Global weak conservative solutions of the μ-Camassa–Holm equation

Abstract

In this paper we consider global weak conservative solutions of a μ-Camassa–Holm equation. By employing a technique of change of variables on the Lagrangian variable, we successfully construct global weak conservative solutions of the μ-Camassa–Holm equation.

Introduction

In this paper, we are concerned with the Cauchy problem of the following periodic μ-Camassa–Holm (μCH) equation :

\begin{aligned} \left \{ \textstyle\begin{array}{l} \mu(u_{t})-u_{xxt}=-2\mu(u)u_{x}+2u_{x}u_{xx}+uu_{xxx},\quad t> 0, x\in S^{1}=\mathbb{R}/\mathbb{Z},\\ u(0,x)=u_{0}(x),\quad x\in S^{1}=\mathbb{R}/\mathbb{Z}, \end{array}\displaystyle \right . \end{aligned}
(1)

where $$u(t,x)$$ is a real-valued spatially periodic function and $$\mu (u)=\int_{S^{1}}u(t,x)\,dx$$ denotes its mean. This equation was introduced in  as an integrable equation arising in the study of the diffeomorphism group of the circle, and it can be viewed as a natural generalization of the famous Camassa–Holm (CH) equation 

$$u_{t}-u_{txx}+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}=0.$$
(2)

Moreover, μCH equation (1) has some properties similar to CH equation (2). For instance, like CH equation (2), μCH equation (1) is also completely integrable in the sense that it has a Lax pair representation, a bi-Hamiltonian structure, an infinite sequence of conservation laws, and peaked soliton solutions (called peakons). μCH equation (1) also arises geometrically as equations for geodesic flow in the context of the diffeomorphism group of the circle $$\operatorname{Diff}(S^{1})$$ endowed with a right-invariant Riemannian metric induced by the μ inner product $$\langle u,v\rangle=\mu(u)\mu(v)+\int_{S^{1}}u_{x}v_{x}\,dx$$.

Recently, well-posedness and wave breaking of the CH equation have been studied extensively. More precisely, it was shown in [3, 4] that the CH equation has a unique solution in $$C([0, T ),H^{s}(\mathbb{R}))$$ with $$s>\frac{3}{2}$$. It was proved in  that the CH equation is locally well posed in Besov spaces $$C([0, T ),B^{s}_{p,r}(\mathbb{R}))$$ with $$s > \max(\frac{3}{2} , 1 + \frac{1}{p} )$$, $$1\leq p\leq+\infty$$, $$1\leq r< +\infty$$. It was shown in  that the strong solutions of CH equation will blow up in finite time when the slope of initial data was bounded by a negative quantity. Also, well-posedness and wave breaking of the μCH equation were studied and some results about the local and global existence and uniqueness in Sobolev spaces were established in . Moreover, the μCH equation admits a similar phenomenon as the CH equation, that is, the solutions break down in finite time when the initial data satisfy some proper conditions. In particular, it was shown in  that μCH equation (1) also admits peakons $$u(x,t)=c\phi(x-ct)$$, where $$\phi(x)=\frac{1}{26}(12x^{2}+23)$$ for $$x\in[-\frac{1}{2},\frac{1}{2}]$$, and ϕ is extended periodically to the real line. Chen, Lenells, and Liu  showed that the above periodic peakons are orbitally stable.

In order to study solutions of CH equation (2) after the time of break-down, Bressan and Constantin [11, 12] introduce some new variables to resolve the singularities formed by wave breaking, and they successfully obtain a semigroup of global solutions by returning to the original variables. Holden and Raynaud [13, 14] also study solutions of CH equation (2) after the formation of singularities, but they introduce a set of different new variables, and this can establish a bijective map between Eulerian and Lagrangian coordinates for CH equation (2). Motivated by the works of Lenells , who constructed global weak conservative solutions of the so-called periodic Hunter–Saxton (HS) equation

$$u_{txx}+2u_{x}u_{xx}+uu_{xxx}=0,$$
(3)

Lee  constructs global weak conservative solutions of CH equation (2) by using transformation of variables on the Lagrangian variable η:

$$\rho=\sqrt{\eta_{x}}.$$
(4)

It is interesting that the author obtains the same global results for CH equation (2) as those presented by Bressan and Constantin  and also improves the corresponding results on the persistence of the smoothness of Lagrangian trajectories presented in . More recently, following closely the ideas used in [12, 14], Tiğlay  studies the periodic Cauchy problem for μCH equation (1) and proves the existence and uniqueness of conservative weak solutions.

