Journal of Applied Mathematics

Volume 2013 (2013), Article ID 809824, 10 pages

http://dx.doi.org/10.1155/2013/809824

Research Article

Wave-Breaking Criterion for the Generalized Weakly Dissipative Periodic Two-Component Hunter-Saxton System

Jianmei Zhang and Lixin Tian

Department of Mathematics, Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received 22 May 2013; Accepted 22 July 2013

Academic Editor: Michael Meylan

Copyright © 2013 Jianmei Zhang and Lixin Tian. 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.

This paper studies the wave-breaking criterion for the generalized weakly dissipative two-component Hunter-Saxton system in the periodic setting. We get local well-posedness for the generalized weakly dissipative two-component Hunter-Saxton system. We study a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles.

1. Introduction

In recent years, the Hunter-Saxton equation [1]

models the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal. In Hunter and Saxton [1], is the space variable in a reference frame moving with the linearized wave velocity, is a slow-time variable, and is a measure of the average orientation of the medium locally around at time . In order to be more precise, the orientation of the molecules is described by the field of unit vectors , [2]. The Hunter-Saxton equation also arises in a different physical context as the high-frequency limit [3, 4] of the Camassa-Holm equation for shallow water waves [5, 6] and a reexpression of the geodesic flow on the diffeomorphism group of the circle [7] with a bi-Hamiltonian structure [1, 8] which is completely integrable [4, 9]. Hunter and Saxton [1] explored the initial value problem for the Hunter and Saxton equation on the line (nonperiodic case) and on the unit circle by using the method of characteristics, while Yin [2] studied it by using the Kato semigroup method. In addition, the two classes of admissible weak solutions, dissipative and conservative solutions, and their stability were investigated in [10–12]. Lenells [13] confirmed that the Hunter-Saxton equation also describes the geodesic flows on the quotient space of the infinite-dimensional group modulo the subgroup of rotations  .

The Camassa-Holm equation admits many integrable multicomponent generalizations. So many authors studied the two-component Camassa-Holm system [14, 15]. Inspired by this, recently, the researchers have made a study of the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting as follows:

The authors of [16] have explored the particular choice of the parameter . The authors of [17] have further studied the wave breaking and global existence for the system for the parameter to determine a wave-breaking criterion for strong solutions by using the localization analysis in the transport equation theory.

In general, avoiding energy dissipation mechanisms in a real world is not so easy. Wu and Yin [18, 19] have investigated the blow-up phenomena and the blow-up rate of the strong solutions of the weakly dissipative CH equation and DP equation. Inspired by the results mentioned above, we are going to discuss the initial value problem associated with the generalized weakly dissipative periodic two-component Hunter-Saxton system

where is the new free parameter and  , .

Our major results of this paper are Theorems 11 and 12 (wave-breaking criterion). The remainder of the paper is organized as follows. Section 2 establishes the local well-posedness for (3) with the initial data in , . Section 3 deals with the wave breaking of this new system. Theorem 11, using transport equation theory, states a wave-breaking criterion which says that the wave breaking only depends on the slope of , not the slope of . Theorem 12 improves the blow-up criterion with a more precise condition.

Notation 1. Throughout this paper, will denote the unit circle. By , , we will represent the Sobolev spaces of equivalence classes of functions defined on the unit circle which have square-integrable distributional derivatives up to order . The -norm will be designated by , and the norm of a vector will be written as . Also, the Lebesgue spaces of order will be denoted by , and the norm of their elements will be denoted by . Finally, if , we agree on the convention .

2. Preliminaries

In this part, we will establish the local well-posedness for the Cauchy problem of system (3) by using Kato’s theory. To pursue our goal, we give the results we wanted in brief.

We now provide the framework in which we will reformulate (3). To do this, we observe that we can write the first equation of (3) in the following integrated form:

where and is determined by the periodicity of to be

Integrating both sides of (4) with respect to variable  , we get

where is an arbitrary continuous function. Therefore, (3) can be written in the “transport” form as follows:

where is an arbitrary continuous function.

Next, we apply Kato’s theory to establish the local well-posedness for the system (3). Consider the abstract quasi-linear evolution equation

Proposition 1 (see [20]). Given the evolution equation (8), assume that the Kato conditions hold. For a fixed , there is a maximal depending only on and a unique solution to the abstract quasi-linear evolution equation (8) such that

Moreover, the map is continuous from to

One may follow the similar argument as in [17] to obtain the following local well-posedness for (3).

Theorem 2. Given any , ,   there exist a maximal   and a unique solution to (3) such that

Moreover, the solution depends continuously on the initial data, that is, the mapping   is continuous, and the maximal existence time can be chosen independently of the Sobolev order .

Now, discuss the initial value problem for the Lagrangian flow map as follows:

where is the first component of the solution to (3). Using classical results from ordinary differential equations, one can acquire the following result on which is of vital importance in the proof of the blow-up scenarios.

