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\begin{document}

\title{The lifespan of small data solutions to the KP-I}

\author{Benjamin Harrop-Griffiths}
\address{Department of Mathematics, University of California at Berkeley}
\email{benhg@math.berkeley.edu}
 \thanks{The first author was partially supported by the NSF grant DMS-1266182}

\author{Mihaela Ifrim}
\address{Department of Mathematics, University of California at Berkeley}
\thanks{The second author was supported by the Simons Foundation}
\email{ifrim@math.berkeley.edu}

\author{ Daniel Tataru}
\address{Department of Mathematics, University of California at Berkeley}
 \thanks{The third author was partially supported by the NSF grant DMS-1266182
as well as by a Simons Investigator grant from the Simons Foundation}
\email{tataru@math.berkeley.edu}

\begin{abstract}
We show that for small, localized initial data there exists a global solution to the KP-I equation in a Galilean-invariant space using the method of testing by wave packets.
\end{abstract}

\maketitle

\section{Introduction}
In this paper we consider the Kadomtsev-Petviashvili equation (KP-I) initial-value problem
\begin{equation}\label{eq-1}
\begin{cases}
\partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0\\
u(0,x,y)=u_0(x,y),
\end{cases}
\end{equation}
on ${\mathbb R}_t\times{\mathbb R}^2_{x,y}$. The KP-I equation and
the KP-II equation, in which the sign of the term
$\partial_x^{-1}\partial_y^2u$ in \eqref{eq-1} is $+$ instead of
$-$, were derived in \cite{KadomtsevPetviashvili} as models for the propagation of
dispersive long waves with weak transverse effects.

The Cauchy theory for \eqref{eq-1} has been extensively studied \cite{MR1635416,MR2047648,MR2719895,MR2415308,MR2097033,MR2358259,MR1944575,MR1933858,MR2976047}. In particular, \eqref{eq-1} is known to be locally well-posed \cite{MR2719895} in the anisotropic space \(H^{1,0}\) with
\[
\|u\|_{H^{1,0}}^2 = \|u\|_{L^2}^2 +\|\partial_xu\|_{L^2}^2,
\]
and globally well-posed \cite{MR2415308} in the energy space \(\mathbf E^1\) where
\[
\|u\|_{\mathbf E^1}^2 = \|u\|_{L^2}^2+\|\partial_xu\|_{L^2}^2+\|\partial_x^{-1}\partial_yu\|_{L^2}^2.
\]
Further work was devoted to the generalized KP equation, see for instance
\cite{MR1685890,MR2830489}.

The question at hand is that of establishing global existence and asymptotics for solutions to \eqref{eq-1} with sufficiently small, regular and spatially localized initial data. To state our main result we begin with a discussion of the symmetries
of the equation \eqref{eq-1}:
\begin{enumerate}
\item \textit{Translation:} Translates of $u$ in $t$, $x$ and $y$ are solutions.
\item \textit{Reversal:} If \(u(t,x,y)\) is a solution, then so is
\(
u(-t,-x,\pm y)
\).
\item\textit{Scaling:} If $\lambda > 0$ then
\begin{equation}\label{eq:scaling}
u_\lambda(t,x,y) = \lambda^2 u(\lambda^3 t, \lambda x, \lambda^2 y)
\end{equation}
is also a solution.
\item\textit{Galilean invariance:}
For all $c\in {\mathbb R}$ the function
\begin{equation}\label{eq:galilei}
    u_c(t,x,y) = u(t,x-cy+c^2t,y-2ct)
\end{equation}
is a solution to \eqref{eq-1}.
Note that \(\hat u_c(t,\xi,\eta)=\hat u(t,\xi,\eta+c\xi)e^{-ic^2t\xi}e^{-2ict\eta}\).
\end{enumerate}

We denote by $\mathcal L$ the linear operator
\begin{equation}\label{ML}
\mathcal L:=\partial_t+\partial^3_x-\partial^{-1}_x\partial^2_y.
\end{equation}
To obtain pointwise estimates for solutions we introduce the ``vector fields"
\[
L_x:=x-3t\partial_x^2-t\partial^{-2}_x\partial^2_y,  \qquad L_y :=y+2t\partial^{-1}_x\partial_y,
\]
which commute with $\mathcal L$. We observe that $L_y \partial_x$ is the
generator of the Galilean symmetry for both the linear and the nonlinear
equation. $L_x$, on the other hand, is not directly associated to a symmetry of the
nonlinear equation. However, it arises in the expression  for the generator of the
scaling symmetry, namely
\[
 S:=3t\partial_t+x\partial_x +2y\partial_y + 2 = 3t \mathcal L +  L_x \partial_x + 2 L_y \partial_y + 2.
\]

In this article, following the the spirit of \cite{itNLS},
we seek to obtain a result which is Galilean-invariant. This is very natural,
as one should not expect the global well-posedness to depend on the reference frame of the observer.  For this reason, we will
avoid using the scaling symmetry, as well as the use of conservation laws
which are not Galilean invariant (e.g. the energy). Instead, we will rely on
the homogeneous scaling operator
\[
S_0:= L_x \partial_x + L_y \partial_y.
\]
This commutes with both $\mathcal L$ and the Galilean group,  but is not associated to a symmetry of the nonlinear  equation. For large $t$ we will also use the following Galilean invariant operator
\[
L_z:=z+ 3t\partial_x^2, \qquad z:=-x+\frac{1}{4t}y^2,
\]
which relates to $S_0$ and $L_y$ as follows:
\[
L_z\partial_x= -S_0 +\frac{1}{4t}L_y^2\partial_x-\frac12.
\]
For \(z\geq0\), the symbol of \(L_z\) may be written as a product of the symbols of the operators
\[
L_z^\pm:=\sqrt{z}\pm i\sqrt{3t}\partial_x,
\]
which will be used repeatedly in our analysis. We note that \(L_z^+\) is hyperbolic on positive \(x\)-frequencies and elliptic on negative \(x\)-frequencies (and conversely for \(L_z^-\)).

For our global result we seek to use function spaces which are as simple as
possible.  Toward that goal, we define the time-dependent space $X$ as
\[
\|u\|_{X}^2 = \|u\|_{L^2}^2 + \| u_{xxx}\|_{L^2}^2
+\| L_y^2 \partial_x u\|_{L^2}^2 + \|S_0 u\|_{L^2}^2.
\]

Then, our main result is as follows:

\begin{theorem}
Assume that the initial data $u_0$ at time $0$ satisfies
\begin{equation}\label{data}
\|u_0\|_{X} \leq \epsilon \ll 1.
\end{equation}
Then, there exists a unique global solution $u$ which satisfies the bound
\begin{equation}\label{sln-energy}
\|u(t)\|_{X} \leq \epsilon \langle t \rangle ^{C \epsilon},
\end{equation}
as well as the pointwise bound
\begin{equation}\label{sln-point}
\| u_x(t)\|_{L^\infty} \lesssim \epsilon t^{-\frac12} \langle t \rangle^{-\frac12}.
\end{equation}
Further, the solution $u$ scatters in $L^2$ at infinity, in the sense
that there exists a linear wave $u_{\mathrm{scatter}}$  satisfying $\mathcal L u_{\mathrm{scatter}} = 0$ and
\(
\|u_{\mathrm{scatter}}\|_{L^2} = \|u\|_{L^2}
\)
so that
\begin{equation}\label{est:ScatteringBound}
\|u - u_{\mathrm{scatter}}\|_{L^2} \lesssim \epsilon^2 t^{-\frac{1}{48}+C\epsilon}.
\end{equation}
\end{theorem}

To frame our result, we first note that the KP-I equation is integrable and admits a Lax pair representation. This leads to an infinite number of formally conserved quantities and allows solutions with small initial data to be studied using inverse scattering techniques (for example see the recent survey \cite{KSsurveyIST} and references therein).
\todo[inline]{
First few conservation laws are:
\[
\int u,\,\int (1+i \partial_x^{-1}\partial_y)u,\;\int u^2,\;\dots
\]
For conservation laws \(\#9\) and higher we have terms of the form \(\int(\partial_x^{-1}\partial_y(\phi^2))^2\) which is only well defined if \(\|u\|_{L^2}=0\).
}
\todo[inline]{Scattering problem is \(i\psi_x+\psi_{yy}+u\psi=0\). Can only be solved for small initial data.
}
However, it is of significant interest to develop more robust techniques to analyze the asymptotic behavior of solutions. In a recent paper Hayashi and Naumkin \cite{HayashiNaumkin2014} prove global existence and derive asymptotics for a certain class of rapidly decaying, smooth initial data. Our result presents a significant improvement by not only considering a larger class of initial data that includes the Schwartz functions, but also does so in a space that respects the Galilean invariance. Indeed, we believe this to be the only known global result for \eqref{eq-1} in a Galilean-invariant space. We note that our initial data space has norm
\[
\|u(0)\|_{X}^2 = \|u(0)\|_{L^2}^2 + \| u_{xxx}(0)\|_{L^2}^2 + \|y^2 u_x(0)\|_{L^2}^2 + \| (x \partial_x + y \partial_y) u(0)\|_{L^2}^2.
\]
\todo[inline]{Note Hayashi-Naumkin's data space does not contain \(\mathcal{S}\) as they assume \(\|\langle (x,y)\rangle^4\partial_x^{-1}u_0\|_{L^2}<\infty\).}

To describe the difficulties in this problem, we first note that the linear
evolution $S(t)$ associated to the $\mathcal L$ operator exhibits $t^{-1}$
dispersive decay,
\[
\|S(t)\|_{L^1 \to L^\infty} \lesssim t^{-1}.
\]
This motivates the pointwise decay rate in \eqref{sln-point}. Unfortunately, this decay rate does not suffice in order to obtain uniform $X$ bounds for the nonlinear equation, and in turn close the bootstrap for the pointwise bound. This difficulty is a familiar one, and several methods have been used to bypass it in certain related problems.

