On an $N$-Component Camassa-Holm equation with peakons

In this paper, we are concerned with \begin{document}$N$\end{document} -Component Camassa-Holm equation with peakons. Firstly, we establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory. Secondly, we present a precise blowup scenario and several blowup results for strong solutions to that system, we then obtain the blowup rate of strong solutions when a blowup occurs. Next, we investigate the persistence property for the strong solutions. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.


(Communicated by Adrian Constantin)
Abstract. In this paper, we are concerned with N -Component Camassa-Holm equation with peakons. Firstly, we establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory. Secondly, we present a precise blowup scenario and several blowup results for strong solutions to that system, we then obtain the blowup rate of strong solutions when a blowup occurs. Next, we investigate the persistence property for the strong solutions. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.

YONGSHENG MI, BOLING GUO AND CHUNLAI MU
can itself be derived from the Korteweg-deVries equation by tri-Hamiltonian duality. The Camassa-Holm equation was originally proposed as a model for surface water waves in the shallow water regime (see [5,20]) and has been studied extensively in the last twenty years because of its many remarkable properties. Camassa-Holm equation has traveling wave solutions of the form ce −|x−ct| , called peakons, which capture the main feature of the exact traveling wave solutions of greatest height of the governing equations (see [8,49,10]). Moreover, the shape of some peakons is stable under small perturbations, making these waves recognizable physically (see [22,43]). It was shown in [21,16,9] that the inverse spectral or scattering approach was a powerful tool to handle Camassa-Holm equation and Eq. (2) is a completely integrable and The geometric formulations [18,19,42,45], wellposedness and breaking waves, meaning solutions that remain bounded while its slope becomes unbounded in finite time [7,13,14,15,44] have been discucssed. Moreover, the Camassa-Holm equation has global conservative solutions [3,35] and dissipative solutions [4,36]. An alternative modified Camassa-Holm equation was studied in [28]. Multi-component versions of the Camassa-Holm equation have been introduced and studied in [32,33,38,39,40,17]. For n = 2, Eq. (1) becomes the two-component Camassa-Holm equation introduced in [30,29] m 1,t + m 1,x u 1 + 2m 1 u 1,x + (m 1 u 2 ) x + m 2 u 2,x = 0, m 2,t + m 2,x u 2 + 2u 2 u 2,x + (m 2 u 1 ) x + m 1 u 1,x = 0, where m i = u i − u i,xx , i = 1, 2. The Cauchy problem of the above two-component Camassa-Holm equation has been studied in [30,29]. It has been shown that this system is locally well-posed for initial data (u 0 , v 0 ) ∈ H s × H s with s > 3 2 . Also, it has blowup solutions modeling wave breaking, moreover, an existence result for a class of local weak solutions was also given. In [50], the authors presented some new criteria on blow-up, global existence and blow-up rate of the solution. Moreover, they discussed persistence properties of this system. In [47], the authors studied the global conservative and dissipative solutions of system 1.2. In [48], the authors obtained the compact and bounded absorbing set and the existence of the global attractor for viscous system (3) with the periodic boundary condition in H 2 by uniform prior estimate. For n = 3, it was shown in [31] that the Eq. (1) has two peakon solitons. In [37], the authors establish the local well-posedness of the initial value problem for system (1) with n = 3 and present a precise blowup scenario and several blowup results for strong solutions to that system. Moreover, they determine the blowup rate of strong solutions to the system when a blowup occurs.
Motivated by the references above, we establish firstly the local well-posedness for the strong solutions to the Cauchy problem of system (1). The proof of the local well-posedness is inspired by the argument of approximate solutions by Danchin [23,24,25] in the study of the local wellposedness to the Camassa-Holm equation. Secondly, we then present a precise blowup scenario, several blowup results for strong solutions to that system, we also obtain the blowup rate of strong solutions to the system when a blowup occurs. Next, we investigate the persistence property for the strong solutions to (4) in L ∞ space which asymptotically exponential decay at infinity as their initial profiles. The idea comes from a recent work of Zhou and his collaborators [34] for the standard Camassa-Holm equation (for slower decay rate, we refer to [46] ). Finally, we study initial value boundary problems of the N -Component Camassa-Holm equation on the half-line subject to homogeneous Dirichlet boundary conditions. Our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.
The rest of this paper is organized as follows. In Section 2, we prove the local wellposedness of the Cauchy problem of system (1) in the Besov spaces. In Section 3, the blowup phenomena is considered. Section 4 is devoted to the study of the persistence property for the strong solutions. Finally, we consider the initial boundary value problem.
