Nonuniform Dependence of a Two-Component NOVIKOV System in Besov Spaces

Abstract

Considered herein is the Cauchy problem of the two-component Novikov system. In the periodic case, we first constructed an approximate solution sequence that possesses the nonuniform dependence property; then, by applying the energy methods, we managed to prove that the difference between the approximate and actual solution is negligible, thus succeeding in proving the nonuniform dependence result in both supercritical Besov spaces $B_{p,r}^s\left(\mathbb{T}\right)\times B_{p,r}^s\left(\mathbb{T}\right)$ with $s>max\{\frac{3}{2},1+\frac{1}{p}\},$$1\le p\le \infty ,1\le r<\infty$ and critical Besov space $B_{2,1}^{\frac{3}{2}}\left(\mathbb{T}\right)\times B_{2,1}^{\frac{3}{2}}\left(\mathbb{T}\right)$. In the non-periodic case, we constructed two sequences of initial data with high and low-frequency terms by analyzing the inner structure of the system under investigation in detail, and we proved that the distance between the two corresponding solution sequences is lower-bounded by time t, but converges to zero at initial time. This implies that the solution map is not uniformly continuous both in supercritical Besov spaces $B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)$ with $s>max\{\frac{3}{2},1+\frac{1}{p}\},1\le p\le \infty ,1\le r<\infty$ and critical Besov spaces $B_{p,1}^{1+\frac{1}{p}}\left(\mathbb{R}\right)\times B_{p,1}^{1+\frac{1}{p}}\left(\mathbb{R}\right)$ with $1\le p\le 2$. The proof of nonuniform dependence is based on approximate solutions and Littlewood–Paley decomposition theory. These approaches are widely applicable in the study of continuous properties for shallow water equations.

1. Introduction

In this paper, we considered the Cauchy problem for the following system, which can be viewed as a two-component generalization of the famous Novikov equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{m}_t+m_x(u^2+v^2)+3m(u{u}_x+v{v}_x)+n(u_{x}v-u{v}_x)=0,\hfill & t>0,x\in \mathbb{R}\mathrm{or}x\in \mathbb{T},\hfill \\ n_t+n_x(u^2+v^2)+3n(u{u}_x+v{v}_x)+m(u_{x}v-u{v}_x)=0,\hfill & t>0,x\in \mathbb{R}\mathrm{or}x\in \mathbb{T},\hfill \\ m=u-u_{xx},n=v-v_{xx},\hfill & x\in \mathbb{R}\mathrm{or}x\in \mathbb{T},\hfill \\ u(0,x)=u_0\left(x\right),v(0,x)=v_0\left(x\right),\hfill & x\in \mathbb{R}\mathrm{or}x\in \mathbb{T}.\hfill \end{array}\right.\end{array}$$

(1)

System (1) was proposed recently by Li et al. in [1,2]. This system was derived from the zero-curvature equation:

$$M_t-N_x+[M,N]=0$$

for the vector prolongation of the Lax pair representation of the Geng–Xue system [3] as follows:

$${\mathsf{\Phi }}_x=M\mathsf{\Phi },{\mathsf{\Phi }}_t=N\mathsf{\Phi },$$

where the matrix M and N are given by [1,2,4]

$$\begin{array}{c}\hfill M=\left(\begin{array}{cccc}0& \lambda m& \lambda n& 1\\ 0& 0& 0& \lambda m\\ 0& 0& 0& \lambda n\\ 1& 0& 0& 0\end{array}\right),\lambda \in \mathbb{R},\end{array}$$

$$\begin{array}{c}\hfill N=\left(\begin{array}{cccc}{\displaystyle -(u{u}_x+v{v}_x)\frac{u_x}{\lambda }}& -\lambda (u^2+v^2)m& {\displaystyle \frac{v_x}{\lambda }-\lambda (u^2+v^2)n}& u_x^2+v_x^2\\ {\displaystyle \frac{u}{\lambda }}& {\displaystyle -\frac{1}{{\lambda }^2}}& u{v}_x-u_{x}v& {\displaystyle -\frac{u_x}{\lambda }-\lambda (u^2+v^2)m}\\ {\displaystyle \frac{v}{\lambda }}& u_{x}v-u{v}_x& {\displaystyle -\frac{1}{{\lambda }^2}}& {\displaystyle -\frac{v_x}{\lambda }-\lambda (u^2+v^2)n}\\ -(u^2+v^2)& {\displaystyle \frac{u}{\lambda }}& {\displaystyle \frac{v}{\lambda }}& u{u}_x+v{v}_x\end{array}\right),\lambda \in \mathbb{R},\end{array}$$

and the bi-Hamiltonian structure of (1) is also given in [1,2,4].

System (1) degenerates into the following well-known Novikov equation by taking $u=v$ in (1) up to a scale invariance:

$$\begin{array}{c}\hfill m_t+3mu{u}_x+m_x{u}^2=0,m=u-u_{xx}.\end{array}$$

(2)

The Novikov Equation (2) was firstly proposed in [5] in a symmetry classification of integrable non-evolutionary partial differential equation

$$(1-D_x^2)u_t=F(u,u_x,u_{xx},u_{xxx},\cdots )$$

by applying the perturbative symmetry approach. The first non-trivial higher symmetries were presented, and the integrability and multi-peakon solutions of Equation (2) were also researched, in [6,7]. The well-posedness of (2) has already been well-established by much literature in Sobolev spaces $H^s$ with $s>\frac{3}{2}$—see [8,9,10]—and Equation (2) is not locally well-posed in Sobolev spaces $H^s$ with $s<\frac{3}{2}$ in the sense that the solution does not continuously depend on the initial data [10], which implies that $s=\frac{3}{2}$ is the critical Sobolev index for well-posedness. Later, the well-posedness investigation for the Novikov Equation (2) was generalized to Besov spaces. Ni and Zhou [11] established local well-posedness in critical Besov space $B_{2,1}^{\frac{3}{2}}$, and the local well-posedness in super-critical Besov spaces $B_{p,r}^s$ with $s>\frac{3}{2},p,r\in [1,\infty ]$ was also studied, but this result fails for the Novikov equation in $B_{2,\infty }^{\frac{3}{2}}$ [12,13]. The precise blow-up scenario for the Novikov equation was given in [9,14,15]. The existence of global strong solutions and global weak solutions was also established [9,16,17]. The stability of Novikov peakons was presented in [18,19,20]. For the continuity properties of solutions to Equation (2), it was shown that the solution map for the Novikov equation is Hölder continuous in $H^r$-topology for all $0\le r<s$ with exponent $\alpha$ depending on $s>\frac{3}{2}$ and r [21].

The studies of multi-component Camassa–Holm-type equations have also attracted much attention. Concerning the multi-component generalizations of the Novikov equation, three typical systems are popular. The first one is the following system proposed by Popowicz [22]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{m}_t+3mu{u}_x+m_x{u}^2-\rho {\left(u\rho \right)}_x=0,\hfill \\ {\rho }_t+u^2{\rho }_x+\rho u{u}_x=0,\hfill \\ m=u-u_{xx}.\hfill \end{array}\right.\end{array}$$

(3)

System (3) is locally well-posed both in supercritical Besov spaces $B_{p,r}^s\times B_{p,r}^{s-1},s>max\{1+\frac{1}{p},\frac{3}{2}\}$ and critical Besov space [23]. The solution map of system (3) is not uniformly continuous in Sobolev spaces $H^s\left(\mathbb{R}\right)\times H^{s-1}\left(\mathbb{R}\right)$ with $s>\frac{5}{2}$ [24], in supercritical Besov spaces $B_{2,r}^s\times B_{2,r}^{s-1}$ with $s>\frac{3}{2},1\le r<\infty$ and in critical Besov spaces $B_{2,1}^{\frac{3}{2}}\times B_{2,1}^{\frac{1}{2}}$ [25]. The persistence properties of the strong solution to Equation (3) was also investigated by Zhou et al. [26].

The second system is the following Geng–Xue system [3]

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{m}_t+3mu{u}_x+m_{x}uv=0,\hfill \\ m_t+3mu{u}_x+m_{x}uv=0,\hfill \\ m=u-u_{xx},n=v-v_{xx}.\hfill \end{array}\right.\end{array}$$

(4)

The local well-posedness results of Equation (4) in supercritical Besov spaces $B_{p,r}^s\times B_{p,r}^s,s>max\{2+\frac{1}{p},\frac{5}{2}\}$ and critical Besov space $B_{2,1}^{\frac{5}{2}}\times B_{2,1}^{\frac{5}{2}}$ were established by Mi et al. [27] and Tang et al. [28], respectively. The bi-Hamiltonian structure was given in [29]. The persistence and wave-breaking criterion [30], integrability [3,29] and dynamics and structure of peaked soliotions [31,32] were also studied recently.

The third system is system (1) under discussion. Fu and Qu [4] established the local well-posedness of (1) in supercritical Besov spaces $B_{p,r}^s$ with $1\le p,r\le \infty ,s>max\{1+\frac{1}{p},\frac{3}{2}\}$. The wave-breaking mechanism and existence of single-peakons and muli-peakons were also demonstrated in [4].

The continuity of the data-to-solution map plays a crucial role in the well-posedness theory, and there is an increasing interest in investigating the non-uniformly continuous dependence of the solution on the initial data, which indicates that the continuity of the data-to-solution map is sharp. The key issue that lies in these problems is determining the regularity of the initial data, which guarantees the non-uniform dependence. Himonas et al. [8,33,34] obtained the non-uniform dependence of the solution map for equations such as the Camassa–Holm(CH) equation, the Degasperis–Procesi equation and the Novikov equation from $H^s$ into $C([0,T];H^s)$, where the regularity index s is always assumed as $s>\frac{3}{2}$. The discussions and methodologies have recently been extended to systems, and similar results were obtained for the two-component Novikov system (3) in the product Sobolev spaces $H^s\times H^{s-1}$ with $s>\frac{5}{2}$ [24]. Yang et al. [35] studied a high-order two-component b-family system. Based on the local well-posedness results and prior estimates, two sequences of solutions whose distance initially goes to zero but is later bounded below by a positive constant were constructed by the method of approximate solutions; in other words, the authors proved that the data-to-solution map is not uniformly continuous in Sobolev spaces with an index of more than $5/2$.

Recently, the discussions of the non-uniform dependence of CH-type equations have been extended to Besov spaces. The techniques devised in this literature are Littlewood–Paley decomposition and transport theories, which are different from the discussions in the Sobolev spaces. Li et al. [36,37] proved the non-uniform dependence of the CH equation in supercritical Besov spaces $B_{p,r}^s$ with $s>max\{\frac{3}{2},1+\frac{1}{p}\}$ and critical Beosv space $B_{2,1}^{\frac{3}{2}}$, respectively. Later, this result was generalized to low-regularity Besov spaces $B_{p,r}^s$ with $s>1,1\le p,r<\infty$ [38]. It is worth mentioning that the investigation on nonuniform dependence properties has been generalized to equations other than the CH-type equations. In [39], Li et al. proved the nonuniform dependence result for the Benjamin–Ono (BO) equation, where they showed that the BO equation cannot be uniformly continuous on bounded sets on $B_{2,r}^s\left(\mathbb{R}\right)$ for $s>3/2$ and $r\in [2,\infty )$. Zhou et al. [40] established the local well-posedness and nonuniform dependence result for the hyperbolic Keller–Segel equation in the Besov framework $B_{p,r}^s\left({\mathbb{R}}^d\right)$ with $1\le p,r\le \infty$ and $s>1+\frac{d}{p}$. Holmes et al. [41] considered a generalized rotation b-family system (R-b-family system) that models the evolution of equatorial water waves. They established sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from $B_{p,r}^s\times B_{p,r}^{s-1}$ to $C(0,T;B_{p,r}^s\times B_{p,r}^{s-1})$. It is also worth noting that Wang et al. [42] extended the result of non-uniform dependence to the Geng–Xue system (4) in the product Besov spaces.

It can be observed from the above discussions that the non-uniform dependence results have been well-established for systems (3) and (4) in Sobolev spaces, but, up until now, due to the complex structure caused by the strongly coupled terms, the non-uniform dependence of system (1) is still open to debate even in the Sobolev spaces. By utilizing the techniques devised in [36,37,42], based on the already established local well-posedness results, we employed the delicate energy methods, successfully overcame the difficulties in the estimates brought about by the complex structure of this system and then formulated an overall investigation of the non-uniform dependence of system (1). To be specific, we studied the non-uniform dependence in the periodic case and non-periodic case, respectively. In the periodic case, the results were established in both supercritical Besov spaces $B_{p,r}^s\left(\mathbb{T}\right)\times B_{p,r}^s\left(\mathbb{T}\right)$ with $s>max\{\frac{3}{2},1+\frac{1}{p}\}$ and critical Besov spaces $B_{2,1}^{\frac{3}{2}}\left(\mathbb{T}\right)\times B_{2,1}^{\frac{3}{2}}\left(\mathbb{T}\right)$, respectively. In the non-periodic case, we also proved that the continuity of the data-to-solution map of system (1) is sharp in the sense that the solution map is non-uniformly dependent on the initial data both in the supercritical space $B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)$ with $s>max\{\frac{3}{2},1+\frac{1}{p}\}$ and the critical case $B_{p,1}^{1+\frac{1}{p}}\left(\mathbb{R}\right)\times B_{p,1}^{1+\frac{1}{p}}\left(\mathbb{R}\right)$ with $1\le p\le 2$, respectively.

For $T>0,s\in \mathbb{R}$ and $1\le p,r\le \infty$, we define

$$\begin{array}{c}\hfill E_{p,r}^s\left(T\right)=\left\{\begin{array}{cc}C([0,T];B_{p,r}^s)\bigcap C^1([0,T];B_{p,r}^{s-1}),\hfill & \mathrm{if}r<\infty ,\hfill \\ L^{\infty }(0,T;B_{p,\infty }^s)\bigcap Lip([0,T];B_{p,\infty }^{s-1}),\hfill & \mathrm{if}r=\infty ,\hfill \end{array}\right.\end{array}$$

In order to apply the transport theories, before proceeding any further, we need to reformulate (1). By applying the inverse of Helmholtz operator ${(1-{\partial }_x^2)}^{-1}$ to both sides of the equations in (1), we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle u_t+(u^2+v^2)u_x=-{\partial }_x{(1-{\partial }_x^2)}^{-1}\left(u^3+u{v}^2+\frac{3}{2}u{u}_x^2+u_{x}v{v}_x+\frac{1}{2}u{v}_x^2\right)}\hfill \\ {\displaystyle -\frac{1}{2}{(1-{\partial }_x^2)}^{-1}\left(u_x^3+u_x{v}_x^2\right)\triangleq f_1(u,v)+f_2(u,v),}\hfill \\ {\displaystyle v_t+(u^2+v^2)v_x=-{\partial }_x{(1-{\partial }_x^2)}^{-1}\left(v^3+u^{2}v+\frac{3}{2}v{v}_x^2+u{u}_x{v}_x+\frac{1}{2}{u}_x^{2}v\right)}\hfill \\ {\displaystyle -\frac{1}{2}{(1-{\partial }_x^2)}^{-1}\left(u_x^2{v}_x+v_x^3\right)\triangleq g_1(u,v)+g_2(u,v),}\hfill \\ u(0,x)=u_0\left(x\right),v(0,x)=v_0\left(x\right).\hfill \end{array}\right.\end{array}$$

(5)

The novelty of this paper lies in skilfully devising the error estimates in the proof of the nonuniform dependence result in the critical Besov space $B_{2,1}^{\frac{3}{2}}\left(\mathbb{T}\right)$, where the lack of estimates for errors in $B_{2,1}^{\frac{1}{2}}\left(\mathbb{T}\right)$ has caused great difficulties for us. In order to overcome these difficulties, we first estimated the errors in $B_{2,\infty }^{\frac{1}{2}}\left(\mathbb{T}\right)$ and $B_{2,1}^{\frac{5}{2}}\left(\mathbb{T}\right)$, respectively. Then, we applied the real interpolation formula to obtain the error estimates in $B_{2,1}^{\frac{3}{2}}\left(\mathbb{T}\right)$, thus obtaining the nonuniform dependence result. The novelty of this paper also lies in the subtle choice of approximate solutions in the non-periodic case, since, compared with CH-type equations, the two component Novikov system (1) possesses higher-order terms $u_x^3,v_x^3$ (see Equation (5)) and strongly coupled terms $u_x{v}_x^2,u_x^2{v}_x$ (see Equation (5)). If we choose the low and high-frequency terms of the approximate solutions similar to that of the CH equations, we will not be able to estimate the errors in suitable Besov spaces successively. Therefore, in the non-periodic case, we considered the inner structure of system (1) and found a new sequence of approximate solutions, where we selected more complicated low and high-frequency parts of the approximate solutions (see Section 4).

The pros of the approximate solution technique is that the construction of approximate solutions to prove nonuniform dependence is very natural and easy to understand. The cons of this technique is that the computation is heavy when obtaining the desired error estimates.

We introduce our main results on non-uniform dependence of (5) in Besov spaces as follows.

The first result of non-uniform dependence indicates that the continuity of the data-to-solution map of Equation (5) is sharp in supercritical periodic Besov spaces.

Theorem 1.

Suppose that $(u_0,v_0)\in B_{p,r}^s\left(\mathbb{T}\right)\times B_{p,r}^s\left(\mathbb{T}\right)$ with $s>max\{\frac{3}{2},1+1/p\},1\le p,r<\infty$; then, there exists a time $T_0>0$ such that the solution map $(u_0,v_0)\mapsto (u\left(t\right),v\left(t\right))$ of the Cauchy problem (5) is not uniformly continuous from any bounded subset of $B_{p,r}^s\left(\mathbb{T}\right)\times B_{p,r}^s\left(\mathbb{T}\right)$ into $\mathcal{C}([0,T_0];B_{p,r}^s\left(\mathbb{T}\right)\times B_{p,r}^s\left(\mathbb{T}\right))$. More precisely, there exist two sequences of solutions $(u_1^n\left(t\right),v_1^n\left(t\right))$ and $(u_2^n\left(t\right),v_2^n\left(t\right))$ such that the corresponding initial data satisfy

$$\parallel u_1^n\left(0\right),v_1^n\left(0\right),u_2^n\left(0\right),v_2^n{\left(0\right)\parallel }_{B_{p,r}^s\left(\mathbb{T}\right)}\lesssim 1,$$

and

$$\underset{n\to \infty }{lim}\parallel u_1^n\left(0\right)-u_2^n{\left(0\right)\parallel }_{B_{p,r}^s\left(\mathbb{T}\right)}=\underset{n\to \infty }{lim}{\parallel v_1^n\left(0\right)-v_2^n\left(0\right)\parallel }_{B_{p,r}^s\left(\mathbb{T}\right)}=0.$$

while

$$\underset{n\to \infty }{lim\; inf}\parallel u_1^n\left(t\right)-u_2^n{\left(t\right)\parallel }_{B_{p,r}^s\left(\mathbb{T}\right)}\gtrsim |sin\frac{t}{4}|,\underset{n\to \infty }{lim\; inf}\parallel v_1^n\left(t\right)-v_2^n{\left(t\right)\parallel }_{B_{p,r}^s\left(\mathbb{T}\right)}\gtrsim |sin\frac{t}{4}|,t\in [0,T_0],$$

with small positive time $T_0$ such that $T_0\le T_1$.

In the critical case when $(s,p,r)=(\frac{3}{2},2,1)$, there are no estimates for solutions in $B_{2,1}^{1/2}$; therefore, we carried out the error estimates in $B_{2,\infty }^{\frac{1}{2}}$ instead to establish the following non-uniform dependence result:

Theorem 2.

Suppose that $(u_0,v_0)\in B_{2,1}^{3/2}\left(\mathbb{T}\right)\times B_{2,1}^{3/2}\left(\mathbb{T}\right)$; then, there exists a time $T_2>0$ such that the solution map $(u_0,v_0)\mapsto (u\left(t\right),v\left(t\right))$ of the Cauchy problem (5) is not uniformly continuous from any bounded subset of $B_{2,1}^{3/2}\left(\mathbb{T}\right)\times B_{2,1}^{3/2}\left(\mathbb{T}\right)$ into $\mathcal{C}([0,T_2];B_{2,1}^{3/2}\left(\mathbb{T}\right)\times B_{2,1}^{3/2}\left(\mathbb{T}\right))$. More precisely, there exist two sequences of solutions $(u_1^n\left(t\right),v_1^n\left(t\right))$ and $(u_2^n\left(t\right),v_2^n\left(t\right))$ such that the corresponding initial data satisfy

$$\parallel u_1^n\left(0\right),v_1^n\left(0\right),u_2^n\left(0\right),v_2^n{\left(0\right)\parallel }_{B_{2,1}^{3/2}\left(\mathbb{T}\right)}\lesssim 1,$$

and

$$\underset{n\to \infty }{lim}\parallel u_1^n\left(0\right)-u_2^n{\left(0\right)\parallel }_{B_{2,1}^{3/2}\left(\mathbb{T}\right)}=\underset{n\to \infty }{lim}{\parallel v_1^n\left(0\right)-v_2^n\left(0\right)\parallel }_{B_{2,1}^{3/2}\left(\mathbb{T}\right)}=0.$$

while

$$\underset{n\to \infty }{lim\; inf}\parallel u_1^n\left(t\right)-u_2^n{\left(t\right)\parallel }_{B_{2,1}^{3/2}\left(\mathbb{T}\right)}\gtrsim |sin\frac{t}{4}|,\underset{n\to \infty }{lim\; inf}\parallel v_1^n\left(t\right)-v_2^n{\left(t\right)\parallel }_{B_{2,1}^{3/2}\left(\mathbb{T}\right)}\gtrsim |sin\frac{t}{4}|,t\in [0,T_3],$$

where $exp(-ct)>3/4$, c is a positive constant and $T_3$ is a small positive time such that $T_3\le T_2$.

