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1.   Introduction

This paper is devoted to the existence of weak solutions to the Cauchy problem for the two-component Novikov equation [18]

Note that this system reduces respectively to the Novikov equation [23]

when $v = u$, and the celebrated Camassa-Holm (CH) equation [1]

when $v = 1$.

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 ${\mathbb R}$ and unit circle ${\mathbb S}^1$ [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 ${\mathbb R}$ and unit circle ${\mathbb S}^1$ [14,20], which can be derived by the inverse spectral method. Orbital stability of peaked solitons over the line ${\mathbb R}$ and unit circle ${\mathbb S}^1$ 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 $u_0\in H^s$, $s> 3/2$ [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]

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 $\|\cdot\|_X$ denote the norm of Banach space $X$ and let $\langle\cdot, \cdot\rangle$ denote the duality paring between $H^1(\mathbb R)$ and $H^{-1}(\mathbb R)$. Let ${\mathcal {M}}(\mathbb R)$ be the space of Radon measures on $\mathbb R$ with bounded total variation and ${\mathcal {M}}^+(\mathbb R)$ be the subset of positive Radon measures. Moreover, we write $BV(\mathbb R)$ for the space of functions with bounded variation, $\mathbb{V}(f)$ being the total variation of $f\in BV(\mathbb R)$. Furthermore, for $0<p<\infty$, $s\geq 0$, let $\|\cdot\|_{L^p}$ and $\|\cdot\|_s$ denote the norm of $L^p(\mathbb R)$ space and $H^s(\mathbb R)$ space, respectively.

With $m = u-u_{xx}$ and $n = v-v_{xx}$, the Cauchy problem of equation (1) takes the form:

Note that if $P(x) = {\frac 12}e^{-|x|}$, $x\in \mathbb R$, we have $(1-\partial^2_x)^{-1}f = P*f$ for all the $f\in L^2(\mathbb R)$ 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.

Lemma 2.1. [12] Let $u_0, v_0\in H^s(\mathbb R)$, $s\geq 3$. Assume that $T = T(u_0, v_0)>0$ be the maximal existence time of the corresponding strong solution $(u, v)$. Then the initial value problem of system (1) possesses a strong solution

Moreover, the solution depends continuously on the initial data, i.e. the mapping $(u_0, v_0)\rightarrow(u(\cdot, u_0), v(\cdot, v_0)):H^s(\mathbb R)\times H^s(\mathbb R)\rightarrow C([0, T);H^s(\mathbb R))\cap C^1([0, T); H^{s-1}(\mathbb R))\times C([0, T);H^s(\mathbb R))\cap C^1([0, T);H^{s-1}(\mathbb R))$ is continuous.

Lemma 2.2. [12] Let $u_0, v_0\in H^s(\mathbb R)$, $s\geq 3$, 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

Lemma 2.3. [22] Let $T>0$. If

then $f, g$ are a.e. equal to functions continuous from $[0, T]$ into $L^2(\mathbb R)$ and

for all $s, t \in [0, T]$.

Throughout this paper, let $\{\rho_n\}_{n\geq 1}$ denote the mollifiers

where $\rho\in C^\infty_c(\mathbb R)$ is defined by

Next, we recall two crucial approximation results and two identities.

Lemma 2.4. [7] Let $f:\mathbb R\rightarrow\mathbb R$ be uniformly continuous and bounded. If $\mu\in {\mathcal {M}}(\mathbb R)$, then

Lemma 2.5. [7] Let $f:\mathbb R\rightarrow\mathbb R$ be uniformly continuous and bounded. If $g\in L^\infty(\mathbb R)$, then

Lemma 2.6. [7] Assume that $u(t, \cdot)\in W^{1, 1}(\mathbb R)$ is uniformly bounded in $W^{1, 1}(\mathbb R)$ for all $t\in {\mathbb R}_+$. Then for a.e. $t\in {\mathbb R}_+$, there hold

and

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.

Lemma 2.7. [12] Let $u_0, v_0\in H^s(\mathbb R)$, $s\geq 3$, and $T>0$ be the life-span of the corresponding strong solution $(u, v)$ to equation (5) with the initial data $(u_0, v_0)$. Then equation (8) has a unique solution $q\in C^1([0, T)\times \mathbb R, \mathbb R)$. Moreover, the map $q(t, \cdot)$ is an increasing diffeomorphism over $\mathbb R$ with

Furthermore, setting $m = u-u_{xx}$ and $n = v-v_{xx}$, we obtain

Theorem 2.8. Let $u_0, v_0\in H^s(\mathbb R)$, $s\geq 3$. Assume that $m_0 = u_0-\partial^2_{x}u_0$ and $n_0 = v_0-\partial^2_{x}v_0$ 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 $(u, v)$, where

