Periodic Conservative Solutions for a Modified Two-Component Camassa-Holm System with Peakons

Abstract

This paper investigates the periodically weak conservative solutions for a modified periodic two-component Camassa-Holm system arisen from shallow water waves moving. The underlying approach is based on a system reformulation by introducing a skillfully defined characteristic and a set of newly defined variables. Such treatment makes it possible to overcome the difficulties inherent in the multicomponent system, leading to the establishment of the periodic conservative solutions for the original system.

1. Introduction

We consider in this paper a modified periodic two-component Camassa-Holm system with peakons [1], namely, 𝑚_𝑡+2𝑚𝑢_𝑥+𝑚_𝑥𝑢+(𝑚𝑣)_𝑥+𝑛𝑣_𝑥𝑛=0,𝑡>0,𝑥∈𝑅,_𝑡+2𝑛𝑣_𝑥+𝑛_𝑥𝑣+(𝑛𝑢)_𝑥+𝑚𝑢_𝑥=0,𝑡>0,𝑥∈𝑅, (1) with the initial and periodic conditions 𝑢(0,𝑥)=𝑢₀(𝑥)𝑣(),𝑥∈𝑅,0,𝑥=𝑣₀(𝑥)𝑢()(𝑥)𝑣()(𝑥),𝑥∈𝑅,𝑡,𝑥+1=𝑢,𝑡>0,𝑥∈𝑅,𝑡,𝑥+1=𝑣,𝑡>0,𝑥∈𝑅, (2) where 𝑚=𝑢−𝑢_𝑥𝑥, 𝑛=𝑣−𝑣_𝑥𝑥 are periodic on the 𝑥-variable. System (1) is a modified version of the coupled two-component Camassa-Holm system in [2], 𝑚_𝑡=2𝑚𝑢_𝑥+𝑚_𝑥𝑢+(𝑚𝑣)_𝑥+𝑛𝑣_𝑥𝑛,𝑡>0,𝑥∈𝑅,_𝑡=2𝑛𝑣_𝑥+𝑛_𝑥𝑣+(𝑛𝑢)_𝑥+𝑚𝑢_𝑥,𝑡>0,𝑥∈𝑅, (3) which has been established by Fu and Qu to allow for peakon solitons in the form of a superposition of multipeakons. By parameterizing 𝑡̃=−𝑡 for system (3), it then takes the form of (1), which can be rewritten as a Hamiltonian system, 𝜕𝑚𝑛𝜕𝑡=−𝜕𝑚+𝑚𝜕𝜕𝑚+𝑛𝜕𝜕𝑛+𝑚𝜕𝜕𝑛+𝑛𝜕𝛿𝐻𝛿𝑚=𝑢𝛿𝐻𝛿𝑛=𝑣 (4) with the Hamiltonian 𝐻=(1/2)∫(𝑚𝐺∗𝑚+𝑛𝐺∗𝑛)𝑑𝑥, where 𝐺∗𝑚=𝑢, 𝐺∗𝑛=𝑣, and 𝐺=(1/2)𝑒^−|𝑥|. Furthermore, it has the following conserved quantities: _𝑆𝑢𝑑𝑥,_𝑆𝑣𝑑𝑥,_𝑆𝑚𝑑𝑥,_𝑆𝑛𝑑𝑥,_𝑆𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥𝑑𝑥. (5) A remarkable fact for the two types of coupled two-component Camassa-Holm system is that they have an infinite number of peakons, which take the following form in the periodic case that 𝑢()=1𝑡,𝑥sinh(1/2)^𝑛_𝑖=1𝑝_𝑖(𝑡)cosh𝑥−𝑞_𝑖(𝑡)−𝑥−𝑞_𝑖(𝑡−1)2,𝑣()=1𝑡,𝑥sinh(1/2)^𝑛_𝑖=1𝑟_𝑖(𝑡)cosh𝑥−𝑞_𝑖(𝑡)−𝑥−𝑞_𝑖(𝑡−1)2, (6) where (𝑝_𝑖(𝑡),𝑟_𝑖(𝑡),𝑞_𝑖(𝑡)) satisfy the explicit system of ordinary differential equations 𝑝̇_𝑖=^𝑛_{𝑗=1,𝑖≠𝑗}𝑝_𝑖𝑝_𝑗+𝑟_𝑖𝑟_𝑗sgn𝑞_𝑗−𝑞_𝑖𝑒^{−|𝑞𝑖−𝑞𝑗|},𝑟̇_𝑖=^𝑛_{𝑗=1,𝑖≠𝑗}𝑝_𝑖𝑝_𝑗+𝑟_𝑖𝑟_𝑗sgn𝑞_𝑗−𝑞_𝑖𝑒^{−|𝑞𝑖−𝑞𝑗|},𝑞̇_𝑖=−^𝑛_𝑗=1𝑝_𝑗+𝑟_𝑗𝑒^{−|𝑞𝑖−𝑞𝑗|}. (7) In particular, when 𝑢=0 (or 𝑣=0), system (1) has the same peakon solitons as the Camassa-Holm (CH) equation.

Moreover, when 𝑢=𝑣, system (1) is reduced to the scalar Camassa-Holm equation, 𝑚_𝑡+4𝑚𝑢_𝑥+2𝑚_𝑥𝑢=0. (8) The CH equation which models the unidirectional propagation of shallow water waves over a flat bottom has attracted considerable attention because it has a bi-Hamiltonian structure [3] and is completely integrable [4, 5]. Another interesting property of the CH equation is that the solutions experience wave breaking [4, 6, 7]. The presence of breaking waves means that the solution remains bounded while its slope becomes unbounded in finite time [7, 8]. After wave breaking the solutions of the CH equation can be continued uniquely as either global conservative [9–12] or global dissipative solutions [13].

The important issue to address is the Cauchy problem for system (1) in the periodic case on 𝑆. It has been shown that system (1) is locally well posed with periodic initial value and there exist global strong solutions which blow up in finite time for certain class of initial data [1]. Moreover, an existence result for a class of lower regularity of solutions [1] and the global attractor with the periodic boundary condition [14] were also given.

It should be noted that extending the solutions beyond wave breaking imposes significant challenge, as reflected in the case of multipeakons given by (6). Peakons interact in a way similar to that of solitons of the CH equation, and wave breaking may appear when at least two of the 𝑞_𝑖 coincide. However, the solutions (6) are not smooth even with continuous functions (𝑝_𝑖(𝑡),𝑟_𝑖(𝑡),𝑞_𝑖(𝑡)). One possible way to interpret (6) as a weak solution of system (1) is to rewrite (6) as 𝑢_𝑡+(𝑢+𝑣)𝑢_𝑥+1−𝜕²_𝑥⁻¹×𝑢𝑣_𝑥+𝑢²+12𝑢²_𝑥+𝑢_𝑥𝑣_𝑥+12𝑣²−12𝑣²_𝑥𝑥𝑣=0,𝑡>0,𝑥∈𝑅,_𝑡+(𝑢+𝑣)𝑣_𝑥+1−𝜕²_𝑥⁻¹×𝑢_𝑥𝑣+𝑣²+12𝑣²_𝑥+𝑢_𝑥𝑣_𝑥+12𝑢²−12𝑢²_𝑥𝑥=0,𝑡>0,𝑥∈𝑅. (9) In our recent work [15, 16], we investigate the continuity of the solutions for system (1) beyond collision time on the full line, leading to uniquely to either global conservative solutions where the energy is conserved for almost all times or global dissipative solutions where energy loss occurs only through wave breaking.

Motivated by our recent work [15–17], we study in this paper the continuation of the solutions for system (1) beyond wave breaking in the periodic case and present a novel approach from a Lagrangian point of view. The approach we applied here is mainly based on the reformulation by introducing a newly defined characteristic and a set of variables, where the associated energy is introduced as an additional variable to obtain a well-posed initial-value problem. The argument is inspired mainly by [11] in the study of the periodic conservative solution of the CH equation. However, one problematic issue is that we here deal with a coupled system where the mutual effect between two components makes the analysis more complicated than a single one. The newly introduced characteristic and the skillfully defined variables are crucial to overcome the above-mentioned problems. By overcoming the problems, we obtain the global periodic conservative solutions of (1).

The outline of this paper is as follows. The original system is transformed into a Lagrangian equivalent semilinear system by introducing a set of newly defined variables in Section 2 and the global solutions of the equivalent system are obtained in Section 3. Section 4 is devoted to establishing the periodic conservative solutions for the original system, which construct a global continuous semigroup. The paper is just closed in Section 5.

1.1. Notation

Throughout this paper, we identify all spaces of periodic functions with function spaces over the unit circle 𝑆 in 𝑅²; that is, 𝑆=𝑅/𝑍. We denote by ∗ the spatial convolution.

