Open Access Research

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

Yujuan Wang and Yongduan Song*

Author Affiliations

School of Automation, Chongqing University, Chongqing, 400044, P.R. China

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Boundary Value Problems 2013, 2013:165  doi:10.1186/1687-2770-2013-165


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/165


Received:6 March 2013
Accepted:26 June 2013
Published:10 July 2013

© 2013 Wang and Song; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M1">View MathML</a>

(1.1)

modeling the unidirectional propagation of shallow water waves in irrotational flow over a flat bottom, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M2">View MathML</a> representing the fluid velocity at time t in the horizontal direction, has attracted considerable attention [1-10]. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M4">View MathML</a>, 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 [4-6] or global dissipative solutions [7].

Considered herein is the two-component Camassa-Holm (CH2) shallow water system [14-16]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M5">View MathML</a>

(1.2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M8">View MathML</a> (or in the ‘short wave’ limit, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M9">View MathML</a>), 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M10">View MathML</a> representing the horizontal velocity of the fluid and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M11">View MathML</a> in connection with the free-surface elevation from equilibrium with the boundary assumptions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M13">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M14">View MathML</a>. It is formally integrable [14-16] in the sense that it can be written as a compatibility condition of two linear systems (Lax pair) with a spectral parameter ζ:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M15">View MathML</a>

It also has a bi-Hamiltonian structure corresponding to the Hamiltonian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M16">View MathML</a>

and the Hamiltonian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M17">View MathML</a>

The Cauchy problem for the two-component Camassa-Holm system has been studied extensively [18-24]. It was shown that the CH2 system is locally well posed with initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M19">View MathML</a>[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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M21">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M22">View MathML</a>), 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M23">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M24">View MathML</a>). Let us introduce an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M25">View MathML</a>, which can be expressed by its associated Green’s function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M26">View MathML</a> such as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M28">View MathML</a>. Thus, we can rewrite Eq. (1.2) as a form of a quasi-linear evolution equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M29">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M30">View MathML</a>. If we define P as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M31">View MathML</a>

then Eq. (1.2) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M32">View MathML</a>

(2.1)

Moreover, for regular solutions, we have that the total energy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M33">View MathML</a>

(2.2)

is constant in time. Thus, Eq. (2.1) possesses the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M34">View MathML</a>-norm conservation law defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M35">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M37">View MathML</a>, Young’s inequality ensures <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M38">View MathML</a>.

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

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M39">View MathML</a> denote the solution of system (2.1). For given initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M40">View MathML</a>, we define the corresponding characteristic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M41">View MathML</a> as the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M42">View MathML</a>

(2.3)

and define the Lagrangian cumulative energy distribution H as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M43">View MathML</a>

(2.4)

It is not hard to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M44">View MathML</a>

(2.5)

Then it follows from (2.3) and (2.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M45">View MathML</a>

(2.6)

Throughout the following, we use the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M46">View MathML</a>

After the change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M48">View MathML</a>, we obtain the following expressions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a> and P, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M50">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M51">View MathML</a>, we can rewrite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a> and P in (2.7) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M53">View MathML</a>

(2.8)

From the definition of the characteristic, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M54">View MathML</a>

(2.9)

Let us introduce another variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M55">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M56">View MathML</a> (it will turn out that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M57">View MathML</a>). With these new variables, we now derive an equivalent system of equations (2.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M58">View MathML</a>

(2.10)

where P and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a> are given by (2.8). Differentiating (2.10) w.r.t. ξ yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M60">View MathML</a>

(2.11)

which is semilinear w.r.t. the variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M65">View MathML</a>.

System (2.10) can be regarded as an O.D.E. in the Banach space E given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M66">View MathML</a>

endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M67">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M68">View MathML</a>. Here W is a Banach space defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M69">View MathML</a>

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M70">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M71">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a>.