The first such method is Shatah's normal form method \cite{MR803256}, which relies on the absence of bilinear resonant interactions in order to replace the quadratic nonlinearity with a cubic one. Unfortunately, our problem does admit three wave resonances. The symbol of $\mathcal L$ is
\[
\ell (\tau,\textbf k)= \tau -\xi^3-\xi^{-1}\eta^2,  \qquad \textbf{k}:=(\xi, \eta).
\]
Hence the dispersion relation for \eqref{eq-1} is given by
\[
\omega(\textbf{k})=\xi^3+\xi^{-1}\eta^2.
\]
Thus, resonances in the bilinear interactions correspond to roots  of the system
\begin{equation*}
\left\{
\begin{aligned}
&\omega ({\textbf{k}_1})+\omega ({\textbf{k}_2})=\omega ({\textbf{k}_3})\\
&\textbf{k}_1+\textbf{k}_2=\textbf{k}_3,
\end{aligned}
\right.
\end{equation*}
which gives
\[
\frac{\eta_1}{\xi_1}-\frac{\eta_2}{\xi_2}=\pm\sqrt{3}(\xi_1+\xi_2).
\]
The presence of these three wave interactions prevents a classical normal form analysis: the quadratic nonlinearity is not removable on the set of resonances. Incidentally, we remark that this also prevents any attempts
to obtain global solutions via Strichartz type estimates or $X^{s,b}$ spaces.
We note, however, that for the closely related KP-II equation, where such
resonant interactions do not occur, one can produce global solutions in this
manner, precisely by employing the more robust $U^2$ and $V^2$ spaces, see \cite{MR2526409}.

More recently, a significant improvement over the normal form method was achieved
with the space-time resonance method introduced by Germain-Masmoudi-Shatah \cite{MR2482120} and Gustafson-Nakanishi-Tsai \cite{MR2559713}, which was used to treat
 a good number of two dimensional problems, e.g. \cite{MR2914945,MR2993751,GMS3dCapillary}. This essentially requires a weaker assumption, namely that there are no resonant interactions of parallel waves.  However, in our problem, waves of opposite frequencies are parallel and interact to yield resonant zero frequency output. Further, the symbol of $\mathcal L$ is also singular at zero $x$-frequency.

Instead of pursuing a Fourier based method as above,  our  result makes use of the \textit{method of testing by wave packets} \cite{itNLS,itWW,itCW,hgmKdV}, originally developed in the context of the \(1d\) cubic NLS \cite{itNLS} and \(2d\) water waves \cite{itWW,itCW}, and then applied to the mKdV
in \cite{hgmKdV}.  This relies on an even weaker nonresonance condition, namely that in resonant interactions  it is not possible to have all three waves travel in the same direction.  To describe this in more detail, consider the Hamiltonian flow corresponding to \eqref{eq-1}, which is given by
\begin{equation}\label{eq:HFlow}
\left\{
\begin{aligned}
&(x,y)\mapsto (x-3t\xi^2+t\xi^{-2}\eta^2,y-2t\xi^{-1}\eta)\\
&(\xi,\eta)\mapsto (\xi,\eta).
\end{aligned}
\right.
\end{equation}
In particular, for \(v_1,v_2\) satisfying \(v=-v_1+\frac{1}{4}v^2_2\geq0\), we expect solutions initially localized spatially near zero and in frequency near
$\pm (\xi_v,\eta_v)$, where
\begin{equation}\label{xivetav}
(\xi_v,\eta_v) = \left(\frac{\sqrt{v}}{\sqrt{3}},-\frac{v_2\sqrt v}{2 \sqrt{3}}\right),
\end{equation}
to travel along the ray
\begin{equation}\label{trajectory}
\Gamma_{\mathbf{v}=(v_1,v_2)} := \{x = v_1 t, \ y=v_2t \}.
\end{equation}
This computation also directly leads to the phase function
\begin{equation}\label{phi-def}
\phi:=-\frac{2}{3\sqrt3}t^{-\frac12}\left(\frac{y^2}{4t}-x\right)^{\frac32}=- \frac{2}{3\sqrt3}t^{-\frac12}z^{\frac32},
\end{equation}
associated to the linear propagator $S(t)$. This satisfies $\nabla_{x,y} \phi(x,y) = (\xi_v,\eta_v)$, and also the eikonal equation $\ell(\nabla \phi) = 0$.
We remark that the kernel of the linear propagator $S(t)$ will essentially have the form $t^{-1} \Re e^{i\phi}$ in the propagation region $\{v \geq 0\}$, with rapid decay away from it.

To conclude our discussion, we observe that, on the one hand, waves corresponding to different rays $\Gamma_{\mathbf v}$, $\Gamma_{\mathbf w}$ will have little interaction as they separate in the physical space. On the other hand,
waves corresponding to the same ray $\Gamma_{\mathbf v}$ have a significant interaction, but the frequency of this interaction will correspond to velocities which are away from $\mathbf v$.

We further comment on the scattering result, which is subtly different from standard
linear scattering. Precisely, we remark that, while $u$ approaches
the linear scatterer $u_{\mathrm{scatter}}$ in $L^2$, one property that fails
in this setting is the stronger bound $\mathcal L(u-u_{\mathrm{scatter}}) \in L^1 L^2$, or any other related Strichartz bound. To remedy this, we explicitly compute a quadratic correction
$u_{\mathrm {mod}}$, decaying in $L^2$, so that $\mathcal L(u-u_{\mathrm{scatter}}- u_{\mathrm {mod}}) \in L^1 L^2$.

A natural question in this setting is what is the regularity of the data $u_{\mathrm{scatter}}(0)$
for the scattering solution. One might expect that $u_{\mathrm{scatter}}(0) \in X(0)$, and we conjecture that this is indeed the case. However, our estimates only yield
the slightly weaker interpolation bound
\[
u_{\mathrm{scatter}} (0) \in [L^2,X(0)]_{C\epsilon},
\]
which is close to $X(0)$ but not quite there.

Our strategy of the proof will be to start with the pointwise bound
\eqref{sln-point} as a bootstrap assumption. The goals of the
 subsequent sections in the paper are as follows:

\begin{itemize}
\item{Energy estimates}, proved using the bootstrap assumption.

\item{Initial pointwise bounds}; these are obtained from the
energy estimates using Klainerman-Sobolev type inequalities adapted
to our problem.

\item{Final pointwise bounds}, closing the bootstrap argument using
the  the wave packet testing method.

\item{The scattering result}, whose proof relies on computing
the quadratic correction $u_{\mathrm {mod}}$ mentioned above.
\end{itemize}

\subsection*{Acknowledgement} This research was carried out while the authors were visiting the Hausdorff Research Institute for Mathematics in Bonn.

\section{Energy estimates}

In this section we prove the energy estimates \eqref{sln-energy} under the bootstrap assumption
\begin{equation}\label{est:bootstrap}
|u_x|\leq C \epsilon t^{-\frac12}\langle t\rangle^{-\frac12}.
\end{equation}
Precisely, we have:
\begin{proposition}\label{p:energy}
Let $u$ be a solution for the  KP-I equation in a time interval $[0,T]$,
so that

(i) The initial data $u_0$ for the KP-I equation satisfies
\eqref{data}.

(ii) The solution $u$ satisfies \eqref{est:bootstrap}.

