# Local well-posedness of Dirac equations with nonlinearity derived from
honeycomb structure in 2 dimensions

Kiyeon Lee  Department of Mathematics, Jeonbuk National University, Jeonju
54896, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to show local well-posedness of 2 dimensional Dirac
equations with power type and Hartree type nonlinearity derived from honeycomb
structure in $H^{s}$ for $s>\frac{7}{8}$ and $s>\frac{3}{8}$, respectively. We
also provide the smoothness failure of flows of Dirac equations.

2010 Mathematics Subject Classification. 35Q55, 35Q40.

Key words and phrases. Dirac equations, Honeycomb lattice, local well-
posedness, non-smoothness, Bourgain’s space.

##  1\. Introduction

In this paper we consider following two Cauchy problems for massless honeycomb
lattice power type Dirac equations($\ell=1$) and Hartree type Dirac
equations$(\ell=2)$:

(1.3) $\displaystyle\left\\{\begin{array}[]{c}(\partial_{t}+{\alpha}\cdot
D)\psi=-i\kappa\mathcal{N}_{\ell}(\psi,\psi)\psi\\\
\psi(0)=\psi_{0}\end{array}\right.$

where $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^{2}$ is the spinor field represented
by a column vector, ${\kappa}$ is constant, $D=-i\nabla$, and
${\alpha}=(P_{\\#}{\alpha}^{1},P_{\\#}{\alpha}^{2})$ are the Dirac matrices
defined by

$\displaystyle\alpha^{1}$ $\displaystyle=\left(\begin{array}[]{ll}0&\;\;1\\\
1&\;\;0\end{array}\right),\qquad\alpha^{2}=\left(\begin{array}[]{ll}0&-i\\\
i&\;\;0\end{array}\right),$

with $P_{\\#}=\left(\begin{array}[]{cc}\overline{\eta_{\\#}}&0\\\
0&\eta_{\\#}\end{array}\right)$ for honeycomb lattice constant $\eta_{\\#}\neq
0$ arising from nonlinear Schrödinger equations(NLS) with honeycomb lattice
potentials(see the Section II in [1]). The nonlinearities $N_{\ell}$ are
defined by

$\mathcal{N}_{1}(\psi_{1},\psi_{2})=\left(\begin{array}[]{cc}b_{1}\psi_{11}\overline{\psi_{21}}+2b_{2}\psi_{12}\overline{\psi_{22}}&0\\\
0&b_{1}\psi_{12}\overline{\psi_{22}}+2b_{2}\psi_{11}\overline{\psi_{21}}\end{array}\right),\quad\mathcal{N}_{2}(\psi_{1},\psi_{2})=\left(|x|^{-1}*(\psi_{1}^{\dagger}\psi_{2})\right)$

where $\psi_{j1},\psi_{j2}$ are components of $\psi_{j}$ and the coefficients
$b_{1},b_{2}>0$ which are an amplitude of Bloch waves. The symbol $*$ denotes
convolution operator in $\mathbb{R}^{2}$ and the $\psi^{\dagger}$ is the
complex conjugate transpose of $\psi$.

Our main equations with the nonlinearity $\mathcal{N}_{\ell}$ are derived from
two dimensional Schrödinger equations with honeycomb lattice potential. Its
rigorous derivation appears in [1]. The honeycomb lattice structure has
appeared in the fabrication of graphene, a mono-crystalline graphitic film in
which electrons behave like massless Dirac fermions(see [6]). Also, the
nonlinear optics which model laser beam propagator in particular types of
photonic crystals, have the honeycomb structure(see [2, 14]).

The equation (1.3) for $\ell=1$ has the scaling invariance structure in
$\dot{H}^{\frac{1}{2}}$. That is, for $\psi_{1}$ the solution to (1.3) with
$\ell=1$, the function $\psi_{1,{\lambda}}$ defined by
$\psi_{1,{\lambda}}(t,x)={\lambda}^{\frac{1}{2}}\psi_{1}({\lambda}t,{\lambda}x)$
is also the solution to the equation (1.3) with $\ell=1$ and satisfies that
$\|\psi_{1,{\lambda}}(0,\cdot)\|_{\dot{H}^{\frac{1}{2}}}=\|\psi_{1}(0,\cdot)\|_{\dot{H}^{\frac{1}{2}}}$.
By this reason, the (1.3) for $\ell=1$ is said to be mass-supercritical case.
Also, since
$\|\psi_{2,{\lambda}}(0,\cdot)\|_{L_{x}^{2}}=\|\psi_{2}(0,\cdot)\|_{L_{x}^{2}}$
for $\psi_{2,{\lambda}}(t,x)={\lambda}\psi_{2}({\lambda}t,{\lambda}x)$ where
$\psi_{2}$ is solution to (1.3) with $\ell=2$, the equation (1.3) with
$\ell=2$ has the scaling invariance structure in $L_{x}^{2}$. The (1.3) for
$\ell=2$ is called to be mass-critical case.

Now we state the main theorem of this paper. For simplicity of representation,
we set an index $s(\ell)$ by

$\displaystyle s(\ell)=\left\\{\begin{array}[]{cc}\frac{1}{2}&\;\mbox{ if
}\ell=1,\\\ 0&\;\,\mbox{ if }\ell=2.\end{array}\right.$

######  Theorem 1.1 (Local well-posedness for $H^{s}$ data).

Let $s>s(\ell)+\frac{3}{8}$ for $\ell=1,2$. Then (1.3) is locally well-posed
for initial data in $H^{s}(\mathbb{R}^{2})$.

Here a definition of the fractional Sobolev space $H^{s}(\mathbb{R}^{2})$ is
placed in Notations below. In particular, LWP result of Dirac equations which
have same nonlinearity $\mathcal{N}_{\ell}$ has been studied in [1] for
$H^{s}(\mathbb{R}^{2})$ with $s>1$.

We can prove Theorem 1 for massive cases ($m>0$) in the same way as proof of
Theorem 1. Since the Physical model comes from massless Dirac Fermions, we
only consider the massless case($m=0$) in this paper.

Lemma 3.2 is deduced from Selberg’s estimates and we get a coefficient
$\mu^{\frac{3}{8}-s}$ in the result (3) of Lemma 3.2. Then the condition
$s>s(\ell)+\frac{3}{8}$ is necessary in process of proof of Theorem 1.1 and
the coefficient $\mu^{\frac{3}{8}-s}$ makes the gap between scaling critical
index $s(\ell)$ and our well-posedness index $s(\ell)+\frac{3}{8}$.

In this paper, we consider Dirac equations with some nonlinearity. Related
equations to (1.3) are well known as semi-relativistic equations as follows:

(1.4) $\displaystyle iu_{t}+\sqrt{m^{2}-\Delta}u$
$\displaystyle={\lambda}|u|^{2}u,$ (1.5) $\displaystyle
iu_{t}+\sqrt{m^{2}-\Delta}u$ $\displaystyle={\lambda}(|x|^{-1}*|u|^{2})u.$

The Cauchy problem for semi-relativistic equations with power type
nonlinearity (1.4) has been investigated in [11, 12]. In [11, 12], Dinh([11])
showed local well-posedness(LWP) of (1.4) with massless case($m=0$) for
$H^{s}(\mathbb{R}^{2})$ with $s>\frac{3}{4}$ and Fujiwara, Georgiev, and Ozawa
extended LWP to global well-posedness(GWP) for $H^{1}(\mathbb{R}^{2})$. The
Cauchy problem for 3 dimensional Hartree type semi-relativistic equations
(1.5) has been investigated in [21, 15]. First the result of well-posedness
was obtained by [21] in $H^{s}(\mathbb{R}^{3})$ for $s\geq\frac{1}{2}$. In
[21], global well-posedness holds in $H^{\frac{1}{2}}(\mathbb{R}^{3})$ for
small data in $L_{x}^{2}$. Later this was improved to $s>\frac{1}{4}$ in [15].
Also they([15]) showed ill-posedness result for $H^{s}(\mathbb{R}^{3})$ with
$s<\frac{1}{4}$. For (1.5) with $d$-dimensions($d\geq 2$), Cho and Ozawa([9])
have revealed the Global well-posedness result for $H^{s}(\mathbb{R}^{d})$
with $s\geq\frac{1}{2}$. Further results for semi-relativistic equations, we
refer to [10].

