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# Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion

Boundary Value Problems20172017:186

https://doi.org/10.1186/s13661-017-0919-1

• Received: 8 June 2017
• Accepted: 9 December 2017
• Published:

## Abstract

The goal of this paper is two-fold. Firstly, by using the Fourier restriction norm method and the fixed point theorem, we prove that the Cauchy problem for a generalized Ostrovsky equation
$$\partial_{x} \biggl(u_{t}-\beta\partial_{x}^{3}u +\frac{1}{3}\partial_{x}\bigl(u^{3}\bigr) \biggr) - \gamma u=0,\quad\beta>0,\gamma>0,$$
is locally well-posed in $$H^{s}(\mathbf{R})$$ with $$s\geq\frac{1}{4}$$. Secondly, we prove that the Cauchy problem for a generalized Ostrovsky equation is not well-posed in $$H^{s}(\mathbf{R})$$ with $$s<\frac{1}{4}$$ in the sense that the solution map is $$C^{3}$$.

## Keywords

• generalized Ostrovsky equation with positive dispersion
• Cauchy problem
• sharp well-posedness

• 35G25

## 1 Introduction

In this paper, we are concerned with the Cauchy problem for a generalized Ostrovsky equation with positive dispersion,
$$\partial_{x} \biggl(u_{t}-\beta\partial_{x}^{3}u +\frac{1}{3}\partial_{x}\bigl(u^{3}\bigr) \biggr) - \gamma u=0,\quad\gamma>0,\beta\in \mathbf {R}.$$
(1.1)
Here $$u(x, t)$$ represents the free surface of the liquid and the parameter $$\gamma>0$$ measures the effect of rotation. (1.1) describes the propagation of internal waves of even modes in the ocean; for instance, see the work of Galkin and Stepanyants , Leonov , and Shrira [3, 4]. The parameter β determines the type of dispersion, more precisely, when $$\beta<0$$, (1.1) denotes the generalized Ostrovsky equation with negative dispersion; when $$\beta>0$$, (1.1) denotes the generalized Ostrovsky equation with positive dispersion.

When $$\gamma= 0$$, (1.1) reduces to the modified Korteweg-de Vries equation which has been investigated by many authors; for instance, see . Kenig et al.  proved that the Cauchy problem for the modified KdV equation is locally well-posed in $$H^{s}(\mathbf {R})$$ with $$s\geq\frac{1}{4}$$. Kenig et al.  proved that the Cauchy problem for the modified KdV equation is ill-posed in $$H^{s}(\mathbf {R})$$ with $$s<\frac{1}{4}$$. Colliander et al.  proved that the Cauchy problem for the modified KdV equation is globally well-posed in $$H^{s}(\mathbf {R})$$ with $$s>\frac{1}{4}$$ and globally well-posed in $$H^{s}(\mathbf{T})$$ with $$s\geq\frac{1}{2}$$. Guo  and Kishimoto  proved that the modified KdV equation is globally well-posed in $$H^{\frac{1}{4}}(\mathbf {R})$$ with the aid of the I method and some new spaces.

Now we give a brief review of the Ostrovsky equation,
$$u_{t}-\beta\partial_{x}^{3}u +\frac{1}{3} \partial_{x}\bigl(u^{2}\bigr) -\gamma\partial_{x}^{-1}u=0,\quad \gamma>0.$$
(1.2)
Equation (1.2) was proposed by Ostrovsky in  as a model for weakly nonlinear long waves in a rotating liquid, by taking into account the Coriolis force, to describe the propagation of surface waves in the ocean in a rotating frame of reference. The parameter β determines the type of dispersion, more precisely, $$\beta<0$$ (negative dispersion) for surface and internal waves in the ocean or surface waves in a shallow channel with an uneven bottom and $$\beta >0$$ (positive dispersion) for capillary waves on the surface of liquid or for oblique magneto-acoustic waves in plasma [1, 1315]. Some authors have investigated the stability of the solitary waves or soliton solutions of (1.2); for instance, see .

Many people have studied the Cauchy problem for (1.2), for instance, see [17, 1930]. The result of [23, 25, 31] showed that $$s=-\frac{3}{4}$$ is the critical regularity index for (1.2). Coclite and di Ruvo [32, 33] have investigated the convergence of the Ostrovsky equation to the Ostrovsky-Hunter one and the dispersive and diffusive limits for Ostrovsky-Hunter type equation. Recently, Li et al.  proved that the Cauchy problem for the Ostrovsky equation with negative dispersion is locally well-posed in $$H^{-\frac {3}{4}}(\mathbf {R})$$.