2. The Original System and the Lagrangian Equivalent System

2.1. Part A: The Original System

Let us introduce an operator Λ=(1−𝜕²_𝑥)⁻¹, associated with its Green function 𝐺(𝑥)=(cosh(𝑥−[𝑥]−1/2))/(2sinh(1/2)), where [𝑥] stands for the integer part of 𝑥, such that (𝑥)(𝑥)=1Λ𝑓=𝐺∗𝑓2¹₀cosh𝑥−𝑥^′−𝑥−𝑥^′−1/2()𝑓𝑥sinh1/2^′𝑑𝑥^′,𝑓∈𝐿²(𝑆). (10) Thus we can rewrite (1) as a form of a quasilinear evolution equation, 𝑢_𝑡+(𝑢+𝑣)𝑢_𝑥+𝐺∗𝑢𝑣_𝑥+𝜕_𝑥𝐺∗𝑢²+12𝑢²_𝑥+𝑢_𝑥𝑣_𝑥+12𝑣²−12𝑣²_𝑥𝑣=0,𝑡>0,𝑥∈𝑅,_𝑡+(𝑢+𝑣)𝑣_𝑥+𝐺∗𝑢_𝑥𝑣+𝜕_𝑥𝐺∗𝑣²+12𝑣²_𝑥+𝑢_𝑥𝑣_𝑥+12𝑢²−12𝑢²_𝑥=0,𝑡>0,𝑥∈𝑅. (11) If we define 𝐴(𝑡,𝑥), 𝐵(𝑡,𝑥), 𝐶(𝑡,𝑥), and 𝐷(𝑡,𝑥) as follows: 𝐴(𝑡,𝑥)=𝐺∗𝑢𝑣_𝑥=12¹₀cosh𝑥−𝑥^′−𝑥−𝑥^′−1/2sinh(1/2)𝑢𝑣_𝑥𝑡,𝑥^′𝑑𝑥^′,𝐵(𝑡,𝑥)=𝐺∗𝑢²+12𝑢²_𝑥+𝑢_𝑥𝑣_𝑥+12𝑣²−12𝑣²_𝑥=12¹₀cosh𝑥−𝑥^′−𝑥−𝑥^′−1/2()×𝑢sinh1/2₂+12𝑢²_𝑥+𝑢_𝑥𝑣_𝑥+12𝑣²−12𝑣²_𝑥𝑡,𝑥^′𝑑𝑥^′,𝐶(𝑡,𝑥)=𝐺∗𝑣𝑢_𝑥=12¹₀cosh𝑥−𝑥^′−𝑥−𝑥^′−1/2sinh(1/2)𝑣𝑢_𝑥𝑡,𝑥^′𝑑𝑥^′,𝐷(𝑡,𝑥)=𝐺∗𝑣²+12𝑣²_𝑥+𝑢_𝑥𝑣_𝑥+12𝑢²−12𝑢²_𝑥=12¹₀cosh𝑥−𝑥^′−𝑥−𝑥^′−1/2()×𝑣sinh1/2₂+12𝑣²_𝑥+𝑢_𝑥𝑣_𝑥+12𝑢²−12𝑢²_𝑥𝑡,𝑥^′𝑑𝑥^′, (12) then (1) can be rewritten as 𝑢_𝑡+(𝑢+𝑣)𝑢_𝑥+𝐴+𝐵_𝑥𝑣=0,𝑡>0,𝑥∈𝑅,_𝑡+(𝑢+𝑣)𝑣_𝑥+𝐶+𝐷_𝑥=0,𝑡>0,𝑥∈𝑅. (13) For smooth solutions, we have that the total energy 𝐸(𝑡)=_𝑆𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥(𝑡,𝑥)𝑑𝑥 (14) is constant in time. Indeed, differentiating the two equations in (13), respectively, with respect to 𝑥 and using the identity 𝜕²_𝑥𝐺∗𝑓=𝐺∗𝑓−𝑓, we have 𝑢_𝑥𝑡+𝑢²_𝑥+𝑢_𝑥𝑣_𝑥+𝑢𝑢_𝑥𝑥+𝑣𝑢_𝑥𝑥+𝐴_𝑥+𝐵−𝑢²+𝑢²_𝑥2+𝑢_𝑥𝑣_𝑥+𝑣²2−𝑣²_𝑥2𝑣=0,_𝑥𝑡+𝑣²_𝑥+𝑢_𝑥𝑣_𝑥+𝑣𝑣_𝑥𝑥+𝑢𝑣_𝑥𝑥+𝐶_𝑥+𝐷−𝑣²+𝑣²_𝑥2+𝑢_𝑥𝑣_𝑥+𝑢²2−𝑢²_𝑥2=0. (15) Multiplying the first equation in (13) by 𝑢 and the second by 𝑣 and multiplying the first equation in (15) by 𝑢_𝑥 and the second by 𝑣_𝑥, we obtain the following four equalities:𝑢²2_𝑡+𝑢³3_𝑥+𝑢𝑣𝑢_𝑥+𝑢𝐴+𝑢𝐵_𝑥𝑣=0,₂2_𝑡+𝑣³3_𝑥+𝑢𝑣𝑣_𝑥+𝑣𝐶+𝑣𝐷_𝑥𝑢=0,_2𝑥2_𝑡+𝑢𝑢²_𝑥2−𝑢³3_𝑥+𝑣𝑢_𝑥𝑢_𝑥𝑥+𝑢²𝑣_𝑥−𝑢𝐴+𝑢_𝑥𝑣𝐵−²𝑢_𝑥2+𝑣²_𝑥𝑢_𝑥2𝑣=0,_2𝑥2_𝑡+𝑣𝑣²_𝑥2−𝑣³3_𝑥+𝑢𝑣_𝑥𝑣_𝑥𝑥+𝑣²𝑢_𝑥−𝑣𝐶+𝑣_𝑥𝑢𝐷−²𝑣_𝑥2+𝑢²_𝑥𝑣_𝑥2=0. (16a)(16b)(16c)(16d)

Using the four identities in (16a), (16b), (16c), and (16d), we can get that 𝑑𝐸(𝑡)=𝑑𝑑𝑡𝑑𝑡_𝑆𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥(𝑡,𝑥)𝑑𝑥=0. (17) Thus (13) possesses the 𝐻¹_per-norm conservation law defined as ‖𝑧‖_𝐻¹_per=‖𝑢‖_𝐻¹_per+‖𝑣‖_𝐻¹_per=_[0,1]𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥𝑑𝑥^1/2, (18) where 𝑧(𝑡,𝑥)=(𝑢,𝑣)(𝑡,𝑥) denotes the solution of system (13). Since 𝑧∈𝐻¹_per, Young’s inequality ensures 𝐴,𝐵,𝐶,𝐷∈𝐻¹_per.

2.2. Part B: The Lagrangian Equivalent System

We reformulate system (13) as follows. We first introduce the corresponding characteristic 𝑦∶𝑅→𝑊₁, 𝑡→𝑦(𝑡,⋅) as the solution of 𝑦_𝑡(𝑡,𝜉)=(𝑢+𝑣)𝑡,𝑦(𝑡,𝜉), (19) where the space 𝑊₁ is defined by 𝑊₁={𝑓∈𝐻¹_loc(𝑅)∣𝑓(𝜉+1)=𝑓(𝜉)+1forall𝜉∈𝑅}. We define the cumulative energy distribution 𝐻 as 𝐻(𝑡,𝜉)=_𝑦()^{𝑦(𝑡,𝜉)}_𝑡,0𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥(𝑡,𝑥)𝑑𝑥. (20) It is not hard to check that 𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥𝑡+(𝑢+𝑣)𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥𝑥=𝑢³−2𝑢𝐵+𝑣³−2𝑣𝐷_𝑥. (21) Then it follows from (19) and (21) that 𝑑𝐻=𝑢𝑑𝑡³−2𝑢𝐵+𝑣³−2𝑣𝐷𝑡,𝑦(𝑡,𝜉)^𝜉₀. (22) Using the fact that 𝑦∈𝑊₁ and the periodicity of 𝑢 and 𝑣, it follows from (20) and (22) that 𝐻()()()()()()𝑡,𝜉+1−𝐻𝑡,𝜉=𝐻𝑡,1−𝐻𝑡,0=𝐻0,1−𝐻0,0 (23) for all 𝜉∈𝑅. Let us introduce the vector space 𝑊 given by 𝑊=𝑓∈𝐻¹_loc(𝑅)∣𝑓(𝜉+1)=𝑓(𝜉)+𝛼forsome𝛼∈𝑅,all𝜉∈𝑅 (24) endowed with the norm ‖𝑓‖_𝑊=‖𝑓‖_𝐻1([0,1]), which is a Banach space. Then we have that 𝐻∈𝑊.