The difficulty stems from the absence of null-structure of
$\mathcal{N}_{\ell}$. We describe the difference between
$\psi^{\dagger}\beta\psi$ and $|\psi|^{2}$ where
$\beta=\left(\begin{array}[]{ll}1&\;\,\,\,0\\\ 0&-1\end{array}\right)$. The
quadratic term $\psi^{\dagger}\beta\psi$ has a null-structure which represents
like $|\psi_{1}|^{2}-|\psi_{2}|^{2}$. On the other hand, another term
$|\psi|^{2}=|\psi_{1}|^{2}+|\psi_{2}|^{2}$ does not have the null-structure.
Since this structure induces delicate bilinear estimates, Dirac equations with
null-structure lead to better results than the case without null-structure.
However, we do not use this structure, because our nonlinearities
$\mathcal{N}_{1}$ are essentially the same as $|\psi|^{2}$. For this reason,
it is picky to control the nonlinear term $\mathcal{N}_{\ell}$. Hence we
describe the Lemma 3.2 which used crucially in the proof of Theorem 1.1.

Also we consider the Dirac equation with Coulomb type nonlinearity which has
null-structure:

(1.6) $\displaystyle(i\partial_{t}+{\alpha}\cdot D)\psi$
$\displaystyle={\lambda}\left(\psi^{\dagger}\beta\psi\right)\psi,$ (1.7)
$\displaystyle(i\partial_{t}+{\alpha}\cdot D)\psi$
$\displaystyle={\lambda}\Big{(}|x|^{-{\gamma}}*\left(\psi^{\dagger}\beta\psi\right)\Big{)}\psi.$

As known result for the equation (1.6), Bejenaru and Herr([3]) showed the GWP
in $H^{\frac{1}{2}}(\mathbb{R}^{2})$. And the known results for the equation
(1.7) are in [8, 20]. In [20], An author of this paper revealed the LWP in
$H^{s}(\mathbb{R}^{2})$ with $s>{\frac{{\gamma}-1}{2}}$ and $1\leq{\gamma}<2$.
It was studied in [8] that global well-posedness and small data scattering
holds in $H^{s}(\mathbb{R}^{2})$ for $s>{{{\gamma}-1}}$ and $1<{\gamma}<2$. As
related to (1.7), there is a Dirac equation with Yukawa potential. One may
find many results of the Dirac equation which has Yukawa potential
nonlinearity in [7, 25, 26, 27].

In view of scaling we expect that LWP results for (1.3) is optimal in
$H^{s(\ell)}$. For this expectation we introduce the following theorem which
denotes the smooth failure of our main equation (1.3) for $s<s(\ell)$.

######  Theorem 1.2.

Let $s<s(\ell)$ and $T>0$. If the flow map $\phi\mapsto u$ in (1.3) exists as
a map from $H^{s}(\mathbb{R}^{2})$ to $C([-T,T];H^{s}(\mathbb{R}^{2})),$ it
fails to be $C^{3}$ at the origin.

If the equation (1.3) has well-posedness in $[-T,T]$ for some $T>0$, the flows
of (1.3) have the smoothness in $[-T,T]$. Since Theorem 1.2 implies that the
smoothness of flows of (1.3) fails, This yields the ill-posedness of (1.3) for
$H^{s}$ with $s<s(\ell)$.

The smoothness failure of some equations was studied for many authors in [22,
4, 15, 20]. Molinet, Saut, and Tzvetkov([22]), Bejenaru and Tao([4]), and Herr
and Lenzmann([15]) have proved the ill-posedness results similar to Theorem
(1.2) for Benjamin-Ono equations, 1-d Schrödinger equations and semi-
relativistic equations, respectively. For Dirac equation, ill-posedness
results have been shown in [20].

It is still opened the well-posedness of (1.3) in $H^{s}(\mathbb{R}^{2})$ for
$s(\ell)\leq s\leq s(\ell)+\frac{3}{8}$. For filling up this gap, we have to
obtain better bilinear estimates than Lemma 3.2. For this purpose we should
find some structure of nonlinearity of (1.3) like null-structure. Then we may
improve LWP in $H^{s}$ with sobolev index $s$ below $s(\ell)+\frac{3}{8}$.

The paper is organized as follows: In Section 2, we discuss projection
operators. In Section 3, we introduce the function spaces and the bilinear
estimates the most useful on proof of main theorem. Section 4, we prove
Theorem 1.1 via the standard contraction method. In section 5, we establish
the proof of Proposition 4.1 arising in Section 4. In the last section, we
discuss Theorem 1.2 by contradiction argument.

Notations.
$\bullet$ Space and space-time Fourier transform:
$\widehat{f}=\mathcal{F}_{x}(f)$ denotes the space variable Fourier transform
of $f$ and $\mathcal{F}_{\xi}^{-1}(g)$ the inverse Fourier transform of $g$
such that

$\mathcal{F}_{x}(f)(\xi)=\int_{\mathbb{R}^{2}}e^{-ix\cdot\xi}f(x)\,dx,\quad\mathcal{F}_{\xi}^{-1}(g)(x)=(2\pi)^{-2}\int_{\mathbb{R}^{2}}e^{ix\cdot\xi}g(\xi)\,d\xi.$

$\widetilde{f}=\mathcal{F}_{t,x}(f)$ denotes the space-time variables Fourier
transform of $f$ such that

$\mathcal{F}_{t,x}(f)(\tau,\xi)=\int_{\mathbb{R}^{1+2}}e^{-it\cdot\tau-
ix\cdot\xi}f(t,x)\,dtdx.$

$\bullet$ Fractional derivatives and Sobolev spaces:
$D^{s}=(-\Delta)^{\frac{s}{2}}=\mathcal{F}_{x}^{-1}|\xi|^{s}\mathcal{F}_{x}$,
$\Lambda^{s}=(1-\Delta)^{\frac{s}{2}}=\mathcal{F}_{x}^{-1}(1+|\xi|^{2})^{\frac{s}{2}}\mathcal{F}_{x}$
for $s>0$. Let us denote $\dot{H}^{s}=D^{s}L^{2}$ and $H^{s}=\Lambda^{s}L^{2}$
for $s\in\mathbb{R}$.

$\bullet$ Mixed-normed spaces: For a Banach space $X$ and an interval $I$,
$u\in L_{I}^{q}X\cap\mathbb{C}$ iff $u(t)\in X$ for a.e.$t\in I$ and
$\|u\|_{L_{I}^{q}X}:=\|\|u(t)\|_{X}\|_{L_{I}^{q}}<\infty$. Especially, we
denote $L_{I}^{q}L_{x}^{r}=L_{t}^{q}(I;L_{x}^{r}(\mathbb{R}^{2}))$,
$L_{I,x}^{q}=L_{I}^{q}L_{x}^{q}$,
$L_{t}^{q}L_{x}^{r}=L_{\mathbb{R}}^{q}L_{x}^{r}$. For vector-valued function
$\psi\in L_{I}^{q}X\cap\mathbb{C}^{2}$, we also denote that
$\|\psi\|_{L_{I}^{q}X}:=\||\psi|\|_{L_{I}^{q}X}$.

$\bullet$ Littlewood-Paley operators: Let us define $\beta_{1}\in
C^{\infty}_{0}(-2,2)$ such that $\beta_{1}(s)=1$ if $|s|\leq 1$ and
$\beta_{{\lambda}}(s):=\beta(\frac{s}{{\lambda}})-\beta(\frac{2s}{{\lambda}})$
for ${\lambda}>1$. Then we define the frequency projection
$\mathcal{F}(P_{{\lambda}}f)(\xi)=\beta_{{\lambda}}(\xi)\widehat{f}(\xi)$,
$P_{\leq{\lambda}}=\sum_{\mu=1}^{{\lambda}}P_{\mu}$ and
$P_{\geq{\lambda}}=I-P_{\leq\frac{{\lambda}}{2}}$. Also, for measurable set
$S\subset\mathbb{R}^{2},R\subset\mathbb{R}^{1+2}$, we denote that
$\mathcal{F}_{x}(P_{S}f)(\xi)=\chi_{S}(\xi)\widehat{f}(\xi)$ and
$\mathcal{F}_{t,x}(P_{R}f)(\tau,\xi)=\chi_{R}(\tau,\xi)\widetilde{f}(\tau,\xi)$.

$\bullet$ As usual different positive constants depending only on
${\alpha},{\kappa}$ are denoted by the same letter $C$, if not specified.
$A\lesssim B$ and $A\gtrsim B$ means that $A\leq CB$ and $A\geq C^{-1}B$,
respectively for some $C>0$. $A\sim B$ means that $A\lesssim B$ and $A\gtrsim
B$.

##  2\. Preliminaries

In this section, for simplicity of the Cauchy problem, we define the
projection operators and rewrite the equations (1.3) to integral equations.