In the non-periodic case, by choosing an appropriate approximate solution sequence, we managed to show the non-uniform dependence result in the sense that, if the initial data are subtly perturbed, then the approximate solution will no longer converge to the exact solution. The main results are stated as follows:

Theorem 3.

Suppose that $(u_0,v_0)\in B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)$, with $s>max\{\frac{3}{2},1+1/p\},1\le p\le \infty ,1\le r<\infty$; then, there exists a time $T_4>0$ such that the solution map $(u_0,v_0)\mapsto (u\left(t\right),v\left(t\right))$ of the Cauchy problem (5) is not uniformly continuous from any bounded subset of $B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)$ into $\mathcal{C}([0,T_4];B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right))$. More precisely, there exist two sequences of solutions $(u_n\left(t\right),v_n\left(t\right))$ and $({\tilde{u}}_n\left(t\right),{\tilde{v}}_n\left(t\right))$ such that the corresponding initial data satisfy

$$\parallel u_n\left(0\right),v_n\left(0\right),{\tilde{u}}_n\left(0\right),{\tilde{v}}_n{\left(0\right)\parallel }_{B_{p,r}^s\left(\mathbb{R}\right)}\lesssim 1,$$

and

$$\underset{n\to \infty }{lim}\parallel u_n\left(0\right)-{\tilde{u}}_n{\left(0\right)\parallel }_{B_{p,r}^s\left(\mathbb{R}\right)}=\underset{n\to \infty }{lim}{\parallel v_n\left(0\right)-{\tilde{v}}_n\left(0\right)\parallel }_{B_{p,r}^s\left(\mathbb{R}\right)}=0.$$

while

$$\underset{n\to \infty }{lim\; inf}\parallel u_n\left(t\right)-{\tilde{u}}_n{\left(t\right)\parallel }_{B_{p,r}^s\left(\mathbb{R}\right)}\gtrsim t,\underset{n\to \infty }{lim\; inf}{\parallel v_n\left(t\right)-{\tilde{v}}_n\left(t\right)\parallel }_{B_{p,r}^s\left(\mathbb{R}\right)}\gtrsim t,t\in [0,T_5],$$

with small positive time $T_5$ such that $T_5\le T_4$.

The last theorem concerns non-uniform dependence in non-periodic critical Besov spaces:

Theorem 4.

Suppose that $(u_0,v_0)\in B_{p,1}^{1+1/p}\left(\mathbb{R}\right)\times B_{p,1}^{1+1/p}\left(\mathbb{R}\right)$ with $1\le p\le 2$; then, there exists a time $T_6>0$ such that the solution map $(u_0,v_0)\mapsto (u\left(t\right),v\left(t\right))$ of the Cauchy problem (5) is not uniformly continuous from any bounded subset of $B_{p,1}^{1+1/p}\left(\mathbb{R}\right)\times B_{p,1}^{1+1/p}\left(\mathbb{R}\right)$ into $\mathcal{C}([0,T_6];B_{p,1}^{1+1/p}\left(\mathbb{R}\right)\times B_{p,1}^{1+1/p}\left(\mathbb{R}\right))$. More precisely, there exist two sequences of solutions $(u_n\left(t\right),v_n\left(t\right))$ and $({\tilde{u}}_n\left(t\right),{\tilde{v}}_n\left(t\right))$ such that the corresponding initial data satisfy

$$\parallel u_n\left(0\right),v_n\left(0\right),{\tilde{u}}_n\left(0\right),{\tilde{v}}_n{\left(0\right)\parallel }_{B_{p,1}^{1+1/p}\left(\mathbb{R}\right)}\lesssim 1,$$

and

$$\underset{n\to \infty }{lim}\parallel u_n\left(0\right)-{\tilde{u}}_n{\left(0\right)\parallel }_{B_{p,1}^{1+1/p}\left(\mathbb{R}\right)}=\underset{n\to \infty }{lim}{\parallel v_n\left(0\right)-{\tilde{v}}_n\left(0\right)\parallel }_{B_{p,1}^{1+1/p}\left(\mathbb{R}\right)}=0.$$

while

$$\underset{n\to \infty }{lim\; inf}\parallel u_n\left(t\right)-{\tilde{u}}_n{\left(t\right)\parallel }_{B_{p,1}^{1+1/p}\left(\mathbb{R}\right)}\gtrsim t,\underset{n\to \infty }{lim\; inf}{\parallel v_n\left(t\right)-{\tilde{v}}_n\left(t\right)\parallel }_{B_{p,1}^{1+1/p}\left(\mathbb{R}\right)}\gtrsim t,t\in [0,T_5],$$

with small positive time $T_5$ such that $T_7\le T_6$.

The rest of this paper is organized as follows. In Section 2, we list several Lemmas that will be helpful to prove the main Theorems. In Section 3, we demonstrate the non-uniform dependence in periodic Besov spaces. The discussions are split into two subsections concerning the supercritical case and critical case, respectively. In Section 4, we also consider the non-uniform dependence in non-periodic Besov spaces, and the supercritical case and critical case are also studied separately in two subsections.

Notations. For a given Banach space X, we denote its norm by ${\parallel ·\parallel }_X$.

2. Preliminaries

In this section, we shall recall some properties of the Besov spaces and the transport equation theories.

Definition 1 ([43]).

Let $s\in \mathbb{R}$, $p,r\in [1,\infty ]$ and $u\in {\mathcal{D}}^{\prime }\left(\mathbb{T}\right)$. Then, we define the Besov space of functions as

$$B_{p,r}^s=\{u\in {\mathcal{D}}^{\prime }{\left(\mathbb{T}\right):\parallel u\parallel }_{B_{p,r}^s}<\infty \},$$

where

$$\begin{array}{c}\hfill {\parallel u\parallel }_{B_{p,r}^s}\doteq \left\{\begin{array}{cc}({\displaystyle \sum _{q\ge -1}}(2^{sq}\parallel {\Delta }_q{u\parallel }_{L^p}{{)}^r)}^{\frac{1}{r}}\hfill & \mathrm{if}1\le r<\infty ,\hfill \\ \underset{q\ge -1}{sup}{2}^{sq}{\parallel {\Delta }_{q}u\parallel }_{L^p},\hfill & \mathrm{if}r=\infty .\hfill \end{array}\right.\end{array}$$

In particular, $B_{p,r}^{\infty }=\bigcap _{s\in \mathbb{R}}{B}_{p,r}^s.$

The following lemma summarizes some useful properties of the Besov space $B_{p,r}^s$.

Lemma 1 ([43]).

Let $s\in \mathbb{R},1\le p,r,p_j,r_j\le \infty ,j=1,2$; then,

(1) 

Density: $C_c^{\infty }$ is dense in $B_{p,r}^s\iff p,r\in [1,\infty )$.

(2) 

Embedding: $B_{p_1,r_1}^s\hookrightarrow B_{p_2,r_2}^{s-n(\frac{1}{p_1}-\frac{1}{p_2})}$ if $p_1\le p_2$ and $r_1\le r_2$. $B_{p,r_2}^{s_2}\hookrightarrow B_{p,r_1}^{s_1}$ is locally compact if $s_1<s_2$.

(3) 

Algebraic properties: $\forall s>0,B_{p,r}^s\bigcap L^{\infty }$ is a Banach algebra. $B_{p,r}^s$ is a Banach algebra $\iff B_{p,r}^s\hookrightarrow L^{\infty }\iff s>\frac{1}{p}$ or ($s\ge \frac{1}{p}$ and $r=1$). In particular, $B_{2,1}^{1/2}$ is continuously embedded in $B_{2,\infty }^{1/2}\bigcap L^{\infty }$ and $B_{2,\infty }^{1/2}\bigcap L^{\infty }$ is a Banach algebra.

(4) 

1-D Moser-type estimates:

(i) 

For $s>0$,

$${\parallel fg\parallel }_{B_{p,r}^s}\le {C(\parallel f\parallel }_{B_{p,r}^s}{\parallel g\parallel }_{L^{\infty }}+{\parallel f\parallel }_{L^{\infty }}{\parallel g\parallel }_{B_{p,r}^s}).$$

(ii) 

$\forall s_1\le \frac{1}{p}<s_2$($s_2\ge \frac{1}{p}$ if $r=1$) and $s_1+s_2>0$, we have

$${\parallel fg\parallel }_{B_{p,r}^{s_1}}\le {C\parallel f\parallel }_{B_{p,r}^{s_1}}{\parallel g\parallel }_{B_{p,r}^{s_2}}.$$

(5) 

Complex interpolation:

$${\parallel f\parallel }_{B_{p,r}^{\theta s_1+(1-\theta )s_2}}\le {\parallel f\parallel }_{B_{p,r}^{s_1}}^{\theta }{\parallel g\parallel }_{B_{p,r}^{s_2}}^{1-\theta },\forall f\in B_{p,r}^{s_1}\bigcap B_{p,r}^{s_2},\forall \theta \in [0,1].$$

(6) 

Fatou lemma: if ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ is bounded in $B_{p,r}^s$ and $u_n\to u$ in ${\mathcal{D}}^{\prime }\left(\mathbb{T}\right)$, then $u\in B_{p,r}^s$ and

$${\parallel u\parallel }_{B_{p,r}^s}\le \underset{n\to \infty }{lim\; inf}{\parallel u_n\parallel }_{B_{p,r}^s}.$$

(7) 

Real interpolation: if $u\in B_{p,\infty }^{s_1}\cap B_{p,\infty }^{s_2}$ and $s_1<s_2$, then $u\in B_{p,1}^{\theta s_1+(1-\theta )s_2}$ for all $\theta \in (0,1)$ and there exists a universal constant C such that

$${\parallel u\parallel }_{B_{p,1}^{\theta s_1+(1-\theta )s_2}}\le \frac{C}{\theta (1-\theta )(s_2-s_1)}{\parallel u\parallel }_{B_{p,\infty }^{s_1}}^{\theta }{\parallel u\parallel }_{B_{p,\infty }^{s_2}}^{1-\theta }.$$

Now, we state some useful results for the following transport equation that are crucial to the proofs of our main theorems later.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}f+v{\partial }_{x}f=g,\hfill \\ {f|}_{t=0}=f_0\left(x\right).\hfill \end{array}\right.\end{array}$$

(6)

Lemma 2 ([44]).

Let $1\le p\le \infty ,1\le r\le \infty$ and $\theta >-min(\frac{1}{p},\frac{1}{p^{\prime }})$. Then, there exists a constant C such that, for all solutions $f\in L^{\infty }(0,T;B_{p,r}^{\theta })$ of (6) with initial data $f_0$ in $B_{p,r}^{\theta }$ and g in $L^1(0,T;B_{p,r}^{\theta })$, we have, for a.e. $t\in [0,T]$,

$$\begin{array}{c}\hfill {\parallel f\left(t\right)\parallel }_{B_{p,r}^{\theta }}\le \parallel f_0{\parallel }_{B_{p,r}^{\theta }}+{\int }_0^t\parallel g\left(t^{\prime }\right){\parallel }_{B_{p,r}^{\theta }}d{t}^{\prime }+{\int }_0^t{V}^{\prime }\left(t^{\prime }\right){\parallel f\left(t^{\prime }\right)\parallel }_{B_{p,r}^{\theta }}d{t}^{\prime }\end{array}$$

or

$$\begin{array}{c}\hfill {\parallel f\left(t\right)\parallel }_{B_{p,r}^{\theta }}\le e^{CV\left(t\right)}(\parallel f_0{\parallel }_{B_{p,r}^{\theta }}+{\int }_0^t{e}^{-CV\left(t^{\prime }\right)}\parallel g\left(t^{\prime }\right){\parallel }_{B_{p,r}^{\theta }}d{t}^{\prime })\end{array}$$

with

$$\begin{array}{c}\hfill V^{\prime }\left(t\right)=\left\{\begin{array}{c}\parallel {\partial }_x{v\left(t\right)\parallel }_{B_{p,\infty }^{\frac{1}{p}}\cap L^{\infty }},if\theta <1+\frac{1}{p},\hfill \\ \parallel {\partial }_x{v\left(t\right)\parallel }_{B_{p,r}^{\theta }},if\theta =1+\frac{1}{p},r>1,\hfill \\ \parallel {\partial }_x{v\left(t\right)\parallel }_{B_{p,r}^{\theta -1}},if\theta >1+\frac{1}{p},(or\theta =1+\frac{1}{p},r=1).\hfill \end{array}\right.\end{array}$$

If $\theta >0$, then there exists a constant $C=C(p,r,\theta )$ such that the following statement holds:

$$\begin{array}{ccc}& & {\parallel f\left(t\right)\parallel }_{B_{p,r}^{\theta }}\hfill \\ & \le & \parallel f_0{\parallel }_{B_{p,r}^{\theta }}+{\int }_0^t{\parallel g\left(\tau \right)\parallel }_{B_{p,r}^{\theta }}d\tau +C{\int }_0^t{(\parallel f\left(\tau \right)\parallel }_{B_{p,r}^{\theta }}\parallel {\partial }_x{v\left(\tau \right)\parallel }_{L^{\infty }}+\parallel {\partial }_x{f\left(\tau \right)\parallel }_{L^{\infty }}\parallel {\partial }_{x}v\left(\tau \right){\parallel }_{B_{p,r}^{\theta -1}})\mathrm{d}\tau .\hfill \end{array}$$

In particular, if $f=av+b,a,b\in \mathbb{R}$, then, for all $\theta >0,V^{\prime }\left(t\right)={\parallel {\partial }_{x}v\left(t\right)\parallel }_{L^{\infty }}$.

Lemma 3 ([42]).

For any $f\in B_{2,1}^{\frac{1}{2}},g\in B_{2,1}^{-\frac{1}{2}}$, the following estimate holds:

$${\parallel fg\parallel }_{B_{2,\infty }^{-\frac{1}{2}}}\le {\parallel f\parallel }_{B_{2,1}^{\frac{1}{2}}}{\parallel g\parallel }_{B_{2,1}^{-\frac{1}{2}}}.$$

Lemma 4 ([45]).

(Osgood Lemma) Assume that $\rho \ge 0$ is a measurable function, $\gamma >0$ is a locally integrable function and μ is an increasing function. Suppose that, for some non-negative real number c, the function ρ satisfies

$$\rho \le c+{\int }_{t_0}^t\gamma \left(\tau \right)\mu \left(\rho \left(\tau \right)\right)\mathrm{d}\tau .$$

If $C>0$, then $-\mathcal{M}\left(\rho \left(t\right)\right)+\mathcal{M}\le {\int }_{t_0}^t\gamma \left(\tau \right)\mathrm{d}\tau$ with $\mathcal{M}={\int }_x^1\frac{\mathrm{d}r}{\mu \left(r\right)}$. If $c=0$ and μ satisfies ${\int }_0^1\frac{\mathrm{d}r}{\mu \left(r\right)}=+\infty$, then the function $\rho =0$.

Lemma 5 ([46]).

Let $\sigma ,\alpha \in \mathbb{R}$. If $n\in \mathbb{Z}$ and $n\gg 1$; then, for all $1\le p,r\le \infty$, we have

$${\parallel sin(nx-\alpha )\parallel }_{B_{p,r}^{\sigma }\left(\mathbb{T}\right)}={\parallel cos(nx-\alpha )\parallel }_{B_{p,r}^{\sigma }\left(\mathbb{T}\right)}\approx n^{\sigma }.$$

Lemma 6 ([4]).

Assume that $p,r\in [1,\infty ],s>max\{1+\frac{1}{p},\frac{3}{2}\}$. If the initial data $(u_0,v_0)\in B_{p,r}^s\times B_{p,r}^s$, then there exists a time $T>0$ such that the Cauchy problem (5) has a unique solution $(u,v)\in E_{p,r}^s\left(T\right)\times E_{p,r}^s\left(T\right)$, and the solution map $(u_0,v_0)\to (u,v)$ is continuous from a neighborhood of $B_{p,r}^s\times B_{p,r}^s$ into $E_{p,r}^{s^{\prime }}\left(T\right)\times E_{p,r}^{s^{\prime }}\left(T\right)$ for all $s^{\prime }<s$ if $r=\infty$, and $s^{\prime }=s$ otherwise. Moreover, $(u,v)$ satisfies the estimate

$${\parallel u\parallel }_{B_{p,r}^s}+{\parallel v\parallel }_{B_{p,r}^s}\le 2(\parallel u_0{\parallel }_{B_{p,r}^s}+\parallel v_0{\parallel }_{B_{p,r}^s}),0\le t\le \frac{3}{16C(\parallel u_0{\parallel }_{B_{p,r}^s}+\parallel v_0{{\parallel }_{B_{p,r}^s})}^2},$$

where $C>0$ is a constant.

Lemma 7 ([47]).

Let $(u_0,v_0)\in B_{p,1}^{1+\frac{1}{p}}\times B_{p,1}^{1+\frac{1}{p}}$ with $1\le p<\infty$; then, there exists a time $T>0$ such that the Cauchy problem (5) has a unique solution $(u,v)\in E_{p,1}^{1+\frac{1}{p}}\left(T\right)\times E_{p,1}^{1+\frac{1}{p}}\left(T\right)$, and the solution map $(u_0,v_0)\to (u,v)$ is continuous from a neighborhood of $B_{p,1}^{1+\frac{1}{p}}\times B_{p,1}^{1+\frac{1}{p}}$ into $E_{p,1}^{1+\frac{1}{p}}\left(T\right)\times E_{p,1}^{1+\frac{1}{p}}\left(T\right)$. Moreover, $(u,v)$ satisfies the estimate

$${\parallel u\parallel }_{B_{p,r}^s}+{\parallel v\parallel }_{B_{p,r}^s}\le 2(\parallel u_0{\parallel }_{B_{p,r}^s}+\parallel v_0{\parallel }_{B_{p,r}^s}),0\le t\le \frac{3}{16C(\parallel u_0{\parallel }_{B_{p,r}^s}+\parallel v_0{{\parallel }_{B_{p,r}^s})}^2},$$

where $C>0$ is a constant.

3. Non-Uniform Dependence in the Periodic Case

In this section, we will establish the non-uniform dependence results in the periodic case; in other words, we prove Theorems 1 and 2. Since all function spaces are over $\mathbb{T}$, we drop $\mathbb{T}$ in our notations of function spaces for simplicity if there is no ambiguity in this section. Before proceeding further, we first need to construct the following approximate solutions:

$$\begin{array}{c}\hfill u^{\omega ,n}\left(t\right)=\omega n^{-\frac{1}{2}}+n^{-s}cos(nx-\omega t),v^{\omega ,n}\left(t\right)=\omega n^{-\frac{1}{2}}+n^{-s}cos(nx-\omega t),\end{array}$$

where $\omega =0,\frac{1}{2}.s>max\{\frac{3}{2},1+\frac{1}{p}\}$ or $s=\frac{3}{2}$, $n\in \mathbb{Z},n\gg 1$.

3.1. Non-Uniform Dependence in the Supercritical Besov Spaces

Substituting $(u^{\omega ,n}\left(t\right),v^{\omega ,n}\left(t\right))$ into the equations of (5) yields

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t{u}^{\omega ,n}+[{\left(u^{\omega ,n}\right)}^2+{\left(v^{\omega ,n}\right)}^2]{\partial }_x{u}^{\omega ,n}+f_1(u^{\omega ,n},v^{\omega ,n})+f_2(u^{\omega ,n},v^{\omega ,n})=E\left(t\right),\hfill \\ {\partial }_t{v}^{\omega ,n}+[{\left(u^{\omega ,n}\right)}^2+{\left(v^{\omega ,n}\right)}^2]{\partial }_x{v}^{\omega ,n}+g_1(u^{\omega ,n},v^{\omega ,n})+g_2(u^{\omega ,n},v^{\omega ,n})=F\left(t\right).\hfill \end{array}\right.\end{array}$$

(7)

Direct calculation shows that

$$\begin{array}{ccc}\hfill E_1\left(t\right)& =& {\partial }_t{u}^{\omega ,n}+[{\left(u^{\omega ,n}\right)}^2+{\left(v^{\omega ,n}\right)}^2]{\partial }_x{u}^{\omega ,n}\hfill \\ & =& -2\omega n^{-2s+\frac{1}{2}}sin(2nx-2\omega t)-n^{-3s+1}sin(2nx-2\omega )cos(nx-\omega t).\hfill \\ \hfill E_2\left(t\right)& =& f_1(u^{\omega ,n},v^{\omega ,n})\hfill \\ & =& {(1-{\partial }_x^2)}^{-1}[\frac{7}{2}\omega n^{\frac{5}{2}-2s}sin(2nx-2\omega t)+\frac{7}{2}\omega n^{3-3s}sin(nx-\omega t)cos(2nx-2\omega t)\hfill \\ & & +\frac{7}{4}{n}^{3-3s}sin(2nx-2\omega t)cos(nx-\omega t)-\frac{3}{2}{n}^{4-3s}sin(2nx-2\omega t)sin(nx-\omega t)].\hfill \\ \hfill E_3\left(t\right)& =& f_2(u^{\omega ,n},v^{\omega ,n})={(1-{\partial }_x^2)}^{-1}\left[n^{3-3s}{sin}^3(nx-\omega t)\right].\hfill \end{array}$$

Next, we estimate $\parallel E_i{\left(t\right)\parallel }_{B_{p,r}^{\mu }}(max\{\frac{1}{2},\frac{1}{p}\}<\mu <min\{1+\frac{1}{p},s-1\},i=1,2,3)$.

