On the Cauchy Problem for Mildly Nonlinear and Non-Boussinesq Case-(ABC) System

Luhang Zhou

School of Mathematical Sciences, Chongqing Normal University, Chongqing, China.

DOI: 10.4236/jamp.2024.124079                

Abstract

In this paper, we investigate the local well-posedness, ill-posedness, and Gevrey regularity of the Cauchy problem for Mildly Nonlinear and Non-Boussinesq case-(ABC) system. The local well-posedness of the solution for this system in Besov spaces $B_{p\mathrm{,}r}^{s1}\times B_{p\mathrm{,}r}^s$ with $1\le p\mathrm{,}r\le \infty$ and $s>\max \left\{1\frac{1}{p},\frac{3}{2}\right\}$ was firstly established. Next, we consider the continuity of the solution-to-data map, i.e. the ill-posedness of the solution for this system in Besov space $B_{p\mathrm{,}\infty }^{s1}\times B_{p\mathrm{,}\infty }^s$ was derived. Finally, the Gevrey regularity of the system was presented.

Keywords

Local Well-Posedness, Ill-Posedness, Gevrey Regularity

1. Introduction

In this paper, we consider the following Cauchy problem for Mildly Nonlinear and Non-Boussinesq case-(ABC) system (see [1] ):

$\{\begin{array}{ll}{\zeta }_t+A{\bar{\sigma }}_x+\alpha B{\left(\zeta \bar{\sigma }\right)}_x-{\alpha }^{2}C{\left({\zeta }^2\bar{\sigma }\right)}_x=0,\hfill & t>0,x\in ℝ,\hfill \\ {\bar{\sigma }}_t+{\zeta }_x+\alpha B\bar{\sigma }{\bar{\sigma }}_x-{\alpha }^{2}C{\left(\zeta {\bar{\sigma }}^2\right)}_x={\epsilon }^2\bar{\kappa }{\bar{\sigma }}_{xxt},\hfill & t>0,x\in ℝ,\hfill \\ \zeta \left(x,0\right)={\zeta }_0,\bar{\sigma }\left(x,0\right)={\bar{\sigma }}_0,\hfill & t=0,x\in ℝ,\hfill \end{array}$ (1.1)

where $\bar{\sigma }$ is the averaged weighted vorticity, and $\zeta$ is the interface displacement. $\alpha \mathrm{,}\epsilon$ are the small parameters with $\alpha \ll 1$ and ${\epsilon }^2\ll \alpha \ll \epsilon$ , i.e. the Mildly Nonlinear (MNL) case. Moreover, $A\mathrm{,}C\mathrm{,}\bar{\kappa }$ are the nonnegative parameters, and B is a parameter. For more details, we can refer to [1] [2] .

Fluid waves are a ubiquitous phenomenon in marine and atmospheric science. An important cause of fluid internal waves is density stratification. In density-stratified flows, the displacement of a fluid mass from its neutrally buoyant position results in internal wave motion. The dynamics of these internal waves has been of great interest and has been the subject of much research (see e.g. [3] [4] ). The study of these long-wave limit currents approaches the existence of a physical environment of rapidly varying density and produces a variety of mathematical models depending on the relative strength of the different influences. The models obtained can be dispersive or non-dispersive and are weakly or fully nonlinear. Physically, dispersion is controlled by the relative magnitude of the horizontal length scale with respect to the height of the domain, however, the nonlinearity is controlled by the wave amplitude with respect to the height of the fluid domain.

Strongly nonlinear, non-dispersive approximations take the form of hyperbolic or mixed-type first-order PDEs, first derived in this context by Long [5] . Weakly nonlinear dispersive approximations result in Korteweg-de Vries type models [6] and fully nonlinear dispersive approximations lead to the so-called Miyata-Camassa-Choi system [7] [8] . However, in this paper, we focus on the Hamiltonian structure of 2-layer dispersed stratified fluids in the non-Boussinesq case under mildly nonlinear assumptions (i.e. the Mildly Nonlinear and Non-Boussinesq case-(ABC) system (1.1)).

Moreover, in Ref. [1] , solutions of special forms of Equation (1.1), i.e. traveling wave solutions and unidirectional waves were considered and the dispersion relation was calculated.

For the equivalent form of (1.1), we set $\bar{\sigma }\left(x\mathrm{,}t\right)=u\left(\theta x\mathrm{,}t\right)$ and $\zeta \left(x\mathrm{,}t\right)=\rho \left(\theta x\mathrm{,}t\right)$ with $\theta =\frac{1}{\epsilon \sqrt{\bar{\kappa }}}$ , then we get:

${\rho }_t+\left(\alpha B\theta -2{\alpha }^{2}C\theta \rho \right)u{\rho }_x=-A\theta u_x-\alpha B\theta \rho u_x+{\alpha }^{2}C\theta {\rho }^2{u}_x,$

and

$u_t={\Lambda }^{-2}\left(-\theta {\rho }_x-\alpha B\theta u{u}_x+{\alpha }^{2}C\theta {\left(\rho u^2\right)}_x\right)\mathrm{,}$

where ${\Lambda }^{2s}u={\left(1-{\partial }_x^2\right)}^{s}u$ and s is a integer.

In the sequel, we will, for notational convenience, demonstrate local well-posedness of the following initial-value problem with more general coefficients:

$\{\begin{array}{ll}{u}_t={\Lambda }^{-2}\left(b_1{\rho }_x-b_{2}u{u}_x+b_3{\left(\rho u^2\right)}_x\right),\hfill & t>0,x\in ℝ,\hfill \\ {\rho }_t+\left(b_{2}u-2b_3\rho u\right){\rho }_x=b_4{u}_x-b_2\rho u_x+b_3{\rho }^2{u}_x,\hfill & t>0,x\in ℝ,\hfill \\ u\left(x,0\right)=u_0,\rho \left(x,0\right)={\rho }_0,\hfill & t=0,x\in ℝ,\hfill \end{array}$ (1.2)

with $b_1=-\theta ,b_2=\alpha B\theta ,b_3={\alpha }^{2}C\theta ,b_4=-A\theta$ . Where u is the averaged weighted vorticity, and $\rho$ is the interface displacement. Moreover, $b_i$ ( $i=1,2,3,4$ ) are all parameters, the operator $\Lambda$ is a S-multiplier with ${\Lambda }^{-2}={\left(1-{\partial }_x^2\right)}^{-1}$ .

Inspired by the argument of Danchin [9] [10] in the study of the local well-posedness to the CH equation, we first establish the local well-posedness of (1.2) in Besov spaces.

Theorem 1.1. Suppose that $1\le p\mathrm{,}r\le \infty$ and $s>\max \left\{1+\frac{1}{p},\frac{3}{2}\right\}$ and the initial data $\left(u_0\mathrm{,}{\rho }_0\right)\in B_{p\mathrm{,}r}^{s+1}\times B_{p\mathrm{,}r}^s$ . Then, there exists a time $T>0$ such that the Cauchy problem Equation (2.2) has a unique solution $\left(u\mathrm{,}\rho \right)\in E_{p\mathrm{,}r}^{s+1}\times E_{p\mathrm{,}r}^s$ (see Definition 2.3). Moreover, the map $\left(u_0\mathrm{,}{\rho }_0\right)\mapsto \left(u\mathrm{,}\rho \right)$ is continuous from a neighborhood of $\left(u_0\mathrm{,}{\rho }_0\right)$ in $B_{p\mathrm{,}r}^{s+1}\times B_{p\mathrm{,}r}^s$ into:

$\mathcal{C}\left(\left[0,T\right];B_{p,r}^{s^{\prime }+1}\right)\cap {\mathcal{C}}^1\left(\left[0,T\right];B_{p,r}^{s^{\prime }}\right)\times \mathcal{C}\left(\left[0,T\right];B_{p,r}^{s^{\prime }}\right)\cap {\mathcal{C}}^1\left(\left[0,T\right];B_{p,r}^{s^{\prime }-1}\right)$

for every $s^{\prime }<s$ when $r=+\infty$ and $s^{\prime }=s$ when $r<+\infty$ .

Remark 1.2. In the proof of Theorem 1.1, we use the transport equations theory and the transport diffusion equations theory to establish the local well-posedness of Equation (1.2) in nonhomogeneous Besov spaces. It is well-known that the Besov spaces $B_{\mathrm{2,2}}^{s+1}\times B_{\mathrm{2,2}}^s$ coincide with the Sobolev spaces $H^{s+1}\times H^s$ . Theorem 1.1 implies that under the condition $\left(u_0\mathrm{,}{\rho }_0\right)\in H^{s+1}\times H^s$ with $s>\frac{3}{2}$ , we can obtain the local well-posedness for the data-to-solution map in Sobolev spaces.

In Theorem 1.1, we have proved that for every $s^{\prime }<s$ when $r=+\infty$ and $s^{\prime }=s$ when $r<+\infty$ , the data-to-solution map of system (1.2) is continue in $B_{p\mathrm{,}r}^{s^{\prime }+1}\times B_{p\mathrm{,}r}^{s^{\prime }}$ . Now, another natural question raised: whether or not the data-to-solution map of system (1.2) is continue in $B_{p\mathrm{,}r}^{s^{\prime }+1}\times B_{p\mathrm{,}r}^{s^{\prime }}$ for $s^{\prime }=s$ and $r=+\infty$ . In our next theorem, we shall further show that this data-to-solution map is ill-posedness in $B_{p\mathrm{,}\infty }^{s+1}\times B_{p\mathrm{,}\infty }^s$ in the sense that the solutions starting from $\left(u_0\mathrm{,}{\rho }_0\right)$ are discontinuous at $t=0$ in the metric of $B_{p\mathrm{,}\infty }^{s+1}\times B_{p\mathrm{,}\infty }^s$ , this means that:

Theorem 1.3. Suppose that $1\le p\le \infty$ and $s>\max \left\{1+\frac{1}{p},\frac{3}{2}\right\}$ . Then, the system (1.2) is ill-posed in the Besov spaces $B_{p\mathrm{,}\infty }^{s+1}\times B_{p\mathrm{,}\infty }^s$ . More precisely, there exist $\left(u_0\mathrm{,}{\rho }_0\right)\in B_{p\mathrm{,}\infty }^{s+1}\times B_{p\mathrm{,}\infty }^s$ and a positive constant $\delta$ for which the Cauchy problem of system (1.1) has a unique solution $\left(u\mathrm{,}\rho \right)\in L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}\infty }^{s+1}\times B_{p\mathrm{,}\infty }^s\right)$ for some $T=T\left({\Vert u_0\Vert }_{B_{p\mathrm{,}r}^{s+1}}\mathrm{,}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}r}^s}\right)\mathrm{,}$ while

$\underset{t\to 0}{\lim \inf }\left({\Vert u-u_0\Vert }_{B_{p,\infty }^{s+1}}+{\Vert \rho -{\rho }_0\Vert }_{B_{p,\infty }^s}\right)\ge C\delta .$

Motivation by [11] [12] , we use the generalized Ovsyannikov theorem to solve the Gevrey and analytic regularity for (1.2). To begin with, we introduce the Sobolev-Gevery spaces [11] [12] , a suitable scale of Banach spaces, as follows:

$G_{\sigma \mathrm{,}s}^{\delta }\equiv \left\{f\in {\mathcal{C}}^{\infty }\left(ℝ\right)\mathrm{<mo>\; :\; </mo>}{\Vert f\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}^2\doteq {\displaystyle {\int }_{ℝ}}{\left(1+{\left|\xi \right|}^2\right)}^s{\text{e}}^{2\delta {\left|\xi \right|}_{}^{\frac{1}{\sigma }}}{\left|\hat{f}\left(\xi \right)\right|}^2\text{d}\xi <\infty \right\}\mathrm{,}\sigma \mathrm{,}\delta >0.$

It is easy to cheek that $G_{\sigma \mathrm{,}s}^{\delta }$ equipped the norm ${\Vert \cdot \Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}$ is a Banach space by the completeness of $H^s$ . Then, we can define the Gevrey and analytic regularity as follows:

Definition 1.4. Denoting Fourier multiplier ${\text{e}}^{\delta {\left(-\Delta \right)}_{}^{\frac{1}{2\sigma }}}$ by ${\text{e}}^{\delta {\left(-\Delta \right)}_{}^{\frac{1}{2\sigma }}}f={\Im }^{-1}\left({\text{e}}^{\delta {\left|\xi \right|}_{}^{\frac{1}{\sigma }}}\hat{f}\right)$ , i.e. ${\Vert f\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}={\Vert {\text{e}}^{\delta {\left|\xi \right|}_{}^{\frac{1}{\sigma }}}\hat{f}\Vert }_{H^s}$ . If $0<\sigma <1$ , it is called ultra-analytic function. If $\sigma =1$ , it is usual analytic function and $\delta$ is called the radius of analyticity. If $\sigma >1$ , it is the Gevrey function.