Throughout the following, we use the notation 𝑈()()()(),𝑈𝑡,𝜉=𝑢𝑡,𝑦𝑡,𝜉,𝑉𝑡,𝜉=𝑣𝑡,𝑦𝑡,𝜉_𝑥(𝑡,𝜉)=𝑢_𝑥𝑡,𝑦(𝑡,𝜉),𝑉_𝑥(𝑡,𝜉)=𝑣_𝑥𝑡,𝑦(𝑡,𝜉). (25) After the change of variables 𝑥=𝑦(𝑡,𝜉) and 𝑥^′=𝑦(𝑡,𝜉^′), we obtain the following expressions for 𝐴, 𝐴_𝑥, 𝐵, 𝐵_𝑥, 𝐶, 𝐶_𝑥, 𝐷, and 𝐷_𝑥, namely, 𝐴(𝜉)=12¹₀cosh𝑦(𝜉)−𝑦𝜉^′−𝑦(𝜉)−𝑦𝜉^′−1/2()×sinh1/2𝑈𝑉_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐴_𝑥(𝜉)=12¹₀sinh𝑦(𝜉)−𝑦𝜉^′−𝑦(𝜉)−𝑦𝜉^′−1/2()×sinh1/2𝑈𝑉_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐵(𝜉)=14¹₀cosh𝑦(𝜉)−𝑦𝜉^′−𝑦(𝜉)−𝑦𝜉^′−1/2()⋅𝐻sinh1/2_𝜉+𝑈²+2𝑈_𝑥𝑉_𝑥−𝑉²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐵_𝑥(𝜉)=14¹₀sinh𝑦(𝜉)−𝑦𝜉^′−𝑦(𝜉)−𝑦𝜉^′−1/2()⋅𝐻sinh1/2_𝜉+𝑈²+2𝑈_𝑥𝑉_𝑥−𝑉²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐶(𝜉)=12¹₀cosh𝑦(𝜉)−𝑦𝜉^′−𝑦(𝜉)−𝑦𝜉^′−1/2()×sinh1/2𝑉𝑈_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐶_𝑥(𝜉)=12¹₀sinh𝑦(𝜉)−𝑦𝜉^′−𝑦(𝜉)−𝑦𝜉^′−1/2()×sinh1/2𝑉𝑈_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐷(𝜉)=14¹₀cosh𝑦(𝜉)−𝑦𝜉^′−𝑦(𝜉)−𝑦𝜉^′−1/2()⋅𝐻sinh1/2_𝜉+𝑉²+2𝑈_𝑥𝑉_𝑥−𝑈²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐷_𝑥(𝜉)=14¹₀sinh𝑦(𝜉)−𝑦𝜉^′−𝑦(𝜉)−𝑦𝜉^′−1/2()⋅𝐻sinh1/2_𝜉+𝑉²+2𝑈_𝑥𝑉_𝑥−𝑈²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′, (26) where the variable 𝑡 is omitted for simplification and in the above we used the fact that 𝐻_𝜉=(𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥)∘𝑦𝑦_𝜉 and took for granted that 𝑦 is an increasing function for any fixed time 𝑡 (the validity will be proved later). In view of the periodicity of 𝐻_𝜉, 𝑈, and 𝑉, these expressions can be rewritten as 𝐴(𝜉)=1𝑒−1¹₀cosh𝑦(𝜉)−𝑦𝜉^′𝑈𝑉_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′+12¹₀𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}𝑈𝑉_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐴_𝑥(𝜉)=1𝑒−1¹₀sinh𝑦(𝜉)−𝑦𝜉^′𝑈𝑉_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′−12¹₀sgn𝜉−𝜉^′𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}×𝑈𝑉_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐵(𝜉)=12(𝑒−1)×¹₀cosh𝑦(𝜉)−𝑦𝜉^′×𝐻_𝜉+𝑈²+2𝑈_𝑥𝑉_𝑥−𝑉²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′+14¹₀𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}×𝐻_𝜉+𝑈²+2𝑈_𝑥𝑉_𝑥−𝑉²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐵_𝑥(𝜉)=12()×𝑒−1₁₀sinh𝑦(𝜉)−𝑦𝜉^′×𝐻_𝜉+𝑈²+2𝑈_𝑥𝑉_𝑥−𝑉²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′−14¹₀sgn𝜉−𝜉^′𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}×𝐻_𝜉+𝑈²+2𝑈_𝑥𝑉_𝑥−𝑉²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐶(𝜉)=1𝑒−1¹₀cosh𝑦(𝜉)−𝑦𝜉^′𝑉𝑈_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′+12¹₀𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}𝑉𝑈_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐶_𝑥(𝜉)=1𝑒−1¹₀sinh𝑦(𝜉)−𝑦𝜉^′𝑉𝑈_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′−12¹₀sgn𝜉−𝜉^′𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}×𝑉𝑈_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐷(𝜉)=12()×𝑒−1₁₀cosh𝑦(𝜉)−𝑦𝜉^′×𝐻_𝜉+𝑉²+2𝑈_𝑥𝑉_𝑥−𝑈²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′+14¹₀𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}×𝐻_𝜉+𝑉²+2𝑈_𝑥𝑉_𝑥−𝑈²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′,𝐷_𝑥(𝜉)=12()×𝑒−1₁₀sinh𝑦(𝜉)−𝑦𝜉^′×𝐻_𝜉+𝑉²+2𝑈_𝑥𝑉_𝑥−𝑈²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′−14¹₀sgn𝜉−𝜉^′𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}×𝐻_𝜉+𝑉²+2𝑈_𝑥𝑉_𝑥−𝑈²_𝑥𝑦_𝜉𝜉^′𝑑𝜉^′. (27) From the definition of the characteristic, the evolution equations for 𝑈 and 𝑉 in the new variables (𝑡,𝜉) take the form 𝑈_𝑡(𝑡,𝜉)=𝑢_𝑡𝑡,𝑦+𝑢_𝑥𝑡,𝑦𝑦_𝑡(𝑡,𝜉)=−𝐴−𝐵_𝑥(),𝑉∘𝑦𝑡,𝜉_𝑡(𝑡,𝜉)=𝑣_𝑡𝑡,𝑦+𝑣_𝑥𝑡,𝑦𝑦_𝑡(𝑡,𝜉)=−𝐶−𝐷_𝑥∘𝑦(𝑡,𝜉). (28) With these new variables, we now derive an equivalent system of (13), namely, 𝑦_𝑡𝑈=𝑈+𝑉,_𝑡=−𝐴−𝐵_𝑥,𝑉_𝑡=−𝐶−𝐷_𝑥,𝐻_𝑡=𝑈³−2𝑈𝐵+𝑉³−2𝑉𝐷^𝜉₀, (29) with 𝐴, 𝐴_𝑥, 𝐵, 𝐵_𝑥, 𝐶, 𝐶_𝑥, 𝐷, and 𝐷_𝑥 given in (26). Differentiating (29) with respect to 𝜉 yields 𝑦_𝜉𝑡=𝑈_𝜉+𝑉_𝜉,𝑈_𝜉𝑡=12𝐻_𝜉+12𝑈²+𝑈_𝑥𝑉_𝑥−𝑉²_𝑥−𝐵−𝐴_𝑥𝑦_𝜉,𝑉_𝜉𝑡=12𝐻_𝜉+12𝑉²+𝑈_𝑥𝑉_𝑥−𝑈²_𝑥−𝐷−𝐶_𝑥𝑦_𝜉,𝐻_𝜉𝑡=3𝑈²−2𝐵𝑈_𝜉−2𝑈𝐵_𝑥𝑦_𝜉+3𝑉²−2𝐷𝑉_𝜉−2𝑉𝐷_𝑥𝑦_𝜉, (30) which is semilinear with respect to the variables 𝑦_𝜉, 𝑈_𝜉, 𝑉_𝜉, and 𝐻_𝜉.

3. Global Solutions of the Lagrangian Equivalent System

We consider the Cauchy problem for (29) in the following form: 𝜍_𝑡𝑈=𝑈+𝑉,_𝑡=−𝐴−𝐵_𝑥,𝑉_𝑡=−𝐶−𝐷_𝑥,𝜎_𝑡=𝑈³−2𝑈𝐵+𝑉³−2𝑉𝐷^𝜉₀,ℎ_𝑡=0, (31) in the Banach space 𝐸 given by 𝐸=𝐻¹_per×𝐻¹_per×𝐻¹_per×𝐻¹_per×𝑅, (32) where 𝐻¹_per=𝑓∈𝐻¹_loc(𝑅)∣𝑓(𝜉+1)=𝑓(𝜉)∀𝜉∈𝑅 (33) with the norm ‖𝑓‖_𝐻1per=‖𝑓‖_𝐻1 (here we denote 𝐻¹([0,1]) by 𝐻¹ for simplification and the same for other norms in the following), 𝜍=𝑦−𝐼, 𝜎=𝐻−ℎ𝐼, and ℎ=𝐻(𝑡,1)−𝐻(𝑡,0) (𝐼 denotes identity mapping; i.e., 𝐼(𝜉)=𝜉). Note that there is a bijection (𝜍,𝑈,𝑉,𝜎,ℎ)→(𝑦,𝑈,𝑉,𝐻) between 𝐸 and 𝑊₁×𝐻¹_per×𝐻¹_per×𝑊. The advantage of the modification in (31) is that 𝐸 is a Banach space, which is important for the contraction argument, while 𝑊₁×𝐻¹_per×𝐻¹_per×𝑊 is not Banach space. In the following, we use the Banach space 𝐸 and the set of variables (𝜍,𝑈,𝑉,𝜎,ℎ).