The joint application of Lemmas 1(4) and 5 indicates that

$$\begin{array}{ccc}\hfill \parallel E_1{\left(t\right)\parallel }_{B_{p,r}^{\mu }}& \lesssim & n^{-2s+\frac{1}{2}}{\parallel sin(2nx-2\omega t)\parallel }_{B_{p,r}^{\mu }}+n^{1-3s}{(\parallel sin(2nx-2\omega t)\parallel }_{B_{p,r}^{\mu }}{\parallel cos(nx-\omega t)\parallel }_{L^{\infty }}\hfill \\ & & +{\parallel sin(2nx-2\omega t)\parallel }_{L^{\infty }}{\parallel cos(nx-\omega t)\parallel }_{B_{p,r}^{\mu }})\hfill \\ & \lesssim & n^{-2s+\mu +\frac{1}{2}}+n^{1-3s+\mu }\lesssim n^{-2s+\mu +\frac{1}{2}}.\hfill \\ \hfill \parallel E_2{\left(t\right)\parallel }_{B_{p,r}^{\mu }}& \lesssim & n^{-2s+\frac{5}{2}}{\parallel sin(2nx-2\omega t)\parallel }_{B_{p,r}^{\mu -2}}+n^{3-3s}{\parallel sin(nx-\omega t)cos(2nx-2\omega t)\parallel }_{B_{p,r}^{\mu -1}}\hfill \\ & & +n^{3-3s}{\parallel sin(2nx-2\omega t)cos(nx-\omega t)\parallel }_{B_{p,r}^{\mu -1}}\hfill \\ & & +n^{4-3s}{\parallel sin(2nx-2\omega t)sin(nx-\omega t)\parallel }_{B_{p,r}^{\mu -2}}\hfill \\ & \lesssim & n^{\frac{1}{2}+\mu -2s}+n^{2+\mu -3s}\lesssim n^{\frac{1}{2}+\mu -2s}.\hfill \\ \hfill \parallel E_3{\left(t\right)\parallel }_{B_{p,r}^{\mu }}& \lesssim & n^{3-3s}{\parallel {sin}^3(nx-\omega t)\parallel }_{B_{p,r}^{\mu -1}}\lesssim n^{2+\mu -3s}.\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{ccc}\hfill {\parallel E\left(t\right)\parallel }_{B_{p,r}^{\mu }}& \lesssim & \sum _{i=1}^3{\parallel E_i\left(t\right)\parallel }_{B_{p,r}^{\mu }}\lesssim max\{n^{-2s+\mu +\frac{1}{2}},n^{2+\mu -3s}\}=n^{-2s+\mu +\frac{1}{2}}.\hfill \end{array}$$

In a similar way, one has

$$\begin{array}{c}\hfill {\parallel F\left(t\right)\parallel }_{B_{p,r}^{\mu }}\lesssim n^{-2s+\mu +\frac{1}{2}}.\end{array}$$

As a consequence, we have the following lemma:

Lemma 8.

If $s>max\{\frac{3}{2},1+\frac{1}{p}\}$, $1\le p\le \infty ,1\le r<\infty$, $n\in \mathbb{Z},n\gg 1$, $max\{\frac{1}{p},\frac{1}{2}\}<\mu <min\{1+\frac{1}{p},s-1\}$, then we have

$$\begin{array}{c}\hfill {\parallel E\left(t\right)\parallel }_{B_{p,r}^{\mu }},{\parallel F\left(t\right)\parallel }_{B_{p,r}^{\mu }}\lesssim n^{-2s+\mu +\frac{1}{2}},0\le t\le T_0.\end{array}$$

(8)

Let $(u_{\omega ,n}\left(t\right),v_{\omega ,n}\left(t\right))$ be the solution to the Cauchy problem with initial data $(u^{\omega ,n}\left(0\right),v^{\omega ,n}\left(0\right))$; that is, $(u_{\omega ,n}\left(t\right),v_{\omega ,n}\left(t\right))$ solves the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t{u}_{\omega ,n}+[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\partial }_x{u}_{\omega ,n}+f_1(u_{\omega ,n},v_{\omega ,n})+f_2(u_{\omega ,n},v_{\omega ,n})=0,t>0,x\in \mathbb{T},\hfill \\ {\partial }_t{v}_{\omega ,n}+[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\partial }_x{v}_{\omega ,n}+g_1(u_{\omega ,n},v_{\omega ,n})+g_2(u_{\omega ,n},v_{\omega ,n})=0,t>0,x\in \mathbb{T},\hfill \\ u_{\omega ,n}\left(0\right)=u^{\omega ,n}\left(0\right)=w{n}^{-\frac{1}{2}}+n^{-s}cosnx,x\in \mathbb{T},\hfill \\ v_{\omega ,n}\left(0\right)=v^{\omega ,n}\left(0\right)=w{n}^{-\frac{1}{2}}+n^{-s}cosnx,x\in \mathbb{T},\hfill \end{array}\right.\end{array}$$

(9)

where $f_1(u_{\omega ,n},v_{\omega ,n}),f_2(u_{\omega ,n},v_{\omega ,n}),g_1(u_{\omega ,n},v_{\omega ,n}),g_2(u_{\omega ,n},v_{\omega ,n})$ are as defined in 5) once $(u,v)$ is replaced by $(u_{\omega ,n},v_{\omega ,n})$. It can be easily verified that $(u_{\omega ,n}\left(0\right),v_{\omega ,n}\left(0\right))\in B_{p,r}^s\times B_{p,r}^s$. By applying Lemma 6, there exists a unique solution $(u_{\omega ,n}\left(t\right),v_{\omega ,n}\left(t\right))$ to problem (9) with the maximal existence time

$$T_1>T_0=\frac{1}{8C(\parallel u_0{\parallel }_{B_{p,r}^s}^2+\parallel v_0{\parallel }_{B_{p,r}^s}^2)}\gtrsim 1.$$

Next, we will estimate the difference between the approximate and the actual solutions.

Let $w=u^{\omega ,n}-u_{\omega ,n},z=v^{\omega ,n}-v_{\omega ,n}$. It is easy to verify that $\left(w\right(t),z(t\left)\right)$ satisfies the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}w+[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\partial }_{x}w+[(u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n})z]{\partial }_x{u}^{\omega ,n}+f_1(u^{\omega ,n},v^{\omega ,n})\hfill \\ -f_1(u_{\omega ,n},v_{\omega ,n})+f_2(u^{\omega ,n},v^{\omega ,n})-f_2(u_{\omega ,n},v_{\omega ,n})=E\left(t\right),t>0,x\in \mathbb{T},\hfill \\ {\partial }_{t}z+[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\partial }_{x}z+[(u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n})z]{\partial }_x{v}^{\omega ,n}+g_1(u^{\omega ,n},v^{\omega ,n})\hfill \\ -g_1(u_{\omega ,n},v_{\omega ,n})+g_2(u^{\omega ,n},v^{\omega ,n})-g_2(u_{\omega ,n},v_{\omega ,n})=F\left(t\right),t>0,x\in \mathbb{T},\hfill \\ w(0,x)=w_0\left(x\right)=0,z(0,x)=z_0\left(x\right)=0,x\in \mathbb{T},\hfill \end{array}\right.\end{array}$$

(10)

where

$$\begin{array}{ccc}& & f_1(u^{\omega ,n},v^{\omega ,n})-f_1(u_{\omega ,n},v_{\omega ,n})\hfill \\ & =& {\partial }_x{(1-{\partial }_x^2)}^{-1}\{\frac{3}{2}w{\left({\partial }_x{u}_{\omega ,n}\right)}^2+\frac{3}{2}{u}^{\omega ,n}{\partial }_{x}w({\partial }_x{u}_{\omega ,n}+{\partial }_x{u}^{\omega ,n})+{\partial }_{x}w{v}_{\omega ,n}{\partial }_x{v}_{\omega ,n}+{\partial }_x{u}_{\omega ,n}z{\partial }_x{v}_{\omega ,n}\hfill \\ & & +{\partial }_x{u}^{\omega ,n}{v}^{\omega ,n}{\partial }_{x}z+w[{\left(u_{\omega ,n}\right)}^2+u_{\omega ,n}{u}^{\omega ,n}+{\left(u^{\omega ,n}\right)}^2]+w{\left(v_{\omega ,n}\right)}^2+u^{\omega ,n}z(v_{\omega ,n}+v^{\omega ,n})\hfill \\ & & +\frac{1}{2}w{\left({\partial }_x{v}_{\omega ,n}\right)}^2+\frac{1}{2}{u}^{\omega ,n}{\partial }_{x}z({\partial }_x{v}_{\omega ,n}+{\partial }_x{v}^{\omega ,n})\},\hfill \\ & & f_2(u^{\omega ,n},v^{\omega ,n})-f_2(u_{\omega ,n},v_{\omega ,n})\hfill \\ & =& \frac{1}{2}{(1-{\partial }_x^2)}^{-1}\{{\partial }_{x}w[{\left({\partial }_x{u}_{\omega ,n}\right)}^2+{\partial }_x{u}_{\omega ,n}{\partial }_x{u}^{\omega ,n}+{\left({\partial }_x{u}^{\omega ,n}\right)}^2]+{\partial }_x{u}^{\omega ,n}{\partial }_{x}z({\partial }_x{v}_{\omega ,n}+{\partial }_x{v}^{\omega ,n})\hfill \\ & & +{\partial }_{x}w{\left({\partial }_x{v}_{\omega ,n}\right)}^2\}.\hfill \\ & & g_1(u^{\omega ,n},v^{\omega ,n})-g_1(u_{\omega ,n},v_{\omega ,n})\hfill \\ & =& {\partial }_x{(1-{\partial }_x^2)}^{-1}\{\frac{3}{2}z{\left({\partial }_x{v}_{\omega ,n}\right)}^2+\frac{3}{2}{v}^{\omega ,n}{\partial }_{x}z({\partial }_x{v}_{\omega ,n}+{\partial }_x{v}^{\omega ,n})+{\partial }_{x}z{u}_{\omega ,n}{\partial }_x{u}_{\omega ,n}+{\partial }_x{v}_{\omega ,n}w{\partial }_x{u}_{\omega ,n}\hfill \\ & & +{\partial }_x{v}^{\omega ,n}{u}^{\omega ,n}{\partial }_{x}w+z[{\left(v_{\omega ,n}\right)}^2+v_{\omega ,n}{v}^{\omega ,n}+{\left(v^{\omega ,n}\right)}^2]+z{\left(u_{\omega ,n}\right)}^2+v^{\omega ,n}w(u_{\omega ,n}+u^{\omega ,n})\hfill \\ & & +\frac{1}{2}z{\left({\partial }_x{u}_{\omega ,n}\right)}^2+\frac{1}{2}{v}^{\omega ,n}{\partial }_{x}w({\partial }_x{u}_{\omega ,n}+{\partial }_x{u}^{\omega ,n})\},\hfill \\ & & g_2(u^{\omega ,n},v^{\omega ,n})-g_2(u_{\omega ,n},v_{\omega ,n})\hfill \\ & =& \frac{1}{2}{(1-{\partial }_x^2)}^{-1}\{{\partial }_{x}z[{\left({\partial }_x{v}_{\omega ,n}\right)}^2+{\partial }_x{v}_{\omega ,n}{\partial }_x{v}^{\omega ,n}+{\left({\partial }_x{v}^{\omega ,n}\right)}^2]+{\partial }_x{v}^{\omega ,n}{\partial }_{x}w({\partial }_x{u}_{\omega ,n}+{\partial }_x{u}^{\omega ,n})\hfill \\ & & +{\partial }_{x}z{\left({\partial }_x{u}_{\omega ,n}\right)}^2\}.\hfill \end{array}$$

Lemma 9.

Suppose that $s>max\{\frac{3}{2},1+\frac{1}{p}\}$, $1\le p\le \infty ,1\le r<\infty$, $n\in \mathbb{Z},n\gg 1$, $max\{\frac{1}{p},\frac{1}{2}\}<\mu <min\{1+\frac{1}{p},s-1\}$; then, we have the following estimates:

$$\begin{array}{c}\hfill {\parallel w\left(t\right)\parallel }_{B_{p,r}^{\mu }},{\parallel z\left(t\right)\parallel }_{B_{p,r}^{\mu }}\lesssim n^{-2s+\mu +\frac{1}{2}},0\le t\le T_0,\end{array}$$

(11)

$$\begin{array}{c}\hfill {\parallel w\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }},{\parallel z\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim n^{s-\mu },0\le t\le T_0.\end{array}$$

(12)

Proof. 

Note that $\parallel w_0{\parallel }_{B_{p,r}^{\mu }}={\parallel z_0\parallel }_{B_{p,r}^{\mu }}=0$. Applying Lemma 2 to the first and second solution component of (10) yields, respectively,

$$\begin{array}{ccc}& & {\parallel w\left(t\right)\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \le & e^{CV\left(t\right)}\{{\int }_0^t(\parallel [(u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n})z]{\partial }_x{u}^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}+\parallel E\left(\tau \right){\parallel }_{B_{p,r}^{\mu }}\left)\mathrm{d}\tau \right\}\hfill \\ & & +{\int }_0^t\parallel f_1(u^{\omega ,n},v^{\omega ,n})-f_1(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{\mu }}+{\parallel f_2(u^{\omega ,n},v^{\omega ,n})-f_2(u_{\omega ,n},v_{\omega ,n})\parallel }_{B_{p,r}^{\mu }}\mathrm{d}\tau ,\hfill \end{array}$$

(13)

and

$$\begin{array}{ccc}& & {\parallel z\left(t\right)\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \le & e^{CV\left(t\right)}\{{\int }_0^t(\parallel [(u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n})z]{\partial }_x{v}^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}+\parallel F\left(\tau \right){\parallel }_{B_{p,r}^{\mu }}\left)\mathrm{d}\tau \right\}\hfill \\ & & +{\int }_0^t\parallel g_1(u^{\omega ,n},v^{\omega ,n})-g_1(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{\mu }}+{\parallel g_2(u^{\omega ,n},v^{\omega ,n})-g_2(u_{\omega ,n},v_{\omega ,n})\parallel }_{B_{p,r}^{\mu }}\mathrm{d}\tau ,\hfill \end{array}$$

(14)

where $V\left(t\right)={\int }_0^t{\parallel {\partial }_x[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]\parallel }_{B_{p,r}^{\mu -1}}\mathrm{d}\tau$.

Due to Lemma 1(4)(i), for $max\{\frac{1}{p},\frac{1}{2}\}<\mu <min\{1+\frac{1}{p},s-1\}$, we have

$$\begin{array}{ccc}& & \parallel {\partial }_x[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\parallel }_{B_{p,r}^{\mu -1}}\hfill \\ & \lesssim & \parallel [{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\parallel }_{B_{p,r}^{\mu }}\lesssim \parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}^2+{\parallel v_{\omega ,n}\parallel }_{B_{p,r}^{\mu }}^2\lesssim 1.\hfill \\ & & \parallel [(u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n})z]{\partial }_x{u}^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \lesssim & \parallel (u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n}){z\parallel }_{L^{\infty }}{\parallel {\partial }_x{u}^{\omega ,n}\parallel }_{B_{p,r}^{\mu }}\hfill \\ & & +\parallel (u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n}){z\parallel }_{B_{p,r}^{\mu }}{\parallel {\partial }_x{u}^{\omega ,n}\parallel }_{L^{\infty }}\hfill \\ & \lesssim & \left[\right(\parallel u_{\omega ,n}{\parallel }_{L^{\infty }}+\parallel u^{\omega ,n}{\parallel }_{L^{\infty }}{)\parallel w\parallel }_{L^{\infty }}+\parallel v_{\omega ,n}{\parallel }_{L^{\infty }}+\parallel v^{\omega ,n}{\parallel }_{L^{\infty }}{)\parallel z\parallel }_{L^{\infty }}]\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}\hfill \\ & & +\left[\right(\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}{)\parallel w\parallel }_{B_{p,r}^{\mu }}+(\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}+\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}{)\parallel z\parallel }_{B_{p,r}^{\mu }}]\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \lesssim & (\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^s}+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^s}+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^s}+\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^s}{\left)\right(\parallel w\parallel }_{B_{p,r}^{\mu }}+{\parallel z\parallel }_{B_{p,r}^{\mu }})\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^s}.\hfill \\ & & \parallel f_1(u^{\omega ,n},v^{\omega ,n})-f_1(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \lesssim & \parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}{\parallel w\parallel }_{B_{p,r}^{\mu }}+\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}(\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}+\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}{)\parallel w\parallel }_{B_{p,r}^{\mu }}\hfill \\ & & +\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}{\parallel w\parallel }_{B_{p,r}^{\mu }}+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}{\parallel z\parallel }_{B_{p,r}^{\mu }}+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}{\parallel z\parallel }_{B_{p,r}^{\mu }}\hfill \\ & & +(\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}^2+\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}^2{)\parallel w\parallel }_{B_{p,r}^{\mu }}+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu -1}}\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}{\parallel w\parallel }_{B_{p,r}^{\mu }}\hfill \\ & & +(\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu -1}}\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu -1}}\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}{)\parallel z\parallel }_{B_{p,r}^{\mu }}+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}{\parallel w\parallel }_{B_{p,r}^{\mu }}\hfill \\ & & +\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}(\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}+\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}{)\parallel z\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \lesssim & (\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^s}^2{\left)\right(\parallel w\parallel }_{B_{p,r}^{\mu }}+{\parallel z\parallel }_{B_{p,r}^{\mu }}).\hfill \\ & & \parallel f_2(u^{\omega ,n},v^{\omega ,n})-f_2(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \lesssim & \parallel {\partial }_x{w\parallel }_{B_{p,r}^{\mu -1}}(\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}^2+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}^2)+\parallel {\partial }_x{z\parallel }_{B_{p,r}^{\mu -1}}(\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}^2+\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}^2+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{\mu +1}}^2)\hfill \\ & & +\parallel {\partial }_x{w\parallel }_{B_{p,r}^{\mu -1}}{\parallel v_{\omega ,n}\parallel }_{B_{p,r}^{\mu +1}}^2\hfill \\ & \lesssim & (\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^s}^2{\left)\right(\parallel w\parallel }_{B_{p,r}^{\mu }}+{\parallel z\parallel }_{B_{p,r}^{\mu }}).\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}& & \parallel [(u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n})z]{\partial }_x{v}^{\omega ,n}{\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \lesssim & (\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^s}+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^s}+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^s}+\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^s}{\left)\right(\parallel w\parallel }_{B_{p,r}^{\mu }}+{\parallel z\parallel }_{B_{p,r}^{\mu }})\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^s}.\hfill \\ & & \parallel g_1(u^{\omega ,n},v^{\omega ,n})-g_1(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \lesssim & (\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^s}^2{\left)\right(\parallel w\parallel }_{B_{p,r}^{\mu }}+{\parallel z\parallel }_{B_{p,r}^{\mu }}).\hfill \\ & & \parallel g_2(u^{\omega ,n},v^{\omega ,n})-g_2(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{\mu }}\hfill \\ & \lesssim & (\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel u^{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^s}^2+\parallel v^{\omega ,n}{\parallel }_{B_{p,r}^s}^2{\left)\right(\parallel w\parallel }_{B_{p,r}^{\mu }}+{\parallel z\parallel }_{B_{p,r}^{\mu }}).\hfill \end{array}$$

By substituting the above estimates into (13) and (14), respectively, in view of the fact that

$$\parallel u^{\omega ,n},u_{\omega ,n},v^{\omega ,n},v_{\omega ,n}{\parallel }_{B_{p,r}^s}\lesssim 1,$$

we obtain

$$\begin{array}{c}\hfill {\parallel w\parallel }_{B_{p,r}^{\mu }}+{\parallel z\parallel }_{B_{p,r}^{\mu }}\lesssim n^{-2s+\mu +\frac{1}{2}}+{\int }_0^t{(\parallel w\left(\tau \right)\parallel }_{B_{p,r}^{\mu }}+{\parallel z\left(\tau \right)\parallel }_{B_{p,r}^{\mu }})\mathrm{d}\tau .\end{array}$$

It thus follows from the Gronwall inequality that (11) holds.

Next, we prove (12).

