# The existence of global weak solutions for a weakly dissipative Camassa-Holm equation in ${H}^{1}(R)$

- Shaoyong Lai
^{1}Email author, - Nan Li
^{1}and - Yonghong Wu
^{2}

**2013**:26

https://doi.org/10.1186/1687-2770-2013-26

© Lai et al.; licensee Springer 2013

**Received: **15 November 2012

**Accepted: **27 January 2013

**Published: **12 February 2013

## Abstract

The existence of global weak solutions to the Cauchy problem for a weakly dissipative Camassa-Holm equation is established in the space $C([0,\mathrm{\infty})\times R)\cap {L}^{\mathrm{\infty}}([0,\mathrm{\infty});{H}^{1}(R))$ under the assumption that the initial value ${u}_{0}(x)$ only belongs to the space ${H}^{1}(R)$. The limit of viscous approximations, a one-sided super bound estimate and a space-time higher-norm estimate for the equation are established to prove the existence of the global weak solution.

**MSC:**35G25, 35L05.

## Keywords

## 1 Introduction

where $\lambda \ge 0$, $\beta \ge 0$, $f(u)$ is a polynomial with order *n*, *N* and *m* are nonnegative integers. When $f(u)=2ku+\frac{3}{2}{u}^{2}$, $\lambda =0$, $\beta =0$, Eq. (1) is the standard Camassa-Holm equation [1–3]. In fact, the nonlinear term $-\lambda {u}^{2N+1}+\beta {u}^{2m}{u}_{xx}$ can be regarded as a weakly dissipative term for the Camassa-Holm model (see [4, 5]). Here we coin (1) a weakly dissipative Camassa-Holm equation.

To link with previous works, we review several works on global weak solutions for the Camassa-Holm and Degasperis-Procesi equations. The existence and uniqueness results for global weak solutions of the standard Camassa-Holm equation have been proved by Constantin and Escher [6], Constantin and Molinet [7] and Danchin [8, 9] under the sign condition imposing on the initial value. Xin and Zhang [10] established the global existence of a weak solution for the Camassa-Holm equation in the energy space ${H}^{1}(R)$ without imposing the sign conditions on the initial value, and the uniqueness of the weak solution was obtained under certain conditions on the solution [11]. Under the sign condition for the initial value, Yin and Lai [12] proved the existence and uniqueness results of a global weak solution for a nonlinear shallow water equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases. Lai and Wu [13] obtained the existence of a local weak solution for Eq. (1) in the lower-order Sobolev space ${H}^{s}(R)$ with $1\le s\le \frac{3}{2}$. For other meaningful methods to handle the problems relating to dynamic properties of the Camassa-Holm equation and other partial differential equations, the reader is referred to [14–19]. Coclite *et al.* [20] used the analysis presented in [10, 11] and investigated global weak solutions for a generalized hyperelastic-rod wave equation (or a generalized Camassa-Holm equation), namely, $\lambda =0$, $\beta =0$ in Eq. (1). The existence of a strongly continuous semigroup of global weak solutions for the generalized hyperelastic-rod equation with any initial value in the space ${H}^{1}(R)$ was established in [20]. Up to now, the existence result of the global weak solution for the weakly dissipative Camassa-Holm equation (1) has not been found in the literature. This constitutes the motivation of this work.

The objective of this work is to study the existence of global weak solutions for the Eq. (1) in the space $C([0,\mathrm{\infty})\times R)\cap {L}^{\mathrm{\infty}}([0,\mathrm{\infty});{H}^{1}(R))$ under the assumption ${u}_{0}(x)\in {H}^{1}(R)$. The key elements in our analysis include some new *a priori* one-sided upper bound and space-time higher-norm estimates on the first-order derivatives of the solution. Also, the limit of viscous approximations for the equation is used to establish the existence of the global weak solution. Here we should mention that the approaches used in this work come from Xin and Zhang [10] and Coclite *et al.* [20].

The rest of this paper is as follows. The main result is given in Section 2. In Section 3, we present a viscous problem of Eq. (1) and give a corresponding well-posedness result. An upper bound, a higher integrability estimate and some basic compactness properties for the viscous approximations are also established in Section 3. Strong compactness of the derivative of the viscous approximations is obtained in Section 4, where the main result for the existence of Eq. (1) is proved.

