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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 730860, 12 pages
Blow-Up of Solutions to a Novel Two-Component Rod System
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
Received 4 February 2013; Accepted 23 May 2013
Academic Editor: Ziemowit Popowicz
Copyright © 2013 Shengrong Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a novel two-component rod system which is closely connected to the shallow water theory. The present work is mainly concerned with the blow-up mechanism of strong solutions; we establish new conditions in view of some special classes of initial value to guarantee finite time blow-up of solutions.
We consider the following variation of the two-component rod system: If we introduce a momentum , the previous system possesses the following form: where with and depending on a space variable and a time variable . It is obvious that system (1) for reduces to the rod equation studied in [1–12]. Particularly, for and , (1) becomes the celebrated Camassa-Holm equation which was investigated by many authors [13–21]. It reduces to the two-component rod system for studied in . Moreover, system (1) reduces to the two-component Camassa-Holm (CH2) equation for and which was investigated in [23–30]. The two-component rod system which includes both velocity and density variables in the dynamics possesses the following form: We note that the geometric structure of (1) is different from (4) with the incorporation of the term ; this restricts our discussion only on the unit circle. Furthermore, the discussion of this work shows that system (1) possesses wave breaking phenomenon which is described with different blow-up criteria, while system (4) admits not only breaking wave solutions but also global in time solutions. However, we do not know whether the global solutions of (1) exist or not for the time being. It is the structure of (1) that breaks some properties which previously holds for (4). It is also known that for (4) which follows that is an invariant with respect to time. Particularly, if has zero mean initially, then the solution to (4) will preserve zero mean for all time. This motivates us to introduce the term since it always has zero mean. On the other hand, we consider the system (1) in the spaces for on the circle, where denotes the -Sobolev space of regularity, and denote by the space with two functions being identified if they differ by a constant. The basic idea of this variation is the decomposition of into and , where is independent on variable , belongs to , a subspace of containing all zero mean functions. The main purpose of our work is to investigate formation of singularities of solutions to (1) where the conservation laws play crucial roles. The idea is motivated by Guo's recent works [28, 31].
The rest of this paper is organized as follows. In Section 2, we recall the local well-posedness theorem and show some auxiliary results which will be used in the sequel. In Section 3, the detailed blow-up criteria are established via various initial conditions and as a byproduct, the blow-up rate is presented.
In this section, we would like to list some useful results for later use. We now provide the framework in which we shall reformulate (1). Let Then for all and , where is the spatial convolution. With this in hand, we rewrite (1) as follows: Firstly, we recall the elementary result on the local well-posedness theorem for system (7) which was shown in  from the geometric formulation.
Theorem 1. Let . There is an open neighborhood containing such that for any there exists a maximal and a unique solution to the initial value problem for system (7) with . Moreover, the solution depends continuously on the initial data, that is, the mapping is continuous.
We now introduce two characteristics for (7) where denotes the first component of the solution to (7) with certain initial data and is the lifespan of the solution, then is a diffeomorphism of the line. These two characteristics are both increasing diffeomorphisms of . A direct calculation shows Thus, we have This is usually called the particle trajectory method, and it is important in the discussion of blow-up phenomena. We are now in a position to state the following.
Lemma 2. Let , and let be the maximal existence time of the solution to (7) with the initial data . Then for all , we have
Proof. Differentiating the left-hand side of (15) with respect to , we obtain where we have used defined in (10) and the second equation of system (7). Then is independent on time . Now we choose , due to (13) we know . Therefore, this lemma is easily proved.
The application of (15) leads to the following result.
Theorem 3. Let , and let be the maximal existence time of the solution to (7) with the initial data . If there exists such that then -norm of does not blow up on .
Proof. Applying the operator where to the first equation in (7), and multiplying by , then integrating over , we obtain where From the proof in , we get For the third term on the right-hand side of (13), we estimate it as follows: Combining the previous inequalities, we can get where the constant may be different from instance to instance. We do similar estimates for the second component by applying the operator to the second equation in (7), and multiplying by , then integrating over to obtain Similarly, we have Therefore, it follows that By (22) and (25) we have It is easy to obtain by Gronwall's inequality, Lemma 2, and (13) that Then the -norm of does not blow up on .
Next we present the precise blow-up scenario for sufficiently regular solutions to (7).
Theorem 4. Let , and let be the maximal existence time of the solution to (7) with the initial data . Then the solution X blows up in finite time if and only if
Proof. Multiplying the first equation in (2) by and integrating by parts, we get
It then follows that
Differentiating the first equation in (2) with respect to , multiplying by , and integrating by parts yield
Combining (30) and (31) together, we obtain
Similar arguments made on the second equation in (2) yield
It follows by combining (32) and (33) that
Assume that there exist and , such that
By Lemma 2, we have
It then follows that
It is easy to obtain by Gronwall's inequality that
Sobolev imbedding, (38), and Theorem 3 ensure that the solution does not blow up in finite time.
On the other hand, due to the Sobolev imbedding theorem, we observe that or will lead to blow-up of solutions.
After the local well-posedness of strong solutions (see Theorem 1) is established, a natural question is whether this local solution can exist globally. If the solution only exists in finite time, what induces the blow-up phenomenon? On the other hand, finding sufficient conditions to guarantee the finite time singularities or global existence is very interesting, especially for sufficient conditions added on certain initial data. The following results will give some positive answers.
In this section, we pay more attention to the formation of singularities for strong solutions to our system. It will show that wave breaking is one way that singularities arise in smooth solutions. We start this section with the following lemmas.