Applying Lemma 2 to the first equation in (9) yields

$$\begin{array}{ccc}& & \parallel u_{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\hfill \\ & \lesssim & \parallel u_{\omega ,n}{\left(0\right)\parallel }_{B_{p,r}^{2s-\mu }}+{\int }_0^t(\parallel f_1(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{2s-\mu }}+\parallel f_2(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{2s-\mu }})\mathrm{d}\tau \hfill \\ & +& {\int }_0^t(\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu }}\parallel {\partial }_x[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\parallel }_{L^{\infty }}+\parallel {\partial }_x[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\parallel }_{B_{p,r}^{2s-\mu }}\parallel {\partial }_x{u}_{\omega ,n}{\parallel }_{L^{\infty }})\mathrm{d}\tau .\hfill \end{array}$$

(15)

By applying Lemma 1(4)(i) and the embedding theorem, in view of the fact that $\parallel u_{\omega ,n},v_{\omega ,n}\parallel \lesssim 1$, we find that

$$\begin{array}{ccc}& & \parallel f_1(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{2s-\mu }}\hfill \\ & \lesssim & \parallel u_{\omega ,n}{\left({\partial }_x{u}_{\omega ,n}\right)}^2{\parallel }_{B_{p,r}^{2s-\mu -1}}+\parallel {\partial }_x{u}_{\omega ,n}{v}_{\omega ,n}{\partial }_x{v}_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}+{\parallel u_{\omega ,n}^3\parallel }_{B_{p,r}^{2s-\mu -1}}\hfill \\ & & +\parallel u_{\omega ,n}{\left(v_{\omega ,n}\right)}^2{\parallel }_{B_{p,r}^{2s-\mu -1}}+{\parallel u_{\omega ,n}{\left({\partial }_x{v}_{\omega ,n}\right)}^2\parallel }_{B_{p,r}^{2s-\mu -1}}\hfill \\ & \lesssim & \parallel u_{\omega ,n}{\parallel }_{L^{\infty }}\parallel {\left({\partial }_x{u}_{\omega ,n}\right)}^2{\parallel }_{B_{p,r}^{2s-\mu -1}}+\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}\parallel {\left({\partial }_x{u}_{\omega ,n}\right)}^2{\parallel }_{L^{\infty }}+\parallel {\partial }_x{u}_{\omega ,n}{\parallel }_{L^{\infty }}{\parallel v_{\omega ,n}{\partial }_x{v}_{\omega ,n}\parallel }_{B_{p,r}^{2s-\mu -1}}\hfill \\ & & +\parallel {\partial }_x{u}_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}\parallel v_{\omega ,n}{\partial }_x{v}_{\omega ,n}{\parallel }_{L^{\infty }}+\parallel u_{\omega ,n}{\parallel }_{L^{\infty }}\parallel {\left(u_{\omega ,n}\right)}^2{\parallel }_{B_{p,r}^{2s-\mu -1}}+\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}{\parallel u_{\omega ,n}\parallel }_{L^{\infty }}^2\hfill \\ & & +\parallel u_{\omega ,n}{\parallel }_{L^{\infty }}\parallel {\left(v_{\omega ,n}\right)}^2{\parallel }_{B_{p,r}^{2s-\mu -1}}+\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}\parallel {\left(v_{\omega ,n}\right)}^2{\parallel }_{L^{\infty }}+\parallel u_{\omega ,n}{\parallel }_{L^{\infty }}{\parallel {\left({\partial }_x{v}_{\omega ,n}\right)}^2\parallel }_{B_{p,r}^{2s-\mu -1}}\hfill \\ & & +\parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}{\parallel {\left({\partial }_x{v}_{\omega ,n}\right)}^2\parallel }_{L^{\infty }}\hfill \\ & \lesssim & \parallel u_{\omega ,n}{\parallel }_{L^{\infty }}\parallel {\partial }_x{u}_{\omega ,n}{\parallel }_{L^{\infty }}\parallel {\partial }_x{u}_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}+\parallel {\left({\partial }_x{u}_{\omega ,n}\right)}^2{\parallel }_{L^{\infty }}{\parallel u_{\omega ,n}\parallel }_{B_{p,r}^{2s-\mu -1}}\hfill \\ & & +\parallel {\partial }_x{u}_{\omega ,n}{\parallel }_{L^{\infty }}(\parallel v_{\omega ,n}{\parallel }_{L^{\infty }}\parallel {\partial }_x{v}_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}+\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}\parallel {\partial }_x{v}_{\omega ,n}{\parallel }_{L^{\infty }})\hfill \\ & & +\parallel {\partial }_x{u}_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}\parallel v_{\omega ,n}{\parallel }_{L^{\infty }}\parallel {\partial }_x{v}_{\omega ,n}{\parallel }_{L^{\infty }}+\parallel u_{\omega ,n}{\parallel }_{L^{\infty }}^2{\parallel u_{\omega ,n}\parallel }_{B_{p,r}^{2s-\mu -1}}\hfill \\ & & +\parallel u_{\omega ,n}{\parallel }_{L^{\infty }}\parallel v_{\omega ,n}{\parallel }_{L^{\infty }}\parallel v_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}+\parallel v_{\omega ,n}{\parallel }_{L^{\infty }}^2{\parallel u_{\omega ,n}\parallel }_{B_{p,r}^{2s-\mu -1}}\hfill \\ & & +\parallel u_{\omega ,n}{\parallel }_{L^{\infty }}\parallel {\partial }_x{v}_{\omega ,n}{\parallel }_{L^{\infty }}\parallel {\partial }_x{v}_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu -1}}+\parallel {\partial }_x{v}_{\omega ,n}{\parallel }_{L^{\infty }}^2{\parallel u_{\omega ,n}\parallel }_{B_{p,r}^{2s-\mu -1}}\hfill \\ & \lesssim & \parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu }}+{\parallel v_{\omega ,n}\parallel }_{B_{p,r}^{2s-\mu }}.\hfill \end{array}$$

In a similar way, one obtains

$$\begin{array}{c}\hfill \parallel f_2(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{p,r}^{2s-\mu }}\lesssim \parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu }}+{\parallel v_{\omega ,n}\parallel }_{B_{p,r}^{2s-\mu }},\end{array}$$

and

$$\begin{array}{ccc}& & \parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu }}\parallel {\partial }_x(u_{\omega ,n}^2+v_{\omega ,n}^2){\parallel }_{L^{\infty }}+\parallel {\partial }_x{u}_{\omega ,n}{\parallel }_{L^{\infty }}{\parallel {\partial }_x(u_{\omega ,n}^2+v_{\omega ,n}^2)\parallel }_{B_{p,r}^{2s-\mu -1}}\hfill \\ & \lesssim & \parallel u_{\omega ,n}{\parallel }_{B_{p,r}^{2s-\mu }}+{\parallel v_{\omega ,n}\parallel }_{B_{p,r}^{2s-\mu }}.\hfill \end{array}$$

Substituting the above estimates into (15) yields

$$\begin{array}{c}\hfill \parallel u_{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim \parallel u_{\omega ,n}{\left(0\right)\parallel }_{B_{p,r}^{2s-\mu }}+{\int }_0^t(\parallel u_{\omega ,n}{\left(\tau \right)\parallel }_{B_{p,r}^{2s-\mu }}+\parallel v_{\omega ,n}\left(\tau \right){\parallel }_{B_{p,r}^{2s-\mu }})\mathrm{d}\tau .\end{array}$$

(16)

Similarly,

$$\begin{array}{c}\hfill \parallel v_{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim \parallel v_{\omega ,n}{\left(0\right)\parallel }_{B_{p,r}^{2s-\mu }}+{\int }_0^t(\parallel u_{\omega ,n}{\left(\tau \right)\parallel }_{B_{p,r}^{2s-\mu }}+\parallel v_{\omega ,n}\left(\tau \right){\parallel }_{B_{p,r}^{2s-\mu }})\mathrm{d}\tau .\end{array}$$

(17)

Adding (16) and (17) together yields

$$\begin{array}{ccc}& & \parallel u_{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}+{\parallel v_{\omega ,n}\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\hfill \\ & \lesssim & \parallel u_{\omega ,n}{\left(0\right)\parallel }_{B_{p,r}^{2s-\mu }}+\parallel v_{\omega ,n}{\left(0\right)\parallel }_{B_{p,r}^{2s-\mu }}+{\int }_0^t(\parallel u_{\omega ,n}{\left(\tau \right)\parallel }_{B_{p,r}^{2s-\mu }}+\parallel v_{\omega ,n}\left(\tau \right){\parallel }_{B_{p,r}^{2s-\mu }})\mathrm{d}\tau .\hfill \end{array}$$

An application of the Gronwall inequality implies that

$$\begin{array}{c}\hfill \parallel u_{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}+\parallel v_{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim \parallel u^{\omega ,n}{\left(0\right)\parallel }_{B_{p,r}^{2s-\mu }}+{\parallel v^{\omega ,n}\left(0\right)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim n^{s-\mu },0\le t\le T_0.\end{array}$$

Moreover, by the definition of the approximate solutions $(u^{\omega ,n}\left(t\right),v^{\omega ,n}\left(t\right))$, we have

$$\begin{array}{c}\hfill \parallel u^{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }},\parallel v^{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}={\parallel \omega n^{-\frac{1}{2}}+n^{-s}cos(nx-\omega t)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim n^{s-\mu }.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\parallel w\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim \parallel u_{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}+{\parallel u^{\omega ,n}\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim n^{s-\mu },\\ \hfill {\parallel z\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim \parallel v_{\omega ,n}{\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}+{\parallel v^{\omega ,n}\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}\lesssim n^{s-\mu }.\end{array}$$

which completes the proof of Lemma 9.

Now, we are ready to prove Theorem 1. Let $\omega =0,\frac{1}{2}$. It follows from Lemma 6 that $(u_{0,n}\left(t\right),v_{0,n}\left(t\right))$,$(u_{\frac{1}{2},n}\left(t\right),v_{\frac{1}{2},n}\left(t\right))$ are the unique solutions to problem (3) with initial data $(u^{0,n}\left(0\right),v^{0,n}\left(0\right))$, $(u^{\frac{1}{2},n}\left(0\right),v^{\frac{1}{2},n}\left(0\right))$, respectively. It can be deduced from Lemmas 8 and 9 and the complex interpolation formula in Lemma 1(6) that

$$\begin{array}{ccc}\hfill {\parallel w\left(t\right)\parallel }_{B_{p,r}^s}& \lesssim & {\parallel w\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}^{\frac{1}{2}}{\parallel w\left(t\right)\parallel }_{B_{p,r}^{\mu }}^{\frac{1}{2}}\lesssim n^{\frac{s-\mu }{2}}{n}^{-s+\frac{\mu }{2}+\frac{1}{4}}\lesssim n^{\frac{1-2s}{4}}\lesssim n^{-\frac{1}{2}},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {\parallel z\left(t\right)\parallel }_{B_{p,r}^s}& \lesssim & {\parallel z\left(t\right)\parallel }_{B_{p,r}^{2s-\mu }}^{\frac{1}{2}}{\parallel z\left(t\right)\parallel }_{B_{p,r}^{\mu }}^{\frac{1}{2}}\lesssim n^{\frac{s-\mu }{2}}{n}^{-s+\frac{\mu }{2}+\frac{1}{4}}\lesssim n^{-\frac{1}{2}}.\hfill \end{array}$$

(19)

In view of Lemma 6 and the fact that $u^{\omega ,n}\left(t\right)=\omega n^{-\frac{1}{2}}+n^{-s}cos(nx-\omega t),v^{\omega ,n}\left(t\right)=\omega n^{-\frac{1}{2}}+n^{-s}cos(nx-\omega t)$, we find that

$$\begin{array}{c}\hfill \parallel u_{0,n}{\left(t\right)\parallel }_{B_{p,r}^s}+\parallel u_{\frac{1}{2},n}{\left(t\right)\parallel }_{B_{p,r}^s}+\parallel v_{0,n}{\left(t\right)\parallel }_{B_{p,r}^s}+{\parallel v_{\frac{1}{2},n}\left(t\right)\parallel }_{B_{p,r}^s}\lesssim 1,\\ \hfill \parallel u^{\frac{1}{2},n}\left(0\right)-u^{0,n}{\left(0\right)\parallel }_{B_{p,r}^s}={\parallel v^{\frac{1}{2},n}\left(0\right)-v^{0,n}\left(0\right)\parallel }_{B_{p,r}^s}\lesssim n^{-\frac{1}{2}}.\end{array}$$

Since

$$\begin{array}{ccc}& & \parallel u^{\frac{1}{2},n}\left(t\right)-u^{0,n}{\left(t\right)\parallel }_{B_{p,r}^s}={\parallel \frac{1}{2}{n}^{-\frac{1}{2}}+n^{-s}cos(nx-\frac{t}{2})-n^{-s}cosnx\parallel }_{B_{p,r}^s}\hfill \\ & \gtrsim & (n^{-s}\parallel sin(nx-\frac{t}{4}){\parallel }_{B_{p,r}^s}|sin\frac{t}{4}|-n^{-\frac{1}{2}})\gtrsim |sin\frac{t}{4}|-n^{-\frac{1}{2}}.\hfill \end{array}$$

(20)

We obtain from (18)–(20) that, for $0\le t\le T_0$,

$$\begin{array}{ccc}& & \underset{n\to \infty }{lim\; inf}{\parallel u_{\frac{1}{2},n}\left(t\right)-u_{0,n}\left(t\right)\parallel }_{B_{p,r}^s}\hfill \\ & \gtrsim & \underset{n\to \infty }{lim\; inf}(\parallel u^{\frac{1}{2},n}\left(t\right)-u^{0,n}{\left(t\right)\parallel }_{B_{p,r}^s}-\parallel u_{\frac{1}{2},n}\left(t\right)-u^{\frac{1}{2},n}{\left(t\right)\parallel }_{B_{p,r}^s}-\parallel u_{0,n}\left(t\right)-u^{0,n}\left(t\right){\parallel }_{B_{p,r}^s})\hfill \\ & \gtrsim & \underset{n\to \infty }{lim\; inf}\left(\right|sin\frac{t}{4}|-n^{-\frac{1}{2}}-n^{-\frac{1}{2}})\gtrsim |sin\frac{t}{4}|.\hfill \end{array}$$

The proof of Theorem 1 is complete. □

3.2. Non-Uniform Dependence in the Critical Besov Space

In this part, we will prove the non-uniform dependence of the solution to the initial data in the critical periodic Besov space $B_{2,1}^{\frac{3}{2}}\left(\mathbb{T}\right)\times B_{2,1}^{\frac{3}{2}}\left(\mathbb{T}\right)$. Since now $s=\frac{3}{2}$, the approximate solutions can be rewritten as

$$u^{\omega ,n}=\omega n^{-\frac{1}{2}}+n^{-\frac{3}{2}}cos(nx-\omega t),v^{\omega ,n}=\omega n^{-\frac{1}{2}}+n^{-\frac{3}{2}}cos(nx-\omega t),$$

where $\omega =0,\frac{1}{2},n\in \mathbb{Z},n\gg 1$.

However, for critical case $(s,p,r)=(\frac{3}{2},2,1)$, we now consider that there are no estimates for solutions in $B_{p,r}^{s-1}$; therefore, we carry out the estimates in $B_{2,\infty }^{\frac{1}{2}}$.

First, as discussed in Section 3.1, we substitute the approximate solution $(u^{\omega ,n},v^{\omega ,n})$ into (5) and compute the error estimates.

$$\begin{array}{ccc}\hfill \parallel E_1{\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}& \lesssim & n^{-\frac{5}{2}}{\parallel sin(2nx-2\omega t)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}+n^{-\frac{7}{2}}{(\parallel sin(2nx-2\omega t)\parallel }_{L^{\infty }}{\parallel cos(nx-\omega t)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\hfill \\ & & +{\parallel sin(2nx-2\omega t)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}{\parallel cos(nx-\omega t)\parallel }_{L^{\infty }})\hfill \\ & \lesssim & n^{-\frac{5}{2}}{n}^{\frac{1}{2}}+n^{-\frac{7}{2}}{n}^{\frac{1}{2}}\lesssim n^{-2}+n^{-3}\lesssim n^{-2}.\hfill \\ \hfill \parallel E_2{\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}& \lesssim & n^{-\frac{1}{2}}{\parallel sin(2nx-2\omega t)\parallel }_{B_{2,\infty }^{-\frac{1}{2}}}+n^{-\frac{3}{2}}{(\parallel sin(nx-\omega tcos(2nx-2\omega t))\parallel }_{B_{2,\infty }^{-\frac{1}{2}}}\hfill \\ & & +{\parallel sin(2nx-2\omega t)cos(nx-\omega t)\parallel }_{B_{2,\infty }^{-\frac{1}{2}}})+n^{-\frac{1}{2}}{\parallel sin(2nx-2\omega t)sin(nx-\omega t)\parallel }_{B_{2,\infty }^{-\frac{1}{2}}}\hfill \\ & \lesssim & n^{-\frac{1}{2}}{n}^{-\frac{3}{2}}+n^{-\frac{3}{2}}{n}^{-\frac{3}{2}}+n^{-\frac{1}{2}}{n}^{-\frac{3}{2}}\lesssim n^{-2}.\hfill \\ \hfill \parallel E_3{\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}& \lesssim & n^{-\frac{3}{2}}{\parallel {sin}^3(nx-\omega t)\parallel }_{B_{2,\infty }^{-\frac{3}{2}}}\lesssim n^{-\frac{3}{2}}{n}^{-\frac{3}{2}}\lesssim n^{-3}.\hfill \end{array}$$

Therefore, we obtain ${\parallel E\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\lesssim n^{-2}$. Similarly we can find ${\parallel F\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\lesssim n^{-2}$. The following lemma is a direct consequence of the above discussions:

Lemma 10.

Let $(s,p,r)=(\frac{3}{2},2,1),n\in \mathbb{Z},n\gg 1$. We have

$${\parallel E\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}},{\parallel F\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\lesssim n^{-2},0\le t\le T_3.$$

Similar to the above discussions of the non-critical case, let $(u_{\omega ,n}\left(t\right),v_{\omega ,n}\left(t\right))$ be solutions to the Cauchy problem (3) with initial data $(u_{\omega ,n}\left(0\right),v_{\omega ,n}\left(0\right))=(u^{\omega ,n}\left(0\right),v^{\omega ,n}\left(0\right))$. Thus, $(u_{\omega ,n}\left(t\right),v_{\omega ,n}\left(t\right))$ solves the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t{u}_{\omega ,n}+[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\partial }_x{u}_{\omega ,n}+f_1(u_{\omega ,n},v_{\omega ,n})+f_2(u_{\omega ,n},v_{\omega ,n})=0,t>0,x\in \mathbb{T},\hfill \\ {\partial }_t{v}_{\omega ,n}+[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\partial }_x{v}_{\omega ,n}+g_1(u_{\omega ,n},v^{\omega ,n})+g_2(u_{\omega ,n},v_{\omega ,n})=0,t>0,x\in \mathbb{T},\hfill \\ u_{\omega ,n}\left(0\right)=u^{\omega ,n}\left(0\right)=w{n}^{-\frac{1}{2}}+n^{-\frac{3}{2}}cosnx,x\in \mathbb{T},\hfill \\ v_{\omega ,n}\left(0\right)=v^{\omega ,n}\left(0\right)=w{n}^{-\frac{1}{2}}+n^{-\frac{3}{2}}cosnx,x\in \mathbb{T},\hfill \end{array}\right.\end{array}$$

(21)

where $f_1,f_2,g_1,g_2$ are as stated in (5).

From Lemma 5, we obtain

$$\parallel u_{\omega ,n}{\left(0\right)\parallel }_{B_{2,1}^{\frac{3}{2}}}=\parallel v_{\omega ,n}{\left(0\right)\parallel }_{B_{2,1}^{\frac{3}{2}}}=\parallel \omega n^{-\frac{1}{2}}+n^{-\frac{3}{2}}{cosnx\parallel }_{B_{2,1}^{\frac{3}{2}}}\lesssim n^{-\frac{1}{2}}+n^{-\frac{3}{2}}{\parallel cosnx\parallel }_{B_{2,1}^{\frac{3}{2}}}\lesssim 1.$$

Due to Lemma 7 and the discussions in reference [47], we obtain the existence and uniqueness of solution, and the maximal existence time can be estimated as

$$T_2>T_3=\frac{1}{8C(\parallel u_{\omega ,n}\left(0\right){\parallel }_{B_{2,1}^{\frac{3}{2}}}^2+\parallel v_{\omega ,n}\left(0\right){\parallel }_{B_{2,1}^{\frac{3}{2}}}^2)}\gtrsim 1.$$

Next, we estimate the difference between approximate and actual solutions.

Let $w=u^{\omega ,n}-u_{\omega ,n},z=v^{\omega ,n}-v_{\omega ,n}$; then, $(w,z)$ also satisfies (10), and the difference between approximate and actual solutions are estimated as follows:

Lemma 11.

Let $(s,p,r)=(\frac{3}{2},2,1),n\in \mathbb{Z},n\gg 1$; then, we have

$$\begin{array}{ccc}\hfill {\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}},{\parallel z\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}& \lesssim & n^{-2e^{-Ct}},0\le t\le T_3,\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\parallel w\left(t\right)\parallel }_{B_{2,1}^{\frac{5}{2}}},{\parallel z\left(t\right)\parallel }_{B_{2,1}^{\frac{5}{2}}}& \lesssim & n.0\le t\le T_3.\hfill \end{array}$$

(23)

Proof. 