## 2 Main result

Now we introduce the definition of a weak solution to Cauchy problem (2) or (3).

**Definition 1**A continuous function $u:[0,\mathrm{\infty})\times R\to R$ is said to be a global weak solution to Cauchy problem (3) if

- (i)
$u\in C([0,\mathrm{\infty})\times R)\cap {L}^{\mathrm{\infty}}([0,\mathrm{\infty});{H}^{1}(R))$;

- (ii)
${\parallel u(t,\cdot )\parallel}_{{H}^{1}(R)}\le {\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$;

- (iii)
$u=u(t,x)$ satisfies (3) in the sense of distributions and takes on the initial value pointwise.

The main result of this paper is stated as follows.

**Theorem 1**Assume ${u}_{0}(x)\in {H}^{1}(R)$. Then Cauchy problem (2) or (3) has a global weak solution $u(t,x)$ in the sense of Definition 1. Furthermore, the weak solution satisfies the following properties.

- (a)There exists a positive constant ${c}_{0}$ depending on ${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$ and the coefficients of Eq. (1) such that the following one-sided ${L}^{\mathrm{\infty}}$ norm estimate on the first-order spatial derivative holds:$\frac{\partial u(t,x)}{\partial x}\le \frac{4}{t}+{c}_{0},\phantom{\rule{1em}{0ex}}\mathit{\text{for}}\phantom{\rule{0.1em}{0ex}}(t,x)\in [0,\mathrm{\infty})\times R.$(5)
- (b)Let $0<\gamma <1$, $T>0$ and $a,b\in R$, $a<b$. Then there exists a positive constant ${c}_{1}$ depending only on ${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$,
*γ*,*T*,*a*,*b*and the coefficients of Eq. (1) such that the following space higher integrability estimate holds:${\int}_{0}^{t}{\int}_{a}^{b}{\left|\frac{\partial u(t,x)}{\partial x}\right|}^{2+\gamma}\phantom{\rule{0.2em}{0ex}}dx\le {c}_{1}.$(6)

## 3 Viscous approximations

Now we start our analysis by establishing the following well-posedness result for problem (9).

**Lemma 3.1**

*Provided that*${u}_{0}\in {H}^{1}(R)$,

*for any*$\sigma \ge 3$,

*there exists a unique solution*${u}_{\epsilon}\in C([0,\mathrm{\infty});{H}^{\sigma}(R))$

*to Cauchy problem*(9).

*Moreover*,

*for any*$t>0$,

*it holds that*

*Proof* For any $\sigma \ge 3$ and ${u}_{0}\in {H}^{1}(R)$, we have ${u}_{\epsilon ,0}\in C([0,\mathrm{\infty});{H}^{\sigma}(R))$. From Theorem 2.3 in [21], we conclude that problem (9) has a unique solution ${u}_{\epsilon}\in C([0,\mathrm{\infty});{H}^{\sigma}(R))$ for an arbitrary $\sigma >3$.

which completes the proof. □

*x*and writing $\frac{\partial {u}_{\epsilon}}{\partial x}={q}_{\epsilon}$, we obtain

**Lemma 3.2**

*Let*$0<\gamma <1$, $T>0$

*and*$a,b\in R$, $a<b$.

*Then there exists a positive constant*${c}_{1}$

*depending only on*${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$,

*γ*,

*T*,

*a*,

*b*

*and the coefficients of Eq*. (1),

*but independent of ε*,

*such that the space higher integrability estimate holds*

*where* ${u}_{\epsilon}={u}_{\epsilon}(t,x)$ *is the unique solution of problem* (9).

The proof is similar to that of Proposition 3.2 presented in Xin and Zhang [10] (also see Coclite *et al.* [20]). Here we omit it.

**Lemma 3.3**

*There exists a positive constant*

*C*

*depending only on*${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$

*and the coefficients of Eq*. (1)

*such that*

*where* ${u}_{\epsilon}={u}_{\epsilon}(t,x)$ *is the unique solution of system* (9).

