Abstract
Assuming that the initial value belongs to the space , we prove the existence of global weak solutions for a weakly dissipative hyperelastic rod wave equation in the space . The limit of the viscous approximation for the equation is used to establish the existence.
1. Introduction
This paper focuses on the study of the weakly dissipative model where , is a polynomial with order , and is a nonnegative integer. If ,  and  , (1) becomes the Camassa-Holm equation [1]. When    and   , (1) is turned into the hyperelastic rod wave equation [2]. Since the terms and appear in the equation, here we call (1) a weakly dissipative hyperelastic rod wave equation.
In order to link with previous researches in the field, we state here several works on the existence of global weak solution for the Camassa-Holm equation (CH) and a generalized hyperelastic rod wave equation (or a generalized Camassa-Holm equation). Under the sign condition of the initial value, Constantin and Escher [3] obtained the existence and uniqueness results for the global weak solutions of Camassa-Holm equation (also see Constantin and Molinet [4]). Xin and Zhang [5] proved the existence of the global weak solution for the Camassa-Holm equation in the energy space without adding the sign conditions on the initial value. Coclite et al. [6] used the analysis in [5] and established the existence of global weak solutions for a generalized hyperelastic rod wave equation (or a generalized Camassa-Holm equation); namely, in (1). For the global or local solutions of the Camassa-Holm equation and other partial differential equations, the reader is referred to [7–12] and the references therein.
The objective of this paper is to extend parts of works made by Coclite et al. [6]. We investigate the existence of global weak solutions for the weakly dissipative hyperelastic rod wave equation (1) in the space . Using the limit of viscous approximations for the equation and several estimates derived from the equation itself, we establish the existence of the global weak solution by merely assuming the initial value in the space .
This paper is organized as follows. The main result is stated in Section 2. In Section 3, we present the viscous problem and establish several estimates of the problem. Namely, we give proofs of an upper bound, a higher integrability estimate, and some basic compactness properties for the viscous approximations. Strong compactness about the derivative of solutions for the viscous approximations is proved in Section 4, where the proof of the main result is finished.
2. Main Result
We write the Cauchy problem for (1): which is equal to the form where the operator and Using problem (2), we derive the conservation law
Here we adopt the definition of global weak solution presented in [5].
Definition 1. A continuous function is said to be a global weak solution for the Cauchy problem (2) or (3) if(a); (b); (c) satisfies (3) in the sense of distributions and takes on the initial value pointwise.
Now we state the main result of this paper.
Theorem 2. If , then the Cauchy problem (2) or (3) has a global weak solution in the sense of Definition 1. In addition, the weak solution has the following properties.
There exists a positive constant depending on and the coefficients of (1) such that
Assume that , , and . Then there exists a positive constant depending only on , , , , , and the coefficients of (1) such that
3. Viscous Approximations
Consider We choose the mollifier with and . We know that for any , (see [7]). In fact, we have where is a positive constant.
To prove the existence of global weak solutions to the Cauchy problem (3), we will establish compactness of a sequence of smooth functions satisfying the viscous problem
For simplicity, throughout this paper, we let denote any positive constant which is independent of parameter .
Now we give the well-posedness result to problem (10).
Lemma 3. If , for any , there is a unique solution to the Cauchy problem (10). In addition, it holds that or
Proof . If and , we know . Using Theorem 2.1 in [6] or Theorem 2.3 in [10], we derive that problem (10) has a unique solution .
Applying the first equation in problem (10), we derive
from which we obtain
The proof is completed.
Using Lemma 3, we get Differentiating the first equation of problem (10) with respect to and writing , we obtain
Lemma 4. Let , , and . Then there exists a positive constant depending only on , , , and the coefficients of (1), but independent of , such that the space higher integrability estimate holds: where is the unique solution of problem (10).
Proof. The proof is a variant of the proof presented in Xin and Zhang [5] (or see Coclite et al. [6]). Let be a cut-off function such that and
Considering the map , and observing that
we have
Differentiating the first equation of problem (10) with respect to and setting and for simplicity, we obtain
Multiplying (22) by and integrating over , we get
Using (21) yields
Applying the Hölder inequality, (15) and (20) give rise to
Integrating by parts, we have
From (20), (26), and the Hölder inequality, we have
Using (20) and Lemma 3 derives
It follows from (20) that
In fact, we have
Applying (15), the Hölder inequality, Lemma 3, and , we have
where is a constant independent of .
