Wave Breaking for the Rotational Camassa–Holm Equation on the Circle

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.


Introduction
Recently, a rotational Camassa-Holm equation was proposed in [1][2][3][4], which reads as with β � 3c 4 + 8c 2 − 1 6 c 2 + 1 2 , where Ω is a parameter related to the Coriolis effect. In the whole space, Gui et al. [3] established the local well-posedness in H s (R), s > 3/2. 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. ey 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.
McKean [20] (see also [21] for a simple proof ) established a necessary and sufficient condition on the initial datum u 0 , which depends on the shape of y � u − u xx . In [22], the orbital stability of the peakons was proved. In [23,24], they studied persistence properties and unique continuation of solutions. e 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][27][28].
For the convenience of research, in this paper, we consider the rotational Camassa-Holm equation as the following form on the circle: where m � u − u xx and α, β, c, Γ are real constants. x ∈ S, where S denotes the unit circle, i.e., S � R/Z. e paper is organized as follows. In Section 2, we introduce some useful lemmas. e main result and its proof will be shown in Section 3.

Preliminaries
Let Λ � (1 − z 2 x ) 1/2 ; then, the operator Λ − 2 can be expressed by its associated Green's function as en, (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 2 (see [3]). Assume that u 0 ∈ H 2 (S) and let T be the maximal existence time of the solution u(x, t) to equation (3) with the initial data u 0 (x). en, the corresponding solution of the rotational Camassa-Holm equation (3)

blows up in finite time if and only if
en, we introduce some useful inequality in the circle.

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 as where T is the lifespan of the solution. Taking derivative (10) with respect to x, we obtain erefore, which is always positive before the blow-up time.
Theorem 3. Assume that u 0 ∈ H 2 (S) and there exists x 0 ∈ S such that where

then, the corresponding solution u(x, t) to equation blows up at a finite time T bounded by
Proof. Let H(u) � (α + Γ)u + (β/3)u 3 + (c/4)u 4 . Differentiating (5) with respect to x yields en, we have u xt at the point (q(x, t), t) as 2 Journal of Mathematics Without loss of generality, we choose 0 ≤ q(x, t) ≤ 1, and we have Note that We have Similar argument yields that en, we have Combining (21) into (16), we obtain A direct calculation gives

Journal of Mathematics 3
By Lemma 1, we have For c ≥ 0, we have For c < 0, we have By (22) and the definition of C * , we have is is a Riccati type inequality. By the fundamental ODE methods, the proof is completed by the initial condition.

Conflicts of Interest
e author declares that there are no conflicts of interest.