Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system

In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in \begin{document}$ B_{p,r}^s× B_{p,r}^{s-1}$\end{document} with \begin{document}$s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$\end{document} , \begin{document}$p,r∈ [1,∞]$\end{document} by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space \begin{document}$ B_{2,1}^{3/2}× B_{2,1}^{1/2}$\end{document} , and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.


1.
Introduction. This paper considers a rotation-two-component Camassa-Holm system (R2CH), which is derived to model the equatorial water waves with the effect of the Coriolis force in the rotating fluid [22]: Here, the function u(t, x) stands for the fluid velocity in x direction and ρ(t, x) describes the free surface elevation from equilibrium. In (1), the real non-dimensional parameter σ is a parameter which provides the competition, or balance, in fluid convection between nonlinear steepening and amplification due to stretching, the parameter A is related to a linear underlying shear flow, µ ∈ R is a parameter and Ω characterizes the constant rotational speed of the Earth.
It is worth pointing out that the system (1) is closely related to several models describing the motion of waves at the free surface of shallow water under the influence of gravity.
For example, without considering the effect of the Earth's rotation (i.e., Ω = 0), the system (1) reduces to the generalized two-component integrable Dullin-Gottwald-Holm (gDGH) system, see [32] for more details. If we further take σ = 1 in (1), it then recovers the standard two-component Dullin-Gottwald-Holm (2DGH) system in the form of which was equipped with the boundary assumptions u → 0, ρ → 1 as |x| → ∞ [10,29,44]. It is shown that the 2DGH system is completely integrable, and it can be expressed as a compatibility condition of two linear systems (Lax pair) with a spectral parameter ξ [30,48]: where m = u−u xx . After its first appearance, lots of papers are devoted to studying the Cauchy problem of the 2DGH system. Here we only review some relevant results on our topic. By applying the Kato's theorem for quasilinear hyperbolic systems, Guo, etc. [30] obtain the local well-posedness of the system in Sobolev space H s (R)×H s−1 (R) with s ≥ 2, and they shown that the solutions can only have singularities that correspond to wave breaking. Moreover, a sufficient condition was provided to ensure the existence of global solutions. In [6], by virtue of the bi-linear estimate technique to the approximate solutions, Chen, etc. established the local as well as global well-posedness in H s (R) × H s−1 (R) with s ≥ 3 2 , which improves the results in [30]. Later, in [36], Liu and Yin proved that the 2DGH is local well-posed in the nonhomogeneous Besov spaces B s p,r (R) × B s−1 p,r (R), s > max{1 + 1 p , 3 2 , 2 − 1 p } and 1 ≤ p, r ≤ ∞, which extends the results obtained in [6,29]. On the other hand, the authors investigated the orbital stability of the smooth solitary wave solutions in the energy space H 1 (R) × L 2 (R). In [29], Guo and Wang considered the persistence property of solutions to the 2DGH system, which indicates that the corresponding solutions enjoy the same exponential decay property as the initial datum. Moreover, they proved that the strong solutions must identically equal to zero if the solutions and their spacial derivatives decay exponentially initially and at a later time. The wave breaking phenomena of solutions to the 2DGH system in the period case T = R/Z was discussed by Zhu and Xu [48].
By taking Ω = 0, σ = 1 and µ = 0, the system (1) becomes the celebrated two-component integrable Camassa-Holm (2CH) system: where m = u − u xx . The 2CH system was originally introduced by Olver and Rosenau [40] as a bi-Hamiltonian model, which is complete integrable and has a Lax pair formulation. It is worth mentioning that the 2CH system is closely related to the first negative flow of the AKNS hierarchy via a reciprocal transformation [5,21], in which the authors proved that the 2CH system is integrable under the assumption that it has Lax pair, and it also has peakons and multi-kink solitons. The great interest in the 2CH system lies in the fact that Constantin and Ivanov [10] provided the hytrodynamical derivation of the system as a valid approximation to the governing equations for water waves in the shallow water regime without vorticity (i.e., A = 0). In [19,25], the authors studied the well-posedness and wave breaking phenomenon for 2CH system. In [31], Constantin and Escher studied the wave breaking phenomena for the 2CH system. The local well-posedness of the 2CH system in a range of the Besov spaces was studied in [27], wherein the wavebreaking mechanism for strong solutions and the exact blow-up rate of solutions were also investigated. With the initial data satisfying some certain conditions, it was shown that the 2CH system admits global solutions [26,28]. Furthermore, for the Cauchy problem for some high-order 2CH systems, see for example [20,35,47] and the references therein.