Now, we establish that solutions of (1.2) are analytic in both space and time variables.

Theorem 1.5. Let $\sigma \ge 1$ and $s>\frac{1}{2}$ . Assume that $z_0=\left(u_0\mathrm{,}{\rho }_0\right)\in G_{\sigma \mathrm{,}s}^1\times G_{\sigma \mathrm{,}s}^1$ . Therefore, for every $0<\delta <1$ , there exists a $T_0>0$ such that the system (1.2) has a unique solution $\left(u\mathrm{,}\rho \right)$ , which is holomorphic in $\left|t\right|<\frac{T_0{\left(1-\delta \right)}^{\sigma }}{2^{\sigma }-1}$ with values in $G_{\sigma \mathrm{,}s}^{\delta }\times G_{\sigma \mathrm{,}s}^{\delta }$ .

The rest of our paper is organized as follows. In Section 2, we recall several results in the Littlewood-Paley theory and some properties of Besov spaces are reviewed. In Section 3, we establish the local well-posedness result for Equation (1.2). Moreover, the ill-posedness result for this system is presented in Section 4. Finally, we give the Gevrey regularity in Section 5.

2. Preliminaries

In this section, for the convenience of the readers, we will recall some facts on the Littlewood-Paley theory, which will be frequently used in the following arguments. Then, we introduce some properties of the Besov spaces which will play a key role in proving the local well-posedness and other properties for the system (1.2). One may check [10] [13] for more details. First, we introduce some notations.

Notation. For simplicity, the norm ${\Vert \cdot \Vert }_{B_{p\mathrm{,}r}^s}$ means ${\Vert \cdot \Vert }_{B_{p\mathrm{,}r}^s\left(ℝ\right)}$ in the following, the symbol $A\lesssim B$ means that there is a uniform positive constant C independent of A and B such that $A\le CB$ .

Proposition 2.1. (See Proposition 2.10 in [13] ) Let $\mathcal{B}\doteq \left\{\xi \in {ℝ}^d,\left|\xi \right|\le \frac{4}{3}\right\}$ and $\mathcal{C}\doteq \left\{\xi \in {ℝ}^d,\frac{4}{3}\le \left|\xi \right|\le \frac{8}{3}\right\}$ . There exist two radial functions $\chi \in {\mathcal{C}}_c^{\infty }\left(\mathcal{B}\right)$ and $\phi \in {\mathcal{C}}_c^{\infty }\left(\mathcal{C}\right)$ such that:

$\chi \left(\xi \right)+{\displaystyle \underset{q\ge 0}{\sum }}\phi \left(2^{-q}\xi \right)=\mathrm{1,}\text{for}\text{all}\xi \in {ℝ}^d\mathrm{,}$

$\left|q-q^{\prime }\right|\ge 2\Rightarrow Supp\phi \left(2^{-q}\cdot \right)\cap Supp\phi \left(2^{-q^{\prime }}\cdot \right)=\varnothing \mathrm{,}$

$q\ge 1\Rightarrow Supp\chi (\cdot )\cap Supp\phi \left(2^{-q^{\prime }}\cdot \right)=\varnothing \mathrm{,}$

$\frac{1}{3}\le \chi {\left(\xi \right)}^2+{\displaystyle \underset{q\ge 0}{\sum }}\phi {\left(2^{-q}\xi \right)}^2\le \mathrm{1,}\text{for}\text{all}\xi \in {ℝ}^d\mathrm{.}$

Moreover, let $h\doteq {\mathcal{F}}^{-1}\phi$ and $\tilde{h}\doteq {\mathcal{F}}^{-1}\chi$ . Then, for all $f\in {\mathcal{S}}^{\prime }\left({ℝ}^d\right)$ , the dyadic operators ${\Delta }_q$ and $S_q$ can be defined as follows:

${\Delta }_{q}f\doteq \phi \left(2^{-q}D\right)f=2^{qd}{\displaystyle {\int }_{{ℝ}^d}}h\left(2^{q}y\right)f\left(x-y\right)\text{d}y\mathrm{,}\text{for}q\ge \mathrm{0,}$

$S_{q}f\doteq \chi \left(2^{-q}D\right)f={\displaystyle \underset{-1\le k\le q-1}{\sum }}{\Delta }_k=2^{qd}{\displaystyle {\int }_{{ℝ}^d}}\tilde{h}\left(2^{q}y\right)f\left(x-y\right)\text{d}y,$

${\Delta }_{-1}f\doteq S_{0}f\text{and}{\Delta }_{q}f\doteq 0\text{for}q\le -2.$

Therefore,

$f={\displaystyle \underset{q\ge 0}{\sum }}{\Delta }_{q}f\text{in}{S}^{\prime }\left({ℝ}^d\right)\mathrm{,}$

and the right-hand side is called the nonhomogeneous Littlewood-Paley decomposition of f.

Definition 2.2. (See Definition 2.68 in [13] ) Let $s\in ℝ$ , $1\le p\mathrm{,}r\le \infty$ . The inhomogenous Besov space $B_{p\mathrm{,}r}^s\left({ℝ}^d\right)$ ( $B_{p\mathrm{,}r}^s$ for short) is defined by:

$B_{p\mathrm{,}r}^s\doteq \left\{f\in {\mathcal{S}}^{\prime }\left({ℝ}^d\right)\mathrm{<mo>\; ;\; </mo>}{\Vert f\Vert }_{B_{p\mathrm{,}r}^s}<\infty \right\}\mathrm{,}$

where

${\Vert f\Vert }_{B_{p\mathrm{,}r}^s}\doteq \{\begin{array}{ll}{\left({\displaystyle \underset{q\in \mathbb{Z}}{\sum }}{2}^{qsr}{\Vert {\Delta }_{q}f\Vert }_{L_p}^r\right)}^{\frac{1}{r}}\mathrm{,}\hfill & \text{for}r<\infty \mathrm{,}\hfill \\ \underset{q\in \mathbb{Z}}{\sup }{2}^{qs}{\Vert {\Delta }_{q}f\Vert }_{L_p}\mathrm{,}\hfill & \text{for}r=\infty \mathrm{.}\hfill \end{array}$

If $s=\infty$ , $B_{p\mathrm{,}r}^{\infty }\doteq {\displaystyle \underset{s\in ℝ}{\cap }}{B}_{p\mathrm{,}r}^s$ .

Definition 2.3. For $T>0,s\in ℝ$ and $1\le p\le +\infty$ , we define:

$E_{p\mathrm{,}r}^s\left(T\right)\doteq \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\cap {\mathcal{C}}^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)\mathrm{,}\text{if}r<\infty \mathrm{,}$

$E_{p\mathrm{,}\infty }^s\left(T\right)\doteq L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}\infty }^s\right)\cap Lip\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}\infty }^{s-1}\right)\text{and}{E}_{p\mathrm{,}r}^s\doteq \underset{}{{\displaystyle {\displaystyle \underset{T>0}{\cap }}}}{E}_{p\mathrm{,}r}^s\left(T\right)\mathrm{.}$

Proposition 2.4. (See Corollary 2.86 in [13] ) For any positive real number s and any $\left(p\mathrm{,}r\right)$ in ${\left[\mathrm{1,}\infty \right]}^2$ , the space $L^{\infty }\left({ℝ}^d\right)\cap B_{p\mathrm{,}r}^s\left({ℝ}^d\right)$ is an algebra and a constant C exists such that:

${\Vert uv\Vert }_{B_{p\mathrm{,}r}^s\left({ℝ}^d\right)}\le C\left({\Vert u\Vert }_{L^{\infty }\left({ℝ}^d\right)}{\Vert v\Vert }_{B_{p\mathrm{,}r}^s\left({ℝ}^d\right)}+{\Vert u\Vert }_{B_{p\mathrm{,}r}^s\left({ℝ}^d\right)}{\Vert v\Vert }_{L^{\infty }\left({ℝ}^d\right)}\right)\mathrm{.}$

If $s>\frac{d}{p}$ or $s=\frac{d}{p}$ , $r=1$ , then we have:

${\Vert uv\Vert }_{B_{p\mathrm{,}r}^s\left({ℝ}^d\right)}\le C{\Vert u\Vert }_{B_{p\mathrm{,}r}^s\left({ℝ}^d\right)}{\Vert v\Vert }_{B_{p\mathrm{,}r}^s\left({ℝ}^d\right)}\mathrm{.}$

Proposition 2.5. (See Proposition 1.3.5 in [10] ) Suppose that $s\in ℝ$ , $1\le p\mathrm{,}r\mathrm{,}{p}_i\mathrm{,}{r}_i\le \infty$ ( $i=\mathrm{1,2}$ ). We have:

1) Topological properties: $B_{p\mathrm{,}r}^s$ is a Banach space and is continuously embedded in $S^{\prime }$ .

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

3) Embedding: $B_{p_1\mathrm{,}{r}_1}^s$ ↪ $B_{p_2\mathrm{,}{r}_2}^{s-n\left(\frac{1}{p_1}-\frac{1}{p_2}\right)}$ , if $p_1\le p_2$ and $r_1\le r_2$ . $B_{p\mathrm{,}{r}_2}^{s_2}$ ↪ $B_{p\mathrm{,}{r}_1}^{s_1}$ locally compact, if $s_1<s_2$ .

4) Algebraic properties: for all $s>0$ , $B_{p\mathrm{,}r}^s\cap L^{\infty }$ is an algebra. Moreover, $B_{p\mathrm{,}r}^s$ is an algebra $\iff B_{p\mathrm{,}r}^s$ ↪ $L^{\infty }$ $\iff s>\frac{n}{p}$ (or $s=\frac{n}{p}$ and $r=1$ ).

5) Complex interpolation:

${\Vert u\Vert }_{B_{p\mathrm{,}r}^{\theta s_1+\left(1-\theta \right)s_2}}\le C{\Vert u\Vert }_{B_{p\mathrm{,}r}^{s_1}}^{\theta }{\Vert u\Vert }_{B_{p\mathrm{,}r}^{s_2}}^{1-\theta }\mathrm{,}$

for all $u\in B_{p\mathrm{,}r}^{s_1}\cap B_{p\mathrm{,}r}^{s_2}$ , for all $\theta \in \left[\mathrm{0,1}\right]$ .

