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

R E S E A R C H

Open Access

Global conservative and multipeakon

conservative solutions for the

two-component Camassa-Holm system

Yujuan Wang and Yongduan Song

*_{Correspondence:}

[email protected]

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

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

ut–uxxt+ uux= uxuxx+uuxxx, (.)

modeling the unidirectional propagation of shallow water waves in irrotational flow over a flat bottom, withu(t,x) representing the fluid velocity at timetin the horizontal direc-tion, has attracted considerable attention [–]. 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 [] as a bi-Hamiltonian generalization of the KdV equation. It is completely in-tegrable [, ] and possesses an infinite number of conservation laws. A remarkable prop-erty of the CH equation is the existence of the non-smooth solitary wave solutions called peakons [, ]. The peakonu(t,x) =ce–|x–ct|,c= , 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 [, ], which means that the solution remains bounded while its slope becomes unbounded in finite time [, ], setting it apart from the classi-cal soliton equations such as KdV. After wave breaking, the solutions of the CH equation can be continued uniquely as either global conservative [–] or global dissipative solu-tions [].

Considered herein is the two-component Camassa-Holm (CH) shallow water system [–]

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

mt+umx+ uxm+σρρx= , t> ,x∈R,

ρt+ (uρ)x= , t> ,x∈R,

u(,x) =u(x), x∈R,

ρ(,x) =ρ(x), x∈R,

(.)

withσ=  andm=σu–uxx,σ=  (or in the ‘short wave’ limit,σ= ), which is an ex-tension of the CH equation by combining its integrability property with compressibility or free-surface elevation dynamics in its shallow water interpretation [, ]. This system appeared originally in [] as could be identified with the first negative flow of AKNS hi-erarchy, and then it was derived by Constantin and Ivanov [] in the context of shallow water theory, withu(x,t) representing the horizontal velocity of the fluid andρ(x,t) in connection with the free-surface elevation from equilibrium with the boundary assump-tionsu→ andρ→ as|x| → ∞. It is formally integrable [–] in the sense that it

can be written as a compatibility condition of two linear systems (Lax pair) with a spectral parameterζ:

xx=

–σ ζρ+ζm+σ 

, t=

 ζ –u

x+

 ux.

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

H=  

um+ (ρ– )dx

and the Hamiltonian

H=  

u(ρ– )+ u(ρ– ) +u+uu_xdx.

The Cauchy problem for the two-component Camassa-Holm system has been studied extensively [–]. It was shown that the CH system is locally well posed with initial data (u,ρ)∈Hs×Hs–,s>_ []. The system also has global strong solutions which blow-up in finite time [, , ] and a global weak solution []. 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 [], we studied the continua-tion beyond wave breaking by applying an approach that reformulated system (.) as a semilinear system of O.D.E. taking values in a Banach space. Such treatment makes it pos-sible 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.

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 prob-lem can be obtained, making it convenient to study the dynamic behavior of wave break-ing. Because of the introduction of the new variables, we are able to establish the multi-peakon conservative solution from the global conservative solution for the CH system.

Some related earlier works [, ] 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 [, ]. 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 (.). It is shown that the multipeakon structure is preserved by the semigroup of a global conservative solution and the multipeakon solu-tion is obtained by carefully computing the convolusolu-tion equasolu-tionsPiandPxi(i= , . . . ,n),

where, in contrast to the existing works, the inherent mutual effect between the two com-ponents is well reflected.

The remainder of this paper is organized as follows. Section  presents the transforma-tion from the original system to a Lagrangian semilinear system. The global solutransforma-tions of the equivalent semilinear system are obtained in Section , which are transformed into the global conservative solutions of the original system in Section . Finally, we establish the multipeakon conservative solutions for the original system in Section .

2 The original system and the equivalent Lagrangian system

We first present the original system. For simplicity, we consider here the associated evo-lution for positive times (of course, one would get similar results for negative times just by changing the initial conditionuinto –u). Let us introduce an operator= ( –∂x)–,

which can be expressed by its associated Green’s function G=  e

–|x| _{such as f(x) =} G∗f(x) = _Re–|x–x|_{f(x)dxfor allf ∈L}_{(R). Thus, we can rewrite Eq. (.) as a form} of a quasi-linear evolution equation:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ut+uux+∂xG∗(u+_ux+v

_{+v) = , t> ,x∈R,}

vt+ (uv)x+ux= , t> ,x∈R,

u(,x) =u(x), x∈R,

v(,x) =v(x), x∈R,

wherev=ρ– . If we definePas

P(t,x) =G∗

u+ u



x+

 v

_+v

= 

R e–|x–x|

u+ u



x+

 v

_{+v t,xdx,}

then Eq. (.) can be rewritten as

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ut+uux+Px= , t> ,x∈R, vt+uvx+vux+ux= , t> ,x∈R, u(,x) =u(x), x∈R,

v(,x) =v(x), x∈R.

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

E(t) =

R

u+u_x+vdx (.)

is constant in time. Thus, Eq. (.) possesses theH-norm conservation law defined as

zH=u_H+v_L=

R

u+u_x+vdx

/ ,

wherez= (u,v). Sincez= (u,v)∈H_×[L_∩L∞_{], Young’s inequality ensuresP∈H}_. We reformulate system (.) into a Lagrangian equivalent semilinear system as follows. Letz(t,x) = (u,v)(t,x) denote the solution of system (.). For given initial datay(,ξ), we define the corresponding characteristicy(t,ξ) as the solution of

yt(t,ξ) =u t,y(t,ξ)

, (.)

and define the Lagrangian cumulative energy distributionHas

H(t,ξ) =

y(t,ξ) –∞

u+u_x+v(t,x)dx. (.)

It is not hard to check that

u+u_x+v_t+ u u+u_x+v_x= u– uP_x. (.)

