Abstract
The local well-posedness for a generalized periodic coupled Camassa-Holm system is established in the Sobolev space with . A wave-breaking criterion of strong solutions is acquired in the Sobolev space with by employing the localization analysis in the transport equation theory and a sufficient condition of global existence for the system is derived in the Sobolev space with .
MSC: 35D05, 35G25, 35L05, 35Q35.
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1 Introduction
In this article, we consider a generalized periodic coupled Camassa-Holm system on the circle with (the circle of unit length):
where and are periodic on the -variable and is taken as a constant and is the set of real numbers. In fact, system (1) is a generalization of two components for the following equation (if in system (1)):
Equation (2) is firstly derived as the Euler-Poincaré differential equation on the Bott-Virasoro group with respect to the metric [1], and it is known as a modified Camassa-Holm equation and also viewed as a geodesic equation on some diffeomorphism group [1]. It is shown in [1] that the dynamics of Eq. (2) on the unit circle is significant different from those of Camassa-Holm equation. For example, Eq. (2) does not conform with a blow-up solution in finite time.
If and in system (1), system (1) becomes the famous two-component Camassa-Holm system,
where the variable represents the horizontal velocity of the fluid, and is related to the free surface elevation from equilibrium with the boundary assumptions, and as . System (3) was found originally in [2], but it was firstly derived rigorously by Constantin and Ivanov [3]. The system has bi-Hamiltonian structure and is complete integrability. Since the birth of the system, a lot of literature was devoted to the investigation of the two-component Camassa-Holm system, for example, Chen et al.[4] established a reciprocal transformation between the two-component Camassa-Holm system and the first negative flow of the AKNS hierarchy. Escher et al.[5] used Kato theory to establish local well-posedness for the two-component system and presented some precise blow-up scenarios for strong solutions of the system. Gui and Liu [6], [7] established the local well-posedness for the two-component Camassa-Holm system in the Besov spaces and derived the wave-breaking mechanism and the exact blow-up rate. The dynamics in the periodic case for system (3) was considered in [8]. The other results related to the system can be found in [9]–[21].
If in system (1), system (1) becomes a modified version of the two-component Camassa-Holm system,
where denotes the velocity field, and represent the average density (or depth) and pointwise density (or depth). System (4) is introduced by Holm et al. in [22] and is viewed as geodesic motion on the semidirect product Lie group with respect to a certain metric [22]. System (4) admits peaked solutions in the velocity and average density [22], but it is not integrable unlike system (3). For some other recent work one is referred to Refs. [23]–[26] for details.
The motivations of the present paper is to find whether or not system (1) has some different dynamics from system (4) mathematically, such as wave breaking and a global solution. Comparing with the modified two-component Camasssa-Holm equation [23], we investigate the local well-posedness, global existence, and a wave-breaking criterion in the Sobolev space. One of the difficulties is the acquisition of the priori estimates . The difficulty has been overcome by Lemma 4.8. We use the technique of [7], [27] to derive a wave-breaking criterion for strong solutions of the system (1) in the low Sobolev spaces with . It needs to point out that in the Sobolev spaces with the wave-breaking of the solution of system (4) only depends on the slope of the component of solution [7]. However, the wave-breaking of the solution for system (1) is determined only by the slope of the component of solution definitely. It implies that there are differences between system (1) and system (4). On the other hand, we derive a sufficient condition for global solution in the Sobolev space with , which can be done because and can be controlled by and separately if .
The rest of this paper is organized as follows. Section 2 states the main results of present paper. Section 3 is devoted to the study of the local existence and uniqueness of a solution for system (1) by using the Kato theorem. In Section 4, we employ the transport equation theory to prove a wave-breaking criterion in the low Sobolev space with . The global existence result for system (1) is proved in Section 5.
2 The main results
We denote by ∗ the convolution and let denote the commutator between and . Note that if , then for all and (see [1]). We let denote all of different positive constants which depend only on initial data. To investigate dynamics of system (1) for the Cauchy problem on the circle, we rewrite system (1) by taking and :
The main results of the present paper are listed as follows.
Theorem 2.1
Given (), there exist a maximaland a unique solutionto problem (5), such that
Moreover, the solution depends continuously on the initial data, the mapping
is continuous.
The following wave-breaking criterion shows the wave breaking is only determined by the slope of but not the slope of .
Theorem 2.2
Let, andbe the maximal existence time of the solutionto system (5). Assumeand. Then
A sufficient condition of global existence is given in the following.
