On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators

In this paper, we mainly consider the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators: $m=(1-\partial_x^2)^su, s>1$. By Littlewood-Paley theory and transport equation theory, we first establish the local well-posedness of the generalized b-equation with fractional higher-order inertia operators which is the subsystem of the generalized two-component water wave system. Then we prove the local well-posedness of the generalized two-component water wave system with fractional higher-order inertia operators. Next, we present the blow-up criteria for these systems. Moreover, we obtain some global existence results for these systems.

1. Introduction. In this paper we consider the Cauchy problem of the following generalized two-component shallow water wave system with fractional higher-order inertia operators [35]: x ∈ R, ρ(0, x) = ρ 0 (x), x ∈ R, where s > 1, a = 1 is a real parameter, α is a constant which represents the vorticity of underlying flow, and κ > 0 is an arbitrary real parameter. The system (1.1) is the generalization of the case s = 1, namely, m = (1 − ∂ 2 x )u = u − u xx (see [29], [39] and [50]).
The system (1.1) with s > 1 was recently introduced by Escher and Lyons in [35]. It is the generalization of the same model (1.1) with s = 1 in [29]. In [29], the authors proved the local well-posedness of (1.1) with s = 1 by using a geometrical framework and they studied the blow-up scenarios and global strong solutions of (1.1) in the periodic case. In [39], Guan et al. studied the local well-posedness of (1.1) on the line in supercritical Besov spaces, and several blow-up results and the persistence properties. In [50], He and Yin studied the local well-posedness of (1.1) with s = 1 in the critical Besov spaces on the line and the existence of analytic solutions of the system. In [35], for s > 1, by a geometric approach, the authors gave a blow-up criteria to ensure the geodesic completeness on the circle with s > 3 2 , a = 2, κ ≥ 0 for the C ∞ initial data. However, the Cauchy problem for (1.1) with s > 1 on the line for more general initial data has not been studied yet. Also, the local well-posedness of (1.2) with general s > 1 in Besov spaces with low regularity have not been investigated yet. In this paper, using the Littlewood-Paley theory and transport equation theory, we obtain the local well-posedness of both (1.1) and (1.2) for s > 1, with the initial data in certain Besov spaces. Generally speaking, if the initial data (u 0 , ρ 0 ) is of high regularity, i.e., (u 0 , ρ) ∈ B 2s+qm p,r × B qm+1 p,r (the definition and properties of Besov spaces will be presented in Section 2, for easier understanding, the readers can take p = r = 2 to obtain H σ = B σ 2,2 ), with q m > max( 1 2 , 1 p ), we can readily obtain the local well-posedness results of (1.1), by the Littlewood-Paley theory and transport equation theory (see Remark 3.2). However, we will show that u 0 does not need such high regularity. In these cases that u 0 is of regularity less than 2s, we will face the difficulty of the loss of regularity. To overcome such difficulty, we introduce the function denotes the integer part of s), and transform (1.1) or (1.2) into the forms that v, u are unknown functions, rather than m. Then we can proceed the local well-posedness results in Besov spaces with lower regularity (see Theorem 3.3 and Theorem 3.11).
Besides, we will present the blow-up criteria of (1.1) and (1.2), for some special s, or, for the initial data in Sobolev spaces with sufficiently high regularity (i.e., u 0 ∈ H q with q ≥ 2s). With the aid of these blow-up criteria, we can obtain several global existence results for (1.1) and (1.2). We point out here that when dealing with the second component of (1.1) by the classical Kato-Ponce inequality, it will arise the term ρ x L ∞ . However, we can avoid this term by Lemma 2.8, which was introduced by Li and Yin in [55]. Namely, we can obtain the blow-up criteria only involved the term u x .
Our paper is organized as follows. In Section 2, we give some preliminaries which will be used in Section 3. In Section 3, we establish the local well-posedness of the Cauchy problem associated with (1.1) and with (1.2) in Besov spaces. In Section 4, we discuss the blow-up criteria and the global existence of strong solutions to (1.1) and (1.2).
Notations. In the following, we denote by A B to simplify the writing A ≤ CB for some generic constant C which may depend on some certain parameters independent of A and B. Given a Banach space Z, we denote its norm · Z . Since all spaces of functions are over R, for simplicity, we drop R in our notations of function spaces if there is no ambiguity.

