Global conservative and multipeakon conservative solutions for the two-component Camassa-Holm system

  • Yujuan Wang1 and

    Affiliated with

    • Yongduan Song1Email author

      Affiliated with

      Boundary Value Problems20132013:165

      DOI: 10.1186/1687-2770-2013-165

      Received: 6 March 2013

      Accepted: 26 June 2013

      Published: 10 July 2013

      Abstract

      The continuation of solutions for the two-component Camassa-Holm system after wave breaking is studied in this paper. The global conservative solution is derived first, from which a semigroup and a multipeakon conservative solution are established. In developing the solution, a system transformation based on a skillfully defined characteristic and a set of newly introduced variables is used. It is the transformation, together with the associated properties, that allows for the establishment of the results for continuity of the solution beyond collision time.

      Keywords

      two-component Camassa-Holm system Lagrangian system global conservative solutions conservative multipeakon solutions

      1 Introduction

      Because of its capabilities of describing the dynamic behavior of water wave, the following Camassa-Holm (CH) equation
      u t u x x t + 3 u u x = 2 u x u x x + u u x x x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ1_HTML.gif
      (1.1)

      modeling the unidirectional propagation of shallow water waves in irrotational flow over a flat bottom, with u ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq1_HTML.gif representing the fluid velocity at time t in the horizontal direction, has attracted considerable attention [110]. The CH equation is a quadratic order water wave equation in an asymptotic expansion for unidirectional shallow water waves described by the incompressible Euler equations, which was found earlier by Fuchssteiner and Fokas [1] as a bi-Hamiltonian generalization of the KdV equation. It is completely integrable [2, 3] and possesses an infinite number of conservation laws. A remarkable property of the CH equation is the existence of the non-smooth solitary wave solutions called peakons [2, 11]. The peakon u ( t , x ) = c e | x c t | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq2_HTML.gif, c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq3_HTML.gif, is smooth except at its crest and the tallest among all waves of fixed energy. Another remarkable fact for the CH equation is that it can model wave breaking [2, 12], which means that the solution remains bounded while its slope becomes unbounded in finite time [12, 13], setting it apart from the classical soliton equations such as KdV. After wave breaking, the solutions of the CH equation can be continued uniquely as either global conservative [46] or global dissipative solutions [7].

      Considered herein is the two-component Camassa-Holm (CH2) shallow water system [1416]
      { m t + u m x + 2 u x m + σ ρ ρ x = 0 , t > 0 , x R , ρ t + ( u ρ ) x = 0 , t > 0 , x R , u ( 0 , x ) = u 0 ( x ) , x R , ρ ( 0 , x ) = ρ 0 ( x ) , x R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ2_HTML.gif
      (1.2)
      with σ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq4_HTML.gif and m = σ 1 u u x x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq5_HTML.gif, σ 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq6_HTML.gif (or in the ‘short wave’ limit, σ 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq7_HTML.gif), which is an extension of the CH equation by combining its integrability property with compressibility or free-surface elevation dynamics in its shallow water interpretation [11, 17]. This system appeared originally in [14] as could be identified with the first negative flow of AKNS hierarchy, and then it was derived by Constantin and Ivanov [16] in the context of shallow water theory, with u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq8_HTML.gif representing the horizontal velocity of the fluid and ρ ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq9_HTML.gif in connection with the free-surface elevation from equilibrium with the boundary assumptions u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq10_HTML.gif and ρ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq11_HTML.gif as | x | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq12_HTML.gif. It is formally integrable [1416] in the sense that it can be written as a compatibility condition of two linear systems (Lax pair) with a spectral parameter ζ:
      Ψ x x = ( σ ζ 2 ρ 2 + ζ m + σ 1 4 ) Ψ , Ψ t = ( 1 2 ζ u ) Ψ x + 1 2 u x Ψ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equa_HTML.gif
      It also has a bi-Hamiltonian structure corresponding to the Hamiltonian
      H 1 = 1 2 ( u m + ( ρ 1 ) 2 ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equb_HTML.gif
      and the Hamiltonian
      H 2 = 1 2 ( u ( ρ 1 ) 2 + 2 u ( ρ 1 ) + u 3 + u u x 2 ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equc_HTML.gif

      The Cauchy problem for the two-component Camassa-Holm system has been studied extensively [1824]. It was shown that the CH2 system is locally well posed with initial data ( u 0 , ρ 0 ) H s × H s 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq13_HTML.gif, s > 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq14_HTML.gif [18]. The system also has global strong solutions which blow-up in finite time [19, 21, 22] and a global weak solution [23]. However, the problem about continuation of the solutions beyond wave breaking, although interesting and important, has not been explicitly addressed yet. In our recent work [20], we studied the continuation beyond wave breaking by applying an approach that reformulated system (1.2) as a semilinear system of O.D.E. taking values in a Banach space. Such treatment makes it possible to investigate the continuity of the solution beyond collision time, leading to a global conservative solution where the energy is conserved for almost all times.

      It should be stressed that both global conservation and multipeakon conservation are two important aspects worthy of investigation. To our best knowledge, however, little effort has been made in studying the multipeakon conservation associated with the CH2 system in the literature. As a compliment and extension to the previous work [20], we develop a novel approach in this work to construct the multipeakon conservative solution for the CH2 system. Different from the work [20], we reformulate the problem by utilizing a skillfully defined characteristic and a new set of variables, of which the associated energy serves as an additional variable to be introduced such that a well-posed initial-value problem can be obtained, making it convenient to study the dynamic behavior of wave breaking. Because of the introduction of the new variables, we are able to establish the multipeakon conservative solution from the global conservative solution for the CH2 system.

      Some related earlier works [4, 5] studied the global existence of solutions to the CH equation. However, the system considered in this work is a heavily coupled one, in which the mutual effect between the two components makes the analysis quite complicated and involved as compared with the system with a single component as studied in [4, 5]. The key and novel effort made in this work to circumvent the difficulty is the utilization of the skillfully defined characteristic and the new set of variables, as well as careful estimates for each iterative approximate component of the solutions, which allows us to establish the global conservative solutions of system (1.2). It is shown that the multipeakon structure is preserved by the semigroup of a global conservative solution and the multipeakon solution is obtained by carefully computing the convolution equations P i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq15_HTML.gif and P x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq16_HTML.gif ( i = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq17_HTML.gif), where, in contrast to the existing works, the inherent mutual effect between the two components is well reflected.

      The remainder of this paper is organized as follows. Section 2 presents the transformation from the original system to a Lagrangian semilinear system. The global solutions of the equivalent semilinear system are obtained in Section 3, which are transformed into the global conservative solutions of the original system in Section 4. Finally, we establish the multipeakon conservative solutions for the original system in Section 5.

      2 The original system and the equivalent Lagrangian system

      We first present the original system. For simplicity, we consider here the associated evolution for positive times (of course, one would get similar results for negative times just by changing the initial condition u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq18_HTML.gif into u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq19_HTML.gif). Let us introduce an operator Λ = ( 1 x 2 ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq20_HTML.gif, which can be expressed by its associated Green’s function G = 1 2 e | x | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq21_HTML.gif such as Λ f ( x ) = G f ( x ) = 1 2 R e | x x | f ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq22_HTML.gif for all f L 2 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq23_HTML.gif. Thus, we can rewrite Eq. (1.2) as a form of a quasi-linear evolution equation:
      { u t + u u x + x G ( u 2 + 1 2 u x 2 + 1 2 v 2 + v ) = 0 , t > 0 , x R , v t + ( u v ) x + u x = 0 , t > 0 , x R , u ( 0 , x ) = u 0 ( x ) , x R , v ( 0 , x ) = v 0 ( x ) , x R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equd_HTML.gif
      where v = ρ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq24_HTML.gif. If we define P as
      P ( t , x ) = G ( u 2 + 1 2 u x 2 + 1 2 v 2 + v ) = 1 2 R e | x x | ( u 2 + 1 2 u x 2 + 1 2 v 2 + v ) ( t , x ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Eque_HTML.gif
      then Eq. (1.2) can be rewritten as
      { u t + u u x + P x = 0 , t > 0 , x R , v t + u v x + v u x + u x = 0 , t > 0 , x R , u ( 0 , x ) = u 0 ( x ) , x R , v ( 0 , x ) = v 0 ( x ) , x R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ3_HTML.gif
      (2.1)
      Moreover, for regular solutions, we have that the total energy
      E ( t ) = R u 2 + u x 2 + v 2 d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ4_HTML.gif
      (2.2)
      is constant in time. Thus, Eq. (2.1) possesses the H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq25_HTML.gif-norm conservation law defined as
      z H 1 = u H 1 + v L 2 = ( R [ u 2 + u x 2 + v 2 ] d x ) 1 / 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equf_HTML.gif

      where z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq26_HTML.gif. Since z = ( u , v ) H 1 × [ L 2 L ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq27_HTML.gif, Young’s inequality ensures P H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq28_HTML.gif.

