On the Cauchy problem for a modified Camassa–Holm equation

Abstract

In this paper, we first study the local well-posedness for the Cauchy problem of a modified Camassa–Holm equation in nonhomogeneous Besov spaces. Then we obtain a blow-up criteria and present a blow-up result for the equation. Finally, with proving the norm inflation we show the ill-posedness occurs to the equation in critical Besov spaces.

1 Introduction

In this paper, we consider the Cauchy problem for the following modified Camassa–Holm equation:

$$\begin{aligned} \left\{ \begin{array}{l} \gamma _t=\lambda \left( v_x-\gamma -\frac{1}{\lambda }v\gamma \right) _x, \quad t>0,\ x\in {\mathbb {R}}, \\ v_{xx}-v=\gamma _x+\frac{\gamma ^2}{2\lambda }, \quad t\ge 0,\ x\in {\mathbb {R}}, \\ \gamma (0,x)=\gamma _0(x), \quad x\in {\mathbb {R}}, \end{array}\right. \end{aligned}$$

(1.1)

which was called by Gorka and Reyes [19]. Let $G=\partial _x^2-1, m=Gv$. Then, Eq. (1.1) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \gamma _t+G^{-1}m\gamma _x=\frac{\gamma ^2}{2} +\lambda G^{-1}m- \gamma G^{-1}m_x, \quad t>0,\ x\in {\mathbb {R}}, \\ m=\gamma _x+\frac{\gamma ^2}{2\lambda } , \quad t\ge 0,\ x\in {\mathbb {R}}, \\ \gamma (0,x)=\gamma _0(x), \quad x\in {\mathbb {R}}. \end{array}\right. \end{aligned}$$

(1.2)

The Eq. (1.1) was first studied through the geometric approach in [14, 27]. Pseudo-potentials, conservation laws and the existence and uniqueness of weak solutions to the modified Camassa–Holm equation were presented in [19]. We observe that if we solve (1.2), then m will formally satisfy the following physical form of the Camassa–Holm equation [10]:

$$\begin{aligned} m_t=-2vm_x-mv_x+\lambda v_x. \end{aligned}$$

If $\lambda =0$, it is known as the well-known Camassa–Holm(CH) equation. It was derived as a model for shallow water waves [10, 11]. The CH equation is completely integrable [6, 10]and has a bi-Hamiltonian structure [4, 17]. It admits peakon solitons of the form $ce^{-|x-ct|}$ with $c>0$, which are orbitally stable [13]. The local well-posedness for the Cauchy problem of the CH equation in Sobolev spaces and Besov spaces was proved in [7, 8, 15, 26]. It was shown that there exist finite time blow-up strong solutions and global strong solutions to the CH equation [5, 7,8,9]. Recently, norm inflation and ill-posedness for the CH eqution in the critical Sobolev Space and Besov spaces was proved in [15, 16, 18]. The existence and uniqueness of global weak solutions were presented in [12, 29]. The global conservative, dissipative, and algebro-geometric solutions were studied in [2, 3, 25].

In this paper, we investigate the local well-posedness for the Cauchy problem of a modified Camassa-Holm Eq. (1.2) in Besov spaces, present a blow-up result to (1.2) and prove norm inflation and hence ill-posedness for the equation in critical Besov spaces. This paper is organized as follows. In Sect. 2, we introduce some preliminaries which will be used in sequel. In Sect. 3, we prove the local well-posedness of (1.2) in $B^s_{p,r}$ with $s>\max (\frac{1}{2},\frac{1}{p})$ or $(s=\frac{1}{p}, 1\le p\le 2, r=1)$ in the sense of Hadamard (i.e. (1.2) has a unique local solution in $B^s_{p,r}$ with continuity with respect to the initial data). The main approach is based on the Littlewood–Paley theory and transport equations theory. In Sect. 4, we present a blow-up result of the Eq. (1.2) and then prove that (1.2) is ill-posed in $H^\frac{1}{2}$ and in $B^\frac{1}{2}_{2,r}$, $1<r\le \infty$ by a contradiction argument.

2 Preliminaries

In this section, we will recall some propositions about the Littlewood–Paley decomposition and Besov spaces.

Proposition 2.1

[1] Let ${\mathcal {C}}$ be the annulus $\{\xi \in {\mathbb {R}}^d:\frac{3}{4}\le |\xi |\le \frac{8}{3}\}$. There exist radial functions $\chi$ and $\varphi$, valued in the interval [0, 1], belonging respectively to ${\mathcal {D}}(B(0,\frac{4}{3}))$ and ${\mathcal {D}}({\mathcal {C}})$, and such that

$$\begin{aligned}&\forall \xi \in {\mathbb {R}}^d,\ \chi (\xi )+\sum _{j\ge 0}\varphi (2^{-j}\xi )=1, \\&\forall \xi \in {\mathbb {R}}^d\backslash \{0\},\ \sum _{j\in {\mathbb {Z}}}\varphi (2^{-j}\xi )=1, \\&|j-j'|\ge 2\Rightarrow \mathrm {Supp}\ \varphi (2^{-j}\cdot )\cap \mathrm {Supp}\ \varphi (2^{-j'}\cdot )=\emptyset , \\&j\ge 1\Rightarrow \mathrm {Supp}\ \chi (\cdot )\cap \mathrm {Supp}\ \varphi (2^{-j}\cdot )=\emptyset . \end{aligned}$$

The set $\widetilde{{\mathcal {C}}}=B(0,\frac{2}{3})+{\mathcal {C}}$ is an annulus, and we have

$$\begin{aligned} |j-j'|\ge 5\Rightarrow 2^{j}{\mathcal {C}}\cap 2^{j'}\widetilde{{\mathcal {C}}}=\emptyset . \end{aligned}$$

Further, we have

$$\begin{aligned}&\forall \xi \in {\mathbb {R}}^d,\ \frac{1}{2}\le \chi ^2(\xi )+\sum _{j\ge 0}\varphi ^2(2^{-j}\xi )\le 1, \\&\forall \xi \in {\mathbb {R}}^d\backslash \{0\},\ \frac{1}{2}\le \sum _{j\in {\mathbb {Z}}}\varphi ^2(2^{-j}\xi )\le 1. \end{aligned}$$

${\mathcal {F}}$ represents the Fourier transform and its inverse is denoted by ${\mathcal {F}}^{-1}$. Let u be a tempered distribution in ${\mathcal {S}}'({\mathbb {R}}^d)$. For all $j\in {\mathbb {Z}}$, define

$$\begin{aligned} \Delta _j u= & {} 0\,\ \text {if}\,\ j\le -2,\\ \Delta _{-1} u= & {} {\mathcal {F}}^{-1}(\chi {\mathcal {F}}u),\\ \Delta _j u= & {} {\mathcal {F}}^{-1}(\varphi (2^{-j}\cdot ){\mathcal {F}}u)\,\ \text {if}\,\ j\ge 0,\\ S_j u= & {} \sum _{j'<j}\Delta _{j'}u. \end{aligned}$$

Then the Littlewood–Paley decomposition is given as follows:

$$\begin{aligned} u=\sum _{j\in {\mathbb {Z}}}\Delta _j u \quad \text {in}\ {\mathcal {S}}'({\mathbb {R}}^d). \end{aligned}$$

Let $s\in {\mathbb {R}},\ 1\le p,r\le \infty .$ The nonhomogeneous Besov space $B^s_{p,r}({\mathbb {R}}^d)$ is defined by

$$\begin{aligned} B^s_{p,r}=B^s_{p,r}({\mathbb {R}}^d)=\{u\in S'({\mathbb {R}}^d):\Vert u\Vert _{B^s_{p,r}({\mathbb {R}}^d)}=\Big \Vert (2^{js}\Vert \Delta _j u\Vert _{L^p})_j \Big \Vert _{l^r({\mathbb {Z}})}<\infty \}. \end{aligned}$$

There are some properties about Besov spaces.

Proposition 2.2

[1, 20] Let $s\in {\mathbb {R}},\ 1\le p,p_1,p_2,r,r_1,r_2\le \infty .$

• (1) $B^s_{p,r}$ is a Banach space, and is continuously embedded in ${\mathcal {S}}'$.

• (2) If $r<\infty$, then $\lim \nolimits _{j\rightarrow \infty }\Vert S_j u-u\Vert _{B^s_{p,r}}=0$. If $p,r<\infty$, then $C_0^{\infty }$ is dense in $B^s_{p,r}$.

• (3) If $p_1\le p_2$ and $r_1\le r_2$, then $B^s_{p_1,r_1}\hookrightarrow B^{s-d(\frac{1}{p_1}-\frac{1}{p_2})}_{p_2,r_2}.$ If $s_1<s_2$, then the embedding $B^{s_2}_{p,r_2}\hookrightarrow B^{s_1}_{p,r_1}$ is locally compact.

• (4) $B^s_{p,r}\hookrightarrow L^{\infty } \Leftrightarrow s>\frac{d}{p}\ \text {or}\ s=\frac{d}{p},\ r=1.$

• (5) Fatou property: if $(u_n)_{n\in {\mathbb {N}}}$ is a bounded sequence in $B^s_{p,r}$, then an element $u\in B^s_{p,r}$ and a subsequence $(u_{n_k})_{k\in {\mathbb {N}}}$ exist such that

  $$\begin{aligned} \lim _{k\rightarrow \infty }u_{n_k}=u\ \text {in}\ {\mathcal {S}}'\quad \text {and}\quad \Vert u\Vert _{B^s_{p,r}}\le C\liminf _{k\rightarrow \infty }\Vert u_{n_k}\Vert _{B^s_{p,r}}. \end{aligned}$$

• (6) Let $m\in {\mathbb {R}}$ and f be a $S^m$-mutiplier (i.e. f is a smooth function and satisfies that $\forall \alpha \in {\mathbb {N}}^d$, $\exists C=C(\alpha )$, such that $|\partial ^{\alpha }f(\xi )|\le C(1+|\xi |)^{m-|\alpha |},\ \forall \xi \in {\mathbb {R}}^d)$. Then the operator $f(D)={\mathcal {F}}^{-1}(f{\mathcal {F}})$ is continuous from $B^s_{p,r}$ to $B^{s-m}_{p,r}$.