Lemma 3.1 (See [5])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M73">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M74">View MathML</a>, or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M75">View MathML</a>be two locally Lipschitz maps. Then the product<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M76">View MathML</a>is also a locally Lipschitz map fromEto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M34">View MathML</a>, or fromEtoW.

Lemma 3.2For any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M78">View MathML</a>, Pand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a>defined by (2.8) are locally Lipschitz continuous fromEto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80">View MathML</a>. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M81">View MathML</a>

(3.1)

Proof We write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M82">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M83">View MathML</a> denotes the indicator function of a given set Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M85">View MathML</a> are the operators which correspond to the two terms of the last identity in (3.2). We rewrite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M84">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M87">View MathML</a>

(3.3)

where R is the operator from E to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M88">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M89">View MathML</a>

Since the operator Λ (defined as in Section 2) is linear and continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M90">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M88">View MathML</a> is continuously embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M93">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M94">View MathML</a>. It is not hard to know that R is locally Lipschitz from E into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M95">View MathML</a> and therefore from E into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M90">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M97">View MathML</a> is locally Lipschitz from E to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80">View MathML</a>. Since the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M99">View MathML</a> is locally Lipschitz from E to W, it then follows from Lemma 3.1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M84">View MathML</a> is locally Lipschitz from E to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80">View MathML</a>. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M85">View MathML</a> is also locally Lipschitz and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a> is locally Lipschitz from E to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80">View MathML</a>. We can obtain that P defined by (2.8) is locally Lipschitz continuous from E to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M80">View MathML</a> in the same way. By using the chain rule, the formulas in (3.1) are obtained by direct computation, see [[25], p.129]. □

Theorem 3.1Let any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M106">View MathML</a>be given. System (2.10) admits a unique local solution defined on some time interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M107">View MathML</a>, whereTdepends only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M108">View MathML</a>.

Proof To establish the local existence of solutions, one proceeds as in Lemma 3.2, then obtains that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M109">View MathML</a>, which is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M110">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M111">View MathML</a>, is Lipschitz continuous on any bounded set of E. We rewrite the solutions of system (2.10) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M112">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M113">View MathML</a> belongs to the set Γ, which is defined as follows.

Definition 3.1 The set Γ is composed of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M114">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M115">View MathML</a>

(3.5a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M116">View MathML</a>

(3.5b)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M117">View MathML</a>

(3.5c)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M118">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M119">View MathML</a>.

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 3.3Given initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M120">View MathML</a>, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M121">View MathML</a>, we consider the local solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M122">View MathML</a>of system (2.10) given by Theorem 3.1. We have

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M123">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M125">View MathML</a>for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M127">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M128">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>.

Proof (i) For given initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M130">View MathML</a>, to ensure that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M131">View MathML</a> of (2.10) also belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M132">View MathML</a>, we have to specify the initial conditions for (2.11). Let Ω be the following set:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M133">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M134">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M135">View MathML</a>, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M136">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M137">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M138">View MathML</a>. We consider U, P and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a> as given functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M140">View MathML</a>, which is guaranteed by Lemma 3.2 and V, N in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M141">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M107">View MathML</a>. Thus, for the given initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M143">View MathML</a>, the solution of (2.10) given by Theorem 3.1 also belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M144">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M145">View MathML</a> satisfies (3.5a) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M146">View MathML</a>. We claim that (3.5c) holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147">View MathML</a> and therefore almost everywhere. Consider a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147">View MathML</a> and drop it in the notation. On the one hand, it follows from (2.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M149">View MathML</a>