      We reformulate system (2.1) into a Lagrangian equivalent semilinear system as follows.

      Let z ( t , x ) = ( u , v ) ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq29_HTML.gif denote the solution of system (2.1). For given initial data y ( 0 , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq30_HTML.gif, we define the corresponding characteristic y ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq31_HTML.gif as the solution of
      y t ( t , ξ ) = u ( t , y ( t , ξ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ5_HTML.gif
      (2.3)
      and define the Lagrangian cumulative energy distribution H as
      H ( t , ξ ) = y ( t , ξ ) ( u 2 + u x 2 + v 2 ) ( t , x ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ6_HTML.gif
      (2.4)
      It is not hard to check that
      ( u 2 + u x 2 + v 2 ) t + ( u ( u 2 + u x 2 + v 2 ) ) x = ( u 3 2 u P ) x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ7_HTML.gif
      (2.5)
      Then it follows from (2.3) and (2.5) that
      d H d t = [ ( u 3 2 u P ) ( t , y ( t , ξ ) ) ] ξ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ8_HTML.gif
      (2.6)
      Throughout the following, we use the notation
      U ( t , ξ ) = u ( t , y ( t , ξ ) ) , V ( t , ξ ) = v ( t , y ( t , ξ ) ) , N ( t , ξ ) = u x ( t , y ( t , ξ ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equg_HTML.gif
      After the change of variables x = y ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq32_HTML.gif and x = y ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq33_HTML.gif, we obtain the following expressions for P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif and P, namely
      P ( ξ ) = P ( y ( ξ ) ) = 1 2 R e | y ( ξ ) y ( ξ ) | [ ( U 2 + 1 2 U x 2 + 1 2 V 2 + V ) y ξ ] ( ξ ) d ξ , P x ( ξ ) = P x ( y ( ξ ) ) = 1 2 R sgn ( ξ ξ ) e | y ( ξ ) y ( ξ ) | , [ ( U 2 + 1 2 U x 2 + 1 2 V 2 + V ) y ξ ] ( ξ ) d ξ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ9_HTML.gif
      (2.7)
      where we have dropped the variable t for simplicity and taken that y is an increasing function for any fixed time t for granted (the validity will be proved later). Using H ξ = ( u 2 + u x 2 + v 2 ) y y ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq35_HTML.gif, we can rewrite P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif and P in (2.7) as
      P x ( ξ ) = 1 4 R sgn ( ξ ξ ) e | y ( ξ ) y ( ξ ) | [ H ξ + ( U 2 + 2 V ) y ξ ] ( ξ ) d ξ , P ( ξ ) = 1 4 R e | y ( ξ ) y ( ξ ) | [ H ξ + ( U 2 + 2 V ) y ξ ] ( ξ ) d ξ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ10_HTML.gif
      (2.8)
      From the definition of the characteristic, it follows that
      U t ( t , ξ ) = u t ( t , y ) + u x ( t , y ) y t ( t , ξ ) = P x y ( t , ξ ) , V t ( t , ξ ) = v t ( t , y ) + v x ( t , y ) y t ( t , ξ ) = [ ( v + 1 ) u x ] y ( t , ξ ) , N t ( t , ξ ) = u x t ( t , y ) + u x x ( t , y ) y t ( t , ξ ) = ( u 2 1 2 u x 2 + 1 2 v 2 + v P ) y ( t , ξ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ11_HTML.gif
      (2.9)
      Let us introduce another variable ς ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq36_HTML.gif such that ς ( t , ξ ) = y ( t , ξ ) ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq37_HTML.gif (it will turn out that ς L ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq38_HTML.gif). With these new variables, we now derive an equivalent system of equations (2.1),
      { ς t = U , U t = P x , V t = ( V + 1 ) N , N t = U 2 1 2 N 2 + 1 2 V 2 + V P , H t = U 3 2 U P , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ12_HTML.gif
      (2.10)
      where P and P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif are given by (2.8). Differentiating (2.10) w.r.t. ξ yields
      { ς ξ t = U ξ , U ξ t = 1 2 H ξ + ( 1 2 U 2 + V P ) y ξ , V ξ t = V ξ N ( V + 1 ) N ξ , N ξ t = 2 U U ξ + ( V + 1 ) V ξ N N ξ P x y ξ , H ξ t = ( 3 U 2 2 P ) U ξ 2 U P x y ξ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ13_HTML.gif
      (2.11)

      which is semilinear w.r.t. the variables y ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq39_HTML.gif, U ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq40_HTML.gif, V ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq41_HTML.gif, N ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq42_HTML.gif and H ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq43_HTML.gif.

      System (2.10) can be regarded as an O.D.E. in the Banach space E given by
      E = W × H 1 × [ L 2 L ] × [ L 2 L ] × W , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equh_HTML.gif
      endowed with the norm
      X E = ς W + U H 1 + V L 2 + V L + N L 2 + N L + H W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equi_HTML.gif
      for any X = ( ς , U , V , N , H ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq44_HTML.gif. Here W is a Banach space defined as
      W = { f C ( R ) L ( R ) | f ξ L 2 ( R ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equj_HTML.gif

      with the norm f W = f L ( R ) + f ξ L 2 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq45_HTML.gif. Note that H 1 ( R ) W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq46_HTML.gif.

      3 Global solutions of the equivalent system

      In this section, we prove that the equivalent system admits a unique global solution. We first obtain the Lipschitz bounds we need on P and P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif.

      Lemma 3.1 (See [5])

      Let 1 : E W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq47_HTML.gif and 2 : E H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq48_HTML.gif, or 2 : E W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq49_HTML.gif be two locally Lipschitz maps. Then the product X 1 ( X ) 2 ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq50_HTML.gif is also a locally Lipschitz map from E to H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq25_HTML.gif, or from E to W.

      Lemma 3.2 For any given X = ( ς , U , V , N , H ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq51_HTML.gif, P and P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif defined by (2.8) are locally Lipschitz continuous from E to H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq52_HTML.gif. Moreover, we have
      P x ξ = 1 2 H ξ + ( P 1 2 U 2 V ) ( 1 + ς ξ ) , P ξ = P x ( 1 + ς ξ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ14_HTML.gif
      (3.1)
      Proof We write
      P x ( ξ ) = P x 1 ( X ) ( ξ ) + P x 2 ( X ) ( ξ ) = e ς ( ξ ) 4 R χ { ξ < ξ } e | ξ ξ | e ς ( ξ ) [ H ξ + ( U 2 + 2 V ) ( 1 + ς ξ ) ] ( ξ ) d ξ + e ς ( ξ ) 4 R χ { ξ > ξ } e | ξ ξ | e ς ( ξ ) [ H ξ + ( U 2 + 2 V ) ( 1 + ς ξ ) ] ( ξ ) d ξ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ15_HTML.gif
      (3.2)
      where χ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq53_HTML.gif denotes the indicator function of a given set Ω, and P x 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq54_HTML.gif, P x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq55_HTML.gif are the operators which correspond to the two terms of the last identity in (3.2). We rewrite P x 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq54_HTML.gif as
      P x 1 ( X ) ( ξ ) = e ς ( ξ ) 2 Λ R ( X ) ( ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ16_HTML.gif
      (3.3)
      where R is the operator from E to L 2 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq56_HTML.gif given by
      R ( X ) ( ξ ) = χ { ξ < ξ } e ς [ H ξ + ( U 2 + 2 V ) ( 1 + ς ξ ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equk_HTML.gif

      Since the operator Λ (defined as in Section 2) is linear and continuous from H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq57_HTML.gif to H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq52_HTML.gif, and L 2 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq56_HTML.gif is continuously embedded in H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq58_HTML.gif, we have Λ R ( X ) H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq59_HTML.gif. It is not hard to know that R is locally Lipschitz from E into L 2 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq60_HTML.gif and therefore from E into H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq57_HTML.gif. Thus, Λ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq61_HTML.gif is locally Lipschitz from E to H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq52_HTML.gif. Since the mapping X e ς http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq62_HTML.gif is locally Lipschitz from E to W, it then follows from Lemma 3.1 that P x 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq54_HTML.gif is locally Lipschitz from E to H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq52_HTML.gif. Similarly, P x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq55_HTML.gif is also locally Lipschitz and therefore P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif is locally Lipschitz from E to H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq52_HTML.gif. We can obtain that P defined by (2.8) is locally Lipschitz continuous from E to H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq52_HTML.gif in the same way. By using the chain rule, the formulas in (3.1) are obtained by direct computation, see [[25], p.129]. □

      Theorem 3.1 Let any X ¯ = ( ς ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq63_HTML.gif be given. System (2.10) admits a unique local solution defined on some time interval [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq64_HTML.gif, where T depends only on X ¯ E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq65_HTML.gif.