Applying Lemma 2 to the first equation in (10) yields

$$\begin{array}{ccc}& & {\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\hfill \\ & \lesssim & exp\{C{\int }_0^t\parallel {\partial }_x[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\parallel }_{B_{2,\infty }^{-\frac{1}{2}}}\mathrm{d}\tau \}\times ({\int }_0^t{\parallel [(u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n})z]{\partial }_x{u}^{\omega ,n}\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\hfill \\ & & +{\parallel E\left(\tau \right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}})\mathrm{d}\tau +{\int }_0^t{\parallel f_1(u^{\omega ,n},v^{\omega ,n})-f_1(u_{\omega ,n},v_{\omega ,n})\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\mathrm{d}\tau \hfill \\ & & +{\int }_0^t\parallel f_2(u^{\omega ,n},v^{\omega ,n})-f_2(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\mathrm{d}\tau ).\hfill \end{array}$$

(24)

By applying Lemmas 1(4)(ii) and 3, one obtains

$$\begin{array}{ccc}& & \parallel {\partial }_x[{\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2]{\parallel }_{B_{2,\infty }^{-\frac{1}{2}}}\hfill \\ & \lesssim & \parallel {\left(u_{\omega ,n}\right)}^2+{\left(v_{\omega ,n}\right)}^2{\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\lesssim \parallel u_{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}^2+{\parallel v_{\omega ,n}\parallel }_{B_{2,1}^{\frac{3}{2}}}^2\lesssim 1.\hfill \\ & & \parallel [(u_{\omega ,n}+u^{\omega ,n})w+(v_{\omega ,n}+v^{\omega ,n})z]{\partial }_x{u}^{\omega ,n}{\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\hfill \\ & \lesssim & (\parallel u_{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}+\parallel u^{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}+\parallel v_{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}+\parallel v^{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}{\left)\right(\parallel w\parallel }_{B_{2,1}^{\frac{1}{2}}}+{\parallel z\parallel }_{B_{2,1}^{\frac{1}{2}}})\parallel u^{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}},\hfill \\ & & \parallel f_1(u^{\omega ,n},v^{\omega ,n})-f_1(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\hfill \\ & \lesssim & (\parallel u_{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}^2+\parallel u^{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}^2+\parallel v_{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}^2+\parallel v^{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}^2{\left)\right(\parallel w\parallel }_{B_{2,1}^{\frac{1}{2}}}+{\parallel z\parallel }_{B_{2,1}^{\frac{1}{2}}}),\hfill \\ & & \parallel f_2(u^{\omega ,n},v^{\omega ,n})-f_2(u_{\omega ,n},v_{\omega ,n}){\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\hfill \\ & \lesssim & (\parallel u_{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}^2+\parallel u^{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}^2+\parallel v_{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}^2+\parallel v^{\omega ,n}{\parallel }_{B_{2,1}^{\frac{3}{2}}}^2{\left)\right(\parallel w\parallel }_{B_{2,1}^{\frac{1}{2}}}+{\parallel z\parallel }_{B_{2,1}^{\frac{1}{2}}}).\hfill \end{array}$$

The joint application of Lemmas 7 and 10 yields

$${\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\lesssim n^{-2}+{\int }_0^t{(\parallel w\left(\tau \right)\parallel }_{B_{2,1}}^{\frac{1}{2}}+{\parallel z\left(\tau \right)\parallel }_{B_{2,1}}^{\frac{1}{2}})\mathrm{d}\tau .$$

Notice that ${\parallel w\parallel }_{B_{2,1}}^{\frac{3}{2}}+{\parallel z\parallel }_{B_{2,1}}^{\frac{3}{2}}\le M$. By Lemma 4,

$$\begin{array}{c}\hfill {\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\lesssim n^{-2}+{\int }_0^t{(\parallel w\left(\tau \right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}ln(e+\frac{M}{{\parallel w\parallel }_{B_{2,\infty }^{\frac{1}{2}}}})+\parallel z\left(\tau \right){\parallel }_{B_{2,\infty }^{\frac{1}{2}}}ln(e+\frac{M}{{\parallel z\parallel }_{B_{2,\infty }^{\frac{1}{2}}}}))\mathrm{d}\tau .\end{array}$$

(25)

Similarly, we can derive

$$\begin{array}{c}\hfill {\parallel z\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\lesssim n^{-2}+{\int }_0^t{(\parallel w\left(\tau \right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}ln(e+\frac{M}{{\parallel w\parallel }_{B_{2,\infty }^{\frac{1}{2}}}})+\parallel z\left(\tau \right){\parallel }_{B_{2,\infty }^{\frac{1}{2}}}ln(e+\frac{M}{{\parallel z\parallel }_{B_{2,\infty }^{\frac{1}{2}}}}))\mathrm{d}\tau .\end{array}$$

(26)

Adding (25) and (26) together, in view of the fact that the function $xln(e+\frac{M}{x})$ is nondecreasing, we have

$${\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}+{\parallel z\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\lesssim n^{-2}+{\int }_0^t{(\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}+{\parallel z\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}})ln(e+\frac{M}{{\parallel w\left(\tau \right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}+{\parallel z\left(\tau \right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}})\mathrm{d}\tau .$$

Using the fact that $ln(e+\frac{x}{M})\le (1-ln\frac{x}{M})ln(e+1),x\in (0,M]$, the above estimate reduces to

$$\frac{{\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}+{\parallel z\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}}{M}\lesssim \frac{n^{-2}}{M}+{\int }_0^t\left(\frac{{\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}+{\parallel z\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}}{M}(1-ln\frac{{\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}+{\parallel z\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}}{M})\right)\mathrm{d}\tau .$$

Thanks to Lemma 4, we deduce that

$${\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}+{\parallel z\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}\lesssim n^{-2exp(-Ct)},$$

which completes the proof of (22). The proof of (23) is similar to (22), so we omit it here.

Finally, applying Lemmas 11 and 1(7), it follows that

$${\parallel w\left(t\right)\parallel }_{B_{2,1}^{\frac{3}{2}}}\lesssim {\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}^{\frac{1}{2}}{\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{5}{2}}}^{\frac{1}{2}}\lesssim {\parallel w\left(t\right)\parallel }_{B_{2,\infty }^{\frac{1}{2}}}^{\frac{1}{2}}{\parallel w\left(t\right)\parallel }_{B_{2,1}^{\frac{5}{2}}}^{\frac{1}{2}}\lesssim n^{\frac{1-2exp(-Ct)}{2}}.$$

Choosing $T_3$ to ensure that

$$exp(-Ct)>\frac{3}{4},0\le t\le T_3<T_2,$$

we can derive

$${\parallel w\left(t\right)\parallel }_{B_{2,1}^{\frac{3}{2}}},{\parallel z\left(t\right)\parallel }_{B_{2,1}^{\frac{3}{2}}}\lesssim n^{-\frac{1}{4}},0\le t\le T_3.$$

The rest of the proof is very similar to Theorem 1. This completes the proof of Theorem 2. □

4. Non-Uniform Dependence in the Non-Periodic Case

In this section, we will investigate the non-uniform dependence of the solution for Equation (5) on the real line. Since all function spaces are over $\mathbb{R}$, we drop $\mathbb{R}$ in our notations of function spaces for simplicity if there is no ambiguity here. The techniques employed here are different with the periodic case. This section is divided into two subsections, which investigate the non-uniform dependence in supercritical Besov spaces $B_{p,r}^s\times B_{p,r}^s$, where $s>max\{\frac{3}{2},1+\frac{1}{p}\},1\le p\le \infty ,1\le r<\infty$, and critical Besov spaces $B_{p,r}^s\times B_{p,r}^s$ with $s=1+\frac{1}{p},1\le p\le 2,r=1$, respectively.

4.1. Non-Uniform Dependence in Supercritical Besov Spaces

We will study the non-uniform dependence of Equation (5) in supercritical Besov spaces; or, in other words, we prove Theorem 3 in this subsection.

For this purpose, we introduce a pair of even, real-valued and non-negative cut-off functions $\widehat{\varphi },\widehat{\psi }\in C_0^{\infty }\left(\mathbb{R}\right)$ [48] satisfying

$$\begin{array}{c}\hfill \widehat{\varphi }\left(x\right),\widehat{\psi }\left(x\right)=\left\{\begin{array}{c}1,if\left|x\right|\le \frac{1}{4},\hfill \\ 0,if\left|x\right|\ge \frac{1}{2}.\hfill \end{array}\right.\end{array}$$

It can be deduced from the inversion of the Fourier formula that

$$\begin{array}{c}\hfill \varphi \left(x\right)=\frac{1}{2\pi }{\int }_{\mathbb{R}}{e}^{ix\xi }\widehat{\varphi }\left(\xi \right)\mathrm{d}\xi ,\psi \left(x\right)=\frac{1}{2\pi }{\int }_{\mathbb{R}}{e}^{ix\xi }\widehat{\psi }\left(\xi \right)\mathrm{d}\xi .\\ \hfill {\partial }_x\varphi \left(x\right)=\frac{1}{2\pi }{\int }_{\mathbb{R}}i\xi e^{ix\xi }\widehat{\varphi }\left(\xi \right)\mathrm{d}\xi ,{\partial }_x\psi \left(x\right)=\frac{1}{2\pi }{\int }_{\mathbb{R}}i\xi e^{ix\xi }\widehat{\psi }\left(\xi \right)\mathrm{d}\xi .\end{array}$$

Therefore, the Fubini theorem indicates that:

$$\begin{array}{c}\hfill {\parallel \varphi \parallel }_{L^{\infty }}=\underset{x\in \mathbb{R}}{sup}\frac{1}{2\pi }|{\int }_{\mathbb{R}}cos\left(x\xi \right)\widehat{\varphi }\left(\xi \right)\mathrm{d}\xi |\le {\int }_{\mathbb{R}}\widehat{\varphi }\left(\xi \right)\mathrm{d}\xi \le C,\varphi \left(0\right)=\frac{1}{2\pi }{\int }_{\mathbb{R}}\widehat{\varphi }\left(\xi \right)\mathrm{d}\xi .\\ \hfill {\parallel \psi \parallel }_{L^{\infty }}=\underset{x\in \mathbb{R}}{sup}\frac{1}{2\pi }|{\int }_{\mathbb{R}}cos\left(x\xi \right)\widehat{\psi }\left(\xi \right)\mathrm{d}\xi |\le {\int }_{\mathbb{R}}\widehat{\psi }\left(\xi \right)\mathrm{d}\xi \le C,\psi \left(0\right)=\frac{1}{2\pi }{\int }_{\mathbb{R}}\widehat{\psi }\left(\xi \right)\mathrm{d}\xi .\end{array}$$

Define the high-frequency functions

$$f_n^1=2^{-ns}\varphi \left(x\right)sin\left(\frac{33}{24}{2}^{n}x\right),f_n^2=2^{-ns}\psi \left(x\right)sin\left(\frac{33}{24}{2}^{n}x\right),n\gg 1,$$

and the low-frequency functions

$$g_n^1=\frac{33}{24}{2}^{-\frac{n}{2}}\varphi \left(x\right),g_n^2=\frac{33}{24}{2}^{-\frac{n}{2}}\psi \left(x\right),n\gg 1.$$

It is easy to verify that $(f_n^1,f_n^2)$ lies in $B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)$, and so does $(f_n^1+g_n^1,f_n^2+g_n^2)$.

Let $(u_{0,n},v_{0,n})=(f_n^1,f_n^2)$ and $({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})=(f_n^1+g_n^1,f_n^2+g_n^2)$; then, we have the following lemmas.

Lemma 12 ([48]).

Let $s\in \mathbb{R}$ and $1\le p\le \infty ,1\le r<\infty$; then, for any $\theta \in \mathbb{R}$, we have the following estimates:

$$\begin{array}{ccc}\hfill \parallel u_{0,n},v_{0,n}{\parallel }_{L^p}& \lesssim & 2^{-ns},{\parallel {\partial }_x{u}_{0,n},{\partial }_x{v}_{0,n}\parallel }_{L^p}\lesssim 2^{n(1-s)},\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill \parallel u_{0,n},v_{0,n}{\parallel }_{B_{p,r}^{\theta }}& \lesssim & 2^{n(\theta -s)}.\hfill \end{array}$$

(28)

Lemma 13 ([48]).

Let $s\in \mathbb{R}$ and $1\le p\le \infty ,1\le r<\infty$; then,

$$\begin{array}{ccc}\hfill \parallel g_n^1,g_n^2{\parallel }_{L^p}& \lesssim & 2^{-\frac{n}{2}},{\parallel {\partial }_x{g}_n^1,{\partial }_x{g}_n^2\parallel }_{L^p}\lesssim 2^{-\frac{n}{2}},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill \parallel g_n^1,g_n^2{\parallel }_{B_{p,r}^s}& \lesssim & 2^{-\frac{n}{2}-s}.\hfill \end{array}$$

(30)

Lemma 14 ([49]).

Let $s\in \mathbb{R}$ and $1\le p\le \infty ,1\le r<\infty$; then, for any $\theta \ge -\frac{1}{2}$, we have

$$\begin{array}{c}\hfill \parallel {\tilde{u}}_{0,n},{\tilde{v}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-\frac{n}{2}},{\parallel {\tilde{u}}_{0,n},{\tilde{v}}_{0,n}\parallel }_{B_{p,r}^{s+\theta }}\lesssim 2^{n\theta }.\end{array}$$

(31)

Let $(u_n,v_n)$ be the solution of Equation (5) with the initial data $(u_{0,n},v_{0,n})$; then, we have the following estimates:

Proposition 1.

Let $s\in \mathbb{R}$ and $1\le p,r<\infty$, and suppose that $(s,p,r)$ meets the requirements in Theorem 3; then, for $k=-1,1$, we have

$$\begin{array}{ccc}\hfill \parallel u_n,v_n{\parallel }_{B_{p,r}^{s+k}}& \lesssim & 2^{kn},\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill \parallel u_n-u_{0,n},v_n-v_{0,n}{\parallel }_{B_{p,r}^s}& \lesssim & 2^{-\frac{n}{2}(s-1)}.\hfill \end{array}$$

(33)

Proof. 

It can be deduced from (28) that

$$\begin{array}{c}\hfill \parallel u_{0,n},v_{0,n}{\parallel }_{B_{p,r}^{s+k}}\lesssim 2^{kn},k=-1,0,1.\end{array}$$

(34)

From Lemma 6, there exists a $T=T(\parallel u_{0,n},v_{0,n}\parallel )$ such that (5) (with initial data $(u_{0,n},v_{0,n})$) has a unique solution $(u_n,v_n)\in E_{p,r}^s\left(T\right)\times E_{p,r}^s\left(T\right)$ and $T\approx 1$. Moreover,

$$\parallel u_n{\parallel }_{L_T^{\infty }\left(B_{p,r}^s\right)}+\parallel v_n{\parallel }_{L_T^{\infty }\left(B_{p,r}^s\right)}\lesssim \parallel u_{0,n}{\parallel }_{B_{p,r}^s}+{\parallel v_{0,n}\parallel }_{B_{p,r}^s}.$$

Similar to the discussions in [47], we obtain, for $k=-1,1$,

$$\parallel u_n{\left(t\right)\parallel }_{B_{p,r}^{s+k}}+\parallel v_n{\left(t\right)\parallel }_{B_{p,r}^{s+k}}\lesssim \parallel u_{0,n}{\parallel }_{B_{p,r}^{s+k}}+\parallel v_{0,n}{\parallel }_{B_{p,r}^{s+k}}+{\int }_0^t(\parallel u_n{\left(\tau \right)\parallel }_{B_{p,r}^{s+k}}+\parallel v_n\left(\tau \right){{\parallel }_{B_{p,r}^{s+k}})}^3\mathrm{d}\tau .$$

By applying the Gronwall lemma and (34), we have, for all $t\in [0,T]$,

$$\begin{array}{c}\hfill \parallel u_n{\left(t\right)\parallel }_{B_{p,r}^{s-1}}+\parallel v_n{\left(t\right)\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-n},\parallel u_n{\left(t\right)\parallel }_{B_{p,r}^{s+1}}+{\parallel v_n\left(t\right)\parallel }_{B_{p,r}^{s+1}}\lesssim 2^n.\end{array}$$

(35)

Let $\sigma =u_n-u_{0,n},\tau =v_n-v_{0,n}$; then, we can derive from (5) that $(\sigma ,\tau )$ solves the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\sigma }_t+(u_n^2+v_n^2){\partial }_x\sigma =-[(u_n^2+v_n^2)-(u_{0,n}^2+v_{0,n}^2)]{\partial }_x{u}_{0,n}-[f_1(u_n,v_n)-f_1(u_{0,n},v_{0,n})]\hfill \\ -[f_2(u_n,v_n)-f_2(u_{0,n},v_{0,n})]-f_1(u_{0,n},v_{0,n})-f_2(u_{0,n},v_{0,n})-(u_{0,n}^2+v_{0,n}^2){\partial }_x{u}_{0,n},\hfill \\ {\tau }_t+(u_n^2+v_n^2){\partial }_x\tau =-[(u_n^2+v_n^2)-(u_{0,n}^2+v_{0,n}^2)]{\partial }_x{v}_{0,n}-[g_1(u_n,v_n)-g_1(u_{0,n},v_{0,n})]\hfill \\ -[g_2(u_n,v_n)-g_2(u_{0,n},v_{0,n})]-g_1(u_{0,n},v_{0,n})-g_2(u_{0,n},v_{0,n})-(u_{0,n}^2+v_{0,n}^2){\partial }_x{v}_{0,n},\hfill \\ \sigma (0,x)=0,\tau (0,x)=0.\hfill \end{array}\right.\end{array}$$

(36)

Applying Lemma 2 to the first equation of (36) yields

$$\begin{array}{ccc}\hfill {\parallel \sigma \parallel }_{B_{p,r}^{s-1}}& \lesssim & {\int }_0^t\parallel {\partial }_x(u_n^2+v_n^2){\parallel }_{B_{p,r}^{s-1}}{\parallel \sigma \parallel }_{B_{p,r}^{s-1}}+{\int }_0^t{\parallel [(u_n^2+v_n^2)-(u_{0,n}^2+v_{0,n}^2)]{\partial }_x{u}_{0,n}\parallel }_{B_{p,r}^{s-1}}\mathrm{d}\tau \hfill \\ & & +{\int }_0^t\parallel f_1(u_n,v_n)-f_1(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}\mathrm{d}\tau +{\int }_0^t{\parallel f_2(u_n,v_n)-f_2(u_{0,n},v_{0,n})\parallel }_{B_{p,r}^{s-1}}\mathrm{d}\tau \hfill \\ & & +{\int }_0^t(\parallel f_1(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}+\parallel f_2(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}+\parallel (u_{0,n}^2+v_{0,n}^2){\partial }_x{u}_{0,n}{\parallel }_{B_{p,r}^{s-1}})\mathrm{d}\tau .\hfill \end{array}$$

(37)

In view of the fact that $B_{p,r}^{s-1}$ is a Banach algebra when $s>max\{1+\frac{1}{p},\frac{3}{2}\}$, by applying Lemma 1(4)(ii), one obtains

$$\begin{array}{ccc}& & \parallel [(u_n^2+v_n^2)-(u_{0,n}^2+v_{0,n}^2)]{\partial }_x{u}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\hfill \\ & \lesssim & (\parallel u_n-u_{0,n}{\parallel }_{B_{p,r}^{s-1}}\parallel u_n+u_{0,n}{\parallel }_{B_{p,r}^{s-1}}+\parallel v_n-v_{0,n}{\parallel }_{B_{p,r}^{s-1}}\parallel v_n+v_{0,n}{\parallel }_{B_{p,r}^{s-1}})\parallel {\partial }_x{u}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\hfill \\ & \lesssim & (\parallel u_n-u_{0,n}{\parallel }_{B_{p,r}^{s-1}}+\parallel v_n-v_{0,n}{\parallel }_{B_{p,r}^{s-1}}\left)\right(\parallel u_n{\parallel }_{B_{p,r}^s}^2+\parallel u_{0,n}{\parallel }_{B_{p,r}^s}^2+\parallel v_n{\parallel }_{B_{p,r}^s}^2+\parallel v_{0,n}{\parallel }_{B_{p,r}^s}^2).\hfill \\ & & \parallel f_1(u_n,v_n)-f_1(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}\hfill \\ & \lesssim & (\parallel u_n-u_{0,n}{\parallel }_{B_{p,r}^{s-1}}+\parallel v_n-v_{0,n}{\parallel }_{B_{p,r}^{s-1}}\left)\right(\parallel u_n{\parallel }_{B_{p,r}^s}^2+\parallel u_{0,n}{\parallel }_{B_{p,r}^s}^2+\parallel v_n{\parallel }_{B_{p,r}^s}^2+\parallel v_{0,n}{\parallel }_{B_{p,r}^s}^2).\hfill \\ & & \parallel f_2(u_n,v_n)-f_2(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}\hfill \\ & \lesssim & (\parallel u_n-u_{0,n}{\parallel }_{B_{p,r}^{s-1}}+\parallel v_n-v_{0,n}{\parallel }_{B_{p,r}^{s-1}}\left)\right(\parallel u_n{\parallel }_{B_{p,r}^s}^2+\parallel u_{0,n}{\parallel }_{B_{p,r}^s}^2+\parallel v_n{\parallel }_{B_{p,r}^s}^2+\parallel v_{0,n}{\parallel }_{B_{p,r}^s}^2).\hfill \end{array}$$

It follows that, by utilizing the fact that $B_{p,r}^{s-1}$ is a Banach algebra,

$$\begin{array}{ccc}\hfill \parallel f_{11}(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel {\left(u_{0,n}\right)}^3{\parallel }_{B_{p,r}^{s-2}}\lesssim \parallel u_{0,n}{\parallel }_{B_{p,r}^{s-2}}\parallel u_{0,n}{\parallel }_{L^{\infty }}{\parallel u_{0,n}\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-n(s+3)},\hfill \\ \hfill \parallel f_{12}(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel u_{0,n}{\left(v_{0,n}\right)}^2{\parallel }_{B_{p,r}^{s-2}}\lesssim \parallel u_{0,n}{\parallel }_{B_{p,r}^{s-2}}\parallel v_{0,n}{\parallel }_{L^{\infty }}{|v_{0,n}\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-n(s+3)},\hfill \\ \hfill \parallel f_{13}(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel u_{0,n}{\left({\partial }_x{u}_{0,n}\right)}^2{\parallel }_{B_{p,r}^{s-2}}\lesssim \parallel u_{0,n}{\parallel }_{B_{p,r}^{s-2}}\parallel {\partial }_x{u}_{0,n}{\parallel }_{L^{\infty }}{\parallel {\partial }_x{u}_{0,n}\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-n(s+1)}.\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill \parallel f_{14}(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel {\partial }_x\left(u_{0,n}\right)v_{0,n}{\partial }_x\left(v_{0,n}\right){\parallel }_{B_{p,r}^{s-2}}\lesssim 2^{-n(s+1)},\hfill \\ \hfill \parallel f_{15}(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel u_{0,n}{\left({\partial }_x\left(v_{0,n}\right)\right)}^2{\parallel }_{B_{p,r}^{s-2}}\lesssim 2^{-n(s+1)}.\hfill \end{array}$$

It can also be deduced that

$$\parallel f_{21}(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}\lesssim \parallel {\left({\partial }_x{u}_{0,n}\right)}^3{\parallel }_{B_{p,r}^{s-2}}\lesssim \parallel u_{0,n}{\parallel }_{B_{p,r}^{s-1}}\parallel {\partial }_x{u}_{0,n}{\parallel }_{L^{\infty }}{\parallel {\partial }_x{u}_{0,n}\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-ns}.$$

Similarly,

$$\begin{array}{ccc}\hfill \parallel f_{22}(u_{0,n},v_{0,n}){\parallel }_{B_{p,r}^{s-1}}& \lesssim & 2^{-ns}.\hfill \\ \hfill \parallel (u_{0,n}^2+v_{0,n}^2){\partial }_x{u}_{0,n}{\parallel }_{B_{p,r}^{s-1}}& \lesssim & {\partial }_x{u}_{0,n}{\parallel }_{B_{p,r}^{s-1}}(\parallel u_{0,n}{\parallel }_{L^{\infty }}+\parallel u_{0,n}{\parallel }_{B_{p,r}^{s-1}}+\parallel v_{0,n}{\parallel }_{L^{\infty }}+\parallel v_{0,n}{\parallel }_{B_{p,r}^{s-1}})\hfill \\ & \lesssim & 2^{-n(s+1)}.\hfill \end{array}$$

Substituting all of the above estimates back into (37), we find that

$${\parallel \sigma \parallel }_{B_{p,r}^{s-1}}={\parallel u_n-u_{0,n}\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-ns}.$$

In a similar manner, we obtain

$${\parallel \tau \parallel }_{B_{p,r}^{s-1}}={\parallel v_n-v_{0,n}\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-ns}.$$

The application of the interpolation inequality yields

$$\begin{array}{ccc}& & \parallel u_n-u_{0,n}{\parallel }_{B_{p,r}^s}+{\parallel v_n-v_{0,n}\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & (\parallel u_n-u_{0,n}{\parallel }_{B_{p,r}^{s-1}}+\parallel v_n-v_{0,n}{\parallel }_{B_{p,r}^{s-1}}{)}^{\frac{1}{2}}(\parallel u_n-u_{0,n}{\parallel }_{B_{p,r}^{s+1}}+\parallel v_n-v_{0,n}{{\parallel }_{B_{p,r}^{s+1}})}^{\frac{1}{2}}\hfill \\ & \lesssim & 2^{-\frac{ns}{2}}·2^{\frac{n}{2}}=2^{-\frac{n}{2}(s-1)}.\hfill \end{array}$$

The proof of Proposition 1 is complete. □

For the purpose of deducing the non-uniform dependence of the solution on the initial data, we will show that, for the constructed initial data $({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})$ with a small perturbation, the corresponding solution $({\tilde{u}}_n,{\tilde{v}}_n)$ can no longer approximate the exact solution.

