Ill-posedness for the gCH-mCH equation in Besov spaces

Abstract

In this paper, we consider the Cauchy problem to the generalized Fokas–Qiao–Xia–Li/generalized Camassa–Holm-modified Camassa–Holm (gFQXL/gCH-mCH) equation, which includes the Camassa–Holm equation, the generalized Camassa–Holm equation, the Novikov equation, the Fokas–Olver–Rosenau–Qiao/Modified Camassa–Holm equation and the Fokas–Qiao–Xia–Li/Camassa–Holm-modified Camassa–Holm equation. We prove the ill-posedness for the Cauchy problem of the gFQXL/gCH-mCH equation in $B^s_{p,\infty }$ with $s>\max \{2+1/p, 5/2\}$ and $1\le p\le \infty$ in the sense that the solution map to this equation starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^s_{p,\infty }$.

1 Introduction

In this paper, we consider the Cauchy problem of the following generalized Fokas–Qiao–Xia–Li/generalized Camassa–Holm-modified Camassa–Holm (gFQXL/gCH-mCH) equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} m_{t}+k_{1}\left( (u^{2}-u_{x}^{2}) m\right) _{x}+k_{2}\left( u^{k} m_{x}+(k+1) u^{k-1} u_{x} m\right) =0, \\ m=u-u_{x x}, \\ m(0, x)=m_0(x), \end{array}\right. } \end{aligned}$$

(1.1)

where $k_1,k_2\in \mathbb {R}$ and $k \in \mathbb {Z}^+$. $u = u(t, x)$ is a horizontal velocity.

When $k_{1}=0$ and $k_{2}=1$, Eq. (1.1) reduces to the following generalized Camassa–Holm (gCH) equation which was proposed in [1, 20, 24]:

$$\begin{aligned} {\left\{ \begin{array}{ll} m_{t}+u^{k} m_{x}+(k+1) u^{k-1} u_{x} m=0,\\ m=u-u_{x x},\\ m(0, x)=m_0(x). \end{array}\right. } \end{aligned}$$

(1.2)

It is remarkable that Eq. (1.2) possesses peakon solutions $u(t, x)=c^{1/{k}} \textrm{e}^{-|x-c t|}$ with $c>0$ (see [20, 29]). Himonas–Holliman [24] obtained the local well-posedness in Sobolev spaces by means of the Galerkin approximation method, which was generalized to the Besov spaces by Zhang and Liu [54]. It was further proved that the data-to-solution map is continuous [31] but not uniformly continuous [51].

Equation (1.2) can also be a special case of the g-kbCH equation

$$\begin{aligned} {\left\{ \begin{array}{ll} m_{t}+u^{k} m_{x}+b u^{k-1} u_{x} m=0,\quad b\in \mathbb {R},\\ m=u-u_{x x},\\ m(0, x)=m_0(x). \end{array}\right. } \end{aligned}$$

(1.3)

Zhao–Li–Yan [55] obtained the well-posedness of the Cauchy problem (1.3) in Besov space $B_{p, r}^s(\mathbb {R})$ with $s>\max \{1+1/{p}, 3/{2}\}$ and $1\le p, r\le \infty$. However, for $r=\infty$, they established the continuity of the data-to-solution map in a weaker topology. Chen–Li–Yan [5] solved the critical case for $(s, p, r)=(3/{2}, 2, 1)$. Li–Yu–Zhu [38] proved the sharp ill-posedness of the Cauchy problem for the g-kbCH equation in $B^s_{p,\infty }$ with $s>\max \{1+1/p, 3/2\}$ and $1\le p\le \infty$ in the sense that the solution map to this equation starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^s_{p,\infty }$.

When $k=1$, Eq. (1.2) becomes the well-known Camassa–Holm (CH) equation

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t-u_{xxt}+3uu_x=2u_xu_{xx}+uu_{xxx},\\ u(0, x)=u_0(x). \end{array}\right. } \end{aligned}$$

(1.4)

The CH equation was originally derived as a bi-Hamiltonian system by Fokas and Fuchssteiner [17] in the context of the KdV model and gained prominence after Camassa–Holm [3] independently re-derived it from the Euler equations of hydrodynamics using asymptotic expansions. The CH equation is completely integrable [3, 8] with a bi-Hamiltonian structure [7, 17] and infinitely many conservation laws [3, 17]. Also, it admits exact peaked soliton solutions (peakons) of the form $ce^{-|x-ct|}$ with $c>0$, which are orbitally stable [9] and models wave breaking (i.e., the solution remains bounded, while its slope becomes unbounded in finite time [10,11,12]). In 1998, Misiołek [42] showed that the CH equation re-expresses geodesic motion on the Bott–Virasoro group. Subsequently, the CH equation with periodic boundary conditions had been recast as a geodesic flow on the diffeomorphism group of the circle by Kouranbaeva [35] and Constantin–Kolev [13, 14].

When $k=2$, Eq. (1.2) is the famous Novikov equation [23, 43, 44, 49, 51]

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t-u_{xxt}+4u^2u_x=3uu_xu_{xx}+u^2u_{xxx},\\ u(0, x)=u_0(x). \end{array}\right. } \end{aligned}$$

(1.5)

Home–Wang [30] proved that the Novikov equation with cubic nonlinearity shares similar properties with the CH equation, such as a Lax pair in matrix form, a bi-Hamiltonian structure, infinitely many conserved quantities and peakon solutions. The Novikov equation admits multi-peakon traveling wave solutions on both the line and the circle. More precisely, on the line the n-peakon

$$\begin{aligned} u(x, t)=\sum _{j=1}^n p_j(t) e^{-\left| x-q_j(t)\right| } \end{aligned}$$

is a solution to the Novikov equation if and only if the positions $\left( q_1, \ldots , q_n\right)$ and the momenta $\left( p_1, \ldots , p_n\right)$ satisfy the following system of 2n differential equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\textrm{d}q_j}{\textrm{d}t} =u^2(q_j), \\ \frac{\textrm{d}p_j}{\textrm{d}t} =-u(q_j) u_x(q_j) p_j . \end{array}\right. } \end{aligned}$$

Himonas–Holliman–Kenig [26] constructed a 2-peakon solution with an asymmetric antipeakon-peakon initial data and showed the Cauchy problem (1.5) on both the line and the circle is ill-posed in Sobolev spaces $H^s$ with $s<3/2$. One may note that the CH equation has quadratic rather than cubic nonlinearities, and this plays an important role in the analysis of these two equations.