We introduce two useful interpolation inequalities.

Proposition 2.3

[1, 20] (1) If $s_1<s_2$, $\theta \in (0,1)$, and (p, r) is in $[1,\infty ]^2$, then we have

$$\begin{aligned} \Vert u\Vert _{B^{\theta s_1+(1-\theta )s_2}_{p,r}}\le \Vert u\Vert _{B^{s_1}_{p,r}}^{\theta }\Vert u\Vert _{B^{s_2}_{p,r}}^{1-\theta }. \end{aligned}$$

(2) If $s\in {\mathbb {R}},\ 1\le p\le \infty ,\ \varepsilon >0$, a constant $C=C(\varepsilon )$ exists such that

$$\begin{aligned} \Vert u\Vert _{B^s_{p,1}}\le C\Vert u\Vert _{B^s_{p,\infty }}\ln \Big (e+\frac{\Vert u\Vert _{B^{s+\varepsilon }_{p,\infty }}}{\Vert u\Vert _{B^s_{p,\infty }}}\Big ). \end{aligned}$$

Proposition 2.4

[1] Let $s\in {\mathbb {R}},\ 1\le p,r\le \infty .$

$$\begin{aligned}\left\{ \begin{array}{l} B^s_{p,r}\times B^{-s}_{p',r'}\longrightarrow {\mathbb {R}}, \\ (u,\phi )\longmapsto \sum \limits _{|j-j'|\le 1}\langle \Delta _j u,\Delta _{j'}\phi \rangle , \end{array}\right. \end{aligned}$$

defines a continuous bilinear functional on $B^s_{p,r}\times B^{-s}_{p',r'}$. Denote by $Q^{-s}_{p',r'}$ the set of functions $\phi$ in ${\mathcal {S}}'$ such that $\Vert \phi \Vert _{B^{-s}_{p',r'}}\le 1$. If u is in ${\mathcal {S}}'$, then we have

$$\begin{aligned} \Vert u\Vert _{B^s_{p,r}}\le C\sup _{\phi \in Q^{-s}_{p',r'}}\langle u,\phi \rangle . \end{aligned}$$

We then have the following product laws:

Lemma 2.5

[1, 20] (1) For any $s>0$ and any (p, r) in $[1,\infty ]^2$, the space $L^{\infty } \cap B^s_{p,r}$ is an algebra, and a constant $C=C(s,d)$ exists such that

$$\begin{aligned} \Vert uv\Vert _{B^s_{p,r}}\le C(\Vert u\Vert _{L^{\infty }}\Vert v\Vert _{B^s_{p,r}}+\Vert u\Vert _{B^s_{p,r}}\Vert v\Vert _{L^{\infty }}). \end{aligned}$$

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

$$\begin{aligned} \Vert uv\Vert _{B^{s_1}_{p,r}}\le C\Vert u\Vert _{B^{s_1}_{p,r}}\Vert v\Vert _{B^{s_2}_{p,r}}. \end{aligned}$$

(3) If $1\le p\le 2$, there exists $C=C(p,d)$ such that

$$\begin{aligned} \Vert uv\Vert _{B^{\frac{d}{p}-d}_{p,\infty }}\le C \Vert u\Vert _{B^{\frac{d}{p}-d}_{p,\infty }}\Vert v\Vert _{B^{\frac{d}{p}}_{p,1}}. \end{aligned}$$

We now state the so-called Osgood lemma, a generalization of the Gronwall lemma.

Lemma 2.6

[1] Let $\rho$ be a measurable function from $[t_0,T]$ to [0, a], $\gamma$ a locally integrable function from $[t_0,T]$ to ${\mathbb {R}}^+$, and $\mu$ a continuous and nondecreasing function from [0, a] to ${\mathbb {R}}^+$. Assume that for some $c\ge 0$, the function $\rho$ satisfies

$$\begin{aligned} \rho (t)\le c+\int _{t_0}^t \gamma (t')\mu (\rho (t'))dt'\ \ \text {for}\ a.e.\ t\in [t_0,T]. \end{aligned}$$

If $c>0$, then for a.e. $t\in [t_0,T]$,

$$\begin{aligned} -{\mathcal {M}}(\rho (t))+{\mathcal {M}}(c)\le \int _{t_0}^t \gamma (t')dt'\ \ \text {with}\ \ {\mathcal {M}}(x)=\int _x^a \frac{dr}{\mu (r)}. \end{aligned}$$

If $c=0$, and $\mu$ satisfies $\int _0^a \frac{dr}{\mu (r)}=\infty$, then $\rho =0$, a.e.

Remark 2.7

[21] For example, when $\mu (r)=r(1-\ln r),\ r\in [0,1]$, we have ${\mathcal {M}}(x)=\ln (1-\ln x)$, and $\rho (t)\le ec^{\exp {-\int _{t_0}^t\gamma (t')dt'}}$, if $c>0$. We will use this result later.

Now we state some useful estimates in the study of transport equations, which are crucial to the proofs of our main theorem later.

$$\begin{aligned} \left\{ \begin{array}{l} f_t+v\cdot \nabla f=g,\ x\in {\mathbb {R}}^d,\ t>0, \\ f(0,x)=f_0(x). \end{array}\right. \end{aligned}$$

(2.1)

Lemma 2.8

[1, 23] Let $s\in {\mathbb {R}},\ 1\le p,r\le \infty$. There exists a constant C such that for all solutions $f\in L^{\infty }([0,T];B^s_{p,r})$ of (2.1) in one dimension with initial data $f_0$ in $B^s_{p,r}$, and g in $L^1([0,T];B^s_{p,r})$, we have, for a.e. $t\in [0,T]$,

$$\begin{aligned} \Vert f(t)\Vert _{B^s_{p,r}}\le e^{CV(t)}\Big (\Vert f_0\Vert _{B^s_{p,r}}+\int _0^t e^{-CV(t')}\Vert g(t')\Vert _{B^s_{p,r}}dt'\Big ) \end{aligned}$$

with

$$\begin{aligned} V'(t)=\left\{ \begin{array}{ll} \Vert \nabla v\Vert _{B^{s+1}_{p,r}},\ &{}\text {if}\ s>\max (-\frac{1}{2},\frac{1}{p}-1), \\ \Vert \nabla v\Vert _{B^{s}_{p,r}},\ &{}\text {if}\ s>\frac{1}{p}\ \text {or}\ (s=\frac{1}{p},\ p<\infty , \ r=1), \\ \Vert \nabla v\Vert _{B^{\frac{1}{p} }_{p,1}},\ &{}\text {if}\ s=\frac{1}{p}-1, 1\le p\le 2, r=\infty , \end{array}\right. \end{aligned}$$

and when $s=\frac{1}{p}-1,\ 1\le p\le 2,\ r=\infty ,\ \text {and}\ V'(t)=\Vert \nabla v\Vert _{B^{\frac{1}{p}}_{p,1}}$.

Lemma 2.9

[24] Let $s>0,\ 1\le p,r\le \infty$. Define $R_j=[v\cdot \nabla , \Delta _j]f$. There exists a constant C such that

$$\begin{aligned} \Big \Vert (2^{js}\Vert R_j\Vert _{L^p})_j\Big \Vert _{l^r({\mathbb {Z}})}\le C(\Vert \nabla v\Vert _{L^{\infty }}\Vert f\Vert _{B^s_{p,r}}+\Vert \nabla v\Vert _{B^{s}_{p,r}}\Vert f\Vert _{L^{\infty }}). \end{aligned}$$

Hence, if f solves the equation (2.1), we have

$$\begin{aligned} \Vert f(t)\Vert _{B^s_{p,r}}\le \Vert f_0\Vert _{B^s_{p,r}}+C\int _0 ^t (\Vert \nabla v\Vert _{L^{\infty }}\Vert f\Vert _{B^s_{p,r}}+\Vert \nabla v\Vert _{B^{s}_{p,r}}\Vert f\Vert _{L^{\infty }}+\Vert g\Vert _{B^s_{p,r}}dt'). \end{aligned}$$

Lemma 2.10

[1, 23] Let $1\le p\le p_1\le \infty ,\ 1\le r\le \infty ,\ s> -d\min (\frac{1}{p_1}, \frac{1}{p'})$. Let $f_0\in B^s_{p,r}$, $g\in L^1([0,T];B^s_{p,r})$, and let v be a time-dependent vector field such that $v\in L^\rho ([0,T];B^{-M}_{\infty ,\infty })$ for some $\rho >1$ and $M>0$, and

$$\begin{aligned} \begin{array}{ll} \nabla v\in L^1([0,T];B^{\frac{d}{p_1}}_{p_1,\infty }\bigcap L^{\infty }), &{}\ \text {if}\ s<1+\frac{d}{p_1}, \\ \nabla v\in L^1([0,T];B^{1+\frac{d}{p} }_{p,r}), &{}\ \text {if}\ s=1+\frac{d}{p}, \ r>1, \\ \nabla v\in L^1([0,T];B^{s-1}_{p_1,r}), &{}\ \text {if}\ s>1+\frac{d}{p_1}\ or\ (s=1+\frac{d}{p_1}\ and\ r=1). \end{array} \end{aligned}$$

Then the equation (2.1) has a unique solution f in

-the space $C([0,T];B^s_{p,r})$, if $r<\infty$,

-the space $\Big (\bigcap _{s'<s}C([0,T];B^{s'}_{p,\infty })\Big )\bigcap C_w([0,T];B^s_{p,\infty })$, if $r=\infty$.