Proposition 2.

Under the assumptions of Theorem 3, we have

$$\parallel {\tilde{u}}_n-{\tilde{u}}_{0,n}-t{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}+{\parallel {\tilde{v}}_n-{\tilde{v}}_{0,n}-t{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}\parallel }_{B_{p,r}^s}\lesssim t^2+2^{-\frac{n}{2}},$$

where ${\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}=-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n},{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}=-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{v}}_{0,n}$.

Proof. 

It follows from (31) that

$$\begin{array}{c}\hfill \parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^{s+k}}+\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^{s+k}}\lesssim \parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s+k}}+{\parallel {\tilde{v}}_{0,n}\parallel }_{B_{p,r}^{s+k}}\lesssim \left\{\begin{array}{c}{2}^{kn},k\ge -\frac{1}{2},\hfill \\ 2^{-\frac{n}{2}},k=-1.\hfill \end{array}\right.\end{array}$$

(38)

Since ${\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}=-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n}$, we can deduce that

$$\begin{array}{ccc}\hfill \parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s-1}}& =& \parallel ({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\lesssim (\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}^2+\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}^2)\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-n}.\hfill \\ \hfill \parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s+\sigma }}& =& \parallel ({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s+\sigma }}\hfill \\ & \lesssim & \parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}^2\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s+\sigma }}+\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s+\sigma }}{\parallel {\partial }_x{\tilde{u}}_{0,n}\parallel }_{L^{\infty }}\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& \lesssim & {\left(2^{-\frac{n}{2}}\right)}^2·2^{n(\sigma +1)}+2^{-\frac{n}{2}}{2}^{n\sigma }{2}^{-\frac{n}{2}}\lesssim 2^{n\sigma },\hfill \end{array}$$

(40)

where $\sigma \ge -\frac{1}{2}$.

In a similar manner, we have

$$\begin{array}{c}\hfill \parallel {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-n},{\parallel {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}\parallel }_{B_{p,r}^{s+\sigma }}\lesssim 2^{n\sigma }.\end{array}$$

(41)

Let ${\xi }_n={\tilde{u}}_n-{\tilde{u}}_{0,n}-t{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}},{\eta }_n={\tilde{v}}_n-{\tilde{v}}_{0,n}-t{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}$; then, we know from (5) that $({\xi }_n,{\eta }_n)$ solves the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t{\xi }_n+({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\xi }_n=-t({\tilde{u}}_n+{\tilde{u}}_{0,n}){\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{u}}_{0,n}-t({\tilde{v}}_n+{\tilde{v}}_{0,n}){\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{u}}_{0,n}\hfill \\ -{\xi }_n({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\tilde{u}}_{0,n}-{\eta }_n({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\tilde{u}}_{0,n}-f_1({\tilde{u}}_n,{\tilde{v}}_n)-f_2({\tilde{u}}_n,{\tilde{v}}_n)-t({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}},\hfill \\ {\partial }_t{\eta }_n+({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\eta }_n=-t({\tilde{u}}_n+{\tilde{u}}_{0,n}){\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{v}}_{0,n}-t({\tilde{v}}_n+{\tilde{v}}_{0,n}){\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{v}}_{0,n}\hfill \\ -{\xi }_n({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\tilde{v}}_{0,n}-{\eta }_n({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\tilde{v}}_{0,n}-g_1({\tilde{u}}_n,{\tilde{v}}_n)-g_2({\tilde{u}}_n,{\tilde{v}}_n)-t({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}},\hfill \\ {\xi }_n(0,x)={\eta }_n(0,x)=0.\hfill \end{array}\right.\end{array}$$

(42)

By applying Lemma 2 to the first and second solution component in (42), one obtains

$$\begin{array}{ccc}\hfill \parallel {\xi }_n{\left(t\right)\parallel }_{B_{p,r}^s}& \lesssim & e^{CV\left(t\right)}\{{\int }_0^t{\parallel \tau ({\tilde{u}}_n+{\tilde{u}}_{0,n}){\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{u}}_{0,n}+\tau ({\tilde{v}}_n+{\tilde{v}}_{0,n}){\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{u}}_{0,n}\parallel }_{B_{p,r}^s}\mathrm{d}\tau \hfill \\ & & +{\int }_0^t{\parallel {\xi }_n({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\tilde{u}}_{0,n}+{\eta }_n({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\tilde{u}}_{0,n}\parallel }_{B_{p,r}^s}\mathrm{d}\tau \hfill \end{array}$$

(43)

$$\begin{array}{ccc}& & +{\int }_0^t\parallel f_1({\tilde{u}}_n,{\tilde{v}}_n)+f_2({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^s}\mathrm{d}\tau +{\int }_0^t\parallel \tau ({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}\mathrm{d}\tau \},\hfill \\ \hfill \parallel {\eta }_n{\left(t\right)\parallel }_{B_{p,r}^s}& \lesssim & e^{CV\left(t\right)}\{{\int }_0^t{\parallel \tau ({\tilde{u}}_n+{\tilde{u}}_{0,n}){\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{v}}_{0,n}+\tau ({\tilde{v}}_n+{\tilde{v}}_{0,n}){\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{v}}_{0,n}\parallel }_{B_{p,r}^s}\mathrm{d}\tau \hfill \\ & & +{\int }_0^t{\parallel {\xi }_n({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\tilde{v}}_{0,n}+{\eta }_n({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\tilde{v}}_{0,n}\parallel }_{B_{p,r}^s}\mathrm{d}\tau \hfill \end{array}$$

$$\begin{array}{ccc}& & +{\int }_0^t\parallel g_1({\tilde{u}}_n,{\tilde{v}}_n)+g_2({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^s}\mathrm{d}\tau +{\int }_0^t\parallel \tau ({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}\mathrm{d}\tau \},\hfill \end{array}$$

(44)

where $V\left(t\right)={\int }_0^t{\parallel {\partial }_x({\tilde{u}}_n^2+{\tilde{v}}_n^2)\left(t^{\prime }\right)\parallel }_{B_{p,r}^{s-1}}\mathrm{d}{t}^{\prime }$, and

$$\begin{array}{ccc}\hfill f_1({\tilde{u}}_n,{\tilde{v}}_n)& =& -{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n^3\right)-{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\tilde{v}}_n^2\right)-\frac{3}{2}{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\left({\partial }_x{\tilde{u}}_n\right)}^2\right)\hfill \\ & & -{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_n{\tilde{v}}_n{\partial }_x{\tilde{v}}_n\right)-\frac{1}{2}{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\left({\partial }_x{\tilde{v}}_n\right)}^2\right)\hfill \\ & \triangleq & {\mathcal{A}}_1+{\mathcal{A}}_2+{\mathcal{A}}_3+{\mathcal{A}}_4+{\mathcal{A}}_5.\hfill \\ \hfill f_2({\tilde{u}}_n,{\tilde{v}}_n)& =& -\frac{1}{2}{(1-{\partial }_x^2)}^{-1}\left({\left({\partial }_x{\tilde{u}}_n\right)}^3\right)-\frac{1}{2}{(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_n{\left({\partial }_x{\tilde{v}}_n\right)}^2\right)\triangleq {\mathcal{A}}_6+{\mathcal{A}}_7.\hfill \end{array}$$

In order to facilitate the computation, we decompose ${\mathcal{A}}_i(i=3to7)$ as follows:

$$\begin{array}{ccc}& & {\mathcal{A}}_3({\tilde{u}}_n,{\tilde{v}}_n)\hfill \\ & =& -\frac{3}{2}{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\left({\partial }_x{\tilde{u}}_n\right)}^2\right)\hfill \\ & =& -\frac{3}{2}{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\partial }_x({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\xi }_n\right)-\frac{3}{2}\tau {\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\partial }_x({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}\right)\hfill \\ & & -\frac{3}{2}{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\left({\partial }_x{\tilde{u}}_{0,n}\right)}^2\right),\hfill \\ & & {\mathcal{A}}_4({\tilde{u}}_n,{\tilde{v}}_n)=-{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_n{\tilde{v}}_n{\partial }_x{\tilde{v}}_n\right)\hfill \\ & =& -{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{v}}_n{\partial }_x{\tilde{v}}_n{\partial }_x{\xi }_n\right)-\tau {\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{v}}_n{\partial }_x{\tilde{v}}_n{\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}\right)-{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_{0,n}{\tilde{v}}_n{\partial }_x{\tilde{v}}_n\right),\hfill \\ & & {\mathcal{A}}_5({\tilde{u}}_n,{\tilde{v}}_n)=-\frac{1}{2}{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\left({\partial }_x{\tilde{v}}_n\right)}^2\right)\hfill \\ & =& -\frac{1}{2}{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\partial }_x({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\eta }_n\right)-\frac{1}{2}\tau {\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\partial }_x({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}\right)\hfill \\ & & -\frac{1}{2}{\partial }_x{(1-{\partial }_x^2)}^{-1}\left({\tilde{u}}_n{\left({\partial }_x{\tilde{v}}_{0,n}\right)}^2\right),\hfill \\ & & {\mathcal{A}}_6({\tilde{u}}_n,{\tilde{v}}_n)=-\frac{1}{2}{(1-{\partial }_x^2)}^{-1}\left({\left({\partial }_x{\tilde{u}}_n\right)}^3\right)\hfill \\ & =& -\frac{1}{2}{(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_n{\partial }_x({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\xi }_n\right)-\frac{1}{2}\tau {(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_n{\partial }_x({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}\right)\hfill \\ & & -\frac{1}{2}\left({\partial }_x{\tilde{u}}_n{\left({\partial }_x{\tilde{u}}_{0,n}\right)}^2\right),\hfill \\ & & {\mathcal{A}}_7({\tilde{u}}_n,{\tilde{v}}_n)=-\frac{1}{2}{(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_n{\left({\partial }_x{\tilde{v}}_n\right)}^2\right)\hfill \\ & =& -\frac{1}{2}{(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_n{\partial }_x({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\eta }_n\right)-\frac{1}{2}\tau {(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_n{\partial }_x({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}\right)\hfill \\ & & -\frac{1}{2}\tau {(1-{\partial }_x^2)}^{-1}\left({\partial }_x{\tilde{u}}_n{\left({\partial }_x{\tilde{v}}_{0,n}\right)}^2\right).\hfill \end{array}$$

For the purpose of convenience, we list the estimates of the simple terms that will be used later.

$$\begin{array}{ccc}& & \parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }},\parallel {\partial }_x{\tilde{v}}_n{\parallel }_{L^{\infty }}\lesssim 2^{-\frac{n}{2}},\parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^{s-1}},\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-\frac{n}{2}},\parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^s},{\parallel {\tilde{v}}_n\parallel }_{B_{p,r}^s}\lesssim 1,\hfill \\ & & \parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^{s+1}},\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^{s+1}}\lesssim 2^n,\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }},\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}\lesssim 2^{-\frac{n}{2}},\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }},{\parallel {\partial }_x{\tilde{v}}_{0,n}\parallel }_{L^{\infty }}\lesssim 2^{-\frac{n}{2}},\hfill \\ & & \parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s-1}},\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-\frac{n}{2}},\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s},\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,r}^s}\lesssim 1,\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s+1}},{\parallel {\tilde{v}}_{0,n}\parallel }_{B_{p,r}^{s+1}}\lesssim 2^n,\hfill \\ & & \parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{L^{\infty }},\parallel {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{L^{\infty }}\lesssim 2^{-\frac{3}{2}n},\parallel {\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{L^{\infty }},\parallel {\partial }_x{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{L^{\infty }}\lesssim 2^{-n},\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s-1}},{\parallel {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-n},\hfill \\ & & \parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s},\parallel {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}\lesssim 1,\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s+1}},{\parallel {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}\parallel }_{B_{p,r}^{s+1}}\lesssim 2^n.\hfill \end{array}$$

Next, we estimate every integrant on the right-hand side of (43):

$$\begin{array}{ccc}& & \parallel \tau ({\tilde{u}}_n+{\tilde{u}}_{0,n}){\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \tau (\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s}\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{L^{\infty }}\parallel {\partial }_x{\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}\hfill \\ & & +\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}(\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}\parallel {\partial }_x{\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{L^{\infty }}\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,r}^{s+1}}\left)\right)\hfill \\ & \lesssim & \tau ·2^{-\frac{3}{2}n}·2^{-\frac{n}{2}}+\tau ·2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+\tau ·2^{-\frac{n}{2}}·2^{-\frac{3}{2}n}·2^n\lesssim 2^{-n}\tau .\hfill \end{array}$$

(45)

Similarly,

$$\begin{array}{c}\hfill \parallel \tau ({\tilde{v}}_n+{\tilde{v}}_{0,n}){\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s}\lesssim 2^{-n}\tau .\end{array}$$

(46)

and

$$\begin{array}{ccc}& & \parallel {\xi }_n({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^s}\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\xi }_n{\parallel }_{L^{\infty }}\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s}{\parallel {\partial }_x{\tilde{u}}_{0,n}\parallel }_{L^{\infty }}\hfill \\ & & +\parallel {\xi }_n{\parallel }_{L^{\infty }}\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}{\parallel {\partial }_x{\tilde{u}}_{0,n}\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^s}\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}{\parallel {\partial }_x{\tilde{u}}_{0,n}\parallel }_{L^{\infty }}\hfill \\ & & +\parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}(\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s}\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s+1}})\hfill \\ & \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^s}·2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+{\parallel {\xi }_n\parallel }_{B_{p,r}^{s-1}}(2^{-\frac{n}{2}}+2^{-\frac{n}{2}}·2^n)\hfill \\ & \lesssim & 2^{-n}\parallel {\xi }_n{\parallel }_{B_{p,r}^s}+2^{\frac{n}{2}}{\parallel {\xi }_n\parallel }_{B_{p,r}^{s-1}},\hfill \end{array}$$

(47)

$$\begin{array}{c}\hfill \parallel {\eta }_n({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s}\lesssim 2^{-n}\parallel {\eta }_n{\parallel }_{B_{p,r}^s}+2^{\frac{n}{2}}{\parallel {\eta }_n\parallel }_{B_{p,r}^{s-1}}.\end{array}$$

(48)

For terms ${\mathcal{A}}_i(i=1$ to $7)$, we estimate them as follows:

$$\begin{array}{ccc}& & \parallel {\mathcal{A}}_1({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^s}\lesssim {\parallel {\tilde{u}}_n\parallel }_{B_{p,r}^{s-1}}^3\lesssim 2^{-\frac{3}{2}n},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & \parallel {\mathcal{A}}_2({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^s}\lesssim \parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^{s-1}}{\parallel {\tilde{v}}_n\parallel }_{B_{p,r}^{s-1}}^2\lesssim 2^{-\frac{n}{2}}{\left(2^{-\frac{n}{2}}\right)}^2=2^{-\frac{3}{2}n},\hfill \\ & & \parallel {\mathcal{A}}_3({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \parallel {\tilde{u}}_n{\partial }_x({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\tau \parallel {\tilde{u}}_n{\partial }_x({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s-1}}+{\parallel {\tilde{u}}_n{\left({\partial }_x{\tilde{u}}_{0,n}\right)}^2\parallel }_{B_{p,r}^{s-1}}\hfill \\ & \lesssim & \parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^{s-1}}\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s}\parallel {\xi }_n{\parallel }_{B_{p,r}^s}+\tau \parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^{s-1}}\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^s}{\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}\parallel }_{B_{p,r}^s}\hfill \\ & & +\parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^{s-1}}{\parallel {\tilde{u}}_{0,n}\parallel }_{B_{p,r}^{s-1}}^2\hfill \end{array}$$

(50)

$$\begin{array}{ccc}& \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^s}+\tau ·2^{-\frac{n}{2}}+2^{-\frac{3}{2}n},\hfill \\ & & \parallel {\mathcal{A}}_4({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \parallel {\tilde{v}}_n{\partial }_x{\tilde{v}}_n{\partial }_x{\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\tau \parallel {\tilde{v}}_n{\partial }_x{\tilde{v}}_n{\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s-1}}+{\parallel {\partial }_x{\tilde{u}}_{0,n}{\tilde{v}}_n{\partial }_x{\tilde{v}}_n\parallel }_{B_{p,r}^{s-1}}\hfill \\ & \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^s}+\tau (\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^s}\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}+\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^{s-1}}\parallel {\partial }_x{\tilde{v}}_n{\parallel }_{L^{\infty }}{\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}\parallel }_{B_{p,r}^s}\hfill \\ & & +\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^{s-1}}\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^s}\parallel {\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{L^{\infty }})+\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^{s-1}}{\parallel {\tilde{v}}_n\parallel }_{B_{p,r}^s}\hfill \\ & & +\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^s}+\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^{s-1}}{\parallel {\partial }_x{\tilde{v}}_n\parallel }_{L^{\infty }}\hfill \\ & \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^s}+\tau (2^{-\frac{n}{2}}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}})+2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+2^{-\frac{n}{2}}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}}\hfill \end{array}$$

(51)

$$\begin{array}{ccc}& \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^s}+\tau ·2^{-\frac{n}{2}}+2^{-\frac{n}{2}},\hfill \end{array}$$

(52)

$$\begin{array}{ccc}& & \parallel {\mathcal{A}}_5({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \parallel {\tilde{u}}_n{\partial }_x({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\eta }_n{\parallel }_{B_{p,r}^{s-1}}+\tau \parallel {\tilde{u}}_n{\partial }_x({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s-1}}+{\parallel {\tilde{u}}_n{\left({\partial }_x{\tilde{v}}_{0,n}\right)}^2\parallel }_{B_{p,r}^{s-1}}\hfill \end{array}$$

(53)

$$\begin{array}{ccc}& \lesssim & \parallel {\eta }_n{\parallel }_{B_{p,r}^s}+\tau ·2^{-\frac{n}{2}}+2^{-\frac{n}{2}},\hfill \\ & & \parallel {\mathcal{A}}_6({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \parallel {\partial }_x{\tilde{u}}_n{\partial }_x({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\xi }_n{\parallel }_{B_{p,r}^{s-2}}+\tau {\parallel {\partial }_x{\tilde{u}}_n{\partial }_x({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}\parallel }_{B_{p,r}^{s-2}}\hfill \end{array}$$

(54)

$$\begin{array}{ccc}& \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^s}+2^{-n}\tau +2^{-\frac{3}{2}n},\hfill \\ & & \parallel {\mathcal{A}}_7({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \parallel {\partial }_x{\tilde{u}}_n{\partial }_x({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\eta }_n{\parallel }_{B_{p,r}^{s-2}}+\tau \parallel {\partial }_x{\tilde{u}}_n{\partial }_x({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s-2}}+{\parallel {\partial }_x{\tilde{u}}_n{\left({\partial }_x{\tilde{v}}_{0,n}\right)}^2\parallel }_{B_{p,r}^{s-2}}\hfill \\ & \lesssim & \parallel {\eta }_n{\parallel }_{B_{p,r}^s}+\tau \parallel {\partial }_x{\tilde{u}}_n{\parallel }_{B_{p,r}^{s-1}}\parallel {\partial }_x({\tilde{v}}_n+{\tilde{v}}_{0,n}){\parallel }_{B_{p,r}^{s-1}}\parallel {\partial }_x{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s-2}}+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{B_{p,r}^{s-2}}{\parallel {\partial }_x{\tilde{v}}_{0,n}\parallel }_{B_{p,r}^{s-1}}^2\hfill \end{array}$$

