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Boundary Value Problems

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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

©  The Author(s) 2017

Received: 8 June 2017

Accepted: 9 December 2017

Published: 21 December 2017

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 dispersionCauchy problemsharp well-posedness

MSC

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 [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.

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.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Lanzhou University, LanZhou, P.R. China
(2)
College of Mathematics and Information Science, Henan Normal University, Xinxiang, P.R. China

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