Theorem 2.3
Let, . Then system (5) admits a unique solution
3 Local well-posedness
In this section, we establish the local well-posedness by using Kato theory [28].
Set , , , and .
In order to verify Theorem 2.1, we need the following lemmas in which , , , and are constants depending only on .
Lemma 3.1
The operatorbelongs to.
Lemma 3.2
Let, then. Moreover, for all,
Lemma 3.3
For, and, we haveand
The proofs of Lemmas 3.1-3.3 can be found in [5].
Lemma 3.4
([28])
Let, be real numbers such that. Then
whereis a positive constant depending on, .
Lemma 3.5
Let
Thenis bounded on bounded sets inwithand satisfies the following:
-
(a)
, ;
-
(b)
, .
Proof
-
(a)
Let , we have
(6)
Noting that , we have
and
Similarly, for the last two terms on the right-hand side of Eq. (6), we get
and
Therefore, from Eqs. (7)-(14), we obtain
from which we know (a) holds.
Now, we prove (b). We have
Note that . Using Lemma 3.4 with and gives rise to
In an analogous way to Eq. (17), we have
and
For the fourth term on the right-hand side of Eq. (16), one has
where we used Lemma 3.4.
In an analogous way to Eq. (22), we can estimate the last two terms on the right-hand side of Eq. (16):
and
Therefore, from Eqs. (16)-(24), we deduce
This completes the proof of Lemma 3.5. □
Proof of Theorem 2.1
Applying the Kato theorem for abstract quasi-linear evolution equations of hyperbolic type [28], Lemmas 3.1-3.3 and 3.5, we obtain the local well-posedness of system (5) in , , and
□
4 Wave-breaking criterion
In order to prove Theorem 2.2, the following lemmas are crucial.
Lemma 4.1
The following estimates hold:
-
(i)
For ,
(26)
-
(ii)
For ,
(27)
-
(iii)
For ,
(28)
whereis a constant independent ofand.
Lemma 4.2
Suppose that. Letbe a vector field such thatbelongs toifor tootherwise. Suppose also that, and thatsolves the-dimensional linear transport equations
Then. More precisely, there exists a constantdepending only, , and, and such that the following statements hold:
-
(1)
If ,
(30)
or hence
withifandelse.
Lemma 4.3
([7])
Let. Suppose that, , , andsolves the 1-dimensional linear transport equation
Then. More precisely, there exists a constantdepending only on, such that the following statement holds:
or hence
with.
Lemma 4.4
For all, the following statements hold:
and
Proof
Let be the Green’s function for the operator . Then from
we get
Hence,
From the Fourier series, we know
from which we get
On the other hand,
Hence, we have
This completes the proof of (i).
Now, we prove (ii). Let satisfy . Then, for all , we have
from which one finds
□
Lemma 4.5
Letwith. Suppose thatis the maximal existence time of solutionof system (5) with the initial data. Then, for all, the following conservation law holds:
Proof
Multiplying the first equation of system (1) by and integrating by parts, we reach
Multiplying the second equation of system (1) by and integrating by parts, we get
which together with Eq. (38) yields
which implies Eq. (37).
Let us consider the following differential equation.
where denotes the first component of solution to system (5). □
Lemma 4.6
(See [25])
Let, . Then Eq. (41) has a unique solution. Moreover, the mapis an increasing diffeomorphism ofwith
Lemma 4.7
Letwithandbe the maximal existence time of the corresponding solutionto system (5). Then we have
Moreover, for all, we have
Proof
Differentiating the left-hand side of Eq. (42) with respect to and making use of system (5), we get
This proves Eq. (42). From Eq. (42), we obtain for all
which results in
where Lemma 4.5 is used. This completes the proof of Lemma 4.7. □
Lemma 4.8
Letwith. Suppose thatandis the maximal existence time of solutionof system (5) with the initial data.
Proof
Multiplying the first equation of system (1) by and integrating by parts, we have
which results in
By the Hölder inequality, Eq. (47) ensures that
Applying the Gronwall inequality, we get
which, together with Eqs. (37) and (43), yields
On the other hand, from Lemma 4.4, we deduce
Therefore, from Eq. (48) we deduce that Eq. (45) holds. This completes the proof of Lemma 4.8. □
Next, we give the proof of Theorem 2.2.
Proof of Theorem 2.2
We split the proof of Theorem 2.2 into five steps.