In this paper, we investigate the global weak conservative solutions of μCH equation (1). We closely follow the main ideas of Lee  in the study of conservative weak solutions of CH equation (2). We show that the wave breaking singularities of μCH equation (1) can be removed by introducing the same transformation (4) about the new Lagrangian variable ρ, even though the structure of the μCH equation is in some sense different from the CH equation and the HS equation. Also, we obtain the global results of the μCH equation due to Tiğlay  in much simpler way.

The paper is organized as follows. In Sect. 2, we use the Lagrangian variable ρ defined by (4) to change μCH equation (1) into an abstract ODE system which admits global solutions. In Sect. 3, we construct global weak conservative solutions of the original μCH equation by using this global solution in ρ variables. Moreover, by putting together all the previous results obtained, we give and prove the main result of the paper. Finally, some conclusions and discussions are presented in Sect. 4.

The setting

Firstly, the Cauchy problem for periodic μCH equation (1) can be rewritten in the weak form

\begin{aligned} \left \{ \textstyle\begin{array}{lll} & u_{t}+uu_{x}=-(\mu-\partial_{x}^{2})^{-1}\partial_{x} (2\mu(u)u+\frac {1}{2}u_{x}^{2} ), \\ & u(0,x)=u_{0}(x). \end{array}\displaystyle \right . \end{aligned}
(5)

By taking methods analogous to those in , we define the following flow equation in terms of Lagrangian variable η:

$$\frac{\partial\eta}{\partial t}=u\bigl(t, \eta(t,x)\bigr),$$
(6)

then Cauchy problem (5) can be written in η variable as follows:

\begin{aligned} \left \{ \textstyle\begin{array}{l} \eta_{tt}=- [\varLambda^{-1}\partial_{x} (2\mu\eta_{t}\circ\eta ^{-1}+\frac{1}{2}(\frac{\eta_{tx}}{\eta_{x}}\circ\eta^{-1})^{2} ) ]\circ\eta, \\ \eta(0,x)=x,\qquad\eta_{t}(0,x)=u_{0}(x), \end{array}\displaystyle \right . \end{aligned}
(7)

where $$\varLambda^{-1}=(\mu-\partial_{x}^{2})^{-1}$$ is the operator defined by

$$\varLambda^{-1}u(x)= \int_{S^{1}}g(x-y)u(y)\,dy,$$

and the kernel $$g(x)$$ of the elliptic operator $$\varLambda^{-1}$$ is given by $$g(x)=\frac {1}{2}(x^{2}-|x|)+\frac{13}{12}$$.

Now our main aim is to write Eq. (7) in terms of the new variable ρ defined by (4). For this purpose, we take full advantage of the conserved quantities $$\mu(u)$$ of the μCH equation to finish our work. More precisely, note that $$\mu(u)=\int_{0}^{1}u(t,x)\,dx=\int_{0}^{1}u_{0}(x)\, dx$$ is a constant determined by the initial condition $$u_{0}$$, so in view of (4) and (6) we have

$$\mu= \int_{0}^{1}(u\circ\eta)\cdot\eta_{x}\,dx= \int_{0}^{1}\eta_{t}\eta_{x} \,dx= \int _{0}^{1}\eta_{t}\rho^{2} \,dx.$$
(8)

By carrying out exactly similar arguments as those given in , we can determine $$\eta_{t}$$ in terms of μ, ρ, and $$\rho _{t}$$. That is,

$$\eta_{t}:= G(\mu,\rho,\rho_{t}) (t,x)= \int_{0}^{x}2\rho\rho_{t}\,dy+\mu- \int _{0}^{1} \int_{0}^{y}2\rho\rho_{t}\,dz \rho^{2}\,dy.$$
(9)

Also, notice that $$\rho=\sqrt{\eta_{x}}$$ and $$\eta_{tx}=2\rho\rho_{t}$$, then we have $$\frac{\eta_{tx}^{2}}{2\eta_{x}}=2\rho_{t}^{2}$$.

