Global conservative solutions for a modified periodic coupled Camassa-Holm system

1.   Introduction

In this paper, we investigate the Cauchy problem of the modified periodic coupled Camassa-Holm system that has the following form:

where $m = u - u_{xx}$ and $n = v - v_{xx}$ are periodic function respect to $x$. The system (1) is a generalization of the Camassa-Holm equation with peakon solitons in the form of a superposition of multipeakons which was firstly proposed by Fu and Qu in [13]. The well-posedness and blow-up solutions of periodic case for the system (1) were discussed by Fu et al. in [14]. In [30], the authors established the local well posedness and blow-up solutions in Besov spaces. The attractor, non-uniform dependence and persistence properties of the system (1) were discussed in [29,33]. Furthermore, Tian el at. in [28] investigated the global conservative and dissipative solutions of system (1) by tansforming it into an equivalent semilinear ODE system. However, there is gap in [28,31], i.e., the second derivative terms $u_{xx}$ and $v_{xx}$ appeared in the equivalent ODE system cannot be controlled by the $H^1(\mathbb{R})$ conservation law. This paper repair the gap in the following sections.

It is known to us, when $u = v$ and let $\tilde{t} = 2t$, system (1) can be degenerated to the famous Camassa-Holm equation:

The Camassa-Holm equation (2) was first implicitly contained in a bi-Hamiltonian generalization of the Korteweg-de Vries equation by Fuchssteiner and Fokas [15], and later deduced as a model for unidirectional propagation of shallow water over a flat bottom by Camassa and Holm [4]. Similar to the KdV equation, the Camassa-Holm equation has a bi-Hamilton structure [15,24], and is completely integrable [4,5,23]. The equation (2) not only holds an infinity of conservative laws, but also can be solved by its corresponding inverse scattering transform [2,9]. The solitary waves of equation (2) are solitons (i.e., it can keep their shape and velocity after interacted by the same type nonlinear wave). Compared to the KdV equation, the Camassa-Holm equation has many advantages, such as, it has both the finite time wave-breaking solutions (i.e. the solution keeps bounded but the slope becomes unbounded in finite time) and the global strong solutions [6,7,11,25]. The solitary wave solutions are peaked waves and a specific case was given in [4], it is not a classical solution because it has a peak at their crest. The local well-posedness of equation (2) with the inial data $u_{0} \in H^{s}, s > \frac{3}{2}$ was studied in [1,7,25]. Many papers have investigated the weak solutions for the equation (2). Especially, in the non-periodic case, Bressan and Constantin in [3] has done many works, they developed a new method to investigate a conservative solution's semigroup. In the periodic case, Holden and Raynaud in [22] applied the semigroup theory to investigate the periodic Camassa-Holm equation, and proved its conservative solutions, depending continuously on the initial data, which can also construct a semigroup.

Of course, the Camassa-Holm equation has many generalizations such as the modified two-component Camassa-Holm equation (M2CH) and the coupled Camassa-Holm equation. The M2CH equation was firstly introduced by Holm et al. in [21] as a modified version of the two-component Camassa-Holm equation (2CH) that was proposed by Constantin and Ivanov [10] in the context of shallow water theory. The M2CH equation is integrable and its form as follow;

where $m = u - u_{xx}$. The equations (3) was testified that admits singular solutions in both of the variables $m$ and $\rho$ in [21]. The well-posedness, blow-up phenomena, Lipschitz metric and the global weak solution for the equations (3) were studied by Guan et al. in [17,18,19,20]. Tan et al. investigated the global conservative solutions for both of the periodic case and the non-periodic case in [26,27]. Guan proved the Cauchy problem of the M2CH with the initial data $z_{0} = (u_{0}, \rho_{0})\in H^{1}(\mathbb{R})\times(H^{1}\cap W^{1, \infty}))$ has a unique global conservative weak solution in [16].