When $k_{1}=1$ and $k_{2}=0$, Eq. (1.1) reduces to the $\textrm{FORQ} / \textrm{MCH}$ equation

$$\begin{aligned} {\left\{ \begin{array}{ll} m_{t}+\left( (u^{2}-u_{x}^{2}) m\right) _{x}=0, \\ m=u-u_{x x},\\ m(0, x)=m_0(x). \end{array}\right. } \end{aligned}$$

(1.6)

Equation (1.6) written in a slightly different form was first derived by Fokas [16] as an integrable generalisation of the modified KdV equation. Fuchssteiner [18] and Olver–Rosenau [45] independently obtained similar versions of this equation by performing a simple explicit algorithm based on the bi-Hamiltonian representation of the classically integrable system. Later, the concise form written above was recovered by Qiao [46] from the two-dimensional Euler equations by using an approximation procedure. The entire integrable hierarchy related to the FORQ equation was proposed by Qiao [47]. It also has bi-Hamiltonian structure, which was first derived in [45] and then in [46], admits Lax pair [46] and peakon travelling wave solutions that are orbitally stable [21, 39, 48]. It should be mentioned that this equation is also referred as the modified Camassa–Holm equation in [4] (we may call it the FORQ/MCH hierarchy). The local well-posedness and ill-posedness of the Cauchy problem for the FORQ equation (1.6) in Sobolev spaces and Besov spaces were studied in the series of papers [19, 24, 25, 27]. Himonas-Mantzavinos [28] showed that the Cauchy problem (1.6) is well-posed in Sobolev space $H^s$ with $s>5/{2}$. Fu et al. [19] established the local well-posedness in Besov space $B_{p, r}^s$ with $s>\max \{2+1/{p}, 5/{2}\}$ and $1\le p, r\le \infty$. It was further proved the data-to-solution map is continuous [28] but not uniformly continuous [32, 50].

When $k=1$, Eq. (1.1) reduces to the FQXL/CH-mCH equation [6, 22, 40, 41] as an extension of both the $\textrm{CH}$ and $\mathrm {FORQ/MCH}$ equations

$$\begin{aligned} {\left\{ \begin{array}{ll} m_{t}+k_{1}\left( (u^{2}-u_{x}^{2}) m\right) _{x}+k_{2}\left( u m_{x}+2 u_{x} m\right) =0, \\ m=u-u_{x x},\\ m(0, x)=m_0(x), \end{array}\right. } \end{aligned}$$

(1.7)

which was proposed by Fokas [16] from the two-dimensional hydrodynamical equations for surface waves by using tri-Hamiltonian duality to the bi-Hamiltonian Gardner equation. Xia–Qiao–Li [52] discussed its integrability, bi-Hamilton structure, and conservation laws. Although these models mentioned above have similar properties in several aspects, we would like to point out that these equations are truly different. In fact, only a few of them have a geometrical interpretation as geodesic flow. One of the distinctive features of the CH equation is that it comes up in the description of the geodesic flow on the Bott–Virasoro group with respect to certain (weak) right invariant Riemannian metrics [13,14,15, 35, 42]. For the CH equation, many interesting results under geometric aspects can be found in Kolev’s papers [33, 34].

Equation (1.1) can be viewed as a generalization to the FQXL equation or a combination of both gCH and mCH equtions. Based on this reason, we call Eq. (1.1) the gFQXL/gCH-mCH equation. Very recently, based on the transport equation and Littlewood–Paley theory, Yang–Han–Wang [53] proved that the gFQXL/gCH-mCH equation is locally well-posed in Besov spaces. More precisely, they established

Lemma 1.1

[53] Let $p, r \in [1,\infty ]$ and $s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\}$. Assume that $u_{0} \in B_{p, r}^{s}$, then there exists a time $T>0$ and a unique solution u to the Cauchy problem (1.7) such that the map $B_{p, r}^{s} \ni u_{0} \mapsto u\in \mathcal {C}([0, T] ; B_{p, r}^{s^{\prime }}) \cap \mathcal {C}^{1}([0, T] ; B_{p, r}^{s^{\prime }-1})$ is continuous for every $s^{\prime }<s$ when $r=\infty$ and $s^{\prime }=s$ when $r<\infty$.

Naturally, we want to ask a question that whether or not the continuity of the data-to-solution map with values in $L^\infty (0, T;B^s_{p,\infty })$ with $s>\max \{2+1/p,5/2\}$ and $1\le p\le \infty$ holds for the gFQXL/gCH-mCH equation. It should be noticed that well-posedness of the Cauchy problem for the g-kbCH equation holds for $s>\max \{1+1/{p}, 3/{2}\}$ while for the gFQXL/gCH-mCH equation holds for $s>\max \{2+1/{p}, 5/{2}\}$. This difference between the well-posedness index of g-kbCH equation and gFQXL/gCH-mCH equation may be explained by the presence of the extra term $u_x^3$ in (1.7). Recently, Li–Yu–Zhu [38] (see also [37]) proved the solution map to the g-kbCH equation starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^s_{p,\infty }(\mathbb {R})$ with $s>\max \{1+1/{p}, 3/{2}\}$, which implies the ill-posedness of the Cauchy problem for this equation in $B^s_{p,\infty }(\mathbb {R})$. However, this ill-posedness result do not cleanly to the gFQXL/gCH-mCH equation. The main difficulty lies in the presence of the extra term $u_x^3$. In this paper, motivated by the idea used in [38], we shall bypass this obstacle and prove the ill-posedness of the Cauchy problem for the gFQXL/gCH-mCH equation. In order to present our main result, let us reformulate (1.1). Setting $\Lambda ^{-2}=(1-\partial ^2_x)^{-1}$, then $\Lambda ^{-2}f=G*f$ where $G(x)=\frac{1}{2}e^{-|x|}$ is the kernel of the operator $\Lambda ^{-2}$. Thus, we can transform (1.1) equivalently into the following transport type equation

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}+\left( k_{1} u^{2}-\frac{k_{1}}{3} u_{x}^{2}+k_{2} u^{k}\right) u_{x}+\Lambda ^{-2}\left( \frac{k_{1}}{3} u_{x}^{3}+\frac{k_{2}(k-1)}{2} u^{k-2} u_{x}^{3}\right) \\ \qquad +\partial _{x} \Lambda ^{-2}\left( \frac{2 k_{1}}{3} u^{3}+k_{1} u u_{x}^{2}+k_{2} u^{k+1}+\frac{k_{2}(2 k-1)}{2} u^{k-1} u_{x}^{2}\right) =0,\\ u(0,x)=u_0(x). \end{array}\right. } \end{aligned}$$

(1.8)

Now, we state our main result.