      Proof To establish the local existence of solutions, one proceeds as in Lemma 3.2, then obtains that F ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq66_HTML.gif, which is defined by
      F ( X ) = ( U , P x , ( V + 1 ) N , U 2 1 2 N 2 + 1 2 V 2 + V P , U 3 2 U P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equl_HTML.gif
      with X = ( ς , U , V , N , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq67_HTML.gif, is Lipschitz continuous on any bounded set of E. We rewrite the solutions of system (2.10) as
      X ( t ) = X ¯ + 0 t F ( X ( τ ) ) d τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ17_HTML.gif
      (3.4)

      Then the theorem follows from the standard contraction argument on Banach spaces. □

      Theorem 3.1 gives us the existence of local solutions to (2.10) for initial data in E. It remains to prove that the local solutions can be extended to global solutions. Note that the global solutions of (2.10) may not exist for all initial data in E. However, they exist when the initial data X ¯ = ( ς ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq68_HTML.gif belongs to the set Γ, which is defined as follows.

      Definition 3.1 The set Γ is composed of all ( ς , U , V , N , H ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq69_HTML.gif such that
      i i ( i ) ( ς , U , V , N , H ) [ W 1 , ( R ) ] 5 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ18_HTML.gif
      (3.5a)
      i ( ii ) y ξ 0 , H ξ 0 , y ξ + H ξ > 0 almost everywhere, and  lim ξ H ( ξ ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ19_HTML.gif
      (3.5b)
      ( iii ) y ξ H ξ = y ξ 2 U 2 + U ξ 2 + y ξ 2 V 2 almost everywhere , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ20_HTML.gif
      (3.5c)

      with ς ( ξ ) = y ( ξ ) ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq70_HTML.gif, where W 1 , ( R ) = { f C ( R ) L ( R ) | f ξ L ( R ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq71_HTML.gif.

      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 3.3 Given initial data X ¯ = ( ς ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq72_HTML.gif, for some T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq73_HTML.gif, we consider the local solution X ( t ) = ( ς , U , V , N , H ) ( t ) C ( [ 0 , T ] , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq74_HTML.gif of system (2.10) given by Theorem  3.1. We have
      1. (i)

        X ( t ) Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq75_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif,

         
      2. (ii)

        y ξ ( t , ξ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq77_HTML.gif for a.e. t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif and a.e. ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq78_HTML.gif,

         
      3. (iii)

        lim ξ ± H ( t , ξ ) = H ( 0 , ± ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq79_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif.

         
      Proof (i) For given initial data X ¯ = ( ς ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) E [ W 1 , ( R ) ] 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq80_HTML.gif, to ensure that the solution X = ( ς , U , V , N , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq81_HTML.gif of (2.10) also belongs to E [ W 1 , ( R ) ] 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq82_HTML.gif, we have to specify the initial conditions for (2.11). Let Ω be the following set:
      Ω = { ξ R | | ς ¯ ξ ( ξ ) | ς ¯ ξ L , | U ¯ ξ ( ξ ) | U ¯ ξ L , | V ¯ ξ ( ξ ) | V ¯ ξ L , | N ¯ ξ ( ξ ) | N ¯ ξ L , | H ¯ ξ ( ξ ) | H ¯ ξ L } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equm_HTML.gif
      Note that meas ( Ω c ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq83_HTML.gif. For ξ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq84_HTML.gif, we take ( ς ξ , U ξ , V ξ , N ξ , H ξ ) ( 0 , ξ ) = ( ς ¯ ξ , U ¯ ξ , V ¯ ξ , N ¯ ξ , H ¯ ξ ) ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq85_HTML.gif. For ξ Ω c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq86_HTML.gif, we define ( ς ξ , U ξ , V ξ , N ξ , H ξ ) ( 0 , ξ ) = ( 0 , 0 , 0 , 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq87_HTML.gif. We consider U, P and P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif as given functions in C ( [ 0 , T ] , H 1 ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq88_HTML.gif, which is guaranteed by Lemma 3.2 and V, N in C ( [ 0 , T ] , [ L 2 ( R ) L ( R ) ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq89_HTML.gif. Thus system (2.11) is affine (it consists of a sum of a linear transformation and a constant) and, therefore, by using a contraction argument, it admits a unique local solution defined on some time interval [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq64_HTML.gif. Thus, for the given initial condition X ¯ E [ W 1 , ( R ) ] 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq90_HTML.gif, the solution of (2.10) given by Theorem 3.1 also belongs to E [ W 1 , ( R ) ] 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq91_HTML.gif, which implies that X ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq92_HTML.gif satisfies (3.5a) for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq93_HTML.gif. We claim that (3.5c) holds for any ξ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq94_HTML.gif and therefore almost everywhere. Consider a fixed ξ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq94_HTML.gif and drop it in the notation. On the one hand, it follows from (2.11) that
      ( y ξ H ξ ) t = y ξ t H ξ + y ξ H ξ t = U ξ H ξ + y ξ [ ( 3 U 2 2 P 2 A U ) U ξ 2 U P x y ξ ] = U ξ H ξ + 3 U 2 U ξ y ξ 2 P U ξ y ξ 2 A U U ξ y ξ 2 U P x y ξ 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equn_HTML.gif
      and on the other hand,
      ( y ξ 2 U 2 + U ξ 2 + y ξ 2 V 2 ) t = 2 y ξ y ξ t U 2 + 2 y ξ 2 U U t + 2 y ξ y ξ t V 2 + 2 y ξ 2 V V t + 2 U ξ U ξ t = 3 U 2 U ξ y ξ 2 U P x y ξ 2 + U ξ H ξ 2 A U U ξ y ξ 2 P U ξ y ξ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equo_HTML.gif
      Thus, ( y ξ H ξ ) t = ( y ξ 2 U 2 + U ξ 2 + y ξ 2 V 2 ) t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq95_HTML.gif. Notice that y ξ H ξ ( 0 ) = ( y ξ 2 U 2 + U ξ 2 + y ξ 2 V 2 ) ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq96_HTML.gif, which implies that y ξ H ξ ( t ) = ( y ξ 2 U 2 + U ξ 2 + y ξ 2 V 2 ) ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq97_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif. Thus, (3.5c) holds. It remains to prove that the inequalities in (3.5b) hold. Set t = sup { t [ 0 , T ] | y ξ ( t ) 0  for all  t [ 0 , t ] } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq98_HTML.gif. Assume that t < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq99_HTML.gif. Since y ξ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq100_HTML.gif is continuous w.r.t. t, we have y ξ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq101_HTML.gif. It follows from (3.5c) that U ξ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq102_HTML.gif. Furthermore, (2.11) implies y ξ t ( t ) = U ξ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq103_HTML.gif and y ξ t t ( t ) = U ξ t ( t ) = 1 2 H ξ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq104_HTML.gif. If H ξ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq105_HTML.gif, then ( y ξ , U ξ , H ξ ) ( t ) = ( 0 , 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq106_HTML.gif, which implies ( y ξ , U ξ , H ξ ) ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq107_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif by the uniqueness of the solution of system (2.11). This contradicts the fact that y ξ ( 0 ) + H ξ ( 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq108_HTML.gif for all ξ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq94_HTML.gif. If H ξ ( t ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq109_HTML.gif, then y ξ t t ( t ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq110_HTML.gif. Since y ξ ( t ) = y ξ t ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq111_HTML.gif, there exists a neighborhood ϖ of t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq112_HTML.gif such that y ξ ( t ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq113_HTML.gif for all t ϖ / { t } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq114_HTML.gif. This contradicts the definition of t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq112_HTML.gif. Hence, H ξ ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq115_HTML.gif. We now have y ξ t t ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq116_HTML.gif, which conversely implies y ξ ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq117_HTML.gif for all t ϖ / { t } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq118_HTML.gif, which contradicts the fact that t < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq99_HTML.gif. Thus, we have proved y ξ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq119_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif. We now prove that H ξ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq120_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif. This follows from (3.5c) when y ξ ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq117_HTML.gif. If y ξ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq121_HTML.gif, then U ξ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq122_HTML.gif from (3.5c). As we have seen, H ξ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq123_HTML.gif would imply that y ξ ( t ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq124_HTML.gif for some t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq125_HTML.gif in a punctured neighborhood of t, which is impossible. Hence, H ξ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq120_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif. Now we have y ξ ( t ) + H ξ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq126_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq127_HTML.gif. If y ξ ( t ) + H ξ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq128_HTML.gif for some t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq125_HTML.gif, it then follows that ( y ξ , U ξ , H ξ ) ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq129_HTML.gif, which implies ( y ξ , U ξ , H ξ ) ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq130_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq127_HTML.gif, which contradicts the fact that y ξ ( 0 ) + H ξ ( 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq131_HTML.gif for all ξ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq94_HTML.gif. Hence, y ξ ( t ) + H ξ ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq132_HTML.gif.
      1. (ii)
        Define the set N = { ( t , ξ ) [ 0 , T ] × R | y ξ ( t , ξ ) = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq133_HTML.gif. It follows from Fubini’s theorem that
        meas ( N ) = R meas ( N ξ ) d ξ = [ 0 , T ] meas ( N t ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ21_HTML.gif
        (3.6)
         