Then it follows from (.) and (.) that

dH dt = u

_{– uP t,y(t,ξ)}ξ

–∞. (.)

Throughout the following, we use the notation

U(t,ξ) =u t,y(t,ξ), V(t,ξ) =v t,y(t,ξ), N(t,ξ) =ux t,y(t,ξ)

.

After the change of variablesx=y(t,ξ) andx=y(t,ξ), we obtain the following expressions forPxandP, namely

P(ξ) =P y(ξ)=  

R

e–|y(ξ)–y(ξ)|

U+ U



x+

 V

_+V

yξ ξ

dξ,

Px(ξ) =Px y(ξ)

= – 

R

sgn ξ–ξe–|y(ξ)–y(ξ)|_,

U+ U



x+

 V

_+V

yξ ξ

dξ,

(.)

Hξ = (u+u_x+v)◦yyξ, we can rewritePxandPin (.) as

Px(ξ) = –

 

R

sgn ξ–ξe–|y(ξ)–y(ξ)|Hξ+ U+ V

yξ ξ

dξ,

P(ξ) =  

R

e–|y(ξ)–y(ξ)|Hξ+ U+ V

yξ ξ

dξ.

(.)

From the definition of the characteristic, it follows that

Ut(t,ξ) =ut(t,y) +ux(t,y)yt(t,ξ) = –Px◦y(t,ξ),

Vt(t,ξ) =vt(t,y) +vx(t,y)yt(t,ξ) = –

(v+ )ux

◦y(t,ξ),

Nt(t,ξ) =uxt(t,y) +uxx(t,y)yt(t,ξ) =

u– u



x+

 v

_+v–P

◦y(t,ξ).

(.)

Let us introduce another variable ς(t,ξ) such thatς(t,ξ) =y(t,ξ) –ξ (it will turn out thatς∈L∞(R)). With these new variables, we now derive an equivalent system of equa-tions (.),

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ςt=U, Ut= –Px, Vt= –(V+ )N,

Nt=U–_N+_V+V–P, Ht=U– UP,

(.)

wherePandPxare given by (.). Differentiating (.) w.r.t.ξ yields

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ςξt=Uξ,

Uξt=_Hξ+ (_U+V–P)yξ, Vξt= –VξN– (V+ )Nξ,

Nξt= UUξ+ (V+ )Vξ–NNξ–Pxyξ, Hξt= (U– P)Uξ– UPxyξ,

(.)

which is semilinear w.r.t. the variablesyξ,Uξ,Vξ,Nξ andHξ.

System (.) can be regarded as an O.D.E. in the Banach spaceEgiven by

E=W×H×L∩L∞×L∩L∞×W, endowed with the norm

XE=ςW+UH+V_L+V_L∞+N_L+N_L∞+H_W

for anyX= (ς,U,V,N,H)∈E. HereWis a Banach space defined as

W=f∈C(R)∩L∞(R)|fξ∈L(R)

,

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 onPandPx.

Lemma .(See []) Let:E→W and :E→H_{,or:E→W be two locally}

Lipschitz maps.Then the product X→ (X)(X)is also a locally Lipschitz map from E to H_{,or from E to W.}

Lemma . For any given X= (ς,U,V,N,H)∈E,P and Pxdefined by(.)are locally Lipschitz continuous from E to H(R).Moreover,we have

Pxξ = –

 Hξ+

P– U

_–V

( +ςξ), Pξ=Px( +ςξ). (.)

Proof We write

Px(ξ) =Px(X)(ξ) +Px(X)(ξ)

= –e –ς(ξ)



R

χ_{ξ<ξ}e–|ξ–ξ

_|

eς(ξ)Hξ + U+ V

( +ςξ) ξ

dξ

+e

ς(ξ)



R

χ{ξ>ξ}e–|ξ–ξ

_|

e–ς(ξ)Hξ+ U+ V

( +ςξ) ξ

dξ, (.)

whereχ denotes the indicator function of a given set, andP_x,P_x are the operators

which correspond to the two terms of the last identity in (.). We rewriteP

xas

P_x(X)(ξ) = –e –ς(ξ)

 ◦R(X)(ξ), (.)

whereRis the operator fromEtoL_{(R) given by}

R(X)(ξ) =χ{ξ<ξ}eς

Hξ+ U+ V

( +ςξ)

.

Since the operator(defined as in Section ) is linear and continuous fromH–_{(R) to}

H(R), andL(R) is continuously embedded inH–(R), we have◦R(X)∈H. It is not hard to know thatRis locally Lipschitz fromEintoL_{(R) and therefore fromEintoH}–_(R). Thus,◦Ris locally Lipschitz fromEtoH_{(R). Since the mappingX→e}–ς _{is locally}

Lipschitz fromEtoW, it then follows from Lemma . thatP

xis locally Lipschitz fromE

toH_{(R). Similarly,P}

xis also locally Lipschitz and thereforePxis locally Lipschitz fromE

toH(R). We can obtain thatPdefined by (.) is locally Lipschitz continuous fromEto

H(R) in the same way. By using the chain rule, the formulas in (.) are obtained by direct

computation, see [, p.].

Theorem . Let anyX¯ = (ς¯,U¯,V¯,N¯,H¯)∈E be given.System(.)admits a unique local solution defined on some time interval[,T],where T depends only on ¯XE.

Proof To establish the local existence of solutions, one proceeds as in Lemma ., then obtains thatF(X), which is defined by

F(X) =

U, –Px, –(V+ )N,U–

 N

₊ V

_+V–P,U_{– UP}

withX= (ς,U,V,N,H), is Lipschitz continuous on any bounded set ofE. We rewrite the solutions of system (.) as

X(t) =X¯ +

t



F X(τ)dτ. (.)