Inspired by [3,22,26,34,35], this paper mainly discusses the global conservative solutions of the modified periodic Camassa-Holm system. As is known to us, equations (1) is a system, so it's more difficult than the single one. Moreover, the interactions between $u$ and $v$ greatly increases the complexity of the research. To overcome these difficulties, we set the characteristics that completely different from the one used in [3,22,26]. Thus, the calculation is largely reduced. By introducing new variables, we transform the system (1) into a equivalent semilinear ODE system. Firstly, we get the global solutions of the equivalent semilinear ODE system. Then we get the global conservative solutions for the system (1) from the global solutions of the equivalent ODE system. Finally, we obtain a semigroup of the solutions depending continuously on inial data for the original system.

The rest of this paper is organized as follows. Section 2 is the basic equation. In section 3, we get a equivalent semilinear system and the global solutions of the semilinear system. In section 4, we obtain the global conservative solution of the system (1) and construct a solution semigroup.

2.   The basic equations

Now, we reformulate the system (1). Let $m = u - u_{xx}$ and $n = v - v_{xx}$. Note that $G: = \frac{1}{2\sinh\frac{1}{2}}\cosh(x-[x]-\frac{1}{2})$, $x \in\mathbb{R}$, and $(1-\partial_{x}^{2})^{-1}\mathit{f} = G*\mathit{f}$ for all $\mathit{f}\in\mathit{L}^2(\mathbb{R})$. Thus, we can rewrite system (1) as follow:

where $P_1, P_2, Q_1, Q_2$ have the following form:

According to the fact that the above representations, $P_{1}$ and $Q_{1}$ are symmetrical, $P_{2}$ and $Q_{2}$ are symmetrical, For convenience, we can set $P_{1}(u, v) = Q_{1}(v, u) = G*(uv_x)$. So do $P_{2}$ and $Q_{2}$.

In fact, for smooth solutions, differentiating the first and the second equations in (4) with respect to $x$, we get

Multiplying the first and the second equations in (4) by $u$ and $v$, and multiplying the first and the second equations in (5) by $u_x$ and $v_{x}$, respectively, we get following conservation laws

For regular solutions, using (6) and integrating by parts, it is clear that the total energy

is a constant in time. If $u$ and $v$ are smooth, from (4)-(6), it is not very hard to check that

3.   A equivalent semilinear system

Firstly, we introduce the space $E_1$:

and define the characteristics $y:\mathbb{R}\mapsto E_1, t\mapsto y(t, \cdot)$ as the solution of

In addition, we denote

and

By (7) and (8), we get

Using (10), the periodicity of $u, v$ and $y \in E_1$, we obtain

According to (11), it is very easy to prove that $H(t, 1) - H(t, 0)$ is a constant in time. Thus, we get that $H(t, 1) - H(t, 0) = H(0, 1) - H(0, 0)$. For every $t > 0$, $H$ belongs to the vector space $E$ defined as follow

We define the norm $\|f\|_{E} = \|f\|_{H_{[0, 1]}^{1}}$ for $E$. For convenience, we will replace $\mathit{H}_{[0, 1]}^{1}$ by $\mathit{H}^{1}$. Later, we will verify that $E$ is complete.

We derive formally a system equivalent to system (4). From the definition of the characteristics, it follows that

Then, we get the explicit expression for $P_{i}$, $Q_{i}$, $P_{i, x}$, $Q_{i, x}$ ($i$ = 1, 2):

and $Q_{1}(u, v) = P_{1}(v, u)$, $Q_{1, x}(u, v) = P_{1, x}(v, u)$, $Q_{2}(u, v) = P_{2}(v, u)$, $Q_{2, x} = P_{2, x}(v, u)$.