Theorem 1.1

Let $k_1\in \mathbb {R}{\setminus }\{0\}$, $k_2\in \mathbb {R}$ and $k \in \mathbb {Z}^+$. Assume that $1\le p\le \infty$ and $s>\max \left\{ 2+\frac{1}{p}, \frac{5}{2}\right\} .$ There exists $u_0\in B^s_{p,\infty }(\mathbb {R})$ and a positive constant $\varepsilon _0$ such that the data-to-solution map $u_0\mapsto \textbf{S}_{t}(u_0)$ of the Cauchy problem (1.8) satisfies

$$\begin{aligned} \limsup _{t\rightarrow 0^+}\Vert \textbf{S}_{t}(u_0)-u_0\Vert _{B^s_{p,\infty }}\ge \varepsilon _0. \end{aligned}$$

Remark 1.1

Theorem 1.1 demonstrates the ill-posedness of the Cauchy problem (1.8) in $B^s_{p,\infty }$ with $s>\max \{2+1/p, 5/2\}$ and $1\le p\le \infty$ in the sense that the solution map to this equation starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^s_{p,\infty }$.

Organization of our paper In Sect. 2 we present some preliminary results and introduce notation used. In Sect. 3 we rewrite the original system (1.1) by introducing a new unknown quantity. In Sect. 4 we give the proof of main Theorem.

2 Littlewood–Paley analysis

Notation $A\le B$ (resp., $A \gtrsim B$) means that there exists a harmless positive constant c such that $A \le cB$ (resp., $A \ge cB$). Given a Banach space X, we denote its norm by $\Vert \cdot \Vert _{X}$. For $I\subset \mathbb {R}$, we denote by $\mathcal {C}(I;X)$ the set of continuous functions on I with values in X. Sometimes we will denote $L^p(0,T;X)$ by $L_T^pX$. Next, we will recall some facts about the Littlewood–Paley (L–P) decomposition, the nonhomogeneous Besov spaces and some of their useful properties. Let $\mathcal {B}:=\{\xi \in \mathbb {R}:|\xi |\le \frac{4}{3}\}$ and $\mathcal {C}:=\{\xi \in \mathbb {R}:\frac{3}{4}\le |\xi |\le \frac{8}{3}\}.$ There exist two radial functions $\chi \in C_c^{\infty }(\mathcal {B})$ and $\varphi \in C_c^{\infty }(\mathcal {C})$ both taking values in [0, 1] such that

$$\begin{aligned}&\chi (\xi )+\sum _{j\ge 0}\varphi (2^{-j}\xi )=1 \quad \forall \; \xi \in \mathbb {R}. \end{aligned}$$

Definition 2.1

[2] For every $u\in \mathcal {S'}(\mathbb {R})$, the L–P dyadic blocks ${\Delta }_j$ are defined as follows

$$\begin{aligned} \Delta _ju= \left\{ \begin{array}{ll} 0, &{}\quad if~j\le -2;\\ \chi (D)u=\mathcal {F}^{-1}(\chi \mathcal {F}u), &{}\quad if~j=-1;\\ \varphi (2^{-j}D)u=\mathcal {F}^{-1}\big (\varphi (2^{-j}\cdot )\mathcal {F}u\big ), &{}\quad if~j\ge 0. \end{array}\right. \end{aligned}$$

Definition 2.2

[2] Let $s\in \mathbb {R}$ and $(p,r)\in [1, \infty ]^2$. The nonhomogeneous Besov space $B^{s}_{p,r}(\mathbb {R})$ is defined by

$$\begin{aligned} B^{s}_{p,r}(\mathbb {R}):=\Big \{f\in \mathcal {S}'(\mathbb {R}):\;\Vert f\Vert _{B^{s}_{p,r}(\mathbb {R})}<\infty \Big \},\quad \text {where} \end{aligned}$$

$$\begin{aligned} \Vert f\Vert _{B^{s}_{p,r}(\mathbb {R})}= \left\{ \begin{array}{ll} \left( \sum _{j\ge -1}2^{sjr}\Vert \Delta _jf\Vert ^r_{L^p(\mathbb {R})}\right) ^{1/r}, &{}\quad if~1\le r<\infty ,\\ \sup _{j\ge -1}2^{sj}\Vert \Delta _jf\Vert _{L^p(\mathbb {R})}, &{}\quad if~r=\infty . \end{array}\right. \end{aligned}$$

Remark 2.1

The fact $B^s_{p,\infty }(\mathbb {R})\hookrightarrow B^t_{p,\infty }(\mathbb {R})$ with $s>t$ will be often used implicity.

We give some important properties which will be also often used throughout the paper.

Lemma 2.1

[2] (1) Let $(p,r)\in [1, \infty ]^2$ and $s>\max \big \{1+\frac{1}{p},\frac{3}{2}\big \}$. Then we have

$$\begin{aligned}&\Vert uv\Vert _{B^{s-2}_{p,r}(\mathbb {R})}\le C\Vert u\Vert _{B^{s-2}_{p,r}(\mathbb {R})}\Vert v\Vert _{B^{s-1}_{p,r}(\mathbb {R})}. \end{aligned}$$

(2) For $(p,r)\in [1, \infty ]^2$, $B^{s-1}_{p,r}(\mathbb {R})$ with $s>1+\frac{1}{p}$ is an algebra. Moreover, for any $u,v \in B^{s-1}_{p,r}(\mathbb {R})$ with $s>1+\frac{1}{p}$, we have

$$\begin{aligned}&\Vert uv\Vert _{B^{s-1}_{p,r}(\mathbb {R})}\le C\Vert u\Vert _{B^{s-1}_{p,r}(\mathbb {R})}\Vert v\Vert _{B^{s-1}_{p,r}(\mathbb {R})}. \end{aligned}$$

(3) Let $m\in \mathbb {R}$ and f be an $S^m$-multiplier (i.e., $f: \mathbb {R}\rightarrow \mathbb {R}$ is smooth and satisfies that $\forall \alpha \in \mathbb {N}$, there exists a constant $\mathcal {C}_\alpha$ such that $|\partial ^\alpha f(\xi )|\le \mathcal {C}_\alpha (1+|\xi |)^{m-|\alpha |}$ for all $\xi \in \mathbb {R}$). Then the operator f(D) is continuous from $B_{p, r}^s(\mathbb {R})$ to $B_{p, r}^{s-m}(\mathbb {R})$.