      where N ξ = { t [ 0 , T ] | y ξ ( t , ξ ) = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq134_HTML.gif and N t = { ξ R | y ξ ( t , ξ ) = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq135_HTML.gif. From the above proof, we know that for all ξ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq94_HTML.gif, N ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq136_HTML.gif consists of isolated points that are countable. This means that meas ( N ξ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq137_HTML.gif. It follows from (3.6), and since meas ( Ω c ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq83_HTML.gif, that
      meas ( N t ) = 0 for almost every  t [ 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equp_HTML.gif
      This implies that y ξ ( t , ξ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq77_HTML.gif for almost all t and therefore y ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq31_HTML.gif is strictly increasing and invertible w.r.t. ξ.
      1. (iii)
        For any given t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif, since H ξ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq138_HTML.gif and H ( t , ξ ) L ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq139_HTML.gif, we know that H ( t , ± ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq140_HTML.gif exist. We have
        H ( t , ξ ) = H ( 0 , ξ ) + 0 t ( U 3 2 P U ) ( τ , ξ ) d τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ22_HTML.gif
        (3.7)
         

      Let ξ ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq141_HTML.gif. Since U, P are bounded in L ( [ 0 , T ] × R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq142_HTML.gif and lim ξ ± U ( t , ξ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq143_HTML.gif as U ( t , ) H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq144_HTML.gif, it then follows from (3.7) that H ( t , ± ) = H ( 0 , ± ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq145_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif. Since X ¯ Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq146_HTML.gif, it follows that H ( 0 , ± ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq147_HTML.gif for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq76_HTML.gif. □

      Theorem 3.2 For any initial data X ¯ = ( y ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq148_HTML.gif, there exists a unique global solution X ( t ) = ( y , U , V , N , H ) ( t ) C 1 ( R + , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq149_HTML.gif for system (2.10). Moreover, for all t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq150_HTML.gif, if we equip Γ with the topology endowed with the E-norm, then the map S t : Γ Γ × R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq151_HTML.gif defined as
      S t ( X ¯ ) = X ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equq_HTML.gif

      is a continuous semigroup.

      Proof Let ( ς , U , V , N , H ) C ( [ 0 , T ] , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq152_HTML.gif be a local solution of (2.10) with initial data ( ς ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq153_HTML.gif. To obtain the global existence of solutions, it suffices to show that
      sup t [ 0 , T ) ς ( t , ) , U ( t , ) , V ( t , ) , N ( t , ) , H ( t , ) E < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ23_HTML.gif
      (3.8)
      Since H ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq154_HTML.gif is an increasing function w.r.t. ξ for all t and lim ξ H ( t , ξ ) = lim ξ H ( 0 , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq155_HTML.gif, we have sup t [ 0 , T ) H ( t , ) L ( R ) = H ¯ L ( R ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq156_HTML.gif. We consider a fixed t [ 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq157_HTML.gif and drop it for simplification. Since U ξ ( ξ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq158_HTML.gif when y ξ ( ξ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq159_HTML.gif, and y ξ ( ξ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq160_HTML.gif for a.e. ξ, it follows from (3.5c) that
      U 2 ( ξ ) = 2 ξ U ( ξ ) U ξ ( ξ ) d ξ = 2 { ξ < ξ | y ξ ( ξ ) > 0 } U ( ξ ) U ξ ( ξ ) d ξ { ξ < ξ | y ξ ( ξ ) > 0 } ( y ξ U 2 + U ξ 2 / y ξ ) ( ξ ) d ξ R H ξ ( ξ ) d ξ = H ( ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equr_HTML.gif
      which implies
      sup t [ 0 , T ) U 2 ( t , ) L sup t [ 0 , T ) H ( t , ) L ( R ) = H ¯ L ( R ) < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equs_HTML.gif
      and therefore
      sup t [ 0 , T ) U ( t , ) L < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equt_HTML.gif
      We can obtain from the governing equation (2.10) that
      | ς ( t , ξ ) | | ς ( 0 , ξ ) | + sup t [ 0 , T ) U ( t , ) L T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equu_HTML.gif

      Thus, sup t [ 0 , T ) ς ( t , ) L < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq161_HTML.gif. The governing equation (2.10) also implies that sup t [ 0 , T ) V ( t , ) L < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq162_HTML.gif and sup t [ 0 , T ) N ( t , ) L < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq163_HTML.gif.

      From the identity H ξ = ( U 2 + U x 2 + V 2 ) y ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq164_HTML.gif, we can deduce that
      | ( U 2 + 2 V ) y ξ | ( U 2 + V 2 + 1 ) y ξ H ξ + y ξ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equv_HTML.gif
      which implies that
      | P x | 1 2 | R e | y ( ξ ) y ( ξ ) | [ H ξ + ( U 2 + 2 V ) y ξ ] ( ξ ) d ξ | 1 2 | R e | y ( ξ ) y ( ξ ) | [ 2 H ξ + y ξ ] ( ξ ) d ξ | C ( sup t [ 0 , T ) H ( t , ) L ( R ) + sup t [ 0 , T ) ς ( t , ) L ) < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equw_HTML.gif
      Therefore, P x L < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq165_HTML.gif. Similarly, we obtain P x L 2 < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq166_HTML.gif and the bounds hold for P. Let
      Z ( t ) = U ( t , ) L 2 + U ξ ( t , ) L 2 + V ( t , ) L 2 + N ( t , ) L 2 + ς ξ ( t , ) L 2 + H ξ ( t , ) L 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equx_HTML.gif
      After taking the L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq167_HTML.gif-norms on both sides of (2.10) and (2.11), we obtain
      Z ( t ) Z ( 0 ) + C 0 t Z ( τ ) d τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equy_HTML.gif

      It follows from Gronwall’s lemma that sup t [ 0 , T ) Z ( t ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq168_HTML.gif, which implies that S t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq169_HTML.gif is a continuous semigroup by the standard O.D.E. theory. □

      4 Global solutions for the original system

      We transform the global solution of the equivalent system (2.10) into the global conservative solution of the original system (2.1) in this section. It suffices to establish the correspondence between the Lagrangian equivalent system and the original system.

      We first introduce a set G as the set of relabeling functions defined by
      G = { f  is invertible | f Id  and  f 1 Id  both belong to  W 1 , ( R ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equz_HTML.gif
      where Id denotes the identity function. For any α > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq170_HTML.gif, we define the subsets G α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq171_HTML.gif of G as
      G α = { f G | f Id W 1 , ( R ) + f 1 Id W 1 , ( R ) α } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equaa_HTML.gif
      with a useful property: If f G α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq172_HTML.gif ( α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq173_HTML.gif), then 1 / ( 1 + α ) f ξ 1 + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq174_HTML.gif almost everywhere. Conversely, if f is absolutely continuous, f Id L ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq175_HTML.gif and there exists c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq176_HTML.gif such that 1 / c f ξ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq177_HTML.gif almost everywhere, then f G α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq172_HTML.gif for some α depending only on c and f Id L ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq178_HTML.gif. We now define the subsets F and F α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq179_HTML.gif of Γ such that
      F = { X = ( y , U , V , H ) Γ | y + H G } , F α = { X = ( y , U , V , H ) Γ | y + H G α } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equab_HTML.gif

      With the above useful property of G α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq171_HTML.gif, it is not hard to prove that the space F is preserved by the governing equation (2.10).