In the above formulae, we can perform the change of variables $y = y(t, \theta')$, and rewrite the convolution respect to $\theta'$. From (9), we get new expressions of $P_{i}$, $Q_{i}$, $P_{i, x}$, $Q_{i, x}$($i = 1, 2$) with the new variable $\theta$

and

Straight computation shows that

From (8), (11)-(12) and (13)-(15), we obtain a new system which is equivalent to system (4). And the Cauchy problem of the new system can be rewritten with respect to the variables $(y, U, V, M, N, H)$ in the following form

Differentiating (16) with respect to $\theta$ and utilizing (15), we have

The system (17) is semilinear for the variables $y_{\theta}$, $U_{\theta}$, $V_{\theta}$ and $H_{\theta}$. By introducing the space $H_{per}^{1}$

with the norm $\|f\|_{H_{per}^{1}} = \|f\|_{H_{[0, 1]}^{1}}$, We define a linear map $\Phi$: $\|f\|_{H_{per}^{1}}\times\mathbb{R}$ $\mapsto$ $E$ as $\Phi:(\sigma, h)$ $\mapsto$ $f = \sigma + hId$.

Lemma 3.1. The map $\Phi$ defined above is homeomorphism from $H_{per}^{1}\times\mathbb{R}$ to E.

It is clear that the space $E$ is a Banach space, because the space $H_{per}^{1}\times\mathbb{R}$ is a Banach space. Let's introduce $\eta$ = $y - Id$ and $(\sigma, h)$ = $\sigma + hId$, i.e. $h = H(t, 1) - H(t, 0)$ and $\sigma = H - hId$. Therefore, the system (16) is equivalent to

In the next section, the well-posedness of system (18) will be proved as an ordinary differential equations in the Banach space $W$. Note that

We have a bijection$(\eta, U, V, M, N, \sigma, h)$ $\mapsto$ $(y, U, V, M, N, H)$ from $W$ to $E_{1}\times H_{per}^{1}\times H_{per}^{1}\times L_{per}^{\infty}\times L_{per}^{\infty}\times E$ with $y = \eta + Id$ and $H = \sigma + hId$.

Theorem 3.2. Let $\bar X$ = $(\bar{\eta}, \bar{U}, \bar{V}, \bar{M}, \bar{N}, \bar{\rho}, \bar{h})\in W$, there exists a $T > 0$ depending only on $\|\bar{X}\|_{W}$ such that the system (18) has a unique solution in $C^{1}([0, T], E)$ with initial data $\bar{X}$.

Proof. To prove the this theorem, the key step is to prove the Lipchitz continuity of the right side of system (18). We define the map $\psi$:$W \mapsto W$

Firstly, we testify that $P_{i}$, $Q_{i}$, $P_{i, x}$, $Q_{i, x}$ are local Lipchitz continuous. Note that $B_{M}$ = $\{X = (\eta, U, V, M, N, \sigma, h)\in W|$ $\|X\|_{W}\leq M\}$. Let $X$ = $(\eta, U, V, M, N, \sigma, h)$ and $\bar{X}$ = $(\bar{\eta}, \bar{U}, \bar{V}, \bar{M}, \bar{N}, \bar{\sigma}, \bar{h})$ belong to $B_{M}$. Then, we get that

and $\|\bar{y}\|_{L^{\infty}}$ $\leq 1 + CM$. It is clear that $\cosh x$ and $\sinh x$ are local Lipchitz continuous on $\{X\in\mathbb{R}|$ $|X| \leq 1 + CM\}$ where C is a constant that depends only on $M$, we yield

and

for all $\theta$, $\theta'$ in [0, 1]. Then, we have

So does $\exp(-sgn(\theta - \theta')(y(\theta) - y(\theta')))[H_{\theta} + (U^{2} + 2MN - N^2)y_{\theta}]$. Thus, we obtain that

The similar computation shows that

Utilizing (15), we have

and

In conclusion, the local Lipchitz continuity from $W$ to $H_{per}^{1}$ of $P_{2}$ and $P_{2, x}$ has been proven, and so do $P_{1}$, $P_{1, x}$, $Q_{1}$, $Q_{1, x}$, $Q_{2}$ and $Q_{2, x}$. For above certify, it is easy to prove that $\psi$ is locally Lipchitz continuous from $W$ to $W$. Therefore, the theorem follows the standard theory of ordinary differential equations on Banach spaces.