Next, we begin to rewrite Eq. (7) in ρ variables. By differentiating Eq. (7) with respect to x, we have

$$\eta_{ttx}=2\mu\eta_{t}\eta_{x}+ \frac{\eta_{tx}^{2}}{2\eta_{x}}- \biggl[ \int _{0}^{1} \biggl(2\mu\eta_{t} \eta_{y}+\frac{\eta_{ty}^{2}}{2\eta_{y}} \biggr)\,dy \biggr]\cdot\eta_{x},$$
(10)

where we have applied the chain rule and the identity $$\varLambda ^{-1}(-\partial_{x}^{2})=1-\mu$$. For convenience, we denote that

$$F(\mu,\rho,\rho_{t}) (t):= \int_{0}^{1}2\mu\rho^{2}G+2 \rho_{t}^{2}\,dx,$$
(11)

so Eq. (10) becomes

$$2\rho\rho_{tt}=2\mu G\rho^{2}-F \rho^{2},$$
(12)

since $$\eta_{ttx}-\frac{\eta_{tx}^{2}}{2\eta_{x}}=2\rho\rho_{tt}$$. Thus, we can derive the following Cauchy problem with respect to ρ variables:

\begin{aligned} \left \{ \textstyle\begin{array}{l} \rho_{tt}=\rho(\mu G-\frac{1}{2}F), \\ \rho(0,x)=1,\qquad\rho_{t}(0,x)=\frac{1}{2}u_{0}'(x). \end{array}\displaystyle \right . \end{aligned}
(13)

If we assume that ρ and $$\rho_{t}$$ are just functions in $$L^{2}(S^{1})$$ satisfying Eq. (13), then this equation can be used to describe the integral curve of the vector field $$(\rho,\rho_{t})\mapsto f(\rho,\rho_{t}):=\rho (\mu G(\rho,\rho_{t})-\frac{1}{2}F(\rho,\rho_{t}) )$$. According to the smoothness of f with respect to ρ and $$\rho_{t}$$ variables, we can conclude that the solutions of (13) are locally well posed.

Proposition 2.1

The planar system

\begin{aligned} \left \{ \textstyle\begin{array}{l} \frac{d\rho}{dt}=\rho_{t}, \\ \frac{d\rho_{t}}{dt}=f(\rho,\rho_{t}):=\rho (\mu G(\rho,\rho_{t})-\frac {1}{2}F(\rho,\rho_{t}) ) \end{array}\displaystyle \right . \end{aligned}
(14)

with initial conditions$$\rho(0,\cdot)=1$$and$$\rho_{t}(0,\cdot)=\frac{1}{2}u_{0}'$$describes the flow of a$$C^{\infty}$$vector field on$$TL^{2}(S^{1}) = L^{2}(S^{1}) \times L^{2}(S^{1})$$, and there exists$$T>0$$such that it has a unique solution$$(\rho,\rho_{t})\in C ([0,T];TL^{2}(S^{1}) )$$.

Proof

Firstly, note that

$$\frac{\partial f}{\partial\rho_{t}}=\rho\biggl(\mu\frac{\partial G}{\partial \rho_{t}}-\frac{1}{2} \frac{\partial F}{\partial\rho_{t}}\biggr).$$

It is easy to see that G is smooth with respect to $$\rho_{t}$$ since it is linear in $$\rho_{t}$$. On the other hand, we have

$$\frac{\partial F}{\partial\rho_{t}}= \int_{0}^{1}2\mu\rho^{2} \frac{\partial G}{\partial\rho_{t}}+4\rho_{t}\,dx,$$

and so we can observe that F is also smooth with respect to $$\rho_{t}$$. Thus, it implies that the vector field f is smooth with respect to $$\rho_{t}$$. Similarly, we can check that the vector field f is smooth with respect to ρ. □

Note that G is periodic on $$S^{1}$$ if and only if $$\int_{0}^{1}\rho^{2}\, dx=0$$. The following proposition shows that we need to restrict ρ to be on the unit sphere $$U:= \{\rho\in L^{2}(S^{1}) : \| \rho\|_{L^{2}} = 1\}$$ if we want to obtain the global well-posedness of Cauchy problem (13).