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

Theorem . gives us the existence of local solutions to (.) for initial data inE. It remains to prove that the local solutions can be extended to global solutions. Note that the global solutions of (.) may not exist for all initial data inE. However, they exist when the initial dataX¯= (ς¯,U¯,V¯,N¯,H¯) belongs to the set, which is defined as follows.

Definition . The setis composed of all (ς,U,V,N,H)∈Esuch that

(i) (ς,U,V,N,H)∈W,∞(R), (.a)

(ii) yξ≥,Hξ≥,yξ+Hξ>  almost everywhere, and lim

ξ→–∞H(ξ) = , (.b) (iii) yξHξ=yξU

_+U

ξ +y



ξV

 _{almost everywhere, (.c)}

withς(ξ) =y(ξ) –ξ, whereW,∞_{(R) ={f∈C(R)∩L}∞_(R)|f

ξ∈L∞(R)}.

The global existence of the solution for initial data inrelies essentially on the fact that the setis preserved by the flow as the next lemma shows.

Lemma . Given initial dataX¯ = (ς¯,U¯,V¯,N¯,H¯)∈,for some T> ,we consider the local solution X(t) = (ς,U,V,N,H)(t)∈C([,T],E)of system(.)given by Theorem..

We have

(i) X(t)∈for allt∈[,T],

(ii) yξ(t,ξ) > for a.e.t∈[,T]and a.e.ξ∈R,

(iii) limξ→±∞H(t,ξ) =H(,±∞) = for allt∈[,T].

Proof (i) For given initial dataX¯ = (ς¯,U¯,V¯,N¯,H¯)∈E∩[W,∞_(R)]_{, to ensure that the} solutionX= (ς,U,V,N,H) of (.) also belongs toE∩[W,∞_(R)]_{, we have to specify} the initial conditions for (.). Letbe the following set:

=ξ∈R|ς¯ξ(ξ)≤ ¯ςξL∞,U¯ξ(ξ)≤ ¯UξL∞,V¯ξ(ξ)≤ ¯VξL∞,

N¯ξ(ξ)≤ ¯NξL∞,H¯ξ(ξ)≤ ¯HξL∞

.

Note that meas(c) = . For ξ ∈, we take (ςξ,Uξ,Vξ,Nξ,Hξ)(,ξ) = (ς¯ξ,U¯ξ,V¯ξ,N¯ξ,

¯

Hξ)(ξ). Forξ ∈c, we define (ςξ,Uξ,Vξ,Nξ,Hξ)(,ξ) = (, , , , ). We considerU,P

(.c) holds for anyξ ∈and therefore almost everywhere. Consider a fixedξ ∈and drop it in the notation. On the one hand, it follows from (.) that

(yξHξ)t=yξtHξ +yξHξt

=UξHξ+yξ U– P– AU

Uξ– UPxyξ

=UξHξ+ UUξyξ– PUξyξ– AUUξyξ– UPxyξ,

and on the other hand,

yξU+Uξ+yξV

t

= yξyξtU+ yξUUt+ yξyξtV+ yξVVt+ UξUξt

= UUξyξ– UPxyξ +UξHξ– AUUξyξ– PUξyξ.

Thus, (yξHξ)t= (yξU+Uξ+yξV)t. Notice thatyξHξ() = (yξU+Uξ+yξV)(), which

implies thatyξHξ(t) = (yξU+Uξ+yξV)(t) for allt∈[,T]. Thus, (.c) holds. It remains

to prove that the inequalities in (.b) hold. Set t∗=sup{t∈[,T]|yξ(t)≥ for allt∈

[,t]}. Assume thatt∗ <T. Sinceyξ(t) is continuous w.r.t.t, we haveyξ(t∗) = . It

fol-lows from (.c) that Uξ(t∗) = . Furthermore, (.) implies yξt(t∗) =Uξ(t∗) =  and yξtt(t∗) =Uξt(t∗) = _Hξ(t∗). If Hξ(t∗) = , then (yξ,Uξ,Hξ)(t∗) = (, , ), which implies

(yξ,Uξ,Hξ)(t) =  for allt∈[,T] by the uniqueness of the solution of system (.). This

contradicts the fact thatyξ() +Hξ() >  for allξ ∈. IfHξ(t∗) < , thenyξtt(t∗) < .

Since yξ(t∗) =yξt(t∗) = , there exists a neighborhood oft∗ such that yξ(t) <  for

all t∈/{t∗}. This contradicts the definition of t∗. Hence, Hξ(t∗) > . We now have yξtt(t∗) > , which conversely impliesyξ(t) >  for allt∈/{t∗}, which contradicts the fact

thatt∗<T. Thus, we have provedyξ(t)≥ for allt∈[,T]. We now prove thatHξ ≥

for allt∈[,T]. This follows from (.c) whenyξ(t) > . Ifyξ(t) = , thenUξ(t) =  from

(.c). As we have seen,Hξ <  would imply thatyξ(t) <  for some t in a punctured

neighborhood oft, which is impossible. Hence,Hξ ≥ for allt∈[,T]. Now we have yξ(t) +Hξ(t)≥ for allt∈[,T]. Ifyξ(t) +Hξ(t) =  for somet, it then follows that

(yξ,Uξ,Hξ)(t) = , which implies (yξ,Uξ,Hξ)(t) =  for allt∈[,T], which contradicts the

fact thatyξ() +Hξ() >  for allξ∈. Hence,yξ(t) +Hξ(t) > .

(ii) Define the setN={(t,ξ)∈[,T]×R|yξ(t,ξ) = }. It follows from Fubini’s theorem

that

meas(N) =

R

meas(Nξ)dξ=

[,T]

meas(Nt)dt, (.)

whereNξ={t∈[,T]|yξ(t,ξ) = }andNt={ξ∈R|yξ(t,ξ) = }. From the above proof, we

know that for allξ∈,Nξ consists of isolated points that are countable. This means that meas(Nξ) = . It follows from (.), and sincemeas(c) = , that

meas(Nt) =  for almost everyt∈[,T].

