Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 5374180, 11 pages
https://doi.org/10.1155/2018/5374180

Research Article

Blow-Up Analysis for the Periodic Two-Component μ-Hunter-Saxton System

Yunxi Guo¹ and Tingjian Xiong²

¹School of Mathematics, Zunyi Normal University, 563000 Zunyi, China
²School of Mathematics and Statistics, Sichuan University of Science and Engineering, 643000 Zigong, China

Correspondence should be addressed to Yunxi Guo; matyunxiguo@126.com

Received 24 June 2018; Accepted 19 August 2018; Published 12 September 2018

Academic Editor: Alessandro Arsie

Copyright © 2018 Yunxi Guo and Tingjian Xiong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The two-component -Hunter-Saxton system is considered in the spatially periodic setting. Firstly, a wave-breaking criterion is derived by employing the localization analysis of the transport equation theory. Secondly, several sufficient conditions of the blow-up solutions are established by using the classic method. The results obtained in this paper are new and different from those in previous works.

1. Introduction

In this article, we will consider the periodic two-component -Hunter-Saxton system derived by Zuo [1]

where and are time-dependent functions on the unit circle ,   denotes its mean, and ,  . It is shown in [1] that system (1) is an Euler equation with bi-Hamilton structure

where , and it also can be viewed as a bivariational equation. Moreover, for ,  , system (1) has a Lax pair given by

where is a spectral parameter (see [1]).

In fact, system (1) is a generalization of the generalized Hunter-Saxton equation [2, 3]

which describes the geodesic flow on with the right-invariant metric given at the identity by the inner product and models the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal with external magnetic nematic field and self-interaction. Here, the solution denotes the director field of a nematic liquid crystal. It was observed in [2–4] that the -Hunter-Saxton equation is formally integrable and has bi-Hamiltonian structure and infinite hierarchy of conservation laws. Further, the development of singularities in finite time and geometric descriptions of the system from nonstretching invariant curve flows in centroequiaffine geometries and pseudospherical surfaces and affine surfaces are described by Fu et al. [5].

For ,  ,   and replacing by , system (1) reduces to periodic two-component Hunter-Saxton equation

which is a generalization of the well-known Hunter-Saxton equation. It was also viewed as a particular case of the Gurevich-Zybin system [6] pertaining to nonlinear one-dimensional dynamics of dark matter as well as nonlinear ion-acoustic waves (cf. [7] and the references therein). The Hunter-Saxton system is formally integrable, possesses a bi-Hamiltonian structure, and admits peakon solutions [8]. The local well-posedness, global existence, blow-up phenomena, solitary wave solutions, and geometric properties of system (5) were recently discussed in literatures (see [9–17] and the references therein). It is worthwhile to mention that Moon and Liu [13] studied the Cauchy problem for the two-component Hunter-Saxton system in the periodic setting and gave some interesting results.

Recently, Liu and Yin [18, 19] investigated the Cauchy problem for system (1). In [18], the local well-posedness and several precise blow-up criteria for the system were obtained. Under the conditions and , the sufficient conditions of blow-up solutions were presented. The global existence for strong solution for system (1) in the Sobolev space with is also given [18], and in [19], existence of global weak solution is established for the periodic two-component -Hunter-Saxton system. The objective of the present paper is to focus mainly on wave-breaking criterion and several sufficient conditions of blow-up solutions.

Motivated by the works in [13, 20], in the present paper, the localization analysis in the transport equation theory is employed to derive a new wave-breaking criterion of solutions for the system (1) in the Sobolev space with . It implies that the wave-breaking criterion is determined only by the slope of the component of solution definitely. Inspired by the work in [21, 22], we use the Lyapunov function of to deduce several new blow-up results for the periodic two-component -Hunter-Saxton system (1), which are different from the ones obtained in [18]. These results obtained in this paper are new and different from those in Liu and Yin’s work [18].

The rest of this paper is organized as follows. Section 2 states several properties for the periodic two-component -Hunter-Saxton system and gives several lemmas. In Section 3, we employ the transport equation theory to prove a wave-breaking criterion in the Soblev space with . Section 4 is devoted to the study of blow-up mechanism.

