Liouville theorems for periodic two-component shallow water systems

We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point \begin{document}$(t_0, x_0)$\end{document} , are the identically zero solutions. Also, we derive new local-in-space blow-up criteria for the dispersive 2CH and 2DP.


(Communicated by Adrian Constantin)
Abstract. We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point (t 0 , x 0 ), are the identically zero solutions. Also, we derive new local-in-space blow-up criteria for the dispersive 2CH and 2DP.
With ρ ≡ 0 in Eq.(1.1), we find the Camassa-Holm equation, which models the wave motion on shallow water, u(t, x) representing the fluid's free surface above a flat bottom (or equivalently the fluid velocity at time t ≥ 0 in the spatial x direction) [6,21,33]. It is remarkable that the Camassa-Holm equation on line has peakons of the form u(t, x) = ce −|x−ct| , x ∈ R, t ≥ 0, c > 0 which interact like solitons [6].The Cauchy problem of the Camassa-Holm equation has been studied extensively. It has been shown that this equation is locally wellposed [9,11,36,42] for initial data u 0 ∈ H s with s > 3 2 . More interestingly, it has not only global strong solutions modeling permanent waves [11,12,16] and but also blowup solutions modeling wave breaking [10,11,12,16,36,42]. On the other hand, it has global weak solutions with initial data u 0 ∈ H 1 [7,17,44]. Moreover, the Camassa-Holm equation has global conservative solutions [4,29] and dissipative solutions [5,30].
The Cauchy problem of Eq.(1.1) on the line has been discussed in [13,27]. In [13], Constantin and Ivanov investigated the global existence and blow-up phenomena of strong solutions of Eq.(1.1). Later, Guan and Yin obtained a new global existence result for strong solutions to Eq.(1.1) and got several blow-up results [27] which improved the results in [13]. After that Hu and Yin studied the global existence and blow-up phenomena for the periodic two-component Camassa-Holm equation Eq.(1.1) in [32].
Regarding to the blow-up criteria, L. Brandolese et. al. recently derived new local-in-space blow-up criteria for a class of nonlinear partial differential equations. In [1], L. Brandolese established new local-in-space blow-up criteria for the equations modeling shallow water waves and nonlinear dispersive waves in elastic rods. Later, using the new blow-up criteria and the computation of several best constants in convolution estimates and weighted Poincaré inequalities, L. Brandolese and M. F. Cortez studied the permanent and breaking waves for the rod equation and the Degasperis-Procesi equation [2,3]. These local-in-space blow-up criteria attract our interests, as they have the specific feature of being purely local in the space variable: indeed the blow-up conditions only involve the values of u 0 (ξ) and u 0 (ξ) in a single point ξ of the real line. In [28], Duc Truc Hoang investigated wave breaking criteria for the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system. The author established a new blow-up criterion for the general case γ + c 0 α 2 ≥ 0 involving local-in-space conditions on the initial data.
Motivated by the works of the above authors (see [1,2,3,28]), we plan to investigate the new local-in-space blow-up criteria for some two-component shallow water systems, including Eqs.(1.2) and (4.2). For earlier blow-up results for the two-component shallow water systems, it is a natural way to control the quantities u(t, ) L ∞ and ρ(t, ) L ∞ by some global quantities, such as the u 0 H 1 −norm and ρ 0 L 2 −norm appeared in [13,27,31,32], or other global conditions like antisymmetry assumptions or sign conditions on the associated potential y 0 . However, we do not use any conservation laws and other global conditions rather than the derivation of new local-in-space blow-up criteria (see Eq.(3.4)) in the paper.