Proposition 2.2

Let$$f(\rho,\rho_{t})$$be the vector field on$$TL^{2}(S^{1})$$defined by (13). If$$\rho\in U$$, then$$\int_{0}^{1}\rho\rho_{t}\,dx=0$$for all time. This implies thatfrestricts on the tangent bundleTU.

Proof

When $$t=0$$, we have $$\int_{0}^{1}\rho(0,x)\rho_{t}(0,x)=\int_{0}^{1}\frac {1}{2}u_{0}'(x)\,dx=\frac{1}{2}(u_{0}(1)-u_{0}(0))=0$$. Note that F does not depend on x and the assumption that $$\rho\in U$$, so we have $$\int _{0}^{1}\rho^{2} F\,dx=F$$. Thus, in view of (11), we have

$$\frac{d}{dt} \int_{0}^{1}\rho\rho_{t}\,dx= \int_{0}^{1}\rho_{t}^{2}+\rho \rho_{tt}\, dx= \int_{0}^{1}\rho_{t}^{2}+ \rho^{2}\biggl(\mu G-\frac{1}{2}F\biggr)\,dx=\frac{1}{2}F- \frac{1}{2}F=0.$$

This proves that $$\int_{0}^{1}\rho\rho_{t}\,dx=0$$ for all time since the integral is initially zero. □

It is well known that the $$\dot{H}^{1}$$ energy $$\int_{0}^{1}u_{x}^{2}\,dx$$ is a conserved quantity for μCH equation (1). Next, we show that the $$\dot{H}^{1}$$ energy for the corresponding ρ equation is also conserved.

Proposition 2.3

For theρequation, we have

$$\frac{d}{dt} \int_{0}^{1}4\rho_{t}^{2}\,dx=0 .$$
(15)

Proof

Recall that

$$G_{x}=2\rho\rho_{t}.$$

Then we have

\begin{aligned} \frac{d}{dt} \int_{0}^{1}4\rho_{t}^{2}\,dx =& \int_{0}^{1}8\rho_{t}\rho_{tt} \,dx \\ =& \int_{0}^{1}4\rho\rho_{t}(2\mu G-F)\,dx \\ =& 2\mu \int_{0}^{1}2GG_{x}\,dx-2F \int_{0}^{1}2\rho\rho_{t}\,dx \\ =& 2\mu \int_{0}^{1}d\bigl(G^{2}\bigr)- 2F \int_{0}^{1}d(G) =0 , \end{aligned}

since G is periodic with respect to x. □

Now, we can show that the solution of Eq. (13) is global when ρ is restricted on the unit sphere.

Proposition 2.4

If$$\rho\in U$$, then the flow described by (13) exists for all timet.

Proof

In view of Hölder’s inequality, we have

$$\vert 2\mu G-F \vert \leq2\mu \vert G \vert + \vert F \vert \leq4\mu \Vert \rho \Vert _{L^{2}} \Vert \rho_{t} \Vert _{L^{2}}+4\mu \Vert \rho \Vert _{L^{2}}^{3} \Vert \rho_{t} \Vert _{L^{2}}+2 \Vert \rho_{t} \Vert _{L^{2}}^{2}.$$
(16)

It is easily seen that $$|2\mu G-F|$$ is uniformly bounded since $$\|\rho\| _{L^{2}}=1$$ and $$\|\rho_{t}\|_{L^{2}}$$ is uniformly bounded by the energy conservation. This implies that the right-hand side of (13) is uniformly bounded in $$(\rho,\rho_{t})$$. Thus, by Wintner’s theorem from the ODE theory , we can conclude that the solutions of Cauchy problem (13) exist for all time. □

Global weak solutions of the μCH equation

We want to construct global weak solutions for the original μCH equation (5) from the global solution ρ. To do this, we first prove the following important lemma.