Now, we prove the existence of a global solution of system (18). As we all know that initial data is very significant to system (18), but, here, we will only consider a particular initial data that belong to

$W_{1}$ is complete subspace of $W$. Let $\bar{X} \in W_{1}$. We investigate the short time solution X = $(\eta, U, V, M, N, \sigma,$ $h)$ $\in ([0, T], E)$ of system (18) given by $Theorem$ 3.2. Because $X \in C([0, T], W)$, $P_{i}$, $Q_{i}$, $P_{i, x}$, $Q_{i, x}$ $\in C([0, T], W)$, we now consider $U$, $V$, $P_{i}$, $Q_{i}$, $P_{i, x}$, $Q_{i, x}$ as functions in $C([0, T], H_{per}^{1})$ and $M, N$ in $C([0, T], L_{per}^{\infty})$. Then, for any fixed $\theta$ in $\mathbb{R}$, we can solve the following system of ordinary differential equations in $\mathbb{R}^{4}$ given by

which is obtained by substituting $\eta_{\theta}, U_{\theta}, V_{\theta}$ and $\sigma_{\theta}$ in system (17) by $\alpha$, $\beta$, $\gamma$ and $\delta$, respectively. We also replaced $h(t)$ by $h$, hence $h(t) = h$ for all $t$. Let

It is not very hard to check that $meas(S^{c}) = 0$. For all $\theta \in {S}$, let

For $\theta \in S^{c}$, we take $(\alpha(0, \theta), \beta(0, \theta), \gamma(0, \theta), \delta(0, \theta)) = (0, 0, 0, 0)$.

Lemma 3.3. Given initial data

Let $X = (\eta, U, V, M, N, \sigma, h) \in C([0, T], W)$ be the solution of system (18) followed Theorem 3.2. Then

and the functions $\alpha(t, \theta), \beta(t, \theta), \gamma(t, \theta), \delta(t, \theta)$, which solve system (19) for any fixed $\theta$ with the specified initial data above, coincide for almost every $\theta$ and for all time with $(\eta_{\theta}, U_{\theta}, V_{\theta}, \sigma_{\theta})$, that is, for all $t \in [0, T]$, for almost every $\theta$,

Proof. Firstly, let's introduce a key space $B_{per}^{\infty}$ in which elements are bounded periodic functions with the norm $\|f\|_{B_{per}^{\infty}}$ = $\sup_{\theta\in [0, 1]}|f(\theta)|$, clearly, this space is complete. According to the initial data conditions in the lemma, the solution for system (19) in space $[B_{per}^{\infty}]^{4}$ can be defined as $(\alpha(t, \theta), \beta(t, \theta), \gamma(t,$ $\theta),$ $\delta(t, \theta))$. It is not very difficult to check that the solutions exists on the interval [0, T] on which $(\eta, U, V, M, N, \sigma, h)$ is defined, since the system (19) is linear in $(\alpha, \beta, \gamma, \delta)$. According system (18), we get that

Since $B_{per}^{\infty}$ imbeds in $L_{per}^{2}$, the integral form of system (19) holds in $L_{per}^{2}$ sense. Now, we know that $(\alpha, \beta, \gamma, \delta)$ and $(\eta_{\theta}, U_{\theta}, V_{\theta}, \sigma_{\theta})$ satisfy the linear system (19) with the same initial data in $L^2$ sense on the interval $[0, T]$. By uniqueness, we get that

in $L_{per}^{2}$ on $[0, T].$

Now, we give the initial data as follow

where $\bar{h} = \int_{0}^{1}(\bar{u}^{2} + \bar{u}_{x}^{2} + \bar{v}^{2} + \bar{v}_{x}^{2}) d x$. By (22), we know that $\bar{y}(\theta)$ is continuous, strictly increasing. In the following work, we will testify that $(\bar{y} - Id , \bar{U}, \bar{V}, \bar{m}, \bar{N}, \bar{H} - \bar{h}Id, \bar{h})$ belongs to $\mathit{G}$ which is defined as follows.