Levandosky and Liu  studied the stability of solitary waves of the generalized Ostrovsky equation,
$$\bigl[u_{t}-\beta u_{xxx}+\bigl(f(u)\bigr)_{x} \bigr]_{x}=\gamma u,\quad x\in \mathbf {R},$$
(1.3)
where f is a $$C^{2}$$ function which is homogeneous of degree $$p\geq 2$$ in the sense that it satisfies $$sf^{\prime}(s)=pf(s)$$. Levandosky  studied the stability of ground state solitary waves of (1.4) with homogeneous nonlinearities of the form $$f(u)=c_{1}|u|^{p}+c_{2}|u|^{p-1}u$$, $$c_{1},c_{2}\in \mathbf {R}$$, $$p\geq2$$.
Equation (1.1) can be written in the following form:
$$u_{t}-\beta\partial_{x}^{3}u +\frac{1}{3} \partial_{x}\bigl(u^{3}\bigr)- \gamma\partial_{x}^{-1} u =0.$$
(1.4)
Let $$w(x,t)=\beta^{-\frac{1}{2}} u(x,\beta^{-1} t)$$, then $$w(x,t)$$ is the solution to
$$w_{t}-w_{xxx}+\frac{1}{3}\partial_{x} \bigl(w^{3}\bigr)-\gamma\beta^{-1}w=0.$$
Without loss of generality, we can assume that $$\beta=\gamma=1$$.
Motivated by , firstly, by using the $$X_{s,b}$$ spaces introduced by  and developed in [8, 41, 42] and the Strichartz estimates established in [19, 43], we prove that (1.3) with initial data
$$u(x,0)=u_{0}(x)$$
(1.5)
is locally well-posed in $$H^{s}(\mathbf {R})$$ with $$s\geq\frac{1}{4}$$, $$\beta >0$$, $$\gamma>0$$; secondly, we prove that the problems (1.3), (1.5) are not quantitatively well-posed in $$H^{s}(\mathbf {R})$$ with $$s<\frac{1}{4}$$, $$\beta\neq0$$, $$\gamma>0$$. Thus, our result is sharp.
We introduce some notations before giving the main result. Throughout this paper, we assume that C is a positive constant which may vary from line to line and $$0<\epsilon<{10^{-4}}$$. $$A\sim B$$ means that $$|B|\leq|A|\leq4|B|$$. $$A\gg B$$ means that $$|A|> 4|B|$$. $$\psi(t)$$ is a smooth function supported in $$[-1,2]$$ and equals 1 in $$[-1,1]$$. We assume that $$\mathscr{F}u$$ is the Fourier transformation of u with respect to both space and time variables and $$\mathscr{F}^{-1}u$$ is the inverse transformation of u with respect to both space and time variables, while $$\mathscr{F}_{x}u$$ denotes the Fourier transformation of u with respect to the space variable and $$\mathscr{F}^{-1}_{x}u$$ denotes the inverse transformation of u with respect to the space variable. Let $$I\subset \mathbf {R}$$, $$\chi_{I}(x)=1$$ if $$x\in I$$; $$\chi_{I}(x)=0$$ if x does not belong to I. Let
$$\langle\cdot\rangle=1+|\cdot|,\qquad \phi(\xi)=\xi^{3}+\frac{1}{\xi},\qquad \sigma=\tau+\phi(\xi),\qquad\sigma_{j}=\tau_{j}+\phi( \xi_{j}) \quad(j=1,2,3).$$
The space $$X_{s, b}$$ is defined by
$$X_{s, b}= \bigl\{ u\in\mathscr{S}{'}\bigl(\mathbf {R}^{2} \bigr) : \|u\|_{X_{s, b}} = \bigl\Vert \langle\xi\rangle^{s} \bigl\langle \tau+\phi(\xi ) \bigr\rangle ^{b}\mathscr{F}u(\xi,\tau) \bigr\Vert _{L_{\tau\xi }^{2}(\mathbf {R}^{2})}< \infty \bigr\} .$$
The space $$X_{s,b}^{T}$$ denotes the restriction of $$X_{s,b}$$ onto the finite time interval $$[-T,T]$$ and is equipped with the norm
$$\|u\|_{X_{s,b}^{T}} =\inf \bigl\{ \|w\|_{X_{s,b}}:w\in X_{s,b}, u(t)=w(t) \text{ for } {-}T\leq t\leq T \bigr\} .$$

The main results of this paper are as follows.

### Theorem 1.1

Let $$s\geq\frac{1}{4}$$ and $$\beta>0$$ and $$\gamma>0$$. Then the problems (1.4), (1.5) are locally well-posed in $$H^{s}(\mathbf {R})$$. More precisely, for $$u_{0} \in H^{s}(\mathbf {R})$$, there exist a $$T>0$$ and a unique solution $$u\in C([-T, T]; H^{s}(\mathbf {R}))$$.

### Remark 1

The result of Theorem 1.1 is optimal in the sense of Theorem 1.2.

### Theorem 1.2

Let $$s<\frac{1}{4}$$ and $$\beta>0$$ and $$\gamma>0$$. Then the problems (1.4), (1.5) are not well-posed in $$H^{s}(\mathbf {R})$$ in the sense that the solution map is $$C^{3}$$.

The rest of the paper is arranged as follows. In Section 2, we give some preliminaries. In Section 3, we establish a trilinear estimate. In Section 4, we prove Theorem 1.1. In Section 5, we prove Theorem 1.2.

## 2 Preliminaries

In this section, we give Lemmas 2.1-2.4.

### Lemma 2.1

Let $$0<\epsilon<\frac{1}{10^{8}}$$ and $$\mathscr{F}(P^{a}f)(\xi )=\chi_{\{|\xi|\geq a\}}(\xi) \mathscr{F}f(\xi)$$ with $$a\geq2$$ and $$\mathscr{F}(D_{x}^{b}f)(\xi)=|\xi|^{b}\mathscr{F}f(\xi)$$ with $$b\in \mathbf {R}$$. Then we have
\begin{aligned}& \|u\|_{L_{xt}^{6}}\leq C\|u\|_{X_{0,\frac{1}{2}+\epsilon}}, \end{aligned}
(2.1)
\begin{aligned}& \bigl\Vert D_{x}^{\frac{1}{6}}P^{a}u \bigr\Vert _{L_{xt}^{6}}\leq C\|u\| _{X_{0,\frac{1}{2}+\epsilon}}, \end{aligned}
(2.2)
\begin{aligned}& \|u\|_{L_{xt}^{4}}\leq C\|u\|_{X_{0,\frac{3}{4} (\frac {1}{2}+\epsilon )}}. \end{aligned}
(2.3)

For the proof of Lemma 2.1, we refer the reader to (2.27) and (2.21) of .