Lemma 1. For any given 𝑋=(𝜍,𝑈,𝑉,𝜎,ℎ)∈𝐸, one has that 𝐴, 𝐴_𝑥, 𝐵, 𝐵_𝑥, 𝐶, 𝐶_𝑥, 𝐷, and 𝐷_𝑥 defined by (26) are locally Lipschitz continuous from 𝐸 to 𝐻¹_𝑝𝑒𝑟. Moreover, one has 𝐴_𝑥𝜉=𝐴−𝑈𝑉_𝑥1+𝜍_𝜉,𝐴_𝜉=𝐴_𝑥1+𝜍_𝜉,𝐵_𝑥𝜉1=−2𝜎_𝜉+1+ℎ𝐵−2𝑈²−𝑈_𝑥𝑉_𝑥+𝑉²_𝑥1+𝜍_𝜉,𝐵_𝜉=𝐵_𝑥1+𝜍_𝜉,𝐶_𝑥𝜉=𝐶−𝑉𝑈_𝑥1+𝜍_𝜉,𝐶_𝜉=𝐶_𝑥1+𝜍_𝜉,𝐷_𝑥𝜉1=−2𝜎_𝜉+1+ℎ𝐷−2𝑉²−𝑈_𝑥𝑉_𝑥+𝑈²_𝑥1+𝜍_𝜉,𝐷_𝜉=𝐷_𝑥1+𝜍_𝜉. (34)

Proof. We firstly prove that 𝐵_𝑥 given in (26) is locally Lipschitz continuous from 𝐸 to 𝐻¹_per, and the others follow in the same way. Let 𝑋=(𝜍,𝑈,𝑉,𝜎,ℎ) and 𝑋=(𝜍,̃𝑈,𝑉,𝜎,ℎ) be two elements of 𝐵_𝑀 where 𝐵_𝑀={𝑋=(𝜍,𝑈,𝑉,𝜎,ℎ)∣‖𝑋‖_𝐸≤𝑀}. We have 𝑦_𝐿∞=‖𝐼+𝜍‖_𝐿^∞≤1+𝐶‖𝜍‖_𝐻¹≤1+𝐶𝑀 (35) and ‖𝑦‖_𝐿∞≤1+𝐶𝑀. Since cosh𝑥 is locally Lipschitz on {𝑥∈𝑅∣|𝑥|≤2(1+𝐶𝑀)}, we have cosh𝑦(𝜉)−𝑦𝜉^′−cosh𝑦(𝜉)−𝑦𝜉^′≤𝐶𝑦(𝜉)−𝑦(𝜉)−𝑦𝜉^′+𝑦𝜉^′≤𝐶‖𝜍−𝜍‖̃_𝐿^∞, (36) where 𝐶 is a generic constant depending only on 𝑀, for all 𝜉,𝜉^′∈[0,1]. It follows that cosh𝑦(𝜉)−𝑦(⋅)𝑈²𝑦_𝜉(⋅)−cosh𝑦(𝜉)−𝑦(⋅)𝑈²𝑦_𝜉(⋅)_𝐿2≤𝐶‖𝜍−𝜍‖̃_𝐿^∞+𝑈−𝑈_𝐿∞+𝜍_𝜉−𝜍̃_𝜉𝐿2, (37) where we have used the fact that ‖𝑈²−𝑈²‖_𝐿^∞≤𝐶‖𝑈−𝑈‖_𝐿^∞ in 𝐵_𝑀. We handle the other terms of 𝐵 in the same way and then prove that 𝐵 is Lipschitz from 𝐵_𝑀 to 𝐿^∞_per. Similarly, one proves that 𝐵_𝑥 is Lipschitz from 𝐵_𝑀 to 𝐿^∞_per. Direct computation by the chain rule gives us the expressions (34) for the derivatives of 𝐴, 𝐴_𝑥, 𝐵, 𝐵_𝑥, 𝐶, 𝐶_𝑥, 𝐷, and 𝐷_𝑥. Using the derivative 𝐵_𝑥𝜉 of 𝐵_𝑥, we have 𝐵_𝑥𝜉−𝐵_{𝑥𝜉𝐿2}≤𝐶𝐵−𝐵_𝐿∞+𝑈−𝑈_𝐿∞+𝑈_𝑥−𝑈_𝑥𝐿2+𝑉_𝑥−𝑉_𝑥𝐿2+𝜍_𝜉−𝜍̃_𝜉𝐿2+𝜎_𝜉−𝜎_𝜉𝐿2+ℎ≤𝑋ℎ−𝑋−_𝐸. (38) Thus we have proved that 𝐵_𝑥 given in (26) is Lipschitz from 𝐵_𝑀 to 𝐻¹_per. Similarly, 𝐴, 𝐴_𝑥, 𝐵, 𝐶, 𝐶_𝑥, 𝐷, and 𝐷_𝑥 defined in (26) are locally Lipschitz continuous from 𝐵_𝑀 to 𝐻¹_per.

Theorem 2. Let any initial data 𝑋=(𝜍,𝑈,𝑉,𝜎,ℎ)∈𝐸 be given. The system (31) admits a unique local solution defined on some time interval [0,𝑇], where 𝑇 depends only on ‖𝑋‖_𝐸.

Proof. To establish the local existence of solutions, one proceeds as in Lemma 1 and obtain that 𝐹(𝑋) given by 𝐹(𝑋)=𝑈+𝑉,−𝐴−𝐵_𝑥,−𝐶−𝐷_𝑥,𝑈³−2𝑈𝐵+𝑉³−2𝑉𝐷^𝜉₀,0, (39) with 𝑋=(𝜍,𝑈,𝑉,𝜎,ℎ), is Lipchitz continuous on any bounded set of 𝐸. The solutions of system (31) then can be rewritten as 𝑋(𝑡)=𝑋+^𝑡₀𝐹(𝑋(𝜏))𝑑𝜏. (40) Since 𝐸 is a Banach space, the theorem then follows from the standard contraction argument.

It remains to show that the local solution can be extended globally in time. Note that global solutions of (31) may not exist for all initial data in 𝐸. However, they exist when the initial data 𝑋=(𝜍,𝑈,𝑉,𝜎,ℎ) belongs to the set Γ which is defined as follows.

Definition 3. The set Γ is composed of all (𝜍,𝑈,𝑉,𝜎,ℎ)∈𝐸 such that(i)(𝜍,𝑈,𝑉,𝜎,ℎ)∈𝑊^1,∞(𝑅)⁴()𝑦×𝑅,ii_𝜉≥0,𝐻_𝜉≥0,𝑦_𝜉+𝐻_𝜉()𝑦>0almosteverywhere,iii_𝜉𝐻_𝜉=𝑦²_𝜉𝑈²+𝑈²_𝜉+𝑦²_𝜉𝑉²+𝑉²_𝜉almosteverywhere, (41a)(41b)(41c)
with 𝑦(𝜉)=𝜍(𝜉)+𝜉, 𝐻=𝜎+ℎ𝐼, where 𝑊^1,∞_per={𝑓∈𝑊^1,∞_loc∣𝑓(𝜉+1)=𝑓(𝜉)forall𝜉∈𝑅}, which is a Banach space with the norm ‖𝑓‖_𝑊1,∞per=‖𝑓‖_𝑊1,∞.
Theorem 2 gives us the existence of local solutions of (31) for initial data in 𝐸. In the following, we will only consider initial data that belongs to 𝐸∩[𝑊^1,∞_per]⁴×𝑅. To obtain that the solution of (31) belongs to 𝐸∩[𝑊^1,∞_per]⁴×𝑅, we have to specify the initial condition for (30). Define Ω=𝜉∈𝑅∣𝜍_𝜉(𝜉)≤𝜍_𝜉𝐿∞,𝑈_𝜉(𝜉)≤𝑈_𝜉𝐿∞,𝑉_𝜉(𝜉)≤𝑉_𝜉𝐿∞,𝜎_𝜉(𝜉)≤𝜎_𝜉𝐿∞. (42) It is clear that meas(Ω^𝑐)=0. We take (𝜍_𝜉,𝑈_𝜉,𝑉_𝜉,𝜎_𝜉)(0,𝜉)=(𝜍_𝜉,𝑈_𝜉,𝑉_𝜉,𝜎_𝜉)(𝜉) for 𝜉∈Ω, while (𝜍_𝜉,𝑈_𝜉,𝑉_𝜉,𝜎_𝜉)(0,𝜉)=(0,0,0,0) for 𝜉∈Ω^𝑐.
The global existence of the solution for initial data in Γ relies essentially on the fact that the set Γ is preserved by the flow as the next lemma shows.

Lemma 4. For initial data 𝑋=(𝜍,𝑈,𝑉,𝜎,ℎ)∈Γ and some 𝑇>0, one considers the local solution 𝑋(𝑡)=(𝜍,𝑈,𝑉,𝜎,ℎ)(𝑡)∈𝐶([0,𝑇],𝐸) of system (31). Then(i)𝑋(𝑡)∈Γ  for all  𝑡∈[0,𝑇],(ii)𝑦_𝜉(𝑡,𝜉)>0  for almost every  𝑡∈[0,𝑇] and almost every  𝜉∈𝑅.