6) Fatou lemma: If ${\left(u_n\right)}_{n\in \mathbb{N}}$ is bounded in $B_{p\mathrm{,}r}^s$ and $u_n\to u$ in $S^{\prime }$ , then $u\in B_{p\mathrm{,}r}^s$ and

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

7) Let $m\in ℝ$ and let f be an $S^m$ -multiplier (i.e. $f\mathrm{<mo>\; :\; </mo>}{ℝ}^d\to ℝ$ is smooth and satisfies that for all $\alpha \in {\mathbb{N}}^d$ , there exists a constant $C_{\alpha }$ , s.t. $\left|{\partial }^{\alpha }f\left(\xi \right)\right|\le C_{\alpha }\left(1+{\left|\xi \right|}^{m-\left|\alpha \right|}\right)$ for all $\xi \in {ℝ}^d$ ). Then, the operator $f\left(D\right)$ is continuous from $B_{p\mathrm{,}r}^s$ to $B_{p\mathrm{,}r}^{s-m}$ .

Lemma 2.6. (See Lemma 2.8 in [14] or [13] ) Suppose that $\left(p\mathrm{,}r\right)\in {\left[\mathrm{1,}+\infty \right]}^2$ and $s>-\min \left\{\frac{1}{p},1-\frac{1}{p}\right\}$ . Assume $f_0\in B_{p\mathrm{,}r}^s$ , $F\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ , and

$\{\begin{array}{ll}{\partial }_{x}v\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)\mathrm{<mo>\; ;\; </mo>}\hfill & \text{if}s>1+\frac{1}{p}\left(s=1+\frac{1}{p},r=1\right),\hfill \\ {\partial }_{x}v\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{\frac{1}{p}}\cap L^{\infty }\right)\mathrm{<mo>\; ;\; </mo>}\hfill & \text{if}s<1+\frac{1}{p}\mathrm{.}\hfill \end{array}$

If $f\in L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\cap \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{\mathcal{S}}^{\prime }\right)$ solves the following 1-D linear transport equation:

$\{\begin{array}{l}{\partial }_{t}f+v\cdot {\partial }_{x}f=F,\\ {f|}_{t=0}=f_0,\end{array}$ (2.1)

then exists a constant C depending only on $p\mathrm{,}r\mathrm{,}s$ , such that the following statements hold:

1) If $r=1$ or $s\ne 1+\frac{1}{p}$ ,

${\Vert f\left(t\right)\Vert }_{B_{p\mathrm{,}r}^s}\le {\Vert f_0\Vert }_{B_{p\mathrm{,}r}^s}+{\displaystyle {\int }_0^t}{\Vert F\left(\tau \right)\Vert }_{B_{p\mathrm{,}r}^s}+C{V}^{\prime }\left(\tau \right){\Vert f\left(\tau \right)\Vert }_{B_{p\mathrm{,}r}^s}\text{d}\tau \mathrm{,}$

or

${\Vert f\left(t\right)\Vert }_{B_{p\mathrm{,}r}^s}\le {\text{e}}^{CV\left(t\right)}\left({\Vert f_0\Vert }_{B_{p\mathrm{,}r}^s}+{\displaystyle {\int }_0^t}{\text{e}}^{-CV\left(\tau \right)}{\Vert F\left(\tau \right)\Vert }_{B_{p\mathrm{,}r}^s}\text{d}\tau \right)\mathrm{,}$

with

$V\left(t\right)=\{\begin{array}{ll}{\displaystyle {\int }_0^t}{\Vert {\partial }_{x}v\Vert }_{B_{p\mathrm{,}r}^{\frac{1}{p}}\cap L^{\infty }}^{}\text{d}\tau \mathrm{,}\hfill & \text{if}s<1+\frac{1}{p}\mathrm{<mo>\; ;\; </mo>}\hfill \\ {\displaystyle {\int }_0^t}{\Vert {\partial }_{x}v\Vert }_{B_{p\mathrm{,}r}^{s-1}}\text{d}\tau \mathrm{,}\hfill & \text{if}s>1+\frac{1}{p}\left(\text{or}s=1+\frac{1}{p}\mathrm{,}r=1\right)\mathrm{.}\hfill \end{array}$

2) If $f=v$ , then for all $s>0$ , 1) holds true with $V\left(t\right)={\displaystyle {\int }_0^t}{\Vert {\partial }_{x}v\Vert }_{L^{\infty }}\text{d}\tau$ .

3) If $r<\infty$ , then $f\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ . If $r=\infty$ , then $f\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{s^{\prime }}\right)$ for all $s^{\prime }<s$ .

Lemma 2.7. (See Theorem 3.3.1 in [10] or [15] ) Let $\left(p\mathrm{,}{p}_1\mathrm{,}r\right)\in {\left[\mathrm{0,}T\right]}^3$ , and let $s>-\min \left(\frac{1}{p_1},\frac{1}{p^{\prime }}\right)$ with $p^{\prime }\doteq {\left(1-\frac{1}{p}\right)}^{-1}$ . Assume that $f_0\in B_{p\mathrm{,}r}^s$ , $F\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ . Let v be a time-dependent vector field such that $v\in L^{\rho }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{\infty \mathrm{,}\infty }^{-M}\right)$ for some $\rho >1$ , $M>0$ and

${\partial }_{x}v\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p_1\mathrm{,}\infty }^{\frac{1}{p_1}}\right)\mathrm{,}\text{if}s<1+\frac{1}{p_1}\mathrm{,}$

and

${\partial }_{x}v\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p_1\mathrm{,}r}^{s-1}\right)\mathrm{,}\text{if}s>1+\frac{1}{p_1}\mathrm{,}\text{or}\left(s=1+\frac{1}{p_1}\mathrm{,}r=1\right)\mathrm{.}$

Then, Equation (2.1) has a unique solution $f\in L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\cap \left({\displaystyle \underset{s^{\prime }<s}{\cap }}\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{s^{\prime }}\right)\right)$ and the inequalities in Lemma 2.6 hold true.

Moreover, if $r<\infty$ , then we have $f\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ .

Lemma 2.8. (See [13] ) Let $s>0$ , $1\le p\le \infty$ , then we have:

${\Vert 2^{js}{\Vert \left[{\Delta }_j\mathrm{,}u\right]{\partial }_{x}v\Vert }_{L^p}\Vert }_{l^{\infty }}\le C\left({\Vert {\partial }_{x}u\Vert }_{L^{\infty }}{\Vert v\Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert {\partial }_{x}v\Vert }_{L^{\infty }}{\Vert u\Vert }_{B_{p\mathrm{,}\infty }^s}\right)\mathrm{,}$

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

3. Local Well-Posedness in Besov Space

In this section, we shall discuss the local well-posedness of the Cauchy problem (1.2) in nonhomogeneous Besov spaces, and prove Theorem 1.1. In the following, we denote $C>0$ a generic constant only depending on $p\mathrm{,}r\mathrm{,}s\mathrm{,}{b}_1\mathrm{,}{b}_2\mathrm{,}{b}_3\mathrm{,}{b}_4$ . Moreover, uniqueness and continuity with respect to the initial data $\left(u_0\mathrm{,}{\rho }_0\right)$ are an immediate consequence of the following lemma.

Lemma 3.1. Let $1\le p\mathrm{,}r\le +\infty$ and $s>\max \left\{1+\frac{1}{p},\frac{3}{2}\right\}$ . Suppose that:

$\left(u_i\mathrm{,}{\rho }_i\right)\in \left\{L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s+1}\right)\cap \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{\mathcal{S}}^{\prime }\right)\right\}\times \left\{L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\cap \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{\mathcal{S}}^{\prime }\right)\right\}\left(i=\mathrm{1,2}\right)$

be two given solutions of the initial-value problem (1.2) with the initial data $\left(u_i\left(0\right)\mathrm{,}{\rho }_i\left(0\right)\right)\in B_{p\mathrm{,}r}^{s+1}\times B_{p\mathrm{,}r}^s\left(i=\mathrm{1,2}\right)$ , and denote $u_{12}=u_1-u_2$ , ${\rho }_{12}={\rho }_1-{\rho }_2$ . Then, for every $t\in \left[\mathrm{0,}T\right]$ , we have:

${\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\le \left({\Vert u_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}\right)\exp \left(C{\displaystyle {\int }_0^t}{\Gamma }_s\left(\tau \right)\text{d}\tau \right)\mathrm{,}$ (3.1)

for $s\ne 2+\frac{1}{p}$ , where

${\Gamma }_s\left(t\right)={\left(1+{\Vert u_1\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert u_2\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert {\rho }_1\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_2\Vert }_{B_{p\mathrm{,}r}^s}\right)}^2\mathrm{.}$

For the critical case $s=2+\frac{1}{p}$ ,

${\Vert u_{12}\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{p}}}^{}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}q}^{1+\frac{1}{p}}}^{}\le C{\left({\Vert u_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{p}}}^{}+{\Vert {\rho }_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}q}^{1+\frac{1}{p}}}^{}\right)}^{\theta }\exp \left(C\theta {\displaystyle {\int }_0^T}{\Gamma }_{3+\frac{1}{p}}\text{d}\tau \right){\Gamma }_{3+\frac{1}{p}}^{\frac{1-\theta }{2}}\mathrm{,}$

where $\theta =1-\frac{1}{2p}\in \left(0,1\right)$ .

Proof (Proof of Lemma 3.1) It is easy to see that $u_{12}\in L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s+1}\right)\cap \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{\mathcal{S}}^{\prime }\right)$ and ${\rho }_{12}\in L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\cap \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{\mathcal{S}}^{\prime }\right)$ which implies that the solution pair $\left(u_{12}\mathrm{,}{\rho }_{12}\right)\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ , and $\left(u_{12}\mathrm{,}{\rho }_{12}\right)$ solves the following transport equations:

$\{\begin{array}{l}{\partial }_t{u}_{12}={\Lambda }^{-2}{\partial }_{x}H\left(u_1,u_2,{\rho }_1,{\rho }_2\right),\\ {\partial }_t{\rho }_{12}+\left(b_2{u}_1-2b_3{\rho }_1{u}_1\right){\partial }_x{\rho }_{12}=G\left(u_1,u_2,{\rho }_1,{\rho }_2\right),\\ {u_{12}|}_{t=0}=u_1\left(0\right)-u_2\left(0\right),{{\rho }_{12}|}_{t=0}={\rho }_1\left(0\right)-{\rho }_2\left(0\right),\end{array}$ (3.2)

where

$H=b_1{\rho }_{12}-\frac{b_2}{2}\left(u_1+u_2\right)u_{12}+b_3\left({\rho }_{12}{u}_1^2+{\rho }_2\left(u_1+u_2\right)u_{12}\right)\mathrm{,}$

and

$\begin{array}{c}G=-b_2{u}_{12}{\partial }_x{\rho }_2+2b_3{\partial }_x{\rho }_2\left({\rho }_1{u}_{12}+u_2{\rho }_{12}\right)+b_4{\partial }_x{u}_{12}\\ -b_2\left({\rho }_1{\partial }_x{u}_{12}+{\rho }_{12}{\partial }_x{u}_2\right)+b_3\left[{\rho }_1^2{\partial }_x{u}_{12}+{\partial }_x{u}_2{\rho }_{12}\left({\rho }_1+{\rho }_2\right)\right]\mathrm{.}\end{array}$

Integrating the first equation of (3.2) with respect to variable t, it is easy to see that:

${\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}\le {\Vert u_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^s}+{\displaystyle {\int }_0^t}{\Vert {\Lambda }^{-2}{\partial }_{x}H\Vert }_{B_{p\mathrm{,}r}^s}\text{d}\tau \mathrm{.}$

Note that the operator $\Lambda$ is a S-multiplier. Applying Proposition 2.4, Proposition 2.5 and the algebraic property for $B_{p\mathrm{,}r}^{s-1}$ for $s>1+\frac{1}{p}$ , we have:

$\begin{array}{c}{\Vert {\Lambda }^{-2}{\partial }_{x}H\Vert }_{B_{p\mathrm{,}r}^s}\le C{\Vert b_1{\rho }_{12}-\frac{b_2}{2}\left(u_1+u_2\right)u_{12}+b_3\left({\rho }_{12}{u}_1^2+{\rho }_2\left(u_1+u_2\right)u_{12}\right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}\\ \le C\left({\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}+{\Vert \left(u_1+u_2\right)u_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}+{\Vert {\rho }_{12}{u}_1^2\Vert }_{B_{p\mathrm{,}r}^{s-1}}+{\Vert {\rho }_2\left(u_1+u_2\right)u_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\right)\\ \le C(\left({\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\right){\Gamma }_s(\; \tau \; ))\end{array}$

Therefore, we obtain:

${\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}\le {\Vert u_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^s}+C{\displaystyle {\int }_0^t}\left({\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\right){\Gamma }_s\left(\tau \right)\text{d}\tau \mathrm{.}$ (3.3)

Applying the Lemma 2.6 for $s>\max \left\{1+\frac{1}{p},\frac{3}{2}\right\}$ and $s\ne 2+\frac{1}{p}$ , we get the following inequality:

${\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\le {\Vert {\rho }_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}+{\displaystyle {\int }_0^t}{\Vert G\left(\tau \right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}+C{V}^{\prime }\left(\tau \right){\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\text{d}\tau \mathrm{,}$ (3.4)

with

$V\left(t\right)=\{\begin{array}{ll}{\displaystyle {\int }_0^t}{\Vert {\partial }_x\left(b_2{u}_1-2b_3{\rho }_1{u}_1\right)\Vert }_{B_{p\mathrm{,}r}^{\frac{1}{p}}\cap L^{\infty }}^{}\text{d}\tau \mathrm{,}\hfill & \text{if}\max \left\{1+\frac{1}{p}\mathrm{,}\frac{3}{2}\right\}<s<\frac{5}{2}\mathrm{,}\hfill \\ {\displaystyle {\int }_0^t}{\Vert {\partial }_x\left(b_2{u}_1-2b_3{\rho }_1{u}_1\right)\Vert }_{B_{p\mathrm{,}r}^{s-2}}\text{d}\tau \mathrm{,}\hfill & \text{if}s\ge \frac{5}{2}\mathrm{.}\hfill \end{array}$

Using Proposition 2.5 (3) ( $B_{p\mathrm{,}{r}_2}^{s_2}$ ↪ $B_{p\mathrm{,}{r}_1}^{s_1}$ locally compact, if $s_1<s_2$ ) and Proposition 2.5 (4) ( $B_{p\mathrm{,}r}^s$ ↪ $L^{\infty }$ $\iff s>\frac{n}{p}$ ), for $s>\max \left\{1+\frac{1}{p},\frac{3}{2}\right\}$ , we get:

$\begin{array}{c}{V}^{\prime }\left(t\right)\le {\Vert {\partial }_x\left(b_2{u}_1-2b_3{\rho }_1{u}_1\right)\Vert }_{B_{p\mathrm{,}r}^{\frac{1}{p}}\cap L^{\infty }}^{}+{\Vert {\partial }_x\left(b_2{u}_1-2b_3{\rho }_1{u}_1\right)\Vert }_{B_{p\mathrm{,}r}^{s-2}}\\ \le 2{\Vert {\partial }_x\left(b_2{u}_1-2b_3{\rho }_1{u}_1\right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}\\ \le 2C\left({\Vert {\rho }_1\Vert }_{B_{p\mathrm{,}r}^s}{\Vert u_1\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert u_1\Vert }_{B_{p\mathrm{,}r}^{s+1}}\right)\end{array}$ (3.5)

For $s>\max \left\{1+\frac{1}{p},\frac{3}{2}\right\}$ and $s\ne 2+\frac{1}{p}$ , applying Proposition 2.4, we can obtain that:

$\begin{array}{c}{\Vert G\Vert }_{B_{p\mathrm{,}r}^{s-1}}\le C({\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}{\Vert {\rho }_2\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_2\Vert }_{B_{p\mathrm{,}r}^s}({\Vert {\rho }_1\Vert }_{B_{p\mathrm{,}r}^s}{\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}\\ +{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}{\Vert u_2\Vert }_{B_{p\mathrm{,}r}^{s+1}})+{\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}{\Vert {\rho }_1\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}{\Vert u_2\Vert }_{B_{p\mathrm{,}r}^{s+1}}\\ +{\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}{\Vert {\rho }_1\Vert }_{B_{p\mathrm{,}r}^s}^2+{\Vert u_2\Vert }_{B_{p\mathrm{,}r}^{s+1}}{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\left({\Vert {\rho }_1\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_2\Vert }_{B_{p\mathrm{,}r}^s}\right))\\ \le C\left(\left({\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\right){\Gamma }_s\left(\tau \right)\right)\mathrm{.}\end{array}$ (3.6)

Submitting (3.5)-(3.6) into (3.4), we may derive:

$\begin{array}{c}{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\le {\Vert {\rho }_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}+{\displaystyle {\int }_0^t}{\Vert G\left(\tau \right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}+C{V}^{\prime }\left(\tau \right){\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\text{d}\tau \\ \le {\Vert {\rho }_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}+C{\displaystyle {\int }_0^t}\left({\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\right){\Gamma }_s\left(\tau \right)\text{d}\tau \end{array}$ (3.7)

for $s>\max \left\{1+\frac{1}{p},\frac{3}{2}\right\}$ and $s\ne 2+\frac{1}{p}$ .

Consequently, combining (3.3) and (3.7), we have:

$\begin{array}{l}{\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\\ \le {\Vert u_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}+C{\displaystyle {\int }_0^t}\left({\Vert u_{12}\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\right){\Gamma }_s\left(\tau \right)\text{d}\tau \mathrm{.}\end{array}$

Applying Gronwall’s lemma to the above inequality leads to (3.1).

For the case $s=2+\frac{1}{p}$ , choosing $\theta =1-\frac{1}{2p}\in \left(0,1\right)$ , then we have:

$1+\frac{1}{p}=\theta \left(1+\frac{1}{2p}\right)+\left(1-\theta \right)\left(2+\frac{1}{2p}\right)\mathrm{,}$

and

$2+\frac{1}{p}=\theta \left(2+\frac{1}{2p}\right)+\left(1-\theta \right)\left(3+\frac{1}{2p}\right)\mathrm{.}$

And according to the interpolation formula in Proposition 2.5 (5) and the obtained result (3.1), we can see that:

$\begin{array}{l}{\Vert u_{12}\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{p}}}^{}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}q}^{1+\frac{1}{p}}}^{}\le {\Vert u_{12}\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{2p}}}^{\theta }{\Vert u_{12}\Vert }_{B_{p\mathrm{,}q}^{3+\frac{1}{2p}}}^{1-\theta }+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}q}^{1+\frac{1}{2p}}}^{\theta }{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{2p}}}^{1-\theta }\\ \le {\left({\Vert u_{12}\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{2p}}}^{}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}q}^{1+\frac{1}{2p}}}^{}\right)}^{\theta }\left({\Vert u_{12}\Vert }_{B_{p\mathrm{,}q}^{3+\frac{1}{2p}}}^{1-\theta }+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{2p}}}^{1-\theta }\right)\\ \le {\left({\Vert u_{12}\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{2p}}}^{}+{\Vert {\rho }_{12}\Vert }_{B_{p\mathrm{,}q}^{1+\frac{1}{2p}}}^{}\right)}^{\theta }{\left({\Vert u_1\Vert }_{B_{p\mathrm{,}q}^{3+\frac{1}{2p}}}^{}+{\Vert u_2\Vert }_{B_{p\mathrm{,}q}^{3+\frac{1}{2p}}}^{}+{\Vert {\rho }_1\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{2p}}}^{}+{\Vert {\rho }_2\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{2p}}}^{}\right)}^{1-\theta }\\ \le C{\left({\Vert u_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}q}^{2+\frac{1}{p}}}^{}+{\Vert {\rho }_{12}\left(0\right)\Vert }_{B_{p\mathrm{,}q}^{1+\frac{1}{p}}}^{}\right)}^{\theta }\exp \left(C\theta {\displaystyle {\int }_0^T}{\Gamma }_{3+\frac{1}{p}}\text{d}\tau \right){\Gamma }_{3+\frac{1}{p}}^{\frac{1-\theta }{2}}\end{array}$

which yields the Lemma 3.1.

Now, let us start the proof of Theorem 1.1, which is motivated by the proof of local existence theorem about Camassa-Holm type equations in [15] [16] . Next, by using the classical Friedrichs regularization method, we construct the approximate solutions to (1.2).

Lemma 3.2. Let p, r and s be as in the statement of Lemma 2.6. Assume that $u\left(0\right)=\rho \left(0\right):=0$ . There exists a sequence of smooth functions $\left(u_k\mathrm{,}{\rho }_k\right)\in \mathcal{C}\left({ℝ}^{+}\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{\infty }\right)\times \mathcal{C}\left({ℝ}^{+}\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{\infty }\right)$ solving:

$\{\begin{array}{l}{\partial }_t{u}_{k+1}={\Lambda }^{-2}{\partial }_x{H}_k\left(u_k,{\rho }_k\right),\\ {\partial }_t{\rho }_{k+1}+\left(b_2{u}_k-2b_3{\rho }_k{u}_k\right){\partial }_x{\rho }_{k+1}=G_k\left(u_k,{\rho }_k\right),\\ u_{k+1}\left(0\right)=S_{k+1}u\left(0\right),{\rho }_{k+1}\left(0\right)=S_{k+1}\rho \left(0\right),\end{array}$ (3.8)

where

$H_k=b_1{\rho }_k-\frac{b_2}{2}{u}_k^2+b_3{\rho }_k{u}_k^2,$

and

$G_k=b_4{\partial }_x{u}_k-b_2{\rho }_k{\partial }_x{u}_k+b_3{\rho }_k^2{\partial }_x{u}_k.$

Moreover, there is a positive time T such that the solutions satisfy the following properties:

1) ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ is uniformly bounded in $E_{p\mathrm{,}r}^{s+1}\left(T\right)\times E_{p\mathrm{,}r}^s\left(T\right)$ .

2) ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ is a Cauchy sequence in $\mathcal{C}\left(\left[0,T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times \mathcal{C}\left(\left[0,T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ .

Proof (Proof of Lemma 3.2) Since all the data $S_{k+1}{u}_0$ and $S_{k+1}{\rho }_0$ belong to $B_{p\mathrm{,}r}^{\infty }$ , Lemma 2.7 indicates that for all $k\in \mathbb{N}$ , Equation (3.8) has a global solution in $\mathcal{C}\left({ℝ}^{+}\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{\infty }\right)\times \mathcal{C}\left({ℝ}^{+}\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{\infty }\right)$ .