Definition 3.4. The set $\mathit{G}$ is made up of all $(\eta, U, V, M, N, \sigma, h)\in W$ such that

where $y = \eta(\theta) + \theta$, $H(\theta) = \sigma(\theta) + h\theta$.

We know that the initial data $(\bar{\eta}, \bar{U}, \bar{V}, \bar{M}, \bar{N}, \bar{\sigma}, \bar{h})$ belongs to $\mathit{G}$. We will certify that the solution of system (18) exists globally in time with any initial data in $\mathit{G}$.

Lemma 3.5. Given the initial data $\bar{X} = (\bar{\eta}, \bar{U}, \bar{V}, \bar{M}, \bar{N}, \bar{\sigma}, \bar{h})$ in $\mathit{G}$. Let

be the local solution of system (18) in $C([0, T], W)$ for some $T > 0$, with the above initial data. Then

(i) $X(t)$ belongs to $\mathit{G}$ for all $t\in[0, T]$;

(ii) almost every $t\in[0, T]$, $y_{\theta}(t, \theta) > 0$ for almost every $\theta$.

Proof. (ⅰ) We continue to use $S$ defined above. Therefore, from $Lemma$ 3.3, the first one of $(23)$ holds for all $t\in[0, T]$ and $X\in C^{1}([0, T], W)$. We will prove that the second and third inequality of (23) hold for all $\theta\in S$. Fixed $\theta\in S$ and dropped it in the notation without ambiguity. By system (17), we get

and on the other hand,

It is not very difficult to check that $(My_{\theta})_{t} = (U_{\theta})_{t}$, $\bar{M}\bar{y}_{\theta} = \bar{U}_{\theta}$, $(Ny_{\theta})_{t} = (V_{\theta})_{t}$ and $\bar{N}\bar{y}_{\theta} = \bar{V}_{\theta}$. We can get that $M(t, \theta)y_{\theta}(t, \theta) = V_{\theta}(t, \theta)$ for $t \in [0, T]$, since the uniqueness of ordinary differential equation. Thus, (23) is defined. Now, we define $t^{*}$ as follow

If $t^{*}\leq T$, we obtain

for the continuity of $y_{\theta}(t)$ on time. From (23), we get that $U_{\theta}(t^{*}) = V_{\theta}(t^{*}) = 0$. Then, utilizing (17), we have

From system (17) and the fact $y_{\theta}(t^{*}) = U_{\theta}(t^{*}) + V_{\theta}(t^{*}) = 0$, we can infer

If $H_{\theta}(t^{*}) = 0$, combining with (23), we can deduce that

This is a contradiction to the uniqueness of system (17). If $H_{\theta}(t^{*}) < 0$, then $y_{\theta t t} < 0$. Thus, $y_{\theta}(t^{*})$ is the strict maximum. This contradicts the definition of $t^{*}$. Hence $H_{\theta}(t^{*}) > 0$ implies that $y_{\theta}(t^{*})$ is the strict maximum. This contradicts the fact $t^{*} < T$. Therefore, we get $y_{\theta}(t^{*}) \geq 0$ for all $t \in [0, T]$. Let us certify that $H_{\theta}(t) \geq 0$. This follows from (23) when $y_{\theta}(t) > 0$. If $y_{\theta}(t) = 0$ then as above, $H_{\theta}(t) < 0$ implies that $y_{\theta}(t) = 0$ is the strict maximum. This contradicts the fact $y_{\theta}(t) \geq 0$ on $[0, T]$. Hence $H_{\theta}(t) \geq 0$ on $[0, T]$. Now we have that $y_{\theta}(t) + H_{\theta}(t) \geq 0$. If the equality holds, it then follows that $y_{\theta}(t) = U_{\theta}(t) = V_{\theta}(t) = H_{\theta}(t) = 0$. This contradicts the uniqueness of system (17) for $\overline{y}_{\theta} > 0$.