(4) For any $s\in \mathbb {R}$, $(1-\partial _x)^{-1}$ is an isomorphic mapping from $B_{p, r}^{s-1}(\mathbb {R})$ into $B_{p, r}^s(\mathbb {R})$.

(5) For $1\le p\le \infty$ and $s>0$, there exists a positive constant C such that

$$\begin{aligned} \Big \Vert 2^{j s}\left\| [\Delta _j,v]\partial _xf\right\| _{L^{p}}\Big \Vert _{\ell ^{\infty }} \le C\big (\Vert \partial _x v\Vert _{L^{\infty }}\Vert f\Vert _{B_{p, \infty }^{s}}+\Vert \partial _x f\Vert _{L^{\infty }}\Vert \partial _xv\Vert _{B_{p, \infty }^{s-1}}\big ), \end{aligned}$$

where we denote the standard commutator $[\Delta _j,v]\partial _xf=\Delta _j(v\partial _xf)-v\Delta _j\partial _xf$.

3 Reformulation of system

Due to the presence of the extra term $u_x^3$ in (1.8), it seems difficult to deal with Eq. (1.8) directly. From Eq. (1.1), we infer that

$$\begin{aligned} \begin{aligned}&\left( 1-\partial _{x}^{2}\right) \left( u_{t}+\big (k_{1}\big (u^{2}-u_{x}^{2}\big )+k_{2} u^{k}\big ) u_{x}\right) \\&\quad =-2 k_{1}\big (u_{x}^2m\big )_{x}-2 k_{1} u u_xm+k_{2}(2 k-1) u^{k-1} u_xm\\&\qquad -k_{2} k(k-1) u^{k-2} u_{x}^{3}-3 k_{2} k u^{k} u_{x}, \end{aligned} \end{aligned}$$

which implies

$$\begin{aligned}&u_{t}+\left( k_{1}\big (u^{2}-u_{x}^{2}\big )+k_{2} u^{k}\right) u_{x}\nonumber \\&\quad =k_{2}\Lambda ^{-2}\left( (2 k-1) u^{k-1} u_xm-k(k-1) u^{k-2} u_{x}^{3}-3 k u^{k} u_{x}\right) \nonumber \\&\qquad -2 k_{1} \partial _{x}\Lambda ^{-2}\big (u_{x}^2m\big )-2 k_{1}\Lambda ^{-2}(u u_xm) . \end{aligned}$$

(3.9)

Differentiating (3.9) with respect to x yields

$$\begin{aligned}&u_{x t}+\left( k_{1}\big (u^{2}-u_{x}^{2}\big )+k_{2} u^{k}\right) u_{x x} \nonumber \\&\quad = k_{2} \partial _{x}\Lambda ^{-2}\left( (2 k-1) u^{k-1} u_xm-k(k-1) u^{k-2} u_{x}^{3}-3 k u^{k} u_{x}\right) \nonumber \\&\qquad -2 k_{1}\Lambda ^{-2}\big (u_{x}^2m\big )-2 k_{1} \partial _{x}\Lambda ^{-2}(u u_xm)-k_{2} k u^{k-1} u_{x}^{2}. \end{aligned}$$

(3.10)

Introducing $v:=(1-\partial _x)u,$ which implies $u^{2}-u_{x}^{2}=(2u-v)v$, then we have from (3.9) and (3.10)

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tv+\left( k_{1}(2u-v)v+k_{2} u^{k}\right) \partial _xv= \Phi _1(u)+\Phi _2(v)+\Phi _3(v),\\ u=(1-\partial _x)^{-1}v,\\ v_0=(1-\partial _x)u_0, \end{array}\right. } \end{aligned}$$

(3.11)

where the terms $\Phi _1(u), \Phi _2(v)$ and $\Phi _3(v)$ are defined by

$$\begin{aligned} \Phi _1(u)&=-k_{2} k u^{k-1} u_{x}^{2},\\ \Phi _2(v)&=2 k_{1} \partial _{x}\Lambda ^{-2}\left( v_{x} u_xm\right) -2 k_{1}\Lambda ^{-2}(v u_xm) ,\\ \Phi _3(v)&=k_{2} (1-\partial _x)\Lambda ^{-2}\left( (2 k-1) u^{k-1} u_xm-k(k-1) u^{k-2} u_{x}^{3}-3 k u^{k} u_{x}\right) . \end{aligned}$$

Since $(1-\partial _x)^{-1}$ is an isomorphic mapping from $B_{p, r}^{s-1}(\mathbb {R})$ into $B_{p, r}^s(\mathbb {R})$, the ill-posedness of u in $B_{p, r}^{s+1}$ can be transformed into that of v in $B_{p, r}^{s}$. Based on this observation, we shall consider Eq. (3.11) satisfied by v in the rest of this paper. Now we restate our main result as follows.

Theorem 3.1

Let $k_1\in \mathbb {R}{\setminus }\{0\}$, $k_2\in \mathbb {R}$ and $k \in \mathbb {Z}^+$. Assume that $1\le p\le \infty$ and $s>\max \left\{ 1+\frac{1}{p}, \frac{3}{2}\right\} .$ There exists $v_0\in B^s_{p,\infty }(\mathbb {R})$ and a positive constant $\varepsilon _0$ such that the data-to-solution map $v_0\mapsto \textbf{S}_{t}(v_0)$ of the Cauchy problem (3.11) satisfies

$$\begin{aligned} \limsup _{t\rightarrow 0^+}\Vert \textbf{S}_{t}(v_0)-v_0\Vert _{B^s_{p,\infty }}\ge \varepsilon _0. \end{aligned}$$

4 Proof of Theorem 3.1

In this section, we will give the proof of Theorem 3.1.