and on the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M150">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M151">View MathML</a>. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M152">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M153">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>. Thus, (3.5c) holds. It remains to prove that the inequalities in (3.5b) hold. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M155">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M156">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M157">View MathML</a> is continuous w.r.t. t, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M158">View MathML</a>. It follows from (3.5c) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M159">View MathML</a>. Furthermore, (2.11) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M161">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M162">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M163">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M164">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a> by the uniqueness of the solution of system (2.11). This contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M166">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M168">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M169">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M170">View MathML</a>, there exists a neighborhood ϖ of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M172">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M173">View MathML</a>. This contradicts the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M175">View MathML</a>. We now have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M176">View MathML</a>, which conversely implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M177">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M178">View MathML</a>, which contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M156">View MathML</a>. Thus, we have proved <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M180">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>. We now prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M182">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>. This follows from (3.5c) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M177">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M185">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M186">View MathML</a> from (3.5c). As we have seen, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M187">View MathML</a> would imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M188">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M189">View MathML</a> in a punctured neighborhood of t, which is impossible. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M182">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>. Now we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M192">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M193">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M194">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M189">View MathML</a>, it then follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M196">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M197">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M193">View MathML</a>, which contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M199">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M201">View MathML</a>.

(ii) Define the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M202">View MathML</a>. It follows from Fubini’s theorem that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M203">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M205">View MathML</a>. From the above proof, we know that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M207">View MathML</a> consists of isolated points that are countable. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M208">View MathML</a>. It follows from (3.6), and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M134">View MathML</a>, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M210">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M125">View MathML</a> for almost all t and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M41">View MathML</a> is strictly increasing and invertible w.r.t. ξ.

(iii) For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M215">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M216">View MathML</a> exist. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M217">View MathML</a>

(3.7)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M218">View MathML</a>. Since U, P are bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M220">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M221">View MathML</a>, it then follows from (3.7) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M222">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M224">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M225">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M124">View MathML</a>. □

Theorem 3.2For any initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M227">View MathML</a>, there exists a unique global solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M228">View MathML</a>for system (2.10). Moreover, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M229">View MathML</a>, if we equip Γ with the topology endowed with theE-norm, then the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M230">View MathML</a>defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M231">View MathML</a>

is a continuous semigroup.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M232">View MathML</a> be a local solution of (2.10) with initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M233">View MathML</a>. To obtain the global existence of solutions, it suffices to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M234">View MathML</a>

(3.8)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M235">View MathML</a> is an increasing function w.r.t. ξ for all t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M236">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M237">View MathML</a>. We consider a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M238">View MathML</a> and drop it for simplification. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M239">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M240">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M241">View MathML</a> for a.e. ξ, it follows from (3.5c) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M242">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M243">View MathML</a>

and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M244">View MathML</a>

We can obtain from the governing equation (2.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M245">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M246">View MathML</a>. The governing equation (2.10) also implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M248">View MathML</a>.

From the identity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M249">View MathML</a>, we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M250">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M251">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M252">View MathML</a>. Similarly, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M253">View MathML</a> and the bounds hold for P. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M254">View MathML</a>

After taking the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M255">View MathML</a>-norms on both sides of (2.10) and (2.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M256">View MathML</a>

It follows from Gronwall’s lemma that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M257">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M258">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M259">View MathML</a>

where Id denotes the identity function. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M260">View MathML</a>, we define the subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M261">View MathML</a> of G as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M262">View MathML</a>

with a useful property: If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M263">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M264">View MathML</a>), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M265">View MathML</a> almost everywhere. Conversely, if f is absolutely continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M266">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M267">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M268">View MathML</a> almost everywhere, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M263">View MathML</a> for some α depending only on c and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M270">View MathML</a>. We now define the subsets F and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M271">View MathML</a> of Γ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M272">View MathML</a>

With the above useful property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M261">View MathML</a>, it is not hard to prove that the space F is preserved by the governing equation (2.10).