Lemma 2.11

[22] Let $1\le p\le \infty ,\ 1\le r<\infty ,\ s>\frac{d}{p}\ (or \ s=\frac{d}{p},\ p<\infty ,\ r=1)$. Denote $\bar{{\mathbb {N}}}={\mathbb {N}}\cup \{\infty \}$. Let $(v^n)_{n\in \bar{{\mathbb {N}}}}\in C([0,T];B^{s+1}_{p,r})$. Assume that $(f^n)_{n\in \bar{{\mathbb {N}}}}$ in $C([0,T];B^s_{p,r})$ is the solution to

$$\begin{aligned} \left\{ \begin{array}{l} f^n_t+v^n\cdot \nabla f^n=g,\ x\in {\mathbb {R}}^d,\ t>0, \\ f^n(0,x)=f_0(x) \end{array}\right. \end{aligned}$$

(2.2)

with initial data $f_0\in B^s_{p,r},\ g\in L^1([0,T];B^s_{p,r})$ and that for some $\alpha \in L^1([0,T])$, $\sup \limits _{n\in \bar{{\mathbb {N}}}}\Vert v^n(t)\Vert _{B^{s+1}_{p,r}}\le \alpha (t)$. If $v^n \rightarrow v^{\infty }$ in $L^1([0,T];B^s_{p,r})$, then $f^n \rightarrow f^{\infty }$ in $C([0,T];B^s_{p,r})$.

3 Local well-posedness

In this section, we will investigate the local well-posedness for (1.2) in Besov spaces.We introduce the following function spaces.

Definition 3.1

Let $T>0,\ s\in {\mathbb {R}},$ and $1\le p,r \le \infty .$ Set

$$\begin{aligned} E^s_{p,r}(T)\triangleq \left\{ \begin{array}{ll} C([0,T];B^s_{p,r})\cap C^1([0,T];B^{s-1}_{p,r}), &{} \text {if}\ r<\infty , \\ C_w([0,T];B^s_{p,\infty })\cap C^{0,1}([0,T];B^{s-1}_{p,\infty }), &{} \text {if}\ r=\infty . \end{array}\right. \end{aligned}$$

In this section, our main theorem is stated as follows.

Theorem 3.2

Let $1\le p,r \le \infty ,\ s\in {\mathbb {R}}$ and let (s, p, r) satisfy the condition $s>\max (\frac{1}{2},\frac{1}{p})$ or $(s=\frac{1}{p},\ 1\le p\le 2,\ r=1).$ Assume that $\gamma _0\in B^s_{p,r}.$ Then there exists a time $T>0$ such that (1.2) has a unique solution $\gamma$ in $E^s_{p,r}(T).$ Moreover the solution depends continuously on the initial data.

We divide it into six steps to prove Theorem 3.2.

Step one: Constructing approximate solutions.

We starts from $\gamma ^0\triangleq 0,$ and define a sequence $(\gamma ^n)_{n\in {\mathbb {N}}}$ of smooth functions by solving the following linear transport equations:

$$\begin{aligned} \left\{ \begin{array}{l} \gamma _t^{n+1}+G^{-1}m^{n}\gamma _x^{n+1}=\frac{(\gamma ^{n})^2}{2}\ +\lambda G^{-1}m^{n}-\gamma G^{-1}m^{n}_x, \\ m^{n}=\gamma ^{n}_x+\frac{(\gamma ^{n})^2}{2\lambda }\ , \\ \gamma ^{n+1}(0,x)=S_{n+1}\gamma _0. \end{array}\right. \end{aligned}$$

(3.1)

Define $G^n=G^{-1}m^{n},\ F^n=\frac{(\gamma ^{n})^2}{2}\ +\lambda G^{-1}m^{n}-\gamma G^{-1}m^{n}_x.$ Assume that $\gamma _n\in L^\infty ([0,T];B^{s}_{p,r})$ for all $T>0$. Note that under the assumptions on (s, p, r), $B^{s}_{p,r}$ is an algebra. We have

$$\begin{aligned} \Vert G_x^n\Vert _{B^{s}_{p,r}}&\le C\Vert m^n\Vert _{B^{s-1}_{p,r}} \nonumber \\&\le C\left( \Vert \gamma ^n\Vert _{B^{s}_{p,r}}+ \Vert \gamma ^n\Vert ^2_{B^{s}_{p,r}}\right) . \end{aligned}$$

(3.2)

$$\begin{aligned} \Vert F^n\Vert _{B^{s}_{p,r}}&\le C\left( \Vert \gamma ^n\Vert ^2_{B^s_{p,r}}+\Vert G^{-1}m^n\Vert _{B^s_{p,r}}+\Vert \gamma ^n\Vert _{B^s_{p,r}}\Vert G^{-1}m_x^n\Vert _{B^s_{p,r}}\right) \nonumber \\&\le C\left( \Vert \gamma ^n\Vert _{B^{s}_{p,r}} +\Vert \gamma ^n\Vert ^2_{B^{s}_{p,r}}+\Vert \gamma ^n\Vert ^3_{B^{s}_{p,r}}\right) . \end{aligned}$$

(3.3)

Therefore $G^n_x,\ F^n\in L^{\infty }([0,T];B^s_{p,r})$. Hence, applying Lemma 2.10 ensures that (3.1) has a global solution $\gamma ^{n+1}$ which belongs to $E^s_{p,r}(T)$ for all $T>0$.

Step two: Uniform bounds.

Define $R_n=\Vert \gamma ^{n}(t)\Vert _{B^{s}_{p,r}}.$Using Lemma 2.8 together with (3.2) and (3.3), we have

$$\begin{aligned} R_{n+1}&\le e^{C\int _0^t \Vert G_x^n\Vert _{B^{s}_{p,r}}dt'} \Big (\Vert S_{n+1}\gamma _0\Vert _{B^{s}_{p,r}}+\int _0^t e^{-C\int _0^{t'} \Vert G_x^n\Vert _{B^{s}_{p,r}} dt''} \Vert F^n\Vert _{B^{s}_{p,r}}dt'\Big ) \nonumber \\&\le Ce^{C\int _0^t R_n+R_n^2 dt'}\Big (\Vert \gamma _0\Vert _{B^{s}_{p,r}}+\int _0^t e^{-C\int _0^{t'} R_n+R_n^2 dt''} (R_n+R_n^2+R_n^3)dt'\Big ). \end{aligned}$$

(3.4)

The case where $\Vert \gamma _0\Vert _{B^{s}_{p,r}}=0$ is trivial, we start with the case where $\Vert \gamma _0\Vert _{B^{s}_{p,r}}\ne 0$. We have known that $R_0=0$. Fix a $T>0$ such that $4C^3 T\Vert \gamma _0\Vert ^2_{B^{s}_{p,r}} <1$ and suppose that

$$\begin{aligned} \forall t\in [0,T],\ R_n\le \frac{C\Vert \gamma _0\Vert _{B^{s}_{p,r}}}{2(1-4C^3 t\Vert \gamma _0\Vert ^2_{B^{s}_{p,r}})^{\frac{1}{2}}}. \end{aligned}$$

(3.5)

Pluge (3.5) into (3.4) and choose $C\ge 2\Vert \gamma _0\Vert _{B^{s}_{p,r}}$. After a simple calculation we derive

$$\begin{aligned} R_{n+1}&\le C\Vert \gamma _0\Vert _{B^{s}_{p,r}}(1-4C^3 t\Vert \gamma _0\Vert ^2_{B^{s}_{p,r}})^{-\frac{1}{4}} \left( 1+C^3\Vert \gamma _0\Vert ^2_{B^{s}_{p,r}} \int _0^t (1\right. \\&\left. \quad -4C^3 t'\Vert \gamma _0\Vert ^2_{B^{s}_{p,r}})^{-\frac{5}{4}} dt'\right) \\&\le \frac{C\Vert \gamma _0\Vert _{B^{s}_{p,r}}}{2(1-4C^3 t\Vert \gamma _0\Vert ^2_{B^{s}_{p,r}})^{\frac{1}{2}}}. \end{aligned}$$

Therefore, $(\gamma ^n)_{n\in {\mathbb {N}}}$ is bounded in $L^{\infty }([0,T];B^{s}_{p,r})$.

Step three: Cauchy sequence.

When $s>\max (\frac{1}{2}, \frac{1}{p})$ or $(s=\frac{1}{p},\ 1\le p\le 2,\ r=1)$, some estimates we need are a little different, so we have to discuss separately.

Case 1 $s>\max (\frac{1}{2}, \frac{1}{p})$.

We are going to prove that $(\gamma ^n)_{n\in {\mathbb {N}}}$ is a Cauchy sequence in $L^{\infty }([0,T];B^{s-1}_{p,r})$. For that purpose, for all $(n,k)\in {\mathbb {N}}^2$, we have

$$\begin{aligned} (\gamma ^{n+k+1}-\gamma ^{n+1})_t+G^{n+k} (\gamma ^{n+k+1}-\gamma ^{n+1})_x =\left( G^n-G^{n+k}\right) \gamma _x^{n+1}+F^{n+k}-F^n, \end{aligned}$$

where

$$\begin{aligned}&G^n-G^{n+k}=G^{-1}(\gamma ^n-\gamma ^{n+k})_x-\frac{1}{2\lambda } G^{-1}\left( \left( \gamma ^n\right) ^2-\left( \gamma ^{n+k}\right) ^2\right) , \\&F^{n+k}-F^n=\frac{1}{2} \left( \gamma ^{n+k}-\gamma ^n\right) \left( \gamma ^{n+k}+\gamma ^n\right) + \lambda G^{-1}\left( \gamma ^{n+k}-\gamma ^n\right) +\frac{1}{2} G^{-1}\left( \left( \gamma ^n\right) ^2 \right. \\&\quad \left. -\left( \gamma ^{n+k}\right) ^2\right) -\gamma ^{n+k}\left( \partial _xG^{-1}m^{n+k}-\partial _xG^{-1}m^n\right) - (\gamma ^{n+k}-\gamma ^n)\partial _xG^{-1}m^n. \end{aligned}$$