4.1 Construction of initial data

We need to introduce smooth, radial cut-off functions to localize the frequency region. Precisely, let $\widehat{\phi }\in \mathcal {C}^\infty _0(\mathbb {R})$ be an even, real-valued and non-negative function on $\mathbb {R}$ and satisfy

$$\begin{aligned} {\widehat{\phi }(\xi )=} \left\{ \begin{array}{ll} 1,&{}\quad if~|\xi |\le \frac{1}{4},\\ 0,&{}\quad if~|\xi |\ge \frac{1}{2}. \end{array}\right. \end{aligned}$$

Remark 4.1

By the Fourier–Plancherel formula, we have $\phi (x)=\mathcal {F}^{-1}(\widehat{\phi }(\xi ))$. Obviously

$$\begin{aligned}&\phi (0)=\frac{1}{2\pi }\int _{\mathbb {R}}\widehat{\phi }(\xi )\textrm{d}\xi >0. \end{aligned}$$

Lemma 4.1

[36] Let $n\gg 1$. Define the function $f_n(x)$ by

$$\begin{aligned} f_n(x)=\phi (x)\sin \left( \frac{17}{12}2^{n}x\right) \end{aligned}$$

or

$$\begin{aligned} f_n(x)=\phi (x)\cos \left( \frac{17}{12}2^{n}x\right) . \end{aligned}$$

Then we have

$$\begin{aligned} {\Delta _j(f_n)=} \left\{ \begin{array}{ll} f_n, &{}\quad if~j=n,\\ 0, &{}\quad if~j\ne n. \end{array}\right. \end{aligned}$$

Lemma 4.2

Define the initial data $u_0(x)$ and $v_0(x)$ as

$$\begin{aligned} u_0(x)&:=\sum \limits ^{\infty }_{n=0}2^{-n(s+1)} \phi (x)\sin \left( \frac{17}{12}2^{n}x\right) ,\\ v_0(x)&:=(1-\partial _x)u_0(x). \end{aligned}$$

Then for any $s>\max \big \{\frac{3}{2},1+\frac{1}{p}\big \}$ and $k\in \mathbb {Z}^+$, we have for some n large enough

$$\begin{aligned}&\Vert v_0\Vert _{B^{s}_{p,\infty }}\approx \Vert u_0\Vert _{B^{s+1}_{p,\infty }}\le C, \end{aligned}$$

(4.1)

$$\begin{aligned}&\left\| \big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )\partial _x\Delta _{n}v_0\right\| _{L^p}\ge \frac{c_0c_1}{2}2^{n(1-s)}, \end{aligned}$$

(4.2)

where C and $c_0,c_1$ are some positive constants. In particular,

$$\begin{aligned} {c_0:=}{\left\{ \begin{array}{ll} \left( \frac{\delta }{\pi }\int ^\pi _0|\sin x|^p\textrm{d}x\right) ^{{1}/{p}}, &{}\text {if}\quad p\in [1,\infty ),\\ 1, &{}\text {if}\quad p=\infty , \end{array}\right. }\quad \text {and}\quad c_1:=\frac{|k_{1}|}{(1-2^{-s})^2}\phi ^3(0). \end{aligned}$$

Proof

Using Lemma 4.1 yields

$$\begin{aligned} \Delta _{n}u_0(x)&=2^{-n(s+1)} \phi (x)\sin \left( \frac{17}{12}2^{n}x\right) . \end{aligned}$$

(4.3)

By the definition of Besov space, we have

$$\begin{aligned} \Vert u_0\Vert _{B^{s+1}_{p,\infty }}&=\sup _{j\ge -1}2^{(s+1) j}\Vert \Delta _{j}u_0\Vert _{L^p}=\sup _{j\ge 0}\left\| \phi (x)\sin \left( \frac{17}{12}2^{j}x\right) \right\| _{L^p}\le C. \end{aligned}$$

Due to the relation $u_0=(1-\partial _x)^{-1}v_0$, we have $\Vert u_0\Vert _{B^{s+1}_{p,\infty }}\lesssim \Vert v_0\Vert _{B^{s}_{p,\infty }}$, which implies (4.1).

From (4.3), we have

$$\begin{aligned} \partial _x\Delta _{n}u_0(x)&=2^{-n(s+1)} \phi '(x)\sin \left( \frac{17}{12}2^{n}x\right) +\frac{17}{12}2^{-ns} \phi (x)\cos \left( \frac{17}{12}2^{n}x\right) . \end{aligned}$$

(4.4)

Then

$$\begin{aligned} \partial _x\partial _x\Delta _{n}u_0(x)&=2^{-n(s+1)} \phi ''(x)\sin \left( \frac{17}{12}2^{n}x\right) +\frac{17}{6}2^{-ns} \phi '(x)\cos \left( \frac{17}{12}2^{n}x\right) \nonumber \\&\quad -\left( \frac{17}{12}\right) ^22^{n}2^{-ns} \phi (x)\sin \left( \frac{17}{12}2^{n}x\right) . \end{aligned}$$

(4.5)

Combining (4.4) and (4.5) directly gives that

$$\begin{aligned} \partial _x\Delta _{n}v_0&=\partial _x\Delta _{n}u_0-\partial _x\partial _x\Delta _{n}u_0\\&=\left( \frac{17}{12}\right) ^22^{n}2^{-ns} \phi (x)\sin \left( \frac{17}{12}2^{n}x\right) \\&\quad +2^{-n(s+1)} (\phi '(x)-\phi ''(x))\sin \left( \frac{17}{12}2^{n}x\right) \\&\quad +\frac{17}{12}2^{-ns} \left( \phi (x)-2\phi '(x)\right) \cos \left( \frac{17}{12}2^{n}x\right) . \end{aligned}$$