Set $m(t, \cdot) = u(t, \cdot)-u_{xx}(t, \cdot)$ and $n(t, \cdot) = v(t, \cdot)-v_{xx}(t, \cdot)$. Then, $E_u(u) = \int_{\mathbb R}\left(u^2+u^2_x\right)dx$, $E_v(v) = \int_{\mathbb R}\left(v^2+v^2_x\right)dx$, $H(u, v) = \int_{\mathbb R}\left(uv+u_xv_x\right)dx$ and $E_0(u, v) = \int_{\mathbb R}\left(mn\right)^{\frac 13}dx$ are four conservation laws and we have for all $t\in {\mathbb R}_+$

Moreover, if $m_0, n_0\in L^1(\mathbb R)$, we obtain

Proof. Let $u_0, v_0\in H^s(\mathbb R)$, $s\geq 3$, 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 $m_0\geq 0$ and $n_0\geq 0$, then Lemma 2.7 ensures that $m(t, \cdot)\geq 0$ and $n(t, \cdot)\geq 0$ for all $t\in [0, \infty)$. By $u = P\ast m$, $v = P\ast n$ and the positivity of $P$, we infer that $u(t, \cdot)\geq 0$ and $v(t, \cdot)\geq 0$ for all $t\geq 0$. Note that $v$ is analogous as $u$ and

and

From the above two relations and $m\geq 0$, 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 $u = P\ast m$ and $u_x = P_x\ast m$. 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

it immediately follows that $E_0(u, v)$ is a conserved density. On the other hand, by equation (5), we have

Since $m_0\in L^1(\mathbb R)$, 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.

Theorem 3.1. Let $u_0, \, v_0 \in H^1(\mathbb R)$. Assume $m_0 = u_0-u_{0xx}\, and\, n_0 = v_0-v_{0xx}\in {\mathcal {M}}^+(\mathbb R)$.Then equation (6) has a pair of unique weak solution $(u, v)$, where

with the initial data $u(0, x) = u_0, \, v(0, x) = v_0$ and such that $m = u-u_{xx}, \, n = v-v_{xx}\in {\mathcal {M}}^+(\mathbb R)$ are bounded on $[0, T)$, for any fixed $T>0$. Moreover, $E_u(u) = \int_{\mathbb R}\left(u^2+u^2_x\right)dx$, $E_v(v) = \int_{\mathbb R}\left(v^2+v^2_x\right)dx$ and $H(u, v) = \int_{\mathbb R}\left(uv+v_xu_x\right)dx$ are conserved densities.

Proof. First, we shall prove $u, v \in W^{1, \infty}({\mathbb R}_x\times\mathbb R)\cap L^{\infty}({\mathbb R}_+;H^1(\mathbb R))$. Let $u_0, v_0\in H^1(\mathbb R)$ and assume that $m_0 = u_0-u_{0, xx}, \, n_0 = v_0-v_{0, xx}\in {\mathcal {M}}^+(\mathbb R)$. Note that $u_0 = P\ast m_0$ and $v_0 = P\ast n_0$. Thus, we have for any $f\in L^{\infty}(\mathbb R)$,

Similarly, we have

We first prove that there exists a corresponding $(u, v)$ with the initial data $(u_0, v_0)$, which belongs to $H^1_{loc}({\mathbb R}_+\times \mathbb R)\cap L^\infty({\mathbb R}_+;H^1(\mathbb R))\times H^1_{loc}({\mathbb R}_+\times \mathbb R)\cap L^\infty({\mathbb R}_+;H^1(\mathbb R))$, satisfying equation (6) in the sense of distributions.

Let us define $u^n_0 = \rho_{n}\ast u_0\in H^{\infty}(\mathbb R)$ and $v^n_0 = \rho_{n}\ast v_0\in H^{\infty}(\mathbb R)$ for $n\geq 1$. Obviously, we have

and for all $n\geq 1$,

in view of Young's inequality. Note that for all $n\geq 1$,

Comparing with the proof of relation (11) and (12), we get

By Theorem 2.8, we obtain that there exists a global strong solution

for every $s\geq 3$, and we have $u^n(t, x)-u^n_{xx}(t, x)\geq 0$ and $v^n(t, x)-v^n_{xx}(t, x)\geq 0$ for all $(t, x)\in {\mathbb R}_+\times \mathbb R$. In view of theorem 2.8 and (14), we obtain for $n\geq 1$ and $t\geq 0$,

