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

models the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal. In Hunter and Saxton [1], x is the space variable in a reference framemoving with the linearized wave velocity, t is a slow-time variable, and u(t, x) is a measure of the average orientation of the medium locally around x at time t. In order to be more precise, the orientation of the molecules is described by the field of unit vectors (cos u(t, x), sin u(t, x)) [2]. The HunterSaxton 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 S = R/Z 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 Ds(S) modulo the subgroup of rotations Rot(S). 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:


Introduction
In recent years, the Hunter-Saxton equation [1]   + 2    +   = 0 (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 (cos (, ), sin (, )) [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][11][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 Rot().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: + 2    +   −   +   = 0,  > 0,  ∈ ,   + ()  = 0,  > 0,  ∈ ,  (,  + 1) =  (, ) , (,  + 1) =  (, ) ,  ≥ 0,  ∈ ,  (0, ) =  0 () , (0, ) =  0 () ,  ∈ . (2) The authors of [16] have explored the particular choice of the parameter  = 1.The authors of [17] have further studied the wave breaking and global existence for the system for the parameter  ∈ R 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  ≥ 0,  < 0.
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 wellposedness for (3) with the initial data in   ×  −1 ,  ≥ 2. Section 3 deals with the wave breaking of this new system.Theorem 11, using transport equation theory, states a wavebreaking 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.Throughout this paper,  = / will denote the unit circle.By   ,  ≥ 0, 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 V ∈   ×  −1 will be written as ‖V‖   × −1 .Also, the Lebesgue spaces of order  ∈ [1, ∞] will be denoted by   (), and the norm of their elements will be denoted by ‖‖   () .Finally, if  = 2, 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.
Next, we apply Kato's theory to establish the local wellposedness for the system (3).Consider the abstract quasilinear evolution equation Proposition 1 (see [20]).Given the evolution equation (8), assume that the Kato conditions hold.For a fixed V 0 ∈ , there is a maximal  > 0 depending only on ‖V 0 ‖  and a unique solution V to the abstract quasi-linear evolution equation (8) such that Moreover, the map One may follow the similar argument as in [17] to obtain the following local well-posedness for (3).

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  ∈ ([0, );   ) ∩ Similarly, we have Lemma 5. Let  0 = ( , and let  be the maximal existence time of the solution  = (   ) to (3) with initial data  0 .Then, for all  ∈ [0, ), 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.
, and let  be the maximal existence time of the solution  = (   ) to (3) with initial data  0 .Then, for all  ∈ [0, ), we have the following results: where Proof.By computing directly, we have where Multiplying ( 6) by  and integrating with respect to , using the periodicity of  and (24), we obtain where  2 =  1 + 4 − 2; note that  2 >  1 .By Gronwall's inequality, we get This completes the proof of Lemma 6.
Then, for  ∈ , by holder equality, we have This implies sup (30)

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: where    are constants that are independent of  and .
Theorem 11.Let  0 = (  0  0 ) ∈   ×  −1 with  ≥ 2, and  = (   ) be the corresponding solution to (3).Assume that  > 0 is the maximal time of existence.Then Our next result describes the necessary and sufficient condition for the blow-up of solutions to (3).
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 sup ∈   (, ) and inf ∈   (, ).
For any given  ∈ , Note that  2 () is also  1 -differentiable function on [0, ) and satisfies We now claim that  2 () ≥ 0, for any  ∈ [0, ).Suppose not, then there is t ∈ [0, ) such that  2 ( t0 ) < 0. Define Then,  2 ( 2 ) = 0 and   2 ( 2 ) < 0, or equivalently, 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 : We know () > 0 for  ∈ [0, ).From the second equation of (81), we obtain that Integrating (88) on [0, ], we prove (47) as follows: To obtain a lower bound for inf ∈   (, ), we use the same argument.
Conversely, the Sobolev embedding theorem   () →  ∞ () with  > 1/2 implies that if (70) holds, the corresponding solution blows up in finite time, which completes the proof of Theorem 12.