Step 1. For , applying Lemma 4.3 to the second equation, we have
From Lemma 4.1(iii), we get
and
From Eqs. (50) and (51), we obtain
On the other hand, using Lemma 4.2, we get from the first equation of system (5)
Thanks to Lemma 4.1(iii), one has
Hence, we obtain
which, together with Eq. (52), ensures that
Using the Gronwall inequality and Lemma 4.4, we have
From Lemmas 4.5 and 4.8, we get
Therefore, if the maximal existence time satisfies , we get from Eq. (56)
which contradicts the assumption on the maximal time . This completes the proof of Theorem 2.2 for .
Step 2. For , applying Lemma 4.2 to the second equation of system (5), we get
Using Eqs. (50) and (51) gives rise to
which, together with Eq. (53), yields
where and we used the fact that .
Using the Gronwall inequality and Lemma 4.4, we have
From Lemmas 4.5 and 4.8, we get
Using the argument as in Step 1 one completes Theorem 2.2 for .
Step 3. For , differentiating once the second equation of system (5) with respect to , we have
Using Lemma 4.3, we get
where we used the following estimates:
and
where Lemma 4.1(iii) was used.
Using Eqs. (62), (53), and (52) (where is replaced by ) yields
Using the Gronwall inequality again, we have
From Lemmas 4.4, 4.5, and 4.8, we get
Using the argument as in Step 1 one completes Theorem 2.2 for .
Step 4. For , , differentiating times the second equation of system (5) with respect to , we obtain
Using Lemma 4.2, we get from Eq. (66)
From Lemma 3.4 and Lemma 4.1, we have
and
Therefore, we deduce that
From the Gagliardo-Nirenberg inequality, we have for
On the other hand, Eq. (52) implies that
which, together with Eq. (68), yields
Note that . Using Lemma 4.1 and 4.2, we get
which, together with Eq. (70), results in
Using the Gronwall inequality, Lemma 4.4, we get
If satisfies , applying Step 2 and induction assumption, we obtain from Lemma 4.5 and Lemma 4.8 that is uniformly bounded. From Eq. (73), we get
which contradicts the assumption that is the maximal existence time. This completes the proof of Theorem 2.2 for and .
Step 5. For , and , differentiating times the second equation of system (5) with respect to , we obtain
Using Lemma 4.3 with , we get from Eq. (74)
For each , using Lemmas 3.4 and 4.1, and the fact that , we have
and
Therefore, from Eqs. (75), (77), and (78), we get
Using Lemma 4.2 in the first equation of system (5) for with , we obtain
which, together with Eqs. (79) and (52) (where is replaced by ), shows that
Using the Gronwall inequality again, we get
Noting that , and , and applying Step 4, we obtain is uniformly bounded. Thus, we complete the proof of Theorem 2.2 for , and .
Therefore, from Step 1 to Step 5, we finish the proof of Theorem 2.2. □
5 Global solution
To prove Theorem 2.3, we need the following lemmas.
Lemma 5.1
([31])
Let. Ifand, then
Lemma 5.2
Let, . Thenis finite for.
Proof
Applying to , where , and multiplying by and the integrating over , we have
From Lemma 5.1 and the Cauchy inequality, we obtain
The Cauchy inequality ensures
and
where we have used Lemma 4.1.
Hence,
where .
Applying to , and multiplying by and the integrating over , we have
We will estimate each of the terms on the right-hand side of Eq. (88). Note that . Using Lemmas 4.5 and 5.1 and the Cauchy inequality, we have
and
It follows from Eqs. (89)-(91) that
which together with Eq. (87) yields
which implies
Note that , and we get from Eq. (94)
which results in
This completes the proof of Lemma 5.2. □
Proof of Theorem 2.3
Theorem 2.3 is a direct consequence of Theorem 2.1 and Lemma 5.2. □
Remark
We have discussed some dynamics of system (1) in the periodic case. In fact, the above results hold true with , in the periodic case. We have
More precisely, the local well-posedness Theorem 2.1 and the global existence result Theorem 2.3 hold true in the Sobolev space with , the wave-breaking criterion Theorem 2.2 is shown to be true under the condition .
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Acknowledgements
We are grateful to the referees for their comments and constructive suggestions. Guo’s work was supported by the key project (No. 2013QZJ02) and FSUSE (No. 2014RC03), Wang’s work was supported by FSUSE (No. 2012KY09) and (No. 2014PY06).
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Guo, Y., Wang, Y. Wave-breaking criterion and global solution for a generalized periodic coupled Camassa-Holm system. Bound Value Probl 2014, 155 (2014). https://doi.org/10.1186/s13661-014-0155-x
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DOI: https://doi.org/10.1186/s13661-014-0155-x