GLOBAL WEAK SOLUTIONS FOR THE TWO-COMPONENT NOVIKOV EQUATION

. The two-component Novikov equation is an integrable generalization of the Novikov equation, which has the peaked solitons in the sense of distribution as the Novikov and Camassa-Holm equations. In this paper, we prove the existence of the H 1 -weak solution for the two-component Novikov equation by the regular approximation method due to the existence of three conserved densities. The key elements in our approach are some a priori esti-mates on the approximation solutions.


1.
Introduction. This paper is devoted to the existence of weak solutions to the Cauchy problem for the two-component Novikov equation [18] m t + uvm x + (2vu x + uv x )m = 0, m = u − u xx , t > 0, n t + uvn x + (2uv x + vu x )n = 0, n = v − v xx .
The CH equation was proposed as a nonlinear model describing the unidirectional propagation of the shallow water waves over a flat bottom [1]. Based on the Hamiltonian theory of integrable systems, it was found earlier by using the method of recursion operator due to Fuchssteiner and Fokas [10]. It can also be obtained by using the tri-Hamiltonian duality approach related to the bi-Hamiltonian representation of the Korteweg-de Vries (KdV) equation [9,25]. The CH equation exhibits several remarkable properties. One is the the existence of the multi-peaked solitons on the line R and unit circle S 1 [1,2], where the peaked solitons are the weak solution in the sense of distribution. Second, it can describes wave breaking phenomena [4], which is different from the classical integrable systems. The existence of H 1 -conservation law to the CH equation enables ones to define the H 1 -weak solution [28]. There have been a number of results concerning about integrability, well-posedness, blow up and wave breaking, orbital stability in the energy space and geometric formulations etc, see for instance [4,5,6,8,28] and references therein.
The Novikov equation (2) can be viewed as a cubic generalization of the CH equation, which was introduced by Novikov [23,24] in the classification for a class of equations while they possesses higher-order generalized symmetries. Eq. (2) was proved to be integrable since it enjoys Lax-pair and bi-Hamiltonian structure [14], and is equivalent to the first equation in the negative flow of the Sawada-Kotera hierarchy via Liouville transformation [16]. The Novikov equation (2) also admits peaked solitons over the line R and unit circle S 1 [14,20], which can be derived by the inverse spectral method. Orbital stability of peaked solitons over the line R and unit circle S 1 of (2) in the energy space were verified [20] based on the conservation laws and the structure of peaked solitons of the Novikov equation (2). The well-posedness and wave breaking of the Novikov equation have been discussed in a number of papers, and it reveals that the Cauchy problem of the Novikov equation (2) has global strong solutions when the initial data u 0 ∈ H s , s > 3/2 [3,15,26,27]. The existence of global weak solutions to the Cauchy problem of the Novikov equation (2) was also discussed in [17].
As the two-component generalization of Novikov equation (2), the so-called Geng-Xue system [11] m t + 3vu x m + uvm x = 0, has been studied extensively [11,13]. The integrability [11,19], dynamics and structure of the peaked solitons of (4) [21] were discussed. In [13], well-posedness and wave breaking phenomena of the Cauchy problem of (4) were discussed. The single peakons and multi-peakons of system (4) were constructed in [21] by using compatibility of Lax-pair, which are not the weak solutions in the sense of distribution. Furthermore, the Geng-Xue system does not have the H 1 -conserved density, this is different from the CH and Novikov equations. The weak solution in H 1 is not well-defined since it does not obey the H 1 -conservation law. The main object in this work is to investigate the existence of weak solutions to system (1). It is of great interest to understand the effect from interactions among the two-components, nonlinear dispersion and various nonlinear terms. More specifically, we shall consider the Cauchy problem of (1) and aim to leverage ideas from previous works on CH and Novikov equations. The weak solution of the Cauchy problem associated with (1) is established in Theorem 3.1.
The remainder of this paper is organized as follows. In the next section 2, we review some basic results and lemmas as well as invariant properties of momentum densities m and n. In Section 3, we establish the existence of weak solutions, our approach is the regular approximation method together with some a priori estimates.
2. Strong solutions and some a priori estimates. In this section, we recall the local well-posedness, some properties of strong and weak solutions to equation (1) and several approximation results.
First, we introduce some notations. Throughout the paper, we denote the convolution by * . Let · X denote the norm of Banach space X and let ·, · denote the duality paring between H 1 (R) and H −1 (R). Let M(R) be the space of Radon measures on R with bounded total variation and M + (R) be the subset of positive Radon measures. Moreover, we write BV (R) for the space of functions with bounded variation, V(f ) being the total variation of f ∈ BV (R). Furthermore, for 0 < p < ∞, s ≥ 0, let · L p and · s denote the norm of L p (R) space and H s (R) space, respectively.