Obviously, we can get the following inequality:

$\begin{array}{c}{\Vert u_{k+1}\Vert }_{B_{p\mathrm{,}r}^{s+1}}\le {\Vert u\left(0\right)\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\displaystyle {\int }_0^t}{\Vert {\Lambda }^{-2}{\partial }_x{H}_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}\text{d}\tau \\ \le {\Vert u\left(0\right)\Vert }_{B_{p\mathrm{,}r}^{s+1}}+C{\displaystyle {\int }_0^t}{\Vert {\rho }_k\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}^2+{\Vert {\rho }_k\Vert }_{B_{p\mathrm{,}r}^s}{\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}^2\text{d}\tau \mathrm{.}\end{array}$

Using Lemma 2.6 again, for $s>\max \left\{1+\frac{1}{p},\frac{3}{2}\right\}$ , we can see that:

${\Vert {\rho }_{k+1}\Vert }_{B_{p\mathrm{,}r}^s}\le e^{CV\left(t\right)}\left({\Vert \rho \left(0\right)\Vert }_{B_{p\mathrm{,}r}^s}+{\displaystyle {\int }_0^t}{e}^{-CV\left(\tau \right)}{\Vert G_k\left(\tau \right)\Vert }_{B_{p\mathrm{,}r}^s}\text{d}\tau \right)\mathrm{,}$

with

$\begin{array}{c}V\left(t\right)={\displaystyle {\int }_0^t}{\Vert {\partial }_x\left(b_2{u}_k-2b_3{\rho }_k{u}_k\right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}\text{d}\tau \\ \le C{\displaystyle {\int }_0^t}{\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert {\rho }_k\Vert }_{B_{p\mathrm{,}r}^s}{\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}\text{d}\tau \mathrm{.}\end{array}$

And with the help of Proposition 2.4 again, we have:

${\Vert G_k\left(\tau \right)\Vert }_{B_{p\mathrm{,}r}^s}\le C\left({\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert {\rho }_k\Vert }_{B_{p\mathrm{,}r}^s}{\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert {\rho }_k\Vert }_{B_{p\mathrm{,}r}^s}^2{\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}\right)\mathrm{.}$

Combining the above inequalities, we get:

$\begin{array}{c}{\Vert {\rho }_{k+1}\Vert }_{B_{p,r}^s}\le \exp \left[C{\displaystyle {\int }_0^t}{\Vert u_k\Vert }_{B_{p,r}^{s+1}}\left(1+{\Vert {\rho }_k\Vert }_{B_{p,r}^s}\right)\text{d}\tau \right]{\Vert \rho \left(0\right)\Vert }_{B_{p,r}^s}\\ +{\displaystyle {\int }_0^t}\exp \left[C{\displaystyle {\int }_{\tau }^t}{\Vert u_k\Vert }_{B_{p,r}^{s+1}}\left(1+{\Vert {\rho }_k\Vert }_{B_{p,r}^s}\right)\text{d}{\tau }^{\prime }\right]\\ \left[{\Vert u_k\Vert }_{B_{p,r}^{s+1}}\left(1+{\Vert {\rho }_k\Vert }_{B_{p,r}^s}+{\Vert {\rho }_k\Vert }_{B_{p,r}^s}^2\right)\right]\text{d}\tau ,\end{array}$

Therefore, if we define $U_k\left(t\right)={\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert {\rho }_k\Vert }_{B_{p\mathrm{,}r}^s}+1$ and $U_0={\Vert u\left(0\right)\Vert }_{B_{p,r}^{s+1}}+{\Vert \rho \left(0\right)\Vert }_{B_{p,r}^s}+1$ , then

$\parallel u_{k+1}{\parallel }_{B_{p\mathrm{,}r}^{s+1}}\le U_0+C{\displaystyle {\int }_0^t}{U}_k^{3}d\tau \mathrm{,}$ (3.9)

and

${\Vert {\rho }_{k+1}\Vert }_{B_{p\mathrm{,}r}^s}\le \exp \left(C{\displaystyle {\int }_0^t}{U}_k^2\left(\tau \right)\text{d}\tau \right)U_0+C{\displaystyle {\int }_0^t}\exp \left(C{\displaystyle {\int }_{\tau }^t}{U}_k^2\left({\tau }^{\prime }\right)\text{d}{\tau }^{\prime }\right)U_k^3\text{d}\tau \mathrm{.}$ (3.10)

Choosing $T^{\ast }=\frac{3}{256C{U}_0^2}$ , by induction, we show that:

$U_k\le \frac{4U_0}{\sqrt{1-64C{U}_0^{2}t}}\mathrm{,}\forall t\in \left[\mathrm{0,}{T}^{\ast }\right)\mathrm{.}$ (3.11)

In fact, suppose that (3.11) is valid for k, then for $0\le \tau <t<T^{\ast }=\frac{3}{256C{U}_0^2}<\frac{1}{64C{U}_0^2}$ , we have:

$\exp \left(C{\displaystyle {\int }_{\tau }^t}{U}_k^2\left({\tau }^{\prime }\right)\text{d}{\tau }^{\prime }\right)\le \exp \left({\displaystyle {\int }_{\tau }^t}\frac{16C{U}_0^2}{1-64C{U}_0^2{\tau }^{\prime }}\text{d}{\tau }^{\prime }\right)={\left(\frac{1-64C{U}_0^2\tau }{1-64C{U}_0^{2}t}\right)}^{\frac{1}{4}}\mathrm{,}$ (3.12)

and it is easy to see that:

$1\le \frac{1}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}\mathrm{.}$

Submitting (3.11) and (3.12) into (3.9) and (3.10), we obtain:

$\begin{array}{c}{\Vert u_{k+1}\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert {\rho }_{k+1}\Vert }_{B_{p\mathrm{,}r}^s}\le \frac{1}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}{\Vert u_{k+1}\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert {\rho }_{k+1}\Vert }_{B_{p\mathrm{,}r}^s}\\ \le \frac{2U_0+C{\displaystyle {\int }_0^t}{U}_k^3\text{d}\tau +C{\displaystyle {\int }_0^t}{\left(1-64C{U}_0^2\tau \right)}^{\frac{1}{4}}{U}_k^3\text{d}\tau }{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}\\ \le \frac{2U_0}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}+\frac{2U_0}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}\left(\frac{1}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}-1\right)\\ +\frac{4U_0}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}\left(\frac{1}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}-1\right)\\ \le \frac{4U_0}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}+\frac{U_0}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}\left(\frac{2}{{\left(1-64C{U}_0^{2}t\right)}^{\frac{1}{4}}}-4\right)\\ \le \frac{4U_0}{\sqrt{1-64C{U}_0^{2}t}}\mathrm{,}\end{array}$

in the last inequality we used that $t<T^{\ast }=\frac{3}{256C{U}_0^2}$ .

Therefore, ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ is uniformly bounded in $\mathcal{C}\left(\left[0,T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s+1}\right)\times \mathcal{C}\left(\left[0,T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ . Using Equation (3.8) and the similar argument in the proof Lemma 3.1, one can easily prove that ${\left({\partial }_t{u}_k\mathrm{,}{\partial }_t{\rho }_k\right)}_{k\in \mathbb{N}}$ is uniformly bounded in $\mathcal{C}\left(\left[0,T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times \mathcal{C}\left(\left[0,T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ , i.e. the sequence ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ is uniformly bounded in $E_{p\mathrm{,}r}^{s+1}\left(T\right)\times E_{p\mathrm{,}r}^s\left(T\right)$ .

Then, let us show that ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ is a Cauchy sequence in $\mathcal{C}\left(\left[0,T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times \mathcal{C}\left(\left[0,T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ . For all $k\mathrm{,}j\in \mathbb{N}$ , from (3.8), we can obtain that:

$\begin{array}{c}{\partial }_t\left(u_{k+j+1}-u_{k+1}\right)={\Lambda }^{-2}{\partial }_x(b_1\left({\rho }_{k+j}-{\rho }_k\right)-\frac{b_2}{2}\left(u_{k+j}+u_k\right)\left(u_{k+j}-u_k\right)\\ \stackrel{}{}+b_3\left({\rho }_{k+j}-{\rho }_k\right)u_{k+j}^2+b_3{\rho }_k\left(u_{k+j}+u_k\right)\left(u_{k+j}-u_k\right)),\end{array}$

and

${\partial }_t\left({\rho }_{k+j+1}-{\rho }_{k+1}\right)+\left(b_2{u}_{k+j}-2b_3{\rho }_{k+j}{u}_{k+j}\right){\partial }_x\left({\rho }_{k+j+1}-{\rho }_{k+1}\right)=G_{k\mathrm{,}j}\mathrm{,}$

with

$\begin{array}{c}{G}_{k,j}=-\left[b_2\left(u_{k+j}-u_k\right)-2b_3\left(\left({\rho }_{k+j}-{\rho }_k\right)u_{k+j}+\left(u_{k+j}-u_k\right){\rho }_k\right)\right]{\partial }_x{\rho }_{k+1}\\ +b_4{\partial }_x\left(u_{k+j}-u_k\right)-b_2\left(\left({\rho }_{k+j}-{\rho }_k\right){\partial }_x{u}_{k+j}+{\partial }_x\left(u_{k+j}-u_k\right){\rho }_k\right)\\ +b_3\left(\left({\rho }_{k+j}-{\rho }_k\right)\left({\rho }_{k+j}+{\rho }_k\right){\partial }_x{u}_{k+j}+{\partial }_x\left(u_{k+j}-u_k\right){\rho }_k^2\right).\end{array}$

Using the fact that $B_{p\mathrm{,}r}^{s-1}$ is an algebra and the operator $\Lambda$ is a S-multiplier, and let $V_{k+1}^j\left(t\right)\doteq {\Vert u_{k+j}-u_k\Vert }_{B_{p,r}^s}+{\Vert {\rho }_{k+j}-{\rho }_k\Vert }_{B_{p,r}^{s-1}}$ , for every $t\in \left[\mathrm{0,}{T}^{\ast }\right)$ , we have:

$\begin{array}{c}{\Vert u_{k+j+1}-u_{k+1}\Vert }_{B_{p\mathrm{,}r}^s}\le {\Vert u_{k+j+1}\left(0\right)-u_{k+1}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^s}+C{\displaystyle {\int }_0^t}{V}_k^j\left(\tau \right)\\ \cdot {\left(1+{\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert u_{k+j}\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert {\rho }_k\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{k+j}\Vert }_{B_{p\mathrm{,}r}^s}\right)}^2\text{d}\tau \mathrm{,}\end{array}$

moreover, using Lemma 2.6 again, we get:

$\begin{array}{c}{\Vert {\rho }_{k+j+1}-{\rho }_{k+1}\Vert }_{B_{p\mathrm{,}r}^{s-1}}\le \exp \left(C{\displaystyle {\int }_0^t}\left({\Vert u_{k+j}\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert u_{k+j}\Vert }_{B_{p\mathrm{,}r}^{s+1}}{\Vert {\rho }_{k+j}\Vert }_{B_{p\mathrm{,}r}^s}\right)\text{d}\tau \right)\\ \cdot {\Vert {\rho }_{k+j+1}\left(0\right)-{\rho }_{k+1}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^{s-1}}\\ +{\displaystyle {\int }_0^t}\exp \left(C{\displaystyle {\int }_{\tau }^t}\left({\Vert u_{k+j}\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert u_{k+j}\Vert }_{B_{p\mathrm{,}r}^{s+1}}{\Vert {\rho }_{k+j}\Vert }_{B_{p\mathrm{,}r}^s}\right)\text{d}{\tau }^{\prime }\right)V_k^j\left(\tau \right)\\ \cdot {\left(1+{\Vert u_k\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert u_{k+j}\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert {\rho }_k\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\rho }_{k+j}\Vert }_{B_{p\mathrm{,}r}^s}\right)}^2\text{d}\tau \mathrm{.}\end{array}$

By using Proposition 2.1, we can obtain that:

$\begin{array}{c}{\Vert {\rho }_{k+j+1}\left(0\right)-{\rho }_{k+1}\left(0\right)\Vert }_{B_{p,r}^{s-1}}={\Vert S_{k+j+1}\rho \left(0\right)-S_{k+1}\rho \left(0\right)\Vert }_{B_{p,r}^{s-1}}\\ ={\Vert {\displaystyle {\sum }_{d=k+1}^{k+j}{\Delta }_d{u}_0}\Vert }_{B_{p,r}^{s-1}}\\ ={\left({\displaystyle {\sum }_{k\ge -1}{2}^{k\left(s-1\right)\sigma }}{\Vert {\Delta }_k\left({\displaystyle {\sum }_{d=k+1}^{k+j}{\Delta }_d{u}_0}\right)\Vert }_{L^p}^{\sigma }\right)}^{\frac{1}{\sigma }}\\ \le C{\left({\displaystyle {\sum }_{d=k}^{k+j+1}{2}^{-d\sigma }{2}^{d\sigma s}}{\Vert {\Delta }_d{u}_0\Vert }_{L^p}^{\sigma }\right)}^{\frac{1}{\sigma }}\\ \le C{2}^{-k}{\Vert {\rho }_0\Vert }_{B_{p,r}^s},\end{array}$

and

${\Vert u_{k+j+1}\left(0\right)-u_{k+1}\left(0\right)\Vert }_{B_{p\mathrm{,}r}^s}\le C{2}^{-k}{\Vert u_0\Vert }_{B_{p\mathrm{,}r}^{s+1}}\mathrm{.}$