Due to strong similarities with the proof of Lemma 5.1 presented in Coclite *et al.* [20], we do not prove Lemma 3.3 here.

**Lemma 3.4**

*Assume that*${u}_{\epsilon}={u}_{\epsilon}(t,x)$

*is the unique solution of*(9).

*For an arbitrary*$T>0$,

*there exists a positive constant*

*C*

*depending only on*${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$

*and the coefficients of Eq*. (1)

*such that the following one*-

*sided*${L}^{\mathrm{\infty}}$

*norm estimate on the first*-

*order spatial derivative holds*:

*Proof*From (15) and Lemma 3.3, we know that there exists a positive constant

*C*depending only on ${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$ and the coefficients of Eq. (1) such that ${\parallel {Q}_{\epsilon}(t,x)\parallel}_{{L}^{\mathrm{\infty}}(R)}\le C$. Therefore,

Letting $F(t)=\frac{4}{t}+2\sqrt{{M}_{0}}$, we have $\frac{dF(t)}{dt}+\frac{1}{4}{F}^{2}(t)-{M}_{0}=\frac{4\sqrt{{M}_{0}}}{t}>0$. From the comparison principle for ordinary differential equations, we get $f(t)\le F(t)$ for all $t>0$. Therefore, by this and (22), the estimate (19) is proved. □

**Lemma 3.5**

*For*${u}_{0}\in {H}^{1}(R)$,

*there exists a sequence*${\{{\epsilon}_{j}\}}_{j\in N}$

*tending to zero and a function*$u\in {L}^{\mathrm{\infty}}([0,\mathrm{\infty});{H}^{1}(R))\cap {H}^{1}([0,T]\times R)$

*such that*,

*for each*$T\ge 0$,

*it holds that*

*where* ${u}_{\epsilon}={u}_{\epsilon}(t,x)$ *is the unique solution of* (9).

**Lemma 3.6**

*There exists a sequence*${\{{\epsilon}_{j}\}}_{j\in N}$

*tending to zero and a function*$Q\in {L}^{\mathrm{\infty}}([0,\mathrm{\infty})\times R)$

*such that for each*$1<p<\mathrm{\infty}$,

The proofs of Lemmas 3.5 and 3.6 are similar to those of Lemmas 5.2 and 5.3 in [20]. Here we omit their proofs.

Throughout this paper, we use overbars to denote weak limits (the space in which these weak limits are taken is ${L}_{\mathrm{loc}}^{r}([0,\mathrm{\infty})\times R)$ with $1<r<\frac{3}{2}$).

**Lemma 3.7**

*There exists a sequence*${\{{\epsilon}_{j}\}}_{j\in N}$

*tending to zero and two functions*$q\in {L}_{\mathrm{loc}}^{p}([0,\mathrm{\infty})\times R)$, $\overline{{q}^{2}}\in {L}_{\mathrm{loc}}^{r}([0,\mathrm{\infty})\times R)$

*such that*

*for each*$1<p<3$

*and*$1<r<\frac{3}{2}$.

*Moreover*,

*and*

*Proof* (28) and (29) are a direct consequence of Lemmas 3.1 and 3.2. Inequality (30) is valid because of the weak convergence in (29). Finally, (31) is a consequence of the definition of ${q}_{\epsilon}$, Lemma 3.5 and (28). □

In the following, for notational convenience, we replace the sequence ${\{{u}_{{\epsilon}_{j}}\}}_{j\in N}$, ${\{{q}_{{\epsilon}_{j}}\}}_{j\in N}$ and ${\{{Q}_{{\epsilon}_{j}}\}}_{j\in N}$ by ${\{{u}_{\epsilon}\}}_{\epsilon >0}$, ${\{{q}_{\epsilon}\}}_{\epsilon >0}$ and ${\{{Q}_{\epsilon}\}}_{\epsilon >0}$, respectively.