(ⅱ) Let

Fubini's theorem infers that

where

From the above proof, we know that for $\theta \in \mathit{A}$, the time points $t$ satisfying $y_{\theta}(t, \theta) = 0$ are isolated. Thus, we have that $meas(\mathit{A}_{\theta}) = 0$. It follows from (24) and $meas(\mathit{A}^{c}) = 0$ that

We denote by $\mathit{K}$ the set of times such that $meas(\mathit{A}_{t}) > 0$, i.e.

Then, $meas(\mathit{K}) = 0$. For all $t\in \mathit{K}^{c}$, $y_{\theta} > 0$ almost everywhere. Therefore, $y(t, \theta)$ in strictly increasing and invertible.

Theorem 3.6. Given any $\bar{X} = (\bar{y}, \bar{U}, \bar{V}, \bar{M}, \bar{N}, \bar{H}) \in \mathit{G}$. Then the system (16) admits a global solution $X(t) = (y(t), U(t), V(t), M(t), N(t), h(t))$ $in\ C^{1}(\mathbb{R}^{+}, W)$ with the initial data $\bar{X} = (\bar{y}, \bar{U}, \bar{V}, \bar{M}, \bar{N}, \bar{H})$ and $X(t) \in \mathit{G}$ for all $t \in \mathbb{R}^{+}$. Moreover, by equipping $\mathit{G}$ with the topology inducted by the E-norm then the map D: $\mathit{G}\times\mathbb{R}^{+} \mapsto \mathit{G}$ defined as

is a continuous semigroup.

Proof. Let us write $(y, U, V, M, N, H)$ to denote $\eta, U, V, M, N, \sigma, h$ with $y = \eta + Id$ and $H = \sigma + hId$. Assuming $(\eta, U, V, M, N, \sigma, h)$ be a solution of system (18) in $C^{1}(\mathbb{R}^{+}, W)$ with the initial data $(\bar{\eta}, \bar{U}, \bar{V}, \bar{M}, \bar{N}, \bar{\sigma}, \bar{h})$, we have

It is clear that $h(t) = \bar{h}$ for all $t\in \mathbb{R}^{+}$. According to system (16) we get that $H(t, 0) = 0$. Because $H_{\theta} \geq 0$, we obtain $\|H\|_{L^{\infty}} \leq H(t, 1) = \bar{h}$. Hence, $\|\sigma\|_{L^{\infty}} \leq 2\bar{h}$, and $\sup_{t \in [0, T)}\|\sigma\|_{L^{\infty}} \leq 2\bar{h}$. For $\theta$ and $\theta'$ in [0, 1], we have that $|y(\theta) - y(\theta')| \leq 1$ for $y_{\theta} \geq 0$ and $y(1) - y(0) = 1$. Thus, we claim that

Indeed, using (23), we obtain that $y_{\theta} = 0$ which deduces $U_{\theta} = 0$. Thus, (26) holds if $y_{\theta} = 0$. Othetwise, if $y_{\theta} \geq 0$, from (23), we have

and

Utilizing (27)-(28), we get (26). From (13)-(22), we have that

where the constant $C$ is independent of $t$ and the initial data. From system (18), we have following estimates

For $\eta_{t} = U + V$, therefore

Now, we have certified that

is finite. Let

Since the system (17) is semilinear, we have that

where $C$ is only depend on $C_{1}$. Applying Gronwall's inequality, we get (25). By the standard theorem of ordinary differential equation, we obtain that $D_{t}$ is a continuous semigroup.

4.   The original system

In this section, we will investigate how to obtain a global conservative solution of system (4) from the global solution of system (17) with the initial variables $(t, x)$. Let $(y$$, U,$$V,$$M,$$N,$$H)$ be the global solution of system (17). Note that

Theorem 4.1. Let $(y, U, V, M, N, H)$ be a global solution of system (17). Then the pair function $(u(t$, $x)$, $v(t,$$x))$ defined by (29) is the global solution to the system (4). Moreover, this solution $(u, v)$ satisfies the following property:

Furthermore, let $(\bar{u}_{n}, \bar{v}_{n})$ be a sequence of the initial data such that

Then the corresponding solutions $(\bar{u}_{n}, \bar{v}_{n})$ converge to $(u(t, x), v(t, x))$ uniformly for all $(t, x)$ in any bounded set.

