Global Conservative Solutions of the Two-Component 
 μ
 -Hunter–Saxton System

In this paper, we establish global conservative solutions of the two-component 
 
 μ
 
 -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).”


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][2][3]: e μ-Hunter-Saxton equation [4] is as follows: e Hunter-Saxton equation [5] is as follows: en, 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][7][8][9][10][11][12][13][14][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. eir 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 H 1 energy (u 2 + u 2 x ) dx remains constant except for the exact time of break down, we call the conservative solutions; if (u 2 + u 2 x ) dx 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 μ-Hunter-Saxton system [16]: where t > 0 is the time vector, and x ∈ R is a space vector. is system is of a bi-Hamilton structure, and it can also be viewed as a bivariational equation set. erefore, equation (4) can be rewritten as where Also, (4) becomes a two-component CH system, which has been studied in [17][18][19][20][21]. e 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 A(t, x) � − u xx , then equation (4) turns to a two-component HS system, which has been looked into in [22,23]. Hence, we can say 2-μHS system equation (4) lies in an intermediate between 2-CH and 2-HS systems. e Cauchy problem for equation (4) has been studied extensively in [24][25][26][27]. In addition, research studies have shown that this system is locally well-posed [26] for (u 0 , ρ 0 ) ∈ H s × H s− 1 , s > (3/2); 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]. e 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 μ(u) in the 2-μHS system brings some difficulties to the calculation of equation (28). Fortunately, we overcame it. Because A(t, x) � − u xx in the 2-HS system, the 2-μHS system is structurally more complex than the 2-HS system. In [23], the author gave the specific expression of y, U, H, and r (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-μHS 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).

Preliminaries
Firstly, we reformulate system equation (4). Assume which is equivalent to (13/12). 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, V 1 is not a Banach space. We define y: R ⟶ V 1 , t ⟶ y(t, ·) as the solution of And then, we define Taking the derivative of both sides of equations (15)-(17) with respect to t and using equation (11), we can obtain a result. Combining this result with equation (14), we have the following semilinear system of (y, U, H, r):

Mathematical Problems in Engineering
After we define a new variable z � y(t, ξ ′ ), we have Here, we will make some explanation about H. Since (u, ρ) is periodic with period 1 and y ∈ V 1 , we can obtain with norm ‖f‖ V � ‖f‖ H 1 ([0,1]) as a Banach space [10] and H ∈ V. Moreover, we introduce the Banach space 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 y and H. One is that the space V 1 in which y belongs to is not a Banach space, and the other is that H is not periodic with period 1. Hence, we let ζ � y − Id and σ � H − eId 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) Theorem 1 (local existence and uniqueness). For initial In order to obtain global existence and uniqueness, we need to make more hypotheses on initial data, so let G be a space consisting of all (ζ, Theorem 2 (global existence and uniqueness). For initial data X 0 � (y 0 , U 0 , H 0 , μ 0 , e 0 , r 0 ) ∈ G, system equation (18) has a unique global solution X(t) ∈ C 1 (R + , E). Moreover, X(t) ∈ G is satisfied at all times. Furthermore, the map S: G × R + ⟶ G defined as S t (X 0 ) � X(t) which is a continuous semigroup.

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. D is the set of all triplets (u, ρ, η) such that u ∈ H 1 per , ρ ∈ L 2 per , and η is a positive and periodic Radon measure whose absolute continuous part is η ac � (u 2 x + ρ 2 ) dx. Note that the variables in Eulerian space are (u, ρ, η) and those in the Lagrangian space are (y, U, H, r). 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 G. Let us start by relabeling invariance first. Let and with s ≥ 1. As is described in many references (for example, Lemma 3.2 in [29]), if f ∈ G s , then (1/1 + s) ≤ f x ≤ 1 + s a.e. and if f ∈ W 1,∞ loc , f is invertible and f(x + 1) � f(x) + 1 for x ∈ R, and there is a c ≥ 1 such that (1/c) ≤ f x ≤ c a.e. and then f ∈ G s for some s is dependent only on c.
Define subsets F and F s of G as F � (y, U, H, μ, e, r) ∈ G: and F s � (y, U, H, μ, e, r) ∈ G: Let G � G × R be a group with its operation defined by , c)(y, U, H, μ, e, r)) � y°f, U°f, H°f + c, μ, e, r°ff x , is an equivalence class map on G. Based on the proof of eorem 4.2 in [14], we have the following theorem by a slight modification.   y(t, ξ)), y(t, ξ))y ξ , where Mathematical Problems in Engineering en (y, U, H, μ, e, r) ∈ F 0 . We denote L: D ⟶ (F/G), and let L(u, ρ, η) ∈ (F/G) denote the equivalence class of (y , U, H, μ, e, r).
Before giving the proof of eorem 4, we give a critical lemma. Define a set Note that (u 2 x + ρ 2 ) dx here is the absolute continuous part of η. By Besicovitch's derivation theorem, one can obtain meas (B c ) � 0.
Proof. Firstly, we claim that for all i ∈ N, there is a 0 < ε < (1/i) such that x − ε and x + ε are in supp(η s ) c , where η s is the singular part of Radon measure η and its support supp (η s ) is a point set with a countable number of elements. If not, then there exists i ∈ N, such that for any 0 Consequently, we may construct an injection between (x − (1/i), x + (1/i))/supp (η s ) and supp (η s ), which is rather impossible because (x − (1/i), x + (1/i))/supp (η s ) is uncountable and supp (η s ) is countable. en, we can construct sequences y(ξ i ) and y(ξ i ′ ) such that By the definition of F η , we have Dividing equation (40) by ξ i ′ − ξ i and taking i ⟶ ∞, we obtain equation (38).

□
Proof of eorem 4. By Lemma 1 and slight modifications of eorem 4.3 in [14], we will establish the map from Lagrangian coordinates to Eulerian ones, which is a generalization of eorem 4.7 in [14]. We only state the results here, as this proof and that of eorem 4.7 in [14] are very similar. □ Theorem 5. Given any [X] ∈ F/G, we define (u, ρ, η) by 1 (B)) for any Borel set B is called the push forward element of ξ by f. en, (u, ρ, η) belongs to D and is independent of the representative X from [X]. We denote M: F/G ⟶ D.
Next, we will clarify the relation between L and M.
Proof. e proof follows the same lines as in eorem 4.8 in [14], so we do not present it here. Now, we obtain the solution map T t � M°S t°L , that is,
Moreover, if this solution (u, ρ) satisfies then we say it is a global conservative solution of equation (8).

Data Availability
e computation data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.