Lemma 3.1

LetGbe defined by (9). Then$$G_{t}=-H$$, where

\begin{aligned} H(t,x)&= \int_{0}^{1} \biggl( \int_{y}^{x}\rho^{2}\,dz \biggr) \bigl(2\mu G\rho^{2}+2\rho _{t}^{2} \bigr)\,dy\\&\quad+ \frac{1}{2} \int_{x}^{1}2\mu\rho^{2}G+2 \rho_{t}^{2}\,dy-\frac {1}{2} \int_{0}^{x}2\mu\rho^{2}G+2 \rho_{t}^{2}\,dy. \end{aligned}

Proof

Note that

$$\frac{\partial H}{\partial x}=\rho^{2}F-\bigl(2\mu\rho^{2}G+2 \rho_{t}^{2}\bigr)$$

and

$$H\rho^{2}=\varPhi_{x},$$

where

$$\varPhi(t,x)= \int_{0}^{1} \biggl[ \biggl( \int_{y}^{x}\rho^{2}\,dz \biggr)^{2}- \biggl\vert \int_{y}^{x}\rho ^{2}\,dz \biggr\vert \biggr] \bigl(\mu\rho^{2}G+\rho_{t}^{2} \bigr)\,dy.$$

Denote

$$c(t):=\mu- \int_{0}^{1} \int_{0}^{x}2\rho\rho_{t}\,dy \rho^{2}\,dx.$$
(17)

By differentiating G with respect to t, we have

\begin{aligned} G_{t} =& \int_{0}^{x}2\rho_{t}^{2}+2\rho \rho_{tt}\,dy+c'(t) \\ =& \int_{0}^{x}2\rho_{t}^{2}+2\mu \rho^{2}G-\rho^{2}F\,dy+c'(t) \\ =& - \int_{0}^{x}\frac{\partial H}{\partial y}\,dy+c'(t) \\ =& -H(t,x)+H(t,0)+c'(t) . \end{aligned}

So we only need to show that $$H(t,0)+c'(t)=0$$. However, we have

\begin{aligned} c'(t) =& - \int_{0}^{1} \int_{0}^{x}2\rho_{t}^{2}+2\rho \rho_{tt}\,dy\rho^{2}\,dx \\ =& - \int_{0}^{1} \int_{0}^{x}2\rho_{t}^{2}+2\mu \rho^{2}G-\rho^{2}F\,dy\rho^{2}\,dx \\ =& - \int_{0}^{1}\bigl(2\rho_{t}^{2}+2 \mu\rho^{2}G-\rho^{2}F\bigr) \int_{y}^{1}\rho^{2}\,dx\,dy \\ =& \int_{0}^{1}\frac{\partial H}{\partial y} \int_{y}^{1}\rho^{2}\,dx\,dy \\ =& \int_{0}^{1} \biggl[\rho^{2} \int_{0}^{x}\frac{\partial H}{\partial y}\,dy \biggr]\,dx \\ =& \int_{0}^{1} \bigl[\rho^{2}H(t,x)- \rho^{2}H(t,0) \bigr]\,dx \\ =& \int_{0}^{1}\varPhi_{x}\,dx-H(t,0) \int_{0}^{1}\rho^{2}\,dx \\ =& -H(t,0) , \end{aligned}

since $$\int_{0}^{1}\rho^{2}\,dx=1$$ and Φ is periodic with respect to x. Therefore, we have $$G_{t}=-H$$. □

Define

$$K(t,x):= \int_{0}^{x}\rho^{2}\,dy+tu_{0}(0)- \int_{0}^{t} \int_{0}^{\tau}H(s,0)\,ds\,d\tau.$$

From Lemma 3.1, we can see that $$-H(t,0)=c'(t)$$, where $$c(t)$$ is defined by (17). So the relation $$\frac{\partial K}{\partial t}=G$$ holds true between K and G. Next, we will show that $$(K,G)$$ satisfies the second order equation (7).