Proof. Firstly, what we have to do is to show that the definition of $u$ and $v$ make sense. Given $x\in [0, 1)$, if $\theta_{1} \leq \theta_{2}$, $x = y(t, \theta_{1}) = y(t, \theta_{2})$, then

We can obtain that $U_{\theta} = V_{\theta} = 0$ in $[\theta_{1}, \theta_{2}]$. Therefore, $U(\theta_{1}) = U(\theta_{2})$, $V(\theta_{1}) = V(\theta_{2})$, and we can get that $(u(t, x), v(t, x))$ from the above definition is meaningful. It is clear that $u(x +1) = u(x)$ and $v(x +1) = v(x)$. Now we will certify $u \in H^{1}$. Obviously, $u\in L^{\infty}$, which yields $u \in L^{2}$. So does $v$. Next, we will show that $u_{x}, v_{x} \in L^{2}$. From (23), we have

It is similar to $v_{x}$. Now, we testify the pair function $(u, v)$ satisfied (4). Given $\phi \in C^{\infty}(\mathbb{R}^{+}\times\mathbb{R})$ with compact support. Let $(y, U, V, M, N, H)$ be the solution of system (16), then

Utilizing $y_{t} = U + V$ and $y_{\theta t} = U_{\theta} = V_{\theta}$, we get

Integrating (32) over $\mathbb{R}^{+}\times\mathbb{R}$, using (23) and taking $x = y(t, \theta)$, we have that

By (31)-(33), we obtain the first two equation of system (4). When $t\in \mathit{K}^{c}$, we have $y_{\theta}(t, \theta) > 0$ a.e. Utilizing (23), we obtain

holds almost everywhere. By taking $x = y(t, \theta)$, we have

Therefore, we get (30).

Finally, let $(\bar{u}_{n}, \bar{v}_{n})$ converge to $(\bar{u}, \bar{v})$ in $H^{1}\times H^{1}$. From (22), it is not very hard to check that

Now, we certify that

Let $g_{n} = u_{n}^{2} + u_{nx}^{2} + v_{n}^{2} + v_{nx}^{2}$ and $g = u^{2} + u_{x}^{2} + v^{2} + v_{x}^{2}$. From (22), we have that

The first item on the right side of (34) can be written as follow

Because $0 \leq y_{\theta} \leq 1 + h$, it follows that

Note that $g\in L^{1}$. For any $\varepsilon \geq 0$, there exists a continuous function $r$ such that $\|g - v\|_{L^{1}} \leq \varepsilon$. Then

The first and third item tend to zero for the boundedness of $y_{n\theta}, y_{\theta}$ and the arbitrariness of $\varepsilon$. The second item tends to zero by utilizing the dominated convergence theorem. Hence $\bar{y}_{n\theta} \to \bar{y}_{\theta}$ in $L^{1}$. We finally obtain that $\bar{y}_{n\theta} \to \bar{y}_{\theta}$ in $L^{2}$, because $y_{n\theta}$ is bounded in $L^{\infty}$. From (22), we get $\bar{H}_{n\theta} \to \bar{H}_{\theta}$ in $L^{2}$,

For $\bar{u}_{nx} \to \bar{u}_{x}$ in $L^{\infty}$, we know that

Given any $\varepsilon > 0$, there exists a continuous function $r(\theta)$ such that

Then

Utilizing (35), we have

and

According to dominated convergence theorem, we get

Thus, we have $\bar{M}_{n}\to\bar{M}$ in $L^{2}$. Similarly, $\bar{N}_{n}\to\bar{N}$ in $L^{2}$.