Lemma 3 (see [17]). Let , . Then, initial value problem (12) admits a unique solution  . Moreover, is increasing diffeomorphism of with

Remark 4. Since is a diffeomorphism of the linear for every , the -norm of any function , is preserved under the family of diffeomorphisms with , that is,

Similarly, we have

Lemma 5. Let , , and let be the maximal existence time of the solution to (3) with initial data . Then, for all , we have the following results:

Proof. On the one hand, integrating the second equation in (3) by parts and using the periodicity of and , we acquire

On the other hand, multiplying (4) by and integrating by parts, considering the periodicity of , we obtain

Multiplying the second equation in (3) by and integrating by parts, we have

Adding the above two equations, we get

We acquire

This completes the proof of Lemma 5.

Lemma 6. Let , , and let be the maximal existence time of the solution to (3) with initial data . Then, for all , we have the following results:

where , .

Proof. By computing directly, we have

where and

Multiplying (6) by and integrating with respect to , using the periodicity of and (24), we obtain

where ; note that .

By Gronwall's inequality, we get

This completes the proof of Lemma 6.

Lemma 7. Assume that , , , and that the corresponding solution of (3) has a zero point for any time . Then, for all we have

Proof. By assumption, there is such that for each .

Then, for , by holder equality, we have

This implies

3. Wave-Breaking Criteria

In this section, by using transport equation theory, we obtain the wave-breaking criteria for solutions to (3). We first recall the following propositions.

Proposition 8 (1D Moser-type estimates). The following estimates hold:

(a)For ,

(b)For ,

(c)For , ,  ,

where are constants that are independent of and .

Proposition 9 (see [21]). Suppose that . Let be a vector field such that belongs to if or to , otherwise. Suppose also that , and that   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

with if and   else.

(2)   If , then for all , estimates (35) and (36) hold with

Proposition 10 (see [21]). Let . Suppose that , , and that solves the 1-dimensional linear transport equation

Then . More precisely, there exists a constant depending only on such that the following statements hold:

or

with .

The above proposition was proved in [8] using Littlewood-Paley analysis for the transport equation and Moser-type estimates. Using this result and performing the same argument, as in [17], we can obtain the following blow-up criterion.

Theorem 11. Let with  , and be the corresponding solution to (3). Assume that is the maximal time of existence. Then

Our next result describes the necessary and sufficient condition for the blow-up of solutions to (3).

Theorem 12. Suppose that . Let , with , and let be the maximal existence time of the solution to (3) with initial data . Then, the solution blows up in finite time if and only if

The approach one takes here is the method of characteristics. Applying the following lemma, we may carry out the estimates along the characteristics which captures and .

Lemma 13 (see [22]). Let and let . Then, for every , there exists at least one point with

and the function is almost everywhere differentiable on with

Lemma 14. Let with , and let be the maximal existence time of the solution to (3) with initial data . Then one has the following:

(1)

(2)

The constants above are defined as follows:

Proof of Lemma 14. By Theorem 2 and a simple density argument, we show that the desired results are valid when  , so we take in the proof.

Let . Using Lemma 13 and the fact that

We can consider and as follows:

Hence,

Take the trajectory defined in (12). Then we know that is a diffeomorphism for every . Therefore, there exists such that

Now, let

Therefore, along the trajectory , (4) and the second equation of (3) become

where the notation denotes the derivative with respect to and represents the function

We first compute the upper and lower bounds for for later use in getting the blow-up result as follows:

Since , (17), we obtain the upper bound for

Now we turn to the lower bound of . Using previous arguments, we get

When , we have a finer estimate

Combining (59) and (60), we obtain

Since , we have . Therefore,

Hence, for . From the second equation of (55), we obtain

Hence,

For any given , define

Observing that is a -differentiable function on and satisfies

We now claim that .

Assume the contrary that there is such that .

Let . Then and , or equivalently,

and a.e. . On the other hand, we have

which is a contradiction. Therefore, for all . Since is arbitrarily chosen, we obtain (45).

To derive (46) in the case of , we consider and as in Lemma 13:

Hence,

Using previous arguments, we take the characteristic defined in (13) and choose such that

Let

Hence, along the trajectory     , (4) and the second equation of (3) become

Define

For any given , Note that is also -differentiable function on and satisfies

We now claim that , for any .

Suppose not, then there is such that . Define

Then, and , or equivalently,

and a.e. . However, we have

Therefore, for any . Since is chosen arbitrarily, we obtain (46).

Let . Using previous arguments, (56) becomes

where the notation denotes the derivative with respect to and represents the function

We first compute the upper and lower bounds for for later use in getting the blow-up result:

Now, we turn to the lower bound of :

Combining (82) and (83), we obtain

We know for . From the second equation of (81), we obtain that

Hence,

Therefore, we have

Integrating (88) on , we prove (47) as follows:

To obtain a lower bound for , we use the same argument.