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

$$\begin{aligned}&|[\big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )\phi ](x)|\nonumber \\&\quad \ge \frac{1}{2}|[\big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )\phi ](0)|=\frac{|k_{1}|}{2}|v_0^2(0)|\phi (0)\nonumber \\&\quad =\frac{|k_{1}|}{2}\left( \frac{17}{12}\phi (0)\sum \limits ^{\infty }_{n=0}2^{-ns}\right) ^2\phi (0)\ge c_1. \end{aligned}$$

(4.6)

Obviously,

$$\begin{aligned} \Vert \big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )(x)\Vert _{L^\infty (\mathbb {R})}&\le C. \end{aligned}$$

(4.7)

Thus, for some n large enough, we have from (4.6) and (4.7)

$$\begin{aligned}&\Vert \big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )\partial _x\Delta _{n}u_0\Vert _{L^p}\\&\quad \ge c_12^{n}2^{-ns} \left\| \sin \left( \frac{17}{12}2^{n}x\right) \right\| _{L^p(B_{\delta }(0))}\\&\qquad -C2^{-ns}\left\| (\phi '(x)-\phi ''(x))\sin \left( \frac{17}{12}2^{n}x\right) \right\| _{L^p}\\&\qquad -C2^{-ns}\left\| \left( \phi (x)-2\phi '(x)\right) \cos \left( \frac{17}{12}2^{n}x\right) \right\| _{L^p}\\&\quad \ge (c_0c_12^{n}-C)2^{-ns}, \end{aligned}$$

where we have used

$$\begin{aligned} \left\| \sin \left( \frac{17}{12}2^{n}x\right) \right\| _{L^p(B_{\delta }(0))}\ge c_0>0. \end{aligned}$$

In fact, for $p\in [1,\infty )$, we have

$$\begin{aligned} \left\| \sin \left( \frac{17}{12}2^{n}x\right) \right\| ^p_{L^p(B_{\delta }(0))}&=\frac{2\delta }{\lambda _n}\int ^{\lambda _n}_{0}|\sin x|^p\textrm{d}x\quad \text {with}\quad \lambda _n:=\frac{17}{12}\delta 2^{n}. \end{aligned}$$

Due to the fact

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\lambda _n}\int _0^{\lambda _n}|\sin x|^p\textrm{d}x=\frac{1}{\pi }\int ^\pi _0|\sin x|^p\textrm{d}x, \end{aligned}$$

then there exists a positive integer number N such that for $n>N$

$$\begin{aligned} \frac{1}{\lambda _n}\int _0^{\lambda _n}|\sin x|^p\textrm{d}x\ge \frac{1}{2\pi }\int ^\pi _0|\sin x|^p\textrm{d}x, \end{aligned}$$

For $p=\infty$, we have for some n large enough

$$\begin{aligned} \left\| \sin \left( \frac{17}{12}2^{n}x\right) \right\| _{L^{\infty }(B_{\delta }(0))}=\left\| \sin x\right\| _{L^{\infty }(B_{\lambda _n}(0))}. \end{aligned}$$

We choose n large enough such that $C\le \frac{c_0c_1}{2}2^{n}$ and then finish the proof of Lemma 4.2. $\square$

4.2 Error estimates

From now on, we denote $v(t)=\textbf{S}_t(v_0)$ for the sake of convenience.

Lemma 4.3

Assume that $\Vert v_0\Vert _{B^{s}_{p,\infty }}\lesssim 1$. Under the assumptions of Theorem 1.1, we have

$$\begin{aligned}&\Vert (v^2-2uv)\partial _xv\Vert _{B^{s-1}_{p,\infty }}+\Vert u^k\partial _xv\Vert _{B^{s-1}_{p,\infty }}\le 1,\\&\Vert \Phi _1(u)\Vert _{B^{s}_{p,\infty }}+\Vert \Phi _2(v)\Vert _{B^{s}_{p,\infty }}+\Vert \Phi _3(v)\Vert _{B^{s}_{p,\infty }}\le 1. \end{aligned}$$

Proof

By the local well-posedness result [53], there exists a short time $T=T(\Vert u_0\Vert _{B^{s+1}_{p,\infty }})$ such that Eq. (1.1) has a unique solution $u(t)\in \mathcal {C}([0, T]; B_{p, r}^{s+1})$. Moreover, for all $t\in [0,T]$, there holds

$$\begin{aligned} \Vert u(t)\Vert _{B^{s+1}_{p,\infty }}\le C\Vert u_0\Vert _{B^{s+1}_{p,\infty }}\;\;\text {or}\;\; \Vert v(t)\Vert _{B^{s}_{p,\infty }}\le C\Vert v_0\Vert _{B^{s}_{p,\infty }}. \end{aligned}$$

(4.8)

It can be inferred from (4.8) that which will be frequently used later

$$\begin{aligned} \Vert v\Vert _{B^{s-1}_{p,\infty }}\lesssim \Vert v_0\Vert _{B^{s-1}_{p,\infty }}\lesssim \Vert v_0\Vert _{B^{s}_{p,\infty }}\lesssim 1. \end{aligned}$$

Using the fact that $B_{p, r}^{s-1}$ is a Banach algebra with $s-1>\max \{\frac{1}{p}, \frac{1}{2}\}$ and Lemma 2.1, one has

$$\begin{aligned}&\Vert (v^2-2uv)\partial _xv\Vert _{B^{s-1}_{p,\infty }}+\Vert u^k\partial _xv\Vert _{B^{s-1}_{p,\infty }}\\&\quad \le \Vert (v^2-2uv)\Vert _{B^{s-1}_{p,\infty }}\Vert \partial _xv\Vert _{B^{s-1}_{p,\infty }}+\Vert u\Vert ^k_{B^{s-1}_{p,\infty }}\Vert v\Vert _{B^{s}_{p,\infty }}\\&\quad \le \Vert v\Vert ^2_{B^{s-2}_{p,\infty }}\Vert v\Vert _{B^{s}_{p,\infty }}+\Vert v\Vert ^3_{B^{s-1}_{p,\infty }} \le 1. \end{aligned}$$