      Notice that the map Φ : G × F F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq180_HTML.gif given by Φ ( f , X ) = X f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq181_HTML.gif defines a group action of G on F, we then consider the quotient space F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq182_HTML.gif of F w.r.t. the group action. The equivalence relation on F is defined as: for any X , X F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq183_HTML.gif, if there exists f G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq184_HTML.gif such that X = X f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq185_HTML.gif, we claim that X and X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq186_HTML.gif are equivalent. We denote the projection Π : F F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq187_HTML.gif by Π ( X ) = [ X ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq188_HTML.gif. For any X = ( y , U , V , N , H ) F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq189_HTML.gif, we introduce the map K : F F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq190_HTML.gif given by K ( X ) = X ( y + H ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq191_HTML.gif. It is not hard to prove that K ( X ) = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq192_HTML.gif when X F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq193_HTML.gif, and K ( X f ) = K ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq194_HTML.gif for any X F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq195_HTML.gif and f G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq196_HTML.gif. Hence, we can define the map K ˜ : F / G F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq197_HTML.gif as K ˜ ( [ X ] ) = K ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq198_HTML.gif for any representative [ X ] F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq199_HTML.gif of X F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq195_HTML.gif. For any X F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq193_HTML.gif, we have K ˜ Π ( X ) = K ( X ) = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq200_HTML.gif. Hence, K ˜ Π | F 0 = Id | F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq201_HTML.gif. Note that any topology defined on F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq202_HTML.gif is naturally transported into F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq203_HTML.gif by this isomorphism, that is, if we equip F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq202_HTML.gif with the metric induced by the E-norm, i.e., d F 0 ( X , X ) = X X E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq204_HTML.gif for all X , X F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq205_HTML.gif, which is complete, then for any [ X ] , [ X ] F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq206_HTML.gif, the topology on F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq182_HTML.gif is defined by a complete metric given by d F / G ( [ X ] , [ X ] ) = K ( X ) K ( X ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq207_HTML.gif.

      For any initial data X ¯ F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq208_HTML.gif, we denote the continuous semigroup with the solution X ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq92_HTML.gif of system (2.10) by S : F × R + F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq209_HTML.gif. As we indicated earlier, Eq. (2.1) is invariant w.r.t. relabeling. That is, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq210_HTML.gif, S t ( X f ) = S t ( X ) f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq211_HTML.gif for any X F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq212_HTML.gif and f G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq213_HTML.gif. Thus, the map S ˜ t : F / G F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq214_HTML.gif defined by S ˜ t ( [ X ] ) = [ S t X ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq215_HTML.gif is valid, which generates a continuous semigroup.

      To derive the correspondence between the Lagrangian equivalent system and the original system, we have to consider the space D, which characterizes the solutions in the original system:
      D = { ( z , μ ) | z H 1 ( R ) × [ L 2 ( R ) × L ( R ) ]  and  μ ac = ( u 2 + u x 2 + v 2 ) d x } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equac_HTML.gif

      where z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq26_HTML.gif and μ is a positive finite Radon measure with μ ac http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq216_HTML.gif as its absolute continuous part.

      We now establish a bijection between F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq182_HTML.gif and D to transport the continuous semigroup obtained in the Lagrangian equivalent system (functions in F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq182_HTML.gif) into the original system (functions in D).

      We first introduce the mapping M, which corresponds to the transformation from 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 ξ d ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq217_HTML.gif in the Lagrangian equivalent system, where the push-forward f # ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq218_HTML.gif of a measure ν by a measurable function f is defined by
      f # ν ( B ) = ν ( f 1 ( B ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equad_HTML.gif
      for all Borel set B. Let ( z , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq219_HTML.gif be defined as
      z ( x ) = Z ( ξ ) for any  ξ  such that  x = y ( ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ24_HTML.gif
      (4.1a)
      μ = y # ( H ξ d ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ25_HTML.gif
      (4.1b)

      where z ( x ) = ( u , v ) ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq220_HTML.gif, Z ( ξ ) = ( U , V ) ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq221_HTML.gif. We have that ( z , μ ) D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq222_HTML.gif, which does not depend on the representative X = ( y , U , V , N , H ) F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq223_HTML.gif of [ X ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq224_HTML.gif we choose. We denote by M : F / G D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq225_HTML.gif the mapping to any [ X ] F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq199_HTML.gif and ( z , μ ) D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq226_HTML.gif given by (4.1a) and (4.1b), which transforms the Lagrangian equivalent system into the original system.

      We are led to the mapping L : D F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq227_HTML.gif, which conversely transforms the original system into the Lagrangian equivalent system defined as follows.

      Definition 4.1 For any ( z , μ ) D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq226_HTML.gif, let
      y ( ξ ) = sup { y | μ ( , y ) + y < ξ } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ26_HTML.gif
      (4.2a)
      U ( ξ ) = u y ( ξ ) , V ( ξ ) = v y ( ξ ) , N ( ξ ) = u x y ( ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ27_HTML.gif
      (4.2b)
      H ( ξ ) = ξ y ( ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ28_HTML.gif
      (4.2c)

      where z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq26_HTML.gif. We define L ( z , μ ) F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq228_HTML.gif as the equivalence class of ( y , U , V , N , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq229_HTML.gif.

      Remark 4.1 Note that X = ( y , U , V , N , H ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq230_HTML.gif, which satisfies (3.5a)-(3.5c) from the definition of y, U, V, N, H in (4.2a)-(4.2c). Moreover, by the definition (4.2c), we have that y + H = Id http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq231_HTML.gif. Thus, X = ( y , U , V , N , H ) F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq232_HTML.gif.

      We claim that the transformation from the original system into the Lagrangian equivalent system is a bijection.

      Theorem 4.1 The maps M and L are invertible, that is,
      L M = Id F / G , M L = Id D . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equae_HTML.gif
      Proof Let [ X ] F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq199_HTML.gif be given. We consider X = ( y , U , V , N , H ) = K ˜ ( [ X ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq233_HTML.gif for a representative of [ X ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq224_HTML.gif and ( z , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq219_HTML.gif given by (4.1a) and (4.1b) for this particular X. From the definition of K ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq234_HTML.gif, we have X F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq193_HTML.gif. Let X ¯ = ( y ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq235_HTML.gif be the representative of L ( z , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq236_HTML.gif in F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq202_HTML.gif given by (4.2a)-(4.2c). To derive L M = Id F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq237_HTML.gif, it suffices to show that ( y ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) = ( y , U , V , N , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq238_HTML.gif. Let
      g ( x ) = sup { ξ R | y ( ξ ) < x } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ29_HTML.gif
      (4.3)
      Using the fact that y is increasing and continuous, it follows that
      y ( g ( x ) ) = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ30_HTML.gif
      (4.4)
      and y 1 ( ( , x ) ) = ( , g ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq239_HTML.gif. From (4.1b) and since H ( ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq240_HTML.gif, for any x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq241_HTML.gif, we get
      μ ( ( , x ) ) = y 1 ( ( , x ) ) H ξ d ξ = g ( x ) H ξ d ξ = H ( g ( x ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equaf_HTML.gif
      Since X F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq193_HTML.gif and y + H = Id http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq231_HTML.gif, we have
      μ ( ( , x ) ) + x = g ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ31_HTML.gif
      (4.5)
      From the definition of y ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq242_HTML.gif, it follows that
      y ¯ ( ξ ) = sup { x R | g ( x ) < ξ } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ32_HTML.gif
      (4.6)

      For any given ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq78_HTML.gif, using the fact that y is increasing and (4.4), it follows that y ¯ ( ξ ) y ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq243_HTML.gif. If y ¯ ( ξ ) < y ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq244_HTML.gif, there then exists x such that y ¯ ( ξ ) < x < y ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq245_HTML.gif and (4.6) implies that g ( x ) ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq246_HTML.gif. Conversely, since y is increasing, we have x = y ( g ( x ) ) < y ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq247_HTML.gif, which implies that g ( x ) < ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq248_HTML.gif. This is a contradiction. Hence, we have that y ¯ = y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq249_HTML.gif. Since y + H = Id http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq231_HTML.gif, it follows directly from the definitions that H ¯ = H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq250_HTML.gif, U ¯ = U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq251_HTML.gif, V ¯ = V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq252_HTML.gif and N ¯ = N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq253_HTML.gif. Hence, L M = Id F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq254_HTML.gif.

      Let ( z , μ ) D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq226_HTML.gif be given and ( y , U , V , N , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq229_HTML.gif be the representative of L ( z , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq236_HTML.gif in F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq202_HTML.gif given by (4.2a)-(4.2c). Then, let ( z ¯ , μ ¯ ) = M L ( z , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq255_HTML.gif. Let g be defined as before by (4.3). The same computation that leads to (4.5) now gives
      μ ¯ ( ( , x ) ) + x = g ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ33_HTML.gif
      (4.7)
      Given ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq78_HTML.gif, we consider an increasing sequence x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq256_HTML.gif converging to y ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq257_HTML.gif, which is guaranteed by (4.2a), and such that μ ( ( , x i ) ) + x i < ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq258_HTML.gif. Let i tend to infinity. Since F ( x ) = μ ( ( , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq259_HTML.gif is lower semi-continuous, we have μ ( ( , y ( ξ ) ) ) + y ( ξ ) ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq260_HTML.gif. Take ξ = g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq261_HTML.gif and then we get
      μ ( ( , x ) ) + x g ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ34_HTML.gif
      (4.8)
      By the definition of g, there exists an increasing sequence ξ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq262_HTML.gif converging to g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq263_HTML.gif such that y ( ξ i ) < x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq264_HTML.gif. It follows from the definition of y in (4.2a) that μ ( ( , x ) ) + x ξ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq265_HTML.gif. Passing to the limit, we obtain μ ( ( , x ) ) + x g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq266_HTML.gif which, together with (4.8), yields
      μ ( ( , x ) ) + x = g ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ35_HTML.gif
      (4.9)

      We obtain that μ ¯ = μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq267_HTML.gif by comparing (4.9) and (4.7). It is clear from the definitions that z ¯ = z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq268_HTML.gif. Hence, ( z ¯ , μ ¯ ) = ( z , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq269_HTML.gif and M L = Id D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq270_HTML.gif. □

      Our next task is to transport the topology defined in F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq203_HTML.gif into D, which is guaranteed by the fact that we have established a bijection between the two equivalent systems and then obtained a continuous semigroup of solutions for the original system.