2. Preliminaries

Lemma 1 (see [20]). The following estimates hold.
(i) For ,

(ii) For ,

where are constants independent of and .

Lemma 2 (see [20]). Suppose that . Let be a vector filed such that belongs to if or to otherwise. Suppose also that , , and solves the dimensional linear transport equations

Then . More precisely, there exists a constant depending only on , and , such that the following statements hold:
(1) If ,

Or hence

with if and else.
(2) If , then, for all , the estimates (9) and (10) hold with .

Lemma 3 (see [20]). Let . Suppose that ,  ,   and solves the dimensional linear transport equation

Then . More precisely, there exists a constant depending only on , and such that the following statement holds:

Or hence

with .

Lemma 4 (see [18]). Given ,  , then there exists a maximal and an unique solution to system (1) such that

Lemma 5 (see [21]). If satisfies , then it has

Lemma 6 (see [23]). For every periodic and with zero average, i.e., such that , it holds that

and equality holds if and only if

Integrating the first equation of system (1) over the circle and noting the periodicity of , we have . Making use of system (1), we have that is conserved in time (see [18]). In what follows we denote

and

Then and are constants and independent of time .

Notice that . From Lemma 5, we get

which implies that the amplitude of wave remains bounded in any time. Namely, we have

which results in

In fact, the initial-value problem (1) can be recast in the following:

where is an isomorphism between and with the inverse given explicitly by

Commuting and , we get

and

Note that if , we have , where we denotes by convolution and is Green’s function of the operator , given by

and the derivative of can be assigned

Now, consider the initial-value problem for the Lagrangian flow map:

where denotes the first component of the solution to system (1). Applying classical results from ordinary differential equations, one can obtain the result.

Lemma 7 (see [18]). Let ,  . Then (29) has a unique solution . Moreover, the map is an increasing diffeomorphism of with

Lemma 8 (see [18]). Let ,  , and let be the maximal existence time of the corresponding solution to system (1). Then it has

Lemma 9 (see [18]). Assume that ,  , and let be the maximal existence time of corresponding solution to system (1) with the initial data . Then the corresponding solution blows up in finite time if and only if

3. Wave-Breaking Criterion

Theorem 10. Let with , and be the corresponding solution to (1). Assume that is the maximal existence time. Then

Proof. We shall complete the proof of the theorem by an inductive argument with respect to the index . Let us first give upper bound for .
From (31), we derive

Next, we split four steps to finish the proof of Theorem 10.
Step 1. For , applying Lemma 3 to the second equation of system (23), we have

Using (6), we get

Therefore, it yields

Differentiating once the second equation of system (23) with respect to , we have

Using Lemma 3, we get

From (7), we have

and

Therefore, we have

From (37) and (42), it has

On the other hand, the first equation of system (23) is equivalent to

Therefore, using Lemma 2, we get from (44)

Note that

where we have used (27), (28), and Young’s inequality.
Using (6) and (46), one has

Thus, we reach

which together with (43) reaches

Using Gronwall’s inequality, we have

From (22) and (34), we get

where .
Hence, if the maximal existence time satisfies , we obtain from (51) that

which contradicts the assumption on the maximal existence time . It completes the Theorem 10 for .
Step 2. For , applying Lemma 2 to the second equation of system (23), we get

From (6), we get

which ensures that

which together with (48) gives rise to

Using Gronwall’s inequality, we have

where and we used the fact that .
Therefore, using Step 1 and arguing by induction assumption, we get that

is uniformly bounded. We obtain from (57) that

which contradicts the assumption on the maximal existence time . It completes the Theorem 10 for .
Step 3. For ,  . Differentiating times the second equation of system (23) with respect to , we obtain

Using Lemma 2, we get from (60) that

Since is an algebra, we have

and

Therefore, we have

Applying Lemma 3 to the second equation of system (23) yields

where we used Lemma 1.
From (64) and (65), it yields

where we used the Gagliardo-Nirenberg inequality for .
Using (48) implies that

Using Gronwall’s inequality, we get

Therefore, if the maximal existence time satisfies , using Step 2, we get that

is uniformly bounded by the induction assumption. From (66), we get

which contradicts the assumption that is the maximal existence time. This completes the proof of Theorem 10 for and .
Step 4. For and . Differentiating times the second equation of system (23) with respect to , we obtain