It is worth pointing out that the system (1) (or (5)) can be regarded as a generalization of the famous Camassa-Holm (CH) equation in the case of ρ = 0: which was first introduced as a bi-Hamiltonian system by Fokas and Fuchssteiner in [24], and then Camassa and Holm [4] independently re-derived it by approximating directly in the Hamiltonian for the Euler equations in the shallow water regime. The CH equation possesses the bi-Hamiltonian structures and is completely integrable [4,7,9,24]. One of the remarkable property for Equ. (6) is that it possesses the peaked solitions (called peakons) in the form of u(x, t) = ce −|x−ct| (c = 0) which is orbitally stable [15,16], and in the periodic case u( . Since the peakons capture an essential feature that is characteristic for the traveling waves of largest amplitude, the CH equation has attracted a lot of attention in the literature over the past decades. For instance, the local well-posedness and blow-up phenomena for (6) in the Sobolev as well as the Besov spaces were investigated in [8,11,12,13,17,44,46]. For the global existence of the weak and strong solutions, we refer the readers to [8,11,12,14,45]. Moreover, the global conservative and dissipative solutions to the CH equation are investigated in [2,3,33].
Recently, Fan, etc. [22] derived the system (1) in the spirit of Ivanov's asymptotic perturbation analysis for the governing equations of two-dimensional rotational gravity water waves [34]. The authors mainly established the precise blow-up mechanism, and it is shown that the model (1) can only have singularities which corresponding to wave breaking. Moreover, some initial conditions are provided to guarantee that the wave breaking phenomena occurs in finite time.
To the best known of our knowledge, the Cauchy problem of (1) in Besov spaces and the Gevrey regularity and analyticity of the solutions have not been studied yet. In this paper, based on the Littlewood-Paley theory and the transport theory, we shall show that the system (1) is locally well-posed in B s p,r (R) × B s−1 p,r (R) with s > max{1 + 1 p , 3 2 , 2 − 1 p }. Then we establish the local well-posedness in the critical space B 2,1 (R). The major difficulties in our discussion are as follows. First, due to the appearance of the Earth's rotation, the system (1) involves two cubic-order nonlinear terms, which makes the proof of several required nonlinear estimates in Besov spaces more delicate than the standard 2CH system and the 2DGH system. Second, it seems that one can not obtain the convergence of the iterated sequence (u (n) , v (n) ) n∈N in C([0, T ]; B 2,∞ (R)) by using a endpoint Moser-type estimate and the Log-type interpolation inequality. Then, we give a blow-up result for the solution to (1) with data in B 2,1 (R) with the help of the conservation law. Finally, we study the Gevrey regularity of the system (1) by using a generalized Cauchy-Kovalevsky theorem, which implies that the system (1) admits analytical solutions locally in time and globally in space. Moreover, we obtain a precise lower bound of the lifespan and the continuity of the solution mapping in the sense of this paper.
The structure of the paper is organized as follows. In section 2, we recall some facts on the Besov space and the transport theory. Section 3 and Section 4 are devoted to the local well-posedness of solutions to (1) in Besov spaces, and a blow-up result is provided. In section 5, we investigate the Gevrey regularity and analyticity of the system (1).
Notation. All function spaces are considered in R, and we shall drop them in our notation if there is no ambiguity. We denote by C the estimates that hold up to some universal constant which may change from line to line but whose meaning is clear throughout the context.

2.