      Let us define the distance d D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq271_HTML.gif on D as
      d D ( ( z , μ ) , ( z ¯ , μ ¯ ) ) = d F / G ( L ( z , μ ) , L ( z ¯ , μ ¯ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equag_HTML.gif
      which makes the bijection L between D and F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq203_HTML.gif into an isometry. Since F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq182_HTML.gif equipped with d F / G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq272_HTML.gif is a complete metric space, it is not hard to know that D equipped with the metric d D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq271_HTML.gif is also a complete metric space. For each t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq273_HTML.gif, we define the mapping T t : D D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq274_HTML.gif as
      T t = M S ˜ t L . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equah_HTML.gif
      Theorem 4.2 Given ( z ¯ , μ ¯ ) D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq275_HTML.gif, if we denote t ( z , μ ) ( t ) = T t ( z ¯ , μ ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq276_HTML.gif the corresponding trajectory, then z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq277_HTML.gif is a weak solution of the two-component Camassa-Holm equations (2.1), which constructs a continuous semigroup. Moreover, μ is a weak solution of the following transport equation:
      μ t + ( u μ ) x = ( u 3 2 P u ) x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ36_HTML.gif
      (4.10)
      Furthermore, we have
      μ ( t ) ( R ) = μ ( 0 ) ( R ) for all t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ37_HTML.gif
      (4.11)
      and
      μ ( t ) ( R ) = μ ac ( t ) ( R ) = z ( t ) H 1 2 = u ( t ) H 1 2 + v ( t ) L 2 2 = μ ( 0 ) ( R ) for almost all t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ38_HTML.gif
      (4.12)

      Thus, the unique solution described here is a conservative weak solution of system (2.1).

      Proof To prove that z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq26_HTML.gif is a weak solution of the original system (2.1), it suffices to show that, for all ϕ C ( R + × R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq278_HTML.gif with compact support,
      R + × R ( u ϕ t + u u x ϕ ) ( t , x ) d x d t = R + × R ( P x ϕ ) ( t , x ) d x d t , R + × R ( v ϕ t + u v x ϕ ) ( t , x ) d x d t = R + × R [ ( v + 1 ) u x ϕ ] ( t , x ) d x d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ39_HTML.gif
      (4.13)
      where P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif is given by (2.1). We denote by the solution ( y , U , V , N , H ) ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq279_HTML.gif of (2.10) a representative of L ( z ( t ) , μ ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq280_HTML.gif. On the one hand, since y ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq31_HTML.gif is Lipschitz and invertible w.r.t. ξ for almost all t, we can use the change of variables x = y ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq32_HTML.gif, then we get
      R + × R ( u ϕ t + u u x ϕ ) ( t , x ) d x d t = R + × R [ ( U y ξ ) ( t , ξ ) ϕ t ( t , y ( t , ξ ) ) + ( U U ξ ) ( t , ξ ) ϕ ( t , y ( t , ξ ) ) ] d ξ d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ40_HTML.gif
      (4.14)
      Since y t = U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq281_HTML.gif and y ξ t = U ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq282_HTML.gif, it then follows from (2.10) that
      R + × R [ U y ξ ϕ t ( t , y ) + U U ξ ϕ ( t , y ) ] d ξ d t = 1 4 R + × R 2 { sgn ( ξ ξ ) e | y ( ξ ) y ( ξ ) | [ H ξ + ( U 2 + 2 V ) y ξ ] } ( ξ ) ϕ ( t , y ( ξ ) ) y ξ ( ξ ) d ξ d ξ d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ41_HTML.gif
      (4.15)
      On the other hand, using the change of variables x = y ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq32_HTML.gif and x = y ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq283_HTML.gif, and since y is an increasing function, we have
      R + × R ( P x ϕ ) ( t , x ) d x d t = 1 2 R + × R 2 [ sgn ( ξ ξ ) e | y ( ξ ) y ( ξ ) | ( u 2 + 1 2 u x 2 + 1 2 v 2 + v ) ] ( t , y ( ξ ) ) ϕ ( t , y ( ξ ) ) y ξ ( ξ ) y ξ ( ξ ) d ξ d ξ d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equai_HTML.gif
      It follows from the identity (3.5c) that
      R + × R ( P x ϕ ) ( t , x ) d x d t = 1 4 R + × R 2 { sgn ( ξ ξ ) e | y ( ξ ) y ( ξ ) | [ H ξ + ( U 2 + 2 V ) y ξ ] } ( ξ ) ϕ ( t , y ( ξ ) ) y ξ ( ξ ) d ξ d ξ d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ42_HTML.gif
      (4.16)
      By comparing (4.15) and (4.16), we know that
      R + × R [ U y ξ ϕ t ( t , y ) + U U ξ ϕ ( t , y ) ] d ξ d t = R + × R ( P x ϕ ) ( t , x ) d x d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equaj_HTML.gif
      Thus, the first identity in (4.13) follows directly from (4.14) and the second identity in (4.13) follows in the same way. It is not hard to check that μ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq284_HTML.gif is the solution of (4.10). From the definition μ in (4.1b), we can get that
      μ ( t ) ( R ) = R H ξ d ξ = H ( t , ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equak_HTML.gif

      which is constant in time from Lemma 3.3(iii). Thus, we have proved (4.11).

      Since y ξ ( t , ξ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq77_HTML.gif a.e. for almost every ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq78_HTML.gif, it then follows from (3.5c) that
      μ ( t ) ( B ) = y 1 ( B ) H ξ d ξ = y 1 ( B ) ( U 2 + U ξ 2 / y ξ 2 + V 2 ) y ξ d ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ43_HTML.gif
      (4.17)
      for any Borel set B. Since y is one-to-one and u x y y ξ = U ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq285_HTML.gif almost everywhere, then (4.17) implies that
      μ ( t ) ( B ) = B ( u 2 + u x 2 + v 2 ) ( t , x ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equal_HTML.gif

      Thus, (4.12) holds and the proof is completed. □

      5 Multipeakon solutions of the original system

      We derive a new system of ordinary differential equations for the multipeakon solutions which is well posed even when collisions occur in this section, and the variables ( y , U , V , N , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq286_HTML.gif are used to characterize multipeakons in a way that avoids the problems related to blowing up.

      Solutions of the two-component Camassa-Holm system may experience wave breaking in the sense that the solution develops singularities in finite time, while keeping the H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq25_HTML.gif norm finite. Extending the solution beyond wave breaking imposes significant challenge as can be illustrated in the case of multipeakons given by
      u ( t , x ) = i = 1 n p i ( t ) e | x q i ( t ) | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ44_HTML.gif
      (5.1)
      where ( p i ( t ) , q i ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq287_HTML.gif satisfy the explicit system of ordinary differential equations
      { p ˙ i = j = 1 n p i p j sgn ( q i q j ) e | q i q j | , q ˙ i = j = 1 n p j e | q i q j | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ45_HTML.gif
      (5.2)

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq288_HTML.gif coincide. Clearly, if the q i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq288_HTML.gif remain distinct, system (5.2) allows for a global smooth solution. In the case where p i ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq289_HTML.gif has the same sign for all i = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq290_HTML.gif, the q i ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq291_HTML.gif remain distinct, and (5.2) admits a unique global solution. In this case, the peakons are traveling in the same direction. However, when two peakons have opposite signs, collisions may occur, and if so, system (5.2) blows up.

      We consider initial data z ¯ = ( u ¯ , v ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq292_HTML.gif given by
      { u ¯ ( x ) = i = 1 n p i e | x ξ i | , v ¯ ( x ) = i = 1 n r i e | x ξ i | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ46_HTML.gif
      (5.3)
      Without loss of generality, we assume that the p i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq293_HTML.gif and r i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq294_HTML.gif are all nonzero, and that the ξ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq262_HTML.gif are all distinct. From Theorem 4.2 we know that there exists a unique and global weak solution with initial data (5.3), and the aim is to characterize this solution explicitly. We consider the following characterization of multipeakons. The multipeakons are given as continuous solutions u defined on intervals [ x i , x i + 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq295_HTML.gif as the solutions of the Dirichlet problem
      u u x x = 0 , u ( x i ) = u i , u ( x i + 1 ) = u i + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equam_HTML.gif

      where the variables x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq256_HTML.gif denote the position of the peaks, and the variables u i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq296_HTML.gif denote the values of u at the peaks. In the following, we will show that this property persists for conservative solutions.