2.
Preliminaries. In this section, we will recall some facts on the Littlewood-Paley decomposition, the nonhomogeneous Besov spaces and their some useful properties. We will also recall the transport equation theory, which will be used in our work. For more details, the readers can refer to [1,25].
Then for all u ∈ S , we can define the nonhomogeneous dyadic blocks as follows.
where the right-hand side is called the nonhomogeneous Littlewood-Paley decomposition of u.
Remark 2.2. [1,25] (1) The low frequency cut-off operator S q is defined by (2) The Littlewood-Paley decomposition is quasi-orthogonal in L 2 in the following sense: for all u, v ∈ S (R).
(3) Thanks to Young's inequality, we get where C is a positive constant independent of q.
In the following lemma, we list some important properties of Besov spaces.
(4) Algebraic properties: ∀s > 0, B s p,r L ∞ is an algebra. Moreover, B s p,r → L ∞ , provided s > 1 p or s ≥ 1 p and r = 1, hence B s p,r is an algebra under such conditions. (5) Complex interpolation: Logarithm interpolation: there exists a constant C such that for all s ∈ R, ε > 0, (1) If s > 0, 1 ≤ p, r ≤ ∞, u, v ∈ B s p,r ∩L ∞ . then there exists a constant C = C(s) such that Proposition 2.6.
[1] For all 1 ≤ p, r ≤ ∞ and s ∈ R, Now we state some useful results in the transport equation theory, which are crucial to the proofs of our main theorems later.
S ) solves the following 1-D linear transport equation: then there exists a constant C depending only on p, r and σ, such that the following statements hold: or hence, Lemma 2.9. [58] For the solution f ∈ L ∞ (0, T ; B we then have, dτ and C = C(p, r).
) and the inequalities of Lemmas 2.7-2.9 3. Local well-posedness. In this section, we are going to study the local wellposedness of the system (1.1) and the equation ( Firstly, we introduce some notations to simplify our presentation. To begin with, we will give a remark, as follows, which states the local wellposedness of (m, ρ) with initial data in Besov spaces with high regularity.

Remark 3.2.
It is readily to see that the system (1.1) is local well-posed for p,r (T ), and (m, ρ) depends continuously on the initial data (m 0 , ρ 0 ).
In the above remark, since m 0 ∈ B qm p,r , then u 0 ∈ B qm+2s p,r . It is of too high the regularity, which is not natural, in compare with that the original Camassa-Holm equation only requires that u 0 ∈ B q p,r for q > 1 + max{ 1 2 , 1 p } or q = 1 + max{ 1 2 , 1 p } if r = 1. We are going to present the local well-posedness result of the system (1.1) in Besov spaces with lower regularity. Before that, we will firstly present a theorem to state the local well-posedness result of (1.2) with s > 1.
or the critical condition Given the initial data u 0 ∈ B q p,r , then the equation (1.2) has the unique solution u ∈ E q p,r (T ) for some positive T , and u depends continuously on the initial data u 0 .
Moreover, suppose T * is the lifespan of the solution, then there exists a positive c, such that We are going to prove Theorem 3.3 by several steps.
which may give some information of the lost regularity by transposing the derivatives. From (1.2), we see, according to Leibniz's formula, For simplicity, we omit the coefficients, and write, Note that every term in the brackets is of odd order, and Similarly, we can write the remain terms ∂ 2 x u∂ 2k−1 x v, ..., ∂ 2k x u∂ x v, and so on, in the forms that the orders of u and v are as close as possible, but the orders of u should not be less than that of v. Now, we can write (1.2) as where "l. o. t." means "lower order terms".
Step 1. Constructing Approximate Solutions and Uniform Bounds. Here we only consider the cases s = [s], since the cases s = [s] are similar. Starting from u 0 := 0, we define by induction a sequence (u n , v n ) n∈N with of smooth functions by solving the following linear system: Since p satisfies the condition (3.1), and Assume v n ∈ E q−2β p,r (T ) for all positive T (the validity for n = 0, n = 1 is obvious). According to Lemma 2.7, we have v n+1 . For simplicity, we only estimate the first and the last highest order term of G(u n , v n ). In fact, the lower order terms can be controlled more easily. We can see Here we applied Lemma 2.5 (2) under the facts that , these inequalities hold true since (p, q, r, s) satisfies the condition (3.1). In summary, we have We may assume C ≥ 1 and fix a T > 0 such that By induction, we can see that, for all t ∈ [0, T ] and n ∈ N, Therefore, (v n ) n∈N is uniformly bounded in L ∞ (0, T ; B q−2β p,r ). Returning to the system (T n ), we can conclude that (v n ) n∈N is uniformly bounded in E q−2β p,r (T ). Now, for all (n, l) ∈ N 2 , taking the difference between the systems (T n+l ) and (T n ) and denoting w n,l := v n+l − v n , we have with w n+1,l (t, x)| t=0 = w n+1,l On the other hand, we need to control the term G(u n+l , v n+l ) − G(u n , v n ). As before, for simplicity, we only estimate the first and the last term of the highest order. At first, we can see . Also, we can obtain, . Hence, we can estimate G(u n+l , v n+l ) − G(u n , v n ) by adding and subtracting terms, for an example, The other terms can be controlled similarly. All in all, Now, we turn to the initial data, where we used the fact that ∆ j ( n+l j =n+1 According to the uniform boundedness of {v n } n∈N in C([0, T ], B q−2β p,r ), there exists a C T = C(s, q, p, r, T, v 0 B q−2β p,r ) such that, for all t ∈ [0, T ], ). We then conclude that v ∈ E q−2β p,r (T ).