This implies thatyξ(t,ξ) >  for almost alltand thereforey(t,ξ) is strictly increasing and

(iii) For any givent∈[,T], sinceHξ(t)≥ andH(t,ξ)∈L∞(R), we know thatH(t,±∞)

exist. We have

H(t,ξ) =H(,ξ) +

t



U– PU(τ,ξ)dτ. (.)

Let ξ → ±∞. Since U, P are bounded in L∞([,T]×R) and limξ→±∞U(t,ξ) =  as U(t,·)∈H_{(R), it then follows from (.) thatH(t,±∞) =H(,±∞) for allt∈[,T]. Since}

¯

X∈, it follows thatH(,±∞) =  for allt∈[,T].

Theorem . For any initial data X¯ = (¯y,U¯,V¯,N¯,H¯)∈,there exists a unique global solution X(t) = (y,U,V,N,H)(t)∈C_(R+_{,E)for system(.).Moreover,for all t≥,if we}

equipwith the topology endowed with the E-norm,then the map St:→×R+defined as

St(X¯) =X(t)

is a continuous semigroup.

Proof Let (ς,U,V,N,H)∈ C([,T],E) be a local solution of (.) with initial data (ς¯,U¯,V¯,N¯,H¯). To obtain the global existence of solutions, it suffices to show that

sup t∈[,T)

ς(t,·),U(t,·),V(t,·),N(t,·),H(t,·)_E<∞. (.)

SinceH(t,ξ) is an increasing function w.r.t.ξfor alltandlimξ→∞H(t,ξ) =limξ→∞H(,ξ),

we havesup_t∈[,T)H(t,·)L∞(R)= ¯HL∞(R)<∞. We consider a fixedt∈[,T) and drop it for simplification. SinceUξ(ξ) =  whenyξ(ξ) = , andyξ(ξ) >  for a.e.ξ, it follows from

(.c) that

U(ξ) = 

ξ

–∞

U ξUξ ξ

dξ= 

{ξ<ξ|yξ(ξ)>}

U ξUξ ξ

dξ

≤

{ξ<ξ|yξ(ξ)>}

yξU+Uξ/yξ ξ

dξ≤

R Hξ ξ

dξ=H(ξ),

which implies

sup t∈[,T)

U(t,·)_L∞≤ sup

t∈[,T)

H(t,·)_L∞_(R)= ¯HL∞(R)<∞,

and therefore

sup t∈[,T)

U(t,·)_L∞<∞.

We can obtain from the governing equation (.) that

ς(t,ξ)≤ς(,ξ)+ sup t∈[,T)

Thus, sup_t∈[,T)ς(t,·)L∞ < ∞. The governing equation (.) also implies that sup_t∈[,T)V(t,·)L∞<∞andsupt∈[,T)N(t,·)L∞<∞.

From the identityHξ= (U+Ux+V)yξ, we can deduce that

U+ Vyξ≤ U+V+ 

yξ≤Hξ+yξ,

which implies that

|Px| ≤

 

_Re–|y(ξ)–y(ξ)|_H

ξ+ U+ V

yξ ξ

dξ ≤ 



_Re–|y(ξ)–y(ξ)|[Hξ+yξ]ξ

dξ

≤C

sup t∈[,T)

H(t,·)_L∞_(R)+ sup

t∈[,T)

ς(t,·)_L∞

<∞.

Therefore,PxL∞<∞. Similarly, we obtainPxL<∞and the bounds hold forP. Let

Z(t) =U(t,·)_L+Uξ(t,·)_L+V(t,·)_L+N(t,·)_L+ςξ(t,·)_L+Hξ(t,·)_L. After taking theL_{-norms on both sides of (.) and (.), we obtain}

Z(t)≤Z() +C

t



Z(τ)dτ.

It follows from Gronwall’s lemma thatsup_t∈[,T)Z(t) <∞, which implies thatSt is a

con-tinuous semigroup by the standard O.D.E. theory.

4 Global solutions for the original system

We transform the global solution of the equivalent system (.) into the global conserva-tive solution of the original system (.) in this section. It suffices to establish the corre-spondence between the Lagrangian equivalent system and the original system.

We first introduce a setGas the set of relabeling functions defined by

G=f is invertible|f –Id andf––Idboth belong toW,∞(R),

whereIddenotes the identity function. For anyα> , we define the subsetsGαofGas

Gα=

f∈G|f–Id_W,∞_(R)+f––Id

W,∞_(R)≤α

,

with a useful property: Iff ∈Gα(α≥), then /( +α)≤fξ ≤ +αalmost everywhere.

Conversely, iff is absolutely continuous,f –Id∈L∞(R) and there existsc≥ such that /c≤fξ ≤calmost everywhere, thenf ∈Gα for someα depending only oncandf – IdL∞(R). We now define the subsetsFandFαofsuch that

F=X= (y,U,V,H)∈|y+H∈G,

Fα=

X= (y,U,V,H)∈|y+H∈Gα

With the above useful property ofGα, it is not hard to prove that the spaceFis preserved

by the governing equation (.).