More specifically, we first exploite the characteristic ODE related to the system (1.1) (see Eq.(2.1)) to construct some invariant properties of the solutions and sufficiently utilizing the structure of the system itself to deduce the fine properties of the second component ρ of the system (1.1) (see Lemma 2.2). Then, by the introduction of the two auxiliary maps (see Eq.(3.1) and Eq.(3.3)), we obtain the new local-in-space blow-up criteria (see Eq.(3.4)) contrary to the previous ones as in [13,27,31,32]. Finally, we prove the following Liouville-type theorem for strong solutions to Eq.(1.1) by using Eq.(3.4) and introducing the two families of Lyapunov functions (see Eqs.(3.7) and (3.8)). For the two-component Camassa-Holm equation with dispersion, we reformulate the above theorem as follows: Remark 1.1. In fact, Theorem 1.2 can be reduced to Theorem 1.1, as We now conclude this introduction by outlining the rest of the paper. In Section 2, we briefly give some needed results including the local well-posedness of Eq.(1.2), the precise blow-up scenarios and some useful lemmas to prove Theorem 1.1. In Section 3, we give the detailed proof of Theorem 1.1. In Section 4, we establish the similar Liouville-type theorem for the two-component Degasperis-Procesi equation.

2.
Preliminaries. In the section, we briefly give the needed results to pursue our goal. We first present the local well-posedness for the Cauchy problem of Eq. [32]. Applying the Kato's semigroup theorem [34], we obtain the following local well-posedness theorem for Eq.(1.2).
Moreover, the solution depends continuously on the initial data, i.e. the mapping By the local well-posedness in Theorem 2.1 and the energy method, one can get the following precise blow-up scenario of strong solutions to Eq.(2.1).
where u denotes the first component of the solution z to Eq.(1.2) with the initial data z 0 . Since u(t, .) ∈ H 2 (S) ⊂ C m (S) with 0 ≤ m ≤ 3 2 , it follows that u ∈ C 1 ([0, T )×R, R). Applying the classical results in the theory of ordinary differential equations, one can obtain the following results of q which are useful in the proof of Theorem 1.1.
3. Proof of Theorem 1.1. In the section we prove Theorem 1.1.
We deduce from this the existence of ξ ∈ (a, b) such that Hence, we get the contradiction φ(a) ≤ φ(b).
Next we consider the case u 0 (a) < 0: introducing now the map and arguing as before, we get in this case the existence of a point ξ such that u 0 (ξ) < √ 2u 0 (ξ) < 0. Notice that in both cases we get there exists ξ ∈ S such that Finally, we suffice to establish the finite time blow-up under the above condition Differentiating the first equation in (1.2) with respect to x, we get Substituting (t, q(t, x 0 )) into Eq.(3.5), we obtain (3.6) Setting L(x) = u 2 + 1 2 u 2 x + 1 2 ρ 2 , we now introduce the C 1 functions, defined on (0, T ),
Factorizing the right-hand side of (3.9) leads to the following differential inequality A similar computation yields We first observe that Moreover, we deduce from the system (3.10)-(3.11), applying the geometric-arithmetic mean inequality, that This immediately implies T ≤ 2 H(0) < ∞, hence u(t, x) ≡ 0 for all (t, x). By using u(t, x) ≡ 0 and the second equation of (1.1), one can get that ρ t = 0. Thus ρ(t, x) = ρ(t, 0) = h(t) = 0 for all (t, x).
The proof of Theorem 1.1 is completely finished.
Remark 3.1. The "local-in-space" blowup results investigated in [28] look more general and more precise than Theorem 1.1. On the other hand, our proof of blowup criteria is shorter and sufficient for our purpose.
As a byproduct of proof of Theorem 1.1, we get the following new blow-up criterion for periodic solutions the two-component Camassa-Holm equation with or without dispersion:    2). Then, by our theorem, sign(u) = 1, 0 or −1 is well defined and independent on (t, x). Moreover, u (t, x) ≥ − √ 2|u(t, x)| for all t ≥ 0 and x ∈ S. Then, arguing as in (3.5), we deduce that, for all t ≥ 0, the map x → e sign(u) √ 2x u(t, x) is increasing.

