Ill-Posedness of a Three-Component Novikov System in Besov Spaces

Abstract

In this paper, we consider the Cauchy problem for a three-component Novikov system on the line. We give a construction of the initial data $({\rho }_0,u_0,v_0)\in B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$ with $\sigma >max\left\{3+\frac{1}{p},\frac{7}{2}\right\},1\le p\le \infty$, such that the corresponding solution to the three-component Novikov system starting from $({\rho }_0,u_0,v_0)$ is discontinuous at $t=0$ in the metric of $B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$, which implies the ill-posedness for this system in $B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$.

1. Introduction

In this paper, we investigate the Cauchy problem of a three-component Novikov system which takes the form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_t+{\left(\rho uv\right)}_x=0,t>0,x\in \mathbb{R},\hfill \\ m_t+3m{u}_{x}v+m_{x}uv+{\rho }^{2}u=0,t>0,x\in \mathbb{R},\hfill \\ n_t+3nu{v}_x+n_{x}uv-{\rho }^{2}v=0,t>0,x\in \mathbb{R},\hfill \\ m=u-u_{xx},n=v-v_{xx},\hfill \\ \rho (0,x)={\rho }_0\left(x\right),u(0,x)=u_0\left(x\right),v(0,x)=v_0\left(x\right),x\in \mathbb{R}.\hfill \end{array}\right.\end{array}$$

(1)

System (1) was proposed by Li [1]; it can be derived from the following spectral problem:

$${\varphi }_x=U\varphi ,\varphi =\left(\begin{array}{c}{\varphi }_1\\ {\varphi }_2\\ {\varphi }_3\end{array}\right),M=\left(\begin{array}{ccc}0& 1& 0\\ 1+\lambda {\rho }^2& 0& m\\ \lambda n& 0& 0\end{array}\right),$$

where $\lambda$ is a constant spectral parameter. The authors showed that system (1) is equivalent to the zero-curvature equation

$$U_t-V_x+[U,V]=0,$$

where the spacial and temporal $3\times 3$ matrices U and V are

$$U=\left(\begin{array}{ccc}0& 1& 0\\ 1+\lambda {\rho }^2& 0& m\\ \lambda n& 0& 0\end{array}\right),V=\left(\begin{array}{ccc}\frac{1}{3\lambda }+u{v}_x& -uv& \frac{u}{\lambda }\\ u_x{v}_x-\lambda {\rho }^{2}uv& \frac{1}{3\lambda }-u_{x}v& \frac{u_x}{\lambda }-muv\\ -\lambda nuv-v_x& v& u_{x}v-u{v}_x-\frac{2}{3\lambda }\end{array}\right).$$

The authors also constructed a bi-Hamiltonian structure and infinitely many conserved quantities. Li-Hu [2] established the local well-posedness of the system in Besov spaces ${\left(B_{p,r}^s\left(\mathbb{R}\right)\right)}^3(s>max\left\{\frac{1}{p},\frac{1}{2}\right\},1\le p,r\le \infty )$, and derived two blow-up criteria for the system.

For $\rho =0$, system (1) degenerates into the following Geng-Xue (GX) system:

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

(2)

Mi-Mu-Tao [3] and Tang-Liu [4] established the local well-posedness of a GX system (2) in the supercritical Besov spaces $B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)(s>max\{2+\frac{1}{p},\frac{5}{2}\},1\le p,r\le \infty )$ and the critical Besov space $B_{2,1}^{\frac{5}{2}}\left(\mathbb{R}\right)\times B_{2,1}^{\frac{5}{2}}\left(\mathbb{R}\right)$. Based on the local well-posedness results of a GX system (2) in Besov spaces, Wang-Chong-Wu [5] further established nonuniform continuous dependence on initial data for a GX system (2) in Besov spaces: $B_{p,r}^s\left(\mathbb{T}\right)\times B_{p,r}^s\left(\mathbb{T}\right)$$(s>max\{2+\frac{1}{p},\frac{5}{2}\},1\le p,r\le \infty )$, $B_{2,1}^{\frac{5}{2}}\left(\mathbb{T}\right)\times B_{2,1}^{\frac{5}{2}}\left(\mathbb{T}\right)$ and $B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)$$(s>max\{2+\frac{1}{p},\frac{5}{2}\},$$1\le p,r\le \infty )$.

For $v=1$, the GX system (2) degenerates into the following Degasperis–Procesi (DP) equation:

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

(3)

The DP Equation (3) can be regarded as a nonlinear shallow water wave dynamics model, and its asymptotic accuracy is the same as that of the Camassa–Holm (CH) shallow water system [6]. By constructing the Lax pair [7], the formal integrability of DP equation was proved, which has a bi-Hamiltonian structure and infinitely many conserved quantities. The Cauchy problem of DP Equation (3) is locally well-posed in certain Sobolev spaces and Besov spaces in [8,9,10]. Li-Yu-Zhu studied the local existence and uniqueness of the solution of the DP Equation (3) on the critical Besov space $B_{\infty ,1}^1\left(\mathbb{R}\right)$ in [11], and further proved the nonuniform continuous dependence of the data-to-solution map. Zhou [12] gave the local well-posedness of the CH-DP system in the nonhomogeneous Besov spaces $B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)$ $(s>max\{2+\frac{1}{p},\frac{5}{2}\},1\le p,r\le \infty )$ and the critical Besov space $B_{2,1}^{\frac{5}{2}}\left(\mathbb{R}\right)\times B_{2,1}^{\frac{5}{2}}\left(\mathbb{R}\right)$ by using the transport equation theory and the classical Friedrichs regularization method. Zhu-Li-Li [13] proved the ill-posedness of the two-component DP system in the Besov space $B_{\infty ,1}^1\left(\mathbb{R}\right)\times B_{\infty ,1}^0\left(\mathbb{R}\right)$.

When $u=v$, the GX system (2) degenerates into the following Novikov equation:

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

(4)

Novikov Equation (4) is a new integrabe equation with cubic nonlinearities derived by Novikov [14]. It has similar properties to the CH equation, such as a Lax pair in matrix form and a bi-Hamiltonian structure, it is integrable in the with infinitely many conserved quantities, and admits peakon solutions given by the formula $u(x,t)=\pm \sqrt{C}{e}^{|x-ct|}$ [15]. The local well-posedness, global existence, and asymptotic behavior of the Novikov Equation (4) in Sobolev spaces and Besov spaces were established in [16,17,18,19,20]. Ni-Zhou established the local well-posedness in the critical Besov space $B_{2,1}^{\frac{3}{2}}\left(\mathbb{R}\right)$ in [17]. Yan-Li-Zhang generalized well-posed spaces to a larger class of Besov spaces $B_{p,r}^s\left(\mathbb{R}\right)$$(s>max\left\{1+\frac{1}{p},\frac{3}{2}\right\},$$1\le p\le \infty ,$$1\le r\le \infty )$, and the local well-posedness was proved to be invalid in Besov space $B_{2,\infty }^{\frac{3}{2}}\left(\mathbb{R}\right)$ by using the peakon traveling wave solution in [20]. Li-Yu-Zhu [21] established the ill-posedness of the Novikov Equation (4) in Besov spaces $B_{2,\infty }^s\left(\mathbb{R}\right)(s>\frac{7}{2})$. Subsequently, Wu-Li extended the above results to the following two-component Novikov system in [22].

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_t={\rho }_x{u}^2+\rho u{u}_x,t>0,x\in \mathbb{R},\hfill \\ m_t=3u_{x}um+u^2{m}_x-\rho {\left(u\rho \right)}_x,t>0,x\in \mathbb{R},\hfill \\ m=u-u_{xx},\hfill \\ \rho (0,x)={\rho }_0,u(0,x)=u_0,x\in \mathbb{R}.\hfill \end{array}\right.\end{array}$$

(5)

Wu-Li proved that the two-component Novikov system (5) is ill-posed in Besov spaces $B_{p,\infty }^{s-1}\left(\mathbb{R}\right)\times B_{p,\infty }^s\left(\mathbb{R}\right)(s>max\left\{1+\frac{1}{p},\frac{3}{2}\right\},1\le p,r\le \infty )$ by constructing the initial data $({\rho }_0,u_0)$ in [22]. The result of the nonuniform continuous dependence of the two-component Novikov system (5) was established by Wu-Cao in [23].