Proof. (i) Let Ω be defined as before. For any given initial data 𝑋∈𝐸∩[𝑊^1,∞_per]⁴×𝑅, we have that the local solution 𝑋(𝑡)∈𝐶([0,𝑇],𝐸) of (31) belongs to [𝑊^1,∞_per]⁴×𝑅, which satisfies (41a) for all 𝑡∈[0,𝑇]. We now show that (41c) holds for any 𝜉∈Ω and therefore almost everywhere. Consider a fixed 𝜉∈Ω and drop it in the notation. On the one hand, it follows from (30) that 𝑦_𝜉𝐻_𝜉𝑡=𝑦_𝜉𝑡𝐻_𝜉+𝑦_𝜉𝐻_𝜉𝑡=𝑈_𝜉+𝑉_𝜉𝐻_𝜉+𝑦_𝜉×3𝑈²−2𝐵𝑈_𝜉+3𝑉²−2𝐷𝑉_𝜉−2𝑈𝐵_𝑥𝑦_𝜉−2𝑉𝐷_𝑥𝑦_𝜉=𝑈_𝜉𝐻_𝜉+𝑉_𝜉𝐻_𝜉+3𝑈²𝑈_𝜉𝑦_𝜉−2𝐵𝑈_𝜉𝑦_𝜉+3𝑉²𝑉_𝜉𝑦_𝜉−2𝐷𝑉_𝜉𝑦_𝜉−2𝑈𝐵_𝑥𝑦²_𝜉−2𝑉𝐷_𝑥𝑦²_𝜉, (43) and on the other hand, 𝑦²_𝜉𝑈²+𝑈²_𝜉+𝑦²_𝜉𝑉²+𝑉²_𝜉𝑡=2𝑦_𝜉𝑦_𝜉𝑡𝑈²+2𝑦²_𝜉𝑈𝑈_𝑡+2𝑈_𝜉𝑈_𝜉𝑡+2𝑦_𝜉𝑦_𝜉𝑡𝑉²+2𝑦²_𝜉𝑉𝑉_𝑡+2𝑉_𝜉𝑉_𝜉𝑡=𝑈_𝜉𝐻_𝜉+𝑉_𝜉𝐻_𝜉+3𝑈²𝑈_𝜉𝑦_𝜉+3𝑉²𝑉_𝜉𝑦_𝜉−2𝐵𝑈_𝜉𝑦_𝜉−2𝐷𝑉_𝜉𝑦_𝜉−2𝑈𝐵_𝑥𝑦²_𝜉−2𝑉𝐷_𝑥𝑦²_𝜉. (44) Thus (𝑦_𝜉𝐻_𝜉)_𝑡=(𝑦²_𝜉𝑈²+𝑈²_𝜉+𝑦²_𝜉𝑉²+𝑉²_𝜉)_𝑡. Notice that 𝑦_𝜉𝐻_𝜉(0)=(𝑦²_𝜉𝑈²+𝑈²_𝜉+𝑦²_𝜉𝑉²+𝑉²_𝜉)(0), and then 𝑦_𝜉𝐻_𝜉(𝑡)=(𝑦²_𝜉𝑈²+𝑈²_𝜉+𝑦²_𝜉𝑉²+𝑉²_𝜉)(𝑡) for all 𝑡∈[0,𝑇] and (41c) has been proved. It remains to prove that the inequalities in (41b) hold. Define 𝑡^∗=sup{𝑡∈[0,𝑇]∣𝑦_𝜉(𝑡^′)≥0forall𝑡^′∈[0,𝑡]}. Assume that 𝑡^∗<𝑇. Since 𝑦_𝜉(𝑡) is continuous with respect to 𝑡, we have 𝑦_𝜉(𝑡^∗)=0. It follows from (41c) that 𝑈_𝜉(𝑡^∗)=𝑉_𝜉(𝑡^∗)=0. Furthermore, (30) implies 𝑦_𝜉𝑡(𝑡^∗)=𝑈_𝜉(𝑡^∗)+𝑉_𝜉(𝑡^∗)=0 and 𝑦_𝜉𝑡𝑡(𝑡^∗)=(𝑈_𝜉𝑡+𝑉_𝜉𝑡)(𝑡^∗)=𝐻_𝜉𝑡(𝑡^∗). If 𝐻_𝜉𝑡(𝑡^∗)=0, then (𝑦_𝜉,𝑈_𝜉,𝑉_𝜉,𝐻_𝜉)(𝑡^∗)=(0,0,0,0) which implies (𝑦_𝜉,𝑈_𝜉,𝑉_𝜉,𝐻_𝜉)(𝑡)=0 for all 𝑡∈[0,𝑇] by the uniqueness of the solution of system (30). This contradicts the fact that 𝑦_𝜉(0)+𝐻_𝜉(0)>0 for all 𝜉∈Ω. If 𝐻_𝜉𝑡(𝑡^∗)<0, then 𝑦_𝜉𝑡𝑡(𝑡^∗)<0. Since 𝑦_𝜉(𝑡^∗)=𝑦_𝜉𝑡(𝑡^∗)=0, there exists a neighborhood 𝜛 of 𝑡^∗ such that 𝑦_𝜉(𝑡)<0 for all 𝑡∈𝜛/{𝑡^∗}. This contradicts the definition of 𝑡^∗. Hence, 𝐻_𝜉𝑡(𝑡^∗)>0. We now have 𝑦_𝜉𝑡𝑡(𝑡^∗)>0, which conversely implies 𝑦_𝜉(𝑡)>0 for all 𝑡∈𝜛/{𝑡^∗}, which contradicts the fact that 𝑡^∗<𝑇. Thus we have proved 𝑦_𝜉(𝑡)≥0 for all 𝑡∈[0,𝑇]. We now prove that 𝐻_𝜉≥0 for all 𝑡∈[0,𝑇]. This follows from (41c) when 𝑦_𝜉(𝑡)>0. If 𝑦_𝜉(𝑡)=0, then 𝑈_𝜉(𝑡)=𝑉_𝜉(𝑡)=0 from (41c). As we have seen, 𝐻_𝜉<0 would imply that 𝑦_𝜉(𝑡^′)<0 for some 𝑡^′ in a punctured neighborhood of 𝑡, which is impossible. Hence, 𝐻_𝜉≥0 for all 𝑡∈[0,𝑇]. Now we have that 𝑦_𝜉(𝑡)+𝐻_𝜉(𝑡)≥0 for all 𝑡∈[0,𝑇]. If 𝑦_𝜉(𝑡^′)+𝐻_𝜉(𝑡^′)=0 for some 𝑡^′, it then follows that (𝑦_𝜉,𝑈_𝜉,𝑉_𝜉,𝐻_𝜉)(𝑡^′)=0 which implies (𝑦_𝜉,𝑈_𝜉,𝑉_𝜉,𝐻_𝜉)(𝑡)=0 for all 𝑡∈[0,𝑇], which contradicts the fact that 𝑦_𝜉(0)+𝐻_𝜉(0)>0 for all 𝜉∈Ω. Thus 𝑦_𝜉(𝑡)+𝐻_𝜉(𝑡)>0.
(ii) Define the set Θ={(𝑡,𝜉)∈[0,𝑇]×𝑅∣𝑦_𝜉(𝑡,𝜉)=0}. It follows from Fubini’s theorem that meas(Θ)=_𝑅measΘ_𝜉𝑑𝜉=_[0,𝑇]measΘ_𝑡𝑑𝑡, (45) where Θ_𝜉={𝑡∈[0,𝑇]∣𝑦_𝜉(𝑡,𝜉)=0} and Θ_𝑡={𝜉∈𝑅∣𝑦_𝜉(𝑡,𝜉)=0}.  From the above proof, we know that, for all 𝜉∈Ω, Θ_𝜉 consists of isolated points that are countable. This means that meas(Θ_𝜉)=0. It follows from (45) and since meas(Ω^𝑐)=0 that measΘ_𝑡=0foralmostevery𝑡∈[0,𝑇]. (46) This implies 𝑦_𝜉(𝑡,𝜉)>0 for almost all 𝑡 and therefore 𝑦(𝑡,𝜉) is strictly increasing and invertible w. r. t. 𝜉.

Theorem 5. For any initial data 𝑋=(𝑦,𝑈,𝑉,𝜎,ℎ)∈Γ, system (31) admits a unique global solution 𝑋(𝑡)=(𝑦,𝑈,𝑉,𝜎,ℎ)(𝑡)∈𝐶¹(𝑅⁺,𝐸). Furthermore, for all 𝑡≥0, one has 𝑋(𝑡)∈Γ, which constructs a continuous semigroup.