2. Local well-posedness in B s p,r , p, r ∈ [1, ∞], s > max{ 3 2 , 1+ 1 p }. In this section, we shall establish local well-posedness of the initial value problem (1) in the Besov spaces. Note that G(x) : (1) takes the form of a quasi-linear evolution equation of hyperbolic type Introducing the Fourier integral operators First, for the convenience of the readers, we recall some facts on the Littlewood-Paley decomposition and some useful lemmas.
Notation. S stands for the Schwartz space of smooth functions over R d whose derivatives of all order decay at infinity. The set S of temperate distributions is the dual set of S for the usual pairing. We denote the norm of the Lebesgue space L p (R) by || · || L p with 1 ≤ p ≤ ∞, and the norm in the Sobolev space H s (R) with s ∈ R by || · || H s .

YONGSHENG MI, BOLING GUO AND CHUNLAI MU
Then for all f ∈ S (R d ), the dyadic operators ∆ q and S q can be defined as follows where the right-hand side is called the nonhomogeneous Littlewood-Paley decomposition of f . Lemma 2.1.(Bernstein's inequality [24]) Let B be a ball with center 0 in R d and C a ring with center 0 in R d . A constant C exists so that, for any positive real number λ, any non negative integer k, any smooth homogeneous function σ of degree m and any couple of real numbers (a, b) with b ≥ a ≥ 1, there hold = ∩ s∈R B s p,r . Proposition 2.2. (see [24]) Suppose that s ∈ R, 1 ≤ p, r, p i , r i ≤ ∞(i = 1, 2). We have (1) Topological properties: B s p,r is a Banach space which is continuously embedded in S .
(2) Density: , if p 1 ≤ p 2 and r 1 ≤ r 2 . B s2 p,r2 → B s1 p,r1 locally compact, if s 1 < s 2 . (4) Algebraic properties: ∀s > 0, B s p,r ∩ L ∞ is an algebra. Moreover, B s p,r is an algebra, provided that s > n p or s ≥ n p and r = 1. (5) Complex interpolation: (6) Fatou 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 ||u|| B s p,r ≤ lim inf n→∞ u n B s p,r .
(7) Let m ∈ R and f be an S m -multiplier (i.e., f : R d → R is smooth and satisfies that ∀α ∈ N d , there exists a constant C α , s.t. Then the operator f (D) is continuous from B s p,r to B s−m p,r . Now we state some useful results in the transport equation theory, which are crucial to the proofs of our main theorems later. [23,24]) Suppose that (p, r) ∈ [1, +∞] 2 and s > − d p . Let v be a vector field such that ∇v belongs to Then there exists a constant C depending only on s, p and d such that the following statements hold: (1) If r = 1 or s = 1 + d p , then  [23,24]). Let (p, p 1 , r) ∈ [1, +∞] 3 and  [23,24]) Assume that 1 ≤ p, r ≤ +∞, the following estimates hold: (i) For s > 0, (iii) In Sobolev spaces H s = B s 2,2 , we have for s > 0, where C is a positive constant independent of f and g.
. We now have the following local well-posedness result.
n . There exists a time T > 0 such that the initial-value problem (4) has a unique solution (u 1 , u 2 , · · · , u n ) ∈ E s p,r (T ) n and the map (u 1,0 , u 2,0 , · · · , ) n for every s < s when r = ∞ and s = s whereas r < ∞.
In the following, we denote C > 0 a generic constant only depending on p, r, s. Uniqueness and continuity with respect to the initial data are an immediate consequence of the following result.
2 , · · · , u (2) n ) be two given solutions of the initialvalue problem (4) with the initial data (u Proof. For s = 2 + 1 p , denote u n ) solves the transport equations j,x .
According to Lemma 2.2, we have For s > 1 + 1 p , B s−1 p,r ⊂ L ∞ is an algebra according to Proposition 2.2, so we have Since s > max{ 3 2 , 1 + 1 p }, by Proposition 2.2 and Lemma 2.4, we have Therefore, inserting the above estimates to (9) we obtain Hence, applying the Gronwall inequality, we reach (7). For the critical case s = 2 + 1 p , we here use the interpolation method to deal with it. Now let us start the proof of Theorem 1.1, which is motivated by the proof of local existence theorem about the Camassa-Holm equation in [23]. Firstly, we shall use the classical Friedrichs regularization method to construct the approximate solutions to the Cauchy problem problem (4).