      Let us define X ¯ = ( y ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq297_HTML.gif as
      y ¯ ( ξ ) = ξ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ47_HTML.gif
      (5.4a)
      U ¯ ( ξ ) = u ¯ ( ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ48_HTML.gif
      (5.4b)
      V ¯ ( ξ ) = v ¯ ( ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ49_HTML.gif
      (5.4c)
      N ¯ ( ξ ) = u ¯ x ( ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ50_HTML.gif
      (5.4d)
      H ¯ ( ξ ) = ξ ( u 2 + u x 2 + v 2 ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ51_HTML.gif
      (5.4e)

      which is a representative of z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq26_HTML.gif in the Lagrangian equivalent system, that is, [ X ¯ ] = L ( z ¯ , ( u ¯ 2 + u ¯ x 2 + v ¯ 2 ) d x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq298_HTML.gif. Let A = R { ξ 1 , , ξ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq299_HTML.gif. We claim that the functions U ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq300_HTML.gif, V ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq301_HTML.gif, N ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq302_HTML.gif and H ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq303_HTML.gif belong to C 2 ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq304_HTML.gif and even belong to C ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq305_HTML.gif, as the next lemma shows.

      Lemma 5.1 For given initial data X ¯ = ( y ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq306_HTML.gif such that X ¯ [ C 2 ( A ) ] 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq307_HTML.gif, the associated solution X = ( y , U , V , W , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq308_HTML.gif of (2.10) belongs to C 1 ( R + , [ C 2 ( A ) ] 5 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq309_HTML.gif.

      Proof To prove this lemma, one proceeds as in Theorem 3.1 by using the contraction argument. The Banach space E is replaced by
      E ¯ = E [ C 2 ( A ) ] 5 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equan_HTML.gif
      endowed with the norm
      X E ¯ = X E + y Id W 2 , ( A ) + U W 2 , ( A ) + V W 2 , ( A ) + N W 2 , ( A ) + H W 2 , ( A ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equao_HTML.gif
      It suffices to show that P and P i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq15_HTML.gif are Lipschitz from bounded sets of E ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq310_HTML.gif into H 1 ( R ) C 2 ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq311_HTML.gif. Given a bounded set B = { X E ¯ | X E ¯ C B } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq312_HTML.gif, where C B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq313_HTML.gif is a positive constant, it follows from Lemma 3.2 that
      P x ( X ) P x ( X ¯ ) L ( R ) C X X ¯ E C X X ¯ E ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equap_HTML.gif
      for a constant C depending only on C B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq313_HTML.gif. From the derivative of P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif given by (3.1) and Lemma 3.1, we have that P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif is locally Lipschitz from E ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq310_HTML.gif into C 1 ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq314_HTML.gif. Similarly, we obtain the same result for P. We compute the derivative of P ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq315_HTML.gif and P x ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq316_HTML.gif on A as follows:
      P x ξ ξ = H ξ ξ / 2 + ( P x y ξ U U ξ V ξ ) y ξ + ( P U 2 / 2 V ) y ξ ξ , P ξ ξ = H ξ y ξ / 2 + ( P U 2 / 2 V ) y ξ 2 + P x y ξ ξ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ52_HTML.gif
      (5.5)
      Since P ξ ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq317_HTML.gif and P x ξ ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq318_HTML.gif are locally Lipschitz maps from E ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq310_HTML.gif into C ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq319_HTML.gif, we have that P and P x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq34_HTML.gif are locally Lipschitz from E ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq310_HTML.gif into C 2 ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq320_HTML.gif. The local solution of (2.10) in E ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq310_HTML.gif then can be obtained by the standard contraction argument. As we know, as far as global existence is concerned, X W 1 , ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq321_HTML.gif does not blow up with initial data in W 1 , ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq322_HTML.gif (see Lemma 3.3(i)). For any ξ A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq323_HTML.gif, we have that
      y ξ ξ t = U ξ ξ , U ξ ξ t = 1 2 H ξ ξ ( P x y ξ U U ξ V ξ ) y ξ ( P 1 2 U 2 V ) y ξ ξ , V ξ ξ t = V ξ ξ N 2 V ξ N ξ V N ξ ξ N ξ ξ , N ξ ξ t = 2 U ξ 2 + 2 U U ξ N ξ 2 N N ξ ξ + V ξ 2 + V V ξ ξ + V ξ ξ + 1 2 H ξ y ξ N ξ ξ t = ( P 1 2 U 2 V ) y ξ 2 + P x y ξ ξ , H ξ ξ t = ( 3 U 2 2 P ) U ξ ξ 2 U P x y ξ ξ + 6 U U ξ 2 4 U ξ P x y ξ H ξ ξ t = + U H ξ y ξ 2 U P y ξ 2 + U 3 y ξ 2 + 2 U V y ξ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ53_HTML.gif
      (5.6)
      System (5.6) is affine w.r.t. y ξ ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq324_HTML.gif, U ξ ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq325_HTML.gif, V ξ ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq326_HTML.gif, N ξ ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq327_HTML.gif, H ξ ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq328_HTML.gif. Hence, on any interval [ 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq329_HTML.gif, we have
      X ξ ξ ( t , ) L ( A ) X ξ ξ ( 0 , ) L ( A ) + C + C 0 t X ξ ξ ( τ , ) L ( A ) d τ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equaq_HTML.gif

      where C is a constant depending only on sup t [ 0 , T ) X W 1 , ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq330_HTML.gif, which is bounded. Thus X W 2 , ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq331_HTML.gif does not blow up from Gronwall’s lemma, and therefore the solution is globally defined in E ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq310_HTML.gif. □

      Theorem 5.1 Let the initial data be given in (5.3). The solution given by Theorem  4.2 satisfies u u x x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq332_HTML.gif between the peaks.

      Proof Assuming that y ξ ( t , ξ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq333_HTML.gif, we have
      u x y = U ξ y ξ , u x x y = ( U ξ y ξ ) ξ 1 y ξ = U ξ ξ y ξ y ξ ξ U ξ y ξ 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equar_HTML.gif
      Hence,
      ( u u x x ) y = ( U y ξ 3 U ξ ξ y ξ + y ξ ξ U ξ ) / y ξ 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ54_HTML.gif
      (5.7)
      Let
      M = U y ξ 3 U ξ ξ y ξ + y ξ ξ U ξ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ55_HTML.gif
      (5.8)
      For a given ξ A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq323_HTML.gif, differentiating (5.8) w.r.t. t, we obtain, by using (2.10), (2.11) and (5.6), that
      d M / d t = 3 U y ξ 2 y ξ t + U t y ξ 3 U ξ ξ t y ξ U ξ ξ y ξ t + y ξ ξ t U ξ + y ξ ξ U ξ t = P x y ξ 3 + 3 y ξ 2 U U ξ H ξ ξ y ξ / 2 + ( Q y ξ U U ξ + A U ξ V ξ ) y ξ 2 + ( P U 2 / 2 V ) y ξ ξ y ξ U ξ ξ U ξ + U ξ ξ U ξ + y ξ ξ [ H ξ / 2 + ( U 2 / 2 + V P ) y ξ ] = 2 U U ξ y ξ 2 H ξ ξ y ξ / 2 + H ξ y ξ ξ / 2 V ξ y ξ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ56_HTML.gif
      (5.9)
      We differentiate (3.5c) w.r.t. ξ and get
      y ξ ξ H ξ + y ξ H ξ ξ = 2 y ξ y ξ ξ U 2 + 2 y ξ 2 U U ξ + 2 U ξ U ξ ξ + 2 y ξ y ξ ξ V 2 + 2 y ξ 2 V V ξ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ57_HTML.gif
      (5.10)
      After inserting the value of y ξ H ξ ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq334_HTML.gif given by (5.10) into (5.9) and multiplying the equation by y ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq39_HTML.gif, we obtain that
      y ξ d M / d t = U U ξ y ξ 3 + U ξ 2 y ξ ξ U ξ U ξ ξ y ξ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equas_HTML.gif
      It follows from (3.5c) and since y ξ t = U ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq282_HTML.gif that
      y ξ d M / d t = y ξ t M . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ58_HTML.gif
      (5.11)
      We claim that M / y ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq335_HTML.gif is C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq336_HTML.gif in time. Indeed, we have
      M y ξ = U y ξ 2 U ξ ξ + y ξ ξ U ξ y ξ = U y ξ 2 U ξ ξ + y ξ ξ U ξ y ξ + H ξ + y ξ ξ N H ξ y ξ + H ξ = J ( X , X ξ , X ξ ξ ) y ξ + H ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equat_HTML.gif
      for some polynomial J. Since X C 1 ( R , E ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq337_HTML.gif, we have X, X ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq338_HTML.gif and X ξ ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq339_HTML.gif are C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq336_HTML.gif in time. Since X ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq340_HTML.gif remains in Γ for all t, from (3.5b), we have y ξ + H ξ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq341_HTML.gif and therefore 1 / ( y ξ + H ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq342_HTML.gif is C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq336_HTML.gif in time, which implies that M / y ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq335_HTML.gif is C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq336_HTML.gif in time. For any time t such that y ξ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq343_HTML.gif, we have
      d d t ( M y ξ ) = M t y ξ y ξ t M y ξ 2 = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equau_HTML.gif
      Hence,
      M ( t , ξ ) = K ( ξ ) y ξ ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equav_HTML.gif
      for some constant K ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq344_HTML.gif independent of time. This leads to
      y ξ 2 ( u u x x ) y = K ( ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equaw_HTML.gif
      which corresponds to the conservation of spatial angular momentum. For the multipeakons at time t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq345_HTML.gif, we have y ( 0 , ξ ) = ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq346_HTML.gif and ( u u x x ) ( 0 , ξ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq347_HTML.gif for all ξ A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq348_HTML.gif. Hence,
      M / y ξ ( t , ξ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ59_HTML.gif
      (5.12)