###  2.1. Projection operator

We first define the projections about (1.3) as following:

$\displaystyle\Pi^{\pm}(D):=\frac{1}{2}\left(I\pm\frac{{\alpha}\cdot
D}{|\eta_{\\#}||\nabla|}\right).$

Then we get

${\alpha}\cdot D=|\eta_{\\#}||\nabla|\left(\Pi^{+}(D)-\Pi^{-}(D)\right).$

Using these projection operators, we decompose

$\psi=\psi_{+}+\psi_{-}$

where $\psi_{\pm}:=\Pi^{\pm}(D)\psi$. Also, these projection operators satisfy
that

$\Pi^{\pm}(D)\Pi^{\pm}(D)=\Pi^{\pm}(D),\qquad\Pi^{\pm}(D)\Pi^{\mp}(D)=0.$

Applying these operator to (1.3) we see that

(2.1)
$\displaystyle(\partial_{t}\pm|\eta_{\\#}||\nabla|)\psi_{\pm}=-i\kappa\Pi^{\pm}(D)\mathcal{N}_{\ell}(\psi,\psi)\psi$

for $\ell=1,2$ with initial data

$\psi_{\pm}(0)=:\psi_{0,\pm}\in H^{s}.$

To simplify the representation of (2.1), we set the spinner
$\phi_{\pm}(t,x)=\psi_{\pm}\left(\frac{t}{|\eta_{\\#}|},x\right)$. Hence
$\phi$ satisfies that

$(i\partial_{t}\pm|\nabla|)\phi_{\pm}=\frac{1}{|\eta_{\\#}|}(i\partial_{t}\pm|\eta_{\\#}||\nabla|)\psi_{\pm}=-\frac{i\kappa}{|\eta_{\\#}|}\Pi^{\pm}(D)\mathcal{N}_{\ell}(\phi,\phi)\phi$

for $\ell=1,2$. We still call the spinner to $\psi$. Then we finally get the
second main equation

(2.2)
$\displaystyle(i\partial_{t}\pm|\nabla|)\psi_{\pm}=-i\kappa_{\\#}\Pi^{\pm}(D)\mathcal{N}_{\ell}(\psi,\psi)\psi.$

where $\kappa_{\\#}=\frac{\kappa}{|\eta_{\\#}|}$.

By Duhamel’s formula, we can represent the equations (2.2) written as an
integral equation

(2.3) $\displaystyle\psi_{\pm}(t)$ $\displaystyle=$ $\displaystyle
S_{\pm}(t)\psi_{0,\pm}+\kappa_{\\#}\int_{0}^{t}S_{\pm}(t-t^{\prime})\Pi^{\pm}(D)\left[\mathcal{N}_{\ell}\Big{(}\psi(t^{\prime}),\psi(t^{\prime})\Big{)}\psi(t^{\prime})\right]dt^{\prime}$

for $\ell=1,2$. Here we define the linear propagator $S_{\pm}(t)$ as
following:

(2.4) $\displaystyle S_{\pm}(t)f=e^{\mp it|\nabla|}f.$

###  2.2. Fractional Leibniz rule

The following lemma which is called fractional Leibniz rule is useful in the
proof of LWP.

######  Lemma 2.1 ([16, 18, 17]).

Let $0<s<1,\;1<p<\infty$. Then

$\|D^{s}(fg)-fD^{s}g-gD^{s}f\|_{L^{p}}\lesssim\|D^{s_{1}}f\|_{L^{p_{1}}}\|D^{s_{2}}g\|_{L^{p_{2}}}$

provided $s=s_{1}+s_{2}$ with $0\leq s_{1},s_{2}\leq 1$ and
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$.

The proof of Lemma 2.1 is in [16, 18, 17].

##  3\. Function spaces and Bilinear estimates

###  3.1. Functions spaces: $X^{s,b}$-space

We first introduce $X_{\pm}^{s,b}$ space which will be useful in local
theories. (See e.g. [5, 19, 24].) Let us define the norm for
$s,b\in\mathbb{R}$ as follows:

$X_{\pm}^{s,b}(T):=\Big{\\{}\psi:\Big{\|}\chi_{[-T,T]}\psi\Big{\|}_{X_{\pm}^{s,b}}<\infty\Big{\\}}$

with a norm

$\|\psi\|_{X_{\pm}^{s,b}}:=\left(\int_{\mathbb{R}^{1+2}}\left|\left<\xi\right>^{s}\left<\tau\mp|\xi|\right>^{b}\widetilde{\psi}(\tau,\xi)\right|^{2}d\tau
d\xi\right)^{\frac{1}{2}}.$

In particular, we denote that $X_{\pm_{j}}^{s,b}$ is $X_{+}^{s,b}$ for
$\pm_{j}=+$ and $X_{\pm_{j}}^{s,b}$ is $X_{-}^{s,b}$ for $\pm_{j}=-$. These
function spaces satisfy the embedding for $b>\frac{1}{2}$

$X_{\pm}^{s,b}(T)\hookrightarrow C([-T,T];H^{s}).$

###  3.2. Bilinear estimates

######  Lemma 3.1 (Theorem 2.1 of [23]).

Let ${\lambda}>0$ and $L\geq 1$. Let us define the thickened cones

$\displaystyle
K_{{\lambda},L}^{\pm}=\Big{\\{}(\tau,\xi):|\xi|\lesssim{\lambda},\;\tau\mp|\xi|=O(L)\Big{\\}}$

Then

$\|P_{K_{{\lambda},L}^{\pm}\cap(\mathbb{R}\times
B_{\mu})}u\|_{L_{t,x}^{4}}\lesssim\mu^{\frac{1}{4}}{\lambda}^{\frac{1}{8}}L^{\frac{3}{8}}\|P_{K_{{\lambda},L}^{\pm}\cap(\mathbb{R}\times
B_{\mu})}u\|_{L_{t,x}^{2}}$

for $u:\mathbb{R}^{1+2}\to\mathbb{C}$ and any ball
$B_{\mu}\subset\mathbb{R}^{2}$ with radius $\mu>0$.

The following lemma is readily obtained by Lemma 3.1.

######  Lemma 3.2.

Let $s>\frac{3}{8}$, $b>\frac{1}{2}$, and $u:\mathbb{R}^{1+2}\to\mathbb{C}$.
Then the following holds:

  1. $(1)$

$\|P_{B_{\mu}}u\|_{L_{t,x}^{4}}\lesssim\mu^{\frac{1}{4}}\|P_{B_{\mu}}u\|_{X_{\pm}^{\frac{1}{8},b}}$
for $u\in X_{\pm}^{\frac{1}{8},b}$ and any ball $B_{\mu}$ with radius $\mu>0$,

  2. $(2)$

$\|u\|_{L_{t,x}^{4}}\lesssim\|u\|_{X_{\pm}^{\frac{3}{8},b}}$ for $u\in
X_{\pm}^{\frac{3}{8},b}$,

  3. $(3)$

$\|P_{\mu}(u_{1}\overline{u_{2}})\|_{L_{t,x}^{2}}\lesssim\mu^{\frac{3}{8}-s}\|u_{1}\|_{X_{\pm_{1}}^{s,b}}\|u_{2}\|_{X_{\pm_{2}}^{s,b}}$
for $\mu>0$, $u_{j}\in X_{\pm_{j}}^{s,b}$.

###### Proof.

We first prove (1). Lemma 3.1 yields that, for ${\lambda}\geq 1$,

$\displaystyle\|P_{B_{\mu}}P_{{\lambda}}u\|_{L_{t,x}^{4}}$
$\displaystyle\lesssim\sum_{L\geq
1}\mu^{\frac{1}{4}}{\lambda}^{\frac{1}{8}}L^{\frac{3}{8}}\|P_{K_{{\lambda},L}^{\pm}\cap(\mathbb{R}\times
B_{\mu})}u\|_{L_{t,x}^{2}}$ $\displaystyle\lesssim\sum_{L\geq
1}\mu^{\frac{1}{4}}{\lambda}^{\frac{1}{8}}L^{\frac{3}{8}}\|P_{K_{{\lambda},L}^{\pm}\cap(\mathbb{R}\times
B_{\mu})}u\|_{L_{t,x}^{2}}\lesssim\sum_{L\geq
1}\mu^{\frac{1}{4}}{\lambda}^{\frac{1}{8}}L^{\frac{3}{8}-b}\|L^{b}\chi_{K_{{\lambda},L}^{\pm}\cap(\mathbb{R}\times
B_{\mu})}\widetilde{u}\|_{L_{\tau,\xi}^{2}}$ $\displaystyle\lesssim\sum_{L\geq
1}\mu^{\frac{1}{4}}{\lambda}^{\frac{1}{8}}L^{\frac{3}{8}-b}\|\left<\tau\mp|\xi|\right>^{b}P_{B_{\mu}}\widetilde{u}\|_{L_{\tau,\xi}^{2}}\lesssim\mu^{\frac{1}{4}}{\lambda}^{\frac{1}{8}}\|P_{B_{\mu}}u\|_{X_{\pm}^{0,b}}.$

