Abstract
In this paper, we establish global conservative solutions of the two-component -Hunter–Saxton system by the methods developed in “A. Bressan, A. Constantin, Global Conservative Solutions of the Camassa-Holm Equation, Arch. Ration. Mech. Anal. 183 (2), 215–239 (2007)” and “H. Holden, X. Raynaud, Periodic Conservative Solutions of the Camassa-Holm Equation, Ann. Inst. Fourier (Grenoble) 58(3), 945–988 (2008).”
1. Introduction
If a solution remains bounded pointwise and its slope becomes unbounded in finite time, we say this solution breaks down in finite time. Blow-up is a highly interesting property exhibited in a lot of nonlinear dispersive-wave equations, e.g., the Camassa–Holm equation [1–3]:
The -Hunter–Saxton equation [4] is as follows:
The Hunter–Saxton equation [5] is as follows:
Then, what will happen after wave breaking is an interesting problem, which has received considerable attention in the past decade. Several methods have been developed to study this issue, including the vanishing viscosity approach, initial data mollification, and coordinate transformation [6–15]. Among all the methods, two of them will be used in this paper. One was introduced by Bressan and Constantin in [6], and the other was proposed by Holden and Raynaud in [10]. Both methods converted the problem to solving a corresponding semilinear system by the application of new variables. Their mutual difference lies in the fact that Holden and Raynaud used a different set of variables and constructed a bijective map between Eulerian and Lagrangian coordinates for (CH). In addition, if the energy remains constant except for the exact time of break down, we call the conservative solutions; if decreases to zero at the breakdown time, we call the solutions dissipative.
In this paper, we discuss the conservative solutions of the following periodic two-component -Hunter–Saxton system [16]:where , , where is the time vector, and is a space vector. This system is of a bi-Hamilton structure, and it can also be viewed as a bivariational equation set. Therefore, equation (4) can be rewritten aswhere
Also,with . It is a generalization of HS equation (where ). If , then equation (4) becomes a two-component CH system, which has been studied in [17–21]. The system exhibits local well-posedness, and it has finite-time blowup solutions and global strong solutions in time. Global conservative weak solutions can be obtained by coordinate transformation in [19, 20], and admissible weak solutions can be found in [21] by mollifying the initial data. If , then equation (4) turns to a two-component HS system, which has been looked into in [22, 23]. Hence, we can say 2-HS system equation (4) lies in an intermediate between 2-CH and 2-HS systems. The Cauchy problem for equation (4) has been studied extensively in [24–27]. In addition, research studies have shown that this system is locally well-posed [26] for , ; besides, its global classical solutions [26] and finite-time blowup solutions [25, 27] have also been found, and its geometric background has been comprehensively given by Escher in [24]. The global admissible weak solution of system equation (4) has been obtained in [28] by mollifying the initial date. Here, we will follow previous research studies [6, 10, 14] and demonstrate the existence of global conservative weak solutions. However, compared to the 2-CH system, the existing in the 2-HS system brings some difficulties to the calculation of equation (28). Fortunately, we overcame it. Because in the 2-HS system, the 2-HS system is structurally more complex than the 2-HS system. In [23], the author gave the specific expression of (one can find them in equations (14)–(17)), which is very helpful to the proof of the main theorem. However, this practice is almost impossible for the 2-HS system, so our proof is a little bit more difficult. To sum up, although we refer to the methods in [6, 10, 14], our results and the proofs are quite different.
Our paper is organized as follows. In Section 2, we reformulate system equation (4) and give an equivalent system in Lagrangian coordinates. We also try to illustrate the existence and uniqueness of solutions to the equivalent system to Banach contraction arguments. In Section 3, we establish maps between Lagrangian and Eulerian coordinates, which can connect conservative weak solutions of equation (4) and solutions of a semilinear system together. In Section 4, we give the existence of global conservative weak solutions to equation (4).
2. Preliminaries
Firstly, we reformulate system equation (4). Assume in equation (4); we havewhich is equivalent towhere and for all with . By differentiating the first equation in equation (9), we obtain
Based on the second equation in equations (9) and (10), a direct computation implies
For smooth solutions, we combine the first equation in equations (8) and (11), and we find the following conservation laws:
Since system equation (4) is periodic with period 1, we define a space
However, is not a Banach space. We define as the solution of
And then, we define
Taking the derivative of both sides of equations (15)–(17) with respect to and using equation (11), we can obtain a result. Combining this result with equation (14), we have the following semilinear system of :where
After we define a new variable , we have
Here, we will make some explanation about . Since is periodic with period 1 and , we can obtain . By equation (11), a direct computation implies that , which follows that . Define space aswith norm as a Banach space [10] and . Moreover, we introduce the Banach spacewith the norm and .