The inverse scattering approach is a powerful tool to study the CH equation and analyze its dynamics [24]. The inverse scattering transform (IST) problem of the CH equation is a complicated issue [25]. An algorithm was proposed by Constantin and Lenells in [26] and slightly modificated in [27] to solve the inverse scattering problem for the CH equation; while other approaches were subsequently introduced [28], this method turns out to be quite effective in giving a closed form to the CH solitary waves. The IST method for the DP Equation (3) was presented in [29]; the basic aspects of the IST such as the construction of fundamental analytic solutions, the formulation of a Riemann–Hilbert (RH) problem, and the implementation of the dressing method were introduced. The authors in [30] developed the IST method for the Novikov Equation (4) in the case of nonzero constant background; the approach was based on the analysis of an associated RH problem, which in this case was a $3\times 3$ matrix problem. A new IST method corresponding to a RH problem was also formulated for the two-component generalization of the CH equation [31].

From the above discussion, it can be seen that the nonuniform dependence and well-posedness results of the GX system (2), DP Equation (3), Novikov equation (4), and the two-component Novikov system (5), have been established in Sobolev spaces and Besov spaces. However, the ill-posedness for the three-component Novikov system (1) has yet to be systematically addressed. Using the ideas in [22,32,33], based on the existing results, we investigate ill-posedness of the solution for the system (1) in Besov spaces.

To facilitate the proof of the main result, we rewrite (1) into the following equivalent nonlocal form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_t=uv{\rho }_x+\rho v{u}_x+\rho u{v}_x,t>0,x\in \mathbb{R},\hfill \\ u_t=uv{u}_x+P_1(u,v)+R_1(u,\rho ),t>0,x\in \mathbb{R},\hfill \\ v_t=uv{v}_x+P_2(u,v)+R_2(v,\rho ),t>0,x\in \mathbb{R},\hfill \\ \rho (0,x)={\rho }_0\left(x\right),u(0,x)=u_0\left(x\right),v(0,x)=v_0\left(x\right),x\in \mathbb{R},\hfill \end{array}\right.\end{array}$$

(6)

where

$$\begin{array}{cc}& P_1(u,v)=P_{11}(u,v)+P_{12}(u,v)+P_{13}(u,v)+P_{14}(u,v),\hfill \\ & P_2(u,v)=P_{21}(u,v)+P_{22}(u,v)+P_{23}(u,v)+P_{24}(u,v),\hfill \\ & P_{11}(u,v)=\mathcal{P}\ast \left(3u{u}_{x}v\right),P_{12}(u,v)=\mathcal{P}\ast \left(u_x^2{v}_x\right),P_{13}(u,v)=\mathcal{P}\ast \left(u{u}_{xx}{v}_x\right),P_{14}(u,v)={\partial }_x\mathcal{P}\ast \left(u{u}_x{v}_x\right),\hfill \\ & P_{21}(u,v)=\mathcal{P}\ast \left(3uv{v}_x\right),P_{22}(u,v)=\mathcal{P}\ast \left(u_x{v}_x^2\right),P_{23}(u,v)=\mathcal{P}\ast \left(u_{x}v{v}_{xx}\right),P_{24}(u,v)={\partial }_x\mathcal{P}\ast \left(u_{x}v{v}_x\right),\hfill \\ & R_1(u,\rho )=\mathcal{P}\ast \left(u{\rho }^2\right),R_2(v,\rho )=-\mathcal{P}\ast \left({\rho }^{2}v\right).\hfill \end{array}$$

where we denote by ∗ the convolution, for all $f\in L^2$,

$$\mathcal{P}\ast f={(1-{\partial }_x^2)}^{-1}f,\mathcal{P}\left(x\right)=\frac{1}{2}{e}^{-\left|x\right|}.$$

We can now state our main result as follows.

Theorem 1.

Let $\sigma >max\left\{3+\frac{1}{p},\frac{7}{2}\right\},1\le p\le \infty$. Then the three-component Novikov system (1) is ill-posed in the Besov spaces $B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$. More precisely, there exists $({\rho }_0,u_0,v_0)\in B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$ and a positive constant ${\epsilon }_0>0$ such that the data-to-solution map $({\rho }_0,u_0,v_0)⟼(\rho ,u,v)$ of the Cauchy problem (6) satisfies

$$\underset{t\to 0^{+}}{lim\; sup}(\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)}+\parallel u-u_0{\parallel }_{B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)}+\parallel v-v_0{\parallel }_{B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)})\ge {\epsilon }_0.$$

Remark 1.

Theorem 1 demonstrates the ill-posedness of the three-component Novikov system in $B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$. More precisely, there exists initial data $({\rho }_0,u_0,v_0)$$\in B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$ such that corresponding solution to the Cauchy problem (6) that starts from $({\rho }_0,u_0,v_0)$ does not converge back to $({\rho }_0,u_0,v_0)$ in the metric of $B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$ as time goes to zero.

This paper is structured as follows. In Section 2, we list some notations and known results and recall some lemmas which will be used in the sequel. In Section 3, by establishing some technical lemmas and propositions, we present the proof of Theorem 1.

2. Preliminaries

In this section, we list some basic concepts and useful lemmas, which will be frequently used in proving our main results.

Denote $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ by the Fourier transform and the Fourier inverse transform, respectively, as follows:

$$\begin{array}{cc}\hfill & \mathcal{F}u\left(\xi \right)=\widehat{u}\left(\xi \right)={\int }_{{\mathbb{R}}^d}{e}^{-ix\xi }u\left(x\right)\mathrm{d}x,\hfill \\ \hfill & u\left(x\right)={\mathcal{F}}^{-1}\widehat{u}\left(x\right)=\frac{1}{2\pi }{\int }_{{\mathbb{R}}^d}{e}^{ix\xi }\widehat{u}\left(\xi \right)\mathrm{d}\xi .\hfill \end{array}$$

For any $u\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right)$ and all $j\in \mathbb{Z}$, define ${\mathsf{\Delta }}_{j}u=0$ for $j\le -2$; ${\mathsf{\Delta }}_{-1}u={\mathcal{F}}^{-1}\left(\chi \mathcal{F}u\right)$; ${\mathsf{\Delta }}_{j}u={\mathcal{F}}^{-1}\left(\phi (2^{-j}·)\mathcal{F}u\right)$ for $j\ge 0$; and $S_{j}u={\sum }_{j^{\prime }<j}{\mathsf{\Delta }}_{j^{\prime }}u$.

Let $s\in \mathbb{R},1\le p,r\le \infty$. We define the nonhomogeneous Besov space $B_{p,r}^s\left({\mathbb{R}}^d\right)$ as

$$B_{p,r}^s=B_{p,r}^s\left({\mathbb{R}}^d\right)=\left\{u\in S^{\prime }\left({\mathbb{R}}^d\right):{\parallel u\parallel }_{B_{p,r}^s}={∥(2^{js}\parallel {\mathsf{\Delta }}_{j}u{{\parallel }_{L^p})}_j∥}_{l^r\left(\mathbb{Z}\right)}<\infty \right\}.$$

Lemma 1

(See [34]). The following estimates hold:

(1) 

For any $s>0$ and any $p,r\in [1,\infty ]$, the space $L^{\infty }\cap B_{p,r}^s$ is an algebra and a constant $C=C\left(s\right)$ exists such that

$${\parallel uv\parallel }_{B_{p,r}^s}\le {C(\parallel u\parallel }_{L^{\infty }}{\parallel v\parallel }_{B_{p,r}^s}+{\parallel u\parallel }_{B_{p,r}^s}{\parallel v\parallel }_{L^{\infty }}).$$

(2) 

If $1\le p,r\le \infty ,s_1\le s_2,s_2>\frac{1}{p}$ ($s_2\ge \frac{1}{p}$ if $r=1$) and $s_1+s_2>max\{0,\frac{2}{p}-1\}$, there exists $C=C(s_1,s_2,p,r)$, such that

$${\parallel uv\parallel }_{B_{p,r}^{s_1}}\le {C\parallel u\parallel }_{B_{p,r}^{s_1}}{\parallel v\parallel }_{B_{p,r}^{s_2}}.$$

Lemma 2

(See [34,35]). Let $1\le p,r\le \infty ,\theta >-min(\frac{1}{p},\frac{1}{p^{\prime }})$, Then there exists a constant C such that for all solutions $f\in L^{\infty }(0,T;B_{p,r}^{\theta })$ of the following problem:

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

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. $t\in [0,T]$,

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

or

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

with

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

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

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

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

(See [22]). Let $s>0,1\le p\le \infty$, then we have

$${∥2^{js}{\parallel [{\mathsf{\Delta }}_j,u]{\partial }_{x}v\parallel }_{L^p}∥}_{L^{\infty }}\le C(\parallel {\partial }_x{u\parallel }_{L^{\infty }}{\parallel v\parallel }_{B_{p,r}^s}+\parallel {\partial }_x{v\parallel }_{L^{\infty }}{\parallel u\parallel }_{B_{p,r}^s}),$$

with

$$[{\mathsf{\Delta }}_j,u]{\partial }_{x}v={\mathsf{\Delta }}_j\left(u{\partial }_{x}v\right)-u{\mathsf{\Delta }}_j\left({\partial }_{x}v\right).$$

Next we present the local well-posedness result of system (1) which was given in [2].