From (30) and (31), we know that there exists a positive constant depending on , but independent of , such that
which results in
From inequalities (22)–(29) and (33), we obtain (17).
Lemma 5. There exists a positive constant depending only on and the coefficients of (1) such that
Proof. Writing , we get
Inequality (34) is proved in Lemma 4 (see (32)). Now we prove (35). Since
we know that (35) holds.
Applying the Tonelli theorem, (34), and (35), we get
The proof of Lemma 5 is completed.
Lemma 6. Assume that is the unique solution of (10). For an arbitrary , there exists a positive constant depending only on and the coefficients of (1) such that the following one-sided norm estimate on the first order spatial derivative holds:
Proof. From (16) and Lemma 5, we know that there exists a positive constant depending only on and the coefficients of (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 (15) and suitably choosing , we have and . Furthermore, we have
Setting , we obtain
Letting , we have . From the comparison principle for ordinary differential equations, we get for all . Therefore, by this and (43), the estimate (40) is proved.
Lemma 7. There exists a sequence tending to zero and a function such that, for each , it holds that where is the unique solution of (10).
The proof of this lemma is fully similar to that of Lemma 5.2 in [6]. Here we omit its proof.
Lemma 8. There exists a sequence tending to zero and a function such that for each
Proof. Using the same arguments presented in Coclite et al. [6], we obtain which together with Lemma 5 derives that (47) holds.
In this paper we use overbars to denote weak limits which are taken in with .
Lemma 9. There exists a sequence tending to zero and two functions , such that for each and . Moreover,
Proof. Using Lemmas 3 and 4 derives that (48) and (49) hold. It follows from (49) that (50) holds. From Lemma 7 and (48), we know that (51) is valid.
Using (48), for any convex function with being bounded and Lipschitz continuous on and for any , we conclude that Multiplying (16) by yields
Lemma 10. For the convex with being bounded and Lipschitz continuous on , it has in the sense of distributions on . Here and denote the weak limits of and in , , respectively.
Proof . In (53), by the convexity of , (15), Lemmas 7, 8, and 9, taking limit for gives rise to the desired result.
Remark 11. Using Lemma 9, we know that almost everywhere in , where for . From Lemma 6 and (48), we have where is a constant depending only on and the coefficients of (1).
Lemma 12. The identity holds in the sense of distributions on .
Proof . Applying (16), Lemmas 7 and 8, (48), (49), and (51), the conclusion (57) holds by taking limit for in (16).
Lemma 13. If with , then the identity holds in the sense of distributions on .
Proof . Let be a family of mollifiers defined on . Consider where the is the convolution with respect to variable. Multiplying (57) by gives rise to
Applying the boundedness of , and letting in (59), we derive that (58) holds.
4. Strong Convergence of and Existence of Global Weak Solutions
Using the methods in [5] or [6], in this section, we will prove that the weak convergence of in (48) is strong convergence. This results in the existence of global weak solutions for problem (10).
Lemma 14. Assume that . Then
Lemma 15. If , for constant , then where and , .
Lemma 16. Let constant . Then for each
The proofs of Lemmas 14, 15, and 16 can be found in [5] or [6].
Lemma 17. Assume that . Then for almost all
Lemma 18. For any and constant , it holds that
Using the same techniques in [5] or [6], because of (54), (58), and Lemma 6, we can prove Lemmas 17 and 18. Here we omit their proofs.
Lemma 19. It holds that
Directly applying the approaches in [5] or [6] and Lemmas 17 and 18, we obtain that Lemma 19 holds. Here we omit its proof.
Proof of the Main Result. From (9), (11), and Lemma 7, we conclude that conditions (a) and (b) in Definition 1 are satisfied. We need to prove (c). Using Lemma 19, we get From Lemma 7, (47), and (67), we know that is a distributional solution to problem (3). The inequalities (6) and (7) are consequences of Lemmas 4 and 6. The proof is completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Authors’ Contribution
The paper is a joint work of three authors who contributed equally to the final version of the paper. All the authors read and approved the final paper.