Abstract

In this paper, we consider the rotational Camassa–Holm equation on the circle. Sufficient conditions on the initial data to guarantee wave breaking are established.

1. Introduction

Recently, a rotational Camassa–Holm equation was proposed in [1–4], which reads aswithwhere is a parameter related to the Coriolis effect. In the whole space, Gui et al. [3] established the local well-posedness in , . Sufficient conditions to guarantee wave breaking phenomena were also studied in [3]. Tu et al. [4] established the global existence and uniqueness of the energy conservative weak solutions. In [5], they found some explicit solutions by elliptic integrals. They also classified the members of the equation describing pseudo-spherical surfaces. Generic regularity of conservative solutions was investigated in [6]. Moon [7] studied the question of nonexistence of periodic peaked traveling wave solution for rotational Camassa–Holm equation.

If , it follows , and then system (1) reduces to the Dullin–Gottwald–Holm equation [8]. Some mathematical studies can be found in [9–12].

Another highly related model is the well-known Camassa–Holm equation [13] . This equation has a physical background with shallow water propagation. The Camassa–Holm equation [14,15] has infinitely many conservation laws. In [16,17], they established the local well-posedness. Wave breaking phenomena were widely studied in [16–19]. McKean [20] (see also [21] for a simple proof) established a necessary and sufficient condition on the initial datum , which depends on the shape of . In [22], the orbital stability of the peakons was proved. In [23, 24], they studied persistence properties and unique continuation of solutions. The long-time behavior for the support of momentum density of the Camassa–Holm equation was discussed in [25]. Mathematical studies for the related models can been found in [26–28].

For the convenience of research, in this paper, we consider the rotational Camassa–Holm equation as the following form on the circle:where and are real constants. , where denotes the unit circle, i.e., .

The paper is organized as follows. In Section 2, we introduce some useful lemmas. The main result and its proof will be shown in Section 3.

2. Preliminaries

Let ; then, the operator can be expressed by its associated Green’s function as

Then, (3) can be rewritten as

By applying Kato’s semigroup theory [29] and similar arguments in [3], we can also have the local well-posedness on the circle.

Theorem 1 (see [3]). Given , , then there exist a maximal and a unique solution to the rotational Camassa–Holm equation (3) such that

Then, the precise blow-up scenario will been shown as follows.

Theorem 2 (see [3]). Assume that and let be the maximal existence time of the solution to equation (3) with the initial data . Then, the corresponding solution of the rotational Camassa–Holm equation (3) blows up in finite time if and only if

Then, we introduce some useful inequality in the circle.

Lemma 1 (see [30]). For all , the following inequality holds:where .

Lemma 2 (see [30]). Let . If , then the following inequalities hold:

3. Main Results

In this section, we firstly establish a sufficient condition to guarantee the blow up of the solution to the rotational Camassa–Holm equation (3). We give the particle trajectory aswhere is the lifespan of the solution. Taking derivative (10) with respect to , we obtain

Therefore,which is always positive before the blow-up time.

Theorem 3. Assume that and there exists such thatwhere ; then, the corresponding solution to equation blows up at a finite time bounded by

Proof. Let . Differentiating (5) with respect to yieldsThen, we have at the point asWithout loss of generality, we choose , and we haveNote thatWe haveSimilar argument yields thatThen, we haveCombining (21) into (16), we obtainA direct calculation givesBy Lemma 1, we haveFor , we haveFor , we haveBy (22) and the definition of , we haveThis is a Riccati type inequality. By the fundamental ODE methods, the proof is completed by the initial condition.

Theorem 4. Assume that , , and there exists such thatwhere ; then, the corresponding solution to equation blows up at a finite time bounded by

Proof. Recall thatWith the initial condition , by Lemma 2, we haveFor , we haveFor , we haveCombining the above arguments into (30), we havewhere is defined in Theorem 4. The proof of Theorem 4 is completed by the fundamental ODE methods.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This study was partially supported by the National Natural Science Foundation of China (grant no. 12 001 220).