Notice that the map:G×F→Fgiven by(f,X) =X◦f defines a group action ofG

onF, we then consider the quotient spaceF/GofFw.r.t. the group action. The equivalence relation onFis defined as: for anyX,X∈F, if there existsf ∈Gsuch thatX=X◦f, we claim thatXandXare equivalent. We denote the projection:F→F/Gby(X) = [X]. For anyX= (y,U,V,N,H)∈F, we introduce the mapK:F→Fgiven byK(X) =X◦ (y+H)–. It is not hard to prove thatK(X) =XwhenX∈F, andK(X◦f) =K(X) for any

X∈F andf ∈G. Hence, we can define the mapK˜ :F/G→FasK˜([X]) =K(X) for any representative [X]∈F/GofX∈F. For anyX∈F, we haveK˜ ◦(X) =K(X) =X. Hence,

˜

K◦|F=Id|F. Note that any topology defined onFis naturally transported intoF/G by this isomorphism, that is, if we equipFwith the metric induced by theE-norm,i.e.,

dF(X,X) =X–XEfor allX,X∈F, which is complete, then for any [X], [X]∈F/G, the topology onF/Gis defined by a complete metric given bydF/G([X], [X]) =K(X) – K(X)E.

For any initial dataX¯ ∈F, we denote the continuous semigroup with the solutionX(t) of system (.) byS:F×R+_{→F. As we indicated earlier, Eq. (.) is invariant w.r.t.} relabeling. That is,t> ,St(X◦f) =St(X)◦f for anyX∈Fandf ∈G. Thus, the mapS˜t: F/G→F/Gdefined byS˜t([X]) = [StX] is valid, which generates a continuous semigroup.

To derive the correspondence between the Lagrangian equivalent system and the orig-inal system, we have to consider the spaceD, which characterizes the solutions in the original system:

D=(z,μ)|z∈H(R)×L(R)×L∞(R)andμac= u+ux+v

dx,

wherez= (u,v) andμis a positive finite Radon measure withμacas its absolute continuous part.

We now establish a bijection betweenF/GandDto transport the continuous semigroup obtained in the Lagrangian equivalent system (functions inF/G) into the original system (functions inD).

We first introduce the mappingM, which corresponds to the transformation from the Lagrangian equivalent system into the original system. In the other direction, we obtain the energy densityμin the original system, by pushing forward byythe energy density

Hξdξin the Lagrangian equivalent system, where the push-forwardf#νof a measureνby a measurable functionf is defined by

f#ν(B) =ν f–(B)

for all Borel setB. Let (z,μ) be defined as

z(x) =Z(ξ) for anyξsuch thatx=y(ξ), (.a)

μ=y#(Hξdξ), (.b)

the mapping to any [X]∈F/Gand (z,μ)∈Dgiven by (.a) and (.b), which transforms the Lagrangian equivalent system into the original system.

We are led to the mappingL:D→F/G, which conversely transforms the original system into the Lagrangian equivalent system defined as follows.

Definition . For any (z,μ)∈D, let

y(ξ) =supy|μ(–∞,y) +y<ξ, (.a)

U(ξ) =u◦y(ξ), V(ξ) =v◦y(ξ), N(ξ) =ux◦y(ξ), (.b)

H(ξ) =ξ–y(ξ), (.c)

wherez= (u,v). We defineL(z,μ)∈F/Gas the equivalence class of (y,U,V,N,H).

Remark . Note thatX= (y,U,V,N,H)∈E, which satisfies (.a)-(.c) from the def-inition ofy,U,V,N,Hin (.a)-(.c). Moreover, by the definition (.c), we have that

y+H=Id. Thus,X= (y,U,V,N,H)∈F.

We claim that the transformation from the original system into the Lagrangian equiva-lent system is a bijection.

Theorem . The maps M and L are invertible,that is,

L◦M=IdF/G, M◦L=IdD.

Proof Let [X]∈F/Gbe given. We considerX= (y,U,V,N,H) =K˜([X]) for a represen-tative of [X] and (z,μ) given by (.a) and (.b) for this particularX. From the defini-tion of K˜, we haveX∈F. LetX¯ = (¯y,U¯,V¯,N¯,H¯) be the representative ofL(z,μ) inF given by (.a)-(.c). To deriveL◦M=IdF/G, it suffices to show that (y¯,U¯,V¯,N¯,H¯) =

(y,U,V,N,H). Let

g(x) =supξ∈R|y(ξ) <x. (.)

Using the fact thatyis increasing and continuous, it follows that

y g(x)=x (.)

andy–((–∞,x)) = (–∞,g(x)). From (.b) and sinceH(–∞) = , for anyx∈R, we get

μ (–∞,x)=

y–_((–∞,x))Hξd ξ=

g(x) –∞ Hξd

ξ=H g(x).

SinceX∈Fandy+H=Id, we have

μ (–∞,x)+x=g(x). (.)

From the definition ofy¯, it follows that

¯

For any givenξ∈R, using the fact thatyis increasing and (.), it follows thaty¯(ξ)≤y(ξ). Ify¯(ξ) <y(ξ), there then existsxsuch that¯y(ξ) <x<y(ξ) and (.) implies thatg(x)≥ξ. Conversely, sinceyis increasing, we havex=y(g(x)) <y(ξ), which implies thatg(x) <ξ. This is a contradiction. Hence, we have thaty¯=y. Sincey+H=Id, it follows directly from the definitions thatH¯ =H,U¯ =U,V¯ =V andN¯ =N. Hence,L◦M=Id_F/G.

Let (z,μ)∈Dbe given and (y,U,V,N,H) be the representative ofL(z,μ) inFgiven by (.a)-(.c). Then, let (¯z,μ¯) =M◦L(z,μ). Letgbe defined as before by (.). The same computation that leads to (.) now gives

¯

μ (–∞,x)+x=g(x). (.)

Givenξ∈R, we consider an increasing sequencexiconverging toy(ξ), which is guaranteed

by (.a), and such thatμ((–∞,xi)) +xi<ξ. Letitend to infinity. SinceF(x) =μ((–∞,x))

is lower semi-continuous, we haveμ((–∞,y(ξ))) +y(ξ)≤ξ. Takeξ=g(x) and then we get

μ (–∞,x)+x≤g(x). (.)