Then we have

$\|P_{B_{\mu}}u\|_{L_{t,x}^{4}}\lesssim\sum_{{\lambda}\geq
1}\|P_{B_{\mu}}P_{{\lambda}}u\|_{L_{t,x}^{4}}\lesssim\sum_{{\lambda}\geq
1}\mu^{\frac{1}{4}}{\lambda}^{\frac{1}{8}}\|P_{B_{\mu}}P_{{\lambda}}u\|_{X_{\pm}^{0,b}}\lesssim\mu^{\frac{1}{4}}\|P_{B_{\mu}}u\|_{X^{\frac{1}{8},b}_{\pm}}.$

For (2), by (1) we obtain

$\displaystyle\|u\|_{L_{t,x}^{4}}\lesssim\left(\sum_{\mu\geq
1}\|P_{\mu}u\|_{L_{t,x}^{4}}^{2}\right)^{\frac{1}{2}}\lesssim\left(\sum_{\mu\geq
1}\mu^{\frac{1}{4}}\|P_{\mu}u\|_{X_{\pm}^{\frac{1}{8},b}}^{2}\right)^{\frac{1}{2}}\lesssim\|u\|_{X_{\pm}^{\frac{3}{8},b}}.$

Let us now prove (3). Using frequency localization and (2), we see that

$\displaystyle\left\|P_{\mu}(u_{1}\overline{u_{2}})\right\|_{L_{t,x}^{2}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{l}{\lambda}_{1},{\lambda}_{2}\geq
1\\\
\mu\lesssim{\lambda}_{1}\sim{\lambda}_{2}\end{subarray}}\left\|P_{\mu}(u_{1,{\lambda}_{1}}\overline{u_{2,{\lambda}_{2}}})\right\|_{L_{t,x}^{2}}+\sum_{\begin{subarray}{l}\;\;\;{\lambda}_{1},{\lambda}_{2}\geq
1\\\
{\lambda}_{\min}\lesssim{\lambda}_{\max}\sim\mu\end{subarray}}\left\|P_{\mu}(u_{1,{\lambda}_{1}}\overline{u_{2,{\lambda}_{2}}})\right\|_{L_{t,x}^{2}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{l}{\lambda}_{1},{\lambda}_{2}\geq
1\\\
\mu\lesssim{\lambda}_{1}\sim{\lambda}_{2}\end{subarray}}\|u_{1,{\lambda}_{1}}\|_{L_{t,x}^{4}}\|u_{2,{\lambda}_{2}}\|_{L_{t,x}^{4}}+\sum_{\begin{subarray}{l}\;\;\;{\lambda}_{1},{\lambda}_{2}\geq
1\\\
{\lambda}_{\min}\lesssim{\lambda}_{\max}\sim\mu\end{subarray}}\|u_{1,{\lambda}_{1}}\|_{L_{t,x}^{4}}\|u_{2,{\lambda}_{2}}\|_{L_{t,x}^{4}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{l}{\lambda}_{1},{\lambda}_{2}\geq
1\\\
\mu\lesssim{\lambda}_{1}\sim{\lambda}_{2}\end{subarray}}{\lambda}_{1}^{\frac{3}{8}-s}{\lambda}_{2}^{\frac{3}{8}-s}\|u_{1,{\lambda}_{1}}\|_{X_{\pm_{1}}^{s,b}}\|u_{2,{\lambda}_{2}}\|_{X_{\pm_{2}}^{s,b}}+\sum_{\begin{subarray}{l}\;\;\;{\lambda}_{1},{\lambda}_{2}\geq
1\\\
{\lambda}_{\min}\lesssim{\lambda}_{\max}\sim\mu\end{subarray}}{\lambda}_{\max}^{\frac{3}{8}-s}\|u_{1,{\lambda}_{1}}\|_{X_{\pm_{1}}^{s,b}}\|u_{2,{\lambda}_{2}}\|_{X_{\pm_{2}}^{s,b}}$
$\displaystyle\lesssim\mu^{\frac{3}{8}-s}\|u_{1}\|_{X_{\pm_{1}}^{s,b}}\|u_{2}\|_{X_{\pm_{2}}^{s,b}}.$

Here we used ${\lambda}_{\max}=\max({\lambda}_{1},{\lambda}_{2})$,
${\lambda}_{\min}=\min({\lambda}_{1},{\lambda}_{2})$. ∎

##  4\. Local Well-posedness: Proof of Theorem 1.1

Let us define a complete Banach metric space
$(\mathcal{M}^{s,b}(T,{\delta}),d)$ as follows:

$\displaystyle\mathcal{M}^{s,b}(T,{\delta})$ $\displaystyle=\left\\{\psi\in
C\left([-T,T]:L_{x}^{2}\right)\cap
X_{\pm}^{s,b}(T):\|\psi\|_{\mathcal{M}^{s,b}}:=\|\psi_{+}\|_{X_{+}^{s,b}}+\|\psi_{-}\|_{X_{+}^{s,b}}<{\delta}\right\\},$
$\displaystyle d(\psi,\phi)$
$\displaystyle=\|\psi-\phi\|_{\mathcal{M}^{s,b}}.$

We now consider a map $\mathcal{D}$ on $\mathcal{M}^{s,b}(T,{\delta})$ by

$\displaystyle\mathcal{D}(\psi)=\sum_{\pm_{0}\in\\{\pm\\}}S_{\pm_{0}}(t)\psi_{0,\pm_{0}}+\sum_{\begin{subarray}{l}\pm_{j}\in\\{\pm\\}\\\
j=0,1,2,3\end{subarray}}\kappa_{\\#}\int_{0}^{t}S_{\pm_{0}}(t-t^{\prime})\Pi^{\pm_{0}}(D)\left[\mathcal{N}_{\ell}(\psi_{\pm_{1}},\psi_{\pm_{2}})\psi_{\pm_{3}}\right]dt^{\prime}.$

where $\sum_{\pm_{0}\in\\{\pm\\}}F_{\pm_{0}}=F_{+}+F_{-}$. Then we first show
the map $\mathcal{D}$ is self-mapping on $\mathcal{M}^{s,b}(T,{\delta})$. By
Lemma 2.1 of [13] we see that

$\Big{\|}\chi_{[-T,T]}S_{\pm}(t)\psi_{0,\pm}\Big{\|}_{X_{\pm}^{s,b}}\lesssim
T^{\frac{1}{2}-b}\|\psi_{0}\|_{H^{s}}$

and

$\left\|\chi_{[-T,T]}\int_{0}^{t}S_{\pm}(t-t^{\prime})f(t^{\prime})dt^{\prime}\right\|_{X_{\pm}^{s,b}}\lesssim
T^{1-b+b^{\prime}}\|f\|_{X^{s,b^{\prime}}}$

for $-\frac{1}{2}<b^{\prime}<0<\frac{1}{2}<b\leq b^{\prime}+1$.

######  Proposition 4.1.

Let $s>s(\ell)+\frac{3}{8}$ for $\ell=1,2$. Then there exists
$-\frac{1}{2}<b^{\prime}<-\frac{1}{4}<\frac{1}{2}<b\leq b^{\prime}+1$ and
${\varepsilon}>0$, such that

$\Big{\|}\mathcal{N}_{\ell}(\psi_{1},\psi_{2})\psi_{3}\Big{\|}_{X_{\pm}^{s,b^{\prime}}}\leq
T^{{\varepsilon}}\prod_{j=1}^{3}\|\psi_{j}\|_{X_{\pm_{j}}^{s,b}}$

for all $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^{2}$ and $\psi_{j}\in
X_{\pm_{j}}^{s,b}$ with ${\it supp}(\psi_{j})\subset\\{(t,x):|t|\leq T\\}.$

Proposition 4.1 will be proved in the next section. We now assume the validity
of Proposition 4.1. Then we estimate

$\displaystyle\|\mathcal{D}(\psi)\|_{\mathcal{M}^{s,b}}\leq
C_{1}\|\psi_{0}\|_{H^{s}}+C_{2}T^{\varepsilon}\sum_{\pm}\|\psi_{\pm}\|_{X_{\pm}^{s,b}}^{3}\leq
C_{1}\|\psi_{0}\|_{H^{s}}+C_{2}T^{\varepsilon}{\delta}^{3}.$

Set $C_{1}\|\psi_{0}\|_{H^{s}}<\frac{{\delta}}{2}$ and choose the time $T$
that satisfies $C_{2}T^{{\varepsilon}}{\delta}^{3}<\frac{{\delta}}{2}$. Hence
we see that $\|\mathcal{D}(\psi)\|_{\mathcal{M}^{s,b}}<{\delta}$. Therefore
$\mathcal{D}$ is self-mapping on $\mathcal{M}^{s,b}(T,{\delta})$. We now
describe the fact that $\mathcal{D}$ is contraction mapping on
$\mathcal{M}^{s,b}(T,{\delta})$:

$\displaystyle d(\mathcal{D}(\psi),\mathcal{D}(\phi))$
$\displaystyle=\|\mathcal{D}(\psi)-\mathcal{D}(\phi)\|_{\mathcal{M}^{s,b}}\leq
C\left(\|\psi\|_{\mathcal{M}^{s,b}}^{2}+\|\phi\|_{\mathcal{M}^{s,b}}^{2}\right)\|\psi-\phi\|_{\mathcal{M}^{s,b}}$
$\displaystyle\leq
2C{\delta}^{2}\|\psi-\phi\|_{\mathcal{M}^{s,b}}<\frac{1}{2}d(\mathcal{D}(\psi),\mathcal{D}(\phi))$

for ${\delta}$ satisfying that $4C{\delta}^{2}<\frac{1}{2}$.

