at the origin in . Very recently, Guo et al. [3] studied the Cauchy problem for
and he proved that it is locally well-posed in with with the aid of a short time Bourgain space.
In this paper, inspired by [1–5], by using the Fourier restriction norm method, we establish that (1.1)-(1.2) is locally well-posed in Sobolev space with .
Now we give some notations and definitions. Throughout this paper, we always assume that ψ is a smooth function, , satisfying , when , and , (),
for any , and denotes the Fourier transformation of u with respect to its all variables. denotes the Fourier inverse transformation of u with respect to its all variables. denotes the Fourier transformation of u with respect to its space variable. denotes the Fourier inverse transformation of u with respect to its space variable. is the Schwarz space and is its dual space. is the Sobolev space with norm . For any , is the Bourgain space with phase function . That is, a function in belongs to iff
For any given interval L, is the space of the restriction of all functions in on , and for its norm is
When , is abbreviated as .
The main result of this paper is as follows.
Theorem 1.1 Assume that with . Then the Cauchy problem for (1.1) is locally well-posed.
The remainder of paper is arranged as follows. In Section 2, we make some preliminaries. In Section 3, we give an important bilinear estimate. In Section 4, we establish Theorem 1.1.
2 Preliminaries
Lemma 2.1 Let . Then
Proof For the proof of (2.1)-(2.5), we refer the readers to Lemma 2.1 of [5].
We have completed the proof of Lemma 2.1. □
Lemma 2.2 Assume that . Then
where
Lemma 2.2 is the case of of Lemma 3.1 of [5].
Lemma 2.3 For any , and , for , we have
For , we have
Lemma 2.3 can be found as Lemma 2.4 of [6].
3 Bilinear estimates
In this section, we will give an important bilinear estimate.
We give an important relation before proving the bilinear estimate.
where
Lemma 3.1 Let , , where , . Then
Proof Let
To establish (3.2), it is sufficient to derive the following inequality:
where
Without loss of generality, we assume that , (). To derive (3.3), it suffices to prove that
By using the symmetry between and , without loss of generality, we can assume that . Obviously,
where
We will denote the integrals in (3.5) corresponding to () by (), respectively. Let , , .
-
(1)
Subregion . Since , we have , which yields
Then, by the Plancherel identity, the Hölder inequality, and , we derive
-
(2)
Subregion . In this subregion, obviously, .
It is easily checked that
Consequently, by the Cauchy-Schwarz inequality and Lemma 2.2, we have
-
(3)
Subregion . In this subregion, we derive . Thus,
-
(i)
Case . By (3.1), we derive
If , then
If , then
This case can be proved similarly to .
-
(ii)
Case . Since , we have
If , we have
consequently, by using the Cauchy-Schwarz inequality and (2.5) and (2.4), we have
If , since , we have
This case can be proved similarly to the above case.
-
(iii)
Case . This case is similar to (ii) case .
-
(4)
Subregion . In this subregion, , and it is easy to obtain
-
(i)
Case . By using, , when , we have
This case can be proved similarly to Subregion . When , we have
If , since , then
By using the Cauchy-Schwarz inequality, we have
-
(ii)
Case . Since , by using , we have
This case can be proved similarly to .
-
(iii)
Case .
This case can be proved similarly to .
-
(5)
Subregion . In this region , thus, we have
-
(i)
If , by using (3.1) and , we have
By using the Plancherel identity, the Hölder inequality, and as well as (2.5), we have
-
(ii)
If , then . By using (3.1), we have
By using the Plancherel identity, the Hölder inequality, (2.5) and , we have
-
(iii)
If .
This case can be proved similarly to the case .
-
(6)
Subregion . In this region, we have .
This case can be proved similarly to the Subregion .
We have completed the proof of Lemma 3.1. □
4 Proof of Theorem 1.1
The system (1.1)-(1.2) is equivalent to the following integral equation:
We define
Combining Lemmas 2.3 and 3.1 with the fixed point theorem, we easily obtain Theorem 1.1.
References
Kenig CE, Ponce G, Vega L: Higher-order nonlinear dispersive equations. Proc. Am. Math. Soc. 1994, 122: 157-166.
Pilod D: On the Cauchy problem for higher-order nonlinear dispersive equations. J. Differ. Equ. 2008, 245: 2055-2077.
Guo ZH, Kwak C, Kwon S: Rough solutions of the fifth-order KdV equations. J. Funct. Anal. 2013, 265: 2791-2829.
Bourgain J: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: the KdV equation. Geom. Funct. Anal. 1993, 3: 209-262.
Li Y, Yan W, Yang X: Well-posedness of a higher order modified Camassa-Holm equation in spaces of low regularity. J. Evol. Equ. 2010, 10: 465-486.
Li Y, Li S, Yan W: Sharp well-posedness and ill-posedness of a higher-order modified Camassa-Holm equation. Differ. Integral Equ. 2012, 25: 1053-1074.
Acknowledgements
We would like to thank reviewers for a careful reading and valuable comments on the original draft. The first author is supported by Foundation and Frontier of Henan Province under grant Nos. 122300410414, 132300410432.
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Wang, H., Zheng, Y. The Cauchy problem for the seventh-order dispersive equation in Sobolev space. Bound Value Probl 2014, 122 (2014). https://doi.org/10.1186/1687-2770-2014-122
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DOI: https://doi.org/10.1186/1687-2770-2014-122
Keywords
- Cauchy problem
- well-posedness
- Sobolev spaces