Lemma 4

(See [2]). Let $\sigma >max\left\{2+\frac{1}{p},\frac{5}{2}\right\},1\le p\le \infty ,1\le r<\infty ,$ and $({\rho }_0,u_0,v_0)\in B_{p,r}^{s-1}$$\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right).$ Then, there exists some $T>0$ such that system (1) has a unique solution $(\rho ,u,v)$ in $C([0,T];B_{p,r}^{s-1}\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right)\times B_{p,r}^s\left(\mathbb{R}\right))\bigcap C^1([0,T];B_{p,r}^{s-2}\left(\mathbb{R}\right)\times B_{p,r}^{s-1}\left(\mathbb{R}\right)\times B_{p,r}^{s-1}\left(\mathbb{R}\right)).$

Let $\widehat{\varphi }\in {\mathcal{C}}_0^{\infty }\left(\mathbb{R}\right)$ be an even, real-valued and non-negative function and satisfy

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

Define the function $g_n\left(x\right)$ by

$$g_n\left(x\right):=\varphi \left(x\right)cos\left(\frac{17}{12}{2}^{n}x\right),n\ge 2.$$

Easy computations give that

$$\mathrm{supp}\varphi (·)\widehat{cos(\frac{17}{12}}{2}^n·)\subset \left\{\xi :-\frac{1}{2}+\lambda 2^n\le \left|\xi \right|\le \frac{1}{2}+\lambda 2^n\right\},$$

and

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_j\left(g_n\right)=\left\{\begin{array}{cc}{g}_n,\hfill & \mathrm{if}j=n,\hfill \\ 0,\hfill & \mathrm{if}j\ne n.\hfill \end{array}\right.\end{array}$$

(7)

3. Proof of Theorem 1

In this section, we prove Theorem 1. Similar to the discussion in [21], the following lemma can be illustrated.

Lemma 5.

Let $l\ge 4,n\in {\mathbb{N}}^{+}$, and define the function $h_{m,n}^l\left(x\right)$ by

$$h_{m,n}^l\left(x\right):=\varphi \left(x\right)cos\left(\frac{17}{12}(2^{ln}\pm 2^{lm})x\right),0\le m\le n-1.$$

Then we have

$${\mathsf{\Delta }}_j\left(h_{m,n}^l\right)=\left\{\begin{array}{cc}{h}_{m,n}^l,\hfill & \mathrm{if}j=ln,\hfill \\ 0,\hfill & \mathrm{if}j\ne ln.\hfill \end{array}\right.$$

The proof of the above lemma is similar to that of [21], which is omitted here.

For $1\le p,r\le \infty$, define the initial data $({\rho }_0,u_0,v_0)$ as

$$\begin{array}{cc}\hfill u_0\left(x\right)=v_0\left(x\right):=& \sum _{n=0}^{\infty }{2}^{-ln\sigma }\varphi \left(x\right)cos\left(\frac{17}{12}{2}^{ln}x\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\rho }_0\left(x\right):=& \sum _{n=0}^{\infty }{2}^{-ln(\sigma -1)}\varphi \left(x\right)cos\left(\frac{17}{12}{2}^{ln}x\right).\hfill \end{array}$$

(9)

In order to prove Theorem 1, we first give some estimation results.

Lemma 6.

Let $\sigma >0$. Then for the above constructed initial data $({\rho }_0,u_0,v_0)$, we have

$$\begin{array}{cc}& {\parallel u_0{v}_0{\partial }_x{\mathsf{\Delta }}_{ln}{u}_0\parallel }_{L^p}\ge C{2}^{-ln(\sigma -1)},\hfill \end{array}$$

(10)

$$\begin{array}{cc}& {\parallel u_0{v}_0{\partial }_x{\mathsf{\Delta }}_{ln}{v}_0\parallel }_{L^p}\ge C{2}^{-ln(\sigma -1)},\hfill \end{array}$$

(11)

$$\begin{array}{cc}& {\parallel u_0{v}_0{\partial }_x{\mathsf{\Delta }}_{ln}{\rho }_0\parallel }_{L^p}\ge C{2}^{-ln(\sigma -2)},\hfill \end{array}$$

(12)

for some large enough n.

Proof. 

We just show (10) here, since (11) and (12) can be performed in a similar way. Firstly, according to Lemma 5, we derive that

$${\mathsf{\Delta }}_{ln}{u}_0\left(x\right)=2^{-ln\sigma }\varphi \left(x\right)cos\left(\frac{17}{12}{2}^{ln}x\right),$$

therefore,

$${\partial }_x{\mathsf{\Delta }}_{ln}{u}_0\left(x\right)=2^{-ln\sigma }{\varphi }^{\prime }\left(x\right)cos\left(\frac{17}{12}{2}^{ln}x\right)-\frac{17}{12}{2}^{-ln(\sigma -1)}\varphi \left(x\right)sin\left(\frac{17}{12}{2}^{ln}x\right).$$

Thus, we have

$$u_0{v}_0{\partial }_x{\mathsf{\Delta }}_{ln}{u}_0\left(x\right)=2^{-ln\sigma }{u}_0{v}_0{\varphi }^{\prime }\left(x\right)cos\left(\frac{17}{12}{2}^{ln}x\right)-\frac{17}{12}{2}^{-ln(\sigma -1)}{u}_0{v}_0\varphi \left(x\right)sin\left(\frac{17}{12}{2}^{ln}x\right).$$

Since $u_0\left(x\right)$ and $v_0\left(x\right)$ are real-valued continuous functions on $\mathbb{R}$, then there exists some $\delta >0$, such that

$$\begin{array}{c}\hfill |u_0\left(x\right)v_0\left(x\right)|=|u_0^2\left(x\right)|\ge \frac{1}{2}\left|u^2\left(0\right)\right|=\frac{1}{2}{\left(\sum _{n=0}^{\infty }{2}^{-ln\sigma }\varphi \left(0\right)\right)}^2=\frac{2^{l\sigma -1}{\varphi }^2\left(0\right)}{{(2^{l\sigma }-1)}^2}.\end{array}$$

(13)

Thus, we have from (13)

$$\begin{array}{cc}\hfill \parallel u_0{v}_0{\partial }_x{\mathsf{\Delta }}_{ln}{u}_0{\parallel }_{L_p}\ge & C{2}^{-ln(\sigma -1)}\parallel \varphi \left(x\right)sin\left(\frac{17}{12}{2}^{ln}x\right){\parallel }_{L^p\left(B_{\delta }\left(0\right)\right)}\hfill \\ & -C_1{2}^{-ln\sigma }\parallel {\varphi }^{\prime }\left(x\right)cos\left(\frac{17}{12}{2}^{ln}x\right){\parallel }_{L^p\left(B_{\delta }\left(0\right)\right)}\hfill \\ \hfill \ge & (C{2}^{ln}-C_1)2^{-ln\sigma }.\hfill \end{array}$$

We choose n large enough such that $C_1<\frac{C}{2}{2}^{ln}$ and then finish the proof of (10). □

Proposition 1.

Let $s=\sigma -2,({\rho }_0,u_0,v_0)\in B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$ with $\sigma >$$max\left\{3+\frac{1}{p},\frac{7}{2}\right\},1\le p\le \infty$. Assume that $(\rho ,u,v)\in L^{\infty }\left(0,T;B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\right)$ is the solution to the Cauchy problem (6); we have

$$\begin{array}{cc}\hfill \parallel \rho \left(t\right)-{\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}\hfill \\ & +\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \parallel u\left(t\right)-u_0{\parallel }_{B_{p,\infty }^{s-1}}\le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}\hfill \\ & +\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel v_0{\parallel }_{B_{p,\infty }^s}),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \parallel v\left(t\right)-v_0{\parallel }_{B_{p,\infty }^{s-1}}\le & Ct(\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}\parallel u_0{\parallel }_{B_{p,\infty }^s}\hfill \\ & +\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel u_0{\parallel }_{B_{p,\infty }^s}),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \parallel \rho \left(t\right)-{\rho }_0{\parallel }_{B_{p,\infty }^s}\le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s+1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s+1}}\hfill \\ & +\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}{\parallel v_0\parallel }_{B_{p,\infty }^s}\hfill \\ & +\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \parallel u\left(t\right)-u_0{\parallel }_{B_{p,\infty }^s}\le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^s}^2\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \parallel v\left(t\right)-v_0{\parallel }_{B_{p,\infty }^s}\le & Ct(\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel v_0{\parallel }_{B_{p,\infty }^s}^2\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ & +\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}\parallel u_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \parallel u\left(t\right)-u_0{\parallel }_{B_{p,\infty }^{s+1}}\le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^{s+2}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s+1}}+\parallel u_0{\parallel }_{B_{p,\infty }^s}^2{\parallel v_0\parallel }_{B_{p,\infty }^s}\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \parallel v\left(t\right)-v_0{\parallel }_{B_{p,\infty }^{s+1}}\le & Ct(\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s+2}}\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel v_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ & +\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}+\parallel v_0{\parallel }_{B_{p,\infty }^s}^2{\parallel u_0\parallel }_{B_{p,\infty }^s}\hfill \\ & +\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}).\hfill \end{array}$$

(21)

Proof. 