Applying Lemma 2.8, for any t in [0, T], we get

$$\begin{aligned} \Vert (\gamma ^{n+k+1}-\gamma ^{n+1})(t)\Vert _{B^{s-1}_{p,r}}&\le e^{C\int _0^t \Vert G_x^{n+k}\Vert _{B^{s}_{p,r}}dt'} \Big (\Vert S_{n+k+1}\gamma _0-S_{n+1}\gamma _0\Vert _{B^{s-1}_{p,r}} \nonumber \\&\quad +\int _0^t e^{-C\int _0^{t'} \Vert G_x^{n+k}\Vert _{B^{s}_{p,r}} dt''} \left( \Vert \left( G^n-G^{n+k}\right) m_x^{n+1}\Vert _{B^{s-1}_{p,r}}\right. \nonumber \\&\quad \left. +\Vert F^{n+k}-F^n\Vert _{B^{s-1}_{p,r}}\right) dt'\Big ). \end{aligned}$$

(3.6)

Using the fact $B^s_{p,r}$ is an algebra and applying Lemma 2.5 (2), we have

$$\begin{aligned}&\Vert \left( G^n-G^{n+k}\right) \gamma _x^{n+1}\Vert _{B^s_{p,r}}\nonumber \\&\quad \le C\Vert \gamma ^{n+1}\Vert _{B^{s}_{p,r}}\left( 1+\Vert \gamma ^{n}\Vert _{B^{s}_{p,r}}\Vert \gamma ^{n+k}\Vert _{B^{s}_{p,r}}\right) \Vert \gamma ^{n+k}-\gamma ^n\Vert _{B^{s-1}_{p,r}}, \end{aligned}$$

(3.7)

and

$$\begin{aligned} \Vert F^{n+k}-F^n\Vert _{B^{s-1}_{p,r}}&\le C\left( 1+\Vert \gamma ^{n+k}\Vert _{B^{s}_{p,r}}+\Vert \gamma ^{n+k}\Vert _{B^{s}_{p,r}}\Vert \gamma ^{n}\Vert _{B^{s}_{p,r}}+\Vert \gamma ^{n+k}\Vert ^2_{B^{s}_{p,r}} \right. \nonumber \\&\quad \left. +\Vert \gamma ^{n}\Vert _{B^{s}_{p,r}}+\Vert \gamma ^{n}\Vert ^2_{B^{s}_{p,r}}\right) \Vert \gamma ^{n+k}-\gamma ^n\Vert _{B^{s-1}_{p,r}}. \end{aligned}$$

(3.8)

Since $(\gamma ^n)_{n\in {\mathbb {N}}}$ is bounded in $L^{\infty }([0,T];B^s_{p,r})$ for all t in [0, T], we finally get

$$\begin{aligned}&\Vert \left( \gamma ^{n+k+1}-\gamma ^{n+1}\right) (t)\Vert _{B^{s-1}_{p,r}} \\&\quad \le C\Big (\Vert S_{n+k+1}\gamma _0-S_{n+1}\gamma _0\Vert _{B^{s-1}_{p,r}}+\int _0^t \Vert \gamma ^{n+k}-\gamma ^n\Vert _{B^{s-1}_{p,r}}dt'\Big ). \end{aligned}$$

Taking an upper bound on [0, t], we have

$$\begin{aligned}&\Vert \gamma ^{n+k+1}-\gamma ^{n+1}\Vert _{L_t^{\infty }({B^{s-1}_{p,r}})} \nonumber \\&\quad \le C\Big (\Vert S_{n+k+1}\gamma _0-S_{n+1}\gamma _0\Vert _{B^{s-1}_{p,r}}+\int _0^t \Vert \gamma ^{n+k}-\gamma ^n\Vert _{L_{t'}^{\infty }({B^{s-1}_{p,r}})}dt'\Big ). \end{aligned}$$

(3.9)

Let $g_n(t)=\sup \nolimits _k\Vert \gamma ^{n+k}-\gamma ^n\Vert _{L_t^{\infty }(B^{s-1}_{p,r})}$. Then (3.9) becomes

$$\begin{aligned} g_{n+1}(t) \le C\Big (\sup _k \Vert S_{n+k+1}\gamma _0-S_{n+1}\gamma _0\Vert _{B^{s-1}_{p,r}}+\int _0^t g_n(t')dt'\Big ). \end{aligned}$$

Since $(S_n \gamma ^0)_{n\in {\mathbb {N}}}$ is a Cauchy sequence in $B^{s-1}_{p,r}$, applying Fatou’s lemma, we have

$$\begin{aligned} g(t) \triangleq \limsup _{n\rightarrow \infty } g_{n+1}(t) \le C\int _0^t g(t')dt'. \end{aligned}$$

The Gronwall lemma implies that $g(t)=0$ for all $t\in [0,T]$. Therefore $(\gamma ^n)_{n\in {\mathbb {N}}}$ is a Cauchy sequence in $C([0,T];B^{s-1}_{p,r})$ and converges to some limit function $\gamma \in C([0,T];B^{s-1}_{p,r})$.

Case 2 $s=\frac{1}{p},\ 1\le p\le 2,\ r=1.$

From Lemma 2.8, comparing with Case 1, we do not have the estimate for the norm $B^{s-1}_{p,1}$ but $B^{s-1}_{p,\infty }$. In fact, we only have

$$\begin{aligned}&\Vert (\gamma ^{n+k+1}-\gamma ^{n+1})(t)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }} \le e^{C\int _0^t \Vert G_x^{n+k}\Vert _{B^{\frac{1}{p}}_{p,1}}dt'} \Big (\Vert S_{n+k+1}\gamma _0-S_{n+1}\gamma _0\Vert _{B^{\frac{1}{p}-1}_{p,\infty }} \nonumber \\&\quad +\int _0^t e^{-C\int _0^{t'} \Vert G_x^{n+k}\Vert _{B^{\frac{1}{p}}_{p,1}} dt''} (\Vert (G^n-G^{n+k}) \gamma _x^{n+1}\Vert _{B^{\frac{1}{p}-1}_{p,\infty }} +\Vert F^{n+k}-F^n\Vert _{B^{\frac{1}{p}-1}_{p,\infty }})dt'\Big ). \end{aligned}$$

(3.10)

Applying Lemma 2.5 (3), we deduce that

$$\begin{aligned}&\Vert (G^n-G^{n+k})\gamma _x^{n+1}\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\le C\Vert \gamma ^{n+1}\Vert _{B^{\frac{1}{p}}_{p,1}}(1+\Vert \gamma ^{n}\Vert _{B^{\frac{1}{p}}_{p,1}}+\Vert \gamma ^{n+k}\Vert _{B^{\frac{1}{p}}_{p,1}})\Vert \gamma ^{n+k}\nonumber \\&\quad -\gamma ^n\Vert _{B^{\frac{1}{p}-1}_{p,1}}. \end{aligned}$$

(3.11)

$$\begin{aligned}&\Vert F^{n+k}-F^n\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\le C(1+\Vert \gamma ^{n}\Vert _{B^{\frac{1}{p}}_{p,1}}+\Vert \gamma ^{n+k}\Vert _{B^{\frac{1}{p}}_{p,1}}+\Vert \gamma ^{n+k}\Vert ^2_{B^{\frac{1}{p}}_{p,1}} \nonumber \\&\quad +\Vert \gamma ^{n}\Vert ^2_{B^{\frac{1}{p}}_{p,1}}+\Vert \gamma ^{n}\Vert _{B^{\frac{1}{p}}_{p,1}}\Vert \gamma ^{n+k}\Vert _{B^{\frac{1}{p}}_{p,1}})\Vert \gamma ^{n+k}-\gamma ^n\Vert _{B^{\frac{1}{p}-1}_{p,1}}. \end{aligned}$$

(3.12)

Plugging (3.11), (3.12) into (3.10), and using uniform bounds of $(\gamma ^n)_{n\in {\mathbb {N}}}$, we have

$$\begin{aligned}&\Vert (\gamma ^{n+k+1}-\gamma ^{n+1})(t)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\nonumber \\&\quad \le C\Big (\Vert S_{n+k+1}\gamma _0-S_{n+1}\gamma _0\Vert _{B^{\frac{1}{p}}_{p,1}}+\int _0^t \Vert \gamma ^{n+k}-\gamma ^n\Vert _{B^{\frac{1}{p}-1}_{p,1}}dt'\Big ). \end{aligned}$$

(3.13)

Applying Proposition 2.3 (2), we find

$$\begin{aligned}&\Vert \gamma ^{n+k}-\gamma ^n\Vert _{B^{\frac{1}{p}-1}_{p,1}} \nonumber \\&\quad \le C\Vert \gamma ^{n+k}-\gamma ^n\Vert _{B^{\frac{1}{p}-1}_{p,\infty }} \ln \Big (e+\frac{\Vert \gamma ^{n+k}-\gamma ^n\Vert _{B^{\frac{1}{p}}_{p,1}}}{\Vert \gamma ^{n+k}-\gamma ^n\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}}\Big ). \end{aligned}$$

(3.14)

Since the function $x\ln (e+\frac{C}{x})$ is nondecreasing in $(0,\infty )$, from (3.13) and (3.14), we have

$$\begin{aligned}&\Vert \gamma ^{n+k+1}-\gamma ^{n+1}\Vert _{L_t^{\infty }(B^{\frac{1}{p}-1}_{p,\infty })} \le C\Big (\Vert S_{n+k+1}\gamma _0-S_{n+1}\gamma _0\Vert _{B^{\frac{1}{p}-1}_{p,1}}\\&\quad +\int _0^t \Vert \gamma ^{n+k}-\gamma ^n\Vert _{L_{t'}^{\infty }(B^{\frac{1}{p}-1}_{p,\infty })} \ln \Big (e+\frac{C}{\Vert \gamma ^{n+k}-\gamma ^n\Vert _{L_{t'}^{\infty }(B^{\frac{1}{p}-1}_{p,\infty })}}\Big )dt'\Big ). \end{aligned}$$

Let $g(t) \triangleq \limsup \nolimits _{n\rightarrow \infty }\sup \nolimits _k\Vert \gamma ^{n+k}-\gamma ^n\Vert _{L_t^{\infty }(B^{\frac{1}{p}-1}_{p,\infty })}$. The above inequality can be written as

$$\begin{aligned} g(t) \le C\int _0^t g(t')\ln \left( e+\frac{C}{g(t')}\right) dt'. \end{aligned}$$

Hence Lemma 2.6 implies that $g(t)\equiv 0$, and $(\gamma ^n)_{n\in {\mathbb {N}}}$ is a Cauchy sequence in $C([0,T];B^{\frac{1}{p}-1}_{p,\infty })$ and converges to some limit function $\gamma$ in $C([0,T];B^{\frac{1}{p}-1}_{p,\infty })$.