In view of ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ being uniformly bounded in $E_{p\mathrm{,}r}^{s+1}\left(T\right)\times E_{p\mathrm{,}r}^s\left(T\right)$ , and combining the above inequalities, one may find a positive constant $C_T$ independent of $k\mathrm{,}j$ such that:

$V_{k+1}^j\left(t\right)\le C_T\left(2^{-k}+{\displaystyle {\int }_0^t}{V}_k^j\left(\tau \right)\text{d}\tau \right)\mathrm{,}$

for all $t\in \left[\mathrm{0,}T\right)$ . Moreover, using the induction procedure with respect to the index k, we have:

$\begin{array}{l}{\Vert u_{k+j+1}-u_{k+1}\Vert }_{L_T^{\infty }\left(B_{p,r}^s\right)}+{\Vert {\rho }_{k+j+1}-{\rho }_{k+1}\Vert }_{L_T^{\infty }\left(B_{p,r}^{s-1}\right)}\\ \le \frac{{\left(T{C}_T\right)}^{k+1}}{\left(k+1\right)!}\left({\Vert u_j\Vert }_{L_T^{\infty }\left(B_{p,r}^{s+1}\right)}+{\Vert {\rho }_j\Vert }_{L_T^{\infty }\left(B_{p,r}^s\right)}\right)+{\displaystyle {\sum }_{n=0}^k{2}^{-\left(k-n\right)}}\frac{{\left(T{C}_T\right)}^{n+1}}{\left(n+1\right)!}.\end{array}$

Since that ${\Vert u_j\Vert }_{L_T^{\infty }\left(B_{p\mathrm{,}r}^{s+1}\right)}\mathrm{,}{\Vert {\rho }_j\Vert }_{L_T^{\infty }\left(B_{p\mathrm{,}r}^s\right)}$ may be bounded independently of j, we conclude to the existence of some new constant ${C^{\prime }}_T$ independent of $k\mathrm{,}j$ such that:

${\Vert u_{k+j+1}-u_{k+1}\Vert }_{L_T^{\infty }\left(B_{p\mathrm{,}r}^s\right)}+{\Vert {\rho }_{k+j+1}-{\rho }_{k+1}\Vert }_{L_T^{\infty }\left(B_{p\mathrm{,}r}^{s-1}\right)}\le {C^{\prime }}_T{2}^{-k}\mathrm{.}$

Thus, we have proved that ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ is a Cauchy sequence in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ . Hence, the proof of Lemma 3.2 is complete.

Finally, we prove the existence and uniqueness for (1.2) in Besov space.

Proof (Proof of Theorem 1.1) By using lemma 3.2, ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ is a Cauchy sequence in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ , so it converges to some limit function $\left(u\mathrm{,}\rho \right)\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ . Next, we have to prove that $\left(u\mathrm{,}\rho \right)\in E_{p\mathrm{,}r}^{s+1}\left(T\right)\times E_{p\mathrm{,}r}^s\left(T\right)$ and solves (1.2). Using lemma 3.2 again, we can see that ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ is uniformly bounded in $L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s+1}\right)\times L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ . Fatou property for Besov spaces (Proposition 2.5 (6)) insures that $\left(u\mathrm{,}\rho \right)$ also belongs to $L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s+1}\right)\times L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ .

On the other hand, as ${\left(u_k\mathrm{,}{\rho }_k\right)}_{k\in \mathbb{N}}$ converges to $\left(u\mathrm{,}\rho \right)$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ , an interpolation argument guarantees that the convergence holds in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }+1}\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }}\right)$ , for any $s^{\prime }<s$ .

Passing limit in (3.8) reveals that $\left(u\mathrm{,}\rho \right)$ satisfy system (1.2). In view of the fact that $\left(u\mathrm{,}\rho \right)$ belongs to $L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s+1}\right)\times L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ , for $s>1+\frac{1}{p}$ , $B_{p\mathrm{,}r}^{s+1}$ and $B_{p\mathrm{,}r}^s$ are algebras, we obtain that the right-hand side of the equation:

$u_t={\Lambda }^{-2}\left(b_1{\rho }_x-b_{2}u{u}_x+b_3{\left(\rho u^2\right)}_x\right)\mathrm{,}$

belongs to $L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s+1}\right)$ , and the right-hand side of the second equation:

${\rho }_t+\left(b_2-2b_3\rho \right)u{\rho }_x=b_4{u}_x-b_2\rho u_x+b_3{\rho }^2{u}_x\mathrm{,}$

belongs to $L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ .

In particular, for the case $r<\infty$ , Lemma 2.6 implies that $\left(u\mathrm{,}\rho \right)\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }+1}\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }}\right)$ for any $s^{\prime }<s$ . Finally, by using the equation again, we see that $\left({\partial }_{t}u\mathrm{,}{\partial }_t\rho \right)\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ if $r<\infty$ , and in $L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\times L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)$ otherwise. Therefore, the pair $\left(u\mathrm{,}\rho \right)\in E_{p\mathrm{,}r}^{s+1}\left(T\right)\times E_{p\mathrm{,}r}^s\left(T\right)$ .

The continuity with respect to the initial data in:

$\left[\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }+1}\right)\cap {\mathcal{C}}^{\prime }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }}\right)\right]\times \left[\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }-1}\right)\cap {\mathcal{C}}^{\prime }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }-2}\right)\right]\mathrm{,}$

for all $s^{\prime }<s$ , can be obtained by Lemma 3.1 and a simple interpolation argument. The case $s^{\prime }=s$ can be proved through the use of a sequence of viscosity approximation solutions ${\left(u_{\epsilon },{\rho }_{\epsilon }\right)}_{\epsilon >0}$ for System (1.2) which converges uniformly in:

$\left[\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s+1}\right)\cap {\mathcal{C}}^{\prime }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\right]\times \left[\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)\cap {\mathcal{C}}^{\prime }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-2}\right)\right]\mathrm{,}$

gives the continuity of solution $\left(u\mathrm{,}\rho \right)$ in $E_{p\mathrm{,}r}^{s+1}\left(T\right)\times E_{p\mathrm{,}r}^s\left(T\right)$ . Hence, the proof of Theorem 1.1 is complete.

4. Ill-Posedness in Besov Space

Next, we select the appropriate initial data to complete the proof of the Theorem 1.3. And we choose the initial data:

${\rho }_0\left(x\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{2}^{-ns}\varphi \left(x\right)\cos \left(\lambda 2^{n}x\right),$

$u_0\left(x\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{2}^{-n\left(s+1\right)}\varphi \left(x\right)\cos \left(\lambda 2^{n}x\right),$

here $\lambda \in \left[\frac{67}{48}\mathrm{,}\frac{69}{48}\right]$ and $\hat{\varphi }\in {\mathcal{C}}_0^{\infty }\left(ℝ\right)$ is an non-negative, even and real-valued function satisfying:

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

It is easy to prove that:

$\text{supp}\varphi (\cdot )\widehat{\cos \left(\lambda 2^n\cdot \right)}\subset \left\{\xi \mathrm{<mo>\; :\; </mo>}-\frac{1}{2}+\lambda 2^n\le \left|\xi \right|\le \frac{1}{2}+\lambda 2^n\right\}\mathrm{,}$

then it can be verified for $j\ge 3$ ,

${\Delta }_j\left(\varphi \left(x\right)\cos \left(\lambda 2^{n}x\right)\right)=\{\begin{array}{ll}\varphi \left(x\right)\cos \left(\lambda 2^{n}x\right)\mathrm{,}\hfill & j=n\mathrm{,}\hfill \\ \mathrm{0,}\hfill & j\ne n\mathrm{,}\hfill \end{array}$ (4.1)

so we have:

$\begin{array}{c}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^{s+1}}={\Vert {\left(2^{j\left(s+1\right)}{\Vert {\Delta }_j{u}_0\Vert }_{L^p}\right)}_{j\in \mathbb{Z}}\Vert }_{l^{\infty }}\\ ={\Vert {\left(2^{j\left(s+1\right)}{\Vert {\Delta }_j{\displaystyle \underset{n\mathrm{<mo>\; =\; </mo>0}}{\overset{\infty }{\sum }}}{2}^{-n\left(s+1\right)}\varphi \left(x\right)\cos \left(\lambda 2^{n}x\right)\Vert }_{L^p}\right)}_{j\in \mathbb{Z}}\Vert }_{l^{\infty }}\\ ={\Vert {\left(2^{j\left(s+1\right)}{\Vert 2^{-j\left(s+1\right)}\varphi \left(x\right)\cos \left(\lambda 2^{j}x\right)\Vert }_{L^p}\right)}_{j\in \mathbb{Z}}\Vert }_{l^{\infty }}\\ \le {\Vert {\Vert \varphi \left(x\right)\cos \left(\lambda 2^{j}x\right)\Vert }_{L^p}\Vert }_{l^{\infty }}\\ \le C\mathrm{.}\end{array}$

Similarly, we have:

${\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\le C\mathrm{,}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^{s+1}}\le C\mathrm{.}$ (4.2)

Next, we give several estimates that play an important role in the proof of the Theorem 1.3.

Lemma 4.1. Let $s>0$ . Then, for the above constructed initial data $\left(u_0\mathrm{,}{\rho }_0\right)$ , we have:

${\Vert u_0{\Delta }_n{\partial }_x{u}_0\Vert }_{L^p}\ge C{2}^{-ns}\mathrm{,}$ (4.3)

and

${\Vert u_0{\Delta }_n{\partial }_x{\rho }_0\Vert }_{L^p}\ge C{2}^{-n\left(s-1\right)}\mathrm{,}$ (4.4)

for n large enough.

Proof We only show (4.4), because (4.3) can be obtained in a similar way.

According to (4.1), we get:

${\Delta }_n{\rho }_0={\Delta }_n\left({\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{2}^{-ns}\varphi \left(x\right)\cos \left(\lambda 2^{n}x\right)\right)=2^{-ns}\varphi \left(x\right)\cos \left(\lambda 2^{n}x\right),$

therefore, one has:

$\begin{array}{c}{u}_0{\Delta }_n{\partial }_x{\rho }_0=u_0{2}^{-ns}{\partial }_x\left(\varphi \left(x\right)\cos \left(\lambda 2^{n}x\right)\right)\\ =u_0{2}^{-ns}\left[{\partial }_x\varphi \cos \left(\lambda 2^{n}x\right)-\varphi \left(x\right)\sin \left(\lambda 2^{n}x\right)\lambda 2^n\right]\\ =2^{-ns}{u}_0{\partial }_x\varphi \cos \left(\lambda 2^{n}x\right)-\lambda 2^{-n\left(s-1\right)}{u}_0\varphi \left(x\right)\sin \left(\lambda 2^{n}x\right).\end{array}$

Since $u_0\left(x\right)$ is a real valued continuous function on $ℝ$ , then there exists $\sigma >0$ , we have:

$\left|u_0\left(x\right)\right|\ge \frac{1}{2}{u}_0\left(0\right)=\frac{1}{2}{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{2}^{-n\left(s+1\right)}\varphi \left(0\right)=\frac{2^s\varphi \left(0\right)}{2^{s+1}-1},$ (4.5)

for any $x\in B_{\sigma }\left(0\right)$ .