Prelimineries. The investigation of the well-posedness of the system (1) strongly depends on the transport theory in Besov spaces. To begain with, let us recall some necessary results on the Besov space. We refer the readers to [1,18] for more details of the proof. Definition 2.1. Let 1 ≤ p, r ≤ ∞, s ∈ R, the 1-D nonhomogeneous Besov space B s p,r is defined by where {∆ j } j≥−1 are the nonhomogeneous Littlewood-Paley decomposition operators defined through a almost-orthogonality partition of unity (see, e.g., [1]).
Lemma 2.2. Let s ∈ R, 1 ≤ p, r, p 1 , r i ≤ ∞, i = 1, 2, then (1) Fatou's lemma: if {u n } n∈N + is bounded in B s p,r and u n → u in S , then u ∈ B s p,r and The operator f (D) is continuous from B s p,r to B s−m p,r , for all s ∈ R and 1 ≤ p, r ≤ ∞. Lemma 2.3. (Interpolation inequality) (1) Let s 2 > s 1 , θ ∈ (0, 1), we have

2659
(2) For ∀s ∈ R, > 0 and 1 ≤ p ≤ ∞, there exists a constant C > 0 such that Let v be a vector field over R. Then the following estimates holds,  (2) If a = 0 and µ satisfies the condition 1 0 dr µ(r) dr = +∞, then ρ ≡ 0. Lemma 2.6. Let v be the smooth vector field with bounded first order space derivatives. Let ψ t (x) be the flow generalized by the vector filed v(t, x), i.e., Then for all t ≥ 0, ψ t (·) ∈ C 1 is a diffeomorphism over R, and the following estimates hold: with (1) If r = 1 or s = 1 + 1 p , there exists C > 0 depending only on s, p and r such that where 3. Well-posedness in Besov space. This section is devoted to studying the local well-posedness of the system (1) in Besov spaces. To this end, let us first introduce the following notation: For ∀T > 0, s ∈ R and 1 ≤ p ≤ +∞, define is the Geen's function 1 2 e −|x| , so one can rewrite the system (1) in the following form: We are now ready to state the first main result of the paper.
p,r ) > 0 such that the system (10) admits a unique solution (u, ρ) ∈ E s p,r (T ). Moreover, the data-to-solution mapping The uniqueness and continuity with respect to the initial data in some sense can be guaranteed by the following priori estimates. (2) and ρ (12) = ρ (1) − ρ (2) , then the following estimates hold: (1) for s > max{ 3 2 , 2 − 1 p , 1 + 1 p } but s = 2 + 1 p and s = 3 + 1 p , we have (3) for s = 3 + 1 p , we have Proof. It is easy to verify that the function (u (12) , ρ (12) solves the following transport system: where F (t, x) := −σu (12) x − (ρ (1) u (12) ) x . By applying the prior estimate for the transport equation (Lemma 2.7) to system (14) with respect to u (12) where we used the fact that ∂ x (σu (2) In order to get the estimation for u (12) (t) B s−1 p,r , we are now in a position to estimate the nonlinear term F (s, ·) B s−1 p,r . Noting that the operator For the cubic nonlinearities, by means of the algebra property in Besov spaces, one can deduce that and Moreover, one observe that (20) together and using the Cauchy-Schwarz inequality, we get On the other hand, due to the fact that B s−2 p,r is a Banach algebra for ∀s > 2 + 1 p , it is easier to derive the similar estimation for F (s, ·) B s−1 p,r for s > 2 + 1 p , we omit the details here. Therefore, combining (15) and (21), we obtain Since s = 3 + 1 p , by applying the Lemma 2.7 to the second equation in (10), we get Let us estimate the term G(t, x) B s−2 p,r for 1 + 1 p < s ≤ 2 + 1 p , otherwise one can easily show these inequalities hold true in view of the fact that B s−2 p,r is a Banach algebra as s > 2 + 1 p . Indeed, using Lemma 2.2, we have which combined with (23) yields that From the inequalities (22) and (25), we deduce that Then the desired estimation (11) can be obtained by taking advantage of the Gronwall inequality.