Next, we will give the existence and uniqueness of solution to equation (18) based on the Banach contraction argument. However, two important issues are noteworthy about and . One is that the space in which belongs to is not a Banach space, and the other is that is not periodic with period 1. Hence, we let and to be transient in order to use Banach contraction argument. And, equation (18) becomes
Since the first four equations in both equations (18) and (23) are independent of and is preserved with respect to time , by following closely the proofs of Theorems 3 and 4 in [14], we have the following results. Let be equipped with the norm
Theorem 1 (local existence and uniqueness). For initial data , there exists a time such that system equation (23) has a unique solution in .
In order to obtain global existence and uniqueness, we need to make more hypotheses on initial data, so let be a space consisting of all in such that
Theorem 2 (global existence and uniqueness). For initial data , system equation (18) has a unique global solution . Moreover, is satisfied at all times. Furthermore, the map defined as which is a continuous semigroup.
Proof. The proof follows the same clue as Theorem 4 in [14], so we prove only equation (26) here. Firstly, by equation (18), we haveIt satisfies thatThus, . Since initial data , equation (26) can be obtained.
3. Bijective Maps between Eulerian and Lagrangian Coordinates
Since the energy is concentrated on the zero measure sets when wave breaking occurs, we must consider a periodic positive Radon measure. Consequently, we make the following definition.
Definition 1. is the set of all triplets such that , , and is a positive and periodic Radon measure whose absolute continuous part is .
Note that the variables in Eulerian space are and those in the Lagrangian space are . As we prefer to get one-to-one correspondence between Eulerian and Lagrangian coordinates, we define equivalence of the latter by establishing an equivalence class map on . Let us start by relabeling invariance first.
Letandwith . As is described in many references (for example, Lemma 3.2 in [29]), if , then and if , is invertible and for , and there is a such that and then for some is dependent only on .
Define subsets and of asandLet be a group with its operation defined by . The map defined asis an equivalence class map on . Based on the proof of Theorem 4.2 in [14], we have the following theorem by a slight modification.
Theorem 3. Define on as ; then, generates a continuous semigroup.
Theorem 4. For any , letwhereThen . We denote , and let denote the equivalence class of .
Before giving the proof of Theorem 4, we give a critical lemma. Define a setNote that here is the absolute continuous part of . By Besicovitch’s derivation theorem, one can obtain .
Lemma 1. For , we have
Proof. Firstly, we claim that for all , there is a such that and are in , where is the singular part of Radon measure and its support is a point set with a countable number of elements. If not, then there exists , such that for any , or and then for any , . Consequently, we may construct an injection between and , which is rather impossible because is uncountable and is countable.
Then, we can construct sequences and such thatBy the definition of , we haveDividing equation (40) by and taking , we obtain equation (38).
Proof of Theorem 4. By Lemma 1 and slight modifications of Theorem 4.3 in [14], we will establish the map from Lagrangian coordinates to Eulerian ones, which is a generalization of Theorem 4.7 in [14]. We only state the results here, as this proof and that of Theorem 4.7 in [14] are very similar.
Theorem 5. Given any , we define bybelonging to , where for any Borel set B is called the push forward element of by . Then, belongs to and is independent of the representative from . We denote .
Next, we will clarify the relation between and .
Theorem 6. The maps and are invertible and
Proof. The proof follows the same lines as in Theorem 4.8 in [14], so we do not present it here.
Now, we obtain the solution map , that is, .
4. Weak Solutions
Definition 2. Let and . Assume that and satisfy the following:(i) and .(ii)If the equationswhere ,andhold for all spatial periodic functions , then we say is a global weak solution of equation (8).Moreover, if this solution satisfiesthen we say it is a global conservative solution of equation (8).
Theorem 7. Given , if , then is a global conservative solution of equation (8).
Proof. Theorem 2 and Definition 1 imply that in Definition 2 holds. In the following section, we will prove equations (43)–(46) one by one for any spatial periodic function . Let and we have . Since , we obtainByand , we haveIntegrating this formula into equation (47), we obtainAnd, the proof for equation (43) completes here. Using equation (28), a direct computation implies thatThis completes the proof for equation (44). By , we haveIt satisfies thatAnd, this completes the proof of equation (45). Similarly, let in the left side of equation (46); we haveThis completes the proof of equation (46).
Data Availability
The computation data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (Nos. 11701525, 11971446, and 51609087).