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 [1], Leonov [2], 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 [5–11]. Kenig et al. [9] 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. [10] 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. [6] 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 [7] and Kishimoto [11] 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 [12] 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, 13–15]. Some authors have investigated the stability of the solitary waves or soliton solutions of (1.2); for instance, see [16–18].

Many people have studied the Cauchy problem for (1.2), for instance, see [17, 19–30]. 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. [34] 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 [16] 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 [18] 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 [35], firstly, by using the \(X_{s,b}\) spaces introduced by [36–40] 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 [19].

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 [43].

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 [6], 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 [35], 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.