Uniqueness.
Set p,r . Along the similar computations as previous, we can get the following estimates which gives the uniqueness: (3.8) Case (2). This is the critical case of the integer s. Applying Lemmas 2.7-2.9 to , we obtain, for any t ∈ [0, T ], dt . For simplicity, we only estimate the following worst term, other terms are simpler. We see On the other hand, since B The initial data can be estimated similarly as the non-critical cases We finally have Similar to the statement of non-critical cases, we obtain that {u n } n∈N is a Cauchy Remark 3.4. Different from the critical case of s = 1, here we do not need to apply the logarithm interpolation and Osgood's Lemma, this is because B is a Banach algebra, but B 1 p p,∞ is not. Proposition 3.5. If u 0 belongs to a small neighbourhood of u 0 in B q p,r , then we can proceed the existence of the unique solution u ∈ E q p,r (T ) of (3.3) with initial data u 0 for a certain positive T . The estimates (3.8) and (3.9), with the interpolations (2.1) and (2.2) in Lemma 2.4, ensures the Hölder continuity of the flow map from the initial data spaces B q p,r to the space E q p,r (T ), for any q < q. Remark 3.6. From Steps 1-2, we see that, when s = [s], to deal with the term , we need the additional regularity, namely, q > s+max(1, 1 p + β, 1 p + 1 − β). This means that there defects the regularity at least of 1 2 when s in not an integer. Denote F (u, v) for q > s + max( 1 2 , 1 p ) (or with " = " if r = 1), then we do not need the additional regularity. We know that (3.10) and (3.11) hold true for s is an integer and s ≥ 1.
Here we conjecture that (3.10) and (3.11) hold true for any s > 1.
Step 3. Continuity with respect to the initial data. To gain the continuity with respect to the initial data, we introduce the following lemma.
As before, we only focus on the cases s = [s]. Now, suppose v n ∈ C([0, T ]; B q−2β p,r ) is the solution to (3.3) with the initial data v n 0 ∈ B q−2β p,r , namely, for all n ∈N, we have To prove the continuity of in C([0, T ]; B q−2β p,r ) with respect to the initial data in B q−2β p,r for r < ∞, we need to prove the following lemma.
Here the positive T satisfies the condition similar to (3.5), namely ).

HUIJUN HE AND ZHAOYANG YIN
We decompose v n x into v n x = y n + z n such that ∂ t y n + u n ∂ x y n = f ∞ , Similar to the proof of the existence, we can obtain (3.12) Since q − 1 < q and B q−1 p,r → B q−2β−1 p,r , we can proceed that u n tends to u ∞ in L 1 (0, T ; B q−2β−1 p,r ), which ensures that, with the application of Lemma 3.7, (3.13) To control z n , we need to estimate On the one hand, we can see, On the other hand, take the derivative in G(u n , v n ) − G(u ∞ , v ∞ ) term by term. For an example, has the similar estimates as in (3.14). Hence, Applying Lemmas 2.7-2.9 to the equation of z n in B q−2β−1 p,r , we have

TWO-COMPONENT SHALLOW WATER SYSTEM 1523
From (3.12) and (3.13), for any ε > 0, we can choose a sufficiently large n such that Thanks to Gronwall's inequality, we get for some constantC =C(s, q, p, r, M, T ). Hence we gain the continuity of in C([0, T ]; B q−2β p,r ) with respect to the initial data in B q−2β p,r for r < ∞.
When r = ∞, by Proposition 3.5 , we see that