      for all time t and all ξ A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq323_HTML.gif. Thus, ( u u x x ) ( t , ξ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq349_HTML.gif. □

      For solutions with multipeakon initial data, we have the following result: If y ξ ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq350_HTML.gif vanishes at some point ξ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq351_HTML.gif in the interval ( ξ i , ξ i + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq352_HTML.gif, then y ξ ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq350_HTML.gif vanishes everywhere in ( ξ i , ξ i + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq352_HTML.gif. Furthermore, for the given initial multipeakon solution z ¯ ( x ) = ( u ¯ , v ¯ ) ( x ) = ( i = 1 n p i e | x ξ i | , i = 1 n r i e | x ξ i | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq353_HTML.gif, let ( y , U , V , N , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq354_HTML.gif be the solution of system (2.10) with initial data ( y ¯ , U ¯ , V ¯ , N ¯ , H ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq355_HTML.gif given by (5.4a)-(5.4e), then between adjacent peaks, if x i = y ( t , ξ i ) x i + 1 = y ( t , ξ i + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq356_HTML.gif, the solution z ( t , x ) = ( u , v ) ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq357_HTML.gif is twice differentiable with respect to the space variable, and for x ( x i , x i + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq358_HTML.gif, we have that ( u u x x ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq359_HTML.gif.

      We now start the derivation of a system of ordinary differential equations for multipeakons.

      For each i = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq360_HTML.gif, we have, from (2.10), that
      { d y i / d t = u i , d u i / d t = P x i , d v i / d t = ( v i + 1 ) u x i , d u x i / d t = u i 2 u x i 2 / 2 + v i 2 / 2 + v i P i , d H i / d t = u i 3 2 u i P i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ60_HTML.gif
      (5.13)
      where ( y i , u i , v i , u x i , H i ) = ( y , U , V , N , H ) ( t , ξ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq361_HTML.gif, P i = P ( t , ξ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq362_HTML.gif, P x i = P x ( t , ξ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq363_HTML.gif, respectively. By using the change of variables x = y ( t , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq32_HTML.gif, P i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq15_HTML.gif and P x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq16_HTML.gif can be rewritten as
      P i = 1 / 2 R e | y i x | ( u 2 + u x 2 / 2 + v 2 / 2 + v ) d x , P x i = 1 / 2 R sgn ( y i x ) e | y i x | ( u 2 + u x 2 / 2 + v 2 / 2 + v ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ61_HTML.gif
      (5.14)
      For x [ y i , y i + 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq364_HTML.gif, i = 1 , 2 , , n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq365_HTML.gif, we write z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq277_HTML.gif as
      z ( x ) = ( u ( x ) v ( x ) ) = ( A i e x + B i e x C i e x + D i e x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ62_HTML.gif
      (5.15)
      The constants A i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq366_HTML.gif, B i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq367_HTML.gif, C i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq368_HTML.gif and D i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq369_HTML.gif depend on u i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq296_HTML.gif, u i + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq370_HTML.gif, v i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq371_HTML.gif, v i + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq372_HTML.gif, y i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq373_HTML.gif and y i + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq374_HTML.gif and read
      A i = e y ¯ i 2 [ u ¯ i cosh ( δ y i ) + δ u i sinh ( δ y i ) ] , B i = e y ¯ i 2 [ u ¯ i cosh ( δ y i ) δ u i sinh ( δ y i ) ] , C i = e y ¯ i 2 [ v ¯ i cosh ( δ y i ) + δ v i sinh ( δ y i ) ] , D i = e y ¯ i 2 [ v ¯ i cosh ( δ y i ) δ v i sinh ( δ y i ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ63_HTML.gif
      (5.16)
      where
      y ¯ i = 1 2 ( y i + y i + 1 ) , δ y i = 1 2 ( y i y i + 1 ) , u ¯ i = 1 2 ( u i + u i + 1 ) , δ u i = 1 2 ( u i u i + 1 ) , v ¯ i = 1 2 ( v i + v i + 1 ) , δ v i = 1 2 ( v i v i + 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ64_HTML.gif
      (5.17)
      The constants A i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq366_HTML.gif, B i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq367_HTML.gif, C i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq368_HTML.gif and D i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq369_HTML.gif uniquely determine z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq375_HTML.gif on the interval [ y i , y i + 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq376_HTML.gif. Thus, we can compute
      δ H i = H i + 1 H i = y i y i + 1 ( u 2 + u x 2 + v 2 ) d x = 2 u ¯ i 2 tanh ( δ y i ) + 2 δ u i 2 coth ( δ y i ) + v ¯ i 2 tanh ( δ y i ) + δ v i 2 coth ( δ y i ) + 4 C i D i δ y i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equ65_HTML.gif
      (5.18)
      At this point, we can get some more understanding of what is happening at the time of collision. Let t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq112_HTML.gif be the time when the two peaks located at y i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq373_HTML.gif and y i + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq374_HTML.gif collide, i.e., such that lim t t δ y i ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq377_HTML.gif. The function z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq375_HTML.gif remains continuous because the solution z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq375_HTML.gif remains in H 1 × [ L 2 L ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq378_HTML.gif for all time, thus we have lim t t δ u i ( t ) = lim t t δ v i ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq379_HTML.gif. Still, A i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq366_HTML.gif, B i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq367_HTML.gif, C i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq368_HTML.gif and D i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq369_HTML.gif may have a finite limit when t tends to t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq112_HTML.gif. However, the first derivative blows up, which implies lim t t B i = lim t t A i = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq380_HTML.gif and lim t t D i = lim t t C i = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq381_HTML.gif. Thus δ u i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq382_HTML.gif and δ v i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq383_HTML.gif tend to zero but slower than δ y i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq384_HTML.gif. In fact, if we let t tend to t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq112_HTML.gif in (5.18), to first order in δ y i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq384_HTML.gif, we obtain
      2 δ u i 2 + δ v i 2 = δ H i δ y i + ( δ y i ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equax_HTML.gif
      which implies that δ u i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq382_HTML.gif and δ v i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq383_HTML.gif tend to zero at the same rate as δ y i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq385_HTML.gif. We now turn to the computation of P i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq15_HTML.gif given by (5.14). Let us write z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq26_HTML.gif as
      z ( t , x ) = ( u ( t , x ) , v ( t , x ) ) = ( j = 0 n ( A j e x + B j e x ) χ ( y j , y j + 1 ) ( x ) , j = 0 n ( C j e x + D j e x ) χ ( y j , y j + 1 ) ( x ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_Equay_HTML.gif
      We have set y 0 = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq386_HTML.gif, y n + 1 = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq387_HTML.gif, u 0 = u n + 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq388_HTML.gif, v 0 = v n + 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq389_HTML.gif, A 0 = u 1 e y 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq390_HTML.gif, B 0 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq391_HTML.gif, A n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq392_HTML.gif, B n = u n e y n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq393_HTML.gif and C 0 = v 1 e y 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq394_HTML.gif, D 0 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq395_HTML.gif, C n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq396_HTML.gif, D n = v n e y n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-165/MediaObjects/13661_2013_Article_420_IEq397_HTML.gif. We have
      u 2 + 1 2 u x 2 + 1 2 v 2 + v = j = 0 n ( 3 2 A j 2 e 2 x + A j B j + 3 2 B j 2 e 2 x + 1 2 C j 2 e 2 x + C