Notice that the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M274">View MathML</a> given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M275">View MathML</a> defines a group action of G on F, we then consider the quotient space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276">View MathML</a> of F w.r.t. the group action. The equivalence relation on F is defined as: for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M277">View MathML</a>, if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M278">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M279">View MathML</a>, we claim that X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M280">View MathML</a> are equivalent. We denote the projection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M281">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M282">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M283">View MathML</a>, we introduce the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M284">View MathML</a> given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M285">View MathML</a>. It is not hard to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M286">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M287">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M288">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M290">View MathML</a>. Hence, we can define the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M291">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M292">View MathML</a> for any representative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M293">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M289">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M287">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M296">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M297">View MathML</a>. Note that any topology defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M298">View MathML</a> is naturally transported into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M299">View MathML</a> by this isomorphism, that is, if we equip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M298">View MathML</a> with the metric induced by the E-norm, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M301">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M302">View MathML</a>, which is complete, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M303">View MathML</a>, the topology on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276">View MathML</a> is defined by a complete metric given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M305">View MathML</a>.

For any initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M306">View MathML</a>, we denote the continuous semigroup with the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M145">View MathML</a> of system (2.10) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M308">View MathML</a>. As we indicated earlier, Eq. (2.1) is invariant w.r.t. relabeling. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M310">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M311">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M312">View MathML</a>. Thus, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M313">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M314">View MathML</a> 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M315">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36">View MathML</a> and μ is a positive finite Radon measure with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M317">View MathML</a> as its absolute continuous part.

We now establish a bijection between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276">View MathML</a> and D to transport the continuous semigroup obtained in the Lagrangian equivalent system (functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276">View MathML</a>) 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M320">View MathML</a> in the Lagrangian equivalent system, where the push-forward <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M321">View MathML</a> of a measure ν by a measurable function f is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M322">View MathML</a>

for all Borel set B. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M323">View MathML</a> be defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M324">View MathML</a>

(4.1a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M325">View MathML</a>

(4.1b)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M326">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M327">View MathML</a>. We have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M328">View MathML</a>, which does not depend on the representative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M329">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M330">View MathML</a> we choose. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M331">View MathML</a> the mapping to any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M293">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M333">View MathML</a> given by (4.1a) and (4.1b), which transforms the Lagrangian equivalent system into the original system.

We are led to the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M334">View MathML</a>, which conversely transforms the original system into the Lagrangian equivalent system defined as follows.

Definition 4.1 For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M333">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M336">View MathML</a>

(4.2a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M337">View MathML</a>

(4.2b)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M338">View MathML</a>

(4.2c)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36">View MathML</a>. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M340">View MathML</a> as the equivalence class of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M341">View MathML</a>.

Remark 4.1 Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M342">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M343">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M344">View MathML</a>.

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

Theorem 4.1The mapsMandLare invertible, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M345">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M293">View MathML</a> be given. We consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M347">View MathML</a> for a representative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M330">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M323">View MathML</a> given by (4.1a) and (4.1b) for this particular X. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M350">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M287">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M352">View MathML</a> be the representative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M353">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M298">View MathML</a> given by (4.2a)-(4.2c). To derive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M355">View MathML</a>, it suffices to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M356">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M357">View MathML</a>

(4.3)

Using the fact that y is increasing and continuous, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M358">View MathML</a>

(4.4)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M359">View MathML</a>. From (4.1b) and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M360">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M361">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M362">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M287">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M343">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M365">View MathML</a>

(4.5)

From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M366">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M367">View MathML</a>

(4.6)

For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M127">View MathML</a>, using the fact that y is increasing and (4.4), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M369">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M370">View MathML</a>, there then exists x such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M371">View MathML</a> and (4.6) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M372">View MathML</a>. Conversely, since y is increasing, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M373">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M374">View MathML</a>. This is a contradiction. Hence, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M375">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M343">View MathML</a>, it follows directly from the definitions that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M377">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M379">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M380">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M381">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M333">View MathML</a> be given and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M341">View MathML</a> be the representative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M353">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M298">View MathML</a> given by (4.2a)-(4.2c). Then, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M386">View MathML</a>. Let g be defined as before by (4.3). The same computation that leads to (4.5) now gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M387">View MathML</a>

(4.7)

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M127">View MathML</a>, we consider an increasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M389">View MathML</a> converging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M390">View MathML</a>, which is guaranteed by (4.2a), and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M391">View MathML</a>. Let i tend to infinity. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M392">View MathML</a> is lower semi-continuous, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M393">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M394">View MathML</a> and then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M395">View MathML</a>