By the above inequality, we have

Similarly, we have

By Young's inequality and (16), for all $t\geq 0$ and $n\geq 1$, we obtain

Similarly, we get

Combining (17)-(20) with equation (6) for all $t\geq 0$ and $n\geq 1$, we find

For fixed $T>0$, by Theorem 2.8 and (21), we have

It follows that the sequence $\{u^n\}_{n\geq 1}$ is uniformly bounded in the space $H^1((0, T)\times\mathbb R)$.Thus we can extract a subsequence such that

and

for some $u\in H^1((0, T)\times\mathbb R)$. By Theorem 2.8, (11) and (14), we have that for fixed $t\in (0, T)$, the sequence $u^{n_k}_x(t, \cdot)\in BV(\mathbb R)$ satisfies

and

Applying Helly's theorem, we obtain that there exists a subsequence, denoted again by $\{u^{n_k}_x(t, \cdot)\}$, which converges at every point to some function $\hat{u}(t, \cdot)$ of finite variation with

Since for almost all $t\in (0, T)$, $u^{n_k}_x(t, \cdot)\rightarrow u_x(t, \cdot)$ in $D'(\mathbb R)$ in view of (24), it follows that $\hat{u}(t, \cdot) = u_x(t, \cdot)$ for a.e. $t\in (0, T)$. Therefore, we have

and for a.e. $t\in (0, T)$,

We can analogously extract a subsequence of $\{v^{n_k}\}$, denote again by $\{v^{n_k}\}$ such that

By Theorem 2.8 $(ii)-(iii)$ and (16), we have

For fixed $t\in (0, T)$, it follows that the sequence $\left\{{\frac 12}(u^n_x)^2v^n+u^nu^n_xv^n_x+(u^n)^2v^n+\right. \left.{\frac 12}(u^n_x)^2v^n_x\right\}$ is uniformly bounded in $L^1(\mathbb R)$. Therefore, it has a subsequence converging weakly in $L^1(\mathbb R)$, denoted again by $\left\{{\frac 12}(u^n_x)^2v^n+u^nu^n_xv^n_x+(u^n)^2v^n+{\frac 12}(u^n_x)^2v^n_x\right\}$. By (24) and (25), we deduce that the weak $L^1(\mathbb R)$-limit is ${\frac 12}(u_x)^2v^n+uu_xv_x+u^2v+{\frac 12}(u_x)^2v_x$. Note that $P, \, P_x\in L^{\infty}(\mathbb R)$. It follows that

We can analogously obtain that

Combining (24)-(26) with (27) and (28), we deduce that $(u, v)$ satisfies equation (6) in $D'((0, T)\times \mathbb R)$.

Since $u^{n_k}_t(t, \cdot)$ is uniformly bounded in $L^2(\mathbb R)$ for all $t\in {\mathbb R}_+$ and $\|u^{n_k}(t, \cdot)\|_1$ has a uniform bound as $t\in {\mathbb R}_+$ and all $n\geq 1$. Hence the family $t\mapsto u^{n_k}(t, \cdot)\in H^1(\mathbb R)$ is weakly equicontinuous on $[0, T]$ for any $T>0$. An application of the Arzela-Ascoli theorem yields that $\left\{u^{n_k}\right\}$ contains a subsequence, denoted again by $\left\{u^{n_k}\right\}$, which converges weakly in $H^1(\mathbb R)$, uniformly in $t\in [0, T]$. The limit function is $u$. Because $T$ is arbitrary, we have that $u$ is locally and weakly continuous from $[0, \infty)$ into $H^1(\mathbb R)$, i.e.

For a.e. $t\in {\mathbb R}_+$, since $u^{n_k}(t, \cdot)\rightharpoonup u(t, \cdot)$ weakly in $H^1(\mathbb R)$, in view of (15) and (16), we obtain

for a.e. $t\in {\mathbb R}_+$. 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

Please note that we use the subsequence of $\left\{v^{n_k}\right\}$ 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. $t\in {\mathbb R}_+$, $n\geq 1$,

By differentiation of the first equation of (31), we obtain

Note that $\partial^2(P\ast f) = P\ast f-f$, $f\in L^2(\mathbb R)$. We can rewrite (32) as

Take these two equation (32) and (33) into the integration below, we obtain

Note that

Therefore, by using H$\ddot{\rm{o}}$lder inequality, we have for a.e. $t\in \mathbb R^{+}$

Similarly, for a.e. $t\in \mathbb R$

as $u(t, \cdot), v(t, \cdot)\in H^1(\mathbb R)$ and $u_x, v_x\in L^{\infty}({\mathbb R}_+\times \mathbb R)$. Furthermore, note that

Observe that

On the other hand

As $u(t, \cdot), v(t, \cdot)\in H^1(\mathbb R)$ and $u_x, v_x\in L^{\infty}({\mathbb R}_+\times \mathbb R)$, 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. $t\in [0, T)$,

Therefore, we get

By Young's inequality and H$\ddot{\rm{o}}$lder's inequality, it follows that there is a $K^u(T)>0$ such that

In view of (38) and (39), an application of Lebesgue's dominated convergence theorem yields that for fixed a.e. $t\in {\mathbb R}_+$,

By (24) and the above relation, for fixed $t\in {\mathbb R}_+$, we can get

By Theorem 2.8, we infer that for all fixed $t\in {\mathbb R}_+$, $E_u(u)$ is conserved. Similarly, we can show that $E_v(v)$ is also conserved.