Therefore, we derive from (4.5) that:

$\begin{array}{c}{\Vert u_0{\Delta }_n{\partial }_x{\rho }_0\Vert }_{L^p}\ge C\left(2^{-n\left(s-1\right)}{\Vert \varphi (\cdot )\sin \left(\lambda 2^n\cdot \right)\Vert }_{L^p\left(B_{\sigma }\left(0\right)\right)}-2^{-ns}{\Vert u_0{\partial }_x\varphi \cos \left(\lambda 2^{n}x\right)\Vert }_{L^p}\right)\\ \ge \left(C{2}^n-C_1\right)2^{-ns}\mathrm{.}\end{array}$

By choosing n large enough such that $C_1\le C{2}^{n-1}$ , then we can yield (4.4). So, Lemma 4.1 has been proved.

Lemma 4.2. Let $s>\max \left\{1+\frac{1}{p},\frac{3}{2}\right\}$ . For the above constructed initial data $\left(u_0\mathrm{,}{\rho }_0\right)$ , then there exists some $T\mathrm{<mo>\; =\; </mo>}T\left({\Vert u_0\Vert }_{B_{p\mathrm{,}r}^{s+1}}\mathrm{,}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}r}^s}\right)$ , for $0\le t\le T$ , we have:

${\Vert u\left(t\right)-u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\le Ct\mathrm{,}{\Vert \rho \left(t\right)-{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le Ct\mathrm{.}$ (4.6)

Proof Since $\left(u_0\mathrm{,}{\rho }_0\right)\in B_{p\mathrm{,}\infty }^{s+1}\times B_{p\mathrm{,}\infty }^s$ and according to the local existence result (see Theorem 1.1), hence the system (1.2) has a unique solution $\left(u\mathrm{,}\rho \right)\in L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}\infty }^{s+1}\times B_{p\mathrm{,}\infty }^s\right)$ for some $T\mathrm{<mo>\; =\; </mo>}T\left({\Vert u_0\Vert }_{B_{p\mathrm{,}r}^{s+1}}\mathrm{,}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}r}^s}\right)$ , and

$\underset{0\le t\le T}{\sup }\left({\Vert u\left(t\right)\Vert }_{B_{p\mathrm{,}\infty }^{s+1}}+{\Vert \rho \left(t\right)\Vert }_{B_{p\mathrm{,}\infty }^s}\right)\le C\left({\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^{s+1}}+{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\right)\mathrm{.}$ (4.7)

Using the differential mean value theorem, the Minkowski inequality, Proposition 2.4 together with (4.7) for $t\in \left[\mathrm{0,}T\right]$ , we have:

$\begin{array}{c}{\Vert \rho -{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le {\displaystyle {\int }_0^t}{\Vert {\partial }_{\tau }\rho \Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\text{d}\tau \\ \le C{\displaystyle {\int }_0^t}{\Vert u\Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert u\rho \Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert {\rho }^{2}u\Vert }_{B_{p\mathrm{,}\infty }^s}\text{d}\tau \\ \le C{\displaystyle {\int }_0^t}{\Vert u\Vert }_{B_{p\mathrm{,}\infty }^{s+1}}\left(1+{\Vert \rho \Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert \rho \Vert }_{B_{p\mathrm{,}\infty }^s}^2\right)\text{d}\tau \\ \le Ct\mathrm{,}\end{array}$

and

${\Vert u-u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\le C{\displaystyle {\int }_0^t}{\Vert \rho \Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert u^2\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert \rho u^2\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\text{d}\tau \le Ct\mathrm{.}$

Therefore, we have completed the proof of Lemma 4.2.

Lemma 4.3. Under the assumption of Theorem 1.3, for all $0\le t\le T$ , we have:

${\Vert \rho \left(t\right)-{\rho }_0-t{v}_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le C{t}^2\mathrm{,}{\Vert u\left(t\right)-u_0-t{w}_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le C{t}^2\mathrm{.}$ (4.8)

where

$v_0=b_4{\partial }_x{u}_0-b_2{\partial }_x\left({\rho }_0{u}_0\right)+b_3{\partial }_x\left({\rho }_0^2{u}_0\right)\mathrm{,}$

$w_0={\Lambda }^{-2}\left(b_1{\partial }_x{\rho }_0-b_2{u}_0{\partial }_x{u}_0+b_3{\partial }_x\left({\rho }_0{u}_0^2\right)\right)\mathrm{.}$

Proof We denote that:

$\{\begin{array}{l}\tilde{\rho }=\rho \left(t\right)-{\rho }_0-t{v}_0,\\ \tilde{u}=u\left(t\right)-u_0-t{w}_0.\end{array}$

Firstly, using the differential mean value theorem and the Minkowski inequality for $t\in \left[\mathrm{0,}T\right]$ , using Proposition 2.4, Proposition 2.5 and (4.7), we get:

$\begin{array}{c}{\Vert \tilde{\rho }\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le {\displaystyle {\int }_0^t}{\Vert {\partial }_{\tau }\rho -v_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\text{d}\tau \\ \le C{\displaystyle {\int }_0^t}{\Vert u-u_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert u\rho -u_0{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert u{\rho }^2-u_0{\rho }_0^2\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\text{d}\tau \\ \le C{\displaystyle {\int }_0^t}\tau \text{d}\tau \le C{t}^2\mathrm{,}\end{array}$ (4.9)

and

${\Vert \tilde{u}\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le {\displaystyle {\int }_0^t}{\Vert {\partial }_{\tau }u-w_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\text{d}\tau \le C{\displaystyle {\int }_0^t}\tau \text{d}\tau \le C{t}^2\mathrm{,}$ (4.10)

Thus, the proof of Lemma 4.3 is completed.

Proof (Proof of Theorem 1.3.) Based on the definition of the Besov norm, we have:

$\begin{array}{c}{\Vert \rho -{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\ge 2^{ns}{\Vert {\Delta }_n\left(\rho -{\rho }_0\right)\Vert }_{L^p}=2^{ns}{\Vert {\Delta }_n\left(\tilde{\rho }+t{v}_0\right)\Vert }_{L^p}\\ \ge t{2}^{ns}{\Vert {\Delta }_n{v}_0\Vert }_{L^p}-2^{ns}{\Vert {\Delta }_n\tilde{\rho }\Vert }_{L^p}\\ \ge t{2}^{ns}{\Vert {\Delta }_n\left(u_0{\partial }_x{\rho }_0\right)\Vert }_{L^p}-Ct{2}^{ns}{\Vert {\Delta }_n{\rho }_0{\partial }_x{u}_0\Vert }_{L^p}-2^{ns}{\Vert {\Delta }_n\tilde{\rho }\Vert }_{L^p}\\ \ge t{2}^{ns}{\Vert {\Delta }_n\left(u_0{\partial }_x{\rho }_0\right)\Vert }_{L^p}-Ct{\Vert {\rho }_0{\partial }_x{u}_0\Vert }_{B_{p\mathrm{,}\infty }^s}-C{2}^{2n}{\Vert \tilde{\rho }\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\mathrm{.}\end{array}$ (4.11)

Since

$\begin{array}{c}{\Delta }_n\left(u_0{\partial }_x{\rho }_0\right)={\Delta }_n\left(u_0{\partial }_x{\rho }_0\right)-u_0{\Delta }_n\left({\partial }_x{\rho }_0\right)+u_0{\Delta }_n\left({\partial }_x{\rho }_0\right)\\ =\left[{\Delta }_n,u_0\right]{\partial }_x{\rho }_0+u_0{\Delta }_n\left({\partial }_x{\rho }_0\right),\end{array}$

Using Proposition 2.5, Lemmas 2.8 and (4.2), we deduce that:

${\Vert {\rho }_0{\partial }_x{u}_0\Vert }_{B_{p\mathrm{,}\infty }^s}\le C{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^{s+1}}\le C\mathrm{,}$

and

${\Vert 2^{ns}{\Vert \left[{\Delta }_n\mathrm{,}{u}_0\right]{\partial }_x{\rho }_0\Vert }_{L^p}\Vert }_{l^{\infty }}\le C\left({\Vert {\partial }_x{u}_0\Vert }_{L^{\infty }}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert {\partial }_x{\rho }_0\Vert }_{L^{\infty }}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\right)\le C\mathrm{.}$

Taking above estimates into (4.11), we have:

${\Vert \rho -{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\ge t{2}^{ns}{\Vert u_0{\Delta }_n\left({\partial }_x{\rho }_0\right)\Vert }_{L^p}-C_{1}t-C_2{2}^{2n}{\Vert \tilde{\rho }\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\mathrm{.}$

And using Lemma 4.1 and Lemma 4.3, we have:

${\Vert \rho -{\rho }_0\Vert }_{B_{p,\infty }^s}\ge C_{3}t{2}^{ns}{2}^{-n\left(s-1\right)}-C_{1}t-C_2{2}^{2n}{t}^2\ge C_{3}t{2}^n-C_{1}t-C_2{2}^{2n}{t}^2.$

Choosing n large enough such that $C_3{2}^n>2C_1$ , then we get:

${\Vert \rho -{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\ge \frac{C_{3}t{2}^n}{2}-C_2{2}^{2n}{t}^2\mathrm{.}$

Picking $t{2}^n\approx \delta <\frac{C_3}{4C_2}$ when $t\to 0$ , we have:

${\Vert \rho -{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\ge \frac{C_3}{2}\delta -C_2{\delta }^2\ge \frac{C_3}{4}\delta \mathrm{.}$

Likewise, we have:

${\Vert u-u_0\Vert }_{B_{p\mathrm{,}\infty }^{s+1}}\ge \frac{C_4}{4}\delta \mathrm{.}$

This completes the proof of Theorem 1.3.

5. Gevrey Regularity for System (1.2)

In this section, we will apply nonlinear Cauchy-Kowalevski theory to establish the existence of analytic solutions to system (1.2). To this purpose, it is necessary to introduce the Cauchy-Kowalevski theorem.

Theorem 5.1. (See [11] [12] ) Let ${\left(X_{\delta }\mathrm{,}{\Vert \cdot \Vert }_{\delta }\right)}_{0<\delta \le 1}$ be a scale of decreasing Banach spaces, such that for any $0<{\delta }^{\prime }<\delta \le 1$ , we have $X_{\delta }\subset X_{{\delta }^{\prime }}$ with ${\Vert \cdot \Vert }_{{\delta }^{\prime }}<{\Vert \cdot \Vert }_{\delta }$ . Consider the Cauchy problem:

$\{\begin{array}{l}\frac{\text{d}u}{\text{d}t}=F\left(t,u\left(t\right)\right),\\ u\left(0\right)=0.\end{array}$ (5.1)

Let $T,R>0$ and $\sigma \ge 1$ . For given $u_0\in X_1$ , assume that F satisfies the following conditions:

(H₁) If for any $0<{\delta }^{\prime }<\delta <1$ , the function $t\mapsto u\left(t\right)$ is holomorphic on $\left|t\right|<T$ and continuous on $\left|t\right|\le T$ with values in $X_{\delta }$ and

$\underset{\left|t\right|<T}{\sup }{\Vert u\left(t\right)\Vert }_{\delta }<R,$

then $t\mapsto F\left(t\mathrm{,}u\left(t\right)\right)$ is a holomorphic function on $\left|t\right|<T$ with values in $X_{{\delta }^{\prime }}$ .