### Lemma 2.2

Let $$\phi(\xi)=\xi^{3}+\frac{1}{\xi}$$ and
$$\mathscr{F} \bigl(I^{s}(u,v)\bigr) (\xi,\tau)= \underset{\tau=\tau_{1}+\tau_{2}}{\int_{ \xi=\xi_{1}+\xi_{2} }}\big|\phi^{\prime}(\xi_{1})- \phi^{\prime}(\xi_{2})\big|^{s}\mathscr {F}u_{1}( \xi_{1},\tau_{1}) \mathscr{F}u_{2}( \xi_{2},\tau_{2})\,d\xi_{1}\,d\tau_{1}.$$
Then we have
$$\bigl\Vert I^{\frac{1}{2}}(u_{1},u_{2}) \bigr\Vert _{L_{xt}^{2}} \leq C\prod_{j=1}^{2} \|u_{j}\|_{X_{0,\frac{1}{2}+\epsilon}}.$$
(2.4)

For the proof of Lemma 2.2, we refer the reader to Lemma 2.5 of .

### Lemma 2.3

Let $$T\in(0,1)$$ and $$b\in(\frac{1}{2},\frac{3}{2})$$. Then, for $$s\in \mathbf {R}$$ and $$\theta\in[0,\frac{3}{2}-b)$$, we have
$$\begin{gathered} \big\| \eta_{T}(t)S(t)\phi\big\| _{X_{s,b}(\mathbf {R}^{2})}\leq CT^{\frac {1}{2}-b}\|\phi \|_{H^{s}(\mathbf {R})}, \\ \biggl\Vert \eta_{T}(t) \int_{0}^{t}S(t-\tau)F(\tau)\,d\tau \biggr\Vert _{X_{s,b}(\mathbf {R}^{2})} \leq CT^{\theta}\|F\|_{X_{s,b-1+\theta}(\mathbf {R}^{2})}.\end{gathered}$$

For the proof of Lemma 2.3, we refer the reader to [8, 39, 44].

### Lemma 2.4

Let $$a_{j}\in \mathbf {R}$$ ($$j=1,2,3$$) and $$\prod _{j=1}^{3}a_{j}\neq 0$$. Then we have
\begin{aligned}[b] & \Biggl(\sum _{j=1}^{3}a_{j} \Biggr)^{3}+\frac{1}{\sum _{j=1}^{3}a_{j}} -\sum _{j=1}^{3} \biggl(a_{j}^{3}+\frac{1}{a_{j}} \biggr) \\ &\quad=3(a_{1}+a_{2}) (a_{1}+a_{3}) (a_{2}+a_{3}) \biggl[1-\frac {1}{3\prod _{j=1}^{3}a_{j} (\sum _{j=1}^{3}a_{j} )} \biggr].\end{aligned}
(2.5)

### Proof

By using the following two identities:
$$\begin{gathered} \Biggl(\sum _{j=1}^{3}a_{j} \Biggr)^{3}- \Biggl(\sum _{j=1}^{3}a_{j}^{3} \Biggr) =3(a_{1}+a_{2}) (a_{1}+a_{3}) (a_{2}+a_{3}), \\ \Biggl(\sum _{j=1}^{3}a_{j} \Biggr) (a_{1}a_{2}+a_{1}a_{3}+a_{2}a_{3}) -\prod _{j=1}^{3}a_{j}=(a_{1}+a_{2}) (a_{1}+a_{3}) (a_{2}+a_{3}),\end{gathered}$$
which can be found in , we have
$$\begin{gathered} \Biggl(\sum _{j=1}^{3}a_{j} \Biggr)^{3}+\frac{1}{\sum _{j=1}^{3}a_{j}} -\sum _{j=1}^{3} \biggl(a_{j}^{3}+\frac{1}{a_{j}} \biggr)\\ \quad= \Biggl(\sum _{j=1}^{3}a_{j} \Biggr)^{3}- \sum _{j=1}^{3}a_{j}^{3}- \Biggl[\sum _{j=1}^{3}\frac {1}{a_{j}}-\frac{1}{\sum _{j=1}^{3}a_{j}} \Biggr] \\ \quad=3(a_{1}+a_{2}) (a_{1}+a_{3}) (a_{2}+a_{3})- \biggl[\frac{ (\sum _{j=1}^{3}a_{j} )(a_{1}a_{2}+a_{1}a_{3}+a_{2}a_{3}) +\prod _{j=1}^{3}a_{j}}{\prod _{j=1}^{3}a_{j} (\sum _{j=1}^{3}a_{j} )} \biggr] \\ \quad=3(a_{1}+a_{2}) (a_{1}+a_{3}) (a_{2}+a_{3})- \biggl[\frac {(a_{1}+a_{2})(a_{1}+a_{3})(a_{2}+a_{3})}{\prod _{j=1}^{3}a_{j} (\sum _{j=1}^{3}a_{j} )} \biggr] \\ \quad=3(a_{1}+a_{2}) (a_{1}+a_{3}) (a_{2}+a_{3}) \biggl[1-\frac{1}{3\prod _{j=1}^{3}a_{j} (\sum _{j=1}^{3}a_{j} )} \biggr].\end{gathered}$$
Thus, (2.5) is valid.

This ends the proof of Lemma 2.4. □

## 3 The trilinear estimate

In this section, by using Lemmas 2.1-2.2, we give the proof of Lemma 3.1.

