GLOBAL EXISTENCE FOR A TWO-COMPONENT CAMASSA–HOLM SYSTEM WITH AN ARBITRARY SMOOTH FUNCTION

. This paper is concerned with a two-component integrable Camassa-Holm type system with arbitrary smooth function H . If the function H belongs to a set H (deﬁned in Section 4), then we obtain the existence and uniqueness of global strong solutions and global weak solutions to the system. Our obtained results generalize and improve considerably recent results in [38, 39].

1. Introduction. In this paper, we consider the following two-component system proposed by Xia, Qiao and Zhou in [33]: where H is an arbitrary function of u, v and there derivatives. They proved that this system is integrable through its Lax pair and infinitely many conservation laws. By choosing suitable H, they also investigated the bi-Hamiltonian structure and the interaction of multi-peakons.
Since H is an arbitrary function of u, v and their derivatives, (1) contains a large number of equations. For example, as v = 2 and H = −u, (1) is reduced to the standard Camassa-Holm (CH) equation which was derived by Camassa and Holm [3] as a shallow water model. The CH equation has a bi-Hamiltonian structure [26,21] and is completely integrable [3,5]. The CH equation possesses peakon solutions of the form Ce −|x−Ct| , which are orbitally stable [14,15]. It is worth pointing out that the peakons, as solitary waves with a peak at the crest, model accurately the famous wave of greatest height pattern of the free-boundary incompressible Euler equations [6,9,7,32]. The Cauchy problem and initial boundary problem of the CH equation has been studied extensively. It was shown in [8,11,31] that the CH equation has a unique global solution in the spaces C([0, T ); H s ) if the initial data u 0 ∈ H s with s > 3 2 . The local well-posedness in Besov spaces C([0, T ); B s p,r ) with s > max{ 3 2 , 1 + 1 p } was studied in [16]. Global strong solutions of the CH equation under some sign conditions were discussed in [4,8,11,17,18], and blow-up solutions in finite time were established in [4,8,10,12,27,17,18]. The global existence of weak solutions of the CH equation were established in [13,34], the global conservative and dissipative solutions were studied in [1,2,24].
One can refer to [30,20,23] for the Lax pair, peakon and soliton solutions, local well-posedness and blow-up phenomena of (3). If H is any an arbitrary polynomial in u, v and their derivatives, Zhang and Yin [37] proved the local well-posedness for (1) in Besov spaces; Hu and Qiao [25] studied the analyticity and the Gevrey regularity for (1). Recently, for some special choices of H : f, g a one-order polynomial of (u, v, u x , v x ), blow-up phenomena were studied in [35,36,37,39]. If H = av 2 + bv + c, under some sign conditions, global strong solutions and global weak solutions were obtained under some sign conditions on the initial data [38]. It is not clear whether or not there exists global solutions to (1) with more general functions H.
In this paper, H is a polynomial function of (u, u x , v, v x ). We aim to find out a function set H, with H ∈ H, the system (1) may have global strong solutions and global weak solutions. For the strong solutions, by taking advantage of the sign-preserving property and two conservation laws of (1), we observe that with a large class of H, defined by H, we can control the H 1 -norm of (u, v) and the L ∞norm of H, and can get upper bounds for H x in finite time, under some suitable sign condition assumption on the initial data. Therefore, the L ∞ -norm of (m, n) is bounded in finite time, and the solution (m, n) exists globally in time. For the weak solutions, the main idea is based on the approximation of smooth global solutions (m k , n k ). Weak convergence for the high-order nonlinear terms, such as m k H k , can be obtained on account of the strong convergence for u k , u k x , v k , v k x . By a regularization technique, we then show that the obtained solution depends continuously on time in some sense. The uniqueness is proved by estimating the W 1,q -norm of the difference of two solutions.
The rest of our paper is then organized as follows. In Section 2, we recall some properties about the strong solutions of (1). In Section 3, we establish the global existence of strong solutions of (1). In Section 4, we introduce the definition of the set H and list some elements in H. The last section is devoted to establishing the existence and uniqueness of the global weak solutions of (1).