Step four Convergence.

We have to prove that $\gamma$ belongs to $E^s_{p,r}(T)$ and satisfies (1.2). Since $(\gamma ^n)_{n\in {\mathbb {N}}}$ is bounded in $L^{\infty }([0,T];B^{s}_{p,r})$, we can apply the Fatou property for the Besov spaces to show that $\gamma$ also belongs to $L^{\infty }([0,T];B^{s}_{p,r})$. Now, applying interpolation inequalities implies that $(\gamma ^n)_{n\in {\mathbb {N}}}$ converges to $\gamma$ in $C([0,T];B^{s'}_{p,r})$ for any $s'<s$. Then it is easy to pass to the limit in (3.1) and to conclude that $\gamma$ is indeed a solution of (1.2) in the sense of distributions.

Finally, since $\gamma$ belongs to $L^{\infty }([0,T];B^{s}_{p,r})$, the right-hand side of (1.2) also belongs to $L^{\infty }([0,T];B^{s}_{p,r})$. According to Lemma 2.10, we can deduce that $\gamma$ belongs to $C([0,T];B^{s}_{p,r})$ (resp., $C_w([0,T];B^s_{p,r})$) if $r<\infty$ (resp., $r=\infty$). Again using the equation (1.2), we prove that $\gamma _t$ is in $C([0,T];B^{s-1}_{p,r})$ if r is finite, and in $L^{\infty }([0,T];B^{s-1}_{p,r})$ otherwise. Hence, $\gamma$ belongs to $E^s_{p,r}(T)$.

Step five Uniqueness.

Then, we will prove the uniqueness of solutions to (1.2). The proof follows almost exactly the proofs which we use in Step 3. Suppose that $\gamma _1, \gamma _2$ are two solutions of (1.2). We obtain

$$\begin{aligned} \partial _t(\gamma _1-\gamma _2)+G_1\partial _x(\gamma _1-\gamma _2)=(G_2-G_1)\partial _x \gamma _2+F_1-F_2, \end{aligned}$$

where for $i=1,2$,

$$\begin{aligned} m_i=\partial _x \gamma _i+\frac{\gamma ^2_i}{2\lambda } , \ G_i=G^{-1}m_i , \ F_i=\frac{1}{2} \gamma ^2_i+\lambda G^{-1}m_i-\gamma _i\partial _x G^{-1}m_i. \end{aligned}$$

Case 1 $s>\max (\frac{1}{2}, \frac{1}{p})$.

Applying Lemma 2.8, we get

$$\begin{aligned}&\Vert (\gamma _1-\gamma _2)(t)\Vert _{B^{s-1}_{p,r}}\le e^{C\int _0^t \Vert \partial _x G_1\Vert _{B^{s}_{p,r}}dt'} \Big ( \Vert (\gamma _1-\gamma _2)(0)\Vert _{B^{s-1}_{p,r}} \nonumber \\&\quad +\int _0^t e^{-C\int _0^{t'} \Vert \partial _x G_1\Vert _{B^{s}_{p,r}} dt''} (\Vert (G_2-G_1)\partial _x \gamma _2\Vert _{B^{s-1}_{p,r}} +\Vert F_1-F_2\Vert _{B^{s-1}_{p,r}})dt'\Big ).\nonumber \\ \end{aligned}$$

(3.15)

After a similar calculation as in Step 3, we have

$$\begin{aligned}&\Vert (G_2-G_1)\partial _x \gamma _2\Vert _{B^{s-1}_{p,r}}\le C\Vert \gamma _1-\gamma _2\Vert _{B^{s-1}_{p,r}} (1+\Vert \gamma _1\Vert ^2_{B^{s}_{p,r}}+\Vert \gamma _2\Vert ^2_{B^{s}_{p,r}}), \end{aligned}$$

(3.16)

$$\begin{aligned}&\Vert F_1-F_2\Vert _{B^{s-1}_{p,r}}\le C\Vert \gamma _1-\gamma _2\Vert _{B^{s-1}_{p,r}} (1+\Vert \gamma _1\Vert ^2_{B^{s}_{p,r}}+\Vert \gamma _2\Vert ^2_{B^{s}_{p,r}}). \end{aligned}$$

(3.17)

Plugging (3.16), (3.17) into (3.15) yields that

$$\begin{aligned}&\Vert (\gamma _1-\gamma _2)(t)\Vert _{B^{s-1}_{p,r}} \le e^{C\int _0^t (\Vert \gamma _1\Vert ^2_{B^{s}_{p,r}}+1)dt'} \Big ( \Vert (\gamma _1-\gamma _2)(0)\Vert _{B^{s-1}_{p,r}} \nonumber \\&\quad +C\int _0^t e^{-C\int _0^{t'} (\Vert \gamma _1\Vert ^2_{B^{s}_{p,r}}+1) dt''} (1+\Vert \gamma _1\Vert ^2_{B^{s}_{p,r}}+\Vert \gamma _2\Vert ^2_{B^{s}_{p,r}})\Vert \gamma _1-\gamma _2\Vert _{B^{s-1}_{p,r}}dt'\Big ). \end{aligned}$$

(3.18)

Appling Gronwall’s inequality, we finally get

$$\begin{aligned} \Vert \gamma _1(t)-\gamma _2(t)\Vert _{B^{s-1}_{p,r}} \le \Vert \gamma _1(0)-\gamma _2(0)\Vert _{B^{s-1}_{p,r}}e^{C\int _0^t (1+\Vert \gamma _1\Vert ^2_{B^s_{p,r}}+\Vert \gamma _2\Vert ^2_{B^s_{p,r}})dt'}. \end{aligned}$$

(3.19)

Case 2 $s=\frac{1}{p},\ 1\le p\le 2,\ r=1.$

According to Lemma 2.8, we get

$$\begin{aligned}&\Vert (\gamma _1-\gamma _2)(t)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\le e^{C\int _0^t \Vert \partial _x G_1\Vert _{B^{\frac{1}{p}}_{p,1}}dt'} \Big ( \Vert (\gamma _1-\gamma _2)(0)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }} \nonumber \\&\quad +\int _0^t e^{-C\int _0^{t'} \Vert \partial _x G_1\Vert _{B^{\frac{1}{p}}_{p,1}} dt''} (\Vert (G_2-G_1)\partial _x \gamma _2\Vert _{B^{\frac{1}{p}-1}_{p,\infty }} +\Vert F_1-F_2\Vert _{B^{\frac{1}{p}-1}_{p,\infty }})dt'\Big ). \end{aligned}$$

(3.20)

Similarly, we deduce that

$$\begin{aligned}&\Vert (G_2-G_1)\partial _x \gamma _2\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\le C(1+\Vert \gamma _1\Vert ^2_{B^{\frac{1}{p} }_{p,\infty }}+ \Vert \gamma _2\Vert ^2_{B^{\frac{1}{p} }_{p,\infty }})\Vert \gamma _1-\gamma _2\Vert _{B^{\frac{1}{p}-1}_{p,1}}, \end{aligned}$$

(3.21)

$$\begin{aligned}&\Vert F_1-F_2\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\le C(1+\Vert \gamma _1\Vert ^2_{B^{\frac{1}{p} }_{p,\infty }}+ \Vert \gamma _2\Vert ^2_{B^{\frac{1}{p}}_{p,\infty }})\Vert \gamma _1-\gamma _2\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}. \end{aligned}$$

(3.22)

Plugging (3.21), (3.22) into (3.20), and using the uniform bounds of $\gamma _i$, we have

$$\begin{aligned} \Vert (\gamma _1-\gamma _2)(t)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }} \le C\Big (\Vert (\gamma _1-\gamma _2)(0)\Vert _{B^{\frac{1}{p}}_{p,\infty }}+\int _0^t \Vert \gamma _1-\gamma _2\Vert _{B^{\frac{1}{p}-1}_{p,1}}dt'\Big ). \end{aligned}$$

(3.23)

Applying Proposition 2.3 (2), it follows that

$$\begin{aligned}&\Vert (\gamma _1-\gamma _2)(t)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\le C \Big ( \Vert (\gamma _1-\gamma _2)(0)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\nonumber \\&\quad +\int _0^t \Vert \gamma _1-\gamma _2\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\ln \Big (e+\frac{C}{\Vert \gamma _1-\gamma _2\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}}\Big )dt'\Big ). \end{aligned}$$

(3.24)

Now let $h(t)=\Vert (\gamma _1-\gamma _2)(t)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}$. From above, h satisfies

$$\begin{aligned} h(t)\le & {} C\Big (h(0)+\int _0^t h(t')\ln \Big (e+\frac{C}{h(t')}\Big )dt'\Big )\\\le & {} C\Big (h(0)+\int _0^t h(t')\Big (1-\ln h(t')\Big )dt'\Big ). \end{aligned}$$

By virtue of Remark 2.7, we finally get

$$\begin{aligned} \Vert \gamma _1(t)-\gamma _2(t)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}\le C \Vert \gamma _1(0)-\gamma _2(0)\Vert _{B^{\frac{1}{p}-1}_{p,\infty }}^{e^{-Ct}}. \end{aligned}$$

(3.25)

Therefore, the uniqueness is obvious in view of (3.19) and (3.25). Moreover, an interpolation argument ensures that the continuity with respect to the initial data holds for the norm $C([0,T];B^{s'}_{p,r})$ whenever $s'<s$.