With m = u − u xx and n = v − v xx , the Cauchy problem of equation (1) takes the form: for all the f ∈ L 2 (R) and P * m = u, P * n = v. Then we can rewrite the equation (5) as follows: Next we recall the local well-posedness and the conservation laws. Moreover, the solution depends continuously on the initial data, i.e. the mapping Lemma 2.2. [12] Let u 0 , v 0 ∈ H s (R), s ≥ 3, and let (u(t, x), v(t, x)) be the corresponding solution to equation (1) with the initial data (u 0 , v 0 ). Then we have Moreover, we have Note that equation (1) has the solitary waves with corner at their peaks. Obviously, such solitons are not strong solutions to equation (6). In order to provide a mathematical framework for the study of these solitons, we define the notion of weak solutions to equation (6). Let Then equation (6) can be written as then f, g are a.e. equal to functions continuous from [0, T ] into L 2 (R) and Throughout this paper, let {ρ n } n≥1 denote the mollifiers for |x| ≥ 1.
Next, we recall two crucial approximation results and two identities.
[7] Let f : R → R be uniformly continuous and bounded. If g ∈ L ∞ (R), then Consider the flow governed by (uv)(t, x): Applying classical results in the theory of ODEs, one can obtain the following useful result on the above initial value problem.
x v 0 are nonnegative, and T > 0 be the maximal existence time of the corresponding strong solution (u, v). Then the initial value problem of system (1) possesses a pair of unique strong solution Proof. Let u 0 , v 0 ∈ H s (R), s ≥ 3, and let T > 0 be the maximal existence time of the solution (u, v) to equation (5) with the initial data (u 0 , v 0 ). If m 0 ≥ 0 and n 0 ≥ 0, then Lemma 2.7 ensures that m(t, ·) ≥ 0 and n(t, ·) ≥ 0 for all t ∈ [0, ∞). By u = P * m, v = P * n and the positivity of P , we infer that u(t, ·) ≥ 0 and v(t, ·) ≥ 0 for all t ≥ 0. Note that v is analogous as u and and From the above two relations and m ≥ 0, we deduce that In view of Lemma 2.2, we obtain that E u (u) and E v (v) are conserved and Since m(t, x) = u − u xx , it follows that u = P * m and u x = P x * m. Note that P L 1 = P x L 1 = 1. Applying Young's inequality, one can easily obtain (i) − (iii). Since equation (1) can be used to derive the following form (mn) On the other hand, by equation (5), we have Since m 0 ∈ L 1 (R), in view of Gronwall's inequality, we can get Similarly, we find This completes the proof of Theorem 2.8.
3. Global weak solutions. In this section, we will prove that there exists a unique global weak solution to equation (6), provided the initial data (u 0 , v 0 ) satisfy certain sign-invariant conditions.
. Note that u 0 = P * m 0 and v 0 = P * n 0 . Thus, we have for any f ∈ L ∞ (R), We first prove that there exists a corresponding (u, v) with the initial data (u 0 , v 0 ), which belongs to H 1 , satisfying equation (6) in the sense of distributions.
Let us define u n 0 = ρ n * u 0 ∈ H ∞ (R) and v n 0 = ρ n * v 0 ∈ H ∞ (R) for n ≥ 1. Obviously, we have and for all n ≥ 1, in view of Young's inequality. Note that for all n ≥ 1, Comparing with the proof of relation (11) and (12), we get By Theorem 2.8, we obtain that there exists a global strong solution u n = u n (·, u n 0 ), v n = v n (·, v n 0 ) ∈ C([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)) for every s ≥ 3, and we have u n (t, x) − u n xx (t, x) ≥ 0 and v n (t, x) − v n xx (t, x) ≥ 0 for all (t, x) ∈ R + × R. In view of theorem 2.8 and (14),we obtain for n ≥ 1 and t ≥ 0, By the above inequality, we have By Young's inequality and (16), for all t ≥ 0 and n ≥ 1, we obtain Similarly, we get Combining (17)- (20) with equation (6) for all t ≥ 0 and n ≥ 1, we find For fixed T > 0, by Theorem 2.8 and (21), we have It follows that the sequence {u n } n≥1 is uniformly bounded in the space H 1 ((0, T ) × R).Thus we can extract a subsequence such that and for some u ∈ H 1 ((0, T ) × R). By Theorem 2.8, (11) and (14), we have that for fixed t ∈ (0, T ), the sequence u n k and u n k x (t, ·) L ∞ ≤ u n k (t, ·) 1 = u n k 0 (t, ·) 1 ≤ u 0 1 . Applying Helly's theorem, we obtain that there exists a subsequence, denoted again by {u n k x (t, ·)}, which converges at every point to some functionû(t, ·) of finite variation with Since for almost all t ∈ (0, T ), u n k x (t, ·) → u x (t, ·) in D (R) in view of (24), it follows thatû(t, ·) = u x (t, ·) for a.e. t ∈ (0, T ). Therefore, we have and for a.e. t ∈ (0, T ), We can analogously extract a subsequence of {v n k }, denote again by {v n k } such that v n k −→ v a.e. on (0, T ) × R for n k → ∞ and v n k x −→ v x a.e. on (0, T ) × R for n k → ∞.