Then $u$ also satisfies the following energy estimate in $[0,T]$:
\begin{equation}\label{en}
\|u(t)\|_{X} \lesssim \epsilon \langle t \rangle ^{C^* \epsilon}, \qquad C^* \lesssim C.
\end{equation}
\end{proposition}

\begin{proof}
We first observe that the $L^2$ norm of the solution $\|u\|_{L^2}$
is a conserved quantity. Secondly, we note that we have good $L^2$ bounds
for the linearized equation
\begin{equation}\label{eq-lin}
\begin{cases}
\partial_t w+\partial_x^3 w-\partial_x^{-1}\partial_y^2w +\partial_x(u w)=0\\
w(0,x)=w_0(x).
\end{cases}
\end{equation}
Indeed, we have
\[
\frac{d}{dt} \|w\|_{L^2}^2 =2\int - w \partial_x (uw) dx =  -\int u_x w^2 dx \leq \|u_x\|_{L^\infty}\|w\|^2_{L^2}.
\]
By Gronwall's inequality this yields energy bounds for $u_x$, $\partial_x L_y u$ and also $Su$ (not needed).

The function $\partial_x^3 u $ solves a perturbed linear equation,
\[
\mathcal L (\partial_x^3 u) + \partial_x (u \partial_x^3 u) = - 3 u_{x}u_{xxx} - 3 (u_{xx})^2
\]
But the term on the right is bounded in $L^2$ by
\[
\| u_{x}u_{xxx} + (u_{xx})^2\|_{L^2} \lesssim \|u_x\|_{L^\infty} \|u_{xxx}\|_{L^2} ,
\]
so the energy estimate for $u_{xxx}$ closes in the same way as in the case of the linearized
equation.

A similar argument applies for the energy bound for $\partial_x L_y^2 u$. To show this, we first prove the following interpolation inequality:

\smallskip
\begin{lemma}
For \(t\neq0\), we have the estimate
\begin{equation}
\|\partial_x L_y u\|_{L^4}^2\lesssim \|u_x\|_{L^\infty}\|\partial_xL_y^2u\|_{L^2}.\label{est:LySobolev}
\end{equation}
\end{lemma}
\begin{proof}
For \(t\neq0\) we write
\(
f(t,x,y) = u(t,x+\frac{1}{4t}y^2,y),
\)
and observe that
\[
\|\partial_xL_yu\|_{L^4} = 2t\|f_y\|_{L^4}.
\]

For dyadic \(\lambda\in 2^{\mathbb Z}\), we take the projection \(P_\lambda\) to act in the \(x\)-variable. Integrating by parts in the \(y\)-variable we obtain
\[
\|P_\lambda f_y\|_{L^4}^4 \leq 3\|P_\lambda f\|_{L^\infty}\|P_\lambda f_{yy}\|_{L^2}\|P_\lambda f_y\|_{L^4}^2\lesssim \|P_\lambda f_x\|_{L^\infty}\|P_\lambda \partial_x^{-1}f_{yy}\|_{L^2}\|P_\lambda f_y\|_{L^4}^2.
\]

Replacing \(L^4\) by the Lorentz space \(L^{4,4}\) and summing over dyadic \(x\)-frequencies using the Cauchy-Schwarz inequality, we have
\[
\|f_y\|_{L^4}^4 \sim \|f_y\|_{L^{4,4}_xL^4_y}^4\lesssim \|f_x\|_{L^\infty}\|\partial_x^{-1}f_{yy}\|_{L^2}\|f_y\|_{L^{4,4}_xL^4_y}^2\sim \|f_x\|_{L^\infty}\|\partial_x^{-1}f_{yy}\|_{L^2}\|f_y\|_{L^4}^2,
\]
from which the estimate \eqref{est:LySobolev} follows.
\end{proof}
\smallskip

We then consider the equation solved by \(\partial_xL_y^2u\),
\[
\mathcal L ( \partial_x L_y^2 u ) = -\partial_x L_y^2 \partial_x (u^2/2) =
- u \partial_x^2 L_y^2 u - (\partial_x L_y u)^2 .
\]
Thus, using the estimate \eqref{est:LySobolev} and integration by parts, we obtain
\[
\begin{split}
\frac{d}{dt} \|\partial_x L_y^2 u \|_{L^2}^2 = & \ \int - \partial_x L_y^2 u
(u \partial_x^2 L_y^2 u + (\partial_x L u)^2) dx
= \int \left( \frac12 u_x |\partial_x L_y^2 u|^2 - \partial_x L_y^2 u (\partial_x L_y u)^2 \right) dx \\
\lesssim  & \ \|u_x\|_{L^\infty} \|\partial_x L_y^2 u\|^2_{L^2}.
\end{split}
\]
and conclude again via Gronwall's inequality.

It remains to obtain $L^2$ bounds for the expression $S_0 u$.
We do this differently for small $t$  and for large $t$.  For small $t$ it suffices to consider the following modification,
\[
w = S_0 u - t u u_x.
\]
The function $w$ also solves a perturbed linearized equation,
\[
\mathcal L w= - (uw)_x + 6 t u_{x}u_{xxx}.
\]
for which we directly obtain energy estimates by using the bootstrap
assumption \eqref{est:bootstrap} for $u_x$. By \eqref{est:bootstrap}
we can also estimate the size of the modification $t u u_x$ in $L^1 L^2$.
The above equation for $w$  is easily checked using the relations
\[
\mathcal L(uu_x)=(u\mathcal L u)_x+3(u_xu_{xx})_x-u_y^2+\partial_x^{-1}u_{yy}u_x,
\]
\[
(L_x\partial_x+L_y\partial_y)(uu_x)=(u(L_x\partial_x+L_y\partial_y)u)_x-9t(u_xu_{xx})_x+tu_y^2-t\partial_x^{-1}u_{yy}u_x-uu_x.
\]

For large $t$ we instead use the relation
\[
(L_x \partial_x + L_y \partial_y)u = S u - L_y \partial_y u - 3 t\mathcal L u -2u
\]
to reduce the problem to an estimate for $w = S u - L_y \partial_y u$.
Since $S$ is a generator of a symmetry for the system, it follows that $Su$ solves the linearized equation \eqref{eq-lin}. It remains to compute
\[
\begin{split}
\mathcal L (L_y \partial_y u) = & \ - \frac12 L_y \partial_y \partial_x( u^2 )
\\ =  & \  - \partial_x ( u L_y \partial_y u) - (L_y \partial_x u)( \partial_y u )
+ (L_y \partial_y u )(\partial_x u)
\\ = & \ - \partial_x ( u L_y \partial_y u) + \frac{1}{2t} [ (L_y \partial_x u)^2
- (L_y^2 \partial_x u)( \partial_x u)] + uu_x.
\end{split}
\]
Thus we obtain
\[
\mathcal L w =  - \partial_x ( u w) +\frac{1}{2t} [ (L_y \partial_x u)^2
- (L_y^2 \partial_x u)( \partial_x u)] + uu_x.
\]
Hence the energy estimate for $w$ follows using \eqref{est:LySobolev} and the $L^2$ bound for $L_y^2 \partial_x u$.

\end{proof}

\section{Klainerman-Sobolev Estimates}\label{sect:KS}

In this section we prove pointwise bounds for \(u,u_x\). Ignoring the dependence of the energy estimates on \(t,\epsilon\), we assume that
\begin{equation}\label{est:NRG0}
\|u\|_X   \lesssim  1.
\end{equation}
The expression $S_0 u$ is somewhat cumbersome to use directly; instead we use $L_z$, for
which we  have the energy estimate
\begin{equation}\label{est:NRG01}
\|L_z\partial_xu\|_{L^2}\lesssim 1, \qquad t \gtrsim 1.
\end{equation}
By a slight abuse of notation we will consider \(\mathbf v\) to be a function of \((t,x,y)\) in this section, defined via the ray \(\Gamma_{\mathbf v}\) of the Hamiltonian flow as \(\mathbf v = (t^{-1}x,t^{-1}y)\). In particular we will write \(v = t^{-1}z\).