We are now in a position to deal with the critical case s = 2 + 1 p , which can be done by virtue of the interpolation argument. Indeed, let us choose According to the complex interpolation (see (1) of Lemma 2.3) and the estimation (26), we have where the function N 2+ 1 p ,1+ 1 p (t) is the same as in (1). On the other hand, the fact of s − 2 = 1 p = 1 + 1 p implies that the estimation in (1) also holds true, i.e., which combined with (27) yields the desired inequality.
For the critical case s = 3 + 1 p , since s − 1 = 2 + 1 p > 1 + 1 p , the estimation for u (12) in case (1) also holds true. Moreover, by taking the similar argument as that in case (2), one can obtain the estimation for ρ (12) , which leads to the inequality (13). Therefore, the proof of Lemma 3.2 is completed.
Next we focus on the existence of the solution to the system (10) by using the classical iterated method, which is ensured by the following result.
x ), and S n+1 = n q≥−1 ∆ q is the low frequency cut-off operator.
Proof. Due to the fact that the initial data (S n+1 u 0 , S n+1 ρ 0 ) ∈ B ∞ p,r × B ∞ p,r , it thus follows from the Lemma 2.8 and by induction with respect to the index n that problem (T (n) ) admits a global solution. It remains to prove (1) and (2).
Indeed, since ∂ x (1 − ∂ 2 x ) −1 is a Fourier multiplier of degree −1, and B s−1 p,r is a Banach algebra with s − 1 > 1 p , one can obtain the estimation for the nonlinear term F (n) (s, ·) B s−1 p,r as follows: For convenience, we denote Otherwise, the B s−2 p,r is a Banach algebra for s > 2 + 1 p , in which case is easier to deal with. By applying the a priori estimates for the transport equation to the first equation of (T (n) ), and using the inequalities (28) and (29), we get where we used the fact that S n+1 u 0 B s p,r ≤ C u 0 B s p,r for some positive C independent of n. By taking the Gronwall's inequality, we deduce By taking the similar argument for the second equation in (T (n) ), we can also obtain Putting the estimates (30) and (31) together leads to To obtain the uniformly boundedness, let us choose T > 0 satisfying T < 1/8CA 2 0 , and suppose by induction that

And then inserting the above inequality and (34) into (33) yields that
where we chosen the constant C > 1, and it implies that the sequence (u (n) , ρ (n) ) n∈N is uniformly bounded in C([0, T ]; B s p,r ) × C([0, T ]; B s−1 p,r ). Using the system (T (n) ) and the similar argument as above, one can prove that (∂ t u (n) , ∂ t ρ (n) ) n∈N is uniformly bounded in C([0, T ]; B s−1 p,r × B s−2 p,r ). Therefore, (u (n) , ρ (n) ) n∈N is uniformly bounded in E s p,r (T ). Now it suffices to show that (u (n) , ρ (n) ) n∈N is a Cauchy sequence in C([0, T ]; B s−1 p,r × B s−2 p,r ). Indeed, for all n, m ∈ N, it follows from (T (n) ) that −Ω(ρ (n+m) + ρ (m) )ρ (n,m) u (n+m) − Ωu (n,m) (ρ (m) ) 2 +Ω(1 − ∂ 2 x ) −1 ((ρ (n+m) + ρ (m) )ρ (n,m) u (n+m) x + u (n,m) x (ρ (m) ) 2 ) and where we denote u (n,m) := u (n+m) − u (m) and ρ (n,m) := ρ (n+m) − ρ (m) for convenience. Similar to the proof of Lemma 3.2, for s > max{ 3 2 , 2− 1 p , 1+ 1 p } but s = 2+ 1 p , 3+ 1 p , one can deduce from the Lemma 2.7 that Here by the definition of the operator S q , we have used the following fact: Likewise we also have ρ Hence, we get H n,m+1 (0) ≤ C2 −m (u 0 , ρ 0 ) B s p,r ×B s−1 p,r . By induction with respect to the index m in (39) and then taking the limit as m → +∞, we arrive at which implies that (u (n) , ρ (n) ) n∈N is a Cauchy sequence in C([0, T ]; B s−1 p,r × B s−2 p,r ). By using the interpolation argument as we did in the proof of Lemma 3.2, one can also prove the results for the critical cases s = 2 + 1 p and s = 3 + 1 p , we omit the details here. This completes the proof of Lemma 3.3. Now we give the proof of the Theorem 3.1.