Proof. To ensure that the local solution 𝑋=(𝜍,𝑈,𝑉,𝜎,ℎ)∈𝐶([0,𝑇],𝐸) of system (31) can be extended to a global solution, it suffices to show that sup_{[)𝑡∈0,𝑇}‖𝜍(𝑡,⋅),𝑈(𝑡,⋅),𝑉(𝑡,⋅),𝜎(𝑡,⋅),ℎ(𝑡)‖_𝐸<∞. (47) We now consider a fixed 𝑡∈[0,𝑇) and drop it for simplification. We have ‖𝐻‖_𝐿^∞≤𝐻(1)=𝐻(1)−𝐻(0)=ℎ, (48) where we have used the fact that 𝐻(0)=0 and 𝐻 is an increasing function as 𝐻_𝜉≥0. It is clear that ℎ(𝑡)=ℎ for all 𝑡. Since 𝜎=𝐻−ℎ𝐼, we have ‖𝜎‖_𝐿^∞≤2ℎ and therefore sup_{𝑡∈[0,𝑇)}‖𝜎(𝑡,⋅)‖_𝐿^∞_(𝑅)≤2ℎ. For 𝜉,𝜉^′∈[0,1], we have |𝑦(𝜉)−𝑦(𝜉^′)|≤𝑦(1)−𝑦(0)=1 as 𝑦 is increasing. Since 𝐻_𝜉=(𝑈²+𝑈²_𝑥+𝑉²+𝑉²_𝑥)𝑦_𝜉, we have 𝑈²+2𝑈_𝑥𝑉_𝑥−𝑉²_𝑥𝑦_𝜉≤𝑈²+𝑈²_𝑥+𝑉²_𝑥+𝑉²_𝑥𝑦_𝜉≤2𝐻_𝜉, (49) which implies that 𝐵_𝑥≤12(𝑒−1)¹₀sinh𝑦(𝜉)−𝑦𝜉^′3𝐻_𝜉𝜉^′𝑑𝜉^′+14¹₀𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}3𝐻_𝜉𝜉^′𝑑𝜉^′≤𝐶(𝐻(1)−𝐻(0))=𝐶ℎ=𝐶ℎ, (50) for some constant 𝐶 and then sup_{𝑡∈[0,𝑇)}‖𝐵_𝑥(𝑡,⋅)‖_{𝐿∞(𝑅)}<∞. Similarly, one can prove that the bounds hold for 𝐴, 𝐴_𝑥, 𝐵, 𝐶, 𝐶_𝑥, 𝐷, and 𝐷_𝑥. Since 𝑈_𝑡=−𝐴−𝐵_𝑥 and 𝑉_𝑡=−𝐶−𝐷_𝑥, we get sup_{[)𝑡∈0,𝑇}‖𝑈(𝑡,⋅)‖_𝐿^∞_(𝑅)<∞,sup_{[)𝑡∈0,𝑇}‖𝑉(𝑡,⋅)‖_𝐿^∞_(𝑅)<∞, (51) and therefore sup_{𝑡∈[0,𝑇)}‖𝜍(𝑡,⋅)‖_𝐿^∞_(𝑅)<∞ as 𝜍_𝑡=𝑈+𝑉. Let 𝑍(𝑡)=‖𝑈(𝑡,⋅)‖_𝐿²+𝑈_𝜉(𝑡,⋅)_𝐿2+‖𝑉(𝑡,⋅)‖_𝐿²+𝑉_𝜉(𝑡,⋅)_𝐿2+𝜍_𝜉(𝑡,⋅)_𝐿2+𝐻_𝜉(𝑡,⋅)_𝐿2. (52) After taking the 𝐿²-norms on both sides of (29) and (30), we have 𝑍(𝑡)≤𝑍(0)+𝐶^𝑡₀𝑍(𝜏)𝑑𝜏. (53) Gronwall’s lemma implies that sup_{𝑡∈[0,𝑇)}𝑍(𝑡)<∞, and therefore (47) holds as 𝜍_𝜉=𝑦_𝜉−𝐼 and 𝜎=𝐻−ℎ𝐼. It then follows that the map 𝑆_𝑡∶Γ→Γ×𝑅⁺ given by 𝑆_𝑡(𝑋)=𝑋(𝑡) generates a continuous semigroup from standard ODE theory.

4. Global Conservative Solutions of the Original System

In this section, we show that the global solution of the equivalent system (31) yields a global conservative solution of the original system (13), which constructs a continuous semigroup.

It suffices to establish the correspondence between the Lagrangian equivalent system and the original system.

We first introduce the subsets 𝐹 and 𝐹_𝛼 of Γ defined by 1𝐹=𝑋=𝑦,𝑈,𝑉,𝐻∈Γ∣,𝐹1+ℎ𝑦+𝐻∈𝐺_𝛼=1𝑋=𝑦,𝑈,𝑉,𝐻∈Γ∣1+ℎ𝑦+𝐻∈𝐺_𝛼, (54) where ℎ=𝐻(𝜉+1)−𝐻(𝜉)=𝐻(1)−𝐻(0), and the set of relabeling functions 𝐺 is given by 𝐺=𝑓∣𝑓∈𝑊^1,∞_loc(𝑅),𝑓(𝜉+1)=𝑓(𝜉)+1∀𝜉∈𝑅,𝑓−𝐼,𝑓⁻¹−𝐼∈𝑊^1,∞_per. (55) For any 𝛼>1, we denote the subsets of 𝐺 by 𝐺_𝛼 as 𝐺_𝛼=𝑓∈𝐺∣𝑓−𝐼_𝑊1,∞per+𝑓⁻¹−𝐼_𝑊1,∞per≤𝛼, (56) with a useful characterization: if 𝑓∈𝐺_𝛼 (𝛼≥0), then 1/(1+𝛼)≤𝑓_𝜉≤1+𝛼 almost everywhere. Conversely, if 𝑓∈𝐺 and there exists 𝑐≥1 such that 1/𝑐≤𝑓_𝜉≤𝑐 almost everywhere, then 𝑓∈𝐺_𝛼 for some 𝛼 depending only on 𝑐. With this useful characterization of 𝐺_𝛼, it is not hard to prove that the space 𝐹 is preserved by the governing equation (29).

We denote by 𝐺 the product group 𝐺×𝑅, where the group operation is given by (𝑓₁,𝛾₁)⋅(𝑓₂,𝛾₂)=(𝑓₂∘𝑓₁,𝛾₁+𝛾₂) for any (𝑓₁,𝛾₁),(𝑓₂,𝛾₂)∈𝐺. We define the map Φ∶𝐺×𝐹→𝐹 as Φ𝑓,𝛾,𝑦,𝑈,𝑉,𝐻=𝑦∘𝑓,𝑈∘𝑓,𝑉∘𝑓,𝐻∘𝑓+𝛾. (57) The map Φ defines a group action of 𝐺 on 𝐹. We then consider the quotient space 𝐹/𝐺 of 𝐹 with respect to the group action. We denote the projection Π∶𝐹→𝐹/𝐺 by Π(𝑋)=[𝑋]. Let us introduce the subset 𝐹₀ of 𝐹₀ defined as 𝐹₀=𝑦,𝑈,𝑉,𝐻∈𝐹₀∣¹₀𝑦(𝜉)𝑑𝜉=0. (58) It turns out that there exists a bijection between 𝐹₀ and 𝐹/𝐺. To prove this, we introduce the composition map 𝐾∶𝐹→𝐹₀ such as 𝐾(𝑋)=𝐾₂∘(𝐾₁(𝑋)), where 𝐾₁∶𝐹→𝐹₀ is given by 𝐾₁(𝑋)=𝑋∘𝑓⁻¹ (59) with 𝑋=(𝑦,𝑈,𝑉,𝐻) and 𝑓=(1/(1+ℎ))(𝑦+𝐻)∈𝐹, and 𝐾₂∶𝐹₀→𝐹₀ is given by 𝐾₂=𝑦()()()()+()𝑎𝑦,𝑈,𝑉,𝐻𝜉−𝑎,𝑈𝜉−𝑎,𝑉𝜉−𝑎,𝐻𝜉−𝑎1+ℎ (60) with 𝑎=∫¹₀𝑦(𝜉)𝑑𝜉. It is not hard to check that 𝐾₁(𝑋)∈𝐹₀,   that is, (1/(1+ℎ))(𝑦+𝐻)=𝐼, and 𝐾₂(𝑦,𝑈,𝑉,𝐻)∈𝐹₀. Thus 𝐾(𝑋) belongs to the same equivalence class as 𝑋, and there corresponds to 𝐾 a map 𝐾∶𝐹/𝐺→𝐹₀ given by 𝐾([𝑋])=𝐾(𝑋), where [𝑋]∈𝐹/𝐺 denotes the equivalence class of 𝑋∈𝐹. It is not hard to check that 𝐾_|𝐹0=𝐼_|𝐹0. Thus, 𝐾∘Π_|𝐹0=𝐼_|𝐹0 and therefore 𝐾 is a bijection from 𝐹/𝐺 to 𝐹₀. Note that any topology defined on 𝐹₀ is naturally transported into 𝐹/𝐺 by the bijection 𝐾. That is, if we equip 𝐹₀ with the metric induced by the 𝐸-norm, that is 𝑑_𝐹0(𝑋,𝑋^′)=‖𝑋−𝑋^′‖_𝐸 for all 𝑋,𝑋^′∈𝐹₀,   which is complete, then the topology on 𝐹/𝐺 is defined by a complete metric given by 𝑑_𝐺𝐹/([𝑋],[𝑋^′])=‖𝐾(𝑋)−𝐾(𝑋^′)‖_𝐸 for any [𝑋],[𝑋^′]∈𝐹/𝐺.

For any initial data 𝑋∈𝐹, we denote by 𝑆∶𝐹×𝑅⁺→𝐹 the continuous semigroup with the solution 𝑋(𝑡) of (29). Equation (13) is invariant with respect to relabeling. That is, for any 𝑋∈𝐹 and (𝑓,𝛾)∈𝐺, we have that 𝑆_𝑡(Φ{(𝑓,𝛾),𝑋})=Φ{(𝑓,𝛾),𝑆_𝑡(𝑋)}. Therefore, the map 𝑆_𝑡∶𝐹/𝐺→𝐹/𝐺 given by 𝑆_𝑡([𝑋])=[𝑆_𝑡𝑋] is well defined, which generates a continuous semigroup.

To obtain a semigroup of solution for (13), we have to consider the space Ξ, which characterizes the solutions in the original system: Ξ=𝑧,𝜇∣𝑧∈𝐻¹_per×𝐻¹_per𝜇and_ac=𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥𝑑𝑥, (61) where 𝑧=(𝑢,𝑣) and 𝜇 is a positive periodic Radon measure with 𝜇_ac being its absolute continuous part.