(4.8)

By the definition of g, there exists an increasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M396">View MathML</a> converging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M397">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M398">View MathML</a>. It follows from the definition of y in (4.2a) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M399">View MathML</a>. Passing to the limit, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M400">View MathML</a> which, together with (4.8), yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M401">View MathML</a>

(4.9)

We obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M402">View MathML</a> by comparing (4.9) and (4.7). It is clear from the definitions that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M403">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M404">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M405">View MathML</a>. □

Our next task is to transport the topology defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M299">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M407">View MathML</a> on D as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M408">View MathML</a>

which makes the bijection L between D and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M299">View MathML</a> into an isometry. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M276">View MathML</a> equipped with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M411">View MathML</a> is a complete metric space, it is not hard to know that D equipped with the metric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M407">View MathML</a> is also a complete metric space. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M413">View MathML</a>, we define the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M414">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M415">View MathML</a>

Theorem 4.2Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M416">View MathML</a>, if we denote<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M417">View MathML</a>the corresponding trajectory, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M418">View MathML</a>is a weak solution of the two-component Camassa-Holm equations (2.1), which constructs a continuous semigroup. Moreover, μis a weak solution of the following transport equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M419">View MathML</a>

(4.10)

Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M420">View MathML</a>

(4.11)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M421">View MathML</a>

(4.12)

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

Proof To prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36">View MathML</a> is a weak solution of the original system (2.1), it suffices to show that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M423">View MathML</a> with compact support,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M424">View MathML</a>

(4.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a> is given by (2.1). We denote by the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M426">View MathML</a> of (2.10) a representative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M427">View MathML</a>. On the one hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M41">View MathML</a> is Lipschitz and invertible w.r.t. ξ for almost all t, we can use the change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M47">View MathML</a>, then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M430">View MathML</a>

(4.14)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M431">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M432">View MathML</a>, it then follows from (2.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M433">View MathML</a>

(4.15)

On the other hand, using the change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M435">View MathML</a>, and since y is an increasing function, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M436">View MathML</a>

It follows from the identity (3.5c) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M437">View MathML</a>

(4.16)

By comparing (4.15) and (4.16), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M438">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M439">View MathML</a> is the solution of (4.10). From the definition μ in (4.1b), we can get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M440">View MathML</a>

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

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M125">View MathML</a> a.e. for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M127">View MathML</a>, it then follows from (3.5c) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M443">View MathML</a>

(4.17)

for any Borel set B. Since y is one-to-one and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M444">View MathML</a> almost everywhere, then (4.17) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M445">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M446">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M34">View MathML</a> norm finite. Extending the solution beyond wave breaking imposes significant challenge as can be illustrated in the case of multipeakons given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M448">View MathML</a>

(5.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M449">View MathML</a> satisfy the explicit system of ordinary differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M450">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M451">View MathML</a> coincide. Clearly, if the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M451">View MathML</a> remain distinct, system (5.2) allows for a global smooth solution. In the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M453">View MathML</a> has the same sign for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M454">View MathML</a>, the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M455">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M456">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M457">View MathML</a>

(5.3)

Without loss of generality, we assume that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M458">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M459">View MathML</a> are all nonzero, and that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M396">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M461">View MathML</a> as the solutions of the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M462">View MathML</a>

where the variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M389">View MathML</a> denote the position of the peaks, and the variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M464">View MathML</a> denote the values of u at the peaks. In the following, we will show that this property persists for conservative solutions.

Let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M465">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M466">View MathML</a>

(5.4a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M467">View MathML</a>

(5.4b)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M468">View MathML</a>

(5.4c)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M469">View MathML</a>

(5.4d)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M470">View MathML</a>

(5.4e)

which is a representative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36">View MathML</a> in the Lagrangian equivalent system, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M472">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M473">View MathML</a>. We claim that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M474">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M475">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M476">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M477">View MathML</a> belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M478">View MathML</a> and even belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M479">View MathML</a>, as the next lemma shows.