Therefore this completes the proof of the local existence and uniqueness of a
solution to (1.3).

##  5\. Proof of Proposition 4.1

###  5.1. Proof of Proposition 4.1

By duality, it suffices to prove that

$I_{\ell}:=\left|\small\iint\mathcal{N}_{\ell}(\psi_{1},\psi_{2})\psi_{3}\Lambda^{s}\psi_{4}^{\dagger}dtdx\right|\lesssim
T^{{\varepsilon}}\prod_{j=1}^{3}\|\psi_{j}\|_{X_{\pm_{j}}^{s,b}}\|\psi_{4}\|_{X_{\pm_{4}}^{0,-b^{\prime}}}$

for $\psi_{4}\in X_{\pm_{4}}^{0,-b^{\prime}}$ and $\ell=1,2$. We set
$\psi_{j}=\left(\begin{array}[]{c}\psi_{j1}\\\ \psi_{j2}\end{array}\right)$
for $j=1,2,3,4$. Then we have

$\displaystyle I_{1}$
$\displaystyle=\left|\small\iint\left(\begin{array}[]{c}\left(b_{1}\psi_{11}\overline{\psi_{21}}+2b_{2}\psi_{12}\overline{\psi_{22}}\right)\psi_{31}\\\
-\left(b_{1}\psi_{12}\overline{\psi_{22}}+2b_{2}\psi_{11}\overline{\psi_{21}}\right)\psi_{32}\end{array}\right)\Big{(}\begin{array}[]{cc}\Lambda^{s}\overline{\psi_{41}}&\;\;\Lambda^{s}\overline{\psi_{42}}\end{array}\Big{)}dtdx\right|$
$\displaystyle=C\sum_{j,k,l\in\\{1,2\\}}\left|\small\iint\psi_{1j}\overline{\psi_{2j}}\psi_{3k}\Lambda^{s}\overline{\psi_{4l}}dtdx\right|$

and

$\displaystyle
I_{2}=\left|\small\iint|\nabla|^{-1}(\psi_{1}^{\dagger}\psi_{2})\psi_{3}\Lambda^{s}\psi_{4}^{\dagger}dtdx\right|=C\sum_{j,k\in\\{1,2\\}}\left|\small\iint|\nabla|^{-1}\left(\overline{\psi_{1j}}\psi_{2j}\right)\psi_{3k}\Lambda^{s}\overline{\psi_{4k}}dtdx\right|.$

To compute the terms above, we introduce $\mathbb{C}$-valued version estimates
below which will be proved Section 5.2.

######  Lemma 5.1.

The following two estimates hold:

  1. $(i)$

Let $s>s(1)+\frac{3}{8}$. Then there exists
$-\frac{1}{2}<b^{\prime}<-\frac{1}{4}<\frac{1}{2}<b\leq b^{\prime}+1$ and
${\varepsilon}>0$, such that

(5.1)
$\displaystyle\left|\small\iint(u_{1}\overline{u_{2}})u_{3}\Lambda^{s}\overline{u_{4}}dtdx\right|\lesssim
T^{{\varepsilon}}\prod_{j=1}^{3}\|u_{j}\|_{X_{\pm_{j}}^{s,b}}\|u_{4}\|_{X_{\pm_{4}}^{0,-b^{\prime}}}$

for all $u_{j}:\mathbb{R}^{1+2}\to\mathbb{C}$ and $u_{j}\in X_{\pm_{j}}^{s,b}$
with ${\it supp}(u_{j})\subset\\{(t,x):|t|\leq T\\}.$

  2. $(ii)$

Let $s>s(2)+\frac{3}{8}$. Then there exists
$-\frac{1}{2}<b^{\prime}<-\frac{1}{4}<\frac{1}{2}<b\leq b^{\prime}+1$ and
${\varepsilon}>0$, such that

(5.2)
$\displaystyle\left|\small\iint\left[|x|^{-1}*(\overline{u_{1}}u_{2})\right]u_{3}\Lambda^{s}\overline{u_{4}}dtdx\right|\lesssim
T^{{\varepsilon}}\prod_{j=1}^{3}\|u_{j}\|_{X_{\pm_{j}}^{s,b}}\|u_{4}\|_{X_{\pm_{4}}^{0,-b^{\prime}}}$

for all $u_{j}:\mathbb{R}^{1+2}\to\mathbb{C}$ and $u_{j}\in X_{\pm_{j}}^{s,b}$
with ${\it supp}(u_{\ell})\subset\\{(t,x):|t|\leq T\\}.$

By Lemma 5.1, we get

$I_{\ell}\lesssim
T^{{\varepsilon}}\sum_{j,k=1,2}\|\psi_{1j}\|_{X_{\pm_{1}}^{s,b}}\|\psi_{2j}\|_{X_{\pm_{2}}^{s,b}}\|\psi_{3k}\|_{X_{\pm_{3}}^{s,b}}\|\psi_{4k}\|_{X_{\pm_{4}}^{0,-b^{\prime}}}\lesssim
T^{{\varepsilon}}\prod_{j=1}^{3}\|\psi_{j}\|_{X_{\pm_{j}}^{s,b}}\|\psi_{4}\|_{X_{\pm_{4}}^{0,-b^{\prime}}}.$

for $\ell=1,2$. It completes the proof of Proposition 4.1.

###  5.2. Proof of Lemma 5.1

###### Proof of $(i)$ of Lemma 5.1.

We first set $\frac{7}{8}<s\leq 1$. By Hölder inequality and Lemma 2.1, we can
split the left-hand side of (5.1) as follows:

$\displaystyle\left|\small\iint(u_{1}\overline{u_{2}})u_{3}\Lambda^{s}\overline{u_{4}}dtdx\right|$
$\displaystyle\leq\left|\iint\Lambda^{s}(\overline{u_{1}}u_{2}u_{3})\overline{u_{4}}dtdx-\iint\Lambda^{s}(\overline{u_{1}}u_{2})u_{3}\overline{u_{4}}dtdx-\iint\overline{u_{1}}u_{2}\left(\Lambda^{s}u_{3}\right)\overline{u_{4}}dtdx\right|$
$\displaystyle\qquad\qquad+\left|\iint\Lambda^{s}(\overline{u_{1}}u_{2})u_{3}\overline{u_{4}}dtdx\right|+\left|\iint\overline{u_{1}}u_{2}\left(\Lambda^{s}u_{3}\right)\overline{u_{4}}dtdx\right|$
$\displaystyle=:J_{1}^{1}+J_{1}^{2}+J_{1}^{3}.$

We first treat the $J_{1}^{1}$. By Lemma 2.1, we estimate

$\displaystyle J_{1}^{1}$
$\displaystyle\lesssim\left\|\Lambda^{s}(\overline{u_{1}}u_{2}u_{3})-\Lambda^{s}(\overline{u_{1}}u_{2})u_{3}-\overline{u_{1}}u_{2}\left(\Lambda^{s}u_{3}\right)\right\|_{L_{t}^{\frac{4}{3}}L_{x}^{2}}\|u_{4}\|_{L_{t}^{4}L_{x}^{2}}$
$\displaystyle\lesssim\|\Lambda^{s}(u_{1}\overline{u_{2}})\|_{L_{t,x}^{2}}\|u_{3}\|_{L_{t}^{4}L_{x}^{\infty}}\|u_{4}\|_{L_{t}^{4}L_{x}^{2}}$

Let us consider $\|\Lambda^{s}(u_{1}\overline{u_{2}})\|_{L_{t,x}^{2}}$. Like
above estimates, Lemma 2.1 yields that

$\displaystyle\|\Lambda^{s}(u_{1}\overline{u_{2}})\|_{L_{t,x}^{2}}$
$\displaystyle\lesssim\Big{\|}\Lambda^{s}(u_{1}\overline{u_{2}})-(\Lambda^{s}u_{1})\overline{u_{2}}-u_{1}(\Lambda^{s}\overline{u_{2}})\Big{\|}_{L_{t,x}^{2}}+\Big{\|}(\Lambda^{s}u_{1})\overline{u_{2}}\Big{\|}_{L_{t,x}^{2}}+\Big{\|}u_{1}(\Lambda^{s}\overline{u_{2}})\Big{\|}_{L_{t,x}^{2}}$
$\displaystyle\lesssim\Big{\|}\Lambda^{s}u_{1}\Big{\|}_{L_{t}^{4}L_{x}^{2}}\|u_{2}\|_{L_{t}^{4}L_{x}^{\infty}}+\|u_{1}\|_{L_{t}^{4}L_{x}^{\infty}}\Big{\|}\Lambda^{s}u_{2}\Big{\|}_{L_{t}^{4}L_{x}^{2}}.$