In the following, we establish a bijection between 𝐹/𝐺 and Ξ to transport the continuous semigroup obtained in the Lagrangian equivalent system (functions in 𝐹/𝐺) into the original system (functions in Ξ).

Let us first introduce the mapping 𝑀, which transforms the Lagrangian equivalent system into the original system. In the other direction, we obtain the energy density 𝜇 in the original system, by pushing forward by 𝑦 the energy density 𝐻_𝜉𝑑𝜉 in Lagrangian equivalent system, where the pushforward 𝑓_#𝜈 of a measure 𝜈 by a measurable function 𝑓 is defined as 𝑓_#𝜈(𝐵)=𝜈𝑓⁻¹(𝐵) (62) for all Borel set 𝐵. Let (𝑧,𝜇) be defined as𝑧(𝑥)=𝑍(𝜉)forany𝜉suchthat𝑥=𝑦(𝜉),𝜇=𝑦_#𝐻_𝜉𝑑𝜉, (63a)(63b)

with 𝑧(𝑥)=(𝑢,𝑣)(𝑥) and 𝑍(𝜉)=(𝑈,𝑉)(𝜉). We have that (𝑧,𝜇)∈Ξ, which does not depend on the representative 𝑋=(𝑦,𝑈,𝑉,𝐻)∈𝐹 of [𝑋] that we choose. We denote by 𝑀∶𝐹/𝐺→Ξ the mapping to any [𝑋]∈𝐹/𝐺 and (𝑧,𝜇)∈Ξ given by ((63a) and (63b)), which transforms the Lagrangian equivalent system into the original system.

We are led to the mapping 𝐿∶Ξ→𝐹/𝐺, which corresponds to the transformation from the original system into the Lagrangian equivalent system defined as follows.

Definition 6. For any (𝑧,𝜇)∈Ξ, let ,𝑦(𝜉)ℎ=𝜇([0,1))=sup𝑦∣𝐹_𝜇𝑦()𝜉𝑈(𝜉)(𝜉)(𝜉)(𝜉),𝐻(𝜉)=()(𝜉),+𝑦<1+ℎ,=𝑢∘𝑦,𝑉=𝑣∘𝑦1+ℎ𝜉−𝑦 (64)(65)(66)(67) where 𝑧=(𝑢,𝑣). We define 𝐿(𝑧,𝜇)∈𝐹/𝐺 as the equivalence class of (𝑦,𝑈,𝑉,𝐻). Here, the function 𝐹_𝜇 associated with any positive periodic Radon measure 𝜇 is given by 𝐹_𝜇(𝑥)=𝜇([0,𝑥))if𝑥>0,0if𝑥=0,−𝜇([𝑥,0))if𝑥<0, (68) which is lower-semicontinuous, increasing and moreover, 𝐹_𝜇(𝑏)−𝐹_𝜇(𝑎)=𝜇([𝑎,𝑏)) for all 𝑎<𝑏 in 𝑅.

Remark 7. From the definition of 𝑦, 𝑈,𝑉, 𝐻 in ((64), (65), (66), and (67)), we can check that 𝑋=(𝑦,𝑈,𝑉,𝐻)∈𝐸 and 𝑋=(𝑦,𝑈,𝑉,𝐻) satisfies (41a), (41b), and (41c), respectively. Moreover, by definition (67), we have (1/(1+ℎ))(𝑦+𝐻)=𝐼. Thus 𝑋=(𝑦,𝑈,𝑉,𝐻)∈𝐹₀.
We claim that the transformation from the original system into the Lagrangian equivalent system is a bijection.

Theorem 8. The mappings 𝑀 and 𝐿 are invertible; that is, 𝐿∘𝑀=𝐼_𝐺𝐹/,𝑀∘𝐿=𝐼_Ξ. (69)

Proof. Let [𝑋]∈𝐹/𝐺 be given. We denote by 𝑋=(𝑦,𝑈,𝑉,𝑁,𝐻)=𝐾([𝑋]) a representative of [𝑋] and (𝑧,𝜇) given by ((63a) and (63b)) for this particular 𝑋. It follows from the definition of 𝐾 that 𝑋∈𝐹₀. Let 𝑋=(𝑦,𝑈,𝑉,𝐻) be the representative of 𝐿(𝑧,𝜇) in 𝐹₀ given by (64), (65), (66), and (67). It suffices to show that (𝑦,𝑈,𝑉,𝐻)=(𝑦,𝑈,𝑉,𝐻) and therefore 𝐿∘𝑀=𝐼_𝐺𝐹/. Set 𝑔(𝑥)=sup𝜉∈𝑅∣𝑦(𝜉)<𝑥. (70) Using the fact that 𝑦 is increasing and continuous, it is not hard to know that 𝑦𝑔(𝑥)=𝑥 (71) and 𝑦⁻¹([𝑎,𝑏))=[𝑔(𝑎),𝑔(𝑏)) for any 𝑎<𝑏 in 𝑅. It follows from (63b) that 𝜇([𝑎,𝑏))=_{𝑦−1([𝑎,𝑏))}𝐻_𝜉𝑑𝜉=_𝑔(𝑎)^𝑔(𝑏)𝐻_𝜉𝑔(𝑏)𝑔(𝑎).𝑑𝜉=𝐻−𝐻 (72) Since 𝑋∈𝐹₀ and 𝑦+𝐻=(1+ℎ)𝐼, we have 𝐻𝑔(𝑏)=()𝑔(𝑏)𝐻𝑔(𝑎)=()𝑔(𝑎)1+ℎ−𝑏,1+ℎ−𝑎. (73) From (72) and the definition of 𝐹_𝜇, we get 𝐹_𝜇(𝑥)+𝑥=(1+ℎ)𝑔(𝑥)−𝑔(0). (74) It then follows from the definition of 𝑦 that 𝑦(𝜉)=sup𝑥∈𝑅∣𝑔(𝑥)<𝜉+𝑔(0). (75) For any given 𝜉∈𝑅, using the fact that 𝑦 is increasing and (71), we have 𝑦(𝜉)≤𝑦(𝜉+𝑔(0)). Assume 𝑦(𝜉)<𝑦(𝜉+𝑔(0)), and there then exists 𝑥 such that 𝑦(𝜉)<𝑥<𝑦(𝜉+𝑔(0)) and (75) implies that 𝑔(𝑥)≥𝜉+𝑔(0). Therefore, 𝑥≥𝑦(𝜉+𝑔(0)), which contradicts the fact that 𝑥<𝑦(𝜉+𝑔(0)). Hence, we have 𝑦(𝜉)=𝑦𝜉+𝑔(0). (76) Since 𝑋∈𝐹₀, 𝐻(𝜉)=(1+ℎ)𝜉−𝑦(𝜉), and (𝑦+𝐻)(𝜉+𝑔(0))=(1+ℎ)(𝜉+𝑔(0)), we have 𝐻(𝜉)=𝐻𝜉+𝑔(0)−(1+ℎ)𝑔(0). (77) It is not hard to know that 𝑈(𝜉)=𝑈𝜉+𝑔(0),𝑉(𝜉)=𝑉𝜉+𝑔(0). (78) It then follows that 𝐿∘𝑀=𝐼_𝐺𝐹/.

Given (𝑧,𝜇) in Ξ, we denote by (𝑦,𝑈,𝑉,𝐻) the representative of 𝐿(𝑧,𝜇) in 𝐹₀ given by (64), (65), (66), and (67). Then, let (𝑧,𝜇)=𝑀∘𝐿(𝑧,𝜇). Let 𝑔 be defined as before by (70). The same computation that leads to (74) now gives 𝐹_𝜇(𝑥)+𝑥=(1+ℎ)𝑔(𝑥)−𝑔(0). (79)

Given 𝜉∈𝑅, we consider an increasing sequence 𝑥_𝑖 converging to 𝑦(𝜉) which is guaranteed by (65), such that 𝐹_𝜇(𝑥_𝑖)+𝑥_𝑖<(1+ℎ)𝜉. Let 𝑖 tend to infinity and since 𝐹_𝜇 is lower semicontinuous, we have 𝐹_𝜇(𝑦(𝜉))+𝑦(𝜉)≤(1+ℎ)𝜉. Take 𝜉=𝑔(𝑥) and then we get 𝐹_𝜇(𝑥)+𝑥≤(1+ℎ)𝑔(𝑥). (80) By the definition of 𝑔, there exists an increasing sequence 𝜉_𝑖 converging to 𝑔(𝑥) such that 𝑦(𝜉_𝑖)<𝑥. It follows from the definition of 𝑦 in (65) that 𝐹_𝜇(𝑥)+𝑥≥(1+ℎ)𝜉_𝑖. Passing to the limit, we obtain 𝐹_𝜇(𝑥)+𝑥≥(1+ℎ)𝑔(𝑥) which, together with (80), yields 𝐹_𝜇(𝑥)+𝑥=(1+ℎ)𝑔(𝑥). (81) We obtain that 𝜇=𝜇 by comparing (81) and (79). It is clear from the definitions that 𝑧=𝑧. Hence, (𝑧,𝜇)=(𝑧,𝜇) and 𝑀∘𝐿=𝐼_Ξ.

