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Zhaowei Sheng, Shaoyong Lai, Yuan Ma, Xuanjun Luo, "The Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation", Abstract and Applied Analysis, vol. 2012, Article ID 693010, 21 pages, 2012. https://doi.org/10.1155/2012/693010
The Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation
The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the space × ; provided that its initial value belongs to the space . A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.
In , the author investigated the following weakly dissipative Camassa-Holm model where . When , (1.1) becomes the classical Camassa-Holm equation . The authors in  obtained the local well-posedness of the solution for the model by using the Kato theorem. A necessary and sufficient condition of the blow-up of the solution and some criteria guaranteeing the blow-up of the solution are derived. The blow-up rate of the solution is discussed. It is also shown in  that the equation has global strong solutions, and these strong solutions decay to zero as time goes to infinite provided the potentials associated to their initial data are of one sign. However, the existence of global weak solutions in the space is not discussed in paper . This will constitute the objective of this work.
More relevant for the present paper, here we state several works on the global weak solution for the Camassa-Holm and Degasperis-Procesi equations. The existence and uniqueness results for global weak solutions to the Camassa-Holm equation have been proved by Constantin and Escher , Constantin and Molinet , and Danchin [5, 6]. Xin and Zhang  proved that the global existence of the weak solution for the Camassa-Holm equation in the energy space without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under certain conditions on the solution . Coclite et al.  investigated the global weak solutions for a generalized hyperelastic-rod wave equation or a generalized Camassa-Holm equation. 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 . Under the sign condition imposed on the initial value, Yin et al.  proved the existence and uniqueness results of global weak solution for a nonlinear shallow water equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases. For other dynamic properties about various generalized Camassa-Holm models and other partial differential equations, the reader is referred to [11–16].
The aim of this work is to study the existence of global weak solutions for (1.1) in the space under the assumption . The limits of viscous approximations for the equation are used to establish the existence of the global weak solution. Here, we should mention that up to now, there have been no global existence results for weak solutions to the generalized Camassa-Holm equation (1.1).
The rest of this paper is as follows. The main result is given in Section 2. In Section 3, we state the viscous problem and give a corresponding well-posedness result. An upper bound, higher integrability estimate, and 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 (1.1) is proved.
2. Main Result
Consider the Cauchy problem for (1.1) which is equivalent to where operator . For a fixed , it has
In fact, problem (2.1) satisfies the following conservation law:
Definition 2.1. A continuous function is said to be a global weak solution to the Cauchy problem (2.2) if(i);
(iii) satisfies (2.2) in the sense of distributions and takes on the initial value pointwise.
The main result of this paper is stated as follows.
Theorem 2.2. Assume . Then, the Cauchy problem (2.1) or (2.2) has a global weak solution in the sense of Definition 2.1. Furthermore, this weak solution satisfies the following properties.(a)There exists a positive constant depending on and 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 , and such that the space higher integrability estimate holds
3. Viscous Approximations
Defining and setting the mollifier with and , we know that for any (see ). In fact, suitably choosing the mollifier, we have
The existence of a weak solution to the Cauchy problem (2.2) will be established by proving compactness of a sequence of smooth functions solving the following viscous problem:
The beginning point of our analysis is the following well-posedness result for problem (3.3).
Lemma 3.1. Provided that , then for any , there exists a unique solution to the Cauchy problem (3.3). Moreover, for any , it holds that or
Proof. For any and , we have . From Theorem 2.1 in  or Theorem 2.3 in , we conclude that problem (3.3) has a unique solution for an arbitrary .
We know that the first equation in system (3.3) is equivalent to the form from which we derive that which completes the proof.
Lemma 3.2. Let , , and . Then, there exists a positive constant depending only on , , and , but independent of , such that the space higher integrability estimate holds where is the unique solution of problem (3.3).
Proof. The proof is a variant of the proof presented by Xin and Zhang  (also see Coclite et al. ). Let be a cut-off function such that and
Consider the map , , and observe that
Differentiating the first equation of problem (3.3) with respect to the variable and writing and for simplicity, we obtain Multiplying (3.15) by , using the chain rule and integrating over , we have From (3.14), we get Using the Hölder inequality, (3.8) and (3.13), yields Integration by parts gives rise to From (3.13), (3.20), and the Hölder inequality, we have Using (3.13) and Lemma 3.1, we have From (3.13), it has Applying (3.8), the Hölder inequality, Lemma 3.1 and , we have From (3.24), we obtain The inequalities (3.15)–(3.23) and (3.25) derive the desired result (3.10).
