The existence of global weak solutions to the Cauchy problem for a weakly dissipative Camassa-Holm equation is established in the space under the assumption that the initial value only belongs to the space . 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:global weak solution; Camassa-Holm type equation; existence
In this work, we investigate the Cauchy problem for the nonlinear model
where , , is a polynomial with order n, N and m are nonnegative integers. When , , , Eq. (1) is the standard Camassa-Holm equation [1-3]. In fact, the nonlinear term 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 , Constantin and Molinet  and Danchin [8,9] under the sign condition imposing on the initial value. Xin and Zhang  established the global existence of a weak solution for the Camassa-Holm equation in the energy space without imposing the sign conditions on the initial value, and the uniqueness of the weak solution was obtained under certain conditions on the solution . Under the sign condition for the initial value, Yin and Lai  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  obtained the existence of a local weak solution for Eq. (1) in the lower-order Sobolev space with . 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. 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, , 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 was established in . 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 under the assumption . 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  and Coclite et al..
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
Consider the Cauchy problem for Eq. (1)
which is equivalent to
where the operator . For a fixed , one has
In fact, as proved in , problem (2) satisfies the following conservation law:
Now we introduce the definition of a weak solution to Cauchy problem (2) or (3).
Definition 1 A continuous function is said to be a global weak solution to Cauchy problem (3) if
(iii) 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 . Then Cauchy problem (2) or (3) has a global weak solution in the sense of Definition 1. Furthermore, the weak solution satisfies the following properties.
(a) There exists a positive constant depending on and the coefficients of Eq. (1) such that the following one-sided norm estimate on the first-order spatial derivative holds:
(b) Let , and , . Then there exists a positive constant depending only on , γ, T, a, b and the coefficients of Eq. (1) such that the following space higher integrability estimate holds:
3 Viscous approximations
and setting the mollifier with and , we know that for any , (see Lai and Wu ). In fact, choosing the mollifier properly, we have
The existence of a weak solution to Cauchy problem (3) will be established by proving the compactness of a sequence of smooth functions solving the following viscous problem:
Now we start our analysis by establishing the following well-posedness result for problem (9).
Lemma 3.1Provided that , for any , there exists a unique solution to Cauchy problem (9). Moreover, for any , it holds that
Proof For any and , we have . From Theorem 2.3 in , we conclude that problem (9) has a unique solution for an arbitrary .
We know that the first equation in system (9) is equivalent to the form
from which we derive that
which completes the proof. □
From Lemma 3.1 and (8), we have
Differentiating the first equation of problem (9) with respect to x and writing , we obtain
Lemma 3.2Let , and , . Then there exists a positive constant depending only on , γ, T, a, band the coefficients of Eq. (1), but independent of ε, such that the space higher integrability estimate holds
where is the unique solution of problem (9).
Lemma 3.3There exists a positive constantCdepending only on and the coefficients of Eq. (1) such that
where is the unique solution of system (9).
Due to strong similarities with the proof of Lemma 5.1 presented in Coclite et al., we do not prove Lemma 3.3 here.
Lemma 3.4Assume that is the unique solution of (9). For an arbitrary , there exists a positive constantCdepending only on and the coefficients of Eq. (1) such that the following one-sided 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 and the coefficients of Eq. (1) such that . Therefore,
Let be the solution of
where is the value of when . From the comparison principle for parabolic equations, we get
Using (14) and , we derive that
where and . Setting , we obtain
Letting , we have . From the comparison principle for ordinary differential equations, we get for all . Therefore, by this and (22), the estimate (19) is proved. □
Lemma 3.5For , there exists a sequence tending to zero and a function such that, for each , it holds that
where is the unique solution of (9).
Lemma 3.6There exists a sequence tending to zero and a function such that for each ,
The proofs of Lemmas 3.5 and 3.6 are similar to those of Lemmas 5.2 and 5.3 in . 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 with ).
Lemma 3.7There exists a sequence tending to zero and two functions , such that
for each and . Moreover,
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 , Lemma 3.5 and (28). □
In the following, for notational convenience, we replace the sequence , and by , and , respectively.
Using (28), we conclude that for any convex function with being bounded and Lipschitz continuous on R and for any , we get
Multiplying Eq. (15) by yields
Lemma 3.8For any convex with being bounded and Lipschitz continuous on R, it holds that
in the sense of distributions on . Here and denote the weak limits of and in , , respectively.
Proof In (34), by the convexity of η, (14), Lemmas 3.5, 3.6 and 3.7, taking limit for gives rise to the desired result. □
Remark 3.9 From (28) and (29), we know that
almost everywhere in , where , for . From Lemma 3.4 and (28), we have
where C is a constant depending only on and the coefficients of Eq. (1).
Lemma 3.10In the sense of distributions on , it holds that
Proof Using (15), Lemmas 3.5 and 3.6, (28), (29) and (31), the conclusion (38) holds by taking limit for in (15). □
The next lemma contains a generalized formulation of (38).
Lemma 3.11For any with , it holds that
in the sense of distributions on .
Proof Let be a family of mollifiers defined on R. Denote , where the ⋆ is the convolution with respect to x variable. Multiplying (38) by yields
Using the boundedness of η, and letting in the above two equations, we obtain (39). □
4 Strong convergence of
Now, we will prove the strong convergence result, i.e.,
which is one of key statements to derive that is a global weak solution required in Theorem 1.
Lemma 4.1Assume . It holds that
Lemma 4.2If , for each , it holds that
and , , .
Lemma 4.3Let . Then for each ,
Lemma 4.4Assume . Then for almost all ,
Lemma 4.5For any , and , it holds that
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..
Lemma 4.6Assume . Then it has
Applying Lemmas 4.4 and 4.5 gives rise to
From Lemma 3.6, we know that there exists a constant , depending only on , such that
By Remark 3.9 and Lemma 4.3, one has
Thus, by the convexity of the map , we get
Using (51) derives
Since is concave, choosing M large enough, we have
Then, from (50) and (55), we have
By using the Gronwall inequality, for each , we have
By the Fatou lemma, Remark 3.9 and (30), letting , we obtain
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. □
The authors declare that they have no competing interests.
The article is a joint work of three authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
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).
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