Following [15, 16], we decompose $$S^{1}$$ by $$S^{1}=N\cup A\cup Z$$ where

$$\begin{gathered} N:=\bigl\{ x\in S^{1}: K_{x}=0,\text{i.e.}, \rho(t,x)=0\bigr\} , \\ A:=\bigl\{ x\in S^{1}: K_{x}>0, \text{i.e.}, \rho(t,x)>0\bigr\} ,\end{gathered}$$

and $$\operatorname{meas}(Z)=0$$. Performing the same argument as in , we can obtain that the set N has a measure zero for almost all time. And then we have the following lemma.

Lemma 3.2

$$\int_{N}\rho_{t}^{2}\,dy=0$$for almost all time$$t\in\mathbb{R}_{+}$$.

Proof

The proof is very similar to that of Lemma 8 in , and hence is omitted. □

Lemma 3.2 shows that $$\rho=0$$ is satisfied only on a set of measure zero for almost all time. Notice that $$K_{x}=\rho^{2}$$, we can derive that K is generically a homeomorphism except for a set of measure zero. In other words, $$K^{-1}$$ is well defined for almost all time.

Proposition 3.3

For almost all timet, $$(K,G)$$satisfies the following equation:

$$G_{t}=- \biggl[\varLambda^{-1} \partial_{x} \biggl(2\mu G\circ K^{-1}+\frac {1}{2}\biggl( \frac{G_{x}}{K_{x}}\circ K^{-1}\biggr)^{2} \biggr) \biggr]\circ K.$$
(18)

Proof

It suffices to show that $$G_{t}=-H$$ is equivalent to Eq. (7) in the weak sense. Note that K is a diffeomorphism on the set A, so we have

\begin{aligned} G_{t} =& -H=- \int_{S^{1}} \biggl[ \int_{y}^{x}\rho^{2}\,dz-\frac{1}{2} \operatorname {sgn} \biggl( \int_{y}^{x}\rho^{2}\,dz \biggr) \biggr] \bigl(2\mu\rho^{2}G+2\rho _{t}^{2} \bigr)\,dy \\ =& - \int_{A} \biggl[K(x)-K(y)-\frac{1}{2}\operatorname {sgn} \bigl(K(x)-K(y)\bigr) \biggr]\biggl(2\mu K_{x}G+\frac{G_{x}^{2}}{2K_{x}} \biggr)\,dy+ \int_{N}\rho _{t}^{2}\,dy \\ =& - \int_{K(A)} \biggl[K(x)-y-\frac{1}{2}\operatorname{sgn} \bigl(K(x)-y\bigr) \biggr] \biggl(2\mu G\circ K^{-1}+\frac{1}{2} \biggl(\frac{G_{x}}{K_{x}}\circ K^{-1}\biggr)^{2} \biggr)\,dy \\ &{}+ \int_{N}\rho_{t}^{2}\,dy \\ =& -\partial_{x} \biggl[ \int_{K(A)} \biggl(\frac{(x-y)^{2}}{2}-\frac { \vert x-y \vert }{2}+ \frac{13}{12} \biggr) \biggl(2\mu G\circ K^{-1}+\frac {1}{2} \biggl(\frac{G_{x}}{K_{x}}\circ K^{-1}\biggr)^{2} \biggr)\,dy \biggr]\circ K \\ &{}+ \int _{N}\rho_{t}^{2}\,dy . \end{aligned}

In view of that $$\operatorname{meas} (K(A) )=1$$, we can observe that the first integral of the above is equivalent to the right-hand side of Eq. (7). Moreover, we have $$\int_{N}\rho_{t}^{2}\,dy=0$$ for almost all time t from Lemma 3.2. Therefore, for almost all time t, $$(K,G)$$ satisfies Eq. (18). □

Next, we check that the velocity field u satisfying flow equation (6) is well defined in $$H^{1}$$ for all time. More precisely, we can obtain the following proposition. It should be remarked here that the proofs for this proposition are omitted since it is exactly the same argument as in .

Proposition 3.4

Let$$(K,G)$$be a weak solution ofμCH equation (7) in the Lagrangian form. Then the velocity field$$u\in C(\mathbb{R}_{+}, H^{1}(S^{1}))\cap \operatorname{Lip}(\mathbb{R}_{+}, L^{2}(S^{1}))$$is well defined by

$$u\bigl(t, K(t,x)\bigr)=G(t,x),\quad(t,x)\in[0,\infty)\times S^{1}.$$
(19)

Now, we are in a position to state the main result of this paper.

