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

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 structurewhere , and it also can be viewed as a bivariational equation. Moreover, for ,  , system (1) has a Lax pair given bywhere 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 equationwhich 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 equationsThen . More precisely, there exists a constant depending only on , and , such that the following statements hold:
(1) If ,Or hencewith 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 equationThen . More precisely, there exists a constant depending only on , and such that the following statement holds:Or hencewith .

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 thatand 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 denoteandThen and are constants and independent of time .

Notice that . From Lemma 5, we getwhich implies that the amplitude of wave remains bounded in any time. Namely, we havewhich results inIn fact, the initial-value problem (1) can be recast in the following:where is an isomorphism between and with the inverse given explicitly byCommuting and , we getandNote 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 deriveNext, 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 haveUsing (6), we getTherefore, it yieldsDifferentiating once the second equation of system (23) with respect to , we haveUsing Lemma 3, we getFrom (7), we haveandTherefore, we haveFrom (37) and (42), it hasOn the other hand, the first equation of system (23) is equivalent toTherefore, using Lemma 2, we get from (44)Note thatwhere we have used (27), (28), and Young’s inequality.
Using (6) and (46), one hasThus, we reachwhich together with (43) reachesUsing Gronwall’s inequality, we haveFrom (22) and (34), we getwhere .
Hence, if the maximal existence time satisfies , we obtain from (51) thatwhich 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 getFrom (6), we getwhich ensures thatwhich together with (48) gives rise toUsing Gronwall’s inequality, we havewhere and we used the fact that .
Therefore, using Step 1 and arguing by induction assumption, we get thatis uniformly bounded. We obtain from (57) thatwhich 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 obtainUsing Lemma 2, we get from (60) thatSince is an algebra, we haveandTherefore, we haveApplying Lemma 3 to the second equation of system (23) yieldswhere we used Lemma 1.
From (64) and (65), it yieldswhere we used the Gagliardo-Nirenberg inequality for .
Using (48) implies thatUsing Gronwall’s inequality, we getTherefore, if the maximal existence time satisfies , using Step 2, we get thatis uniformly bounded by the induction assumption. From (66), we getwhich 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 obtainApplying Lemma 3 to (71) and noting , we getFor each , using (7) and the fact that , we haveandwhere we used Lemma 1.
Therefore, from (72), (73), and (74), we getwhich together with (48) and (37) (where is replaced by ) gives rise tofrom which we haveNoting that ,   and . Therefore, if the maximal existence time satisfies , using Step 3, we get thatis uniformly bounded by the induction assumption. From (77), we getwhich 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 withwhere , such that(ii) If then the corresponding solution to system (1) must blow up in finite time withwhere 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 knowIt then follows that, for any , has a zero point . Thus we havewhich results in Differentiating the first equation of system (23) with respect to yieldsUsing (26), we getMultiplying (87) by and integrating on with respect to , we obtain for any thatOn the other hand, it follows from Lemma 6 for any thatTherefore, we deduce thatwhere we have used Lemma 8 and (19). From the assumption of the theorem, we know that . Let satisfyIt implies that and . Define and byWe getLetting with deriveswhere we used the inequality , which leads toOn the other hand, we havewhich implies that satisfying From (90), we getLet satisfyit implies that . Define and byand let ; we haveLet . If , then . Therefore, we solve (101) to obtainIt follows that there exists satisfyingsuch 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 obtainIn view of (23) and (87), we get after some integration by parts

where .

Observe thatTherefore, 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 hasApplying the assumption of the theorem and Hölder inequality give rise toDefine ,  . It is clear thatwhere . This implies that increase strictly in . Let ; we assume . Otherwise, and the theorem is proved.
From (112), we haveNow, integrating (113) over yieldsUsing again (112), one haswhich leads toOn the other hand, we havewhich implies that satisfying(ii) Applying the assumption of the theorem, (110) and Hölder inequality give rise toDefine ,  . It is clear thatwhere . Let . If , then . Therefore, we solve (101) to obtainIt follows that there exists satisfyingsuch 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.