### Lemma 3.1

Let $$u_{j}\in X_{s,\frac{1}{2}+\epsilon}$$ with $$s\geq\frac{1}{4}$$ and $$j=1,2,3$$. Then we have
$$\Biggl\Vert \partial_{x} \Biggl(\prod_{j=1}^{3}u_{j} \Biggr) \Biggr\Vert _{X_{s,-\frac{1}{2}+2\epsilon}}\leq C\prod_{j=1}^{3} \|u_{j}\|_{X_{s,\frac{1}{2}+\epsilon}}.$$
(3.1)

### Proof

To prove (3.1), by duality, it suffices to prove that
$$\int_{\mathbf {R}^{2}}\bar{u}(x,t)\partial_{x} \Biggl(\prod _{j=1}^{3}u_{j} \Biggr)\,dx\,dt\leq C \Biggl[\prod_{j=1}^{3}\|u_{j} \|_{X_{s,\frac{1}{2}+\epsilon}} \Biggr] \|u\|_{X_{-s,\frac{1}{2}-2\epsilon}}.$$
(3.2)
Let
$$\begin{gathered} F(\xi,\tau)=\langle\xi\rangle^{-s}\langle\sigma\rangle^{\frac {1}{2}-2\epsilon} \mathscr{F}u(\xi,\tau),\\ F_{j}(\xi_{j},\tau_{j})= \langle\xi_{j}\rangle^{s}\langle\sigma _{j} \rangle^{\frac{1}{2}+\epsilon} \mathscr{F}u_{j}(\xi_{j}, \tau_{j})\quad (j=1,2,3).\end{gathered}$$
(3.3)
To obtain (3.2), from (3.3), it suffices to prove that
\begin{aligned}[b] & \int_{\mathbf {R}^{2}} \underset{\tau=\tau_{1}+\tau_{2}+\tau_{3}}{\int_{ \xi=\xi_{1}+\xi_{2}+\xi_{3} }}\frac{|\xi|\langle\xi\rangle^{s} F(\xi,\tau)\prod _{j=1}^{3}F_{j}(\xi_{j},\tau_{j})}{\langle \sigma\rangle^{\frac{1}{2}-2\epsilon} \prod _{j=1}^{3}\langle\xi_{j}\rangle^{s}\langle\sigma_{j} \rangle^{\frac{1}{2}+\epsilon}}\,d\xi_{1}\,d \tau_{1}\,d\xi_{2}\,d\tau _{2}\,d\xi \,d\tau \\ &\quad\leq C \|F\|_{L_{\xi\tau}^{2}} \Biggl(\prod_{j=1}^{3} \|F_{j}\|_{L_{\xi\tau }^{2}} \Biggr).\end{aligned}
(3.4)
Without loss of generality, by using the symmetry, we assume that $$|\xi_{1}|\geq|\xi_{2}|\geq|\xi_{3}|$$ and $$F(\xi,\tau)\geq 0$$, $$F_{j}(\xi_{j},\tau_{j})\geq0$$ ($$j=1,2$$). We define
$$\begin{gathered} \Omega_{1}=\Biggl\{ (\xi_{1}, \tau_{1},\xi_{2},\tau_{2},\xi ,\tau)\in{\mathrm{R}^{6}},\xi=\sum_{j=1}^{3} \xi_{j},\tau=\sum_{j=1}^{3} \tau_{j}, |\xi_{3}|\leq|\xi_{2}|\leq| \xi_{1}|\leq64 \Biggr\} , \\ \Omega_{2}=\Biggl\{ (\xi_{1}, \tau_{1},\xi_{2},\tau_{2},\xi ,\tau)\in{\mathrm{R}^{6}},\xi=\sum_{j=1}^{3} \xi_{j},\tau=\sum_{j=1}^{3} \tau_{j}, |\xi_{1}|\geq64, |\xi_{1}|\geq4| \xi_{2}|\Biggr\} , \\ \Omega_{3}=\Biggl\{ (\xi_{1}, \tau_{1},\xi_{2},\tau_{2},\xi ,\tau)\in{\mathrm{R}^{6}},\xi=\sum_{j=1}^{3} \xi_{j},\tau=\sum_{j=1}^{3} \tau_{j}, |\xi_{1}|\geq64, |\xi_{1}|\sim| \xi_{2}|,|\xi_{2}|\gg|\xi_{3}|\Biggr\} , \\ \Omega_{4}=\Biggl\{ (\xi_{1}, \tau_{1},\xi_{2},\tau_{2},\xi ,\tau)\in{\mathrm{R}^{6}},\xi=\sum_{j=1}^{3} \xi_{j},\tau=\sum_{j=1}^{3} \tau_{j}, |\xi_{1}|\geq64, |\xi_{1}|\sim| \xi_{2}|\sim|\xi_{3}| \Biggr\} .\end{gathered}$$
Obviously, $$\{(\xi_{1},\tau_{1},\xi_{2},\tau_{2},\xi,\tau)\in{\mathrm{ R}^{6}},\xi=\sum _{j=1}^{3}\xi_{j},\tau=\sum _{j=1}^{3}\tau_{j}, |\xi_{3}|\leq|\xi_{2}|\leq|\xi_{1}|\}\subset\bigcup _{j=1}^{4}\Omega_{j}$$. Let
$$K(\xi_{1},\tau_{1},\xi_{2}, \tau_{2},\xi,\tau)=\frac{|\xi |\langle\xi\rangle^{s}}{\langle\sigma\rangle^{\frac {1}{2}-2\epsilon}\prod _{j=1}^{3}\langle\sigma_{j}\rangle ^{\frac{1}{2}+\epsilon}}$$
(3.5)
and
$$I= \int_{\mathbf {R}^{2}} \underset{\tau=\sum _{j=1}^{3}\tau_{j}}{\int_{ \xi=\sum _{j=1}^{3}\xi_{j} }}K(\xi_{1},\tau_{1}, \xi_{2},\tau_{2},\xi,\tau)F(\xi,\tau)\prod _{j=1}^{3}F_{j}(\xi_{j}, \tau_{j}) \,d\xi_{1}\,d\tau_{1}\,d\xi_{2}\,d \tau_{2}\,d\xi \,d\tau.$$
(1) $$\Omega_{1}$$. In this subregion, we have
$$K(\xi_{1},\tau_{1},\xi_{2},\tau_{2}, \xi,\tau)\leq\frac {C}{\langle\sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}}.$$
(3.6)