QIAOYI HU, ZHIXIN WU AND YUMEI SUN
Combining this with the periodicity, we get the pointwise estimates for u(t, x), for all t ≥ 0, all x 0 ∈ R and x 0 ≤ x ≤ x 0 + 1: From (3.15) one immediately deduces the corresponding estimates for global solutions to the dispersive two-component the Camassa -Holm equation (1.3). 4. Application to the 2DP. In this section, we address the Cauchy problem for the periodic two-component Degasperis-Procesi system (2DP): where y = u − u xx . Eq.(4.1) was recently proposed by Popowicz in [41]. There are two cases about this system: (i) k 2 = 0, k 3 = 1 and k 1 is an arbitrary real constant, or k 2 = 0, k 1 = 1 and k 3 takes an arbitrary real value; (ii) k 2 = 1, k 1 = 2 and k 3 = 1. The construction based on the observation that the second Hamiltonian operator of the Degasperis-Procesi equation could be considered as the Dirac reduced Poisson tensor of the second Hamiltonian operator of the Boussinesq equation in [41]. The well-posedness and the blow-up phenomena for the two-component Degasperis-Procesi system were studied in [45]. Escher et. al. showed that the periodic two-component Degasperis-Procesi equation (4.2) can be regarded as geodesic equations on the semidirect product of diffeomorphism group of the circle Diff(S 1 ) with some smooth functions [22]. For ρ = 0, Eq.(4.1) becomes the Degasperis-Procesi equation [19]. The Degasperis-Procesi equation can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as for the Camassa-Holm shallow water equation [15,21,33]. The formal integrability of the Degasperis-Procesi equation was obtained in [20] by constructing a Lax pair. It has a bi-Hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions [20] which are analogous to the Camassa-Holm peakons [6,18]. An inverse scattering approach for computing N-peakon solutions to the Degasperis-Procesi equation was presented in [18,39]. Its traveling wave solutions was investigated in [35,43].
Although the Degasperis-Procesi equation is very similar to the Camassa-Holm equation in many aspects, especially in the structure of equation, there are some essential differences between the two equations. One of the famous features of DP equation is that it has not only peakon solutions u c (t, x) = ce −|x−ct| with c > 0 [20] and periodic peakon solutions [49], but also shock peakons [38] and the periodic shock waves [26]. Besides, the Camassa-Holm equation is a re-expression of geodesic flow on the diffeomorphism group [14] or on the Bott-Virasoro group [40], while the DP equation can be regarded as a non-metric Euler equation [23]. On the other hand, the isospectral problem in the Lax pair for Degasperis-Procesi equation is the third order equation cf. [20], while the isospectral problem for the Camassa-Holm equation is the second order equation (in both cases y = u − u xx ) cf. [6].
Regarding to the Cauchy problem for DP equation, plenty of works [8,25,26,37,46,47,48,49] have been done. For example, the local well-posedness and blowup phenomena to DP equation with initial data u 0 ∈ H s (R), s > 3 2 on the line and on the circle were studied in [47] and [46], respectively. The global existence of strong solutions and global weak solutions to DP equation were shown in [48,49]. Similar to the Camassa-Holm equation, the DP equation has not only global strong solutions [37,48] but also blow-up solutions [25,26,37,24]. Moreover, it has global entropy weak solutions in L 1 (R) ∩ BV (R) and L 2 (R) ∩ L 4 (R), cf. [8].
In this section, we establish the Liouville-type theorem for Eq.(4.1) with k 2 = 0. For our convenience, we rewrite Eq.(4.1) as follows: Or the equivalent form: Now we present some lemmas given in [45]. We first have the following local wellposedness result.  For initial data z 0 = u 0 ρ 0 ∈ H 2 (S) × H 1 (S), we have the following precise blow-up scenario.
where u denotes the first component of the solution z to Eq.(4.2) with the initial data z 0 , we obtain the following useful properties of q.
Using the similar argument discussed in Sec. 3, we establish the following Liouville-type theorem for the two-component Degasperis-Procesi system with k 2 = 0.
Then we have the following new blow-up criterion for periodic solutions the twocomponent Degasperis-Procesi equation with or without dispersion: If there is some ξ ∈ S such that 0 (ξ) = 0 and v 0 (ξ) < −    [2] to the two-component Degasperis-Procesi system. For the comparison with some earlier results for 2DP and DP, we refer to Sect. 2 in [2].