Similarly, one has

$$\begin{aligned}&\Vert \Phi _1(u)\Vert _{B^{s}_{p,\infty }}\le \Vert u^{k-1} u_{x}^{2}\Vert _{B^{s}_{p,\infty }}\le \Vert u\Vert ^{k-1}_{B^{s}_{p,\infty }}\Vert u\Vert ^2_{B^{s+1}_{p,\infty }}\le 1,\\&\Vert \Phi _2(v)\Vert _{B^{s}_{p,\infty }}\le \Vert v_{x} u_xm\Vert _{B^{s-1}_{p,\infty }}+\Vert v u_xm\Vert _{B^{s-1}_{p,\infty }}\le \Vert v\Vert _{B^{s}_{p,\infty }}\Vert u\Vert _{B^{s}_{p,\infty }}\Vert u-u_{xx}\Vert _{B^{s-1}_{p,\infty }}\le 1, \\&\Vert \Phi _3(v)\Vert _{B^{s}_{p,\infty }}\le \Vert u^{k-1} u_xm\Vert _{B^{s-1}_{p,\infty }}+\Vert u^{k-2} u_x^3\Vert _{B^{s-1}_{p,\infty }}+\Vert u^{k} u_x\Vert _{B^{s-1}_{p,\infty }}\le 1. \end{aligned}$$

Thus, we finish the proof of Lemma 4.3.$\square$

Proposition 4.1

Assume that $\Vert v_0\Vert _{B^{s}_{p,\infty }}\lesssim 1$. Under the assumptions of Theorem 1.1, we have

$$\begin{aligned} \Vert \textbf{S}_{t}(v_0)-v_0\Vert _{B^{s-1}_{p,\infty }}\lesssim t. \end{aligned}$$

Proof

By the mean value theorem and the Minkowski inequality, we obtain

$$\begin{aligned} \Vert v(t)-v_0\Vert _{B^{s-1}_{p,\infty }}&\le \int ^t_0\Vert \partial _\tau v\Vert _{B^{s-1}_{p,\infty }} \textrm{d}\tau \\&\le k_1\int ^t_0\Vert (v^2-2uv)\partial _xv\Vert _{B^{s-1}_{p,\infty }} \textrm{d}\tau + k_2\int ^t_0\Vert u^k\partial _xv\Vert _{B^{s-1}_{p,\infty }} \textrm{d}\tau \\&\quad +\int ^t_0\Vert \Phi _1(u)\Vert _{B^{s-1}_{p,\infty }} \textrm{d}\tau +\int ^t_0\Vert \Phi _1(v)\Vert _{B^{s-1}_{p,\infty }} \textrm{d}\tau +\int ^t_0\Vert \Phi _2(v)\Vert _{B^{s-1}_{p,\infty }} \textrm{d}\tau . \end{aligned}$$

Thus, using Lemma 4.3 enable us to finish the proof of Proposition 4.1. $\square$

Next, we shall establish the key estimate which plays an important role in the proof of main Theorem.

Proposition 4.2

Assume that $\Vert v_0\Vert _{B^{s-1}_{p,\infty }}\lesssim 1$. Under the assumptions of Theorem 1.1, there holds

$$\begin{aligned} \Vert \textbf{w}\Vert _{B^{s-2}_{p,\infty }}\lesssim t^{2}, \end{aligned}$$

where we denote $\textbf{w}:=\textbf{S}_{t}(v_0)-v_0-t\textbf{U}_0$ and

$$\begin{aligned} \textbf{U}_0:=\left( k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\right) \partial _xv_0+\Phi _1(u_0)+\Phi _2(v_0)+\Phi _3(v_0). \end{aligned}$$

(4.9)

Proof

Using differential mean value theorem and (3.11), we obtain

$$\begin{aligned} \Vert \textbf{w}\Vert _{B^{s-2}_{p,\infty }}&\le \int ^t_0\Vert \partial _\tau v-\textbf{U}_0\Vert _{B^{s-2}_{p,\infty }} \textrm{d}\tau \nonumber \\&\le \int ^t_0\Vert v^2\partial _xv-v_0^2\partial _xv_0\Vert _{B^{s-2}_{p,\infty }} \textrm{d}\tau + \int ^t_0\Vert uv\partial _xv-u_0v_0\partial _xv_0\Vert _{B^{s-2}_{p,\infty }} \textrm{d}\tau \nonumber \\&\quad +\int ^t_0\Vert \Phi _1(u)-\Phi _1(u_0)\Vert _{B^{s-2}_{p,\infty }} \textrm{d}\tau +\int ^t_0\Vert \Phi _2(v)-\Phi _2(v_0)\Vert _{B^{s-2}_{p,\infty }} \textrm{d}\tau \nonumber \\&\quad +\int ^t_0\Vert \Phi _3(v)-\Phi _3(v_0)\Vert _{B^{s-2}_{p,\infty }} \textrm{d}\tau . \end{aligned}$$

(4.10)

Now we need to estimate each term on the right hand side of (4.10). Notice that $B_{p, r}^{s-1}$ is a Banach algebra with $s-1>\max \{\frac{1}{p}, \frac{1}{2}\}$, combining with Lemma 2.1 yields

$$\begin{aligned} \Vert v^3-v_0^3\Vert _{B^{s-2}_{p,\infty }}&\le \Vert (v-v_0)(v^2+vv_0+v_0^2)\Vert _{B^{s-1}_{p,\infty }} \lesssim \Vert v-v_0\Vert _{B^{s-1}_{p,\infty }},\\ \Vert uv\partial _xv-u_0v_0\partial _xv_0\Vert _{B^{s-2}_{p,\infty }}&=\Vert (u-u_0)v\partial _xv+u_0(v\partial _xv-v_0\partial _xv_0)\Vert _{B^{s-2}_{p,\infty }}\\&\lesssim \Vert u-u_0\Vert _{B^{s-2}_{p,\infty }}\Vert v\partial _xv\Vert _{B^{s-1}_{p,\infty }} +\Vert u_0\Vert _{B^{s-1}_{p,\infty }}\Vert v^2-v_0^2\Vert _{B^{s-1}_{p,\infty }}\\&\lesssim \Vert v-v_0\Vert _{B^{s-1}_{p,\infty }}. \end{aligned}$$