Next, we prove that $H(u, v)$ is a conserved density.

By differentiation of the second equation of (31), we obtain this relation:

In view of (31), (33) and (40), we obtain

We can analogously get the similar convergence like the case ${\frac d{dt}}\int_{\mathbb R}(\rho_n\ast u)^2+(\rho_n\ast u_x)^2dx$ by using Lemma 2.5, $u(t, \cdot), v(t, \cdot)\in H^1(\mathbb R)$ and $u_x, v_x\in L^{\infty}({\mathbb R}_+\times \mathbb R)$.

It is nature to define

and

And it is easy to get

Similarly, we get this estimate by using Young's inequality and Holder's inequality:

An application of Lebesgue's dominated convergence theorem yields that for fixed a.e. $t\in {\mathbb R}_+$,

By these convergence above, for fixed $t\in {\mathbb R}_+$, we can get

which indicates that $H(u, v)$ is a conserved density.

Since $L^1(\mathbb R)\subset(L^\infty(\mathbb R))^*\subset(C_0(\mathbb R))^* = {\mathcal {M}}(\mathbb R)$. It is not too hard to show that for a.e. $t\in[0, T)$,

For any fixed $T$, $\forall t\in[0, T)$, we have proved

Therefore, in view of (24) and (25), we obtain that for all $t\in[0, T)$, as $n\rightarrow\infty$,

Since $u^{n_k}(t, \cdot)-u^{n_k}_{xx}(t, \cdot)\geq 0$ for all $(t, x)\in{\mathbb R}_+\times \mathbb R$, we deduce that for a.e. $t\in [0, T)$

Similarly, we arrive at the conclusion:

Finally, we show the uniqueness of the weak solutions of equation (6). Let $(u, v)$ and $(\bar{u}, \bar{v})$ be two weak solutions of equation (6) in the class

Note that

Define

Then for fixed $T$, we obtain $M(T)<\infty$. For all $(t, x)\in[0, T)\times \mathbb R$, in view of (11), we find that

On the other hand, from (29) and (30), we have

Let us define

Convoluting equation (6) for $(u, v)$ and $(\bar{u}, \bar{v})$ with $\rho_n$, we have that for a.e. $t\in[0, T)$ and all $n\geq 1$,

Using (46) and Young's inequality, we infer that for a.e. $t\in[0, T)$ and all $n\geq 1$

where $C$ is a constant depending on $N$. Similarly, convoluting equation (6) for $(u, v)$ and $(\bar{u}, \bar{v})$ with $\rho_{n, x}$, it follows that

For the term $I_1$, we have

where C is a constant depending on $M(T)$, $N$, $\|u_0\|_1$ and $\|v_0\|_1$ and $R_n(t)$ satisfies

For the second term $I_2$, we find

For the final term $I_3$, we have

Adding these three terms, we obtain

For these terms ${\frac d{dt}}\int_{\mathbb R}|\rho_n\ast\hat{v}|dx$ and ${\frac d{dt}}\int_{\mathbb R}|\rho_n\ast\hat{v}_x|dx$, we have similar results:

From (48), (54) and (55), we infer that

If $\int_{\mathbb R}|\rho_n\ast\hat{u}|dx+\int_{\mathbb R}|\rho_n\ast\hat{u}_x|dx+\int_{\mathbb R}|\rho_n\ast\hat{v}|dx+\int_{\mathbb R}|\rho_n\ast\hat{v}_x|dx\ne 0$, then by Gronwall's inequality, we obtain

where $\tilde{R}_n(t) = R_n(t) \left(\int_{\mathbb R}|\rho_n\ast\hat{u}|dx+\int_{\mathbb R}|\rho_n\ast\hat{u}_x|dx+\int_{\mathbb R}|\rho_n\ast\hat{v}|dx+\int_{\mathbb R}|\rho_n\ast\hat{v}_x|dx\right)^{-1}$. From Lebesgue's dominated convergence theorem, it follows that

As $T$ is arbitrary, $\hat{u}_0 = \hat{u}_{0x} = \hat{v}_0 = \hat{v}_{0, x} = 0$, we obtain $(u, v) = (\bar{u}, \bar{v})$. This completes the proof of theorem 3.1.