Lemma 3.3. There exists a positive constant depending only on and such that where is the unique solution of system (3.3).
Proof. For simplicity, setting , we have
The inequality (3.26) is proved in Lemma 3.2 (see (3.24)). Now, we prove (3.27). Using
and (3.8) result in (3.27).
Applying the Tonelli theorem, (3.26) and (3.27), we get Since we have which, together with Lemma 3.1, we get (3.29) and (3.30). The proof of Lemma 3.3 is completed.
Lemma 3.4. Assume is the unique solution of problem (3.3). There exists a positive constant depending only on and such that the following one-sided norm estimate on the first order spatial derivative holds
Proof. From (3.9) and Lemma 3.3, we know that there exists a positive constant depending only on and such that . Therefore,
Let be the solution of Letting , due to the comparison principle for parabolic equations, we get
Using , we derive that where and . Setting , we obtain Letting , we have . Due to the comparison principle for ordinary differential equations, we get for all . Therefore, by this and (3.39), the estimate (3.36) is proved.
Lemma 3.5. There exists a sequence tending to zero and a function , such that where is the unique solution of (3.3).
Proof. For fixed , using Lemmas 3.1 and 3.3, and
where depends on . Hence, is uniformly bounded in and (3.42) follows.
Observe that, for each ,
Moreover, is uniformly bounded in and . Then, (3.43) is valid.
Lemma 3.6. There exists a sequence tending to zero and a function such that for each
Throughout this paper we use overbars to denote weak limits (the space in which these weak limits are taken is with ).
Lemma 3.7. There exists a sequence tending to zero and two functions , such that for each and . Moreover,
Proof. (3.48) and (3.49) are direct consequence of Lemmas 3.1 and 3.2. Inequality (3.50) is valid because of the weak convergence in (3.49). Finally, (3.51) is a consequence of the definition of , Lemma 3.5 and (3.48).
In the following, for notational convenience, we replace the sequence , and by , and , separately.
Using (3.48), we conclude that for any convex function with bounded, Lipschitz continuous on and any we get
Multiplying (3.9) by yields
Lemma 3.8. For any convex with bounded, Lipschitz continuous on , it holds that in the sense of distributions on . Here, and denote the weak limits of and in , , respectively.
Lemma 3.10. In the sense of distributions on , it holds that
The next lemma contains a generalized formulation of (3.57).
Lemma 3.11. For any with , it has in the sense of distributions on .
Proof. Let be a family of mollifiers defined on . Denote . The is the convolution with respect to the variable. Multiplying (3.57) by , it has Using the boundedness of and letting in the above two equations, we obtain (3.58).
4. Strong Convergence of and Existence for (1.1)
Following the ideas in  or , in this section, we improve the weak convergence of in (3.48) to strong convergence, and then we have an existence result for problem (3.3). Generally speaking, we will derive a “transport equation” for the evolution of the defect measure . Namely, we will prove that the measure is zero initially then it will continue to be zero at all later times .
Lemma 4.1. Assume . It holds that
Lemma 4.3 (Coclite et al. ). Let . Then, for each , it holds that
Lemma 4.4. Assume . Then, for almost all
Proof. For an arbitrary , we let be sufficiently large (see Lemma 3.4). Using (3.54) minus (3.58), and the entropy (see Lemma 4.2) results in By the increasing property of and the convexity of , from (3.50), we have It follows from Lemma 4.3 that In view of Remark 3.9. Let . Applying (3.56) gives rise to In , it has From (4.6)–(4.10), we know that the following inequality holds in Integrating the resultant inequality over yields for almost all . Sending and using Lemma 4.2, we complete the proof.
Lemma 4.5. For any and , it holds that
Proof. Let . Subtracting (3.58) from (3.54) and using entropy , we deduce Since , we get By the convexity of , it has Using Remark 3.9 and Lemma 4.3 yields Inserting the inequalities from (4.15) to (4.18) into (4.14) gives Integrating the above inequality over , we obtain