(55)

$$\begin{array}{ccc}& \lesssim & \parallel {\eta }_n{\parallel }_{B_{p,r}^s}+2^{-\frac{3}{2}n}\tau +2^{-\frac{3}{2}n},\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \parallel \tau ({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \tau (\parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^s}^2+\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^s}^2)\parallel {\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{L^{\infty }}+\tau (\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}^2)\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s+1}}\lesssim 2^{-n}\tau +\tau \lesssim \tau .\hfill \end{array}$$

(56)

Collecting all of the estimates from (45)–(56) and plugging all of them back into (43), we obtain

$$\begin{array}{c}\hfill \parallel {\xi }_n{\parallel }_{B_{p,r}^s}\lesssim {\int }_0^t(\parallel {\xi }_n{\parallel }_{B_{p,r}^s}+\parallel {\eta }_n{\parallel }_{B_{p,r}^s})\mathrm{d}\tau +{\int }_0^t{2}^{\frac{n}{2}}(\parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\parallel {\eta }_n{\parallel }_{B_{p,r}^{s-1}})\mathrm{d}\tau +t^2+2^{-\frac{n}{2}}.\end{array}$$

(57)

Similarly, we find that

$$\begin{array}{c}\hfill \parallel {\eta }_n{\parallel }_{B_{p,r}^s}\lesssim {\int }_0^t(\parallel {\xi }_n{\parallel }_{B_{p,r}^s}+\parallel {\eta }_n{\parallel }_{B_{p,r}^s})\mathrm{d}\tau +{\int }_0^t{2}^{\frac{n}{2}}(\parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\parallel {\eta }_n{\parallel }_{B_{p,r}^{s-1}})\mathrm{d}\tau +t^2+2^{-\frac{n}{2}}.\end{array}$$

(58)

Adding (57) and (58) together, we find that

$$\begin{array}{ccc}& & \parallel {\xi }_n{\parallel }_{B_{p,r}^s}+{\parallel {\eta }_n\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & {\int }_0^t(\parallel {\xi }_n{\parallel }_{B_{p,r}^s}+\parallel {\eta }_n{\parallel }_{B_{p,r}^s})\mathrm{d}\tau +{\int }_0^t{2}^{\frac{n}{2}}(\parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\parallel {\eta }_n{\parallel }_{B_{p,r}^{s-1}})\mathrm{d}\tau +t^2+2^{-\frac{n}{2}}.\hfill \end{array}$$

(59)

We will now measure the $B_{p,r}^{s-1}\times B_{p,r}^{s-1}$ norm of $({\xi }_n,{\eta }_n)$. For this purpose, we list the following estimates:

$$\begin{array}{c}\hfill \parallel \tau ({\tilde{u}}_n+{\tilde{u}}_{0,n}){\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\lesssim \tau (2^{-\frac{n}{2}}·2^{-\frac{3}{2}n}·2^{-\frac{n}{2}}+2^{-\frac{n}{2}}(2^{-n}·2^{-\frac{n}{2}}+2^{-\frac{3}{2}n}))\lesssim 2^{-2n}\tau .\end{array}$$

(60)

Similarly, we can derive

$$\begin{array}{c}\hfill \parallel \tau ({\tilde{v}}_n+{\tilde{v}}_{0,n}){\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\partial }_x{\tilde{v}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-2n}\tau ,\end{array}$$

(61)

$$\begin{array}{ccc}& & \parallel {\xi }_n({\tilde{u}}_n+{\tilde{u}}_{0,n}){\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\hfill \\ & \lesssim & 2^{-n}\parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\parallel {\xi }_n{\parallel }_{L^{\infty }}(2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}})\lesssim 2^{-n}{\parallel {\xi }_n\parallel }_{B_{p,r}^{s-1}},\hfill \end{array}$$

(62)

and

$$\begin{array}{c}\hfill \parallel {\eta }_n({\tilde{v}}_n+{\tilde{v}}_{0,n}){\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-n}{\parallel {\eta }_n\parallel }_{B_{p,r}^{s-1}},\end{array}$$

(63)

$$\begin{array}{c}\hfill \parallel {\mathcal{A}}_1({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^{s-1}},{\parallel {\mathcal{A}}_2({\tilde{u}}_n,{\tilde{v}}_n)\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-\frac{3}{2}n}.\end{array}$$

(64)

Further, we have

$$\begin{array}{ccc}\hfill \parallel {\mathcal{A}}_3({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\tau (2^{-\frac{n}{2}}·2^{-\frac{n}{2}}·2^{-n})+2^{-\frac{n}{2}}·2^{-n}\hfill \end{array}$$

$$\begin{array}{ccc}& \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+2^{-2n}\tau +2^{-\frac{3}{2}n},\hfill \\ \hfill \parallel {\mathcal{A}}_4({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\tau (2^{-\frac{n}{2}}·2^{-\frac{n}{2}}·2^{-n}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}}·2^{-n}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}}·2^{-n}\hfill \\ & & +2^{-\frac{n}{2}}·2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}}·2^{-\frac{n}{2}})\hfill \end{array}$$

(65)

$$\begin{array}{ccc}& \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+2^{-\frac{3}{2}n}\tau ,\hfill \end{array}$$

(66)

$$\begin{array}{ccc}\hfill \parallel {\mathcal{A}}_5({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel {\eta }_n{\parallel }_{B_{p,r}^{s-1}}+2^{-2n}\tau +2^{-n},\hfill \end{array}$$

(67)

$$\begin{array}{ccc}\hfill \parallel {\mathcal{A}}_6({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+2^{-2n}\tau +2^{-\frac{3}{2}n},\hfill \end{array}$$

(68)

$$\begin{array}{ccc}\hfill \parallel {\mathcal{A}}_7({\tilde{u}}_n,{\tilde{v}}_n){\parallel }_{B_{p,r}^{s-1}}& \lesssim & \parallel {\eta }_n{\parallel }_{B_{p,r}^{s-1}}+2^{-2n}\tau +2^{-\frac{3}{2}n},\hfill \end{array}$$

(69)

and

$$\begin{array}{ccc}\hfill \parallel \tau ({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^{s-1}}& \lesssim & \tau (\parallel {\tilde{u}}_n{\parallel }_{B_{p,r}^{s-1}}^2+\parallel {\tilde{v}}_n{\parallel }_{B_{p,r}^{s-1}}^2)\parallel {\partial }_x{\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{L^{\infty }}+\tau (\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}^2)\parallel {\mathbf{u}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & 2^{-n}·2^{-n}·\tau +2^{-n}·\tau \lesssim 2^{-n}\tau .\hfill \end{array}$$

(70)

Collecting all of the estimates from (60)–(70), we obtain

$$\parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}\lesssim {\int }_0^t(\parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\parallel {\eta }_n{\parallel }_{B_{p,r}^{s-1}})\mathrm{d}\tau +2^{-n}{t}^2+2^{-n}.$$

Therefore,

$$\begin{array}{c}\hfill \parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\parallel {\eta }_n{\parallel }_{B_{p,r}^{s-1}}\lesssim {\int }_0^t(\parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+\parallel {\eta }_n{\parallel }_{B_{p,r}^{s-1}})\mathrm{d}\tau +2^{-n}{t}^2+2^{-n},\end{array}$$

(71)

which implies that

$$\begin{array}{c}\hfill \parallel {\xi }_n{\parallel }_{B_{p,r}^{s-1}}+{\parallel {\eta }_n\parallel }_{B_{p,r}^{s-1}}\lesssim 2^{-n}{t}^2+2^{-n}.\end{array}$$

(72)

Thus, (72) together with (57) imply that

$$\parallel {\xi }_n{\parallel }_{B_{p,r}^s}+{\parallel {\eta }_n\parallel }_{B_{p,r}^s}\lesssim t^2+2^{-\frac{n}{2}}.$$

The proof of Proposition 2 is complete. □

Proof of Theorem 3.

Thanks to (30), we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel {\tilde{u}}_{0,n}-u_{0,n}{\parallel }_{B_{p,r}^s}=\underset{n\to \infty }{lim}{\parallel 2^{-\frac{n}{2}}\varphi \left(x\right)\parallel }_{B_{p,r}^s}=0,\\ \hfill \underset{n\to \infty }{lim}\parallel {\tilde{v}}_{0,n}-v_{0,n}{\parallel }_{B_{p,r}^s}=\underset{n\to \infty }{lim}{\parallel 2^{-\frac{n}{2}}\psi \left(x\right)\parallel }_{B_{p,r}^s}=0.\end{array}$$

Moreover, in view of the definition ${\tilde{v}}_n={\eta }_n+{\tilde{v}}_{0,n}+t{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}},{\tilde{v}}_{0,n}=v_{0,n}+g_n^2$, it follows from Propositions 1 and 2 that

$$\begin{array}{ccc}& & \parallel {\tilde{v}}_n-v_n{\parallel }_{B_{p,r}^s}={\parallel {\eta }_n+{\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}+g_n^2+v_{0,n}-v_n\parallel }_{B_{p,r}^s}\hfill \\ & \gtrsim & t\parallel {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}-\parallel {\eta }_n{\parallel }_{B_{p,r}^s}-\parallel g_n^2{\parallel }_{B_{p,r}^s}-{\parallel v_{0,n}-v_n\parallel }_{B_{p,r}^s}\hfill \\ & \gtrsim & t\parallel {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}-t^2-2^{-\frac{n}{2}}-2^{-\frac{n}{2}}-2^{-\frac{n}{2}(s-1)}\hfill \\ & \gtrsim & t\parallel {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}{\parallel }_{B_{p,r}^s}-t^2-2^{-\frac{n}{2}}-2^{-\frac{n}{2}(s-1)}.\hfill \end{array}$$

(73)

By definition,

$$\begin{array}{ccc}\hfill {\mathbf{v}}_{\mathbf{0}}^{\mathbf{n}}& =& -({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{v}}_{0,n}\hfill \\ & =& -[{\left(f_n^1\right)}^2+2f_n^1{g}_n^1+{\left(g_n^1\right)}^2]({\partial }_x{f}_n^1+{\partial }_x{g}_n^1)-[{\left(f_n^2\right)}^2+2f_n^2{g}_n^2+{\left(g_n^2\right)}^2]({\partial }_x{f}_n^1+{\partial }_x{g}_n^1)\hfill \\ & \triangleq & I_1+I_2,\hfill \end{array}$$

where $I_1=-[{\left(f_n^1\right)}^2+2f_n^1{g}_n^1+{\left(g_n^1\right)}^2]({\partial }_x{f}_n^1+{\partial }_x{g}_n^1),I_2=-[{\left(f_n^2\right)}^2+2f_n^2{g}_n^2+{\left(g_n^2\right)}^2]$$({\partial }_x{f}_n^1+{\partial }_x{g}_n^1)$.

For every term in $I_2$, we have the following estimates:

$$\begin{array}{ccc}& & \parallel {\left(f_n^2\right)}^2{\partial }_x{f}_n^1{\parallel }_{B_{p,r}^s}\hfill \end{array}$$

$$\begin{array}{ccc}& \lesssim & \parallel f_n^2{\parallel }_{L^{\infty }}^2\parallel {\partial }_x{f}_n^1{\parallel }_{B_{p,r}^s}+\parallel f_n^2{\parallel }_{B_{p,r}^s}^2{\parallel {\partial }_x{f}_n^1\parallel }_{L^{\infty }}\lesssim 2^{-ns}·2^n+2^{n(1-s)}\lesssim 2^{n(1-s)}.\hfill \\ & & \parallel {\left(f_n^2\right)}^2{\partial }_x{g}_n^1{\parallel }_{B_{p,r}^s}\hfill \end{array}$$

(74)

$$\begin{array}{ccc}& \lesssim & \parallel f_n^2{\parallel }_{L^{\infty }}^2\parallel {\partial }_x{g}_n^1{\parallel }_{B_{p,r}^s}+\parallel f_n^2{\parallel }_{B_{p,r}^s}^2{\parallel {\partial }_x{g}_n^1\parallel }_{L^{\infty }}\lesssim 2^{-ns}·2^{-\frac{n}{2}}+2^{-\frac{n}{2}}\lesssim 2^{-\frac{n}{2}}.\hfill \\ & & \parallel 2f_n^2{g}_n^2{\partial }_x{f}_n^1{\parallel }_{B_{p,r}^s}\hfill \\ & \lesssim & \parallel f_n^2{\parallel }_{L^{\infty }}\parallel g_n^2{\partial }_x{f}_n^1{\parallel }_{B_{p,r}^s}+\parallel f_n^2{\parallel }_{B_{p,r}^s}{\parallel g_n^2{\partial }_x{f}_n^1\parallel }_{L^{\infty }}\hfill \\ & \lesssim & \parallel f_n^2{\parallel }_{L^{\infty }}\parallel g_n^2{\parallel }_{L^{\infty }}\parallel {\partial }_x{f}_n^1{\parallel }_{B_{p,r}^s}+\parallel f_n^2{\parallel }_{L^{\infty }}\parallel \parallel g_n^2{\parallel }_{B_{p,r}^s}\parallel {\partial }_x{f}_n^1{\parallel }_{L^{\infty }}+\parallel f_n^2{\parallel }_{B_{p,r}^s}\parallel g_n^2{\parallel }_{L^{\infty }}{\parallel {\partial }_x{f}_n^1\parallel }_{L^{\infty }}\hfill \end{array}$$

(75)

$$\begin{array}{ccc}& \lesssim & 2^{-ns}·2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+2^{-ns}·2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}}\lesssim 2^{-n-ns}+2^{-n-ns}+2^{-n}\lesssim 2^{-n}.\hfill \\ & & \parallel 2f_n^2{g}_n^2{\partial }_x{g}_n^1{\parallel }_{B_{p,r}^s}\hfill \end{array}$$

(76)

$$\begin{array}{ccc}& \lesssim & 2^{-ns}·2^{-\frac{n}{2}}·2^{-\frac{n}{2}-s-1}+2^{-ns}·2^{-\frac{n}{2}}·2^{-\frac{n}{2}}+2^{-\frac{n}{2}}·2^{-\frac{n}{2}}\lesssim 2^{-n}.\hfill \\ & & \parallel {\left(g_n^2\right)}^2{\partial }_x{g}_n^1{\parallel }_{B_{p,r}^s}\hfill \end{array}$$

(77)

$$\begin{array}{ccc}& \lesssim & \parallel g_n^2{\parallel }_{L^{\infty }}^2\parallel {\partial }_x{g}_n^2{\parallel }_{B_{p,r}^s}+\parallel g_n^2{\parallel }_{B_{p,r}^s}^2{\parallel {\partial }_x{g}_n^2\parallel }_{L^{\infty }}\lesssim 2^{-\frac{3}{2}n}.\hfill \end{array}$$

(78)

Note that

$$\mathrm{supp}\widehat{{\left(g_n^2\right)}^2{\partial }_x{f}_n^1}\subset \{\xi \in \mathbb{R}:\frac{33}{24}{2}^n-\frac{3}{2}\le \xi \le \frac{33}{24}{2}^n+\frac{3}{2}\},$$

and we have ${\Delta }_j\left({\left(g_n^2\right)}^2{\partial }_x{f}_n^1\right)=0(j\ne n),{\Delta }_n\left({\left(g_n^2\right)}^2{\partial }_x{f}_n^1\right)={\left(g_n^2\right)}^2{\partial }_x{f}_n^1$. Furthermore, direct calculations and the application of Riemann theorem show that, as $n\to \infty$,

$$\begin{array}{ccc}& & \parallel {\left(g_n^2\right)}^2{\partial }_x{f}_n^1{\parallel }_{B_{p,r}^s}=2^{ns}{\parallel {\left(g_n^2\right)}^2{\partial }_x{f}_n^1\parallel }_{L^p}\hfill \\ & =& \parallel 2^{-n}{\left(\frac{24}{33}\right)}^2{\varphi }^{\prime }\left(x\right)\psi \left(x\right)sin\left(\frac{33}{24}{2}^{n}x\right)+\frac{24}{33}{\varphi }^2\left(x\right){\psi }^{\prime }\left(x\right)cos\left(\frac{33}{24}{2}^{n}x\right){\parallel }_{L^p}\hfill \\ & \gtrsim & \parallel \frac{24}{33}{\varphi }^2\left(x\right){\psi }^{\prime }\left(x\right)cos\left(\frac{33}{24}{2}^{n}x\right){\parallel }_{L^p}-2^{-n}\to \frac{24}{33}{\left(\frac{{\int }_0^t{|cosx|}^p\mathrm{d}x}{\pi }\right)}^{\frac{1}{p}}{\parallel {\varphi }^3\left(x\right)\parallel }_{L^p}\triangleq C^{\ast }\gtrsim 1.\hfill \end{array}$$

(79)

In view of the estimates from (74)–(79), letting $n\to \infty$ in (73) yields, for sufficiently small t,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; inf}\parallel {\tilde{v}}_n-v_n{\parallel }_{B_{p,r}^s}\gtrsim t{\parallel {\left(g_n^2\right)}^2{\partial }_x{f}_n^1\parallel }_{B_{p,r}^s}-2^{-\frac{n}{2}}t-t^2-2^{-\frac{n}{2}}-2^{-\frac{n}{2}(s-1)}\gtrsim t.\end{array}$$

Similarly, we can prove that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; inf}{\parallel {\tilde{u}}_n-u_n\parallel }_{B_{p,r}^s}\gtrsim t.\end{array}$$

The proof of Theorem 3 is complete. □

4.2. Non-Uniform Dependence in Critical Besov Spaces

We will investigate the non-uniform dependence property in the non-periodic critical Besov spaces $B_{p,1}^{1+\frac{1}{p}}\left(\mathbb{R}\right)\times B_{p,1}^{1+\frac{1}{p}}\left(\mathbb{R}\right)$ with $1\le p\le 2$; or, in other words, we prove Theorem 4 in this subsection.

For the purpose of proving the main result, we list the following lemma.

Lemma 15.

Let $\zeta (u,v)=-(u^2+v^2){\partial }_{x}u-f_1(u,v)-f_2(u,v)$; then, we have the following estimates:

$$\begin{array}{ccc}\hfill {\parallel \zeta (u,v)\parallel }_{L^{\infty }}& \lesssim & {(\parallel u\parallel }_{L^{\infty }}+{\parallel v\parallel }_{L^{\infty }}+\parallel u_x{\parallel }_{L^{\infty }}{\left)\right(\parallel u\parallel }_{L^{\infty }}^2+{\parallel v\parallel }_{L^{\infty }}^2+\parallel u_x{\parallel }_{L^{\infty }}^2+\parallel v_x{\parallel }_{L^{\infty }}^2).\hfill \\ \hfill {\parallel \zeta (u,v)\parallel }_{B_{p,1}^{1+\frac{1}{p}}}& \lesssim & {(\parallel u\parallel }_{L^{\infty }}^2+{\parallel v\parallel }_{L^{\infty }}^2{)\parallel u\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+{(\parallel u\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\parallel v\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\parallel u\parallel }_{B_{p,1}^{2+\frac{1}{p}}})\hfill \end{array}$$

(80)

$$\begin{array}{ccc}& & \times (\parallel u{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}^2+\parallel v{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}^2+\parallel u{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2+\parallel v{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2).\hfill \\ \hfill {\parallel \zeta (u,v)\parallel }_{B_{p,1}^{2+\frac{1}{p}}}& \lesssim & {(\parallel u\parallel }_{L^{\infty }}^2+{\parallel v\parallel }_{L^{\infty }}^2{)\parallel u\parallel }_{B_{p,1}^{3+\frac{1}{p}}}+{(\parallel u\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+{\parallel v\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+{\parallel u\parallel }_{B_{p,1}^{3+\frac{1}{p}}})\hfill \end{array}$$

(81)

$$\begin{array}{ccc}& & \times (\parallel u{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2+\parallel v{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2+\parallel u{\parallel }_{B_{p,1}^{3+\frac{1}{p}}}^2+\parallel v{\parallel }_{B_{p,1}^{3+\frac{1}{p}}}^2).\hfill \end{array}$$

(82)

The proof of Lemma 15 is straightforward, so we omit it here.

From Lemma 7 and the product property, the following result is obvious:

$$\begin{array}{c}\hfill \parallel f_1(u,v)-f_1(u_0,v_0){\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel f_2(u,v)-f_2(u_0,v_0){\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\lesssim \parallel u-u_0{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\parallel v-v_0\parallel }_{B_{p,1}^{1+\frac{1}{p}}}.\end{array}$$

(83)

Denote $\parallel {\tilde{X}}_{0,n}{\parallel }_Y=\parallel {\tilde{u}}_{0,n}{\parallel }_Y+{\parallel {\tilde{v}}_{0,n}\parallel }_Y$. The following proposition is crucial for proving the main theorem in this part.

Proposition 3.