2.
Preliminaries. H is a polynomial function of (u, u x , v, v x ). We begin with recalling the local well-posedness result and a blow-up criteria, a conservation law and the sign-preserving property for (1).
Consider the following initial value problem Then it has a unique solution q ∈ C 1 ([0, T ] × R; R). Moreover, the mapping q(t, ·) Furthermore, for any t ∈ [0, T ), we have We then recall a partial integration result for Bochner spaces.
then f, g are a.e. equal to a function from [0, T ] into L 2 (R) and for all s, t ∈ [0, T ], where ·, · is the H −1 and H 1 duality bracket.
3. Global strong solutions. For simplicity, denote G(t, x) = 1 2 (u − u x )(v + v x ). Proposition 1. Let (m 0 , n 0 ) ∈ H s with s > 1 2 , and let T be the maximal existence time of the corresponding solution (m, n) to (1). If the polynomial H(u, u x , v, v x ) satisfies (1) there exists a monotone increasing function g(t) on R + , such that t ∈ [0, T ), there exists a monotone increasing function f (t) on R + , such that t ∈ [0, T ), then T = ∞ and the solution (m, n) exists globally in time.
Proof. If H satisfies the conditions (1) and (2), then we deduce from (4) in Lemma 2.2 that Since the mapping q(t, ·) is an increasing diffeomorphism of R, we get According to the continuation criterion in Lemma 2.1, (m, n) exists globally in time.
According to (4)-(5) in Lemma 2.2, the solution (m, n) has sign-preserving property. The following lemma will be frequently used throughout the paper.
Proof. We only consider the case m 0 ≥ 0 since the others are similar. Applying (4), we have m(x, t) ≥ 0. Noticing the following relations Next we present another conservation law for (1).
Integration by parts, we deduce that If the initial data (m 0 , n 0 ) do not change sign, then we have the following global well-posedness result.
where C(t) = C(t, K 1 , K 2 , K 3 ) depends on K 1 , K 2 , K 3 and is increasing with t. Then (H − G)(t) L ∞ ≤ C(C(t), t, l), and the solution exists globally in time.
Proof. From (1), we obtain 1 2 Since m 0 , n 0 do not change sign, applying Lemma 3.1, we get If H Satisfies the condition (1'), then direct calculation gives Then, for all t ∈ [0, T ), ZENG ZHANG AND ZHAOYANG YIN Due to Lemma 3.1, Since H is l-order polynomial, we get Combining this inequality with the condition (2) in Theorem 3.3, and by virtue of Proposition 1, we see that the solution (m, n) exists globally. Next, if p = ∞, by the estimates for H x and H − G L ∞ , we readily obtain from (4)- (5), If 1 ≤ p < ∞, then by virtue of (1) and integration by parts, we have Applying Gronwall's inequality then gives m L p ≤ m 0 L p C(C(t), t, l). Similar arguments shows n L p ≤ n 0 L p C(C(t), t, l). This completes the proof of the theorem.
We mention that in Theorem 3.3, m 0 , n 0 do not change sign, Lemma 3.1 can be used. We also mention that for some H ∈ H, additional initial condition such as m 0 n 0 ≤ 0 should also be added in Theorem 3.3. Then we claim that H is not empty. In fact, it contains many functions. Here we lists a few examples.
First, similar arguments as that in Case 1, we get that v Some deformations and generalizations of Case 2, for example, , also belong to the set H.
Applying the conservation law in Lemma 2.2, we get Thus, applying Gronwall's lemma to (7)- (8) shows Some deformations and generalizations of Case 3, for example,  We infer from 4-5 in Lemma 2.2 that Since supp m 0 ∈ [b, +∞), supp n 0 ∈ (−∞, a], and noticing the mapping q(t, ·) is an increasing diffeomorphism of R, we get m(t, y) = 0, if y < q(t, b), n(t, y) = 0, if y > q(t, a). Hence, Since a ≤ b, we readily get q(t, a) ≤ q(t, b). We then deduce that H = (u−u x ) k1 (v+ v x ) k2 = 0 on R. Remark 1. We can verify that (i) H = av 2 + bv + c with a > 0, and (ii) H = bv + c with b ≥ 0, n 0 ≥ 0 or b ≤ 0, n 0 ≤ 0, which were studied in [37], belong to the set H (refer to Case 1). Therefore, our obtained global result in Theorem 3.3, to a large extent, generalizes and covers the recent global result of Theorem 3.8 in [37].