For $\delta >0$, according to the local existence result [33], the Cauchy problem (6) has a unique solution:

$$(\rho ,u,v)\in L^{\infty }(0,T;B_{p,\infty }^{\delta }\left(\mathbb{R}\right)\times B_{p,\infty }^{\delta }\left(\mathbb{R}\right)\times B_{p,\infty }^{\delta }\left(\mathbb{R}\right)),$$

and satisfies the following estimate:

$$\begin{array}{c}\hfill \underset{0\le t\le T}{sup}{(\parallel \rho \left(t\right)\parallel }_{B_{p,\infty }^{\delta }}+{\parallel u\left(t\right)\parallel }_{B_{p,\infty }^{\delta }}+{\parallel v\left(t\right)\parallel }_{B_{p,\infty }^{\delta }})\le C(\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{\delta }}+\parallel u_0{\parallel }_{B_{p,\infty }^{\delta }}+\parallel v_0{\parallel }_{B_{p,\infty }^{\delta }}).\end{array}$$

(22)

Using the differential mean value theorem, the Minkowski inequality, (1) and (2) of Lemma 1, we deduce that

$$\begin{array}{cc}& \parallel \rho \left(t\right)-{\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\le {\int }_0^t{\parallel {\partial }_{\tau }\rho \parallel }_{B_{p,\infty }^{s-1}}d\tau \hfill \\ \hfill \le & {\int }_0^t(\parallel uv{\partial }_x{\rho \parallel }_{B_{p,\infty }^{s-1}}+\parallel u\rho {\partial }_x{v\parallel }_{B_{p,\infty }^{s-1}}+\parallel \rho v{\partial }_{x}u{\parallel }_{B_{p,\infty }^{s-1}})d\tau \hfill \\ \hfill \le & {\int }_0^t{(\parallel u\parallel }_{L_{\infty }}\parallel v{\partial }_x{\rho \parallel }_{B_{p,\infty }^{s-1}}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}\parallel v{\partial }_x{\rho \parallel }_{L_{\infty }}+{\parallel u\parallel }_{L_{\infty }}{\parallel \rho {\partial }_{x}v\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ & +{\parallel u\parallel }_{B_{p,\infty }^{s-1}}\parallel \rho {\partial }_x{v\parallel }_{L_{\infty }}+{\parallel \rho \parallel }_{L_{\infty }}\parallel v{\partial }_x{u\parallel }_{B_{p,\infty }^{s-1}}+{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}\parallel {\partial }_{x}uv{\parallel }_{L_{\infty }})d\tau \hfill \\ \hfill \le & {\int }_0^t{(\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^s}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^s}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^s}\hfill \\ & +{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^s}+{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s}+{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^{s-1}})d\tau \hfill \\ \hfill \le & C{\int }_0^t{(\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^s}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^s}+{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s})d\tau \hfill \\ \hfill \le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}).\hfill \end{array}$$

Thus, we complete the proof of (14). Next we will present the proof of (15).

$$\begin{array}{cc}& \parallel u\left(t\right)-u_0{\parallel }_{B_{p,\infty }^{s-1}}\le {\int }_0^t{\parallel {\partial }_{\tau }u\parallel }_{B_{p,\infty }^{s-1}}d\tau \hfill \\ \hfill \le & {\int }_0^t(\parallel u{\partial }_x{uv\parallel }_{B_{p,\infty }^{s-1}}+\parallel u{\partial }_x{uv\parallel }_{B_{p,\infty }^{s-3}}+\parallel u_x^2{v}_x{\parallel }_{B_{p,\infty }^{s-3}}+\parallel u{\partial }_x^{2}u{\partial }_x{v\parallel }_{B_{p,\infty }^{s-3}}+\parallel {\rho }^2{u\parallel }_{B_{p,\infty }^{s-3}}+\parallel u{\partial }_{x}u{\partial }_{x}v{\parallel }_{B_{p,\infty }^{s-2}})d\tau \hfill \\ \hfill \le & C{\int }_0^t{(\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^s}+{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}^2{\parallel u\parallel }_{B_{p,\infty }^{s-1}}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}^2{\parallel v\parallel }_{B_{p,\infty }^s})d\tau \hfill \\ \hfill \le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel v_0{\parallel }_{B_{p,\infty }^s}).\hfill \end{array}$$

Thus, (15) is proved and (16) can be proved in a similar manner. Next, we will give the proof of (17).

$$\begin{array}{cc}& \parallel \rho \left(t\right)-{\rho }_0{\parallel }_{B_{p,\infty }^s}\le {\int }_0^t{\parallel {\partial }_{\tau }\rho \parallel }_{B_{p,\infty }^s}d\tau \hfill \\ \hfill \le & C{\int }_0^t{(\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^{s+1}}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s+1}}+{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^{s+1}}\hfill \\ & +{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^s}+{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^s}+{\parallel \rho \parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s})d\tau ,\hfill \\ \hfill \le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s+1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s+1}}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}).\hfill \end{array}$$

Thereafter, we shall present the proof of (18).

$$\begin{array}{cc}& \parallel u\left(t\right)-u_0{\parallel }_{B_{p,\infty }^s}\le {\int }_0^t{\parallel {\partial }_{\tau }u\parallel }_{B_{p,\infty }^s}d\tau \hfill \\ \hfill \le & {\int }_0^t(\parallel u{\partial }_x{uv\parallel }_{B_{p,\infty }^s}+\parallel u{\partial }_x{uv\parallel }_{B_{p,\infty }^{s-2}}+\parallel {\left({\partial }_{x}u\right)}^2{\partial }_x{v\parallel }_{B_{p,\infty }^{s-2}}+\parallel u{\partial }_x^{2}u{\partial }_x{v\parallel }_{B_{p,\infty }^{s-2}}+\parallel {\rho }^2{u\parallel }_{B_{p,\infty }^{s-2}}+\parallel u{\partial }_{x}u{\partial }_{x}v{\parallel }_{B_{p,\infty }^{s-1}})d\tau \hfill \\ \hfill \le & C{\int }_0^t{(\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}+{\parallel u\parallel }_{B_{p,\infty }^s}^2{\parallel v\parallel }_{B_{p,\infty }^{s-1}}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^s}+{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}^2{\parallel u\parallel }_{B_{p,\infty }^{s-1}})d\tau \hfill \\ \hfill \le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^s}^2\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}).\hfill \end{array}$$

Similarly, we can obtain (19). Following this, we will furnish the evidence for (20).

$$\begin{array}{cc}& \parallel u\left(t\right)-u_0{\parallel }_{B_{p,\infty }^{s+1}}\le {\int }_0^t{\parallel {\partial }_{\tau }u\parallel }_{B_{p,\infty }^{s+1}}d\tau \hfill \\ \hfill \le & {\int }_0^t(\parallel u{\partial }_x{uv\parallel }_{B_{p,\infty }^{s+1}}+\parallel u{\partial }_x{uv\parallel }_{B_{p,\infty }^{s-1}}+\parallel {\left({\partial }_{x}u\right)}^2{\partial }_x{v\parallel }_{B_{p,\infty }^{s-1}}+\parallel u{\partial }_x^{2}u{\partial }_x{v\parallel }_{B_{p,\infty }^{s-1}}+\parallel {\rho }^2{u\parallel }_{B_{p,\infty }^{s-1}}+\parallel u{\partial }_{x}u{\partial }_{x}v{\parallel }_{B_{p,\infty }^s})d\tau \hfill \\ \hfill \le & C{\int }_0^t{(\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^{s+2}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}+{\parallel u\parallel }_{B_{p,\infty }^{s+1}}{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^{s+1}}\hfill \\ & +{\parallel u\parallel }_{B_{p,\infty }^s}^2{\parallel v\parallel }_{B_{p,\infty }^s}+{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^{s+1}}{\parallel v\parallel }_{B_{p,\infty }^s}+{\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}^2{\parallel u\parallel }_{B_{p,\infty }^{s-1}})d\tau \hfill \\ \hfill \le & Ct(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^{s+2}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s+1}}\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^s}^2\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}^2\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}).\hfill \end{array}$$

Equation (21) can be verified similarly. Thus, we complete the proof of Proposition 1. □

Proposition 2.