p,r )) n solving the following linear transport equation by induction (10) Moreover, there is a positive T such that the solutions satisfying the following prop- Proof. Since all the data S n+1 u j,0 , j = 1, 2, · · · , n belong to B ∞ p,r , Lemma 2.3 enables us to show by induction that for all l ∈ N, the equation (10) has a global solution which belongs to (C(R + ; B ∞ p,r )) n . And from Lemma 2.2 and the proof of Proposition 2.3 and the above inequality, we have the following inequality for all l ∈ N Hence, we have Let us choose a T > 0 such that 2CT n j=1 u j,0 B s p,r < 1, and suppose by induction One obtains from (12) that And then inserting the inequalities (13) and (12) into (11) Hence, one can see that which implies that (u p,r )) n . In fact, for all l, k ∈ N, from (10), we have  Applying Lemma 2.2 again, then for every t ∈ [0, T ], we obtain Similar to the proof of Proposition 2.3, in the case of s > max{ 3 2 , 1 + 1 p } with s = 2 + 1 p , one can deduce that Arguing by induction with respect to the index l, one can easily prove that p,r ) , i = 1, 2, · · · , n and C are bounded independently of k, there exists constant C T independent of l, k such that On the other hand, for the critical points s = 2 + 1 p , we can apply the interpolation method to show that (u p,r )) n for this critical case. Therefore, we have completed the proof of Lemma 2.5.
3. Blowup phenomena. In the section, we present a precise scenario that ensure strong solutions to Eq. (4) blowup in finite time.We first give two useful results which will be used in the sequel.
In view of the above conservation law, we have This completes the proof of Lemma 3.1. Next, we present the precise blowup scenarios for solutions to Eq. (4). We first recall the following two lemmas.
where c is a constant depending only on s.
Theorem 3.1. Let z 0 = (u 1,0 , u 2,0 , · · · , u n,0 ) ∈ (H s ) n , s > 3 2 , and let T be the maximal existence time of the solution z = (u 1 , u 2 , · · · , u n ) to (4) with the initial data z 0 . If there exists M > 0 such that then the (H s ) n − norm of the solution z does not blowup in finite time.
Proof. Let z be the solution to Eq. (4) with the initial data z 0 ∈ H s , s > 3 2 , and let T be the maximal existence time of the corresponding solution z, which is guaranteed by Theorem 2.1. Throughout this proof, C > 0 stands for a generic constant depending only on s.
Applying the operator Λ s to the equation in (4), multiplying by Λ s u i and integrating over R, we obtain Let us estimate A i1 , A i2 and A i3 .
where we use Lemma 3.3 with r = s.
where we use Lemma 3.3 with r = s − 1.
where we use Lemma 3.1 and Lemma 3.2 with r = s − 1. Therefore An application of Gronwall's inequality and the assumption of the theorem yields Next, we present the precise blowup scenario.
Theorem 3.2. Let z 0 = (u 1,0 , u 2,0 , · · · , u n,0 ) ∈ (H s ) n , s > 3 2 , and let T be the maximal existence time of the solution z = (u 1 , u 2 , · · · , u n ) to (2.1) with the initial data z 0 . Then the corresponding solution blows up in finite time if and only if there exists an i (i ∈ {1, 2, · · · , n}) such that Proof. Assume that z 0 ∈ (H s ) n for some s ∈ N, s ≥ 2. Multiplying the No.i equation in (2.1) by m i = u i − u i,xx , integrating by parts, we have From the above equality, we see that if u i,x , i = 1, 2, · · · , n are bounded from below on [0, T ), i.e. there exist the positive constants M i , i = 1, 2, · · · , n such that u i,x ≥ −M i , then we have By the definition of m i , i = 1, 2, · · · , n, we have Hence, we obtain By the above inequalities and Gronwall's inequality, we get On the other hand, if there exists an i (i ∈ {1, 2, · · · , n}) such that then the solution will blowup in finite time. Applying Theorem 2.1 and a simple density argument, we deduce that Theorem 3.2 is true for all s > 3 2 . We now study the problem of finite blowup of solutions to Eq. (4). To pursue our goals, we give a useful lemma.

The function m(t) is absolutely continuous on
, a.e., on (0, T ).
, the corresponding solution to Eq. (4) blows up in finite time.
Substituting (t, 0) into (20) and using (23), we have K, for all t ∈ [0, T ), Using the same argument in Theorem 3.4, we deduce that the solution z does not exist globally in time Next, we give more insight into the blow-up rate for the wave breaking solutions to Equation (17).