(H₂) For any $0<{\delta }^{\prime }<\delta <1$ and $u\mathrm{,}v\in \overline{B\left(u_0\mathrm{,}R\right)}\subset X_{\delta }$ , there exists a positive constant L depending on $u_0$ and R such that:

$\underset{\left|t\right|<T}{\sup }{\Vert F\left(t\mathrm{,}u\right)-F\left(t\mathrm{,}v\right)\Vert }_{{\delta }^{\prime }}\le \frac{L}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}{\Vert u-v\Vert }_{\delta }\mathrm{.}$

(H₃) There exists a $M>0$ depending on $u_0$ and R such that for any $0<\delta <1$ ,

$\underset{\left|t\right|<T}{\sup }{\Vert F\left(t,0\right)\Vert }_{\delta }\le \frac{M}{{\left(1-\delta \right)}^{\sigma }}.$

Then, there exists a $T_0\in \left(\mathrm{0,}T\right)$ and a unique function $u\left(t\right)$ to the Cauchy problem (5.1), which is holomorphic in $\left|t\right|<\frac{{\left(1-\delta \right)}^{\sigma }{T}_0}{2^{\sigma }-1}$ with values in $X_{\delta }$ for every $\delta \in \left(\mathrm{0,1}\right)$ .

We would like to include four properties of the spaces $G_{\sigma \mathrm{,}s}^{\delta }$ , which will be used in the proof of Theorem 1.5 (the proofs of these properties can be find in [12] ).

Proposition 5.2. Let $0<{\delta }^{\prime }<\delta$ , $0<{\sigma }^{\prime }<\sigma$ and $s^{\prime }<s$ . From Definition 1.3, one can check that $G_{\sigma \mathrm{,}s}^{\delta }$ ↪ $G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}$ , $G_{{\sigma }^{\prime }\mathrm{,}s}^{\delta }$ ↪ $G_{\sigma \mathrm{,}s}^{\delta }$ and $G_{\sigma \mathrm{,}s}^{\delta }$ ↪ $G_{\sigma \mathrm{,}{s}^{\prime }}^{\delta }$ .

Proposition 5.3. Let s be a real number and $\sigma >0$ . Assume that $0<{\delta }^{\prime }<\delta$ . Then, we have:

${\Vert {\partial }_{x}f\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}\le \frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}{\Vert f\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}\mathrm{.}$ (5.2)

Proposition 5.4. (Product acts on Sobolev-Gevrey spaces with $d=1$ ) Let $s>\frac{1}{2},\sigma \ge 1$ and $\delta >0$ . Then, $G_{\sigma \mathrm{,}s}^{\delta }$ is an algebra. Moreover, there exists a constant $\overline{C_s}$ such that:

${\Vert fg\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}\le \overline{C_s}{\Vert f\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}{\Vert g\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}\mathrm{.}$ (5.3)

Proposition 5.5. Let $s>\frac{1}{2},\sigma \ge 1$ and $\delta >0$ . There exists a constant $C_s$ such that”

${\Vert fg\Vert }_{G_{\sigma \mathrm{,}s-1}^{\delta }}\le C_s{\Vert f\Vert }_{G_{\sigma \mathrm{,}s-1}^{\delta }}{\Vert g\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}\mathrm{.}$ (5.4)

Now, we use all the above tools to prove Theorem 1.5.

Proof (Proof of Theorem 1.5) Assume that:

$F\left(z\right)\doteq \left(\begin{array}{c}{F}_1\left(z\right)\\ F_2\left(z\right)\end{array}\right)=\left(\begin{array}{c}{\Lambda }^{-2}\left(b_1{\rho }_x-b_{2}u{u}_x+b_3{u}^2{\rho }_x+2b_3\rho u{u}_x\right)\\ b_4{u}_x-b_2\rho u_x-b_{2}u{\rho }_x+b_3{\rho }^2{u}_x+2b_3\rho u{\rho }_x\end{array}\right).$

By using Definition 1.3 and Proposition 5.2, we can see that $G_{{\sigma }^{\prime }\mathrm{,}s}^{\delta }$ is a scale of decreasing Banach spaces for a fixed $\sigma \ge 1$ , $s>\frac{1}{2}$ and $0<\delta \le 1$ . Moreover, for $0<{\delta }^{\prime }<\delta$ , $\sigma \ge 1$ , $s>\frac{1}{2}$ and the estimates (5.2)-(5.4), we have:

$\begin{array}{c}{\Vert F_2\left(u\mathrm{,}\rho \right)\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}\le {\Vert b_4{u}_x-b_2\rho u_x-b_{2}u{\rho }_x+b_3{\rho }^2{u}_x+2b_3\rho u{\rho }_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}\\ \le C_{b_2\mathrm{,}{b}_3\mathrm{,}{b}_4}\left({\Vert u_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}+{\Vert \rho u_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}+{\Vert u{\rho }_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}+{\Vert {\rho }^2{u}_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}+{\Vert \rho u{\rho }_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}\right)\\ \le C_{s\mathrm{,}{b}_2\mathrm{,}{b}_3\mathrm{,}{b}_4}\frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}\left({\Vert \rho \Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}\right){\left({\Vert \rho \Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}+1\right)}^2\mathrm{,}\end{array}$

and

$\begin{array}{c}{\Vert F_1\left(u,\rho \right)\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}\le {\Vert b_1{\rho }_x-b_{2}u{u}_x+b_3{u}^2{\rho }_x+2b_3\rho u{u}_x\Vert }_{G_{\sigma ,s-2}^{{\delta }^{\prime }}}\\ \le C_{s,b_1,b_2,b_3}\frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}\left({\Vert \rho \Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma ,s}^{\delta }}\right){\left({\Vert \rho \Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma ,s}^{\delta }}+1\right)}^2,\end{array}$

where $C_{s\mathrm{,}{b}_1\mathrm{,}{b}_2\mathrm{,}{b}_3\mathrm{,}{b}_4}$ is a constant depends only on $s\mathrm{,}{b}_1\mathrm{,}{b}_2\mathrm{,}{b}_3\mathrm{,}{b}_4$ . Therefore, we can obtain that:

$\begin{array}{l}{\Vert F_1\left(u,\rho \right)\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}+{\Vert F_2\left(u,\rho \right)\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}\\ \le C_{s,b_1,b_2,b_3,b_4}\frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}\left({\Vert \rho \Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma ,s}^{\delta }}\right){\left({\Vert \rho \Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma ,s}^{\delta }}+1\right)}^2,\end{array}$

Similarly, for any $0<\delta <1$ , we also can obtain that:

$\begin{array}{l}{\Vert F_1\left(u_0\mathrm{,}{\rho }_0\right)\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}+{\Vert F_2\left(u_0\mathrm{,}{\rho }_0\right)\Vert }_{G_{\sigma \mathrm{,}s}^{\delta }}\\ \le C_{s\mathrm{,}{b}_1\mathrm{,}{b}_2\mathrm{,}{b}_3\mathrm{,}{b}_4}\frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(1-\delta \right)}^{\sigma }}\left({\Vert {\rho }_0\Vert }_{G_{\sigma \mathrm{,}s}^1}+{\Vert u_0\Vert }_{G_{\sigma \mathrm{,}s}^1}\right){\left({\Vert {\rho }_0\Vert }_{G_{\sigma \mathrm{,}s}^1}+{\Vert u_0\Vert }_{G_{\sigma \mathrm{,}s}^1}+1\right)}^2\mathrm{.}\end{array}$

Therefore, these spaces and $F\left(u\mathrm{,}\rho \right)$ satisfy condition (H₁) and (H₃) in Theorem 5.1.

Next, in order to prove our desire result, it suffices to show that $F\left(u\mathrm{,}\rho \right)$ satisfies the condition (H₂) in Theorem 5.1. Assume that ${\Vert z-z_0\Vert }_{\delta }\le R$ and ${\Vert z^{\prime }-{z^{\prime }}_0\Vert }_{\delta }\le R$ with ${\Vert z\Vert }_{\delta }={\Vert \rho \Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma ,s}^{\delta }}$ , for $0<{\delta }^{\prime }<\delta$ , $s>\frac{1}{2}$ , we can see that:

$\begin{array}{l}{\Vert F_2\left(z\right)-F_2\left(z^{\prime }\right)\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}\\ \le {\Vert b_4{u}_x-b_2\rho u_x-b_{2}u{\rho }_x+b_3{\rho }^2{u}_x+2b_3\rho u{\rho }_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}\\ \le C_{s,b_2,b_3,b_4}({\Vert u_x-{u^{\prime }}_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}+{\Vert \left(\rho -{\rho }^{\prime }\right)u_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}+{\Vert {\rho }^{\prime }\left(u_x-{u^{\prime }}_x\right)\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}\\ +{\Vert \left(u-u^{\prime }\right){\rho }_x\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}+{\Vert u^{\prime }\left({\rho }_x-{{\rho }^{\prime }}_x\right)\Vert }_{G_{\sigma \mathrm{,}s}^{{\delta }^{\prime }}}+{\Vert \left({\rho }^2-{\left({\rho }^{\prime }\right)}^2\right)u_x\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}\\ +{\Vert {\left({\rho }^{\prime }\right)}^2\left(u_x-{u^{\prime }}_x\right)\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}+{\Vert \left(\rho u-{\rho }^{\prime }{u}^{\prime }\right){\rho }_x\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}+{\Vert {\rho }^{\prime }{u}^{\prime }\left({\rho }_x-{{\rho }^{\prime }}_x\right)\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}})\end{array}$

$\begin{array}{l}\le C_{s,b_2,b_3,b_4}\frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}\left({\Vert u-u^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert \rho -{\rho }^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}\right)\\ {\left({\Vert \rho \Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert {\rho }^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}+1\right)}^2,\end{array}$

and

$\begin{array}{c}{\Vert F_1\left(z\right)-F_1\left(z^{\prime }\right)\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}\le C_{s,b_1,b_2,b_3}\frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}\left({\Vert u-u^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert \rho -{\rho }^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}\right)\\ {\left({\Vert \rho \Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert {\rho }^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}+1\right)}^2.\end{array}$

Thus,

$\begin{array}{l}{\Vert F\left(z\right)-F\left(z^{\prime }\right)\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}\\ \le {\Vert F_1\left(z\right)-F_1\left(z^{\prime }\right)\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}+{\Vert F_2\left(z\right)-F_2\left(z^{\prime }\right)\Vert }_{G_{\sigma ,s}^{{\delta }^{\prime }}}\\ \le C_{s,b_1,b_2,b_3,b_4}\frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}\left({\Vert u-u^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert \rho -{\rho }^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}\right)\\ {\left({\Vert \rho \Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert {\rho }^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u\Vert }_{G_{\sigma ,s}^{\delta }}+{\Vert u^{\prime }\Vert }_{G_{\sigma ,s}^{\delta }}+1\right)}^2\\ \le C_{s,b_1,b_2,b_3,b_4}\frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}\left({\Vert z-z^{\prime }\Vert }_{\delta }\right){\left({\Vert z\Vert }_{\delta }+{\Vert z^{\prime }\Vert }_{\delta }+1\right)}^2\\ \le C_{s,b_1,b_2,b_3,b_4}\frac{e^{-\sigma }{\sigma }^{\sigma }}{{\left(\delta -{\delta }^{\prime }\right)}^{\sigma }}\left({\Vert z-z^{\prime }\Vert }_{\delta }\right){\left(2{\Vert z_0\Vert }_{\delta }+2R+1\right)}^2,\end{array}$

where in the last inequality we apply the fact that ${\Vert z-z_0\Vert }_{\delta }\le R$ and ${\Vert z^{\prime }-{z^{\prime }}_0\Vert }_{\delta }\le R$ . From the above inequality, for $0<{\delta }^{\prime }<\delta$ , $\sigma \ge 1$ , $s>\frac{1}{2}$ , we verify that $F\left(z\right)$ satisfies the condition (H₂) of Theorem 5.1 with $L=C_{s,b_1,b_2,b_3,b_4}{e}^{-\sigma }{\sigma }^{\sigma }{\left(2{\Vert z_0\Vert }_{\delta }+2R+1\right)}^2$ . This completes the proof of Theorem 1.5.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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