By using (3.6) and the Cauchy-Schwartz inequality and the Plancherel identity and the Hölder inequality as well as (2.1), we have
\begin{aligned} I&\leq C \int_{\mathbf {R}^{2}} \underset{\tau=\sum _{j=1}^{3}\tau_{j}}{\int_{ \xi=\sum _{j=1}^{3}\xi_{j} }}\frac{ F(\xi,\tau)\prod _{j=1}^{3}F_{j}(\xi_{j},\tau_{j})}{\langle \sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3} \langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}} \,d\xi_{1}\,d \tau_{1}\,d\xi_{2}\,d\tau_{2}\,d\xi \,d\tau \\ &\leq C \biggl\Vert \frac{F(\xi,\tau)}{\langle\sigma\rangle^{\frac {1}{2}-2\epsilon}} \biggr\Vert _{L_{\xi\tau}^{2}} \biggl\Vert \underset{\tau =\sum _{j=1}^{3}\tau_{j}}{\int_{ \xi=\sum _{j=1}^{3}\xi_{j} }}\frac{\prod _{j=1}^{3}F_{j}(\xi_{j},\tau_{j})}{\prod _{j=1}^{3} \langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}}\,d\xi_{1}\,d\tau _{1}\,d\xi_{2}\,d\tau_{2} \biggr\Vert _{L_{\xi\tau}^{2}} \\ &\leq C\|F\|_{L_{\xi\tau}^{2}} \Biggl(\prod _{j=1}^{3} \biggl\Vert \mathscr{F}^{-1} \biggl(\frac {F_{j}}{\langle\sigma_{j} \rangle^{\frac{1}{2}+\epsilon}} \biggr) \biggr\Vert _{L_{xt}^{6}} \Biggr) \\ &\leq C\|F\|_{L_{\xi\tau}^{2}} \Biggl(\prod _{j=1}^{3} \| F_{j}\|_{L_{\xi\tau}^{2}} \Biggr).\end{aligned}
(2) $$\Omega_{2}$$. In this subregion, since $$\vert \phi^{\prime}(\xi_{1})-\phi^{\prime}(\xi_{2}) \vert =3|\xi_{1}^{2}-\xi_{2}^{2}| \vert 1+\frac{1}{3\xi_{1}^{2}\xi _{2}^{2}} \vert \geq3|\xi_{1}^{2}-\xi_{2}^{2}|\geq C|\xi|^{2}$$ and $$|\xi|\sim|\xi _{1}|$$, we have
\begin{aligned}[b] K(\xi_{1},\tau_{1},\xi_{2},\tau_{2}, \xi,\tau)&\leq\frac{C|\xi |}{\langle\sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon }} \\ & \leq C\frac{C|\xi_{1}^{2}-\xi_{2}^{2}|^{\frac{1}{2}} \vert 1+\frac {1}{3\xi_{1}^{2}\xi_{2}^{2}} \vert ^{\frac{1}{2}}}{ \langle\sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}} = \frac{C|\phi^{\prime}(\xi_{1})-\phi^{\prime}(\xi_{2})|^{\frac{1}{2}}}{ \langle\sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}} .\end{aligned}
(3.7)
By using (3.7) and the Cauchy-Schwartz inequality and the Plancherel identity and the Hölder inequality as well as (2.3)-(2.4), since $$\frac{3}{4} (\frac{1}{2}+\epsilon )<\frac {1}{2}-2\epsilon$$, we have
\begin{aligned} I\leq{}& C \int_{\mathbf {R}^{2}} \underset{\tau=\sum _{j=1}^{3}\tau_{j}}{\int_{ \xi=\sum _{j=1}^{3}\xi_{j} }}\frac{|\phi^{\prime}(\xi_{1})-\phi^{\prime}(\xi_{2})|^{\frac{1}{2}} F(\xi,\tau)\prod _{j=1}^{3}F_{j}(\xi_{j},\tau_{j})}{\langle \sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3} \langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}} \,d\xi_{1}\,d \tau_{1}\,d\xi_{2}\,d\tau_{2}\,d\xi \,d\tau \\ \leq{}& C \biggl\Vert \mathscr{F}^{-1} \biggl(\frac{F}{\langle\sigma \rangle^{\frac{1}{2}-2\epsilon}} \biggr) \biggr\Vert _{L_{xt}^{4}} \biggl\Vert I^{\frac{1}{2}} \biggl( \mathscr{F}^{-1} \biggl(\frac {F_{1}}{\langle\sigma_{1} \rangle^{\frac{1}{2}+\epsilon}} \biggr),\mathscr{F}^{-1} \biggl(\frac{F_{1}}{\langle\sigma_{2} \rangle ^{\frac{1}{2}+\epsilon}} \biggr) \biggr) \biggr\Vert _{L_{xt}^{2}} \\ &\times \biggl\Vert \mathscr{F}^{-1} \biggl(\frac{F_{3}}{\langle\sigma _{3}\rangle^{\frac{1}{2}+\epsilon}} \biggr) \biggr\Vert _{L_{xt}^{4}} \\ \leq{}& C\|F\|_{L_{\xi\tau}^{2}} \Biggl(\prod_{j=1}^{3} \|F_{j}\| _{L_{\xi\tau}^{2}} \Biggr).\end{aligned}
(3) $$\Omega_{3}$$. In this subregion, since $$\vert \phi^{\prime}(\xi_{2})-\phi^{\prime}(\xi_{3}) \vert =3|\xi_{2}^{2}-\xi_{3}^{2}| \vert 1+\frac{1}{3\xi_{2}^{2}\xi _{3}^{2}} \vert \geq3|\xi_{2}^{2}-\xi_{3}^{2}|\geq C|\xi_{1}|^{2}$$, we have
