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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 809824, 10 pages
Wave-Breaking Criterion for the Generalized Weakly Dissipative Periodic Two-Component Hunter-Saxton System
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.
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.
In recent years, the Hunter-Saxton equation  models the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal. In Hunter and Saxton , 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 , . 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  with a bi-Hamiltonian structure [1, 8] which is completely integrable [4, 9]. Hunter and Saxton  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  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  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  have explored the particular choice of the parameter . The authors of  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 .
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 ). 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  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.
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
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
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
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 ). 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 ). 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  using Littlewood-Paley analysis for the transport equation and Moser-type estimates. Using this result and performing the same argument, as in , 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 ). 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:
Take the trajectory defined in (12). Then we know that is a diffeomorphism for every . Therefore, there exists such that
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
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:
Using previous arguments, we take the characteristic defined in (13) and choose such that
Hence, along the trajectory , (4) and the second equation of (3) become
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
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
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.
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).
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