Since  , (80) becomes

Because of , we get from the second equation of (90) that

This means that

Then,

Integrating (94) on , we prove (48). This completes the proof of Lemma 14.

Lemma 15. Suppose that . Suppose with , and let be the maximal existence time of the solution to (3) with initial data . Then we have

Moreover, if there exists such that

Then

where and is given in (50).

Proof. Differentiating the left hand side of (95) with respect to , in view of the relations (12) and (3), we obtain

This completes the proof of (95). In view of the assumption (96) and , we obtain .

By Lemma 3 and (95), we have

To obtain (98), we use a similar argument as before. Using (13) and the lower bound for in (46), it follows that

which proves (98). This completes the proof of Lemma 15.

Proof of Theorem 12. Suppose that and that (42) is not valid. Then, there is some positive number such that

It now follows from Lemma 14 that , where . Therefore, Theorem 11 implies that the maximal existence time , which contradicts with the assumption that .

Conversely, the Sobolev embedding theorem with implies that if (70) holds, the corresponding solution blows up in finite time, which completes the proof of Theorem 12.

Acknowledgments

The authors would like to thank the referee for comments and suggestions. This work is supported by the National Nature Science Foundation of China (no. 11171135), the Nature Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), and the high-level talented person special subsidizes of Jiangsu University (no. 05JDG047).

References

J. K. Hunter and R. Saxton, “Dynamics of director fields,” SIAM Journal on Applied Mathematics, vol. 51, no. 6, pp. 1498–1521, 1991.

Z. Yin, “On the structure of solutions to the periodic Hunter-Saxton equation,” SIAM Journal on Mathematical Analysis, vol. 36, no. 1, pp. 272–283, 2004.

H.-H. Dai and M. Pavlov, “Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation,” Journal of the Physical Society of Japan, vol. 67, no. 11, pp. 3655–3657, 1998.

J. K. Hunter and Y. X. Zheng, “On a completely integrable nonlinear hyperbolic variational equation,” Physica D, vol. 79, no. 2–4, pp. 361–386, 1994.

R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993.

R. S. Johnson, “Camassa-Holm, Korteweg-de Vries and related models for water waves,” Journal of Fluid Mechanics, vol. 455, pp. 63–82, 2002.

A. Constantin and B. Kolev, “On the geometric approach to the motion of inertial mechanical systems,” Journal of Physics A, vol. 35, no. 32, pp. R51–R79, 2002.

P. J. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,” Physical Review E, vol. 53, no. 2, pp. 1900–1906, 1996.

R. Beals, D. H. Sattinger, and J. Szmigielski, “Inverse scattering solutions of the Hunter-Saxton equation,” Applicable Analysis, vol. 78, no. 3-4, pp. 255–269, 2001.

A. Bressan and A. Constantin, “Global solutions of the Hunter-Saxton equation,” SIAM Journal on Mathematical Analysis, vol. 94, pp. 68–92, 2010.

A. Bressan, H. Holden, and X. Raynaud, “Lipschitz metric for the Hunter-Saxton equation,” Journal de Mathématiques Pures et Appliquées, vol. 94, no. 1, pp. 68–92, 2010.

J. K. Hunter and Y. X. Zheng, “On a nonlinear hyperbolic variational equation. I. Global existence of weak solutions,” Archive for Rational Mechanics and Analysis, vol. 129, no. 4, pp. 305–353, 1995.

J. Lenells, “The Hunter-Saxton equation describes the geodesic flow on a sphere,” Journal of Geometry and Physics, vol. 57, no. 10, pp. 2049–2064, 2007.

J. B. Li and Y. S. Li, “Bifurcations of travelling wave solutions for a two-component Camassa-Holm equation,” Acta Mathematica Sinica (English Series), vol. 24, no. 8, pp. 1319–1330, 2008.

W. Rui and Y. Long, “Integral bifurcation method together with a translation-dilation transformation for solving an integrable 2-component Camassa-Holm shallow water system,” Journal of Applied Mathematics, vol. 2012, Article ID 736765, 21 pages, 2012.

H. Wu and M. Wunsch, “Global existence for the generalized two-component Hunter-Saxton system,” Journal of Mathematical Fluid Mechanics, vol. 14, no. 3, pp. 455–469, 2012.

B. Moon and Y. Liu, “Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system,” Journal of Differential Equations, vol. 253, no. 1, pp. 319–355, 2012.

S. Wu and Z. Yin, “Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation,” Journal of Mathematical Physics, vol. 47, no. 1, Article ID 013504, 2006.

S. Wu, J. Escher, and Z. Yin, “Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation,” Discrete and Continuous Dynamical Systems B, vol. 12, no. 3, pp. 633–645, 2009.

T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations, vol. 448 of Lecture Notes in Mathematics, pp. 25–70, Springer, Berlin, Germany, 1975.

G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010.

A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998.