\begin{aligned}[b] K(\xi_{1},\tau_{1},\xi_{2},\tau_{2}, \xi,\tau)&\leq\frac{C|\xi_{1}|}{ \langle\sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3} \langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}} \\ &\quad\leq C C\frac{C|\xi_{2}^{2}-\xi_{3}^{2}|^{\frac{1}{2}} \vert 1+\frac {1}{3\xi_{2}^{2}\xi_{3}^{2}} \vert ^{\frac{1}{2}}}{ \langle\sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}}\leq \frac{C|\phi^{\prime}(\xi_{2})-\phi^{\prime}(\xi_{3})|^{\frac{1}{2}}}{ \langle\sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}} .\end{aligned}
(3.8)
By using (3.8) and the Cauchy-Schwartz inequality and the Plancherel identity and the Hölder inequality as well as (2.3)-(2.4), since $$\frac{3}{4} (\frac{1}{2}+\epsilon )<\frac {1}{2}-2\epsilon$$, we have
\begin{aligned} I\leq{}& C \int_{\mathbf {R}^{2}} \underset{\tau=\sum _{j=1}^{3}\tau_{j}}{\int_{ \xi=\sum _{j=1}^{3}\xi_{j} }}\frac{|\phi^{\prime}(\xi_{2})-\phi^{\prime}(\xi_{3})|^{\frac{1}{2}} F(\xi,\tau)\prod _{j=1}^{3}F_{j}(\xi_{j},\tau_{j})}{\langle \sigma\rangle^{\frac{1}{2}-2\epsilon} \prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac {1}{2}+\epsilon}} \,d\xi_{1}\,d \tau_{1}\,d\xi_{2}\,d\tau_{2}\,d\xi \,d\tau \\ \leq{}& C \biggl\Vert \mathscr{F}^{-1} \biggl(\frac{F}{\langle\sigma \rangle^{\frac{1}{2}-2\epsilon}} \biggr) \biggr\Vert _{L_{xt}^{4}} \biggl\Vert I^{\frac{1}{2}} \biggl( \mathscr{F}^{-1} \biggl(\frac {F_{2}}{\langle\sigma_{2} \rangle^{\frac{1}{2}+\epsilon}} \biggr),\mathscr{F}^{-1} \biggl(\frac{F_{3}}{\langle\sigma_{3} \rangle ^{\frac{1}{2}+\epsilon}} \biggr) \biggr) \biggr\Vert _{L_{xt}^{2}} \\ & \times \biggl\Vert \mathscr{F}^{-1} \biggl(\frac{F_{1}}{\langle\sigma _{1}\rangle^{\frac{1}{2}+\epsilon}} \biggr) \biggr\Vert _{L_{xt}^{4}} \\ \leq{}& C\|F\|_{L_{\xi\tau}^{2}} \Biggl(\prod _{j=1}^{3} \| F_{j}\|_{L_{\xi\tau}^{2}} \Biggr).\end{aligned}
(4) $$\Omega_{4}$$. In this subregion, since $$s\geq\frac {1}{4}$$ and $$|\xi_{1}|\sim|\xi_{2}|\sim|\xi_{3}|$$, we have
$$K(\xi_{1},\tau_{1},\xi_{2},\tau_{2}, \xi,\tau)\leq\frac{C|\xi _{1}|^{1-2s}}{\langle\sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon }}\leq\frac{C\prod _{j=1}^{3}|\xi_{j}|^{\frac{1}{6}}}{ \langle\sigma\rangle^{\frac{1}{2}-2\epsilon}\prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac{1}{2}+\epsilon}} .$$
(3.9)
By using (3.9) and the Cauchy-Schwartz inequality and the Plancherel identity and the Hölder inequality as well as (2.2), since $$\frac{3}{4} (\frac{1}{2}+\epsilon )<\frac {1}{2}-2\epsilon$$, we have
\begin{aligned} I&\leq C \int_{\mathbf {R}^{2}} \underset{\tau=\sum _{j=1}^{3}\tau_{j}}{\int_{ \xi=\sum _{j=1}^{3}\xi_{j} }}\frac{ F(\xi,\tau)\prod _{j=1}^{3}|\xi_{j}|^{\frac{1}{6}}F_{j}(\xi _{j},\tau_{j})}{\langle\sigma\rangle^{\frac{1}{2}-2\epsilon} \prod _{j=1}^{3}\langle\sigma_{j}\rangle^{\frac {1}{2}+\epsilon}} \,d\xi_{1}\,d \tau_{1}\,d\xi_{2}\,d\tau_{2}\,d\xi \,d\tau \\ &\leq C \biggl\Vert \frac{F}{\langle\sigma\rangle^{\frac {1}{2}-2\epsilon}} \biggr\Vert _{L_{\xi\tau}^{2}} \Biggl( \prod_{j=1}^{3} \biggl\Vert D_{x}^{\frac{1}{6}}P^{2}\mathscr{F}^{-1} \biggl( \frac {F_{j}}{\langle\sigma_{j} \rangle^{\frac{1}{2}+\epsilon}} \biggr) \biggr\Vert _{L_{xt}^{6}} \Biggr) \\ &\leq C\|F\|_{L_{\xi\tau}^{2}} \Biggl(\prod _{j=1}^{3} \| F_{j}\|_{L_{\xi\tau}^{2}} \Biggr).\end{aligned}