Since u n k t (t, ·) is uniformly bounded in L 2 (R) for all t ∈ R + and u n k (t, ·) 1 has a uniform bound as t ∈ R + and all n ≥ 1. Hence the family t → u n k (t, ·) ∈ H 1 (R) is weakly equicontinuous on [0, T ] for any T > 0. An application of the Arzela-Ascoli theorem yields that {u n k } contains a subsequence, denoted again by {u n k }, which converges weakly in H 1 (R), uniformly in t ∈ [0, T ]. The limit function is u. Because T is arbitrary, we have that u is locally and weakly continuous from [0, ∞) into H 1 (R), i.e. u ∈ C w,loc (R + ; H 1 (R)).
For a.e. t ∈ R + , since u n k (t, ·) u(t, ·) weakly in H 1 (R), in view of (15) and (16), we obtain for a.e. t ∈ R + . The previous relation implies that Note that by Theorem 2.8 and (15), we have Combining this with (25), we deduce that This shows that Taking the same way as u, we get v ∈ W 1,∞ (R + × R) ∩ L ∞ (R + ; H 1 (R)).
Please note that we use the subsequence of {v n k } which is determined after using the Arzela-Ascoli theorem. Now, by a regularization technique, we prove that E u (u), E v (v) and H(u, v) are conserved densities. As (u, v) solves equation (6) in the sense of distributions, we see that for a.e. t ∈ R + , n ≥ 1, By differentiation of the first equation of (31), we obtain . We can rewrite (32) as Take these two equation (32) and (33) into the integration below, we obtain Note that Therefore, by using Hölder inequality, we have for a.e. t ∈ R + R (ρ n * u)(ρ n * (uvu x ))dx −→ R u 2 vu x dx, as n → ∞.
Similarly, for a.e. t ∈ R R (ρ n * u) ρ n * P x * 1 2 as u(t, ·), v(t, ·) ∈ H 1 (R) and u x , v x ∈ L ∞ (R + × R). Furthermore, note that Observe that On the other hand As u(t, ·), v(t, ·) ∈ H 1 (R) and u x , v x ∈ L ∞ (R + × R), by Lemma 2.5, it follows that Therefore, In view of the above relations and (35), we obtain Let us define and We have proved that for fixed T > 0, for a.e. t ∈ [0, T ), Therefore, we get By Young's inequality and Hölder's inequality, it follows that there is a K u (T ) > 0 such that |G u n (t)| ≤ K u (T ), n ≥ 1.
By these convergence above, for fixed t ∈ R + , we can get . It is not too hard to show that for a.e. t ∈ [0, T ), For any fixed T , ∀t ∈ [0, T ), we have proved (u(t, ·) − u xx (t, ·)) ∈ M(R).
Therefore, in view of (24) and (25), we obtain that for all t ∈ [0, T ), as n → ∞, Similarly, we arrive at the conclusion: Finally, we show the uniqueness of the weak solutions of equation (6). Let (u, v) and (ū,v) be two weak solutions of equation (6) in the class Note that Then for fixed T , we obtain M (T ) < ∞. For all (t, x) ∈ [0, T ) × R, in view of (11), we find that On the other hand, from (29) and (30), we have Let us definê Convoluting equation (6) for (u, v) and (ū,v) with ρ n , we have that for a.e. t ∈ [0, T ) and all n ≥ 1, Using (46) and Young's inequality, we infer that for a.e. t ∈ [0, T ) and all n ≥ 1 where C is a constant depending on N . Similarly, convoluting equation (6) for (u, v) and (ū,v) with ρ n,x , it follows that where C is a constant depending on M (T ), N , u 0 1 and v 0 1 and R n (t) satisfies R n (t) −→ 0, n → ∞, |R n (t)| ≤ κ(T ), n ≥ 1, t ∈ [0, T ).