By Sobolev embedding and Lemma 3.2 we get

(5.3)
$\displaystyle\Big{\|}u_{j}\Big{\|}_{L_{t}^{4}L_{x}^{\infty}}\lesssim\Big{\|}\Lambda^{s-\frac{3}{8}}u_{j}\Big{\|}_{L_{t,x}^{4}}\lesssim\|u_{j}\|_{X_{\pm_{j}}^{s,b}}$

for $s>\frac{7}{8},\;b>\frac{1}{2}$, and $j=1,2$. By embedding
$X^{0,\frac{1}{4}}\hookrightarrow L_{t}^{4}L_{x}^{2}$, the estimate (5.3)
leads us that

(5.4)
$\displaystyle\|\Lambda^{s}(u_{1}\overline{u_{2}})\|_{L_{t,x}^{2}}\lesssim\|u_{1}\|_{X_{\pm_{1}}^{s,b}}\|u_{2}\|_{X_{\pm_{2}}^{s,b}}.$

In particular, by (5.3), we have

(5.5)
$\displaystyle\|u_{3}\|_{L_{t}^{4}L_{x}^{\infty}}\lesssim\|u_{3}\|_{X_{\pm_{3}}^{s,b}}.$

Using the estimates (5.4), (5.5), and embedding
$X^{0,\frac{1}{4}}\hookrightarrow L_{t}^{4}L_{x}^{2}$ , we see taht

$\displaystyle J_{1}^{1}$
$\displaystyle\lesssim\|u_{1}\|_{X_{\pm_{1}}^{s,b}}\|u_{2}\|_{X_{\pm_{2}}^{s,b}}\|u_{3}\|_{X_{\pm_{3}}^{s,\frac{1}{4}}}\|u_{4}\|_{X_{\pm_{4}}^{0,\frac{1}{4}}}$
$\displaystyle\lesssim
T^{{\delta}}\prod_{j=1}^{3}{\|u_{j}\|_{X_{\pm_{j}}^{s,b}}}\|u_{4}\|_{X_{\pm_{4}}^{0,-b^{\prime}}}.$

On the other hand, for $J_{1}^{2},J_{1}^{3}$, we obtain

(5.6) $\displaystyle J_{1}^{2}$
$\displaystyle\lesssim\|\Lambda^{s}(u_{1}\overline{u_{2}})\|_{L_{t,x}^{2}}\|u_{3}\|_{L_{t}^{4}L_{x}^{\infty}}\|u_{4}\|_{L_{t}^{4}L_{x}^{2}},$
(5.7) $\displaystyle J_{1}^{3}$
$\displaystyle\lesssim\|u_{1}\|_{L_{t}^{4}L_{x}^{\infty}}\|u_{2}\|_{L_{t}^{4}L_{x}^{\infty}}\|\Lambda^{s}u_{3}\|_{L_{t}^{4}L_{x}^{2}}\|u_{4}\|_{L_{t}^{4}L_{x}^{2}}.$

The estimates for (5.6) are obtained in a similar way to estimates of
$J_{1}^{1}$. Hence we consider the (5.7). Using (2) of Lemma 3.2 and Sobolev
embedding, we estimate

(5.8) $\displaystyle\begin{aligned}
\|u_{j}\|_{L_{t}^{4}L_{x}^{\infty}}&\lesssim\|\Lambda^{s-\frac{3}{8}}u_{j}\|_{L_{t,x}^{4}}\lesssim\|u_{j}\|_{X_{\pm_{j}}^{s,b}},\;\;\mbox{for}\;\;j=1,2,\\\
\|\Lambda^{s}u_{3}\|_{L_{t}^{4}L_{x}^{2}}&\lesssim\|u_{3}\|_{X^{s,\frac{1}{4}}}\lesssim\|u_{3}\|_{X_{\pm_{3}}^{s,b}}.\end{aligned}$

Then the estimate (5.8) yields that

$J_{1}^{3}\lesssim
T^{{\delta}}\prod_{j=1}^{3}\|u_{j}\|_{X_{\pm_{j}}^{s,b}}\|u_{4}\|_{X_{\pm_{4}}^{0,-b^{\prime}}}.$

Therefore this completes the proof of (5.1). ∎

###### Proof of $(ii)$ of Lemma 5.1.

The LHS of (5.2) is bounded by

$\displaystyle\left|\small\iint\left[|x|^{-1}*(\overline{u_{1}}u_{2})\right]u_{3}\Lambda^{s}\overline{u_{4}}dtdx\right|$
$\displaystyle\leq\left|\iint\Lambda^{s}(|\nabla|^{-1}(\overline{u_{1}}u_{2})u_{3})\overline{u_{4}}dtdx-\iint\Lambda^{s}|\nabla|^{-1}(\overline{u_{1}}u_{2})u_{3}\overline{u_{4}}dtdx-\iint|\nabla|^{-1}(\overline{u_{1}}u_{2})(\Lambda^{s}u_{3})\overline{u_{4}}dtdx\right|$
$\displaystyle\qquad\qquad+\left|\iint\Lambda^{s}|\nabla|^{-1}(\overline{u_{1}}u_{2})u_{3}\overline{u_{4}}dtdx\right|+\left|\iint|\nabla|^{-1}(\overline{u_{1}}u_{2})(\Lambda^{s}u_{3})\overline{u_{4}}dtdx\right|$
$\displaystyle=:J_{2}^{1}+J_{2}^{2}+J_{2}^{3}.$

We first consider the $J_{2}^{1}$. Lemma 2.1 yields that

$\displaystyle J_{2}^{1}$
$\displaystyle\lesssim\left\|\Lambda^{s}\left[|\nabla|^{-1}(\overline{u_{1}}u_{2})u_{3}\right]-\Lambda^{s}|\nabla|^{-1}(\overline{u_{1}}u_{2})u_{3}-|\nabla|^{-1}(\overline{u_{1}}u_{2})\left(\Lambda^{s}u_{3}\right)\right\|_{L_{t}^{\frac{4}{3}}L_{x}^{2}}\|u_{4}\|_{L_{t}^{4}L_{x}^{2}}$
$\displaystyle\lesssim\|\Lambda^{s}|\nabla|^{-1}(u_{1}\overline{u_{2}})\|_{L_{t}^{2}L_{x}^{4}}\|u_{3}\|_{L_{t,x}^{4}}\|u_{4}\|_{L_{t}^{4}L_{x}^{2}}$
$\displaystyle\lesssim\sum_{\mu}\|P_{\mu}\Lambda^{s}|\nabla|^{-1}(u_{1}\overline{u_{2}})\|_{L_{t}^{2}L_{x}^{4}}\|u_{3}\|_{L_{t,x}^{4}}\|u_{4}\|_{L_{t}^{4}L_{x}^{2}}.$

Using Hardy-Littlewood-Sobolev and Young’s convolution inequality we see that

$\displaystyle\|P_{\leq
1}\Lambda^{s}|\nabla|^{-1}(u_{1}\overline{u_{2}})\|_{L_{t}^{2}L_{x}^{4}}$
$\displaystyle\lesssim\|P_{\leq
1}(u_{1}\overline{u_{2}})\|_{L_{t}^{2}L_{x}^{\frac{4}{3}}}=\|\check{\beta_{1}}*(u_{1}\overline{u_{2}})\|_{L_{t}^{2}L_{x}^{\frac{4}{3}}}\lesssim\|\check{\beta_{1}}\|_{L_{x}^{\frac{4}{3}}}\|u_{1}\|_{L_{t}^{4}L_{x}^{2}}\|u_{2}\|_{L_{t}^{4}L_{x}^{2}}$
$\displaystyle\lesssim\|u_{1}\|_{X_{\pm_{1}}^{0,\frac{1}{4}}}\|u_{2}\|_{X_{\pm_{2}}^{0,\frac{1}{4}}},$

In third inequality, we used the $\|\check{\beta_{1}}\|_{L_{x}^{p}}\lesssim 1$
for $p>1$. And, by Lemma 3.2, we estimate