Applying Lemma 3 to (71) and noting , we get

For each , using (7) and the fact that , we have

and

where we used Lemma 1.
Therefore, from (72), (73), and (74), we get

which together with (48) and (37) (where is replaced by ) gives rise to

from which we have

Noting that ,   and . Therefore, if the maximal existence time satisfies , using Step 3, we get that

is uniformly bounded by the induction assumption. From (77), we get

which contradicts the assumption that is the maximal existence time. This completes the proof of Theorem 10 for and .
Therefore, from Steps 1–4, we complete the proof of Theorem 10.

4. Wave-Breaking Phenomena

In this section, we will present several blow-up results by using the Lyapunov function of (see [21]). Let with and let be the maximal existence time of the corresponding solution to system (1) with the initial data .

Now, we give the first blow-up result.

Theorem 11. Let with and be a sufficient large positive real number and let be the maximal existence time of the corresponding solution to system (1) with the initial data and .
(i) If , then the corresponding solution to system (1) must blow up in finite time with

where , such that

(ii) If then the corresponding solution to system (1) must blow up in finite time with

where and .

Proof. If the statement is not true, then from Lemma 9 it implies that there exists a large enough real number , such that and for all . From the assumption , we know

It then follows that, for any , has a zero point . Thus we have

which results in

Differentiating the first equation of system (23) with respect to yields

Using (26), we get

Multiplying (87) by and integrating on with respect to , we obtain for any that

On the other hand, it follows from Lemma 6 for any that

Therefore, we deduce that

where we have used Lemma 8 and (19). From the assumption of the theorem, we know that . Let satisfy

It implies that and . Define and by

We get

Letting with derives

where we used the inequality , which leads to

On the other hand, we have

which implies that satisfying

From (90), we get

Let satisfy

it implies that . Define and by

and let ; we have

Let . If , then . Therefore, we solve (101) to obtain

It follows that there exists satisfying

such that . This contradicts the assumption for all . From Lemma 9 we obtain that the solution blows up in finite time.
It completes the proof of Theorem 11.

Before giving the second blow-up result, we firstly claim that, under the condition , is conserved in time. Indeed, like two-component Camassa-Holm equation (see [24–26]), we can easy obtain

In view of (23) and (87), we get after some integration by parts

where .

Observe that

Therefore, we have

Theorem 12. Let with and be a sufficiently large positive real number and let be maximal existence time of corresponding solution to system (1) with the initial data and .
(i) Assume that . Then the corresponding solution to system (1) must blow up in finite time with

(ii) Assume that . Then the corresponding solution to system (1) must blow up in finite time with

where .

Proof. Applying a simple density argument, we only need to consider the case . From (88), one has

Applying the assumption of the theorem and Hölder inequality give rise to

Define ,  . It is clear that

where . This implies that increase strictly in . Let ; we assume . Otherwise, and the theorem is proved.
From (112), we have

Now, integrating (113) over yields

Using again (112), one has

which leads to

On the other hand, we have

which implies that satisfying

(ii) Applying the assumption of the theorem, (110) and Hölder inequality give rise to

Define ,  . It is clear that

where . Let . If , then . Therefore, we solve (101) to obtain

It follows that there exists satisfying

such that . This contradicts the assumption for all . From Lemma 9 we obtain that the solution blows up in finite time. It finishes the proof of the Theorem 12.

Remark 13. If we let , then we can obtain easily. Then system (1) is degenerated into -version Camassa-Holm equation under some transform. The blow-up results related to -version Camassa-Holm equation; the reader is referred to [5].

Remark 14. It is worthwhile to mention that, comparing with the results in [18], our blow-up results are new and quite different from the ones in [18].

Data Availability

No data were used to support this study.

Conflicts of Interest

There are no conflicts of interest.

Authors’ Contributions

The two authors contributed to the work equally.

Acknowledgments

Guo’s work is supported by Zunyi Normal University Doctoral Program Fund [grant number BS201710], Department of Sichuan Province Education Fund [grand number 17ZB0314], and grant number 16ZA0265.

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