Step six Continuity with respect to the initial data.

Finally, we end up with a proposition about continuity until the exponent s.

Proposition 3.3

Let (s, p, r) be the statement of Theorem 3.2. Denote $\bar{{\mathbb {N}}}={\mathbb {N}}\cup \{\infty \}$. Suppose that $(\gamma ^n)_{n\in \bar{{\mathbb {N}}}}$ is the corresponding solution to (1.2) given by Theorem 3.2 with the initial data $\gamma _0^n\in B^s_{p,r}$. If $\gamma _0^n \rightarrow \gamma _0^{\infty }$ in $B^s_{p,r}$, then $\gamma ^n \rightarrow \gamma ^{\infty }$ in $C([0,T];B^s_{p,r})\ (resp., C_w([0,T];B^s_{p,r}))$ if $r<\infty \ (resp., r=\infty )$ with $4C^3 T\sup _{n\in \bar{{\mathbb {N}}}}\Vert \gamma ^n_0\Vert ^2_{B^{s}_{p,r}}<1.$

Proof

According to the proof of the existence, we find for all $n\in \bar{{\mathbb {N}}},\ t\in [0,T]$,

$$\begin{aligned} \Vert \gamma ^n(t)\Vert _{B^s_{p,r}}\le \frac{C\Vert \gamma ^n_0\Vert _{B^{s}_{p,r}}}{2\left( 1-4C^3 t\Vert \gamma ^n_0\Vert ^2_{B^{s}_{p,r}}\right) ^{\frac{1}{4}}}. \end{aligned}$$

Then $(\gamma ^n)_{n\in \bar{{\mathbb {N}}}}$ is bounded in $L^{\infty }([0,T];B^s_{p,r})$. We split $\gamma ^n=y^n+z^n$ with $(y^n, z^n)$ satisfying

$$\begin{aligned} \left\{ \begin{array}{l} y_t^n+G^ny_x^n=F^{\infty }, \\ y^n|_{t=0}=\gamma _0^{\infty }, \\ \end{array}\right. \text {and}\quad \left\{ \begin{array}{l} z_t^n+G^nz_x^n=F^n-F^{\infty }, \\ z^n|_{t=0}=\gamma _0^n-\gamma _0^{\infty }. \\ \end{array}\right. \end{aligned}$$

Obviously we have

$$\begin{aligned}&\Vert G^n\Vert _{B^{s+1}_{p,r}}\le C(\Vert \gamma ^n\Vert _{B^s_{p,r}}+\Vert \gamma ^n\Vert ^2_{B^s_{p,r}}), \end{aligned}$$

(3.26)

$$\begin{aligned}&\Vert G^n-G^{\infty }\Vert _{B^{s}_{p,r}}\le C(\Vert \gamma ^n\Vert _{B^s_{p,r}}+\Vert \gamma ^{\infty }\Vert _{B^s_{p,r}}+1)\Vert \gamma ^n-\gamma ^{\infty }\Vert _{B^{s-1}_{p,r}}. \end{aligned}$$

(3.27)

We have already known $\gamma ^n \rightarrow \gamma ^{\infty }$ in $L^{\infty }([0,T];B^{s-1}_{p,r})$. At the same time, according to (3.27), $G^n$ satisfy the condition of Lemma 2.11. Then we deduce that $y^n\rightarrow y^{\infty }$ in $C([0,T];B^{s}_{p,r})$ if $r<\infty$.

According to Lemma 2.8, we have for all $n\in {\mathbb {N}}$ and $t\in [0,T]$,

$$\begin{aligned}&\Vert z^n(t)\Vert _{B^{s}_{p,r}} \le e^{C\int _0^t \Vert G_x^n\Vert _{B^{s}_{p,r}} dt'}\Big (\Vert \gamma _0^n-\gamma _0^{\infty }\Vert _{B^{s}_{p,r}}\nonumber \\&\quad +\int _0^t e^{-C\int _0^{t'} \Vert G_x^n\Vert _{B^{s}_{p,r}} dt''} \Vert F^n-F^{\infty }\Vert _{B^{s}_{p,r}}dt'\Big ). \end{aligned}$$

(3.28)

We get

$$\begin{aligned} \Vert G_x^n\Vert _{B^{s}_{p,r}}\le C(\Vert \gamma ^n\Vert ^2_{B^{s}_{p,r}}+\Vert \gamma ^n\Vert _{B^{s}_{p,r}}), \end{aligned}$$

$$\begin{aligned} \Vert F^n-F^{\infty }\Vert _{B^{s}_{p,r}}\le C(\Vert \gamma ^n\Vert ^2_{B^{s}_{p,r}}+\Vert \gamma ^{\infty }\Vert ^2_{B^{s}_{p,r}}+1) \Vert \gamma ^n-\gamma ^{\infty }\Vert _{B^{s}_{p,r}}. \end{aligned}$$

(3.29)

Plugging (3.29) into (3.28), and using the uniform bounds of $\gamma ^n$, we obtain

$$\begin{aligned} \Vert z^n(t)\Vert _{B^{s}_{p,r}} \le C\Big (\Vert \gamma _0^n-\gamma _0^{\infty }\Vert _{B^{s}_{p,r}}+\int _0^t \Vert \gamma ^n-\gamma ^{\infty }\Vert _{B^{s}_{p,r}}dt'\Big ). \end{aligned}$$

Observing that $y^{\infty }=\gamma ^{\infty },\ z^{\infty }=0$, we can easily deduce that

$$\begin{aligned} \Vert z^n(t)\Vert _{B^{s}_{p,r}} \le C\Big (\Vert \gamma _0^n-\gamma _0^{\infty }\Vert _{B^{s}_{p,r}}+\int _0^t (\Vert y^n-y^{\infty }\Vert _{B^{s}_{p,r}}+\Vert z^n\Vert _{B^{s}_{p,r}})dt'\Big ). \end{aligned}$$

Appling Gronwall’s inequality yields that

$$\begin{aligned} \Vert z^n(t)\Vert _{B^s_{p,r}} \le e^{Ct}\Big (\Vert \gamma _0^n-\gamma _0^{\infty }\Vert _{B^{s}_{p,r}}+\int _0^t e^{-Ct}\Vert y^n-y^{\infty }\Vert _{B^{s}_{p,r}}dt'\Big ). \end{aligned}$$

Therefore when $r<\infty ,\ z^n \rightarrow 0$ in $C([0,T];B^{s}_{p,r})$, and hence $\gamma ^n\rightarrow \gamma ^{\infty }$ in $C([0,T];B^{s}_{p,r})$.

Considering the case $r=\infty$, we have weak continuity. In fact, for fixed $\phi \in B^{-s}_{p',1}$, we write

$$\begin{aligned} \langle \gamma ^n(t)-\gamma ^{\infty }(t),\phi \rangle =\langle \gamma ^n(t)-\gamma ^{\infty }(t),S_j\phi \rangle +\langle \gamma ^n(t)-\gamma ^{\infty }(t),\phi -S_j\phi \rangle . \end{aligned}$$

According to the duality, we have

$$\begin{aligned} |\langle \gamma ^n(t)-\gamma ^{\infty }(t),\phi \rangle |\le & {} \Vert \gamma ^n(t)-\gamma ^{\infty }(t)\Vert _{B^{s-1}_{p,\infty }}\Vert S_j\phi \Vert _{B^{1-s}_{p',1}}+\Vert \gamma ^n(t)\\&-\gamma ^{\infty }(t)\Vert _{B^s_{p,\infty }}\Vert \phi -S_j\phi \Vert _{B^{-s}_{p',1}}. \end{aligned}$$

Using the fact that $\gamma ^n\rightarrow \gamma ^{\infty }$ in $L^{\infty }([0,T];B^{s-1}_{p,\infty })$, and $S_j \phi \rightarrow \phi$ in $B^{-s}_{p',1}$ and $(\gamma ^n)_{n\in \bar{{\mathbb {N}}}}$ is bounded in $L^{\infty }([0,T];B^s_{p,r})$, it is easy to conclude that $\langle \gamma ^n(t)-\gamma ^{\infty }(t),\phi \rangle \rightarrow 0$ uniformly on [0, T].

$\square$

4 Blow-up and ill-posedness

First we prove a conservation law for (1.2).

Lemma 4.1

Let $\gamma _0\in H^s, s>\frac{1}{2}$ and let $T^*$ be the the maximal existence time of the corresponding solution $\gamma$ to (1.2). For any $t\in [0,T^*)$, then we have

$$\begin{aligned} \Vert \gamma (t)\Vert _{L^2}= \Vert \gamma _0\Vert _{L^2}. \end{aligned}$$

Proof

Arguing by density, it suffices to consider the case where $\gamma \in C_0^{\infty }({\mathbb {R}})$. The Eq. (1.2) can be rewritten as a conservation law

$$\begin{aligned} \gamma _t=[(v_x-\gamma )\lambda -\gamma v]_x. \end{aligned}$$

(4.1)

Using the fact that $v_{xx}-v=\gamma _x+\frac{\gamma ^2}{2\lambda }$ and then multiplying (4.1) with $\gamma$ and integrating by parts, we deduce that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _{{\mathbb {R}}}\gamma ^2dx&=\int _{{\mathbb {R}}} \lambda \gamma (v_{xx}-\gamma _x)-\gamma \gamma _x v-\gamma ^2 v_x dx \\&=\int _{{\mathbb {R}}} -\lambda \gamma _x v_x-\gamma ^2 v_x-\gamma \gamma _x v dx \\&=\int _{{\mathbb {R}}} vv_x-v_xv_{xx}-\frac{1}{2} \gamma ^2 v_x-\gamma \gamma _x v dx \\&=0. \end{aligned}$$

$\square$

Next we state a blow-up criterion for (1.2).