By the definition ofg, there exists an increasing sequenceξiconverging tog(x) such that y(ξi) <x. It follows from the definition ofyin (.a) thatμ((–∞,x)) +x≥ξi. Passing to the

limit, we obtainμ((–∞,x)) +x≥g(x) which, together with (.), yields

μ (–∞,x)+x=g(x). (.)

We obtain thatμ¯ =μby comparing (.) and (.). It is clear from the definitions that ¯

z=z. Hence, (¯z,μ¯) = (z,μ) andM◦L=IdD.

Our next task is to transport the topology defined inF/GintoD, 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 distancedDonDas

dD (z,μ), (z¯,μ¯)

=dF/G L(z,μ),L(z¯,μ¯)

,

which makes the bijectionLbetweenDandF/Ginto an isometry. SinceF/Gequipped withdF/Gis a complete metric space, it is not hard to know thatDequipped with the metric dDis also a complete metric space. For eacht∈R, we define the mappingTt:D→Das

Tt=MS˜tL.

Theorem . Given(¯z,μ¯)∈D,if we denote t→(z,μ)(t) =Tt(¯z,μ¯)the corresponding tra-jectory,then z= (u,v)is a weak solution of the two-component Camassa-Holm equations

(.),which constructs a continuous semigroup.Moreover,μis a weak solution of the fol-lowing transport equation:

μt+ (uμ)x= u– Pu

x. (.)

Furthermore,we have

and

μ(t)(R) =μac(t)(R) =z(t) 

H

=u(t)_H+v(t) 

L=μ()(R) for almost all t. (.)

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

Proof To prove thatz= (u,v) is a weak solution of the original system (.), it suffices to show that, for allφ∈C∞(R+×R) with compact support,

R+_×R(–u

φt+uuxφ)(t,x)dx dt= –

R+_×R(Px

φ)(t,x)dx dt,

R+_×R

(–vφt+uvxφ)(t,x)dx dt= –

R+_×R

(v+ )uxφ

(t,x)dx dt,

(.)

wherePxis given by (.). We denote by the solution (y,U,V,N,H)(t) of (.) a

represen-tative ofL(z(t),μ(t)). On the one hand, sincey(t,ξ) is Lipschitz and invertible w.r.t.ξ for almost allt, we can use the change of variablesx=y(t,ξ), then we get

R+_×R(–u

φt+uuxφ)(t,x)dx dt

=

R+_×R

–(Uyξ)(t,ξ)φt t,y(t,ξ)

+ (UUξ)(t,ξ)φ t,y(t,ξ)

dξdt. (.)

Sinceyt=Uandyξt=Uξ, it then follows from (.) that

R+_×R

–Uyξφt(t,y) +UUξφ(t,y)

dξdt

= 

R+_×R

sgn ξ–ξe–|y(ξ)–y(ξ)|_H

ξ+ U+ V

yξ ξ

·φ t,y(ξ)yξ(ξ)dξdξdt. (.)

On the other hand, using the change of variablesx=y(t,ξ) andx=y(t,ξ), and sinceyis an increasing function, we have

–

R+_×R(Px

φ)(t,x)dx dt

= 

R+_×R

sgn ξ–ξe–|y(ξ)–y(ξ)|_u₊ u



x+

 v

_{+v t,y ξ}

·φ t,y(ξ)yξ ξ

yξ(ξ)dξdξdt.

It follows from the identity (.c) that

–

R+_×R(Pxφ)(t,x)dx dt

= 

R+_×R

sgn ξ–ξe–|y(ξ)–y(ξ)|Hξ+ U+ V

yξ ξ

By comparing (.) and (.), we know that

R+_×R

–Uyξφt(t,y) +UUξφ(t,y)

dξdt= –

R+_×R

(Pxφ)(t,x)dx dt.

Thus, the first identity in (.) follows directly from (.) and the second identity in (.) follows in the same way. It is not hard to check thatμ(t) is the solution of (.). From the definitionμin (.b), we can get that

μ(t)(R) =

R

Hξdξ=H(t,∞),

which is constant in time from Lemma .(iii). Thus, we have proved (.). Sinceyξ(t,ξ) >  a.e. for almost everyξ∈R, it then follows from (.c) that

μ(t)(B) =

y–_(B)Hξd ξ=

y–_(B) U _+U

ξ/yξ+V

yξdξ (.)

for any Borel setB. Sinceyis one-to-one andux◦yyξ =Uξ almost everywhere, then (.)

implies that

μ(t)(B) =

B

u+u_x+v(t,x)dx.

Thus, (.) 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 solu-tions which is well posed even when collisions occur in this section, and the variables (y,U,V,N,H) 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 theH

norm finite. Extending the solution beyond wave breaking imposes significant challenge as can be illustrated in the case of multipeakons given by

u(t,x) =

n

i=

pi(t)e–|x–qi(t)|, (.)

where (pi(t),qi(t)) satisfy the explicit system of ordinary differential equations

⎧ ⎨ ⎩

˙

pi=

n

j=pipjsgn(qi–qj)e–|qi–qj|,

˙

qi=

n

j=pje–|qi–qj|.

(.)

Peakons interact in a way similar to that of solitons of the CH equation, and wave breaking may appear when at least two of theqicoincide. Clearly, if theqiremain distinct, system

case, the peakons are traveling in the same direction. However, when two peakons have opposite signs, collisions may occur, and if so, system (.) blows up.

We consider initial dataz¯= (u¯,v¯) given by

⎧ ⎨ ⎩

¯

u(x) =n_i=pie–|x–ξi|,

¯

v(x) =n_i=rie–|x–ξi|.

(.)