Here, we need only to estimate $\Vert v_{x} u_xm-v_{0x} u_{0x}m_0\Vert _{B^{s-2}_{p,\infty }}$ since the other terms can be processed in a similar more relaxed way. Using Lemma 2.1 yields

$$\begin{aligned}&\Vert v_{x} u_xm-v_{0x} u_{0x}m_0\Vert _{B^{s-2}_{p,\infty }}\\&\quad \le \Vert (v-v_{0})_{x} u_xm\Vert _{B^{s-2}_{p,\infty }}+\Vert v_{0x} (u-u_{0})_xm\Vert _{B^{s-2}_{p,\infty }}\\&\qquad +\Vert v_{0x} u_{0x}(m-m_0)\Vert _{B^{s-2}_{p,\infty }}\\ {}&\le \Vert v-v_{0}\Vert _{B^{s-1}_{p,\infty }}\Vert u\Vert _{B^{s}_{p,\infty }}\Vert m\Vert _{B^{s-1}_{p,\infty }}+\Vert v_{0}\Vert _{B^{s}_{p,\infty }} \Vert u-u_{0}\Vert _{B^{s}_{p,\infty }}\Vert m\Vert _{B^{s-1}_{p,\infty }}\\&\qquad +\Vert v_{0}\Vert _{B^{s}_{p,\infty }} \Vert u_{0}\Vert _{B^{s}_{p,\infty }}\Vert m-m_0\Vert _{B^{s-2}_{p,\infty }}\\&\quad \le \Vert v-v_0\Vert _{B^{s-1}_{p,\infty }}. \end{aligned}$$

Putting the above estimates into (4.10) and using Proposition 4.1 yield

$$\begin{aligned} \Vert \textbf{w}\Vert _{B^{s-1}_{p,\infty }} \lesssim \int _0^t\Vert v(\tau )-v_0\Vert _{B^{s-1}_{p,\infty }}\textrm{d}\tau \lesssim t^{2}. \end{aligned}$$

Thus, we complete the proof of Proposition 4.2.$\square$

Now we present the proof of Theorem 3.1.

Proof of Theorem 3.1

Notice that $\textbf{S}_{t}(v_0)-v_0=t\textbf{U}_0+\textbf{w}$ where $\textbf{U}_0$ is given by (4.9), and

$$\begin{aligned} \Delta _{n}\left( \big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )\partial _xv_0\right)&=\left( k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\right) \partial _x\Delta _{n}v_0\\&\quad +[\Delta _{n},\big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )]\partial _xv_0, \end{aligned}$$

by the triangle inequality and Proposition 4.2, we deduce that

$$\begin{aligned} \Vert \textbf{S}_{t}(v_0)-v_0\Vert _{B^{s}_{p,\infty }}&\ge 2^{{ns}}\left\| \Delta _{n}\left( \textbf{S}_{t}(v_0)-v_0\right) \right\| _{L^p}\nonumber \\&=2^{{ns}}\left\| \Delta _{n}\left( t\textbf{U}_0+\textbf{w}\right) \right\| _{L^p}\nonumber \\&\ge t2^{{ns}}\Vert \Delta _{n}\textbf{U}_0\Vert _{L^p} -2^{{2n}}2^{{n(s-2)}}\Vert \Delta _{n}\textbf{w}\Vert _{L^p}\nonumber \\&\ge t2^{{n}s}\left\| \Delta _{n}\left( (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\right) \partial _xv_0)\right\| _{L^p}\nonumber \\ {}&\quad - t2^{{n}s}\left\| \Delta _{n}\left( \Phi _1(u_0)+\Phi _2(v_0)+\Phi _3(v_0)\right) \right\| _{L^p}-C2^{2{n}}\Vert \textbf{w}\Vert _{B^{s-2}_{p,\infty }} \nonumber \\&\ge t2^{{n}s}\left\| \left( k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\right) \partial _x\Delta _{n}v_0\right\| _{L^p}\nonumber \\&\quad -t2^{{n}s}\left\| [\Delta _{n},\left( k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\right) ]\partial _xu_0\right\| _{L^p}\nonumber \\&\quad -t\left\| \Phi _1(u_0)+\Phi _2(v_0)+\Phi _3(v_0)\right\| _{B^{s}_{p,\infty }}-C2^{2{n}}t^2\nonumber \\&\ge t2^{{n}s}\left\| \left( k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\right) \partial _x\Delta _{n}v_0\right\| _{L^p}\nonumber \\&\quad -Ct\left\| 2^{{n}s}[\Delta _{n},\big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )]\partial _xu_0\Vert _{L^p}\right\| _{\ell ^\infty }\nonumber \\&\quad -Ct-C2^{2{n}}t^2, \end{aligned}$$

(4.11)

where we have used Lemma 4.2 and the commutator estimate

$$\begin{aligned}&\big \Vert 2^{{n}s}\Vert [\Delta _{n},\big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )]\partial _xv_0\Vert _{L^p}\big \Vert _{\ell ^\infty }\\&\le \Vert \partial _x\big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )\Vert _{L^\infty }\Vert v_0\Vert _{B^{s}_{p,\infty }}\\&\quad +\Vert \partial _xv_0\Vert _{L^\infty }\Vert \partial _x\big (k_{1}(2u_0-v_0)v_0+k_{2} u_0^{k}\big )\Vert _{B^{s-1}_{p,\infty }}\le 1. \end{aligned}$$

Gathering all the above estimates and Lemma 4.2 together with (4.11), we obtain

$$\begin{aligned} \Vert \textbf{S}_{t}(v_0)-v_0\Vert _{B^{s}_{p,\infty }}\ge \frac{c_0c_1}{2}t2^{{n}}-Ct-C2^{2{n}}t^2. \end{aligned}$$

Taking large n such that $\frac{c_0c_1}{2}2^{{n}}\ge 2C$, we have

$$\begin{aligned} \Vert \textbf{S}_{t}(v_0)-v_0\Vert _{B^{s}_{p,\infty }}\ge \frac{c_0c_1}{4}t2^{{n}}-C2^{2{n}}t^2. \end{aligned}$$

Thus, picking $t2^{n}\approx \varepsilon$ with small $\varepsilon$, we have

$$\begin{aligned} \Vert \textbf{S}_{t}(v_0)-v_0\Vert _{B^{s}_{p,\infty }}\gtrsim \frac{c_0c_1}{4}\varepsilon -\varepsilon ^2\gtrsim \frac{c_0c_1}{8}\varepsilon . \end{aligned}$$

This completes the proof of Theorem 3.1.$\square$