Lemma 4.2

Let $\gamma _0\in B^s_{p,r}$ with (s, p, r) being as in Theorem 3.2, and let $T^*$ be the maximal existence time of the corresponding solution $\gamma$ to (1.2). Then $\gamma$ blows up in finite time $T^*<\infty$ if and only if

$$\begin{aligned} \int _0^{T^*} \Vert \gamma (t')\Vert ^2_{L^{\infty }}dt'=\infty . \end{aligned}$$

Proof

Applying Lemma 2.9,

$$\begin{aligned} \Vert \gamma (t)\Vert _{B^s_{p,r}}\le C(\Vert \gamma _0\Vert _{B^s_{p,r}}+\int _0^t \Vert G^1_x\Vert _{L^{\infty }}\Vert \gamma \Vert _{B^s_{p,r}}+\Vert G^1_x\Vert _{B^s_{p,r}}\Vert \gamma \Vert _{L^{\infty }}+\Vert F\Vert _{B^s_{p,r}}dt'),\nonumber \\ \end{aligned}$$

(4.2)

where $G^1=G^{-1}m, \ F=\frac{1}{2} \gamma ^2+\lambda G^{-1}m-\gamma G^{-1}m_x, \ m=G^{-1}(\gamma _x+\frac{\gamma ^2}{2\lambda }).$

Note that the operator $G^{-1}$ coincides with the convolution by the function $x\mapsto \frac{-1}{2} e^{-|x|}$, which implies that $\Vert G^{-1}\gamma \Vert _{L^{\infty }},\ \Vert G^{-1}\gamma _x\Vert _{L^{\infty }}$ and $\Vert G^{-1}\gamma _{xx}\Vert _{L^{\infty }}$ can be bounded by $\Vert \gamma \Vert _{L^{\infty }}$. Then

$$\begin{aligned} \Vert G^1_x\Vert _{L^{\infty }}\le C(\Vert \gamma \Vert _{L^{\infty }}+\Vert \gamma \Vert ^2_{L^{\infty }}). \end{aligned}$$

(4.3)

As $s>0$, by Lemma 2.5, we have

$$\begin{aligned} \Vert G^1_x\Vert _{B^s_{p,r}}\le C\Vert \gamma \Vert _{B^{s}_{p,r}}(1+\Vert \gamma \Vert _{L^{\infty }}), \end{aligned}$$

(4.4)

and

$$\begin{aligned} \Vert F\Vert _{B^s_{p,r}}&\le C(\Vert \gamma \Vert _{B^{s}_{p,r}}\Vert \gamma \Vert _{L^{\infty }}+\Vert \gamma \Vert _{B^{s}_{p,r}}(1+\Vert \gamma \Vert _{L^{\infty }})+\Vert G^1_x\Vert _{B^s_{p,r}}\Vert \gamma \Vert _{L^{\infty }}\nonumber \\&\quad + \Vert G^1_x\Vert _{L^{\infty }}\Vert \gamma \Vert _{B^{s}_{p,r}}) \nonumber \\&\le C\Vert \gamma \Vert _{B^s_{p,r}}(1+\Vert \gamma \Vert _{L^{\infty }}+\Vert \gamma \Vert ^2_{L^{\infty }}). \end{aligned}$$

(4.5)

Plugging (4.3), (4.4) and (4.5) into (4.2), we get

$$\begin{aligned} \Vert \gamma (t)\Vert _{B^s_{p,r}}\le C(\Vert \gamma _0\Vert _{B^s_{p,r}}+\int _0^t (1+\Vert \gamma \Vert _{L^{\infty }}+\Vert \gamma \Vert ^2_{L^{\infty }})\Vert \gamma \Vert _{B^s_{p,r}}dt'). \end{aligned}$$

(4.6)

Appling Gronwall’s inequality yields that

$$\begin{aligned} \Vert \gamma (t)\Vert _{B^s_{p,r}}\le \Vert \gamma _0\Vert _{B^s_{p,r}}e^{C\int _0^t (1+\Vert \gamma \Vert _{L^{\infty }}+\Vert \gamma \Vert ^2_{L^{\infty }})dt'}. \end{aligned}$$

If $T^*$ is finite, and $\int _0^{T^*} \Vert \gamma \Vert ^2_{L^{\infty }}dt'<\infty$, then $\gamma \in L^{\infty }([0,T^*);B^s_{p,r})$, which contradicts the assumption that $T^*$ is the maximal existence time.

On the other hand, by Theorem 3.2 and the fact that $B^s_{p,r}\hookrightarrow L^{\infty }$, if $\int _0^{T^*} \Vert \gamma \Vert ^2_{L^{\infty }}dt'=\infty$, then $\gamma$ must blow up in finite time. $\square$

Let us consider the ordinary differential equation:

$$\begin{aligned} \left\{ \begin{array}{l} q_t(t,x)=G^{-1}\left( \gamma _x+\frac{\gamma ^2}{2\lambda }\right) (t,q(t,x)),\quad t\in [0,T), \\ q(0,x)=x,\quad x\in {\mathbb {R}}. \end{array}\right. \end{aligned}$$

(4.7)

If $\gamma \in B^s_{p,r}$ with (s, p, r) being as in Theorem 3.2, then $G^{-1}(\gamma _x+\frac{\gamma ^2}{2\lambda } )\in C([0,T); C^{0,1})$. According to the classical results in the theory of ordinary differential equations, we can easily infer that (4.7) have a unique solution $q\in C^1([0,T)\times {\mathbb {R}};{\mathbb {R}})$ such that the map $q(t,\cdot )$ is an increasing diffeomorphism of ${\mathbb {R}}$ with

$$\begin{aligned} q_x(t,x)=\exp \Big (\int _0^t G^{-1}(\gamma _x+\frac{\gamma ^2}{2\lambda } )(t',q(t',x)\Big )dt'>0,\quad \forall (t,x)\in [0,T)\times {\mathbb {R}}. \end{aligned}$$

We prove the following theorem which shows that the corresponding solution of (1.2) will blow up by giving negative condition for the initial data.

Theorem 4.3

Let $\gamma _0\in H^s$, $s>\frac{1}{2}$. Assume $\gamma _0(x_0)< -2\sqrt{d}$, with $d=C(\Vert \gamma _0\Vert _{L^2}+\frac{1}{2|\lambda |} \Vert \gamma _0\Vert ^2_{L^2})(C\Vert \gamma _0\Vert _{L^2}+C\frac{1}{2|\lambda |} \Vert \gamma _0\Vert ^2_{L^2}+|\lambda |)$. Then the corresponding solution $\gamma$ of (1.2) blows up in finite time.

Proof

Arguing by density, now we assume $s>\frac{3}{2}$. Then applying Lemma 4.1 and Young’s inequality, we get

$$\begin{aligned} \Vert G^{-1}\gamma (t)\Vert _{L^{\infty }},\Vert G^{-1}\gamma _x(t)\Vert _{L^{\infty }}\le \Vert \gamma (t)\Vert _{L^{2}}=\Vert \gamma _0\Vert _{L^{2}}, \end{aligned}$$

and

$$\begin{aligned} \Vert G^{-1}\gamma ^2(t)\Vert _{L^{\infty }},\Vert G^{-1}(\gamma ^2)_x(t)\Vert _{L^{\infty }}\le \Vert \gamma ^2(t)\Vert _{L^{1}}=\Vert \gamma (t)\Vert ^2_{L^{2}}=\Vert \gamma _0\Vert ^2_{L^{2}}. \end{aligned}$$

Denote $c=C(\Vert \gamma _0\Vert _{L^2}+\frac{1}{2|\lambda |} \Vert \gamma _0\Vert ^2_{L^2})$. Then we have:

$$\begin{aligned} \gamma _t(t,q(t,x_0))&=-\frac{1}{2} \gamma ^2-\gamma \left( G^{-1}\gamma +\frac{1}{2\lambda } \partial _xG^{-1}\gamma ^2\right) +\lambda G^{-1}\left( \gamma _x+\frac{\gamma ^2}{2\lambda }\right) \\&\le -\frac{1}{2} \gamma ^2+c|\gamma |+|\lambda |c \\&\le -\frac{1}{4} \gamma ^2+d. \end{aligned}$$

Denote $f(t)=\frac{-\sqrt{d}-\frac{1}{2} \gamma _0(x_0)}{\sqrt{d}-\frac{1}{2} \gamma _0(x_0)} e^{dt}$. Solving the above inequality, we finally get

$$\begin{aligned} \gamma (t,q(t,x_0))\le \frac{2\sqrt{d}f(t)+2\sqrt{d}}{f(t)-1} . \end{aligned}$$

(4.8)

As $\gamma _0(x_0)< -2\sqrt{d}$, we get

$$\begin{aligned} \gamma _t(t,q(t,x_0))< 0. \end{aligned}$$

(4.9)

Then we can deduce that $\gamma (t)$ decreases monotonly and is less than zero at the point $q(t,x_0)$ along the flow. Finally, we prove that the solution $\gamma (t)$ blows up in finite time. It is obvious that $0<f(0)< 1 \ and \ f(\infty )=\infty$, which implies that exists $T>0, \ f(T)=1$. Denote the maximal time of the solution by $T^*$. So we can easily deduce that $T^*\le T=\frac{1}{d} \ln (\frac{\sqrt{d}-\frac{1}{2} \gamma _0(x_0)}{-\sqrt{d}-\frac{1}{2} \gamma _0(x_0)} )$. Therefore, from (4.8) we know $\gamma (t,q(t,x_0))\rightarrow -\infty$ as $t\rightarrow T^*$. Applying Lemma (4.2), the solution $\gamma$ must blow up in finite time. $\square$

Lemma 4.4

Assume $\gamma \in H^1$ to (1.2). We have

$$\begin{aligned} \Vert \gamma \Vert _{L^{\infty }}\le C(\Vert \gamma \Vert _{B^0_{\infty ,\infty }}\cdot \log _2(2+\Vert \gamma \Vert _{H^1})+1) \end{aligned}$$

Proof

Fixing an integer $N>0$, we get

$$\begin{aligned} \Vert \gamma \Vert _{L^{\infty }}&\le \sum \limits _{k\le N-1}\Vert \Delta _k\gamma \Vert _{L^{\infty }}+\sum \limits _{k\ge N}\Vert \Delta _k\gamma \Vert _{L^{\infty }} \\&\le CN\Vert \gamma \Vert _{B^0_{\infty ,\infty }}+C\sum \limits _{k\ge N}2^{\frac{k}{2} }\Vert \Delta _k\gamma \Vert _{L^2} \\&\le CN\Vert \gamma \Vert _{B^0_{\infty ,\infty }}+C2^{-N}\Vert \gamma \Vert _{H^1}. \end{aligned}$$

Setting $N=\log _2(2+\Vert \gamma \Vert _{H^1})$, we complete the proof. $\square$

We need another blow-up criterion for (1.2) to prove norm inflation in the critical Besov Spaces.