Parity considerations and the symbol singularity at $\xi = 0$ lead us to decompose
$u$ into positive and negative $x$-frequencies,
\[
u = u^+ + u^- = 2 \Re u^+,\qquad u^+ = \overline{u^-}.
\]
The $X$ norm bound commutes with this decomposition,
\[
\| u^+\|_{X} = \|u^-\|_X = \frac{1}{\sqrt{2}} \|u\|_{X}.
\]

We now divide \(u^+\) into a hyperbolic and an elliptic part. The corresponding decomposition of \(u^-\) follows by taking complex conjugates. We first fix a constant \(\delta>0\) and take an almost orthogonal decomposition in \(x\)-frequency adapted to the lattice \(2^{\delta{\mathbb Z}}\),
\[
u^+=\sum\limits_{\lambda\in 2^{\delta{\mathbb Z}}}u_\lambda^+.
\]
Here $\delta$ is a small universal constant, which we will only need in order to control the ``resolution" of our decomposition in Section \ref{sect:WP}. The implicit constant in our estimates will depend on $\delta$, but this has no impact on our analysis. This decomposition is compatible with the $X$-norm, in that
\[
\|u^+\|_{X}^2 \approx \sum_{\lambda} \|u^+_\lambda\|_X^2.
\]

For $t \geq 1$ we further  decompose $u^+$ into the hyperbolic and elliptic parts,
\[
u^{{\mathrm{hyp}},+}=\sum\limits_\lambda u^{{\mathrm{hyp}},+}_\lambda,\qquad u^{{\mathrm{ell}},+}=u^+-u^{{\mathrm{hyp}},+},
\]
where, for \(\lambda\geq t^{-\frac13}\), we define
\[
u^{{\mathrm{hyp}},+}_\lambda=\chi_\lambda^{\mathrm{hyp}} u_\lambda^+,\qquad u^{{\mathrm{ell}},+}_\lambda=u_\lambda^+-u^{{\mathrm{hyp}},+}_\lambda,
\]
for a compactly supported, smooth function \(\chi_\lambda^{\mathrm{hyp}}(t,z)\) localized spatially in the hyperbolic region
\[
B^{\mathrm{hyp}}_\lambda=\{v =3\lambda^2(1+O(\delta))\},
\]
which corresponds to the frequencies associated to $u_\lambda$. We remark that here we prefer to take $\chi_\lambda^{\mathrm{hyp}}$ to have compact spatial support. Then, $u_\lambda^{{\mathrm{hyp}},+}$ and $u_\lambda^{{\mathrm{ell}},+}$ are only
localized  at \(x\)-frequencies \(\lambda(1+O(\delta))\) modulo rapidly decaying tails. These tails have size $O((t^\frac13 \lambda)^{-N})$, and play a negligible role in our analysis.

We further note that, as defined above, the hyperbolic component $u^{{\mathrm{hyp}}}$ is supported in the region $\{ v \gtrsim t^{-\frac23}\}$,
and in particular sits outside the parabola $z = 0$. With the above decomposition of $u$, we can now state the pointwise bounds
on $u$ and $u_x$ as follows:

\begin{figure}
\caption{The hyperbolic region \(B^{\mathrm{hyp}}_\lambda\).}
\begin{tikzpicture}[scale=1.3]
\begin{scope}[rotate around={-90:(0,3.5)}]
\begin{axis}[stack plots=y,axis lines=none, no markers]
\addplot[black,smooth,thick] {x^2/4-1};
\addplot[black!10,fill=black!10,line width=0,smooth] {-3} \closedcycle;
\addplot[black,smooth,very thick] {-0.05};
\end{axis}
\end{scope}
\draw[->] (-4,0)--(6,0) node[below left] {\(v_1\)};
\draw[->] (4,-3.1)--(4,3.1) node[below right] {\(v_2\)};
\draw[very thick,<-] (-3,-0.25)--(-2.25,-0.25) node[below] {\(O(\lambda^2\delta)\)};
\filldraw(-2.25,0) circle (1.5pt) node[above] {\(-3\lambda^2\)};
\draw[very thick,->] (-2.25,-0.25)--(-1.4,-0.25);
\node at (-0.75,2) {\(B_\lambda^{\mathrm{hyp}}\)};
\end{tikzpicture}
\end{figure}

\vspace{0.3cm}
\begin{proposition}\label{propn:Pointwise}
For \(0<t<1\) we have the pointwise estimates
\begin{equation}\label{est:LessThan1}
|u|,|u_x|\lesssim t^{-\frac12}.
\end{equation}
For \(t\geq1\) we have the hyperbolic estimates
\begin{equation}\label{est:Hyperbolic}
\begin{aligned}
|u^{\mathrm{hyp}}|&\lesssim t^{-1}\min\{v^{-\frac34},v^{-\frac38}\},\\
|u^{\mathrm{hyp}}_x|&\lesssim t^{-1}\min\{v^{-\frac14},v^{\frac18}\},
\end{aligned}
\end{equation}
and the elliptic improvements
\begin{equation}\label{est:Elliptic}
\begin{aligned}
|u^{\mathrm{ell}}|&\lesssim t^{-\frac34}\langle t^{\frac23}v\rangle^{-\frac34}(1+\log\langle t^{\frac23}v\rangle),\\
|u^{\mathrm{ell}}_x|&\lesssim t^{-\frac{13}{12}}\langle t^{\frac23}v\rangle^{-\frac14}.
\end{aligned}
\end{equation}
\end{proposition}

\begin{proof}
It suffices to assume that \(\delta=1\) and prove bounds for \(u^+\). To simplify notation we drop the superscript and take \(u=u^+\).

\medskip

\textbf{A. Small times $t \leq 1$.}
We recall the Sobolev estimate
\begin{equation}\label{est:UsefulSobolev}
|f|\lesssim\|f\|_{L^2}^{\frac14}\|\partial_xf\|_{L^2}^{\frac12}\|\partial_y^2f\|_{L^2}^{\frac14}.
\end{equation}
Taking \(f(t,x,y)=u_\lambda(t,x+\frac{1}{4t}y^2,y)\) in \eqref{est:UsefulSobolev}, we have
\[
|u_\lambda|\lesssim t^{-\frac12}\|u_\lambda\|_{L^2}^{\frac14}\|\partial_xu_\lambda\|_{L^2}^{\frac12}\|L_y^2\partial_x^2u_\lambda\|_{L^2}^{\frac14}.
\]
From the localization, we then have the estimate
\begin{equation}\label{est:SmallTimes}
|u_\lambda|\lesssim t^{-\frac12}\min\{\lambda^{-\frac32},\lambda^{\frac34}\}.
\end{equation}
For \(0<t<1\) we may sum with respect to  \(\lambda\) to get \eqref{est:LessThan1}.

\medskip

\textbf{B. Large times $t \geq 1$.} Here we split our analysis into low
frequencies and high frequencies, depending on the uncertainty principle
threshold $\lambda = t^{-\frac13}$ for the Airy type operator $L_z$.

\medskip

\textbf{B1. Low frequencies.} Here we consider times \(t\geq 1\) and frequencies $\lambda \leq t^{-\frac13}$. With $f_\lambda$ defined as above,
\[
f_\lambda(t,x,y)=u_\lambda(t,x+\frac{1}{4t}y^2,y)
\]
the $L^2$ bounds for $u$ and $L_z \partial_x u$ yield
\begin{equation}\label{b1}
\|f_{\lambda}\|_{L^2}\lesssim 1, \qquad \| x \partial_x f_{\lambda }\|_{L^2} \lesssim 1,
\end{equation}
while the bound on $L_y^2\partial_x u$ yields
\begin{equation}\label{b2}
\|\partial_y^2 f_{\lambda} \|_{L^2} \lesssim t^{-2} \lambda.
\end{equation}
We claim that the above three bounds imply the pointwise estimate
\begin{equation}\label{f-low}
|f_\lambda| \lesssim t^{-\frac34} \langle \lambda x \rangle^{-\frac34}.
\end{equation}
Summing this over $\lambda < t^{-\frac13}$ yields
\[
| f_{<t^{-\frac13}}| \lesssim t^{-\frac34}  \langle t^{-\frac13} x \rangle^{-\frac34} \log(1+ \langle t^{-\frac13} x \rangle).
\]
For $\partial_x f_\lambda$ we similarly obtain the same bound as \eqref{f-low} but with another factor of $\lambda$. Here summation
over $\lambda < t^{-\frac13}$ is better than above, so we obtain
\[
| \partial_x f_{<t^{-\frac13}}| \lesssim t^{-\frac{13}{12}}  \langle t^{-\frac13} x \rangle^{-\frac34}.
\]
Returning to $u$, the last two bounds imply, as desired, that
\begin{equation}
|u_{\leq t^{-\frac13}}|\lesssim t^{-\frac34}\langle t^{\frac23}v\rangle^{-\frac34}(1+\log\langle t^{\frac23}v\rangle),\qquad |\partial_x u_{\leq t^{-\frac13}}|\lesssim t^{-\frac{13}{12}}\langle t^{\frac23}v\rangle^{-\frac34}.\label{est:ReFix1}
\end{equation}

It remains to show that the bounds \eqref{b1} and \eqref{b2} imply \eqref{f-low}. We first observe that by scaling $(x,y) \to (\lambda x, \lambda^\frac12 y)$ the problem reduces to the case $\lambda = 1$.
For $f := f_1$ at frequency $1$ the bounds for $f$ and $\partial_x f$
are equivalent; precisely, from \eqref{b1} and \eqref{b2} one easily obtains the
equivalent form
\begin{equation*}
\|\langle x \rangle f\|_{L^2}\lesssim 1, \qquad \| \langle x \rangle\partial_x f\|_{L^2} \lesssim 1,\qquad  \| \partial_y^2 f \|_{L^2} \lesssim t^{-2}.
\end{equation*}
At this point we localize $f$ to dyadic spatial regions $\langle x \rangle \approx r$, discarding the frequency localization. For $f_r = \chi_{\langle x \rangle \approx r} f$ we obtain
\begin{equation*}
\| f_r\|_{L^2}\lesssim r^{-1}, \qquad \| \langle x \rangle\partial_x f_r\|_{L^2} \lesssim r^{-1},\qquad \| \partial_y^2 f_r \|_{L^2} \lesssim t^{-2}.
\end{equation*}
Applying \eqref{est:UsefulSobolev} to $f_r$ yields
\[
|f_r| \lesssim t^{-\frac34} r^{-\frac34}
\]
and \eqref{f-low} follows.