Let $s=\sigma -2,({\rho }_0,u_0,v_0)\in B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$ with $\sigma >$$max\left\{3+\frac{1}{p},\frac{7}{2}\right\},1\le p\le \infty$. Assume that $(\rho ,u,v)\in L^{\infty }\left(0,T;B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\right)$ is the solution of the Cauchy problem (6); we have

$$\begin{array}{cc}\hfill \parallel \rho \left(t\right)-{\rho }_0-t{\rho }_0{\parallel }_{B_{p,r}^{s-1}}\le & C{t}^2(\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v_0\parallel }_{B_{p,\infty }^s}\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^s}{\parallel {\rho }_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ & +\parallel v_0{\parallel }_{B_{p,\infty }^s}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \parallel u\left(t\right)-u_0-t{\mathbf{u}}_0{\parallel }_{B_{p,r}^s}\le & C{t}^2(\parallel u_0{\parallel }_{B_{p,\infty }^s}^2+\parallel v_0{\parallel }_{B_{p,\infty }^s}^2+{\parallel {\rho }_0\parallel }_{B_{p,\infty }^s}^2\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s+1}}),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \parallel v\left(t\right)-v_0-t{\mathbf{u}}_0{\parallel }_{B_{p,r}^s}\le & C{t}^2(\parallel u_0{\parallel }_{B_{p,\infty }^s}^2+\parallel v_0{\parallel }_{B_{p,\infty }^s}^2+{\parallel {\rho }_0\parallel }_{B_{p,\infty }^s}^2\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s+1}}).\hfill \end{array}$$

(25)

where

$$\begin{array}{cc}& {\rho }_0=u_0{v}_0{\partial }_x{\rho }_0+u_0{\rho }_0{\partial }_x{v}_0+{\rho }_0{v}_0{\partial }_x{u}_0,\hfill \\ & {\mathbf{u}}_0=u_0{v}_0{\partial }_x{u}_0+P_1(u_0,v_0)+R_1(u_0,{\rho }_0),\hfill \\ & {\mathbf{v}}_0=u_0{v}_0{\partial }_x{v}_0+P_2(u_0,v_0)+R_2(v_0,{\rho }_0).\hfill \end{array}$$

Proof. 

For simplicity, denote

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\rho =\rho \left(t\right)-{\rho }_0-t{\rho }_0,\hfill \\ \mathbf{u}=u\left(t\right)-u_0-t{\mathbf{u}}_0,\hfill \\ \mathbf{v}=v\left(t\right)-v_0-t{\mathbf{v}}_0.\hfill \end{array}\right.\end{array}$$

Combining the differential mean value theorem and the Minkowski inequality, we arrive at

$$\begin{array}{cc}& \parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\le {\int }_0^t{\parallel {\partial }_{\tau }\rho -{\rho }_0\parallel }_{B_{p,\infty }^{s-1}}d\tau \hfill \\ \hfill \le & {\int }_0^t(\parallel uv{\partial }_x\rho -u_0{v}_0{\partial }_x{\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u\rho {\partial }_{x}v-u_0{v}_0{\partial }_x{\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel \rho v{\partial }_{x}u-{\rho }_0{v}_0{\partial }_x{u}_0{\parallel }_{B_{p,\infty }^{s-1}})d\tau \hfill \\ \hfill \stackrel{\mathsf{\Delta }}{=}& {\int }_0^t(I_1\left(\tau \right)+I_2\left(\tau \right)+I_3\left(\tau \right))d\tau .\hfill \end{array}$$

(26)

Note that

$$\begin{array}{cc}\hfill uv{\partial }_x\rho -u_0{v}_0{\partial }_x{\rho }_0& =uv{\partial }_x\rho -u_0{v}_0{\partial }_x\rho +u_0{v}_0{\partial }_x\rho -u_0{v}_0{\partial }_x{\rho }_0\hfill \\ & =(uv-u_0{v}_0){\partial }_x\rho +u_0{v}_0{\partial }_x(\rho -{\rho }_0)\hfill \\ & =(uv-u_{0}v+u_{0}v-u_0{v}_0){\partial }_x\rho +u_0{v}_0{\partial }_x(\rho -{\rho }_0)\hfill \\ & =(u-u_0)v{\partial }_x\rho +u_0(v-v_0){\partial }_x\rho +u_0{v}_0{\partial }_x(\rho -{\rho }_0).\hfill \end{array}$$

We can deduce

$$\begin{array}{cc}\hfill I_1\left(\tau \right)\le & \parallel (u-u_0)v{\partial }_x{\rho \parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0(v-v_0){\partial }_x{\rho \parallel }_{B_{p,\infty }^{s-1}}+{\parallel u_0{v}_0{\partial }_x(\rho -{\rho }_0)\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \stackrel{\mathsf{\Delta }}{=}& {\int }_0^t(I_{11}\left(\tau \right)+I_{12}\left(\tau \right)+I_{13}\left(\tau \right))d\tau ,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill I_{11}\left(\tau \right)\le & \parallel u-u_0{\parallel }_{L^{\infty }}\parallel v{\partial }_x{\rho \parallel }_{B_{p,\infty }^{s-1}}+\parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel {\partial }_x\rho \parallel }_{L^{\infty }}\hfill \\ \hfill \le & \parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}{(\parallel v\parallel }_{L^{\infty }}\parallel {\partial }_x{\rho \parallel }_{B_{p,\infty }^{s-1}}+{\parallel v\parallel }_{B_{p,\infty }^{s-1}}\parallel {\partial }_x\rho {\parallel }_{L^{\infty }})\hfill \\ & +\parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}{(\parallel v\parallel }_{L^{\infty }}\parallel {\partial }_x{\rho \parallel }_{B_{p,\infty }^{s-1}}+{\parallel v\parallel }_{B_{p,\infty }^{s-1}}\parallel {\partial }_x\rho {\parallel }_{L^{\infty }})\hfill \\ \hfill \le & \parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^s}\le C\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel u-u_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \le & C\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}{\parallel v_0\parallel }_{B_{p,\infty }^{s-1}}\tau .\hfill \end{array}$$

(28)

Analogously,

$$\begin{array}{cc}\hfill I_{12}\left(\tau \right)\le & \parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel \rho \parallel }_{B_{p,\infty }^s}\parallel v-v_0{\parallel }_{B_{p,\infty }^{s-1}}\le \parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel {\rho }_0\parallel }_{B_{p,\infty }^s}\tau ,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill I_{13}\left(\tau \right)\le & \parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^s}\le \parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v_0\parallel }_{B_{p,\infty }^{s-1}}\tau .\hfill \end{array}$$

(30)

For the second term $I_2\left(\tau \right)$, since

$$\begin{array}{cc}\hfill u\rho {\partial }_{x}v-u_0{\rho }_0{\partial }_x{v}_0=& u\rho {\partial }_{x}v-u_0{\rho }_0{\partial }_{x}v+u_0{\rho }_0{\partial }_{x}v-u_0{\rho }_0{\partial }_x{v}_0\hfill \\ \hfill =& (u\rho -u_0{\rho }_0){\partial }_{x}v+u_0{\rho }_0{\partial }_x(v-v_0)\hfill \\ \hfill =& (u-u_0)\rho {\partial }_{x}v+u_0(\rho -{\rho }_0){\partial }_{x}v+u_0{\rho }_0{\partial }_x(v-v_0),\hfill \end{array}$$

there holds

$$\begin{array}{cc}\hfill I_2\left(\tau \right)\le & {\parallel \rho \parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^s}\parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^s}{\parallel \rho -{\rho }_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v-v_0\parallel }_{B_{p,\infty }^s}\hfill \\ \hfill \le & C(\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}})\tau .\hfill \end{array}$$

(31)

For the third term $I_3\left(\tau \right)$, since

$$\rho v{\partial }_{x}u-{\rho }_0{v}_0{\partial }_x{u}_0=(\rho -{\rho }_0)v{\partial }_{x}u+{\rho }_0(v-v_0){\partial }_{x}u+{\rho }_0{v}_0{\partial }_x(u-u_0).$$

We obtain

$$\begin{array}{cc}\hfill I_3\left(\tau \right)\le & {\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v-v_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ & +\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel u-u_0\parallel }_{B_{p,\infty }^s}\hfill \\ \hfill \le & C(\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}})\tau .\hfill \end{array}$$

(32)