Lemma 5. Let , , and let be the maximal existence time of the solution to (7) with the initial data . Then we obtain the following conservation laws:
Remark 6. The conservation of guarantees the uniform bound of , then Theorem 4 is also interpreted as wave breaking.
Lemma 7 (see ). For all , the following inequality holds: with Moreover, is the optimal constant obtained by the function
Lemma 8 (see ). For all , the following inequality holds: where Moreover, is the minimum value, so in this sense, is the optimal constant which is obtained by the associated Green's function
Lemma 9 (see ). For any function , the following inequality holds:
Lemma 10 (see ). Assume , . If , then
It is now to state our result.
Theorem 11. Let , , and let be the maximal existence time of solution to (7) with the initial data . If the following inequality holds: where is the initial value of in Lemma 5, and constant is determined later. Then the corresponding solution blows up in finite time.
Proof. Differentiating the first equation in system (7) with respect to , we obtain
Applying the relation to (49) gives
If multiplying (50) with and integrating by parts subsequently, we obtain
where we have used the following identity:
Since , and
In the following, we estimate the three terms on the right-hand side of (51) one by one. The Cauchy-Schwarz inequality implies that hence Using Lemma 8 and the invariant property of , we have Note the fact that This implies that for any , has at least a zero point , that is, . Therefore, we have Suppose that the solution does not blow-up in finite time, it follows that there exists a constant such that and is bounded by ; thus Then By (55)–(60), we have For convenience, we denote where we noticed the fact that That is, Note that if the initial quantity satisfies the standard argument on the Riccati type inequality and the initial hypothesis ensure that there exists a finite time such that Since This implies that Then it contradicts the assumption . By Theorem 4, we know that the solution must blow up in finite time.
If , we get We also denote that It follows that where we note . Similar to case (1), it is easy to see that under the corresponding condition (2) of our theorem and the previous arguments, blow-up phenomenon occurs. We complete the proof.
As we know, the key issue for partial differential equations lies in the estimates. In the following results, we apply different strategies to derive suitable bounds for solution, then blow-up phenomenon occurs while some special initial values are involved. Precisely, we show the following.
Theorem 12. Suppose that , does not vanish identically. If , and one of the following conditions is satisfied for some constant and function is defined as previously, then the corresponding solution to initial data of (7) blows up in finite time.
Proof. Differentiating both sides of the first equation of (7) with respect to variable , we obtain
Applying the relation to (73) gives
Multiplying (74) with and integrating by parts subsequently, we obtain
where we have used the following identity:
On the other hand, we know that in view of the hypothesis, and the following inequality holds
Using Lemma 10 and (77), we obtain
In view of (60), we obtain that
If , it is easy to know that
If there exists a constant such that the initial energy , then there is some such that
On the other hand, we have by Lemma 10
Therefore, the previous inequality yields
In view of Hölder's inequality, there holds
For simplicity of notations, we denote by and the following quantities:
respectively. Therefore we have
First, we can easily get , and it is not difficult to find that there exists a such that . Then for , we get
Solving this inequality yields
which approaches as arrives at ; that is, there exists a time such that
Therefore, it follows that
Then it contradicts the assumption . By blow-up scenario, we know that the solution must blow up in finite time.
If , then we have where We also use as previously to get Similar arguments to (64) in Theorem 11 and condition (2) guarantee the finite time blow-up of solution to (7).
The zero mean of in the previous theorem can be substituted by ; blow-up still occurs with the aid of different estimate from (83).
Theorem 13. Assume that , , does not vanish identically, , if one of the following conditions is satisfied: for some constant , then the corresponding solution to (7) blows up in finite time.
Proof. By assumption and the invariance property of , we have
Therefore, must change sign, so there exists at least one zero point on . Then for each , suppose that there is a such that , for we have
Thus, the previous relation and an integration by parts yield
Doing a similar estimate on , we obtain
In view of (97), we also have
where we use the fact , then
For , we have
If there is a constant such that the initial energy , there exists some such that
On the other hand, we have . Therefore, the previous inequality yields
The remaining part is very close to Theorem 12, so we omit it. So the solution must blow up in finite time.
For , then where . Note that if the initial quantity satisfies then condition (2) can conclude that the solution to (7) goes to in finite time. This completes the proof.
Theorem 14. Suppose that , is the solution to system (7) with the initial data . If there is some point such that one of the following conditions is satisfied: where the constants and are given in Lemmas 7 and 8, is the initial value of , then the solution must blow up in finite time.
Proof. Differentiating the first equation in system (7) with respect to and noticing that , we have
When , this equation, in combination with (11), yields
We can deduce that there exists at least one point such that for . Let us consider this problem at . For convenience, we denote . Then we have
Using the notation , where , we have
In view of the initial condition, it is not difficult to obtain
with determined by . Then, by using the standard arguments for this type of inequality and our hypothesis, it is easy to conclude that the lifespan of the solution is finite; that is, blow-up phenomenon occurs.
When , then where . Similar to the arguments about (113), under the condition (2), the corresponding solution blows up in finite time. We complete the proof.
Theorem 15. Suppose that , is the solution to system (7) with the initial data . If there holds one of the following conditions for some point where the constant is the best constant given by Lemma 7 and Let be the maximal existence time of the corresponding solution to (7) with the initial data . Then is finite.
In the following, as a byproduct, we examine the blow-up rate while the solution blows up in finite time.
Proof. The conclusion follows from the theory of ordinary differential equations to inequality (113). Indeed, when , by (113) we have
In view of Lemma 8 and the conservation of , there holds for all that
It follows that
Since by Theorem 14, it implies that for any there exists a such that for all . Therefore,
Direct integration from to gives
and the arbitrariness of leads to our result.
When , by (116) we obtain where