\medskip

\textbf{B2. High frequencies.} Here we consider times \(t\geq 1\) and frequencies $\lambda \geq t^{-\frac13}$. The key step in the analysis is to carry out a careful
analysis of the operator $L_z$, depending on the balance of $v=t^{-1}z$ and $\lambda$.
Precisely, in the region $v \approx \lambda^{2}$, which corresponds to $u^{{\mathrm{hyp}}}$
the operator $L_z$ is hyperbolic. Elsewhere, $L_z$ is elliptic.

\medskip
\begin{lemma} For \(t\geq1\) and \(\lambda\geq t^{-\frac13}\) we have the estimates
\begin{gather}
\|L_z^+u^{{\mathrm{hyp}},+}_\lambda\|_{L^2}\lesssim \lambda^{-2}t^{-\frac12}
(\|u_\lambda\|_{L^2} + \|L_z \partial_x u_\lambda\|_{L^2})
\label{est:LzHyp},\\
\|\langle \lambda^{-2}v\rangle u^{\mathrm{ell}}_\lambda\|_{L^2}\lesssim\lambda^{-3}t^{-1}(\|u_\lambda\|_{L^2} + \|L_z \partial_x u_\lambda\|_{L^2})
\label{est:LzEll}.
\end{gather}
\end{lemma}
\begin{proof}
We remark that this is a one dimensional estimate, which applies for fixed $y$.
For simplicity we set $y=0$. By rescaling, it suffices to consider \(\lambda=1\). We note that the relation $\lambda > t^{-\frac13}$ is scale
invariant, so after rescaling we still have $t > 1$.

Integrating by parts, we observe that for a smooth, compactly supported function \(f\),
\[
\|\sqrt vf\|_{L^2}^2+3\|\partial_xf\|_{L^2}^2=t^{-1}\|L_z^-f\|_{L^2}^2+2\sqrt3\int\sqrt{v}\Im(f\partial_x\overline f)\, dx.
\]
We apply this to \(f=L_z^+u_1^{{\mathrm{hyp}},+}\).
Then $L_z^-f = L_z^-L_z^+u_1^{{\mathrm{hyp}},+}$
can be directly estimated
\[
\|L_z^- f\|_{L^2}\lesssim\|L_z u_1^+\|_{L^2}+\|u_1^+\|_{L^2}
\lesssim \|L_z\partial_x u_1\|_{L^2}+\|u_1\|_{L^2}
.
\]
where at the first step we use the spatial localization of the
the cutoff function $\chi_1^{hyp}$ to the region $\{ z \approx |t|\}$
and at the second step we have used the localization
of $u_1$ and thus $L_z u_1$ at frequency $1$.

On the other hand, $u_1$ is localized to positive
unit frequencies. Hence $v^\frac14 L_z^+u_1^{{\mathrm{hyp}},+}$ is also localized to positive frequencies modulo $O_{L_2}(t^{-N})$ errors.
Hence distributing the powers of $v$ we have
\[
\int\sqrt{v}\Im(L_z^+u_1^{{\mathrm{hyp}},+}\partial_x\overline{(L_z^+u_1^{{\mathrm{hyp}},+})})\, dx = \int\Im(v^{\frac14} L_z^+u_1^{{\mathrm{hyp}},+}\partial_x\overline{(v^\frac14 L_z^+u_1^{{\mathrm{hyp}},+})})\, dx
\lesssim t^{-N} \|u_1\|_{L^2}^2.
\]
The estimate \eqref{est:LzHyp} then follows from the last two bounds.

For \eqref{est:LzEll} we will use the ellipticity of the operator $L_z$
in the support of $1-\chi_1^{hyp}$. For that we decompose
\[
u_1^{\mathrm{ell}}=\chi_{\{|v|\ll1\}}u_1+\chi_{\{|v|\gg1\}}u_1+\chi_{\{v\sim-1\}}u_1
\]
for smooth cutoff functions \(\chi_{\{|v|\ll1\}},\chi_{\{|v|\gg1\}},\chi_{\{v\sim-1\}}\) localized to the corresponding regions.
Integrating by parts, we have the identity
\[
\|v\partial_xf\|_{L^2}^2+9\|\partial_x^3f\|_{L^2}^2=t^{-2}\|L_z\partial_xf\|_{L^2}^2+3\int v|\partial_x^2f|^2 \, dx.
\]
We then apply this for \(f=\chi_{\{|v|\ll1\}}u_1,\chi_{\{|v|\gg1\}}u_1,\chi_{\{v\sim-1\}}u_1\) respectively,  using
Garding's inequality and the localization of $u_1$ to unit frequency in order to derive the estimates
\[
\begin{gathered}
\int v|\partial_x^2(\chi_{\{|v|\ll1\}}u_1)|^2 \, dx \ll\|\chi_{\{|v|\ll1\}}\partial_x^3 u_1\|_{L^2}^2 +  \|u_1\|_{L^2}^2, \\
\int v|\partial_x^2(\chi_{\{|v|\gg1\}}u_1)|^2 dx \ll\|\chi_{\{|v|\gg1\}} v \partial_x u_1\|_{L^2}^2
 + \|u_1\|_{L^2}^2
,\\
\int v|\partial_x^2(\chi_{\{v\sim-1\}}u_1)|^2 dx\leq 0.
\end{gathered}
\]
Then the bound \eqref{est:LzEll} follows.

\end{proof}

\medskip

\textbf{B2(a) The hyperbolic part.}
Applying \eqref{est:UsefulSobolev} with
\({f(t,x,y)=e^{-\frac{2}{3\sqrt3}it^{-\frac12}|x|^{\frac32}}u^{\mathrm{hyp}}_\lambda(t,x+\frac{1}{4t}y^2,y)}\)
we have
\[
|u^{\mathrm{hyp}}_\lambda|\lesssim t^{-\frac34}\|u^{\mathrm{hyp}}_\lambda\|_{L^2}^{\frac14}\|L_z^+u^{\mathrm{hyp}}_\lambda\|_{L^2}^{\frac12}\|L_y^2\partial_x^2u^{\mathrm{hyp}}_\lambda\|_{L^2}^{\frac14}.
\]
Hence by \eqref{est:LzHyp} we obtain
\[
|u_\lambda^{\mathrm{hyp}}|\lesssim t^{-1}\min\{\lambda^{-\frac34},\lambda^{-\frac32}\}.
\]
The pointwise estimate \eqref{est:Hyperbolic} then follows from the fact that the \(u^{\mathrm{hyp}}_\lambda\) are supported in the essentially disjoint regions \(\{v\approx 3\lambda^2\}\).

\medskip

\textbf{B2(b) The elliptic part.}
Applying \eqref{est:UsefulSobolev} with \({f(t,x,y) = u_\lambda^{\mathrm{ell}}(t,x+\frac{1}{4t}y^2,y)}\), we have
\[
|u_\lambda^{\mathrm{ell}}|\lesssim t^{-\frac12}\|\partial_xu_\lambda^{\mathrm{ell}}\|_{L^2}^{\frac34}\|L_y^2\partial_xu_\lambda^{\mathrm{ell}}\|_{L^2}^{\frac14}.
\]

Estimating as for the low frequencies, we apply the elliptic estimate \eqref{est:LzEll} on dyadic \(v\)-intervals to obtain
\[
|u^{\mathrm{ell}}_\lambda|\lesssim t^{-\frac54}\lambda^{-\frac32}\langle\lambda^{-2}v\rangle^{-\frac34}.
\]
We may then sum over \(\lambda\geq t^{-\frac13}\), to get
\begin{equation}
\sum\limits_\lambda|u^{\mathrm{ell}}_{\lambda}|\lesssim t^{-\frac34}\langle t^{\frac23}v\rangle^{-\frac34}(1+\log\langle t^{\frac23}v\rangle),\qquad\sum\limits_{\lambda}|\partial_xu^{\mathrm{ell}}_{\lambda}|\lesssim t^{-\frac{13}{12}}\langle t^{\frac23}v\rangle^{-\frac14}.\label{est:ReFix2}
\end{equation}
The estimate \eqref{est:Elliptic} then follows from \eqref{est:ReFix1} and \eqref{est:ReFix2}.
\end{proof}

As a consequence of the localization of \(u^{\mathrm{hyp}}\) and the compatibility of the localization with the \(X\)-norm, we have the following corollary:

\begin{corollary}\label{WavePacketHelper}For \(t>1\), we have the estimates
\begin{equation}
\|v^{\frac12}L_z^+\partial_xu^{{\mathrm{hyp}},+}\|_{L^2}\lesssim t^{-\frac12}\|u\|_X,\qquad\|v^{-1}\partial_x^3L_y^2u^{\mathrm{hyp}}\|_{L^2}\lesssim \|u\|_X.
\end{equation}
\end{corollary}

\section{Wave packets}\label{sect:WP}

To study the global decay properties of solutions to \eqref{eq-1} we apply the same idea as in \cite{itNLS,itCW,itWW,hgmKdV}, which is to test the
solution $u$ with wave packets which travel along the Hamilton flow.
Since we aim to prove uniform bounds on $u_x$, it is simpler to test $u_x$ rather than $u$.