Substituting (28)–(30) into (26), we derive

$$\begin{array}{cc}\hfill \parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}\le & C{t}^2(\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel {\rho }_0\parallel }_{B_{p,\infty }^s}\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\rho }_0{|}_{B_{p,\infty }^s}+\parallel v_0{\parallel }_{B_{p,\infty }^s}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}).\hfill \end{array}$$

Thus, we finish the proof of (23). Next, we intend to establish the proof of (24).

$$\begin{array}{cc}\hfill \parallel {\mathbf{u}}_0{\parallel }_{B_{p,\infty }^s}\le & {\int }_0^t{\parallel {\partial }_{\tau }u-{\mathbf{u}}_0\parallel }_{B_{p,\infty }^s}d\tau \hfill \\ \hfill \le & {\int }_0^t\parallel u{u}_{x}v-u_0{\partial }_x{u}_0{v}_0{\parallel }_{B_{p,\infty }^s}d\tau +{\int }_0^t{\parallel P_1(u,v)-P_1(u_0,v_0)\parallel }_{B_{p,\infty }^s}d\tau \hfill \\ & +{\int }_0^t{\parallel R_1(u,\rho )-R_1(u_0,{\rho }_0)\parallel }_{B_{p,\infty }^s}d\tau \hfill \\ \hfill \stackrel{\mathsf{\Delta }}{=}& {\int }_0^t(J_1\left(\tau \right)+J_2\left(\tau \right)+J_3\left(\tau \right))d\tau .\hfill \end{array}$$

(33)

For the first term $J_1\left(\tau \right)$, since

$$\begin{array}{cc}\hfill uv{\partial }_{x}u-u_0{v}_0{\partial }_x{u}_0=& uv{\partial }_{x}u-u_{0}v{\partial }_{x}u+u_{0}v{\partial }_{x}u-u_0{v}_0{\partial }_x{u}_0\hfill \\ \hfill =& (u-u_0)v{\partial }_{x}u-u_0(v{\partial }_{x}u-v{\partial }_x{u}_0+v{\partial }_x{u}_0-v_0{\partial }_x{u}_0)\hfill \\ \hfill =& (u-u_0)v{\partial }_{x}u-u_{0}v{\partial }_x(u-u_0)-u_0(v-v_0){\partial }_x{u}_0.\hfill \end{array}$$

We can perform the following estimate:

$$\begin{array}{cc}\hfill J_1\left(\tau \right)\le & \parallel (u-u_0)v{\partial }_x{u\parallel }_{B_{p,\infty }^s}+\parallel u_{0}v{\partial }_x(u-u_0){\parallel }_{B_{p,\infty }^s}+{\parallel u_0(v-v_0){\partial }_x{u}_0\parallel }_{B_{p,\infty }^s}\hfill \\ \hfill \stackrel{\mathsf{\Delta }}{=}& {\int }_0^t(J_{11}\left(\tau \right)+J_{12}\left(\tau \right)+J_{13}\left(\tau \right))d\tau .\hfill \end{array}$$

(34)

By applying Lemma 1, one obtains

$$\begin{array}{cc}\hfill J_{11}\left(\tau \right)\le & \parallel u-u_0{\parallel }_{L^{\infty }}\parallel v{\partial }_x{u\parallel }_{B_{p,\infty }^s}+\parallel u{u}_o{\parallel }_{B_{p,\infty }^s}{\parallel v{\partial }_{x}u\parallel }_{L^{\infty }}\hfill \\ \hfill \le & \parallel u-u_0{\parallel }_{L^{\infty }}\parallel {\partial }_x{u\parallel }_{L^{\infty }}{\parallel v\parallel }_{B_{p,\infty }^s}+\parallel u-u_0{\parallel }_{L^{\infty }}\parallel {\partial }_x{u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{L^{\infty }}+\parallel u-u_0{\parallel }_{B_{p,\infty }^s}\parallel {\partial }_x{u\parallel }_{L^{\infty }}{\parallel v\parallel }_{L^{\infty }}\hfill \\ \hfill \le & {\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^s}\parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}+{\parallel u\parallel }_{B_{p,\infty }^{s+1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}\parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}+{\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel u-u_0\parallel }_{B_{p,\infty }^s}\hfill \\ \hfill \le & C(\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}})\tau ,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill J_{12}\left(\tau \right)\le & \parallel u_0{\parallel }_{L^{\infty }}\parallel v{\partial }_x(u-u_0){\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^s}{\parallel v{\partial }_x(u-u_0)\parallel }_{L^{\infty }}\hfill \\ \hfill \le & \parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\partial }_x(u-u_0){\parallel }_{L^{\infty }}{\parallel v\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel {\partial }_x(u-u_0){\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{L^{\infty }}+\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel u-u_0{\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{L^{\infty }}\hfill \\ \hfill \le & \parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^s}\parallel u-u_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^{s-1}}{\parallel u-u_0\parallel }_{B_{p,\infty }^{s+1}}\hfill \\ & +\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel u\parallel }_{B_{p,\infty }^{s-1}}{\parallel u-u_0\parallel }_{B_{p,\infty }^s}\hfill \\ \hfill \le & C\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v_0\parallel }_{B_{p,\infty }^s}\tau ,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill J_{13}\left(\tau \right)\le & \parallel u_0{\parallel }_{L^{\infty }}\parallel (v-v_0){\partial }_x{u}_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^s}{\parallel (v-v_0){\partial }_x{u}_0\parallel }_{L^{\infty }}\hfill \\ \hfill \le & \parallel u_0{\parallel }_{L^{\infty }}(\parallel {\partial }_x{u}_0{\parallel }_{L^{\infty }}\parallel v-v_0{\parallel }_{B_{p,\infty }^s}+\parallel {\partial }_x{u}_0{\parallel }_{B_{p,\infty }^s}\parallel v-v_0{\parallel }_{L^{\infty }})+\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel u_0{\parallel }_{B_{p,\infty }^s}{\parallel v-v_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \le & \parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v-v_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v-v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^s}^2{\parallel v-v_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \le & C(\parallel u_0{\parallel }_{B_{p,\infty }^s}^2+\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}})\tau .\hfill \end{array}$$

(37)

For the term $P_{12}(u,v)-P_{12}(u_0,v_0)$, it can be resolved into

$${\left({\partial }_{x}u\right)}^2{\partial }_{x}v-{\left({\partial }_x{u}_0\right)}^2{\partial }_x{v}_0=({\partial }_{x}u+{\partial }_x{u}_0){\partial }_x(u-u_0){\partial }_{x}v+{\left({\partial }_x{u}_0\right)}^2{\partial }_x(v-v_0).$$

We can find that

$$\begin{array}{cc}& \parallel {\left({\partial }_{x}u\right)}^2{\partial }_{x}v-{\left({\partial }_x{u}_0\right)}^2{\partial }_x{v}_0{\parallel }_{B_{p,\infty }^{s-2}}\hfill \\ \hfill \le & \parallel {\partial }_x(u+u_0){\partial }_x(u-u_0){\partial }_x{v\parallel }_{B_{p,\infty }^{s-2}}+{\parallel {\left({\partial }_x{u}_0\right)}^2{\partial }_x(v-v_0)\parallel }_{B_{p,\infty }^{s-2}}\hfill \\ \hfill \le & \parallel {\partial }_x(u+v){\partial }_x{v\parallel }_{B_{p,\infty }^{s-1}}\parallel {\partial }_x(u-u_0){\parallel }_{B_{p,\infty }^{s-2}}+\parallel {\left({\partial }_x{u}_0\right)}^2{\parallel }_{B_{p,\infty }^{s-1}}{\parallel {\partial }_x(v-v_0)\parallel }_{B_{p,\infty }^{s-2}}\hfill \\ \hfill \le & {(\parallel u\parallel }_{B_{p,\infty }^s}+{\parallel v\parallel }_{B_{p,\infty }^s}{)\parallel v\parallel }_{B_{p,\infty }^s}\parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^s}^2{\parallel v-v_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \le & C(\parallel u_0{\parallel }_{B_{p,\infty }^s}^2+\parallel v_0{\parallel }_{B_{p,\infty }^s}^2)\tau .\hfill \end{array}$$

(38)