According to (23), $\|\bar{U}_{n} - \bar{U}\|_{L^{\infty}}\to 0$, $\|\bar{V}_{n} - \bar{V}\|_{L^{\infty}}\to 0$, with $M_{n} \to M$, $y_{n\theta} \to y_{\theta}$ in $L^{2}$, we get $\|U_{n\theta}\|_{L^{2}}\to\|U_{\theta}\|_{L^{2}}$. So does $\|V_{n\theta}\|_{L^{2}}\to\|V_{\theta}\|_{L^{2}}$. What we only need to do to prove $U_{n\theta}\to U_{\theta}$ and $V_{n\theta}\to N_{\theta}$ in $L^{2}$ is to show that for any continuous function $\Psi$ with compact support, we have

Thus,

Then, we obtain $U_{n\theta}\to U_{\theta}$ in $L^{2}$. With the same calculation we have $V_{n\theta}\to V_{\theta}$ in $L^{2}$. Now we have

Combining (16)-(17) and (36), we get

According to Gronwall's inequality, we conclude that $y_{n}\to y$, $U_{n}\to U$ and $V_{n}\to V$ in $L^{\infty}$ on any bounded time interval. This yields that

are uniformly Hölder continuous on any bounded time interval.

Lastly, we shall certify that the solutions obtained in $Theorem$ 4.1 construct a semigroup.

Theorem 4.2. Given initial data $(\bar{u}, \bar{v})\in H_{per}^{1}\times H_{per}^{1}$. Let $(u(t), v(t)) = F_{t}(\bar{u}, \bar{v})$ be the corresponding global solution of system (4) constructed in $Theorem$ 4.1. Then the map $F:H_{per}^{1}\times H_{per}^{1}\times[0, \infty]\mapsto H_{per}^{1}\times H_{per}^{1}$ is semigroup.

Proof. Fix $(\bar{u}, \bar{v})\in H_{per}^{1}\times H_{per}^{1}$ and $\tau > 0$. For every $t>0$, what we need to do is prove

Let $(y(\tau, \theta)$, $U(\tau, \theta)$, $V(\tau, \theta)$, $M(\tau, \theta)$, $N(\tau, \theta)$, $H(\tau, \theta))$ be the solution of system (16) with the initial data given by (22). For any time $\tau$, we have the new initial data as follows

and

Let $(\hat{y}(t + \tau, \theta), \hat{U}(t + \tau, \theta), \hat{V}(t + \tau, \theta), \hat{M}(t + \tau, \theta), \hat{N}(t + \tau, \theta), \hat{H}(t + \tau, \theta))$ be a solution of the system (16) with the initial data (37)-(38). We claim that

where $\tilde{\theta}$ is defined as $\hat{y}(\tau + t, \tilde{\theta}) = y(\tau + t, \theta)$.

Actually, the equality $\hat{y}(\tau + t, \tilde{\theta}) = y(\tau + t, \theta)$ yields that

Utilizing (39), we claim that

For $P_i, P_{i, x}$($i$ = 1, 2), we can get the similar results. At the time $\tau$, $y(\tau, \theta) = \hat{y}(\tau, \hat{\theta})$ implies that

Therefore, we get that $({y}(\tau, \theta(\hat{\theta})), {U}(\tau, \theta(\hat{\theta})), {V}(\tau, \theta(\hat{\theta})), {M}(\tau, \theta(\hat{\theta})), {N}(\tau, \theta(\hat{\theta}))$, ${H}(t + \tau, \theta))$ is a solution of system (16). Utilizing (29) and $\hat{y}(\tau + t, \tilde{\theta}) = y(\tau + t, \theta)$, we obtain that the solution of the system (4) constructs a semigroup.

Acknowledgments

The authors are very grateful to the anonymous reviewers and editors for their careful reading and useful suggestions, which greatly improved the presentation of the paper.

The first author Chen is partially supported by Science and Technology Research Program of Chongqing Municipal Educational Commission. The third author Zhang is partially supported by the National Social Science Fund of China (No.19BJY077, No.18BJY093 and No.17CJY031), the National Natural Science Foud of China (No.71901044), Chongqing Social Science Planning Project (No.2018PY74), Science and Technology Project of Chongqing Education Commission(No.KJQN20180051 and No.KJQN201900537).