Wave breaking and global existence for the periodic rotation-Camassa-Holm system

The rotation-two-componentCamassa-Holm system with the effect of the Coriolis force in therotating fluid is a model in the equatorial water waves. In thispaper we consider its periodic Cauchy problem. The precise blow-upscenarios of strong solutions and several conditions on the initialdata that produce blow-up of the induced solutions are described indetail. Finally, a sufficient condition for global solutions isestablished.

1. Introduction. In this paper, we are concerned with the behavior of solutions to the periodic problem for the following rotation-two-component Camassa-Holm where A characterizes a linear underlying shear flow, the real dimensionless constant σ provides the competition, or balance, in fluid convection between nonlinear steepening and amplification due to stretching, µ is a nondimensional parameter and Ω characterizes the constant rotational speed of the Earth. The system was introduced by Fan-Gao-Liu [19] from the f -plane governing equations for equatorial geophysical water waves which admit a constant underlying current, where u(t, x) is the fluid velocity in the x-direction, ρ(t, x) is related to the free surface elevation from equilibrium.
The R2CH system (1.1) was derived by applying the approach of Ivanov's asymptotic perturbation analysis for the governing equations of two-dimensional rotational gravity water waves [27], which is particularly appealing in comparison to other models that have been studied in the past due to the following reasons. First, the model may be the first approach in incorporating the effect of a current into the asymptotic model to the f -plane equatorial geophysical governing equation although there have been lot of works concerning the asymptotic models to the geophysical governing equation [19,25,6,13,15,24]. Second, the R2CH system is another significant modification model in incorporating the geophysical effects into the generalized Dullin-Gottwald-Holm (DGH) system and two-component Camassa-Holm system [19,27,14,31]. Finally, it should be interesting to see the effects of the constant Equatorial Undercurrent and the Earth's rotational speed on the evolution of water waves by mathematical analysis [19].
The R2CH system (1.1) has significant relationship with several models describing the motion of waves at the free surface of shallow water under the influence of gravity. For instance, if we consider the system in (1.1) without effect of the Earths rotation, i.e., Ω = 0, it becomes the generalized DGH system which was first derived by Han-Guo-Gao [23] in the shallow-water regime following Ivanov's approach [27]. And blow-up mechanism of the generalized DGH system was analyzed in detail in the same paper. Let µ = 0, system (1.2) reduces to the generalized Camassa-Holm (CH) system [4]. Wave-breaking phenomena and existence of global solutions for its Cauchy problem in non-periodic and periodic setting were established in [4] and [5], respectively. See also for [14,17,18,20,21,22,33] and references therein. When σ = 1, and µ = 0, (1.2) recovers the standard two-component integrable Camassa-Holm system [14,31] Moreover, in the case ρ = 0, (1.2) becomes the DGH equation [16] and (1.3) becomes the Camassa-Holm (CH) equation [3]. Well-posedness and blow up phenomena for the CH equation have been studied extensively. Indeed, the local well-posedness of the periodic Camassa-Holm equation with initial data were proved and the global strong solutions for certain class of initial data were also studied [7,9,10]. Existence and uniqueness results for classical solution of the periodic CH equation were established in [30]. The blow up phenomena of the periodic CH equation were investigated in a number of papers (see [3,7,9,10,11,8,12,29] and references therein). After wave breaking the solutions can be continued as either global conservative or global dissipative solutions [1,2,26].
Consider that system (1.1) is a generalization of system (1.2) with the rotation of Earth-these effects feature significantly for such large scale phenomena as currents, and the Coriolis force has introduced a higher order nonlinear term into the generalized two-component CH system, which has interesting implications for the fluid motion, particular in the relation to the wave breaking phenomena and the permanent waves. Fan-Gao-Liu [19] investigated the effects of the Coriolis force caused by the Earth's rotation and nonlocal nonlinearities on blow-up criteria and wave-breaking phenomena. Furthermore, conditions which guarantee the permanent waves were also obtained by using a method of the Lyapunov function when σ = 1, µ = 0.