A wave packet, in the context here, is an approximate solution to the
linear system, with $O(t^{-1})$ errors. Precisely, for each trajectory
$\Gamma_{\mathbf{v}=(v_1,v_2)}$ as in \eqref{trajectory}, we establish decay for $u_x$ along this ray by testing with a wave packet moving along the
ray with velocity $\mathbf{v}$ where, in contrast to Section~3, we now consider \(\mathbf v\) to be independent of \((t,x,y)\).

To motivate the definition of this packet we recall some
useful facts. First, this ray is associated with waves that have
spatial frequencies $\pm  (\xi_{v},\eta_v)$ as in \eqref{xivetav}.
Thus, it is convenient to use the phase function $\pm \phi$, with $\phi$
as in \eqref{phi-def}. Selecting the $+$ sign, which corresponds to positive $x$-frequencies,  it is natural to use as test functions  wave  packets  of the form
\begin{equation}
\label{defPsi}
\Psi_{\mathbf v}(t,x,y)= - i\sqrt{3}v^{-\frac{1}{2}}\partial_{x}\left( \chi(\lambda_1(z-vt),\lambda_2(y-v_2t)) e^{i\phi (t,x,y)}\right),
\end{equation}
where
\[
\lambda_1=t^{-\frac12}v^{-\frac14},\qquad\lambda_2=t^{-\frac12}v^{\frac14}.
\]
 Here we  take $\chi$ smooth with compact support. For normalization purposes we assume that
\[
 \int\chi(\alpha, \beta)\, d\alpha d\beta = 1.
\]
The $t^\frac12$ localization scale is exactly the scale of wave packets which are required to stay coherent on the time scale $t$. The $v$ factors account for the
different dispersion rates in the $x$ and the $y$ directions. Finally, the
$x$ derivative is used in order to simplify the computation of $\mathcal L \Psi_{\mathbf v}$. For other purposes we note that the leading part of $\Psi_{\mathbf{v}}$ is given by
\[
\Psi_{\mathbf{v}}=\chi e^{i\phi}+ O(\lambda_1).
\]

To see that these are reasonable approximate solutions we
observe that we can compute
\begin{equation}
\label{error}
\begin{aligned}
e^{-i\phi}\mathcal{L}\Psi_{\mathbf{v}}=
&\frac12t^{-1}\left[ -\partial_x((z-vt)\chi )+(\partial_y+\frac{y}{2t}\partial_x)((y-v_2t)\chi)\right] \\
&-i\sqrt{3}\left( v^{-\frac{1}{2}}(\partial_y+\frac{y}{2t}\partial_x)^2\chi-v^{\frac12}\partial_x^2\chi\right) +O(t^{-\frac{3}{2}} v^{\frac{1}{4}}).
\end{aligned}
\end{equation}

The explicit terms above  are the leading ones, and, as expected, have size $t^{-1}$
times the size of $\Psi_{\mathbf{v}}$; further, they exhibit some additional
structure, manifested in the presence of the outer
differentiation operators $(\partial_x, \partial_y)$, which we
will take advantage of later on.  The error term at the end has similar localization and regularity, but its size is better by another $t^{\frac12}$ factor, so no further structure information is needed.

The above computation shows that our wave packet $\Psi_v$ is  indeed an approximate
solution for  the linear equation in \eqref{eq-1}.  To be more precise,  as in \cite{itNLS, itCW, itWW,hgmKdV}, our test packet $\Psi_{\mathbf{v}}$ is a good approximate solution for the linear equation associated to our problem only on the dyadic time scale $\Delta t\ll t$. Nevertheless, we are using these packets as test functions
globally in time, and this is where the extra structure above is relevant.

The outcome of testing solutions of \eqref{eq-1} with the wave packet $\Psi_{\mathbf{v}}$ is  the
scalar complex valued function $\gamma(t,\mathbf{v})$, defined in the region $\{v \geq  t^{ -\frac23}\}$ (this is the region where the $O(t^{-\frac{3}{2}} v^{\frac{1}{4}})$ terms in \eqref{error} can actually be treated as error terms):
\[
\gamma(t,\mathbf{v}) := \int u_x \bar \Psi_{\mathbf{v}} \, dxdy,
\]
which we will use as a good measure of the size of $u_x$ along
our chosen ray.

For the purpose of proving global existence of the solutions  we only need to consider $\gamma$ along a single ray. However, in order to obtain a more precise
asymptotics we will think of $\gamma$ as a function $\gamma(t,\mathbf{v})$.

The main purpose of the remaining part of this section is to establish qualitative properties for $\gamma$ and this will be done in the two propositions below. As a prerequisite, we need the following estimates:

\begin{lemma}
\label{puff} Assume $w : {\mathbb R}^2\rightarrow \mathbb C$ is a compactly supported function.  Then the following estimate holds whenever all factors on the right are finite:
\begin{equation}
\label{inegalitatea}
\| w\|_{\dot C^\frac14} \leq (\|w_x\|_{L^2}+\|w_y\|_{L^2})^{\frac14}
\Vert w_x\Vert_{L^2}^{\frac12}\| w_{yy}\|_{L^2}^\frac14.
\end{equation}
\end{lemma}

\begin{proof}
The proof of this lemma is fairly straightforward; we use the embedding of  $\dot{H}^{\frac{3}{4}}$ into $\dot{C}^{\frac14}$ and interpolation to obtain
\[
\Vert w\Vert_{L^{\infty}_x\dot{C}^{\frac14}_y}\leq \Vert w\Vert_{L^{\infty}_x\dot{H}^{\frac34}_y} \lesssim \|w\|^{\frac12}_{L^2_x \dot H^\frac32_y}
\|w_{x}\|^{\frac12}_{L^2_x L^2_y} \lesssim \|w_y\|_{L^2}^{\frac14}
\Vert w_{yy}\Vert_{L^2}^{\frac14}\| w_{x}\|_{L^2}^\frac12.
\]
On the other hand, exchanging the order of the variables, we similarly have
\begin{equation*}
\label{e2}
\| w\|_{L^\infty_y\dot C^\frac14_x} \lesssim
\| w\|_{L^\infty_y \dot H^\frac34_x} \lesssim \|w\|^{\frac34}_{L^2_y \dot H^1_x}
\|w_{yy}\|^{\frac14}_{L^2_y L^2_x}=\|w_x\|^{\frac34}_{L^2}
\|w_{yy}\|^{\frac14}_{L^2 }.
\end{equation*}
The two bounds above complete the proof of \eqref{inegalitatea}.
\end{proof}
Now we are left with two tasks. Firstly, we need to show that $\gamma$ is a good representation of the
pointwise size of $u_x$, and for this we need to  compare $u_x$ to $\gamma(t,\mathbf{v})$ as follows:
\begin{proposition}
\label{lema1}
The function $\gamma$ satisfies the uniform bound
\begin{equation}\label{bd-gamma}
\begin{split}
\|\gamma\|_{L^\infty} \lesssim t \|u_x\|_{L^\infty},
\end{split}
\end{equation}
as well as the approximation error estimate
\begin{equation}
\label{diff-x}
\begin{aligned}
 \| u_x(t,\mathbf{v} t) - 2t^{-1} \Re\{e^{i\phi(t,\mathbf{v}t)} \gamma(t,\mathbf{v})\}\|_{L^{\infty}} \lesssim  &v^{-\frac{1}{16}}t^{-\frac{9}{8}}\Vert u\Vert_X.
 \end{aligned}
\end{equation}
\end{proposition}
\begin{proof}
The first estimate \eqref{bd-gamma} is straightforward as
\[
\int |\Psi_{\mathbf{v}}| dx dy = t.
\]

We turn our attention to \eqref{diff-x}, where we will take advantage of
the dyadic decomposition of $u_x$ in Section~\ref{sect:KS}.