Assume that $(s,p,r)$ is as stated in Theorem 4. $({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\in B_{p,1}^{1+\frac{1}{p}}\times B_{p,1}^{1+\frac{1}{p}}$, and $({\tilde{u}}_n,{\tilde{v}}_n)$ is the solution to Equation (5) with initial data $({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})$; then, we have

$$\begin{array}{ccc}& & \parallel {\tilde{u}}_n-{\tilde{u}}_{0,n}-t{\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}){\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\parallel {\tilde{v}}_n-{\tilde{v}}_{0,n}-t{\zeta }_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & \lesssim & \mathbf{P}({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})t^2+\mathbf{Q}({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})t^3.\hfill \end{array}$$

where

$$\begin{array}{ccc}& & {\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})=-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n}-f_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})-f_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}),\hfill \\ & & {\zeta }_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})=-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{v}}_{0,n}-g_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})-g_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}),\hfill \\ & & \mathbf{P}({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})=C_0{C}_2+C_3+C_1{C}_2^2+C_0^3{C}_4+C_5,\hfill \\ & & \mathbf{Q}({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})=C_1{C}_2+C_0^2{C}_1{C}_4,\hfill \\ & & C_0=\parallel {\tilde{X}}_{0,n}{\parallel }_{L^{\infty }},C_1=(\parallel {\tilde{X}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{X}}_{0,n}{\parallel }_{L^{\infty }}{)}^3,C_2={\parallel {\tilde{X}}_{0,n}\parallel }_{B_{p,1}^{2+\frac{1}{p}}},\hfill \\ & & C_3=(\parallel {\tilde{X}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\tilde{X}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}{)}^3,C_4=\parallel {\tilde{X}}_{0,n}{\parallel }_{B_{p,1}^{3+\frac{1}{p}}},\mathrm{and}{C}_5=(\parallel {\tilde{X}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{X}}_{0,n}{{\parallel }_{B_{p,1}^{3+\frac{1}{p}}})}^3.\hfill \end{array}$$

Proof. 

In view of Lemmas 6 and 7, for any initial data $({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\in B_{p,1}^{\theta }\times B_{p,1}^{\theta },\theta \ge 1+\frac{1}{p}$, there exists a unique solution $({\tilde{u}}_n,{\tilde{v}}_n)\in C([0,T];B_{p,1}^{\theta })\times C([0,T];B_{p,1}^{\theta })$, and it satisfies the following estimate:

$$\begin{array}{c}\hfill \parallel {\tilde{u}}_n{\parallel }_{L_T^{\infty }\left(B_{p,1}^{\theta }\right)}+\parallel {\tilde{v}}_n{\parallel }_{L_T^{\infty }\left(B_{p,1}^{\theta }\right)}\lesssim \parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{\theta }}+{\parallel {\tilde{v}}_{0,n}\parallel }_{B_{p,1}^{\theta }}\lesssim 1.\end{array}$$

(84)

By the embedding theorem $C([0,T];B_{p,1}^{1+\frac{1}{p}})\hookrightarrow C([0,T];C^{0,1})$, we have

$$\begin{array}{c}\hfill {\tilde{u}}_n\left(t\right)-{\tilde{u}}_{0,n}\left(t\right)={\int }_0^t{\partial }_{\tau }{\tilde{u}}_n\mathrm{d}\tau ,{\tilde{v}}_n\left(t\right)-{\tilde{v}}_{0,n}\left(t\right)={\int }_0^t{\partial }_{\tau }{\tilde{v}}_n\mathrm{d}\tau .\end{array}$$

(85)

Let $U_n={\tilde{u}}_n-{\tilde{u}}_{0,n},V_n={\tilde{v}}_n-{\tilde{v}}_{0,n}$; then, we shall prove that $(U_n,V_n)$ solves the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t{U}_n+({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\tilde{u}}_n+f_1(\widetilde{u_n},\widetilde{v_n})+f_2(\widetilde{u_n},\widetilde{v_n})=0,\hfill \\ {\partial }_t{V}_n+({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\tilde{v}}_n+g_1(\widetilde{u_n},\widetilde{v_n})+g_2(\widetilde{u_n},\widetilde{v_n})=0,\hfill \\ U_n(0,x)=V_n(0,x)=0.\hfill \end{array}\right.\end{array}$$

(86)

In view of Equations (5), (80) and (85), we obtain

$$\begin{array}{ccc}& & \parallel U_n{\left(t\right)\parallel }_{L^{\infty }}\lesssim {\int }_0^t\parallel {\partial }_{\tau }{\tilde{u}}_n{\parallel }_{L^{\infty }}\mathrm{d}\tau \lesssim {\int }_0^t{\parallel \zeta ({\tilde{u}}_n,{\tilde{v}}_n)\parallel }_{L^{\infty }}\mathrm{d}\tau \hfill \\ & \lesssim & (\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}\left)\right(\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}^2+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}^2)t.\hfill \end{array}$$

(87)

In a similar manner, one can derive

$$\begin{array}{ccc}& & \parallel V_n{\left(t\right)\parallel }_{L^{\infty }}\lesssim {\int }_0^t\parallel {\partial }_{\tau }{\tilde{u}}_n{\parallel }_{L^{\infty }}\mathrm{d}\tau \lesssim {\int }_0^t{\parallel \zeta ({\tilde{u}}_n,{\tilde{v}}_n)\parallel }_{L^{\infty }}\mathrm{d}\tau \hfill \\ & \lesssim & (\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{v}}_n{\parallel }_{L^{\infty }}\left)\right(\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}^2+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}^2)t.\hfill \end{array}$$

(88)

Adding (87) and (88) together yields

$$\begin{array}{ccc}& & \parallel U_n{\left(t\right)\parallel }_{L^{\infty }}+{\parallel V_n\left(t\right)\parallel }_{L^{\infty }}\hfill \\ & \lesssim & (\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{v}}_n{\parallel }_{L^{\infty }}\left)\right(\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}^2+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\partial }_x{\tilde{v}}_n{\parallel }_{L^{\infty }}^2)t\hfill \\ & \lesssim & (\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{v}}_n{{\parallel }_{L^{\infty }})}^{3}t\hfill \\ & \lesssim & (\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{v}}_{0,n}{{\parallel }_{L^{\infty }})}^{3}t=C_{1}t.\hfill \end{array}$$

(89)

Therefore, we have

$$\begin{array}{c}\hfill \parallel {\tilde{u}}_n{\left(t\right)\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_n{\left(t\right)\parallel }_{L^{\infty }}\lesssim \parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+{\parallel {\tilde{v}}_{0,n}\parallel }_{L^{\infty }}+C_{1}t.\end{array}$$

(90)

It can be deduced from (81) and (84) that

$$\begin{array}{ccc}& & \parallel U_n{\left(t\right)\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel V_n{\left(t\right)\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\lesssim {\int }_0^t\parallel {\partial }_{\tau }{\tilde{u}}_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\parallel {\partial }_{\tau }{\tilde{v}}_n\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\mathrm{d}\tau \hfill \\ & \lesssim & {\int }_0^t(\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}^2\left)\right(\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}})\hfill \\ & & +{\int }_0^t(\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}})\hfill \\ & & \times (\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}^2+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}^2+\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2)\mathrm{d}\tau \hfill \\ & \lesssim & [\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}+(\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}{)}^{3}t]\hfill \\ & & \times (\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}})t+(\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{v}}_{0,n}{{\parallel }_{B_{p,1}^{2+\frac{1}{p}}})}^{3}t\hfill \\ & \triangleq & (C_0+C_{1}t)C_{2}t+C_{3}t.\hfill \end{array}$$

(91)

In addition, it can be derived from (82) and (84) that

$$\begin{array}{ccc}& & \parallel U_n{\left(t\right)\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel V_n{\left(t\right)\parallel }_{B_{p,1}^{2+\frac{1}{p}}}\lesssim {\int }_0^t\parallel {\partial }_{\tau }{\tilde{u}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+{\parallel {\partial }_{\tau }{\tilde{v}}_n\parallel }_{B_{p,1}^{2+\frac{1}{p}}}\mathrm{d}\tau \hfill \\ & \lesssim & {\int }_0^t(\parallel {\tilde{u}}_n{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_n{\parallel }_{L^{\infty }}^2\left)\right(\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{3+\frac{1}{p}}}+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{3+\frac{1}{p}}})\mathrm{d}\tau \hfill \\ & & +{\int }_0^t(\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{3+\frac{1}{p}}}+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{3+\frac{1}{p}}})\hfill \\ & & \times (\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2+\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{3+\frac{1}{p}}}^2+\parallel {\tilde{v}}_n{\parallel }_{B_{p,1}^{3+\frac{1}{p}}}^2)\mathrm{d}\tau \hfill \\ & \lesssim & [\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}+(\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}+\parallel {\partial }_x{\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}{)}^{3}t]\hfill \\ & & \times (\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{3+\frac{1}{p}}}+\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{3+\frac{1}{p}}})t+(\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{3+\frac{1}{p}}}+\parallel {\tilde{v}}_{0,n}{{\parallel }_{B_{p,1}^{3+\frac{1}{p}}})}^{3}t\hfill \\ & \triangleq & (C_0+C_{1}t)C_{4}t+C_{5}t.\hfill \end{array}$$

(92)

Since

$$\begin{array}{ccc}\hfill {\tilde{u}}_n-{\tilde{u}}_{0,n}-t{\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})& =& {\int }_0^t({\partial }_{\tau }\tilde{u}-{\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}))\mathrm{d}\tau ,\hfill \\ \hfill {\tilde{v}}_n-{\tilde{v}}_{0,n}-t{\zeta }_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})& =& {\int }_0^t({\partial }_{\tau }\tilde{v}-{\zeta }_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}))\mathrm{d}\tau ,\hfill \end{array}$$

$$\begin{array}{ccc}& & \parallel ({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\tilde{u}}_n-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & \lesssim & \parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}(\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}\parallel U_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\parallel U_n{\parallel }_{L^{\infty }})\hfill \\ & & +\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}{\parallel U_n\parallel }_{L^{\infty }}\hfill \\ & & +\parallel {\tilde{v}}_n+{\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}(\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}\parallel V_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\parallel V_n{\parallel }_{L^{\infty }})\hfill \\ & & +\parallel {\tilde{v}}_n+{\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}{\parallel V_n\parallel }_{L^{\infty }}\hfill \\ & & +\parallel {\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2{\parallel }_{L^{\infty }}\parallel {\partial }_x{U}_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2{\parallel }_{L^{\infty }}{\parallel {\partial }_x{U}_n\parallel }_{L^{\infty }}\hfill \\ & \lesssim & \parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}\parallel U_n{\parallel }_{L^{\infty }}+\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}{\parallel U_n\parallel }_{L^{\infty }}\hfill \\ & & +\parallel {\tilde{u}}_n+{\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}\parallel U_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}{\parallel U_n\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & & +\parallel {\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2{\parallel }_{L^{\infty }}\parallel U_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2{\parallel }_{L^{\infty }}\parallel U_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel {\tilde{v}}_n+{\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}\parallel {\tilde{u}}_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}{\parallel V_n\parallel }_{L^{\infty }}\hfill \\ & & +\parallel {\tilde{v}}_n+{\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}\parallel V_n{\parallel }_{L^{\infty }}+\parallel {\tilde{v}}_n+{\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}\parallel {\partial }_x{\tilde{u}}_n{\parallel }_{L^{\infty }}{\parallel V_n\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & \lesssim & \parallel U_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel V_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2\parallel U_n{\parallel }_{L^{\infty }}+(\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}^2)\parallel U_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}},\hfill \end{array}$$

(93)

and, similarly,

$$\begin{array}{ccc}& & \parallel ({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\tilde{v}}_n-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & \lesssim & \parallel U_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel V_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2\parallel V_n{\parallel }_{L^{\infty }}+(\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}^2)\parallel V_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}.\hfill \end{array}$$

(94)

Adding (93) and (94) yields

$$\begin{array}{ccc}& & \parallel ({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\tilde{u}}_n-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\parallel ({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\tilde{v}}_n-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{v}}_{0,n}\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & \lesssim & \parallel U_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel V_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+(\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2+\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2\left)\right(\parallel U_n{\parallel }_{L^{\infty }}+\parallel V_n{\parallel }_{L^{\infty }})\hfill \\ & & +(\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}^2\left)\right(\parallel U_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel V_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}).\hfill \end{array}$$

(95)

Equations (83), (89), (91) and (92) together with (95) imply that

$$\begin{array}{ccc}& & \parallel {\tilde{u}}_n-{\tilde{u}}_{0,n}-t{\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}){\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\parallel {\tilde{v}}_n-{\tilde{v}}_{0,n}-t{\zeta }_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & \lesssim & {\int }_0^t\parallel {\partial }_{\tau }{\tilde{u}}_n-{\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}){\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\parallel {\partial }_{\tau }{\tilde{v}}_n-{\zeta }_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\mathrm{d}\tau \hfill \\ & \lesssim & {\int }_0^t{\parallel ({\tilde{u}}_n^2+{\tilde{v}}_n^2){\partial }_x{\tilde{u}}_n-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n}\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\mathrm{d}\tau \hfill \\ & & +{\int }_0^t{\parallel f_1({\tilde{u}}_n,{\tilde{v}}_n)+f_2({\tilde{u}}_n,{\tilde{v}}_n)-f_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})-f_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\mathrm{d}\tau \hfill \\ & & +{\int }_0^t{\parallel g_1({\tilde{u}}_n,{\tilde{v}}_n)+g_2({\tilde{u}}_n,{\tilde{v}}_n)-g_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})-g_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\mathrm{d}\tau \hfill \\ & \lesssim & {\int }_0^t(\parallel U_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel V_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}})\mathrm{d}\tau +(\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2+\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}^2){\int }_0^t(\parallel U_n{\parallel }_{L^{\infty }}+\parallel V_n{\parallel }_{L^{\infty }})\mathrm{d}\tau \hfill \\ & & +(\parallel {\tilde{u}}_{0,n}{\parallel }_{L^{\infty }}^2+\parallel {\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}^2){\int }_0^t(\parallel U_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}}+\parallel V_n{\parallel }_{B_{p,1}^{2+\frac{1}{p}}})\mathrm{d}\tau \hfill \\ & \lesssim & (C_0+C_{1}t)C_2{t}^2+C_3{t}^2+C_1{C}_2^2{t}^2+C_0^2(C_0+C_{1}t)C_4{t}^2+C_5{t}^2\hfill \\ & \lesssim & (C_0{C}_2+C_3+C_1{C}_2^2+C_0^3{C}_4+C_5)t^2+(C_1{C}_2+C_0^2{C}_1{C}_4)t^3\hfill \\ & =& \mathbf{P}({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})t^2+\mathbf{Q}({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})t^3.\hfill \end{array}$$

which completes the proof of Proposition 3. □

Proof of Theorem 4.

It follows from Lemmas 12–14 and the fact that $1\le p\le 2$ that, for $\theta \ge 1+\frac{1}{p}$,

$$\begin{array}{ccc}& & \parallel {\tilde{u}}_{0,n},{\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{\theta }}\lesssim 2^{(\theta -1-\frac{1}{p})n},\parallel {\tilde{u}}_{0,n},{\tilde{v}}_{0,n}{\parallel }_{L^{\infty }}\lesssim 2^{-\frac{n}{2}},{\parallel {\partial }_x{\tilde{u}}_{0,n},{\partial }_x{\tilde{v}}_{0,n}\parallel }_{L^{\infty }}\lesssim 2^{-\frac{n}{2}},\hfill \\ & & \parallel u_{0,n},v_{0,n}{\parallel }_{L^{\infty }}\lesssim 2^{-n(1+\frac{1}{p})}\lesssim 2^{-n},{\parallel {\partial }_x{u}_{0,n},{\partial }_x{v}_{0,n}\parallel }_{L^{\infty }}\lesssim 2^{-\frac{n}{p}}\lesssim 2^{-n},\hfill \\ & & \parallel u_{0,n},v_{0,n}{\parallel }_{B_{p,r}^{\theta }}\lesssim 2^{(\theta -1-\frac{1}{p})n}.\hfill \end{array}$$

(96)

Therefore, we can conclude that $\mathbf{P}({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}),\mathbf{Q}({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\lesssim 1$. Thus, from Proposition 3,

$$\parallel {\tilde{u}}_n-{\tilde{u}}_{0,n}-t{\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}){\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\parallel {\tilde{v}}_n-{\tilde{v}}_{0,n}-t{\zeta }_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\lesssim t^2+t^3.$$

Since

$$\begin{array}{c}\hfill {\tilde{u}}_n={\tilde{u}}_n-{\tilde{u}}_{0,n}-t{\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})+{\tilde{u}}_{0,n}-t(f_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})+f_2({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})+({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n}),\\ \hfill u_n=u_n-u_{0,n}-t{\zeta }_1(u_{0,n},v_{0,n})+u_{0,n}-t(f_1(u_{0,n},v_{0,n})+f_2(u_{0,n},v_{0,n})+(u_{0,n}^2+v_{0,n}^2){\partial }_x{u}_{0,n}).\end{array}$$

Since

$$\begin{array}{ccc}& & ({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{\tilde{u}}_{0,n}-(u_{0,n}^2+v_{0,n}^2){\partial }_x{u}_{0,n}\hfill \\ & =& [2f_n^1{g}_n^1+{\left(g_n^1\right)}^2]{\partial }_x{f}_n^1+[2f_n^2{g}_n^2+{\left(g_n^2\right)}^2]{\partial }_x{f}_n^1-({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{g}_n^1.\hfill \end{array}$$

It follows from Lemma 13 and (96) that

$$\begin{array}{ccc}& & \parallel ({\tilde{u}}_{0,n}^2+{\tilde{v}}_{0,n}^2){\partial }_x{g}_n^1{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\lesssim (\parallel {\tilde{u}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}^2+\parallel {\tilde{v}}_{0,n}{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}^2)\parallel {\partial }_x{g}_n^1{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\lesssim 2^{-\frac{n}{2}-(2+\frac{1}{p})}\lesssim 2^{-\frac{n}{2}},\hfill \\ & & \parallel 2f_n^1{g}_n^1{\partial }_x{f}_n^1{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\lesssim \parallel f_n^2{\parallel }_{L^{\infty }}\parallel g_n^2{\parallel }_{L^{\infty }}\parallel {\partial }_x{f}_n^1{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+\parallel f_n^2{\parallel }_{L^{\infty }}\parallel g_n^2{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}{\parallel {\partial }_x{f}_n^1\parallel }_{L^{\infty }}\hfill \\ & & +\parallel f_n^2{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\parallel g_n^2{\parallel }_{L^{\infty }}{\parallel {\partial }_x{f}_n^1\parallel }_{L^{\infty }}\lesssim 2^{-n}.\hfill \end{array}$$

In a similar manner, one can derive

$$\begin{array}{c}\hfill \parallel 2f_n^2{g}_n^2{\partial }_x{f}_n^1{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\lesssim 2^{-n}.\end{array}$$

Therefore,

$$\begin{array}{ccc}& & \parallel {\tilde{u}}_n-u_n{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & =& \parallel ({\tilde{u}}_n-{\tilde{u}}_{0,n}-t{\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n}))-(u_n-u_{0,n}-t{\zeta }_1(u_{0,n},v_{0,n}))+g_n^1-t(f_1({\tilde{u}}_{0,n},\widetilde{v_{0,n}})\hfill \\ & & +f_2({\tilde{u}}_{0,n},\widetilde{v_{0,n}})-f_1(u_{0,n},v_{0,n})-f_2(u_{0,n},v_{0,n})+2f_n^1{g}_n^1{\partial }_x{f}_n^1\hfill \\ & & +2f_n^2{g}_n^2{\partial }_x{f}_n^1+{\left(g_n^1\right)}^2{\partial }_x{f}_n^1{)\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+{\left(g_n^2\right)}^2{\partial }_x{f}_n^1{)\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & \gtrsim & t\parallel {\left(g_n^1\right)}^2{\partial }_x{f}_n^1{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+t\parallel {\left(g_n^2\right)}^2{\partial }_x{f}_n^1{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}-{\parallel {\tilde{u}}_n-{\tilde{u}}_{0,n}-t{\zeta }_1({\tilde{u}}_{0,n},{\tilde{v}}_{0,n})\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & & -\parallel u_n-u_{0,n}-t{\zeta }_1(u_{0,n},v_{0,n}){\parallel }_{B_{p,1}^{1+\frac{1}{p}}}-{\parallel g_n^1\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & & -t\parallel f_1({\tilde{u}}_{0,n},\widetilde{v_{0,n}})-f_1(u_{0,n},v_{0,n}){\parallel }_{B_{p,1}^{1+\frac{1}{p}}}-t{\parallel f_2({\tilde{u}}_{0,n},\widetilde{v_{0,n}})-f_2(u_{0,n},v_{0,n})\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & & -\parallel f_n^1{g}_n^1{\partial }_x{f}_n^1{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}-{\parallel f_n^2{g}_n^2{\partial }_x{f}_n^1\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\hfill \\ & \gtrsim & t\parallel {\left(g_n^1\right)}^2{\partial }_x{f}_n^1{\parallel }_{B_{p,1}^{1+\frac{1}{p}}}+t{\parallel {\left(g_n^2\right)}^2{\partial }_x{f}_n^1\parallel }_{B_{p,1}^{1+\frac{1}{p}}}-t^2-t^3-2^{-\frac{n}{2}-(1+\frac{1}{p})}-2^{-\frac{n}{2}}t-2^{-\frac{n}{2}}t-2^{-n}-2^{-n},\hfill \end{array}$$

which, together with (79), indicates that, for sufficiently small t,

$$\underset{n\to \infty }{lim\; inf}{\parallel {\tilde{u}}_n-u_n\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\gtrsim t.$$

Similarly,

$$\underset{n\to \infty }{lim\; inf}{\parallel {\tilde{v}}_n-v_n\parallel }_{B_{p,1}^{1+\frac{1}{p}}}\gtrsim t.$$

This completes the proof of Theorem 4. □