The goal of the present paper is to derive some conditions on the initial data for the periodic Cauchy problem of system (1.1) that guarantees the formulation of singularities in the resulting solution in finite time. The first step is to establish a wave-breaking criterion. The theory of transport equations implies that the solution (u, ρ) will not blow up as long as the slope of the velocity, i.e. u x remain bounded, while the solution blows up in finite time when the slope u x is unbounded from blow. Then we try to find conditions of the initial data which can guarantee the wave breaking in finite time when σ = 1, µ = 0. It is shown that the initial velocity has to decrease much faster at some point in the presence of the Earth's rotation and the positive vorticity in order to ensure the occurrence of the wave-breaking phenomena. Furthermore, we also give a blow-up result if u 0 is odd and ρ 0 is even when σ = 1, µ = 0.
The remainder of the paper is organized as follows. In Section 2, some preliminary estimates and results are recalled and presented. Section 3 is devoted to the proofs to our main wave-breaking results, i.e., Theorem 3.1-Theorem 3.3. In Section 4, we provide a sufficient condition for global solutions.
Notation. Throughout this paper, we identity all spaces of periodic functions with function spaces over the unit circle S in R 2 , i.e. S = R/Z. The norm of the Sobolev space H s (S), s ∈ R, by · H s . Since all space of functions are over S, for simplicity, we drop S in our notations of function spaces if there is no ambiguity.

Preliminaries. Let
Our system (1.1) can be written in the following transport type: We begin by present the local well-posedness result for the periodic Cauchy problem of system (1.1). Concerning the R2CH system (1.1) is suitable for applying Kato's theory [28], one may follow the similar argument as in [19,18] to obtain the following theorem.
Moreover, the solution depends continuously on the initial data, i.e., the mapping Consider the following associated Lagrangian scales of (2.1), namely, where u ∈ C 1 ([0, T ); H s−1 ) is the first component of the solution (u, ρ) to system (2.1).
Lemma 2.1. Let (u, ρ) be the solution of system (2.1) with initial data (u 0 , ρ 0 ) ∈ H s × H s−1 , s > 3/2 and T > 0 is the maximal time of existence. Then equation The above lemma indicates that q(t, ·) is an diffeomorphism of the line for each Now we briefly give the some needed results to pursue our goal.
Using this result and performing the same argument as in [21], we can establish the following blow-up criterion.
with the best possible constant c lying within the range (1, 13 12 ]. Moreover, the best constant c is e+1 2(e−1) . At last, we present some conserved properties. Firstly, we give three conserved quantities which will play an important role in all analysis of the solutions.
Proof. Integrating the second equation of system (1.1) by parts, in view of the periodicity of u and ρ, we have For the proof of the other two conserved quantities, integrating the first equation of system (1.1) by parts, in view of the periodicity of u and ρ, we have Multiplying the first equation of system (1.1) by 2u and integrating by parts, in view of the periodicity of u and ρ, we have Multiplying the second equation of system (1.1) by 2ρ and integrating by parts, in view of the periodicity ofu and ρ, we get x) dx = 0, which completes the proof of the lemma.
By the conservation laws stated in Lemma 2.4 and Lemma 2.3 (i), we have the following useful corollary.

YING ZHANG
3. Wave-breaking phenomena. In this section, We establish a blow up criterion and also derive some sufficient conditions fo r the breaking of waves for the initialvalue problem (2.1). First, we give the wave-breaking criterion.
Proof. By Theorem 2.1 and a simple density argument, we need only to consider this theorem for s ≥ 3. We may also assume u 0 = 0, otherwise it is trivial.
