Abstract
This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev space with . We also establish the ill-posedness for the initial data in with . Thus, the regularity requirement for the fifth-order dispersive equations is sharp.
1. Introduction
In this paper, we consider the Cauchy problem and existence of solitary waves of the following fifth-order dispersive equation: Obviously, (1) can be seen as the higher modification of the following dispersive equation:
We recall that where which has been studied in [1], is the generalized Camassa-Holm equation. In fact, Hakkaev and Kirchev [1] studied the local well-posedness and orbital stability and instability of (4) with the aid of the pseudoparabolic regularization and spectral analysis. Obviously, when , (4) is the well-known Camassa-Holm equation; when and , (4) becomes (3). Some people consider the Cauchy problem for the higher modification of the nonlocal form of Camassa-Holm equation; we refer the readers to [2–5].
By acting on both sides of (1), we obtain the following equivalent form: Obviously, (6) is a modification of mKdV equation which has been intensively studied; we refer the readers to [6–11]. In [8], the authors proved that (7) is locally well-posed for the initial data in with . The regularity requirement for the modified KdV equation is sharp; see [9]. In [11], by using the -method which can be seen in [11–14] and Mirua transformation which can be seen in [15], the authors proved that the modified KdV equation is globally well-posed for the initial data in with . In [16], by using the dyadic bilinear estimates and resolution spaces which can be seen in [17, 18], the author proved that the modified KdV equation is globally well-posed for the initial data in with .
In this paper, by using the Fourier restriction norm method introduced in [19], we prove that (1) and (2) are locally well-posed for the initial data in with . When , , we prove that the Cauchy problem for (1) is globally well-posed in . By using the general well-posedness principle proposed by [20], we establish the ill-posedness for the initial data in with . Thus, the requirement for regularity is sharp.
Before stating the main results, we introduce some notations and definitions. We use when , where ; we use when , where is some positive number which is larger than 2. for any , and is the Fourier transform of with respect to its all variables. is the Fourier transform of with respect to its space variables. Denote . () is the Schwartz space and is its dual space. is the usual Sobolev space with norm . For any , is the Bourgain space with phase function . That is, a function in belongs to if and only if For any given interval , is the space of the restriction of all functions in on , and for its norm is We write for when . Let .
The main results of this paper are as follows.
Theorem 1. The Cauchy problem for (1) is locally well-posed for the initial data in with .
Theorem 2. Let . Then (1) and (2) are ill-posed in in the sense that the solution map of the Cauchy problem for (1) is not Lipschitz continuous at zero. More precisely, for any , the solution map is not Lipschitz continuous at zero.
The rest of this paper is arranged as follows. In Section 2, we make some preliminaries. In Section 3, we establish two crucial trilinear estimates. In Section 4, we prove Theorem 1. In Section 5, we prove Theorem 2.
2. Preliminaries
Lemma 3. Let . Then
Proof. From [21], we have From Plancherel’s identity, we have Interpolating (15) with (16) yields (14). Equations (11)–(13) can be seen in [21, 22].
Lemma 4. For , , and , we have and for , we have
Equations (17) and (18) can be seen in [6, 23].
Lemma 5. Let . Then
Lemma 6. Let denote the Riesz potential of order and , . Then the following estimate holds true: where
In particular, when and , we have
3. Trilinear Estimates
In this section, we will prove two crucial trilinear estimates.
Lemma 7. Let , , , . Then
Proof. Let
By duality and Plancherel’s identity, to derive (23), it suffices to prove
where
Without loss of generality, we may assume that and . By the symmetry among , , , without loss of generality, we may assume that . Obviously, , where
In this lemma, integrals over the subregion ’s are, respectively, denoted as (, ). Consider
(1) Subregion . In this subregion, since and , we have
By using Cauchy Schwartz’s inequality and Plancherel’s identity as well as Hölder’s inequality, (11), and , we derive
(2) Subregion . In this subregion, and , since , we have
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (22) of Lemma 6, (11), , and , since , we obtain
(3) Subregion . In this subregion, since and , we have
This case can be treated similarly to .
(4) Subregion . In this subregion and .
When , since and , we have
This case can be treated similarly to .
When , we consider the case and the case .
When , since , we have
This case can be treated similarly to .
When , From Lemma 5, we have
which yields ; thus one of the following four cases must occur:
When (37) is valid, since
we have
when , since and , we have
when , since and , we have
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), , we obtain
When (38) is valid, since
we have
when , we have
when , since and , we have
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), , and , we obtain
Cases (39) and (40) can be treated similarly to case (38).
(5) Subregion . In this subregion, since , which yields
Thus, applying Cauchy-Schwartz’s inequality and Plancherel’s identity, (12) and (13), we have
(6) Subregion . In this subregion, since , we have and and . From Lemma 5, we have
which yields ; thus one of the following four cases must occur:
When , we have
this case can be handled similarly to .
When , we consider (54), (55), (56), and (57), respectively.
When (37) is valid, since , and as well as , we have
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), , we obtain
When (55) holds, since , and as well as , we have
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), , and , we obtain
Cases (56) and (57) can be treated similarly to case (55).
Putting the estimates of (, ) together, we have (25).
Thus, we complete the proof of Lemma 7.
Lemma 8. Let , , , . Then
Lemma 8 can be proved similarly to Lemma 7.
4. The Proof of Theorem 1
In order to prove Theorem 1, firstly, for and , , we define by where By using Lemmas 4, 7, and 8, we have where , , of (66) concord with , , and of Lemmas 7 and 8. Let where . From (66) and (67), we know that is a mapping from the closed ball into itself. Similarly, we have thus, is a contraction mapping from the closed ball into itself; by using Banach fixed point Theorem, we have .
The rest of local well-posedness of Theorem 1 follows from a standard proof.
5. The Proof of Theorem 2
In this section, we prove Theorem 2.
By contradiction, we assume that the solution map of (1) and (2) is Lipschitz continuous at zero with . From the general well-posedness principle of [20], we must have where We consider the initial data where and . Thus, we have which can be seen in [24], where denotes the characteristic function of a set . It is easy to check that . Let and and . We have and thus where where Since , we define resulting from Lemma 5 and . To estimate , we consider the following three cases. Case 1: . Case 2: . Case 3: , , or , , or , , or , .The integrals in (76) corresponding to Cases 1, 2, and 3 are denoted as , , , respectively.
Case 1. In this case, from Lemma 6, we have and . Since , we have
Case 2. In this case, from Lemma 6, we have and . Since , we have
Case 3. In this case , due to Lemma 6, and . Since , we have From (70), we have which yields When , from (83), we have which yields ; we obtain a contradiction since . When , from (83), we have ; we obtain a contradiction since .
The proof of Theorem 2 is completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research of the first author is supported by Foundation and Frontiers of Henan province under nos. 122300410414 and 132300410432.