$\displaystyle\sum_{\mu\geq
2}\|P_{\mu}\Lambda^{s}|\nabla|^{-1}(u_{1}\overline{u_{2}})\|_{L_{t}^{2}L_{x}^{4}}$
$\displaystyle\lesssim\sum_{\mu\geq
2}\mu^{s-1}\|P_{\mu}(\overline{u_{1}}u_{2})\|_{L_{t}^{2}L_{x}^{4}}\lesssim\sum_{\mu\geq
2}\mu^{s-\frac{1}{2}}\|P_{\mu}(\overline{u_{1}}u_{2})\|_{L_{t,x}^{2}}$
$\displaystyle\lesssim\sum_{\mu\geq
2}\mu^{-\frac{1}{8}}\|u_{1}\|_{X_{\pm_{1}}^{s,b}}\|u_{2}\|_{X_{\pm_{2}}^{s,b}}\lesssim\|u_{1}\|_{X_{\pm_{1}}^{s,b}}\|u_{2}\|_{X_{\pm_{2}}^{s,b}}.$

Also, by second estimate of (5.8), we get

$J_{2}^{1}\lesssim\prod_{j=1}^{3}\|u_{j}\|_{X_{\pm_{j}}^{s,b}}\|u_{4}\|_{L_{t}^{4}L_{x}^{2}}\lesssim
T^{\delta}\prod_{j=1}^{3}\|u_{j}\|_{X_{\pm_{j}}^{s,b}}\|u_{4}\|_{X_{\pm_{4}}^{0,-b^{\prime}}}.$

Estimates for $J_{2}^{2}$ are obtained in almost the same way as estimates for
$J_{2}^{1}$. Hence it is left to deal with $J_{2}^{3}$. By Hardy-Littlewood-
Sobolev and Young’s convolution inequality, we have

$\displaystyle\left|\iint|\nabla|^{-1}P_{\leq
2}\left(\overline{u_{1}}u_{2}\right)\left(\Lambda^{s}u_{3}\right)\overline{u_{4}}dtdx\right|$
$\displaystyle\lesssim\||\nabla|^{-1}P_{\leq
2}(\overline{u_{1}}u_{2})\|_{L_{t}^{2}L_{x}^{4}}\left\|P_{\leq
2}\left[(\Lambda^{s}u_{3})\overline{u_{4}}\right]\right\|_{L_{t}^{2}L_{x}^{\frac{4}{3}}}$
$\displaystyle\lesssim\|P_{\leq
2}(\overline{u_{1}}u_{2})\|_{L_{t}^{2}L_{x}^{\frac{4}{3}}}\|(\Lambda^{s}u_{3})\overline{u_{4}}\|_{L_{t}^{2}L_{x}^{1}}$
$\displaystyle\lesssim\|u_{1}\|_{L_{t}^{4}L_{x}^{2}}\|u_{2}\|_{L_{t}^{4}L_{x}^{2}}\|\Lambda^{s}u_{3}\|_{L_{t}^{4}L_{x}^{2}}\|u_{4}\|_{L_{t}^{4}L_{x}^{2}}$
$\displaystyle\lesssim\|u_{1}\|_{X_{\pm_{1}}^{0,\frac{1}{4}}}\|u_{2}\|_{X_{\pm_{2}}^{0,\frac{1}{4}}}\|u_{3}\|_{X_{\pm_{3}}^{s,\frac{1}{4}}}\|u_{4}\|_{X_{\pm_{4}}^{0,\frac{1}{4}}}.$

Since there is no contribution of $P_{\leq 1}(\overline{u_{1}}u_{2})$, we
assume that $P_{\leq 1}(\overline{u_{1}}u_{2})=0$. Let us consider the high-
frequency part of $J_{2}^{3}$. By Lemma 3.2 and Bernstein’s inequality we
estimate

$\displaystyle J_{2}^{3}$ $\displaystyle\lesssim\sum_{\mu\geq
2}\left|\iint|\nabla|^{-1}P_{\mu}(\overline{u_{1}}u_{2})(\Lambda^{s}u_{3})\overline{u_{4}}dtdx\right|$
$\displaystyle\lesssim\sum_{\mu\geq
2}\mu^{-1}\Big{\|}P_{\mu}(\overline{u_{{\lambda}_{1}}}u_{{\lambda}_{2}})\Big{\|}_{L_{t}^{2}L_{x}^{\infty}}\Big{\|}(\Lambda^{s}u_{3})\overline{u_{4}}\Big{\|}_{L_{t}^{2}L_{x}^{1}}$
$\displaystyle\lesssim\sum_{\mu\geq
2}\Big{\|}P_{\mu}(\overline{u_{{\lambda}_{1}}}u_{{\lambda}_{2}})\Big{\|}_{L_{t,x}^{2}}\Big{\|}\Lambda^{s}u_{3}\Big{\|}_{L_{t}^{4}L_{x}^{2}}\Big{\|}u_{4}\Big{\|}_{L_{t}^{4}L_{x}^{2}}$
$\displaystyle\lesssim\sum_{\mu\geq
2}\mu^{\frac{3}{8}-s}\|u_{1}\|_{X_{\pm_{1}}^{x,b}}\|u_{{\lambda}_{2}}\|_{X_{\pm_{2}}^{s,b}}\Big{\|}\Lambda^{s}u_{3}\Big{\|}_{L_{t}^{4}L_{x}^{2}}\Big{\|}u_{4}\Big{\|}_{L_{t}^{4}L_{x}^{2}}$
$\displaystyle\lesssim
T^{\delta}\prod_{j=1}\|u_{j}\|_{X_{\pm_{j}}^{s,b}}\|u_{4}\|_{X_{\pm_{4}}^{0,-b^{\prime}}}.$

Here we used the assumption $s>\frac{3}{8}$ and $b^{\prime}<-\frac{1}{4}$.
Therefore this completes the proof of the (5.2). ∎

##  6\. The proof of Theorem 1.2

This section aims to show Theorem 1.2. It adopts the argument [22, 15, 20] to
prove of smoothness failure of flows of (1.3) with cubic and Hartree type
nonlinearity. As in proof of [22, 15], if the flow map $\psi\mapsto u$ is
$C^{3}$ at the origin from $H^{s}$ to $C(\left[0,T\right);H^{s})$, we have
(6.6). In [22, 15], they showed smoothness failure of flows of Benjamin-Ono,
semi-relativistic equations, respectively. For the results about Dirac
equation, we refer to [20]. Let us consider the system of
equation($\ell=1,2$):

(6.3) $\displaystyle\left\\{\begin{array}[]{l}(\partial_{t}+{\alpha}\cdot
D)\psi=-i{\kappa}\mathcal{N}_{\ell}(\psi,\psi)\psi,\\\
\psi(0)=\delta\psi_{0}\in H^{s}(\mathbb{R}^{2}).\end{array}\right.$

If the flow is $C^{3}$ at the origin in $H^{s}$, then it follows that

(6.4)
$\displaystyle\partial_{\delta}^{3}\psi(0,t,\cdot)=6C\sum_{\pm_{j},j=1,2,3,4}\int_{0}^{t}S_{\pm_{1}}(t-t^{\prime})\Pi^{\pm_{1}}(D)\left[\mathcal{N}_{\ell}\Big{(}S_{\pm_{2}}(t^{\prime})\psi_{0},S_{\pm_{3}}(t^{\prime})\psi_{0}\Big{)}S_{\pm_{4}}(t^{\prime})\psi_{0}\right](t^{\prime})dt^{\prime}$

where $S_{\pm}(t)=e^{-\pm it|D|}$ for $\ell=1,2$. From the $C^{3}$ smoothness
we have that

(6.5) $\displaystyle\sup_{0\leq t\leq
T}\left\|\sum_{\pm_{j},j=1,2,3,4}\int_{0}^{t}S_{\pm_{1}}(t-t^{\prime})\Pi^{\pm_{1}}(D)\left[\mathcal{N}_{\ell}\Big{(}S_{\pm_{2}}(t^{\prime})\psi_{0},S_{\pm_{3}}(t^{\prime})\psi_{0}\Big{)}S_{\pm_{4}}(t^{\prime})\psi_{0}\right]dt^{\prime}\right\|_{H^{s}}\lesssim\|\psi_{0}\|_{H^{s}}^{3}$

for a local existence time $T$ and $j=1,2$. However we show that (6.5) fails
for $s<s(\ell)$. The explicit statement is as follows:

######  Proposition 6.1.

Let $\ell=1,2$. Assume that $s<s(\ell)$. Then the estimate

(6.6) $\displaystyle\sup_{0\leq t\leq
T}\left\|\mathcal{L}_{\ell}({\varphi})(t)\right\|_{H^{s}}\lesssim\|{\varphi}\|_{H^{s}}^{3}.$

fails to hold for all $\varphi\in H^{s}$, where
$\mathcal{L}_{\ell}({\varphi})(t)=\underset{\pm_{j},j=1,\cdots,4}{\sum}\mathcal{L}_{\ell}^{1,\cdots,4}(\varphi)(t)$
with

$\mathcal{L}_{\ell}^{1,\cdots,4}(\varphi)(t)=\int_{0}^{t}S_{\pm_{1}}(t-t^{\prime})\Pi^{\pm_{1}}(D)\mathcal{N}_{\ell}\Big{(}S_{\pm_{2}}(t^{\prime}){\varphi},S_{\pm_{3}}(t^{\prime}){\varphi},\beta
S_{\pm_{4}}(t^{\prime}){\varphi}\Big{)}dt^{\prime}.$

###### Proof.

The proof of Proposition 6.1 is proven by contradiction. For this purpose, let
us assume that the (6.6) holds. Fix ${\lambda}\gg 1$. We first choose
$\mu={\lambda}^{1-\varepsilon}$ for fixed $0<\varepsilon\ll 1$. Let us define
a box

$\displaystyle B_{\mu}$
$\displaystyle=\\{\xi=(\xi_{1},\xi_{2}):|\xi_{1}-{\lambda}|\lesssim\mu,\;|\xi_{2}|\lesssim\mu\\}$

and consider
${\varphi}=\left(\begin{array}[]{c}\mathcal{F}_{\xi}^{-1}\chi_{B_{\mu}}\\\
0\end{array}\right)$. Then we have $\|\varphi\|_{H^{s}}\sim\mu{\lambda}^{s}$.