*R*and for any $1<p<3$, we get

**Lemma 3.8**

*For any convex*$\eta \in {C}^{1}(R)$

*with*${\eta}^{\prime}$

*being bounded and Lipschitz continuous on R*,

*it holds that*

*in the sense of distributions on* $[0,\mathrm{\infty})\times R$. *Here* $\overline{q\eta (q)}$ *and* $\overline{{\eta}^{\prime}(q){q}^{2}}$ *denote the weak limits of* ${q}_{\epsilon}\eta ({q}_{\epsilon})$ *and* ${q}_{\epsilon}^{2}{\eta}^{\prime}({q}_{\epsilon})$ *in* ${L}_{\mathrm{loc}}^{r}([0,\mathrm{\infty})\times R)$, $1<r<\frac{3}{2}$, *respectively*.

*Proof* In (34), by the convexity of *η*, (14), Lemmas 3.5, 3.6 and 3.7, taking limit for $\epsilon \to 0$ gives rise to the desired result. □

**Remark 3.9**From (28) and (29), we know that

where *C* is a constant depending only on ${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$ and the coefficients of Eq. (1).

**Lemma 3.10**

*In the sense of distributions on*$[0,\mathrm{\infty})\times R$,

*it holds that*

*Proof* Using (15), Lemmas 3.5 and 3.6, (28), (29) and (31), the conclusion (38) holds by taking limit for $\epsilon \to 0$ in (15). □

The next lemma contains a generalized formulation of (38).

**Lemma 3.11**

*For any*$\eta \in {C}^{1}(R)$

*with*${\eta}^{\prime}\in {L}^{\mathrm{\infty}}(R)$,

*it holds that*

*in the sense of distributions on* $[0,\mathrm{\infty})\times R$.

*Proof*Let ${\{{\omega}_{\delta}\}}_{\delta}$ be a family of mollifiers defined on

*R*. Denote ${q}_{\delta}(t,x):=(q(t,\cdot )\star {\omega}_{\delta})(x)$, where the ⋆ is the convolution with respect to

*x*variable. Multiplying (38) by ${\eta}^{\prime}({q}_{\delta})$ yields

Using the boundedness of *η*, ${\eta}^{\prime}$ and letting $\delta \to 0$ in the above two equations, we obtain (39). □

## 4 Strong convergence of ${q}_{\epsilon}$

*i.e.*,

which is one of key statements to derive that $u(t,x)$ is a global weak solution required in Theorem 1.

**Lemma 4.1**

*Assume*${u}_{0}\in {H}^{1}(R)$.

*It holds that*

**Lemma 4.2**

*If*${u}_{0}\in {H}^{1}(R)$,

*for each*$M>0$,

*it holds that*

*where*

*and* ${\eta}_{M}^{+}(\xi ):={\eta}_{M}(\xi ){\chi}_{[0,+\mathrm{\infty})}(\xi )$, ${\eta}_{M}^{-}(\xi ):={\eta}_{M}(\xi ){\chi}_{(-\mathrm{\infty},0]}(\xi )$, $\xi \in R$.

**Lemma 4.3**

*Let*$M>0$.

*Then for each*$\xi \in R$,

The proofs of Lemmas 4.1, 4.2 and 4.3 can be found in [10] or [20].

**Lemma 4.4**

*Assume*${u}_{0}\in {H}^{1}(R)$.

*Then for almost all*$t>0$,

We do not provide the proofs of Lemmas 4.4 and 4.5 since they are similar to those of Lemmas 6.4 and 6.5 in Coclite *et al.* [20].

**Lemma 4.6**

*Assume*${u}_{0}\in {H}^{1}(R)$.

*Then it has*

*Proof*

*M*large enough, we have

which completes the proof. □

*Proof of the main result*Using (8), (10) and Lemma 3.5, we know that the conditions (i) and (ii) in Definition 1 are satisfied. We have to verify (iii). Due to Lemma 4.2 and Lemma 4.6, we have

From Lemma 3.5, (27) and (58), we know that *u* is a distributional solution to problem (3). In addition, inequalities (5) and (6) are deduced from Lemmas 3.2 and 3.4. The proof of Theorem 1 is completed. □

## Declarations

### Acknowledgements

Thanks are given to referees whose comments and suggestions were very helpful for revising our paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).

## Authors’ Affiliations

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