Step 1. Consider the system (T (n) ) in Section 3 and assume that (u (n) , ρ (n) ) ∈ L ∞ (0, T ; B 2,1 is a Banach algebra, it is easy to verify that the right hand side of the system (T (n) ) belongs to L ∞ loc (R + ; B and L ∞ loc (R + ; B 1 2 2,1 ), respectively. Taking advantage of the Lemma 2.8 ensures that (T (n) ) has a global solution in E 3 2 2,1 (T ) for any given T > 0.
Step 2. Similar to the proof of Theorem 3.1, one can find T > 0 such that where A n (t) := u (n) (t) . This shows that the sequence (u (n) , ρ (n) ) n∈N is uniformly bounded in L ∞ (0, T ; B 2,1 ). By using the system (T (n) ), we can verify that the sequence (u (n) , ρ (n) ) n∈N is uniformly bounded in E  (36), it follows from the uniformly boundedness of (u (n) , ρ (n) ) n∈N that 2,∞ ∩ L ∞ , the second term on the right hand side of (43) can be estimated as where u (n,m) := u (n+m) − u (m) and ρ (n,m) := ρ (n+m) − ρ (m) are the same as in (36) and (37). By using the Banach algebra property of B  2,∞ ∩ L ∞ , one can estimate the terms on the right hand side of (44) by , since u (n) (t) L ∞ (0,T ;B 3/2 2,1 ) ≤ W is uniformly bounded. Moreover, by using the Moser-type estimate (see (3) of Lemma 2.2), we have It is not difficult to verify that Plugging the above estimates into (44), one can conclude from (43) that On the other hand, by the same taken for the second equation in (T (n) ), one can deduce that Setting P n,m (t) := u n+m (t) − u m (t) It follows from (45) and (46) that is a nondecreasing function, which combined with (47) yields that Define P m (t) = sup τ ∈[0,t],n∈N P n,m (τ ), the previous inequality implies that Let P = lim sup m∈N P m+1 (t). By taking the superior limit on both sides of the inequality (48) with respect to m, and using the Fatou's Lemma and again the monotone property of function f (x) = x ln(e + C x ), we obtain dτ.
An application of the Osgood's Lemma to the above inequality implies that P (t) ≡ 0, which shows that the sequence (u (n) , ρ (n) ) n∈N is a Cauchy sequence in Step 4. In order to prove the uniqueness of the solution, let (u (i) , ρ (i) ) be two solutions corresponding to the initial data (u (i) 0 , ρ (i) 0 ), i = 1, 2. Consider the system (14) with respect to the difference between two solutions. Using the Lemma 2.7 and following the similar computations as we did in the Step 3, we obtain Inserting the above inequality into (49), and noting that ln(e+ a x ) ≤ ln(e+a)(1−ln x) with x > 0 and a > 0, we get where H(t) = u (12) By using the interpolation inequality and the uniformly boundedness of the solutions, we have for any θ = 3 2 −s ∈ (0, 1), which implies that the data-to-solution mapping is Hölder continuous from B Moreover, the solution will blow up if T * is finite, namely, Proof. The first part is a direct conclusion of Step 2 in the proof of Theorem 4.1, and it remains to prove the blow-up result. To this end, by applying the Littlewood-Paley decomposition operator ∆ j to the system (10), we get with the initial data ∆ j u| t=0 = ∆ j u 0 , . Multiplying both sides of the (52) by ∆ j u, and then integrating by parts on R with respect to x, it follows from the Cauchy-Schwarz inequality that It thus transpires that Multiplying both sides of (54) by 2 3 2 j and taking l 1 -norm with respect to j, we deduce that d dt u(t) By virtue of the commutator estimate (see Lemma 2.4), we have (56) where we used the fact that .