We first observe that we can simplify the
expression of $\Psi_{\mathbf{v}}$ in the formula for $\gamma$: the lower order terms in $\Psi_{\mathbf{v}}$ are better by a factor of $v^\frac14 t^{-\frac12}$, therefore we can readily replace $\Psi_{\mathbf{v}}$ by $\chi e^{i\phi}$. Thus, it suffices to work with
 \begin{equation}
 \label{new-gamma}
 \gamma(t,\mathbf{v})=\int u_x e^{-i\phi} \chi \, dxdy.
 \end{equation}

 Decomposing $u_x=u_x^{{\mathrm{ell}}}+u_x^{{\mathrm{hyp}}, -}+u_x^{{\mathrm{hyp}}, +}$ we observe that only the last term has a nontrivial contribution to $\gamma$
 \begin{equation}
 \label{new-new-gamma}
 \gamma =\int u_{x}^{{\mathrm{hyp}}, +}e^{-i\phi}\chi\, dxdy+ O(t^{-N}),
 \end{equation}
where $N $ is arbitrarily large. The contributions of $u_x^{{\mathrm{hyp}}, -}$  and $u^{{\mathrm{ell}}}$ decay as $t^{-N}$ since there are no resonant frequency interactions ($u_x^{{\mathrm{hyp}},-}$ and $u_x^{{\mathrm{ell}}}$ are frequency localized away from $(\xi_v, \eta_v)$ where $\Psi_{\mathbf{v}}$ is localized). We can further harmlessly replace $ u_{x}^{{\mathrm{hyp}}, +}$  by its component $ u_{v,x}^{{\mathrm{hyp}}, +}$ associated to the dyadic frequency
associated to $\xi_v$ (and its immediate neighbors).

Now we turn our attention to $u_x$ in \eqref{diff-x}. For the elliptic part  $u_x^{{\mathrm{ell}}}$ we already have a satisfactory estimate in \eqref{est:Elliptic}. On the other hand $u_x^{{\mathrm{hyp}}}=2\Re u_{x}^{{\mathrm{hyp}},+}$. Hence, given the above considerations  it suffices to estimate the difference
\[
\mathcal{D}:= u^{{\mathrm{hyp}}, +}_{v,x} (t, \mathbf{v}t)-t^{-1}\int u_{v,x}^{{\mathrm{hyp}}, +}e^{-i\phi}\chi\, dxdy.
\]
 Introducing the notation  $w(t,z,y):=e^{-i\phi(t,x,y)}u^{{\mathrm{hyp}}, +}_{v,x}(t, x,y)$ we compute
 \begin{equation*}
\begin{aligned}
 e^{-i\phi(t,\mathbf{v}t)} \mathcal{D} &=  w(t,v t, v_2t) -t^{-1} \int  w(t, z,y) \chi \, dzdy\\
&= t ^{-1}\int \left[ w(t,v t, v_2t) - w(t,z,y)\right]   \chi(\lambda_1(z-vt),\lambda_2(y-v_2t)) \, dzdy\\
&=-\int  \left[ w(t,v t, v_2t) - w(t,\lambda_1^{-1}\alpha+vt,\lambda_2^{-1}\beta+v_2t)\right]  \chi(\alpha,\beta) \, d\alpha d\beta\\
&=-\int  \left[ \tilde{w}(t,0, 0) - \tilde{w}(t,\alpha, \beta)\right]  \chi(\alpha,\beta) \, d\alpha d\beta ,\\
\end{aligned}
\end{equation*}
where we have used the notation $\tilde{w}(t,\alpha,\beta):=w(t,\lambda_1^{-1}\alpha+vt,\lambda_2^{-1}\beta+v_2t)$. Hence,
 \begin{equation}
 \label{diferenta}
\begin{aligned}
&\vert  \mathcal{D} \vert \lesssim \int  \vert \tilde{w}(t,0, 0) - \tilde{w}(t,\alpha, \beta)\vert  \chi(\alpha,\beta) \, d\alpha d\beta.\\
\end{aligned}
\end{equation}

To estimate the RHS above we use the bound in Lemma~\ref{puff}.  This gives
 \begin{equation}
\begin{aligned}
\label{floarea}
 \vert \tilde{w}(t,0, 0) - \tilde{w}(t,\alpha, \beta)\vert  &\lesssim (|\alpha|+|\beta|)^\frac14 \left(  \Vert \tilde{w}_{\alpha}\Vert_{L^2}+ \Vert \tilde{w}_{\beta}\Vert_{L^2}\right)^{\frac34}\Vert  \tilde{w}_{\beta\beta}\Vert^{\frac{1}{4}}_{L^2}.
 \end{aligned}
\end{equation}

 To conclude the bound for $\mathcal{D}$ it remains to  reinterpret the result of \eqref{floarea} in terms of the original function $u^{{\mathrm{hyp}}, +}_{v,x}(t,x,y)$. For that we compute
\begin{equation*}
\begin{aligned}
&\tilde{w}_{\alpha}(t,\alpha,\beta)=\lambda_1^{-1}\partial_z w(t,z,y)
=\frac{-ie^{i\phi (t,x,y)}v^{\frac{1}{4}}}{\sqrt{3}} L^{+}_z  u_{v,x}^{{\mathrm{hyp}}, +}(t,x,y),\\
&\tilde{w}_{\beta}(t,\alpha,\beta)=\lambda_2^{-1}\partial_{y} w(t,z,y)=\frac{e^{-i\phi}v^{-\frac{1}{4}}}{2t^{\frac{1}{2}}}\left[L_y\partial_x^2  u_{v}^{{\mathrm{hyp}}, +}\right],\\
&\tilde{w}_{\beta\beta}(t,\alpha,\beta)=\lambda_2^{-2}\partial_{yy} w(t,z,y)= \frac{e^{-i\phi}v^{-\frac{1}{2}}}{4t}\left[  L^2_y\partial^2_x u_{v,x}^{{\mathrm{hyp}}, +}\right],\\
\end{aligned}
\end{equation*}
and the corresponding $L^2$ norms in the initial variable:
\begin{equation*}
\begin{aligned}
 \Vert \tilde{w}_{\alpha}\Vert_{L^2_{\alpha\beta}}=&\frac{v^{\frac14}}{\sqrt{3}}t^{-\frac{1}{2}}\Vert L^{+}_z  \partial_x u_{v}^{{\mathrm{hyp}}, +}\Vert_{L^2_{xy}}\lesssim v^{-\frac14}t^{-1}\Vert u\Vert_{X},\\  \Vert \tilde{w}_{\beta}\Vert_{L^2_{\alpha \beta}}=&\frac{v^{-\frac14}}{2}t^{-1}\Vert  L_y\partial_x^2 u_{v}^{{\mathrm{hyp}}, +}\Vert_{L^2_{xy}}\lesssim v^{-\frac{1}{4}}t^{-1}\Vert u\Vert_{X},\\
  \Vert \tilde{w}_{\beta \beta}\Vert_{L^2_{\alpha\beta}}=&\frac{v^{-\frac12}}{4}t^{-\frac32}\Vert L^2_y \partial_x^2 u_{v,x}^{{\mathrm{hyp}}, +}\Vert_{L^2_{xy}}\lesssim v^{\frac{1}{2}}t^{-\frac{3}{2}}\Vert u\Vert_X,\\
\end{aligned}
\end{equation*}
where we have used the bounds for $u^{{\mathrm{hyp}},+}$ in Corollary~\ref{WavePacketHelper}.  Thus from \eqref{floarea} we obtain
\[
\vert \tilde{w}(t,0, 0) - \tilde{w}(t,\alpha, \beta)\vert  \lesssim
v^{-\frac{1}{16}} t^{-\frac98} \|u\|_{X}, \qquad
|\alpha|+|\beta| \lesssim 1,
\]
which leads to a similar bound for $\mathcal D$. We remark that we can rewrite
this bound in terms of $w$ as
\begin{equation}\label{bdiff}
 |w(t,v t, v_2t) - w(t,z,y)|\lesssim
v^{-\frac{1}{16}} t^{-\frac98} \|u\|_X, \qquad (t,z,y) \in \text{supp} \ \Psi_{\mathbf v},
\end{equation}
which will be useful later.