Global weak solutions.
Theorem 5.1. Let (m 0 , n 0 ) ∈ L p for some 1 < p ≤ 2, m 0 , n 0 do not change sign, and H ∈ H (See Section 3). Then the system (1) has a global solution Proof.
Step 2. Denote p : 1 p + 1 p = 1. Note that (1 − ∂ xx ) −1 f = 1 2 e −|x| * f. Applying Young's inequality for the convolution of two functions, we deduce that, for s ∈ [p, ∞], 1 + 1 Similarly, we have K k 2 , K k 3 ≤ C( m 0 L p , n 0 L p ). Then, for fixed T > 0, By (6) in Theorem 1, we find that (m k , n k ) is bounded in the space L ∞ (0, T ; L p (R)). Note that . Applying Youngs inequality for the convolution of two functions, we obtain Thus, it has a subsequence still denote by k, such that (m k , n k ) → (m, n) in C w ([0, T ], L p ). Taking 1 2 e |x−·| ∈ L p (R) and 1 2 sign(x − ·)e |x−·| ∈ L p (R) as a text function, we obtain a.e. on (0, T ) × R, as k → ∞. By (*), we obtain (u, v) ∈ C w (0, T ; W 1,s )∩C 1 (0, T ; L s ) with p ≤ s ≤ ∞ for arbitrary time T . Further more, by using the monotone convergence theorem, we have for all p ≤ s 1 < ∞, Step 3. Since H = H(u, u x , v, v x ) and G = G(u, u x , v, v x ) are polynomials of (u, u x , v, v x ) , we readily obtain H k → H ∈ L s1 loc (R + ×R), G k → G ∈ L s1 loc (R + ×R) for all s 1 ∈ [p, ∞). Thus, product items, such as M k H k , can be well handled, we then claim that are satisfied in the sense of distributions.
Step 4. We show that (u is continuous on R + . As (m, n) solves (9) in the sense of distribution, we see for a.e. t ∈ R + , with ρ k (x) = kρ(kx) the mollifier defined in Step 1. Multiplying with ρ k * u and ρ k * u x respectively, and in view of Lemma 2.3, we have for a.e. t ∈ R + , Note that (m, n) ∈ L ∞ (0, T ; L p (R)), u, v, u x , v x are bounded in L ∞ (0, T ; L s ) with p ≤ s ≤ ∞, and ρ k * f L q → f L q as k → ∞ with 1 ≤ q < ∞. So letting k → ∞, we have This proves u ∈ C([0, T ]; H 1 ). Likewise, v ∈ C([0, T ]; H 1 ). This completes the proof of the theorem.
For the uniqueness, we estimate the W 1,q (q ≤ p < ∞) norm of the difference of two solutions. We have the following theorem. Proof. Following exactly the same line as in Theorem 5.1, and by using (6) in Theorem 3.3 for the approximation of smooth solutions (m k , n k ), the global weak solution (m, n) obtained in Theorem 5.1 also belongs to the space L ∞ (0, T ; L ∞ ). This completes the existence part.
In order to prove (12), we only give details below for special two terms which are underlined above, the other terms can be dealt with analogously. where L q ≤ C(T ). Above we have used (10)- (11). Since ρ k * f − f L s → 0 as k → ∞ with 1 ≤ s < ∞, we have We then readily have R 1k (t) → 0 as k → ∞. Thus, R k (t) belongs to the class (13).
We now deal with the first term on the right hand side of (14). For simplicity, we concentrate on the case H = v 2 , thenH = (v 1 + v 2 )ṽ. For the other choice of H in the set H, one can follow the similar steps to do with and can have the same conclusions.
Letting k → ∞, we obtain by the Lebesgue's dominates convergence theorem that X(t) ≤ e C(T )t X(0) with X(t) = ũ(t) q L q + ũ x (t) q L q + ṽ(t) q L q + ṽ x (t) q L q . Uniqueness is thus obtained.
Remark 2. For the uniqueness, the addition initial condition (m 0 , n 0 ) ∈ L ∞ was required in Theorem 5.2. Under this condition, we get (m, n) ∈ L ∞ (0, T ; L ∞ ), therefore, for a general polynomial function H ∈ H, H 1x L ∞ (0,T ;L ∞ ) can be bounded in (15). This addition condition is technical.

Remark 3.
It is worth mentioning that if H = H(v) (or H = H(u)) ∈ H, the addition initial condition (m 0 , n 0 ) ∈ L ∞ is unnecessary to obtain the uniqueness of weak solutions. It is because H 1x L ∞ (0,T ;L ∞ ) can be controlled by v 1 L ∞ (0,T ;L ∞ ) + v 1x L ∞ (0,T ;L ∞ ) , which is already bounded according to (11).