Lemma 5.1For given initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M480">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M481">View MathML</a>, the associated solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M482">View MathML</a>of (2.10) belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M483">View MathML</a>.

Proof To prove this lemma, one proceeds as in Theorem 3.1 by using the contraction argument. The Banach space E is replaced by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M484">View MathML</a>

endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M485">View MathML</a>

It suffices to show that P and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M20">View MathML</a> are Lipschitz from bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M488">View MathML</a>. Given a bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M489">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M490">View MathML</a> is a positive constant, it follows from Lemma 3.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M491">View MathML</a>

for a constant C depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M490">View MathML</a>. From the derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a> given by (3.1) and Lemma 3.1, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a> is locally Lipschitz from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M496">View MathML</a>. Similarly, we obtain the same result for P. We compute the derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M497">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M498">View MathML</a> on A as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M499">View MathML</a>

(5.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M500">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M501">View MathML</a> are locally Lipschitz maps from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M503">View MathML</a>, we have that P and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M49">View MathML</a> are locally Lipschitz from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M506">View MathML</a>. The local solution of (2.10) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487">View MathML</a> then can be obtained by the standard contraction argument. As we know, as far as global existence is concerned, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M508">View MathML</a> does not blow up with initial data in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M509">View MathML</a> (see Lemma 3.3(i)). For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M510">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M511">View MathML</a>

(5.6)

System (5.6) is affine w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M512">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M513">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M514">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M515">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M516">View MathML</a>. Hence, on any interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M517">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M518">View MathML</a>

where C is a constant depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M519">View MathML</a>, which is bounded. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M520">View MathML</a> does not blow up from Gronwall’s lemma, and therefore the solution is globally defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M487">View MathML</a>. □

Theorem 5.1Let the initial data be given in (5.3). The solution given by Theorem 4.2 satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M522">View MathML</a>between the peaks.

Proof Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M523">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M524">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M525">View MathML</a>

(5.7)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M526">View MathML</a>

(5.8)

For a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M510">View MathML</a>, differentiating (5.8) w.r.t. t, we obtain, by using (2.10), (2.11) and (5.6), that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M528">View MathML</a>

(5.9)

We differentiate (3.5c) w.r.t. ξ and get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M529">View MathML</a>

(5.10)

After inserting the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M530">View MathML</a> given by (5.10) into (5.9) and multiplying the equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M61">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M532">View MathML</a>

It follows from (3.5c) and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M432">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M534">View MathML</a>

(5.11)

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M535">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M536">View MathML</a> in time. Indeed, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M537">View MathML</a>

for some polynomial J. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M538">View MathML</a>, we have X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M539">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M540">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M536">View MathML</a> in time. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M542">View MathML</a> remains in Γ for all t, from (3.5b), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M543">View MathML</a> and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M544">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M536">View MathML</a> in time, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M535">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M536">View MathML</a> in time. For any time t such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M548">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M549">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M550">View MathML</a>

for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M551">View MathML</a> independent of time. This leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M552">View MathML</a>

which corresponds to the conservation of spatial angular momentum. For the multipeakons at time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M553">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M554">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M555">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M556">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M557">View MathML</a>

(5.12)

for all time t and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M510">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M559">View MathML</a>. □

For solutions with multipeakon initial data, we have the following result: If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M560">View MathML</a> vanishes at some point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M561">View MathML</a> in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M562">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M560">View MathML</a> vanishes everywhere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M562">View MathML</a>. Furthermore, for the given initial multipeakon solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M565">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M566">View MathML</a> be the solution of system (2.10) with initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M567">View MathML</a> given by (5.4a)-(5.4e), then between adjacent peaks, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M568">View MathML</a>, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M569">View MathML</a> is twice differentiable with respect to the space variable, and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M570">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M571">View MathML</a>.