For the term $P_{13}(u,v)-P_{13}(u_0,v_0)$, since

$$u{\partial }_x^{2}u{\partial }_{x}v-u_0{\partial }_x^2{u}_0{\partial }_x{v}_0=(u-u_0){\partial }_x^{2}u{\partial }_{x}v+u_0{\partial }_{x}v{\partial }_x^2(u-u_0)+u_0{\partial }_x^2{u}_0{\partial }_x(v-v_0),$$

we arrive at

$$\begin{array}{cc}& \parallel u{\partial }_x^{2}u{\partial }_{x}v-u_0{\partial }_x^2{u}_0{\partial }_x{v}_0{\parallel }_{B_{p,\infty }^{s-2}}\hfill \\ \hfill \le & \parallel (u-u_0){\partial }_x^{2}u{\partial }_x{v\parallel }_{B_{p,\infty }^{s-2}}+\parallel u_0{\partial }_{x}v{\partial }_x^2(u-u_0){\parallel }_{B_{p,\infty }^{s-2}}+{\parallel u_0{\partial }_x^2{u}_0{\partial }_x(v-v_0)\parallel }_{B_{p,\infty }^{s-2}}\hfill \\ \hfill \le & \parallel {\partial }_x^{2}u{\partial }_x{v\parallel }_{B_{p,\infty }^{s-2}}\parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\partial }_x{v\parallel }_{B_{p,\infty }^{s-1}}\parallel u-u_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\partial }_x^2{u}_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v-v_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \le & {\parallel u\parallel }_{B_{p,\infty }^s}{\parallel v\parallel }_{B_{p,\infty }^s}\parallel u-u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v\parallel }_{B_{p,\infty }^s}\parallel u-u_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}{\parallel v-v_0\parallel }_{B_{p,\infty }^{s+1}}\hfill \\ \hfill \le & C(\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s+1}})\tau .\hfill \end{array}$$

(39)

For the term $R_1(u,\rho )-R_1(u_0,{\rho }_0)$, it can be decomposed as

$${\rho }^{2}u-{\rho }_0^2{u}_0=(\rho +{\rho }_0)u(\rho -{\rho }_0)+{\rho }_0^2(u-u_0),$$

which gives us that

$$\begin{array}{cc}& \parallel {\rho }^{2}u-{\rho }_0^2{u}_0{\parallel }_{B_{p,\infty }^{s-2}}\hfill \\ \hfill \le & \parallel (\rho +{\rho }_0)(\rho -{\rho }_0){u\parallel }_{B_{p,\infty }^{s-2}}+{\parallel {\rho }_0^2(u-u_0)\parallel }_{B_{p,\infty }^{s-2}}\hfill \\ \hfill \le & \parallel (\rho +{\rho }_0){u\parallel }_{B_{p,\infty }^{s-2}}\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-2}}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel u-u_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \le & {(\parallel \rho \parallel }_{B_{p,\infty }^s}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}{)\parallel u\parallel }_{B_{p,\infty }^{s-1}}\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}^2{\parallel u-u_0\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \le & C(\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}^2)\tau .\hfill \end{array}$$

(40)

Finally, for the term $P_{14}(u,v)-P_{14}(u_0,v_0)$, since

$$u{\partial }_{x}u{\partial }_{x}v-u_0{\partial }_x{u}_0{\partial }_x{v}_0=(u-u_0){\partial }_{x}u{\partial }_{x}v+u_0{\partial }_{x}v{\partial }_x(u-u_0)+u_0{\partial }_x{u}_0{\partial }_x(v-v_0),$$

we can derive

$$\begin{array}{cc}& \parallel u{\partial }_{x}u{\partial }_{x}v-u_0{\partial }_x{u}_0{\partial }_x{v}_0{\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \le & \parallel (u-u_0){\partial }_{x}u{\partial }_x{v\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\partial }_{x}v{\partial }_x(u-u_0){\parallel }_{B_{p,\infty }^{s-1}}+{\parallel u_0{\partial }_x{u}_0{\partial }_x(v-v_0)\parallel }_{B_{p,\infty }^{s-1}}\hfill \\ \hfill \le & \parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s}\parallel u-u_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}{\parallel v-v_0\parallel }_{B_{p,\infty }^s}\hfill \\ \hfill \le & C(\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}\parallel u_0{\parallel }_{B_{p,\infty }^s}+\parallel u_0{\parallel }_{B_{p,\infty }^s}\parallel v_0{\parallel }_{B_{p,\infty }^s})\tau .\hfill \end{array}$$

(41)

Inserting (35)–(41) into (33), we obtain

$${\parallel \mathbf{u}\parallel }_{B_{p,\infty }^s}\le C{t}^2\left(\parallel u_0{\parallel }_{B_{p,\infty }^s}^2+\parallel v_0{\parallel }_{B_{p,\infty }^s}^2+\parallel {\rho }_0{\parallel }_{B_{p,\infty }^s}^2+\parallel u_0{\parallel }_{B_{p,\infty }^{s+1}}\parallel v_0{\parallel }_{B_{p,\infty }^{s-1}}+\parallel u_0{\parallel }_{B_{p,\infty }^{s-1}}{\parallel v_0\parallel }_{B_{p,\infty }^{s+1}}\right).$$

Thus, the proof of (24) is complete. Similarly, we can prove (25). □

With Lemma 5, Proposition 1, and Proposition 2 at hand, we will proceed to verify Theorem 1.

Proof of Theorem 1.

By the definition of the Besov norm, we have

$$\begin{array}{cc}\hfill \parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\ge & 2^{ln(\sigma -1)}\parallel {\mathsf{\Delta }}_{ln}(\rho -{\rho }_0){\parallel }_{L^p}=2^{ln(\sigma -1)}{\parallel {\mathsf{\Delta }}_{ln}(\mathbf{\rho }+t{\mathbf{\rho }}_0)\parallel }_{L^p}\hfill \\ \hfill \ge & t{2}^{ln(\sigma -1)}\parallel {\mathsf{\Delta }}_{ln}(u_0{v}_0{\partial }_x{\rho }_0+u_0{\rho }_0{\partial }_x{v}_0+{\rho }_0{v}_0{\partial }_x{u}_0){\parallel }_{L^p}-2^{ln(\sigma -1)}{\parallel {\mathsf{\Delta }}_{ln}\mathbf{\rho }\parallel }_{L^p}\hfill \\ \hfill \ge & t{2}^{ln(\sigma -1)}\parallel {\mathsf{\Delta }}_{ln}\left(u_0{v}_0{\partial }_x{\rho }_0\right){\parallel }_{L^p}-t{2}^{ln(\sigma -1)}{\parallel {\mathsf{\Delta }}_{ln}\left({\rho }_0{u}_0{\partial }_x{v}_0\right)\parallel }_{L^p}\hfill \\ & -t{2}^{ln(\sigma -1)}\parallel {\mathsf{\Delta }}_{ln}\left({\rho }_0{v}_0{\partial }_x{u}_0\right){\parallel }_{L^p}-2^{ln(\sigma -3)}·2^{2ln}{\parallel {\mathsf{\Delta }}_{ln}\mathbf{\rho }\parallel }_{L^p}\hfill \\ \hfill \ge & t{2}^{ln(\sigma -1)}\parallel {\mathsf{\Delta }}_{ln}\left(u_0{v}_0{\partial }_x{\rho }_0\right){\parallel }_{L^p}-t{\parallel {\rho }_0{u}_0{\partial }_x{v}_0\parallel }_{B_{p,\infty }^{\sigma -1}}\hfill \\ & -t\parallel {\rho }_0{v}_0{\partial }_x{u}_0{\parallel }_{B_{p,\infty }^{\sigma -1}}-2^{2ln}{\parallel \mathbf{\rho }\parallel }_{B_{p,\infty }^{\sigma -3}}.\hfill \end{array}$$

(42)

Since

$$\begin{array}{cc}\hfill {\mathsf{\Delta }}_{ln}\left(u_0{v}_0{\partial }_x{\rho }_0\right)=& {\mathsf{\Delta }}_{ln}\left(u_0{v}_0{\partial }_x{\rho }_0\right)-u_0{v}_0{\partial }_x{\mathsf{\Delta }}_{ln}{\rho }_0+u_0{v}_0{\partial }_x{\mathsf{\Delta }}_{ln}{\rho }_0\hfill \\ \hfill =& [{\mathsf{\Delta }}_{ln},u_0{v}_0{\partial }_x]{\rho }_0+u_0{v}_0{\partial }_x{\mathsf{\Delta }}_{ln}{\rho }_0.\hfill \end{array}$$