Proof. By Lemma 3.1, we get the uniform bound of u i , i = 1, 2, · · · , n, which implies that the solution remains uniformly bounded. For Theorem 3.3, we set Ψ(t) = inf x∈R n j=1 u j,x , while for Theorem 3.4, we set Θ(t) = n j=1 u j,x (t, 0). Using (3.6) and noticing the proofs of Theorems 3.3 and 3.4, we can find a constant K > 0 such that Choose ∈ (0, 1 2 ). Since by Theorem 3.2, there is some t 0 ∈ (0, T ) with g(t 0 ) + λ < 0 and Θ(t 0 ) > K . Let us first prove that Since Θ is locally Lipschitz, there is some δ > 0 such that Note that Θ is locally Lipschitz and therefore absolutely continuous. Integrating the previous relation on (t 0 , t 0 + δ) yields that It follows from the above inequality that The obtained contradiction completes the proof of the relation (28). By (28)-(29), we infer For t ∈ (t 0 , T ), integrating (30) on (t, T ) to get Since Θ(t) < 0 on [t 0 , T ), it follows that By the arbitrariness of ∈ (0, 1 2 ), the statement of the theorem follows.
4. Persistence properties. In this section, we shall investigate the following property for the strong solutions to (4) in L ∞ space which asymptotically exponential decay at infinity as their initial profiles. The main idea comes from a recent work of Zhou and his collaborators [34] for the standard Camassa-Holm equation (for slower decay rate, we refer to [46] ).
where L is a nonnegative constant. In order to shorten the presentation in the sequel, we introduce Proof. In order to arrive at our result, we first introduce a weighted continuous function which is independent on t as follows where the derivative is with respect to the variable x. From the equations of (4), we have Multiplying (32) by (u i Φ N ) 2p−1 with p ∈ Z + and integrating the result in the x-variable, we get Denoting M = sup t∈[0,T ] (u 1 (t), u 1 (t), · · · , u n (t)) H s and by the Gronwall's inequality, we obtain Taking the limits in (33), we get Next, differentiating the equations in (4) in the x−variable produces the equations Using the weight function, we can rewrite (35) as Multiplying (36) by (u i,x Φ N ) 2p−1 with p ∈ Z + and integrating the results in the x-variable, it follows that For the first term on the right side of (37), we know L 2p . Using the above estimates and Hölder inequality, we deduce that Thanks to the Gronwall's inequality, it holds that Taking the limits in (38), we have Combining (34) and (39) together, it follows that A simple calculation shows that there exists C 0 > 0, depending only on θ ∈ (0, 1), such that for any N ∈ Z + , On the other hand, for a suitable function f and g, one obtains, Similarly, we can get Thus, inserting the above estimates into (40), there exists a constant C = C(M, T, Hence, for any N ∈ Z + and any t ∈ [0, T ], we have by Gronwall's inequality Finally, passing limit as N goes to infinity in (42), we obtain We complete the proof of Theorem 4.1.
Theorem 5.1. Given z 0 = z(x, 0) = (u 1,0 , u 2,0 , · · · , u n,0 ) ∈ (H s ∩ H 1 0 ) n with 3 2 < s < 5 2 , then there exist a maximal T > 0 depending only on the norm of z 0 in (H s ∩ H 1 0 ) n and a unique solution z = (u 1 , u 2 , · · · , u n ) to Eq. (43) such Moreover, the solution depends continuously on the initial data, i.e. the mapping z 0 → z(·, z 0 ) : is continuous and the maximal time of existence T > 0 can be chosen to be independent of s. Here, for simplicity, we denote H p (R + )(p = s, s−1) and H 1 0 (R + ) by H p and H 1 0 , respectively. In order to prove Theorem 5.1, we need the following Lemma.
Proof of Theorem 5.1. We first convert the initial boundary value problem of (43) into the cauchy problem of system (4). In order to do so, we extend the initial data u i,0 (x), i = 1, 2, · · · , n, defined on interval R + into an odd function defined on the line.
Proof. Following a similar argument of as in the proof of Theorem 5.1, we first extend the initial data z 0 (x) defined on the interval R + into an odd function z 0 (x) defined in (5.3) on the line. Since Lemma 5.2 and (45)- (46) show that z 0 ∈ (D s k (R)) n is an odd function. Then, following the similar proof in Theorem 5.1, we can obtain the desired result of the theorem. This completes the proof.