To lead a contradiction we adopt a following estimate:

(6.7)
$\displaystyle\left|\sum_{\pm_{j},j=1,\cdots,4}\mathcal{F}_{x}\Big{[}\mathcal{L}_{\ell}^{1\cdots
4}({\varphi})(t)\Big{]}(\xi)\right|\gtrsim t\mu^{3+2s(j)}.$

We now prove the (6.7). By taking Fourier transform we see that

$\displaystyle\mathcal{F}_{x}\Big{[}\mathcal{L}_{\ell}^{1\cdots
4}({\varphi})(t)\Big{]}(\xi)$ $\displaystyle=\Pi^{\pm}(\xi)\int_{0}^{t}\int
e^{-\pm_{1}i(t-t^{\prime})|\xi|}\mathcal{F}_{x}\left[\mathcal{N}_{\ell}\Big{(}S_{\pm_{2}}(t^{\prime}){\varphi},S_{\pm_{3}}(t^{\prime}){\varphi}\Big{)}\right](\sigma)\mathcal{F}_{x}\left[S_{\pm_{4}}(t^{\prime}){\varphi}\right](\xi-\sigma)d\sigma
dt^{\prime}$
$\displaystyle=-\Pi^{\pm}(\xi)\int_{|\sigma|\lesssim\mu}\int_{-B_{\mu}}\mathbf{p}_{1\cdots
4}(t,\xi,\sigma,\zeta)|\sigma|^{-1+2s(j)}\chi_{B_{\mu}}(-\zeta)\chi_{B_{\mu}}(\sigma-\zeta)\chi_{B_{\mu}}(\xi-\sigma)d\zeta
d\sigma$

where $-B_{\mu}:=\\{\xi=(\xi_{1},\xi_{2}):(-\xi_{1},-\xi_{2})\in B_{\mu}\\}$
and

$\displaystyle\mathbf{p}_{1\cdots 4}(t,\xi,\sigma,\zeta)$
$\displaystyle:=\int_{0}^{t}e^{-i(\pm_{1}(t-t^{\prime})|\xi|\pm_{2}t^{\prime}|\zeta|\pm_{3}t^{\prime}|\sigma-\zeta|\pm_{4}t^{\prime}|\sigma|)}dt^{\prime}$
$\displaystyle=\frac{e^{-\pm_{1}it|\xi|}(e^{it\omega}-1)}{i\omega}$

with

$\displaystyle\omega=\pm_{1}|\xi|\pm_{2}|\zeta|\pm_{3}|\sigma-\zeta|\pm_{4}|\sigma|.$

From the support condition it follows that $|\sigma|\lesssim 2\mu$, provided
$\xi\in B_{3\mu}$. Then $|\omega|\lesssim{\lambda}$.

We set $t=\delta{\lambda}^{-1-\varepsilon}$ for fixed $0<\delta\ll 1$. Since
$|t\omega|\ll 1$ for ${\lambda}$ large enough, we get

$\displaystyle\sum_{\pm_{j},j=1,2,3,4}\mathbf{p}_{1\cdots
4}(t,\xi,\sigma,\zeta)$
$\displaystyle\qquad=\sum_{\pm_{j},j=1,2,3,4}te^{-\pm_{1}it|\xi|}\left(\frac{\cos(t\omega)-1}{it\omega}+i\frac{\sin(t\omega)}{it\omega}\right)$
$\displaystyle\qquad=\sum_{\pm_{j},j=1,2,3,4}te^{-\pm_{1}it|\xi|}(O_{\pm}(\delta)+i)$
$\displaystyle\qquad=\sum_{\pm_{j},j=1,2,3,4}te^{-\pm_{1}it|\xi|}O_{\pm}(\delta)+i\sum_{\pm_{j},j=1,2,3,4}te^{-\pm_{1}it|\xi|}$
$\displaystyle\qquad=\sum_{\pm_{j},j=1,2,3,4}te^{-\pm_{1}it|\xi|}O_{\pm}(\delta)+8it\cos(t|\xi|)$
$\displaystyle\qquad=\sum_{\pm_{j},j=1,2,3,4}te^{-\pm_{1}it|\xi|}O_{\pm}(\delta)+8it(1+O(\delta)).$

Hence we obtain

$\displaystyle\left|\sum_{\pm_{j},j=1,2,3,4}\mathcal{F}_{x}\Big{[}\mathcal{L}_{\ell}^{1\cdots
4}({\varphi})(t)\Big{]}(\xi)\right|$
$\displaystyle\qquad\gtrsim\left|\sum_{\pm_{j},j=1,2,3,4}\int_{|\sigma|\lesssim\mu}\int_{-B_{\mu}}\mathbf{p}_{1\cdots
4}(t,\xi,\sigma,\zeta)|\sigma|^{-1+2s(\ell)}\chi_{B_{\mu}}(-\zeta)\chi_{B_{\mu}}(\sigma-\zeta)\chi_{B_{\mu}}(\xi-\sigma)d\zeta
d\sigma\right|$ $\displaystyle\qquad\gtrsim
t\left|\int_{|\sigma|\lesssim\mu}\int_{-B_{\mu}}|\sigma|^{-1+2s(\ell)}\chi_{B_{\mu}}(-\zeta)\chi_{B_{\mu}}(\sigma-\zeta)\chi_{B_{\mu}}(\xi-\sigma)d\zeta
d\sigma\right|$ $\displaystyle\qquad\gtrsim t\mu^{3+2s(\ell)}.$

Therefore we get (6.7).

We return to the main proof. Since
$\mathcal{F}_{x}\Big{[}\mathcal{L}_{\ell}^{1\cdots
4}({\varphi})(t)\Big{]}(\xi)=0$ for $\xi\notin B_{3\mu}$, the (6.7) yields
that

$\displaystyle\left\|\mathcal{L}_{\ell}({\varphi})(t)\right\|_{H^{s}}=\left\|\langle\xi\rangle^{s}\sum_{\pm_{j},j=1,2,3,4}\mathcal{F}_{x}\Big{[}\mathcal{L}_{\ell}^{1\cdots
4}({\varphi})(t)\Big{]}(\xi)\right\|_{L_{\xi}^{2}}$ $\displaystyle\gtrsim
t\mu^{3+2s(\ell)}\left\|\langle\xi\rangle^{s}\right\|_{L_{\xi}^{2}(B_{3\mu})}\gtrsim
t\mu^{4+2s(\ell)}{\lambda}^{s}.$

This gives us that

(6.8) $\displaystyle
t\mu^{4+2s(\ell)}{\lambda}^{s}\lesssim\left\|\langle\xi\rangle^{s}\sum_{\pm_{j},j=1,\cdots,4}\mathcal{F}_{x}\Big{[}\mathcal{L}_{j}^{1\cdots
4}({\varphi})(t)\Big{]}(\xi)\right\|_{L_{\xi}^{2}}\lesssim\mu^{3}{\lambda}^{3s}.$

Therefore, by (6.8) and $t=\delta{\lambda}^{-1-\varepsilon}$, we have

(6.9)
$\displaystyle\delta\lesssim\mu^{-1-2s(\ell)}{\lambda}^{2s+1+{\varepsilon}}={\lambda}^{2s+2s(\ell)+2\varepsilon\big{(}1+s(\ell)\big{)}}.$

Then since the (6.9) does not hold for $s<s(\ell)$ and ${\lambda}\gg 1$, we
reach a contradiction. This completes the proof of Proposition 6.1. ∎

## Acknowledgments

This work was supported by National University Promotion Development Project
in 2019.

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