With the aid of Lemma 3, we infer that

$$\begin{array}{cc}\hfill \parallel 2^{ln(\sigma -1)}[{\mathsf{\Delta }}_{ln},u_0{v}_0{\partial }_x]{\rho }_0{\parallel }_{L^{\infty }}\le & C\parallel {\partial }_x\left(u_0{v}_0\right){\parallel }_{L^{\infty }}\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}+C\parallel {\partial }_x{\rho }_0{\parallel }_{L^{\infty }}{\parallel u_0{v}_0\parallel }_{B_{p,\infty }^{\sigma -1}}\hfill \\ \hfill \le & C\parallel u_0{\parallel }_{B_{p,\infty }^{\sigma }}\parallel v_0{\parallel }_{B_{p,\infty }^{\sigma }}{\parallel {\rho }_0\parallel }_{B_{p,\infty }^{\sigma -1}}\le C,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill \parallel {\rho }_0{u}_0{\partial }_x{v}_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\le & \parallel {\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\parallel u_0{\parallel }_{B_{p,\infty }^{\sigma -1}}{\parallel v_0\parallel }_{B_{p,\infty }^{\sigma }}\le C,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill \parallel {\rho }_0{v}_0{\partial }_x{u}_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\le & \parallel {\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\parallel v_0{\parallel }_{B_{p,\infty }^{\sigma -1}}{\parallel u_0\parallel }_{B_{p,\infty }^{\sigma }}\le C.\hfill \end{array}$$

(45)

Inserting (43)–(45) into (42), we obtain

$$\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\ge t·2^{ln(\sigma -1)}\parallel u_0{v}_0{\partial }_x{\mathsf{\Delta }}_{ln}{\rho }_0{\parallel }_{L^p}-C_{1}t-2^{2ln}{\parallel \mathbf{\rho }\parallel }_{B_{p,\infty }^{\sigma -3}}.$$

Making full use of (12) of Lemma 6, and (23) of Proposition 2, we draw the conclusion that

$$\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\ge C_{3}t·2^{ln}-C_{1}t-C_2{2}^{2ln}{t}^2.$$

For $l\ge 4$, choosing n large enough such that $C_3·2^{ln}>2C_1$, we have

$$\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\ge \frac{C_3}{2}t·2^{ln}-C_2{2}^{2ln}·t^2.$$

As time t tends to zero, picking t sufficiently small such that $t·2^n\approx {\delta }_0<\frac{C_3}{4C_2}$, one can easily check that

$$\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\ge \frac{C_3}{4}{\delta }_0.$$

Similarly, by the definition of the Besov norm, we have

$$\begin{array}{cc}\hfill \parallel u-u_0{\parallel }_{B_{p,\infty }^{\sigma }}\ge & 2^{ln\sigma }\parallel {\mathsf{\Delta }}_{ln}(u-u_0){\parallel }_{L^p}=2^{ln\sigma }{\parallel {\mathsf{\Delta }}_{ln}(\mathbf{u}+t{\mathbf{u}}_0)\parallel }_{L^p}\hfill \\ \hfill \ge & t·2^{ln\sigma }\parallel {\mathsf{\Delta }}_{ln}\{u_0{v}_0{\partial }_x{u}_0+\mathcal{P}\ast [3u_0{v}_0{\partial }_x{u}_0+{\left({\partial }_x{u}_0\right)}^2{\partial }_x{v}_0+u_0{\partial }_x^2{u}_0{\partial }_x{v}_0+{\rho }_0^2{u}_0]\hfill \\ & +{\partial }_x\mathcal{P}\ast \left(u_0{\partial }_x{u}_0{\partial }_x{v}_0\right){\}\parallel }_{L^p}-2^{ln\sigma }{\parallel {\mathsf{\Delta }}_{ln}\mathbf{u}\parallel }_{L^p}\hfill \\ \hfill \ge & t{2}^{ln\sigma }\parallel {\mathsf{\Delta }}_{ln}\left(u_0{v}_0{\partial }_x{u}_0\right){\parallel }_{L^p}-t·2^{ln\sigma }{\parallel {\mathsf{\Delta }}_{ln}[\mathcal{P}\ast \left(3u_0{v}_0{\partial }_x{u}_0\right)]\parallel }_{L^p}\hfill \\ & -t·2^{ln\sigma }\parallel {\mathsf{\Delta }}_{ln}[\mathcal{P}\ast \left({\left({\partial }_x{u}_0\right)}^2{\partial }_x{v}_0\right)]{\parallel }_{L^p}-t·2^{ln\sigma }{\parallel {\mathsf{\Delta }}_{ln}[\mathcal{P}\ast \left(u_0{\partial }_x^2{u}_0{\partial }_x{v}_0\right)]\parallel }_{L^p}\hfill \\ & -t·2^{ln\sigma }\parallel {\mathsf{\Delta }}_{ln}[\mathcal{P}\ast \left({\rho }_0^2{u}_0\right)]{\parallel }_{L^p}-t·2^{ln\sigma }{\parallel {\mathsf{\Delta }}_{ln}[{\partial }_x\mathcal{P}\ast \left(u_0{\partial }_x{u}_0{\partial }_x{v}_0\right)]\parallel }_{L^p}\hfill \\ & -2^{2ln}·2^{ln(\sigma -2)}{\parallel {\mathsf{\Delta }}_{ln}\mathbf{u}\parallel }_{L^p}.\hfill \end{array}$$

(46)

Since

$$\begin{array}{cc}\hfill 2^{ln\sigma }{\parallel {\mathsf{\Delta }}_{ln}[\mathcal{P}\ast \left(3u_0{v}_0{\partial }_x{u}_0\right)]\parallel }_{L^p}\le & \parallel \mathcal{P}\ast \left(3u_0{v}_0{\partial }_x{u}_0\right){\parallel }_{B_{p,\infty }^{\sigma }}\le C\parallel u_0{v}_0{\partial }_x{v}_0{)\parallel }_{B_{p,\infty }^{\sigma -2}}\hfill \\ \hfill \le & C\parallel u_0{\parallel }_{B_{p,\infty }^{\sigma -1}}^2{\parallel v_0\parallel }_{B_{p,\infty }^{\sigma }}\le C.\hfill \end{array}$$

(47)

By the same token, we have

$$\begin{array}{cc}\hfill 2^{ln\sigma }\parallel {\mathsf{\Delta }}_{ln}{[\mathcal{P}\ast \left({\left({\partial }_x{u}_0\right)}^2{\partial }_x{v}_0\right]\parallel }_{L^p}\le & C\parallel {\left({\partial }_x{u}_0\right)}^2{\partial }_x{v}_0{)\parallel }_{B_{p,\infty }^{\sigma -2}}\hfill \\ \hfill \le & \parallel u_0{\parallel }_{B_{p,\infty }^{\sigma }}^2{\parallel v_0\parallel }_{B_{p,\infty }^{\sigma -1}}\le C,\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill 2^{ln\sigma }{\parallel {\mathsf{\Delta }}_{ln}[\mathcal{P}\ast \left(u_0{\partial }_x^2{u}_0{\partial }_x{v}_0\right)]\parallel }_{L^p}\le & C\parallel u_0{\partial }_x^2{u}_0{\partial }_x{v}_0{\parallel }_{B_{p,\infty }^{\sigma -2}}\hfill \\ \hfill \le & C\parallel u_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\parallel u_0{\parallel }_{B_{p,\infty }^{\sigma -1}}{\parallel v_0\parallel }_{B_{p,\infty }^{\sigma }}\le C,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill 2^{ln\sigma }{\parallel {\mathsf{\Delta }}_{ln}[\mathcal{P}\ast \left({\rho }_0^2{u}_0\right)]\parallel }_{L^p}\le & C\parallel {\rho }_0^2{u}_0{\parallel }_{B_{p,\infty }^{\sigma -2}}\hfill \\ \hfill \le & C\parallel {\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}^2{\parallel u_0\parallel }_{B_{p,\infty }^{\sigma -1}}\le C,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill 2^{ln\sigma }{\parallel {\mathsf{\Delta }}_{ln}[{\partial }_x\mathcal{P}\ast \left(u_0{\partial }_x{u}_0{\partial }_x{v}_0\right)]\parallel }_{L^p}\le & C\parallel u_0{\partial }_x{u}_0{\partial }_x{v}_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\hfill \\ \hfill \le & C\parallel u_0{\parallel }_{B_{p,\infty }^{\sigma -1}}\parallel u_0{\parallel }_{B_{p,\infty }^{\sigma }}{\parallel v_0\parallel }_{B_{p,\infty }^{\sigma }}\le C.\hfill \end{array}$$

(51)

Integrating (47)–(51) into (46), and repeating the discussion on $\parallel \rho -{\rho }_0{\parallel }_{B_{p,\infty }^{\sigma -1}}$, there then exist $C_4,C_5>0$ such that

$$\parallel u-u_0{\parallel }_{B_{p,\infty }^{\sigma }}\ge \frac{C_4}{4}{\delta }_0,{\parallel v-v_0\parallel }_{B_{p,\infty }^{\sigma }}\ge \frac{C_5}{4}{\delta }_0.$$

Picking ${\epsilon }_0=min\{\frac{C_3}{4},\frac{C_4}{4},\frac{C_5}{4}\}{\delta }_0$, this completes the proof of Theorem 1. □