Applying Lemma 2.4, we have
and tends to zero as j → ∞ and v n − v ∞ L ∞ (0,T ;B q−2β−1 p,r ) tends to zero as n → ∞. Then the right hand-side of (3.16) may be arbitrarily small for j large enough. For such fixed j, we let n tend to infinity so that the right hand-side of (3.17) tends to zero. Thus, we conclude that v n (t) − v ∞ (t), φ tends to zero as n → ∞ for the case r = ∞. Hence, finally, according to above three steps, we finish the proof of Theorem 3.3. Remark 3.9. For some special a, the result of Theorem 3.3 may not be the best. For example, when s = 2, the system (1.2) can be transformed to Letting a = 5 3 , we eliminate the worst term (u 2 xx ) x to obtain To ensure the local well-posedness result, we need to have
Theorem 3.11. Suppose (p, q, r, s) satisfies the (3.1) or (3.2), and q 1 ∈ R satisfies Given the initial data (u 0 , ρ 0 ) ∈ B q p,r × B q1 p,r , then the system (3.20) has the unique solution (u, ρ) ∈ E q p,r (T ) × E q1 p,r (T ) for some positive T , and (u, ρ) depends continuously on the initial data (u 0 , ρ 0 ). Moreover, suppose T * is the lifespan of the solution, then there exists a positive c, such that Remark 3.12. The "1" in Proof of Theorem 3.11. For simplicity, here we only deal with the cases that q satisfies (3.1) and q 1 satisfies (3.21). (The proof of the cases that q 1 satisfies (3.22) is similar to Case (2) in the proof of Theorem 3.3, so we omit it here.) Also, we only give here the core of the proof -the convergence of the approximated sequence. In fact, we can obtain the difference equation: (3.23) We have proven that p,r . We only need to estimate the following terms: where we used the facts that q satisfies (3.1) and q 1 satisfies (3.21). Since p,r , according to the uniform boundedness of {u n , ρ n } n∈N in E q p,r (T ) × E q1 p,r (T ), with the application of Lemmas 2.7-2.9, we have, for every t ∈ [0, T ], (1). When s = 1, then q m + 1 = q − 2 + 1 = q − 1. Hence, in the case s = 1, q 1 must equals to q − 1. These are the local well-posedness results in [39] and [50]. (2). If q m > max( 1 2 , 1 p ) (or q m = 1 p if 1 ≤ p ≤ 2 and r = 1), then we can choose q 1 = q m + 1, and this is the case of Remark 3.2.
4. Blow-up criteria and global existence.

4.1.
Blow-up criteria for (1.2). In this subsection, we will study several blow-up criteria for the equation (1.2).
p,r . Let T * be the lifespan of the solution u ∈ C([0, T * ); B q p,r ). If T * is finite, then 1) or equivalently, Proof. Theorem 3.3 and the condition of q ensure that u ∈ C([0, T * ); B q p,r ). Since s = k is an integer, we see that v = u in (3.3): Applying Lemma 2.7 (3) with V (t) = t 0 u x L ∞ dt , we can obtain, for every t ∈ [0, T * ), Since q − 1 > q − 2 > ... > q − k > 0, with the aid of Lemma 2.5, Letting u C k := u L ∞ + u x L ∞ + ... + ∂ k x u L ∞ , then the Gronwall lemma leads to Claim. If the pseudo-differential operator P (D) is S −γ with γ > 0 and f ∈ L ∞ , then P (D)f L ∞ f L ∞ . Indeed, according to the logarithm interpolation (2.3), we have, here we used the facts that L ∞ → B 0 ∞,∞ → B −γ ∞,∞ , and the function x → x log(e + C x ) is increasing. Now, applying the above Claim, we can see that, . Applying the L ∞ estimate for the transport equation (4.3), we have Differentiating (4.3) once with respect to x, we get Expand the highest order term: Thus, applying the Claim again, we obtain Taking such procedure for k times, we get Gronwall's inequality then leads to By Gagliardo-Nirenberg's inequality and Young's inequality, ∂ i x u x L ∞ u x L ∞ + ∂ k x u L ∞ , (i = 1, 2, ..., k − 1). Hence Therefore, (4.4) and (4.5) lead to (4.1). The equivalence of (4.1) and (4.2) is obvious.
Now we can state other blow-up criterion in certain Sobolev spaces for the twocomponent system (1.1). Lemma 4.10. Suppose a = 2, s = [s] = k ≥ 2, q > s + 1 2 . Let T * be the lifespan of the solution (u, ρ) ∈ C([0, T * ); H q × H q1 ) to (1.1) with the initial data (u 0 , ρ 0 ) ∈ H q × H q1 . If T * is finite, then lim sup Proof. Since s = [s] = k, we can write the system (3.20) in the form Applying the Gronwall's lemma, we can obtain, for every t ∈ [0, T * ), At the same time, (4.12) leads to At first, we consider the cases q ≥ k + 1, and we need to control u H k+1 .
Firstly, we point out that, when applying the Littlewood-Paley decomposition ∆ j and taking L 2 -norm to obtain the a priori estimates (2.4), the term