This completes the proof of Lemma 3.1. □

## 4 Proof of Theorem 1.1

In this section, we use Lemmas 2.3, 3.1 to prove Theorem 1.1.

The solution to (1.3), (1.5) can be formally rewritten as follows:
$$u(t)=e^{-t(-\partial_{x}^{3}-\partial_{x}^{-1})}u_{0}+\frac {1}{3} \int_{0}^{t}e^{-(t-s) (-\partial_{x}^{3}-\partial_{x}^{-1})}\partial_{x} \bigl(u^{3}\bigr)\,ds.$$
(4.1)
We define
$$\Phi(u)=\psi(t)e^{-t(-\partial_{x}^{3}-\partial _{x}^{-1})}u_{0}+\frac{1}{3} \psi \biggl( \frac{t}{T} \biggr) \int_{0}^{t}e^{-(t-s) (-\partial_{x}^{3}-\partial_{x}^{-1})}\partial_{x} \bigl(u^{3}\bigr)\,ds.$$
(4.2)
By taking advantaging of Lemmas 2.3, 3.1, we derive that
\begin{aligned}[b] \big\| \Phi(u)\big\| _{X_{s,\frac{1}{2}+\epsilon}} &\leq C\|u_{0}\|_{H^{s}(\mathbf {R})}+C \biggl\Vert \psi \biggl(\frac{t}{T} \biggr) \int_{0}^{t}e^{-(t-s) (-\partial_{x}^{3}-\partial_{x}^{-1})}\partial_{x} \bigl(u^{3}\bigr)\,ds \biggr\Vert _{X_{s,\frac{1}{2}+\epsilon}} \\ &\leq C\|u_{0}\|_{H^{s}(\mathbf {R})}+CT^{\epsilon} \bigl\Vert \partial_{x}\bigl(u^{3}\bigr)\,ds \bigr\Vert _{X_{s,-\frac{1}{2}+2\epsilon }} \\ &\leq C\|u_{0}\|_{H^{s}(\mathbf {R})}+CT^{\epsilon} \Vert u \Vert _{X_{s,\frac{1}{2}+\epsilon}}^{3}.\end{aligned}
(4.3)
We define $$B= \{u\in X_{s,\frac{1}{2}+\epsilon}: \|u\|_{X_{s,\frac {1}{2}+\epsilon}}\leq2C\|u_{0}\|_{H^{s}(\mathbf {R})} \}$$. By using (4.3), by choosing T sufficiently small such that $$24C^{3}T^{\epsilon}\|u_{0}\|_{H^{s}}^{2}<1$$, we have
$$\big\| \Phi(u)\big\| _{X_{s,\frac{1}{2}+\epsilon}} \leq C\|u_{0}\|_{H^{s}(\mathbf {R})}+CT^{\epsilon}\bigl(2C \|u_{0}\|_{H^{s}(\mathbf {R})}\bigr)^{3}\leq2C\|u_{0} \|_{H^{s}(\mathbf {R})},$$
(4.4)
thus, $$\Phi(u)$$ is a mapping on B. By using a proof similar to (4.4), by choosing T sufficiently small such that $$24C^{3}T^{\epsilon}\|u_{0}\|_{H^{s}}^{2}<1$$, we obtain
\begin{aligned}[b] &\big\| \Phi(u_{1})-\Phi(u_{2})\big\| _{X_{s,\frac{1}{2}+\epsilon}} \\ &\quad\leq CT^{\epsilon} \bigl[\|u_{1}\|_{X_{s,\frac{1}{2}+\epsilon }}^{2}+ \|u_{1}\|_{X_{s,\frac{1}{2}+\epsilon}}\|u_{2}\|_{X_{s,\frac {1}{2}+\epsilon}} + \|u_{2}\|_{X_{s,\frac{1}{2}+\epsilon}}^{2} \bigr]\|u_{1}-u_{2} \| _{X_{s,\frac{1}{2}+\epsilon}} \\ &\quad\leq\frac{1}{2}\|u_{1}-u_{2}\|_{X_{s,\frac{1}{2}+\epsilon }},\end{aligned}
(4.5)
thus, $$\Phi(u)$$ is a contraction mapping on the closed ball B. Consequently, Φ have a fixed point u and the Cauchy problem for (1.1) possesses a local solution on $$[-T,T]$$. The uniqueness of the solution is obvious.

This completes the proof of Theorem 1.1.

## 5 Proof of Theorem 1.2

In this section, inspired by [5, 35, 45], we present the proof of Theorem 1.2. We will prove Theorem 1.2 by contradiction.