Without loss of generality, we assume that thepiandriare all nonzero, and that theξiare

all distinct. From Theorem . we know that there exists a unique and global weak solution with initial data (.), and the aim is to characterize this solution explicitly. We consider the following characterization of multipeakons. The multipeakons are given as continuous solutionsudefined on intervals [xi,xi+] as the solutions of the Dirichlet problem

u–uxx= , u(xi) =ui, u(xi+) =ui+,

where the variablesxidenote the position of the peaks, and the variablesuidenote the

values ofuat the peaks. In the following, we will show that this property persists for con-servative solutions.

Let us defineX¯= (y¯,U¯,V¯,N¯,H¯) as

¯

y(ξ) =ξ, (.a)

¯

U(ξ) =u¯(ξ), (.b)

¯

V(ξ) =v¯(ξ), (.c)

¯

N(ξ) =u¯x(ξ), (.d)

¯

H(ξ) =

ξ

–∞

u_+u

x+v

dx, (.e)

which is a representative ofz= (u,v) in the Lagrangian equivalent system, that is, [X¯] =

L(¯z, (¯u_+u¯

x+v¯)dx). LetA=R\{ξ, . . . ,ξn}. We claim that the functionsU¯,V¯,N¯ andH¯

belong toC_{(A) and even belong toC}∞_{(A), as the next lemma shows.}

Lemma . For given initial dataX¯ = (¯y,U¯,V¯,N¯,H¯)∈F such thatX¯ ∈[C_(A)]_,the

asso-ciated solution X= (y,U,V,W,H)of(.)belongs to C_(R+_{, [C}_(A)]_).

Proof To prove this lemma, one proceeds as in Theorem . by using the contraction ar-gument. The Banach spaceEis replaced by

¯

E=E∩C(A), endowed with the norm

XE¯=XE+y–IdW,∞_(A)+UW,∞_(A)+VW,∞_(A)+NW,∞_(A)+HW,∞_(A).

It suffices to show thatPandPiare Lipschitz from bounded sets ofE¯ intoH(R)∩C(A).

from Lemma . that

Px(X) –Px(X¯)_L∞_(R)≤CX–X¯E≤CX–X¯E¯

for a constant C depending only on CB. From the derivative of Px given by (.) and

Lemma ., we have thatPxis locally Lipschitz fromE¯ into C(A). Similarly, we obtain

the same result forP. We compute the derivative ofPξ andPxξ onAas follows:

Pxξ ξ= –Hξ ξ/ + (Pxyξ–UUξ–Vξ)yξ+ P–U/ –V

yξ ξ,

Pξ ξ= –Hξyξ/ + P–U/ –V

yξ +Pxyξ ξ.

(.)

SincePξ ξandPxξ ξare locally Lipschitz maps fromE¯intoC(A), we have thatPandPxare

locally Lipschitz fromE¯intoC_{(A). The local solution of (.) inE}¯_{then can be obtained} by the standard contraction argument. As we know, as far as global existence is concerned, X_W,∞_(R)does not blow up with initial data inW,∞(R) (see Lemma .(i)). For anyξ∈A, we have that

yξ ξt=Uξ ξ,

Uξ ξt=



Hξ ξ– (Pxyξ–UUξ –Vξ)yξ–

P– U

_–V

yξ ξ,

Vξ ξt= –Vξ ξN– VξNξ–VNξ ξ–Nξ ξ,

Nξ ξt= Uξ+ UUξ–Nξ–NNξ ξ+Vξ+VVξ ξ+Vξ ξ+



Hξyξ (.)

–

P– U

_–V

yξ+Pxyξ ξ,

Hξ ξt= U– P

Uξ ξ– UPxyξ ξ+ UUξ– UξPxyξ

+UHξyξ– UPyξ+Uyξ+ UVyξ.

System (.) is affine w.r.t.yξ ξ,Uξ ξ,Vξ ξ,Nξ ξ,Hξ ξ. Hence, on any interval [,T), we have

Xξ ξ(t,·)_L∞_(A)≤Xξ ξ(,·)_L∞_(A)+C+C

t



Xξ ξ(τ,·)_L∞_(A)dτ,

whereC is a constant depending only onsup_t∈[,T)X_W,∞_(R), which is bounded. Thus XW,∞_(A)does not blow up from Gronwall’s lemma, and therefore the solution is globally

defined inE¯.

Theorem . Let the initial data be given in(.).The solution given by Theorem.

satisfies u–uxx= between the peaks.

Proof Assuming thatyξ(t,ξ)= , we have

ux◦y= Uξ

yξ

, uxx◦y=

Uξ yξ

ξ



yξ

=Uξ ξyξ–yξ ξUξ

y

ξ

Hence,

(u–uxx)◦y= Uyξ–Uξ ξyξ+yξ ξUξ

/y_ξ. (.)

Let

M=Uyξ–Uξ ξyξ+yξ ξUξ. (.)

For a givenξ∈A, differentiating (.) w.r.t.t, we obtain, by using (.), (.) and (.), that

dM/dt= Uyξyξt+Utyξ–Uξ ξtyξ–Uξ ξyξt+yξ ξtUξ+yξ ξUξt

= –Pxyξ+ y



ξUUξ –Hξ ξyξ/ + (Qyξ–UUξ+AUξ–Vξ)yξ

+ P–U/ –Vyξ ξyξ–Uξ ξUξ+Uξ ξUξ+yξ ξ

Hξ/ + U/ +V–P

yξ

= UUξyξ–Hξ ξyξ/ +Hξyξ ξ/ –Vξyξ. (.)

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

yξ ξHξ +yξHξ ξ= yξyξ ξU+ yξUUξ+ UξUξ ξ+ yξyξ ξV+ yξVVξ. (.)

After inserting the value ofyξHξ ξ given by (.) into (.) and multiplying the equation

byyξ, we obtain that

yξ·dM/dt=UUξyξ +Uξyξ ξ–UξUξ ξyξ.

It follows from (.c) and sinceyξt=Uξ that

yξ·dM/dt=yξt·M. (.)