Lemma 4.5

Let $\gamma _0\in H^1$, and let $T^*$ be the maximal existence time of the corresponding solution $\gamma$ to (1.2). Then $\gamma$ blows up in finite time $T^*<\infty$ if and only if

$$\begin{aligned} \int _0^{T^*} \Vert \gamma (t')\Vert _{B^0_{\infty ,\infty }}dt'=\infty . \end{aligned}$$

Proof

Applying Lemma 2.9, and since $L^{\infty }\hookrightarrow B^0_{\infty ,\infty }$, we have

$$\begin{aligned} \Vert \gamma (t)\Vert _{H^1}\le C(\Vert \gamma _0\Vert _{H^1}+\int _0^t \Vert G^1_x\Vert _{L^{\infty }}\Vert \gamma \Vert _{H^1}+\Vert G^1_x\Vert _{H^1}\Vert \gamma \Vert _{L^{\infty }}+\Vert F\Vert _{H^1}dt'),\nonumber \\ \end{aligned}$$

(4.10)

where $G^1=G^{-1}m, \ F=\frac{1}{2} \gamma ^2+\lambda G^{-1}m-\gamma G^{-1}m_x, \ m=G^{-1}(\gamma _x+\frac{\gamma ^2}{2\lambda }).$

Note that the operator $G^{-1}$ coincides with the convolution by the function $x\mapsto \frac{-1}{2} e^{-|x|}$, which implies that $\Vert G^{-1}\gamma \Vert _{L^{\infty }},\ \Vert G^{-1}\gamma _x\Vert _{L^{\infty }}$ and $\Vert G^{-1}\gamma _{xx}\Vert _{L^{\infty }}$ can be bounded by $\Vert \gamma \Vert _{L^{\infty }}$. Then applying Lemma 4.1, we get

$$\begin{aligned} \Vert G^1_x\Vert _{L^{\infty }}\le C\left( \Vert \gamma \Vert _{L^{\infty }}+\Vert (\frac{-1}{2} e^{-|x|})_x\Vert _{L^{\infty }}\Vert \gamma ^2\Vert _{L^{1}}\right) \le C(1+\Vert \gamma \Vert _{L^{\infty }}). \nonumber \\\end{aligned}$$

(4.11)

Applying Lemma 2.5, we have

$$\begin{aligned} \Vert G^1_x\Vert _{H^1}\le C(\Vert \gamma \Vert _{H^1}+\Vert \gamma ^2\Vert _{L^2})\le C(\Vert \gamma \Vert _{H^1}+\Vert \gamma \Vert _{L^{\infty }}), \end{aligned}$$

(4.12)

and

$$\begin{aligned} \Vert F\Vert _{H^1}&\le C(\Vert \gamma \Vert _{H^{1}}\Vert \gamma \Vert _{L^{\infty }}+\Vert \gamma \Vert _{H^{1}}+\Vert G^1_x\Vert _{H^1}\Vert \gamma \Vert _{L^{\infty }}+ \Vert G^1_x\Vert _{L^{\infty }}\Vert \gamma \Vert _{H^{1}}) \nonumber \\&\le C\Vert \gamma \Vert _{H^1}(1+\Vert \gamma \Vert _{L^{\infty }}). \end{aligned}$$

(4.13)

Plugging (4.11), (4.12) and (4.13) into (4.2), we get

$$\begin{aligned} \Vert \gamma (t)\Vert _{H^1}\le C(\Vert \gamma _0\Vert _{H^1}+\int _0^t (1+\Vert \gamma \Vert _{L^{\infty }})\Vert \gamma \Vert _{H^1}dt'). \end{aligned}$$

(4.14)

By Lemma 4.4, we have

$$\begin{aligned} \Vert \gamma (t)\Vert _{H^1}\le C(\Vert \gamma _0\Vert _{H^1}+\int _0^t \Big (1+\Vert \gamma \Vert _{B^0_{\infty ,\infty }}\log (e+\Vert \gamma \Vert _{H^1})\Big )\Vert \gamma \Vert _{H^1}dt').\nonumber \\ \end{aligned}$$

(4.15)

Appling Gronwall’s inequality yields that

$$\begin{aligned} \Vert \gamma (t)\Vert _{H^1}\le \Vert \gamma _0\Vert _{H^1}e^{Ct+C\int _0^t \Vert \gamma \Vert _{B^0_{\infty ,\infty }}\log (e+\Vert \gamma \Vert _{H^1})dt'}. \end{aligned}$$

Simplifying the above inequality and appling Gronwall’s inequality, we get

$$\begin{aligned} \log (e+\Vert \gamma (t)\Vert _{H^1})\le \Big (\log (e+\Vert \gamma _0\Vert _{H^1})+Ct\Big )e^{C\int _0^t \Vert \gamma \Vert _{B^0_{\infty ,\infty }}dt'}. \end{aligned}$$

If $T^*$ is finite, and $\int _0^{T^*} \Vert \gamma \Vert _{B^0_{\infty ,\infty }}dt'<\infty$, then $\gamma \in L^{\infty }([0,T^*);H^1)$, which contradicts the assumption that $T^*$ is the maximal existence time.

On the other hand, by Theorem 3.2 and the fact that $H^1\hookrightarrow L^{\infty }\hookrightarrow B^0_{\infty ,\infty }$, if $\int _0^{T^*} \Vert \gamma \Vert _{B^0_{\infty ,\infty }}dt'=\infty$, then $\gamma$ must blow up in finite time. $\square$

We end up with the following theorem which proves the norm inflation and hence the ill-posedness of the modified CH equation (1.2) in $H^{\frac{1}{2} }$ and in $B^{\frac{1}{2} }_{2,r}, 1<r\le \infty$.

Theorem 4.6

Let $1\le p\le \infty$ and $1<r\le \infty$. For any $\epsilon >0$, there exists $\gamma _0\in H^{\infty }$, such that the following holds:

• (1) $\Vert \gamma _0\Vert _{B^{\frac{1}{p} }_{p,r}}\le \epsilon$;

• (2) There is a unique solution $\gamma \in C([0,T);H^{\infty })$ to the equation (1.2) with a maximal lifespan $T<\epsilon$;

• (3) $limsup_{t \rightarrow T^{-}}\Vert \gamma \Vert _{B^{\frac{1}{p} }_{p,r}}\ge limsup_{t \rightarrow T^{-}}\Vert \gamma \Vert _{B^{0}_{\infty ,\infty }}=\infty$.

Proof

Fix $1\le p\le \infty$ and $1<r\le \infty$, and $\epsilon >0$. We define g(x)

$$\begin{aligned} g(x)=\sum \limits _{k\ge 1} \frac{1}{2^k k^{\frac{2}{1+r} }} g_k(x) \end{aligned}$$

(4.16)

with $g_k(x)$ given by the Fourier transform ${\hat{g}}_k(\xi )=-2^{-k}\xi {\widetilde{\chi }}(2^{-k}\xi )$, where ${\widetilde{\chi }}$ is a non-negative, non-zero $C^{\infty }_0$ function such that ${\widetilde{\chi }}\chi _0={\widetilde{\chi }}.$ Directly calculating, we have $\Delta _k g(x)=\frac{1}{2^k k^{\frac{2}{1+r} }} g_k(x)$. We also have $\Vert \Delta _k g(x)\Vert _{L^{p}}\sim \frac{2^{\frac{k}{p^\prime } }}{2^k k^{\frac{2}{1+r} }}$ and

$$\begin{aligned} \Vert g\Vert _{B^{\frac{1}{p} }_{p,q}}\sim \Vert \frac{1}{k^{\frac{2}{1+r} }} \Vert _{l^q}. \end{aligned}$$

Then we get $g\in B^{\frac{1}{p} }_{p,r} {\setminus } B^{\frac{1}{p} }_{p,1}$, and

$$\begin{aligned} g(0)=\int {\hat{g}}(\xi ) d\xi =-c\sum \limits _{k\ge 1} \frac{1}{k^{\frac{2}{1+r} }} =-\infty . \end{aligned}$$

For any $\epsilon >0$, let $\gamma _{0,\epsilon }=\Vert g\Vert ^{-1}_{B^{\frac{1}{p} }_{p,r}}\cdot \epsilon S_{K}(g)$ where K is large enough such that $\gamma _{0,\epsilon }(0)<\frac{-2\sqrt{d}(e^{d\epsilon }+1)}{e^{d\epsilon }-1}.$ Then $\gamma _{0,\epsilon }\in H^\infty$, $\Vert \gamma _{0,\epsilon }\Vert _{B^{\frac{1}{p} }_{p,r}}\le \epsilon$. Applying Theorem 4.3, there is a unique associated solution $\gamma \in C([0,T);H^{\infty })$ with a maximal lifespan $T<\epsilon$. By Lemmas 4.2 and 4.5, we can show that $limsup_{t \rightarrow T^{-}}\Vert \gamma \Vert _{B^{0}_{\infty ,\infty }}=\infty$. $\square$