Theorem 3.5

The functionudefined by (19) is the global weak solution to the Cauchy problem for periodicμCH equation (5). Moreover, this solution satisfies the following property: $$\int _{S^{1}}u_{x}^{2}(t,x)\,dx=\int_{S^{1}}(u_{0}'(x))^{2}\,dx$$.

Proof

We want to show that the function u defined by (19) satisfies Eq. (5) in the weak sense. That is, it suffices to prove that

$$\int_{S^{1}\times R_{+}}(u_{t}+uu_{x})\varphi\,dx\,dt= \int_{S^{1}\times R_{+}}-p_{x}\varphi\,dx\,dt$$
(20)

for all $$\varphi\in C_{0}^{\infty}(S^{1}\times R_{+})$$, where $$p=\varLambda ^{-1}(2\mu(u) u+\frac{1}{2}u_{x}^{2})$$. By virtue of (19), we have

\begin{aligned} \int_{S^{1}\times R_{+}}(u_{t}+uu_{x})\varphi\,dx\,dt =& \int_{S^{1}\times R_{+}}-u\varphi_{t}+uu_{x}\varphi\,dx \,dt \\ =& \int_{S^{1}\times R_{+}}-GK_{x}\varphi_{t}\circ K+GG_{x}\varphi\circ K\,dx\,dt. \end{aligned}

Note that $$K_{t}=G$$ and $$K_{tx}=G_{x}$$, so we have

\begin{aligned}& (GK_{x}\varphi\circ K)_{t}- \bigl(G^{2}\varphi\circ K\bigr)_{x} \\& \quad=G_{t}K_{x}\varphi\circ K+GK_{tx}\varphi\circ K+GK_{x}\varphi_{t}\circ K+GK_{x}( \varphi_{x}\circ K)K_{t}\\& \qquad{}-2GG_{x}\varphi\circ K-G^{2}(\varphi_{x}\circ K)K_{x} \\& \quad= G_{t}K_{x}\varphi\circ K+GG_{x}\varphi\circ K+GK_{x}\varphi_{t}\circ K+G^{2}K_{x}( \varphi_{x}\circ K)\\& \qquad{}-2GG_{x}\varphi\circ K-G^{2}( \varphi_{x}\circ K)K_{x} \\& \quad= G_{t}K_{x}\varphi\circ K-GG_{x}\varphi\circ K+GK_{x}\varphi_{t}\circ K. \end{aligned}

Thus, in view of that φ has a compact support, we can get

$$\int_{S^{1}\times R_{+}}-GK_{x}\varphi_{t}\circ K+GG_{x}\varphi\circ K\,dx\, dt= \int_{S^{1}\times R_{+}}G_{t}K_{x}\varphi\circ K\,dx\,dt.$$

On the other hand, from Proposition 3.3 we know that $$G_{t}=- [\varLambda^{-1}\partial_{x} (2\mu G\circ K^{-1} +\frac {1}{2}(\frac{G_{x}}{K_{x}}\circ K^{-1})^{2} ) ]\circ K$$ for almost all t. Also, note that $$u=G\circ K^{-1}$$ and $$u_{x}=\frac {G_{x}}{K_{x}}\circ K^{-1}$$. Therefore, we have

$$\int_{S^{1}\times R_{+}}G_{t}K_{x}\varphi\circ K\,dx\,dt= \int_{S^{1}\times R_{+}}-p_{x}\varphi\,dx\,dt.$$

So we complete the proof of (20).