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

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M572">View MathML</a>, we have, from (2.10), that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M573">View MathML</a>

(5.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M574">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M575">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M576">View MathML</a>, respectively. By using the change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M21">View MathML</a> can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M580">View MathML</a>

(5.14)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M581">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M582">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M418">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M584">View MathML</a>

(5.15)

The constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M585">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M586">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M587">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M588">View MathML</a> depend on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M464">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M590">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M591">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M592">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M593">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M594">View MathML</a> and read

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M595">View MathML</a>

(5.16)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M596">View MathML</a>

(5.17)

The constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M585">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M586">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M587">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M588">View MathML</a> uniquely determine <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M601">View MathML</a> on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M602">View MathML</a>. Thus, we can compute

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M603">View MathML</a>

(5.18)

At this point, we can get some more understanding of what is happening at the time of collision. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171">View MathML</a> be the time when the two peaks located at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M593">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M594">View MathML</a> collide, i.e., such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M607">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M601">View MathML</a> remains continuous because the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M601">View MathML</a> remains in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M610">View MathML</a> for all time, thus we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M611">View MathML</a>. Still, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M585">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M586">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M587">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M588">View MathML</a> may have a finite limit when t tends to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171">View MathML</a>. However, the first derivative blows up, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M617">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M618">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M619">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M620">View MathML</a> tend to zero but slower than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M621">View MathML</a>. In fact, if we let t tend to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M171">View MathML</a> in (5.18), to first order in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M621">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M624">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M619">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M620">View MathML</a> tend to zero at the same rate as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M627">View MathML</a>. We now turn to the computation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M20">View MathML</a> given by (5.14). Let us write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M630">View MathML</a>

We have set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M631">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M632">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M633">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M634">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M635">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M636">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M637">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M638">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M639">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M640">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M641">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M642">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M643">View MathML</a>

(5.19)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M644">View MathML</a>

Inserting (5.19) into (5.14), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M645">View MathML</a>

(5.20)

From (5.16) and (5.18), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M646">View MathML</a>

(5.21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M647">View MathML</a>

(5.22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M648">View MathML</a>

(5.23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M649">View MathML</a>

(5.24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M650">View MathML</a>

(5.25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M651">View MathML</a>

(5.26)

It then follows from (5.21)-(5.26) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M652">View MathML</a>

(5.27)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M653">View MathML</a>

(5.28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M654">View MathML</a>

(5.29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M655">View MathML</a>

(5.30)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M656">View MathML</a>

(5.31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M657">View MathML</a>

(5.32)

Therefore, the above formulas (5.27)-(5.32) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M658">View MathML</a>

which can also be written in the following form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M659">View MathML</a>

(5.33)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M660">View MathML</a>

We compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M661">View MathML</a> in the same way and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M662">View MathML</a>

(5.34)

Now we can summarize the result as follows.

Theorem 5.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M663">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M664">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M665">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M666">View MathML</a>with a multipeakon initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M456">View MathML</a>given by (5.3). Then with initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M668">View MathML</a>there exists a global solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M669">View MathML</a>of (5.13), (5.33), (5.34). For each timet, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M2">View MathML</a>is defined as the solution of the Dirichlet problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M522">View MathML</a>with boundary conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M672">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M673">View MathML</a>on each interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M674">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/165/mathml/M36">View MathML</a>is a conservative solution of the two-component Camassa-Holm system, which is the multipeakon conservative solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YS conceived of the multipeakon conservative solution of the two-component Camassa-Holm system and participated in its coordination and helped to draft the manuscript. YW participated in the design and coordination of the study, carried out the global conservative solution and the multipeakon conservative solution for the two-component Camassa-Holm system and drafted the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The paper is supported by the Major State Basic Research Development Program 973 (No. 2012CB215202), the National Natural Science Foundation of China (No. 61134001) and the Fundamental Research Funds for the Central Universities (No. CDJXS12170003). The authors would like to thank the referees for constructive suggestions and comments.

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