Now we have established a bijection between the two equivalent systems, and thus the topology defined in 𝐹/𝐺 can be transported into Ξ. Let us define the distance 𝑑_Ξ on Ξ as 𝑑_Ξ𝑧,𝜇,𝑧,𝜇=𝑑_𝐺𝐹/𝐿𝑧,𝜇,𝐿𝑧,𝜇, (82) which makes the bijection 𝐿 between Ξ and 𝐹/𝐺 into an isometry. Since 𝐹/𝐺 equipped with 𝑑_𝐺𝐹/ is a complete metric space, it is not hard to check that Ξ equipped with the metric 𝑑_Ξ is also a complete metric space. For each 𝑡∈𝑅, we define the mapping 𝑇_𝑡∶Ξ→Ξ as 𝑇_𝑡=𝑀𝑆_𝑡𝐿. (83)

Our main results are given as follows.

Theorem 9. Let (𝑧,𝜇)∈Ξ be given. If one denotes by 𝑡→(𝑧(𝑡),𝜇(𝑡))=𝑇_𝑡(𝑧,𝜇) the corresponding trajectory, 𝑧=(𝑢,𝑣) is a weak solution of the modified coupled two-component Camassa-Holm system (13), which constructs a continuous semigroup. Moreover, 𝜇 is a weak solution of the following transport equation for the energy density 𝜇_𝑡+(𝑢+𝑣)𝜇_𝑥=𝑢³−2𝐵𝑢+𝑣³−2𝐷𝑣_𝑥. (84) Furthermore, one has 𝜇(𝑡)([0,1))=𝜇(0)([0,1))∀𝑡, (85) and for almost all 𝑡, there holds 𝜇(𝑡)([0,1))=𝜇_𝑎𝑐(𝑡=‖𝑧(𝑡)([0,1)))‖²_{𝐻1𝑝𝑒𝑟}=‖𝑢(𝑡)‖²_{𝐻1𝑝𝑒𝑟}+‖𝑣(𝑡)‖²_{𝐻1𝑝𝑒𝑟}=𝜇(0)([0,1)). (86) Thus, the unique solution described here is a conservative weak solution of the system (13).

Proof. To prove 𝑧=(𝑢,𝑣) is a weak solution of the original system (13), it suffices to show that,   for all 𝜙∈𝐶^∞(𝑅⁺×𝑅) with compact support, _𝑅+×𝑅−𝑢𝜙_𝑡+(𝑢+𝑣)𝑢_𝑥𝜙(𝑡,𝑥)𝑑𝑥𝑑𝑡=−_𝑅+×𝑅𝐴+𝐵_𝑥𝜙(𝑡,𝑥)𝑑𝑥𝑑𝑡,_𝑅+×𝑅−𝑣𝜙_𝑡+(𝑢+𝑣)𝑣_𝑥𝜙(𝑡,𝑥)𝑑𝑥𝑑𝑡=−_𝑅+×𝑅𝐶+𝐷_𝑥𝜙(𝑡,𝑥)𝑑𝑥𝑑𝑡, (87) where 𝐴, 𝐵_𝑥, 𝐶, and 𝐷_𝑥 are given by (13). Let the solution (𝑦,𝑈,𝑉,𝐻)(𝑡) of (29) be a representative of 𝐿(𝑧(𝑡),𝜇(𝑡)). Since 𝑦(𝑡,𝜉) is Lipschitz and invertible w. r. t. 𝜉 for almost all 𝑡, we can use the change of variables 𝑥=𝑦(𝑡,𝜉) and get _𝑅+×𝑅−𝑢𝜙_𝑡+(𝑢+𝑣)𝑢_𝑥𝜙)=(𝑡,𝑥𝑑𝑥𝑑𝑡_𝑅+×𝑅−𝑈𝑦_𝜉(𝑡,𝜉)𝜙_𝑡𝑡,𝑦(𝑡,𝜉)+(𝑈+𝑉)𝑈_𝜉×(𝑡,𝜉)𝜙𝑡,𝑦(𝑡,𝜉)𝑑𝜉𝑑𝑡. (88) Using the identities 𝑦_𝑡=𝑈+𝑉 and 𝑦_𝜉𝑡=𝑈_𝜉+𝑉_𝜉, it then follows from (29) that _𝑅+×𝑅−𝑈𝑦_𝜉𝜙_𝑡𝑡,𝑦+(𝑈+𝑉)𝑈_𝜉𝜙=1𝑡,𝑦𝑑𝜉𝑑𝑡2_𝑅⁺_×𝑅²−𝑈𝑉_𝑥𝑦_𝜉+12sgn𝜉−𝜉^′×𝐻_𝜉+𝑈²+2𝑈_𝑥𝑉_𝑥−𝑉²_𝑥𝑦_𝜉𝜉^′⋅𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}𝜙𝑡,𝑦(𝜉)𝑦_𝜉(𝜉)𝑑𝜉^′𝑑𝜉𝑑𝑡. (89) On the other hand, using the change of variables 𝑥=𝑦(𝑡,𝜉) and 𝑥^′=𝑦(𝑡,𝜉^′) and since 𝑦 is an increasing function, we obtain that −_𝑅+×𝑅𝐴+𝐵_𝑥𝜙)=1(𝑡,𝑥𝑑𝑥𝑑𝑡2_𝑅⁺_×𝑅²−𝑢𝑣_𝑥+sgn𝜉−𝜉^′×𝑢²+12𝑢²_𝑥+𝑢_𝑥𝑣_𝑥+12𝑣²−12𝑣²_𝑥×𝑡,𝑦𝜉^′⋅𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}𝜙𝑡,𝑦(𝜉)𝑦_𝜉𝜉^′𝑦_𝜉(𝜉)𝑑𝜉^′𝑑𝜉𝑑𝑡. (90) It then follows from (41c) that −_𝑅+×𝑅𝐴+𝐵_𝑥𝜙)=1(𝑡,𝑥𝑑𝑥𝑑𝑡2_𝑅⁺_×𝑅²−𝑈𝑉_𝑥𝑦_𝜉+12sgn𝜉−𝜉^′×𝐻_𝜉+𝑈²+2𝑈_𝑥𝑉_𝑥−𝑉²_𝑥𝑦_𝜉𝜉^′⋅𝑒^{−|𝑦(𝜉)−𝑦(𝜉′)|}𝜙𝑡,𝑦(𝜉)𝑦_𝜉(𝜉)𝑑𝜉^′𝑑𝜉𝑑𝑡. (91) By comparing (89) and (91), we can see that _𝑅+×𝑅−𝑈𝑦_𝜉𝜙_𝑡𝑡,𝑦+(𝑈+𝑉)𝑈_𝜉𝜙𝑡,𝑦𝑑𝜉𝑑𝑡=−_𝑅+×𝑅𝐴+𝐵_𝑥𝜙(𝑡,𝑥)𝑑𝑥𝑑𝑡. (92) Hence, the first identity in (87) holds. The second identity in (87) follows in the same way. One can easily check that 𝜇(𝑡) is solution of (84). It follows from the definitiong in (70) that 𝑦⁻¹([0,1))=𝑔(0),𝑔(1),𝑔(𝑥+1)=𝑔(𝑥)+1. (93) Since 𝐻_𝜉 is periodic, (63b) implies that 𝜇(𝑡)([0,1))=_{[𝑔(0),𝑔(0)+1)}𝐻_𝜉𝑑𝜉=_[0,1)𝐻_𝜉𝑑𝜉=𝐻(𝑡,1)−𝐻(𝑡,0), (94) which is constant in time from (29). Thus, we have proved (85). Since 𝑦_𝜉(𝑡,𝜉)>0 almost everywhere for almost every 𝜉∈𝑅, then it follows from (41c) that 𝜇(𝑡)(𝐵)=_{𝑦−1(𝐵)}𝐻_𝜉=𝑑𝜉_{𝑦−1(𝐵)}𝑈²+𝑈²_𝜉𝑦²_𝜉+𝑉²+𝑉²_𝜉𝑦²_𝜉𝑦_𝜉𝑑𝜉 (95) for any Borel set 𝐵. Using the fact that 𝑦 is one-to-one and 𝑢_𝑥∘𝑦𝑦_𝜉=𝑈_𝜉, 𝑣_𝑥∘𝑦𝑦_𝜉=𝑉_𝜉 almost everywhere, (95) implies that 𝜇(𝑡)(𝐵)=_𝐵𝑢²+𝑢²_𝑥+𝑣²+𝑣²_𝑥(𝑡,𝑥)𝑑𝑥. (96) Hence, (86) holds. This completes the proof.

5. Conclusion

We consider in this paper the periodic conservative property of the solutions for the modified coupled two-component Camassa-Holm system with peakons. By introducing a skillfully defined characteristic, together with a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which periodic conservative solution is obtained. It is the transformation together with the associated properties that overcomes the difficulties inherent in the multicomponent system and allows for the continuity of the solution beyond collision time, leading to the desired results.

Acknowledgments

The authors would like to thank the referees for constructive suggestions and comments. The paper is supported by the Major State Basic Research Development Program 973 (no. 2012CB215202) and the National Natural Science Foundation of China (no.  61134001).