Next, we show that $$\int_{S^{1}}u_{x}^{2}(t,x)\,dx$$ is a conserved quantity for μCH equation (5). Note that $$K_{x}=\rho^{2}$$ and $$G_{x}=2\rho\rho_{t}$$, then

\begin{aligned} \int_{S^{1}}u_{x}^{2} \,dx =& \int_{K(A)} \biggl(\frac{G_{x}}{K_{x}}\circ K^{-1} \biggr)^{2} \,dx \\ =& \int_{A}\frac{G_{x}^{2}}{K_{x}}\,dx \\ =& \int_{S^{1}}4\rho_{t}^{2}\,dx \end{aligned}

is conserved by virtue of Proposition 2.3. This proves that $$\int_{S^{1}}u_{x}^{2}(t,x)\,dx=\int_{S^{1}}(u_{0}'(x))^{2}\,dx$$. □

Conclusions

In this paper, we successfully construct global weak conservative solutions of the μCH equation by applying the so-called Lagrangian change of variable technique. We believe that this technique could be extended to study other generalized μ-type equations with different nonlocal nonlinearities which have similar remarkable properties as the μCH equation. More recently, by virtue of an isometric embedding of the group of diffeomorphism group, Gallouët and Vialard  have rewritten the CH equation as an incompressible Euler equation on the cone manifold. Following this geometric point of view developed in , we think that the μCH equation can be also embedded in the incompressible Euler equation on a non-compact Riemannian manifold. We leave these for future research.

References

1. 1.

Khesin, B., Lenells, J., Misiolek, G.: Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms. Math. Ann. 342(3), 617–656 (2008)

2. 2.

Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993)

3. 3.

Constantin, A., Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51(5), 475–504 (1998)

4. 4.

Rodriguez-Blanco, G.: On the Cauchy problem for the Camassa–Holm equation. Nonlinear Anal. 46(3), 309–327 (2001)

5. 5.

Danchin, R.: A few remarks on the Camassa–Holm equation. Differ. Integral Equ. 14(8), 953–988 (2001)

6. 6.

Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50(2), 321–362 (2000)

7. 7.

Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 26(2), 303–328 (1998)

8. 8.

Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243 (1998)

9. 9.

Lenells, J., Misiołek, G., Tiğlay, F.: Integrable evolution equations on spaces of tensor densities and their peakon solutions. Commun. Math. Phys. 299, 129–161 (2010)

10. 10.

Chen, M., Lenells, J., Liu, Y.: Stability of the μ-Camassa–Holm peakons. J. Nonlinear Sci. 23, 97–112 (2013)

11. 11.

Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183(2), 215–239 (2007)

12. 12.

Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5(1), 1–27 (2007)

13. 13.

Holden, H., Raynaud, X.: Periodic conservative solutions of the Camassa–Holm equation. Ann. Inst. Fourier (Grenoble) 58(3), 945–988 (2008)

14. 14.

Holden, H., Raynaud, X.: Dissipative solutions for the Camassa–Holm equation. Discrete Contin. Dyn. Syst. 24(4), 1047–1112 (2009)

15. 15.

Lenells, J.: Weak geodesic flow and global solutions of the Hunter–Saxton equation. Discrete Contin. Dyn. Syst. 18(4), 643–656 (2007)

16. 16.

Lee, J.M.: Geometric approach on the global conservative solutions of the Camassa–Holm equation. J. Geom. Phys. 142, 137–150 (2019)

17. 17.

McKean, H.: Fredholm determinants and the Camassa–Holm hierarchy. Commun. Pure Appl. Math. 56(5), 638–680 (2003)

18. 18.

Tiğlay, F.: Conservative weak solutions of the periodic Cauchy problem for μHS equation. J. Math. Phys. 56, Article ID 021504 (2015)

19. 19.

Hartman, P.: Ordinary Differential Equations. Birkhäuser, Basel (1982)

20. 20.

Gallouët, T., Vialard, F.X.: The Camassa–Holm equation as an incompressible Euler equation: a geometric point of view. J. Differ. Equ. 264(7), 4199–4234 (2018)

Acknowledgements

The author thanks the referees for their constructive suggestions.

Not applicable.

Funding

This research of the first author was partially supported by the Natural Science Foundation of Hunan Province (No. 2018JJ2272), by the Scientific Research Fund of Hunan Provincial Education Department (No. 2018C0721), and Doctoral Research Fund of Hunan University of Arts and Science. The second author was supported by the National Natural Science Foundation of China (No. 11671107), and the Natural Science Foundation of Guangxi (No. 2015GXNSFGA139004).

Author information

Authors

Contributions

Two authors cooperated to complete the work. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Xijun Deng.

Ethics declarations

Competing interests

The authors declare that there are no competing interests. 