We assume that the solution map of (1.4), (1.5) is $$C^{3}$$ in $$H^{s}(\mathbf {R})$$ with $$s<\frac{1}{4}$$. Then, from Theorem 3 of , we have
$$\sup _{t\in[0,T]} \bigl\Vert B_{3}(u_{0}) \bigr\Vert _{H^{s}}\leq C\| u_{0}\|_{H^{s}}^{3}$$
(5.1)
for $$u_{0}\in H^{s}(\mathbf {R})$$. Here
\begin{aligned}& B_{1}(u_{0})= e^{-t(-\partial_{x}^{3}-\partial _{x}^{-1})}u_{0}, \end{aligned}
(5.2)
\begin{aligned}& B_{3}(u_{0})=\frac{1}{3} \int_{0}^{t}e^{-(t-\tau)(-\partial _{x}^{3}-\partial_{x}^{-1})}\partial_{x} \bigl(\bigl(B_{1}(u_{0})\bigr)^{3} \bigr)\,d \tau. \end{aligned}
(5.3)
We consider the initial data
$$u_{0}(x)=r^{-\frac{1}{2}}N^{-s} \biggl\{ e^{iNx} \int_{0}^{r}e^{ix\xi }\,d\xi+e^{-iNx} \int_{r}^{2r} e^{ix\xi}\,d\xi \biggr\} ,\quad r^{2}N=O(1),N\geq2.$$
(5.4)
By using a direct computation, we have
$$\mathscr{F}_{x}u_{0}(\xi)=Cr^{-\frac{1}{2}}N^{-s} \bigl\{ \chi _{[-N,-N+r]}(\xi)+\chi_{[N+r,N+2r]}(\xi) \bigr\} .$$
Here $$\chi_{I}$$ denotes the characteristic function of a set $$I\subset \mathbf {R}$$. Obviously,
$$\|u_{0}\|_{H^{s}(\mathbf {R})}\sim1.$$
(5.5)
We define $$I_{1}:=[-N,-N+r]$$ and $$I_{2}:=[N+r, N+2r]$$ and $$\Omega _{1}:=I_{1}\cup I_{2}$$. By using a direct computation, we have
$$\mathscr{F}_{x}B_{1}u_{0}( \xi)=Ce^{it\phi(\xi)}\mathscr {F}_{x}u_{0}( \xi).$$
(5.6)
Combining (5.6) with the definition of $$B_{3}(u_{0})$$, we have
$$B_{3}(u_{0}) (x,t)=Cg.$$
(5.7)
Here
$$g=Cr^{-\frac{3}{2}}N^{-3s} \int_{\xi_{1}\in\Omega_{1}} \int_{\xi _{2}\in\Omega_{1}} \int_{\xi_{3}\in\Omega_{1}} \Biggl(\sum _{j=1}^{3} \xi_{j} \Biggr)e^{ix\sum _{j=1}^{3}\xi_{j}}H(\xi _{1},\xi_{2}, \xi_{3})\,d\xi_{1}\,d\xi_{2}\,d\xi_{3},$$
(5.8)
where
$$H(\xi_{1},\xi_{2},\xi_{3})=\frac{e^{it(\phi(\xi_{1})+\phi(\xi _{2})+\phi(\xi_{3}))}-e^{it\phi(\sum _{j=1}^{3}\xi_{j})}}{ \phi(\xi_{1})+\phi(\xi_{2})+\phi(\xi_{3})-\phi(\sum _{j=1}^{3}\xi_{j})}.$$
(5.9)
We define
$$\theta_{1}:=\phi(\xi_{1})+\phi(\xi_{2})+\phi( \xi_{3})-\phi \Biggl(\sum _{j=1}^{3} \xi_{j} \Biggr).$$
(5.10)
From Lemma 2.4, we have
$$\theta_{1}=-3 \bigl[(\xi_{1}+\xi_{2}) ( \xi_{1}+\xi_{3}) (\xi_{2}+\xi_{3}) \bigr] \biggl[1-\frac{1}{3\prod _{j=1}^{3}\xi_{j} (\sum _{j=1}^{3}\xi_{j} )} \biggr].$$
(5.11)
To estimate $$\|g\|_{H^{s}(\mathbf {R})}$$, we need to consider the following three cases:
$$\begin{gathered} \text{Case 1:}\quad \xi_{j}\in I_{1} \quad(j=1,2,3), \\ \text{Case 2:}\quad \xi_{j}\in I_{1} \quad(j=1,2,3), \\ \text{Case 3:}\quad \xi_{j}\in I_{1} \quad(j=1,2),\qquad \xi_{3}\in I_{2}\quad \text{or}\quad \xi_{1}\in I_{1},\qquad\xi_{j}\in I_{2} \quad(j=2,3) \\ \quad \text{or} \quad \xi_{j} \in I_{2} \quad(j=1,2),\qquad \xi_{3}\in I_{1} \quad \text{or} \quad\xi_{1}\in I_{2},\qquad\xi_{j}\in I_{1}\quad (j=2,3).\end{gathered}$$
We assume that $$\|g\|_{H^{s}(\mathbf {R})}$$ corresponding to cases 1, 2, 3 are denoted by $$L_{1}$$, $$L_{2}$$, $$L_{3}$$, respectively.
Case 1. In this case, we have $$|\theta_{1}|\sim N^{3}$$ and $$|\xi_{1}+\xi_{2}+\xi_{3}|\sim N$$. Since $$r^{2}N=O(1)$$, we have
$$L_{1}\leq Cr^{-\frac{3}{2}}N^{-3s}N^{s}r^{\frac{5}{2}}N^{-2} \leq CN^{-2s-\frac{5}{2}}.$$
(5.12)
Case 2. In this case, we have $$|\theta_{1}|\sim N^{3}$$ and $$|\xi_{1}+\xi_{2}+\xi_{3}|\sim N$$. Since $$r^{2}N=O(1)$$, we have
$$L_{2}\leq Cr^{-\frac{3}{2}}N^{-3s}N^{s}r^{\frac{5}{2}}N^{-2} \leq CN^{-2s-\frac{5}{2}}.$$
(5.13)
Case 3. In this case, we have $$|\theta_{1}|\sim r^{2} N$$ and $$|\xi_{1}+\xi_{2}+\xi_{3}|\sim N$$ as well as $$H\leq|t|$$. Since $$r^{2}N=O(1)$$, we have
$$L_{3}\geq C|t|r^{-\frac{3}{2}}N^{-3s}N^{s}r^{\frac{5}{2}}N \geq C|t|N^{-2s+\frac{1}{2}}.$$
(5.14)
Combining (5.1), (5.5) with (5.12)-(5.14), we have
$$|t|N^{-2s+\frac{1}{2}}\leq L_{3}-L_{1}-L_{2}\leq\sup _{t\in[0,T]} \bigl\Vert B_{3}(u_{0}) \bigr\Vert _{H^{s}}\leq C\|u_{0}\|_{H^{s}}^{3} \sim C.$$
(5.15)
For fixed $$t>0$$, when $$s<\frac{1}{4}$$, let $$N\longrightarrow\infty$$, we have $$|t|N^{-2s+\frac{1}{2}}\longrightarrow+\infty$$, and this contradicts (5.15).

This ends the proof of Theorem 1.2.

## Declarations

### Acknowledgements

The second author is supported by the Young core Teachers Program of Henan Normal University and 15A110033.

### Authors’ contributions

All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests. 