We claim thatM/yξisCin time. Indeed, we have

M yξ

=Uy_ξ –Uξ ξ+ yξ ξUξ

yξ

=Uy_ξ–Uξ ξ+ yξ ξUξ yξ+Hξ

+yξ ξNHξ

yξ+Hξ

=J(X,Xξ,Xξ ξ)

yξ+Hξ

for some polynomialJ. SinceX∈C_(R,E¯_{), we haveX,X}

ξ andXξ ξ areCin time. Since X(t) remains infor allt, from (.b), we haveyξ+Hξ>  and therefore /(yξ+Hξ) isC

in time, which implies thatM/yξ isCin time. For any timetsuch thatyξ(t)= , we have

d dt

M yξ

=Mtyξ–yξtM

y

ξ

= .

Hence,

for some constantK(ξ) independent of time. This leads to

y_ξ(u–uxx)◦y=K(ξ),

which corresponds to the conservation of spatial angular momentum. For the multi-peakons at timet= , we havey(,ξ) =ξ and (u–uxx)(,ξ) =  for allξ∈A. Hence,

M/yξ(t,ξ) =  (.)

for all timetand allξ∈A. Thus, (u–uxx)(t,ξ) = .

For solutions with multipeakon initial data, we have the following result: If yξ(t,ξ)

vanishes at some point ξ¯ in the interval (ξi,ξi+), then yξ(t,ξ) vanishes everywhere

in (ξi,ξi+). Furthermore, for the given initial multipeakon solution z¯(x) = (u¯,v¯)(x) =

(n_i=pie–|x–ξi|,

n

i=rie–|x–ξi|), let (y,U,V,N,H) be the solution of system (.) with

ini-tial data (¯y,U¯,V¯,N¯,H¯) given by (.a)-(.e), then between adjacent peaks, ifxi=y(t,ξi)= xi+ =y(t,ξi+), the solution z(t,x) = (u,v)(t,x) is twice differentiable with respect to the

space variable, and forx∈(xi,xi+), we have that (u–uxx) = .

We now start the derivation of a system of ordinary differential equations for multi-peakons.

For eachi= , , . . . ,n, we have, from (.), that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

dyi/dt=ui, dui/dt= –Pxi, dvi/dt= –(vi+ )uxi,

duxi/dt=ui –uxi/ +vi/ +vi–Pi, dHi/dt=ui – uiPi,

(.)

where (yi,ui,vi,uxi,Hi) = (y,U,V,N,H)(t,ξi),Pi=P(t,ξi),Pxi=Px(t,ξi), respectively. By

us-ing the change of variablesx=y(t,ξ),PiandPxican be rewritten as

Pi= /·

R

e–|yi–x| _u_+u

x/ +v/ +v

dx,

Pxi= –/·

R

sgn(yi–x)e–|yi–x| u+ux/ +v/ +v

dx.

(.)

Forx∈[yi,yi+],i= , , . . . ,n– , we writez= (u,v) as

z(x) =

u(x)

v(x)

=

Aiex+Bie–x Ciex+Die–x

. (.)

The constantsAi,Bi,CiandDidepend onui,ui+,vi,vi+,yiandyi+and read

Ai= e–¯yi



¯

ui cosh(δyi)

+ δui

sinh(δyi)

, Bi=

ey¯i 

¯

ui cosh(δyi)

– δui

sinh(δyi)

,

Ci= e–¯yi



_¯

vi cosh(δyi)

+ δvi

sinh(δyi)

, Di=

e¯yi 

_¯

vi cosh(δyi)

– δvi

sinh(δyi)

,

where

¯

yi=



(yi+yi+), δyi= 

(yi–yi+), ¯

ui=



(ui+ui+), δui= 

(ui–ui+), ¯

vi=



(vi+vi+), δvi= 

(vi–vi+).

(.)

The constantsAi,Bi,CiandDiuniquely determinez= (u,v) on the interval [yi,yi+]. Thus,

we can compute

δHi=Hi+–Hi=

yi+

yi

u+u_x+vdx

= u¯_itanh(δyi) + δuicoth(δyi) +v¯itanh(δyi)

+δv_icoth(δyi) + CiDiδyi. (.)

At this point, we can get some more understanding of what is happening at the time of collision. Lett∗be the time when the two peaks located atyiandyi+collide,i.e., such that

lim_t→t∗δyi(t) = . The functionz= (u,v) remains continuous because the solutionz= (u,v)

remains inH_×[L_∩L∞_{] for all time, thus we havelim}

t→t∗δui(t) =limt→t∗δvi(t) = . Still, Ai,Bi,CiandDimay have a finite limit whenttends tot∗. However, the first derivative

blows up, which implieslim_t→t∗Bi= –limt→t∗Ai=∞andlimt→t∗Di= –limt→t∗Ci=∞.

Thusδuiandδvitend to zero but slower thanδyi. In fact, if we letttend tot∗in (.), to

first order inδyi, we obtain

δu_i +δv_i =δHi·

δyi+◦(δyi),

which implies thatδuiandδvitend to zero at the same rate as√δyi. We now turn to the

computation ofPigiven by (.). Let us writez= (u,v) as

z(t,x) = u(t,x),v(t,x) =

_n

j=

Ajex+Bje–x

χ(yj,yj+)(x),

n

j=

Cjex+Dje–x

χ(yj,yj+)(x)

.

We have sety= –∞,yn+=∞,u=un+= ,v=vn+= ,A=ue–y,B= ,An= , Bn=uneynandC=ve–y,D= ,Cn= ,Dn=vneyn. We have

u+ u



x+

 v

_+v=

n

j=

 A



jex+AjBj+

 B



je–x

+ C



jex+CjDj+

 D



je–x+Cjex+Dje–x

χ(yj,yj+). (.)

Let

kij=

⎧ ⎨ ⎩