We first try to prove the blow-up criterion (3.1). If (3.1) holds, the Sobolev embedding theorem H s → L ∞ , s > 1/2 implies that the corresponding solution blows up in finite time. Conversely, assume that T < ∞ and (3.1) is not valid. Then there is some positive number M 0 > 0, such that Therefore, it follows from Theorem 2.2 that the maximal existence time T = ∞, which contradicts the assumption that T < ∞. Now we turn to prove the case in (3.2). using the identity −∂ 2 x G * f = f − G * f for any f ∈ L 2 . Differentiating the first equation in (2.1) with respect to x, we have where q(t, x) is defined by (2.2). Then along the trajectory q(t, x), the above equation and the second equation of (2.1) become for t ∈ [0, T ), where denotes the derivative with respect to t and f (t, q(t, x)) is given by By the definition of M (t), assume that T < ∞ and (3.2) is not valid. Then there is some positive number M 1 > 0, such that Then from the second equation of (3.4), we obtain that for x ∈ S, i.e., which completes the proof of the lemma. Now we will derive the upper bound for f for later use in getting the wavebreaking result. Using that ∂ 2 , applying Young's inequality and Corollary 2.1, we have and where C 1 denotes a positive constant that depends only on A, Ω andẼ(u 0 , ρ 0 ), where we use the following relations Observe P (t) is a C 1 -differentiable function in [0, T ) and satisfies We will show that If not, then suppose there is a t 0 ∈ [0, T ) such that P (t 0 ) > 0. Define Then P (t 1 ) = 0 and P (t 1 ) ≤ 0, or equivalently, On the other hand, we have which is a contradiction to (3.8). This verifies the estimate in (3.7). Therefore, it implies for arbitrary x ∈ S, Recall that we have |u x | < ∞. This contradicts our assumption T < ∞, which completes the proof of Theorem 3.1.
Now we are in position to state the following that provide some cases that wave breaks in finite time.
Theorem 3.2. Assume that 1 − 2ΩA > 0, σ = 1 and µ = 0. Let (u 0 , ρ 0 ) ∈ H s (S) × H s−1 (S) with s > 3/2, and T > 0 be the maximal time of existence of solution (u, ρ) to system (2.1) with initial data (u 0 , ρ 0 ). Assume there exists a x 0 such that ρ 0 (x 0 ) = 0 and u 0, Then the corresponding solution (u, ρ) blows up in finite time in the following sense: there exists a T 1 with where The proof of Theorem 3.2 relies on the following crucial lemma.
Proposition 3.1. [19] The first equation of system (2.1) can be rewritten as (3.10) where K = u + ΩG * ρ 2 . Furthermore, (3.11) Proof. Similar to the arguments in the beginning of the proof of Theorem 3.1, we have just need to consider s ≥ 3. First take σ = 1, µ = 0 in (3.11), we get (3.12) where q(t, x) is defined by (2.2). Then we can write the equation of ρ in (2.1) along the trajectory of q(t, x) as Taking x = x 0 , the assumption γ(0) = ρ 0 (x 0 ) = 0 and the above equation imply Therefore, it follows from (3.12) that at (t, q(t, x 0 )), where is the derivative with respect to t and Since and in view of (3.5)-(3.7) and , we have that (3.17) From (3.17), we deduce If the assumption holds, then We now claim that But then we would have by (3.18) Being locally Lipshitz, the function M 1 (t) is absolutely continuous on [0, t 0 ), and thus an integration of the previous inequality would lead us to which contradicts our assumption M 1 (t 0 ) = −C 3 − Ω 1−2ΩA C 2 . Hence (3.19) holds, implying that M 1 (t) is strictly decrease on [0, T ). Then Solving the above inequality gives where 1 − 2ΩA > 0, then T = +∞, and the solution (u, ρ) is global.
Proof. Define M 1 (t) and γ(t) as in (3.13), then along the characteristics q(t, x) defined in (2.2), system (1.1) leads to the following ordinary differential equations for t ∈ [0, T ), where f is defined in (3.15). The second equation above implies that γ(t) and γ(0) are of the same sign.
Recalling the assumption (4.1) of the theorem, we know γ(0) = ρ 0 (x) > 0 for every x ∈ S. Define the following Lyapunov function by which is always positive for t ∈ [0, T ) in view of the assumption (4.2). Differentiating ω and using (4.3), we obtain where we have used the bound (3.16) and (3.17) for ∂ x G * ρ 2 and f .
It is then inferred by applying the inequality (4.9) that which implying |u x (t, q(t, ξ))| < ∞, t ∈ [t, T 1 ) and then contradicts our assumption (4.5). This completes the proof of Theorem 4.1.