By taking the similar argument to (53), we can get ).
Therefore, we obtain which combined with the Gronwall inequality yields that, for ∀t ∈ [0, T * ), Thanks to the conservation law associated to the system (10) E(u, ρ) = R u 2 + u 2 x + (1 − 2ΩA)(ρ − 1) 2 dx [22]. Using the Sobolev embedding theorem and the assumption 1 − 2ΩA > 0, it follows that Therefore, we deduce from (58) that, for ∀t ∈ [0, T * ), Moreover, by using the embedding C([0, T * ), B 3 2 2,1 ) → C([0, T * ), C 0,1 ), an application of the Lemma 2.6 to the equation ρ t + ρu x = −ρ x u ensures that the flow ψ t (x) = ψ(t, x) is smooth enough, and the corresponding solution can be expressed by Taking L ∞ -norm to the both sides of the above equality and using the Gronwall inequality, we get for ∀t ∈ [0, T * ). Assume that the maximum existence time T * is finite and which indicates that we are able to extend the solution (u, ρ) beyond T * , and this contradicts to the fact that T * is the maximum existence time. Hence the proof of Theorem 4.2 is completed.
The Sobolev embedding theorem and Theorem 4.2 imply the following result. 5. Gevery regularity and analyticity. In this section, we show that the system (10) is locally well-posed in the Gevery-Sobolev spaces in the sense of Hardamard. Especially, the system (10) admits unique analytic solutions locally in time and globally in space. Moreover, we obtain a lower bound of the lifespan.
To begin with, we recall the classical definition of the Gevrey class.
Definition 5.1. A function g ∈ C ∞ (R n ) is said to be in the Gevrey class G τ (R n ) for some τ > 0, if there exists positive constants C and R such that where R represents the radius of Gevrey-class regularity of the function f .
Remark 5.2. The class G 1 (R n ) is equal to the space of real-analytic functions on R n . If 0 < τ < 1, the f is called an ultra-analytic function. Moreover, the functions in G τ (R n ) with τ > 1 is a smooth but not analytic function.
The following definition, which was introduced by Foias and Temam [23] to study the analyticity of solution to the Navier-Stokes equation, is an equivalent characterization of the Gevrey-class. where f is the Fourier coefficient of f in T n .
Combining the above two definitions, we now introduce the framework of the Gevery class spaces G δ τ,s (R n ), which is the working space in the present paper.
Definition 5.4. Given τ, δ > 0 and s ∈ R, the Sobolev-Gevrey spaces G δ τ,s (R n ) is defined by Remark 5.5. In the period case T n , the Gevery-Sobolev norm can be stated as f G δ τ,s (T n ) = ( k∈Z n (1 + |k| 2 ) s e 2δ|k| 1/τ | f (k)| 2 ) 1/2 . In the following paper, we mainly focus on the solutions in the Gevery-Sobolev spaces defined on R n , the similar results still hold true for the period case, and we omit the details here.
The following two lemmas on the property of G δ τ,s (R) can be found in [37]. Lemma 5.6. Let s be a real number and τ > 0. Assume that 0 < δ < δ. Then Moreover, for s ∈ R, τ, δ > 0 and f ∈ G δ τ,s−1 , we have Proof. We just need to prove (61), and the other results can be found in [37]. Indeed, by the Fourier transform implies that ∂ k x f = (iξ) k f , it follows that . This finish the proof of Lemma Theorem 5.6.
(ii) For 0 < δ < δ < 1 and any u, v ∈ X δ with u − u 0 X δ < R, v − v 0 X δ < R, there exists K > 0 only depending on u 0 and R such that The abstract Cauchy-Kovalevsky theorem was studied by many authors, for example [38,39,41,42,43]. Very recently, Luo and Yin proved a generalized Ovsyannikov theorem (see Theorem 5.8), which contains the classical Cauchy-Kovalevsky theorem as a special case. Estimation for A 3 .
Estimation for A 4 .