\end{proof}

Secondly, we need to show that $\gamma$ stays bounded,  which we do by  establishing a differential equation for it:

\begin{proposition}
\label{lema2}
If $u$ solves \eqref{eq-1}, then we have that
\begin{equation}
\label{gamma-dot}
\dot{\gamma}(t, \mathbf{v})=O(t^{-\frac{13}{12}})(\|u\|_X+\|u\|_X^2 ), \quad v>t^{-\frac{1}{3}}.
\end{equation}
\end{proposition}

\begin{proof}
 We obtain the differential equation for $\gamma$ by simply testing  \eqref{eq-1} against our wave packet $\Psi_{\mathbf{v}}$,
\begin{equation}
\label{gdot}
\dot{\gamma}(t,\mathbf{v})=\int  \mathcal{L}\bar{\Psi}_{\mathbf{v}}u_x-\bar{\Psi}_{\mathbf{v}}\partial_x(uu_x)\, dxdy.
\end{equation}
First we measure the error in the linear component of \eqref{gdot}.  We separate $u$ and $u_x$ into hyperbolic and elliptic parts. The  decay is slightly better in the elliptic case
\[
 \left| \int  \mathcal{L}\bar{\Psi}_{\mathbf{v}}u^{{\mathrm{ell}}}_x\, dx dy\right| \lesssim v^{-\frac14}t^{-\frac{5}{4}}\| u\|_X.
\]
For $u_x^{{\mathrm{hyp}}}$, we further decompose into $u_x^{{\mathrm{hyp}}, +}$ and $u_x^{{\mathrm{hyp}}, -}$. The contribution of the last one is of $O(t^{-N})$, due to mismatched frequencies. So we are left with the contribution of $u_x^{{\mathrm{hyp}}, +}$. For $\mathcal{L}\Psi_{\mathbf{v}}$ it suffices to consider its leading term from \eqref{error}, which is of order $O(t^{-1})$. This yields the following integral
\begin{equation*}
\begin{aligned}
 &\int\left[ \frac12t^{-1}\left[ -\partial_x((z-vt)\chi )+(\partial_y+\frac{y}{2t}\partial_x)((y-v_2t)\chi)\right]\right.  \\
 &\hspace*{1.5cm}\left.  +i\sqrt{3}\left( v^{-\frac{1}{2}}(\partial_y+\frac{y}{2t}\partial_x)^2\chi-v^{\frac12}\partial_x^2\chi\right) \right]e^{-i\phi} u^{{\mathrm{hyp}}, +}_x\, dx dy.
 \end{aligned}
\end{equation*}
 Using the bound \eqref{bdiff} for $ w(t,z,y):=e^{-i\phi(t,x,y)} u^{{\mathrm{hyp}}, +}_x(t,x,y)$, we approximate
 \[
 w(t,z,y)=w(t,vt, v_2t)+O(v^{-\frac{1}{16}}t^{-\frac98})\|u\|_{X},
 \]
 and substitute it in the integral above. The contribution of the error term yields a $v^{-\frac{1}{16}}t^{-\frac98}$ bound, and the contribution of the leading term vanishes when we integrate by parts.

For the second term in \eqref{gdot} we integrate by parts and separate $u$ and $u_x$ into hyperbolic and elliptic parts. For example, when we estimate the hyperbolic and elliptic interaction, we make use of bounds obtained in \eqref{est:Hyperbolic} and \eqref{est:Elliptic}
\begin{equation*}
\left| \int \bar{\Psi}_{\mathbf{v}}\partial_x(u^{{\mathrm{hyp}}}u_x^{{\mathrm{ell}}})\, dxdy\right| \lesssim v^{-\frac{1}{2}}t^{-\frac{5}{4}}\Vert u\Vert_X^2.
\end{equation*}
The same argument applies whenever one of the factors is elliptic; so we are left only with the hyperbolic-hyperbolic interaction
\begin{equation*}
\int \bar{\Psi}_{\mathbf{v}}\partial_x(u^{{\mathrm{hyp}}}u_x^{{\mathrm{hyp}}})\, dxdy.
\end{equation*}
By definition, the hyperbolic components are frequency localized near $\pm(\xi_{v}, \eta_v)$, while $\bar{\Psi}_{\mathbf{v}}$ is localized at $-(\xi_v, \eta_v)$. Since the three interacting frequencies cannot add up to zero, it follows that the above integral is rapidly decreasing, i.e., is of order $\epsilon t^{-N}$, for $N$ large enough.
\end{proof}

In the last part of this section we finalize the bootstrap argument and prove \eqref{sln-point}. We already have the estimate for \(0<t<1\), so we consider
a time interval $[1,T]$ where we make the bootstrap assumption
\[
\vert u_x\vert \leq C\epsilon t^{-1},
\]
with a fixed large universal constant $C$. Here $C$ is chosen with the property that
\[
1\ll C \ll \epsilon^{-\frac12}.
\]
Under this assumption, by Proposition~\ref{p:energy}, $u$ satisfies the energy estimate \eqref{en} in the same time interval $ [1,T]$. From Proposition~ \ref{propn:Pointwise} we have
\begin{equation}\label{abc}
\vert u_x\vert \lesssim \epsilon t^{-1+C_*\epsilon}\left[\min\{|v|^{-\frac14},|v|^{\frac18}\}+ t^{-\frac{1}{12}}\right].
\end{equation}
This immediately implies
\begin{equation}\label{abcd}
\vert \gamma(t,\mathbf{v})\vert \lesssim \epsilon t^{C_*\epsilon}\min\{|v|^{-\frac14},|v|^{\frac18}\}.
\end{equation}
Our goal is to prove \eqref{sln-point}. For this we consider the domain $\Omega$
\[
\Omega:= \left\{ v \, : t^{-\alpha}\leq v\leq t^{\alpha}\right\},
\]
where $\alpha $ is a sufficiently small parameter; $\alpha =\frac{1}{6}$ suffices.
Outside $\Omega$, the bound in \eqref{sln-point} follows from \eqref{abc}. Inside $\Omega$ we use
use Proposition~ \ref{lema1} and Proposition~\ref{lema2} to show that  \eqref{sln-point} holds  with an implicit constant which does not depend on $C$.  From the bound \eqref{diff-x} obtained in Proposition~\ref{lema1}, and \eqref{en} we get
\begin{equation*}
 \| u_x(t,\mathbf{v} t) - 2t^{-1} \Re\{e^{i\phi(t,\mathbf{v}t)} \gamma(t,\mathbf{v})\}\|_{L^{\infty}} \lesssim  v^{-\frac{1}{16}}t^{-\frac{9}{8}}\Vert u\Vert_X\lesssim \epsilon v^{-\frac{1}{16}}t^{-\frac{9}{8}+\epsilon C_*}.
\end{equation*}
This estimate implies that inside $\Omega$ we can substitute  the bound \eqref{sln-point} for $u_x$ with its analogue for $\gamma$, namely
\begin{equation}
\label{need}
|\gamma(t,\mathbf{v})| \lesssim \epsilon .
\end{equation}

Our goal now is to use the ODE \eqref{gamma-dot} in order to transition from
\eqref{abcd} to \eqref{need} along rays $\Gamma_{\mathbf{v}}$. From Proposition~\ref{lema2} we have the following bound in $\Omega$
\begin{equation}
\label{gd}
\vert \dot{\gamma}(t, \mathbf{v})\vert \lesssim \epsilon t^{-\frac{13}{12}+2\epsilon C_*}.
\end{equation}

We consider three cases for $v$:

(i)  Suppose first that
$v \approx 1$, i.e., $z\approx t$. Then we initially have
\[
|\gamma(t,\mathbf{v})| \lesssim \epsilon, \qquad t \approx 1.
 \]
Integrating \eqref{gamma-dot} we conclude that
\[
|\gamma(t,\mathbf{v})| \lesssim \epsilon, \qquad t \geq 1,
\]
and \eqref{need} follows.

(ii) Assume now that $v \ll 1$, i.e., $z\ll t$. Then, as $t$ increases, the ray $\Gamma_{\mathbf{v}}$
enters $\Omega$ at some point $t_0$ with $v \approx t_0^{-\alpha}$. Then by
 \eqref{abcd} we obtain
\[
|\gamma(t_0,\mathbf{v})| \lesssim \epsilon t_0^{\epsilon C_*}v^{\frac{1}{8}}\lesssim \epsilon.
\]
We use this to initialize $\gamma$. For larger $t$ we use \eqref{gd}
to conclude that
\[
|\gamma(t, \mathbf{v})| \lesssim \vert \gamma(t_0,\mathbf{v})\vert + \epsilon t_0^{-\frac{1}{12}+2\epsilon C_*} \lesssim \epsilon, \qquad t > t_0.
\]
Then  \eqref{need} follows.

(iii) Finally, consider the case $v \gg 1$, i.e., $z\gg t$.
Again, as $t$ increases, the ray $z = vt$ enters $\Omega$ at some point $t_0$
 $v \approx t_0^{\alpha}$, therefore by \eqref{abcd} we obtain
\[
|\gamma(t_0,\mathbf{v})| \lesssim  \epsilon t_0^{\epsilon C_*}v^{-\frac{1}{4}}  \lesssim \epsilon.
\]
We use this to initialize $\gamma$. For larger $t$ we use \eqref{gd}
to conclude that
\[
|\gamma(t,\mathbf{v})| \lesssim  \vert \gamma(t_0,\mathbf{v})\vert + \epsilon t_0^{-\frac{1}{12}+2\epsilon C_*}  \lesssim \epsilon ,  \qquad t > t_0.
\]
Then  \eqref{need} again follows.

\section{Scattering}

In this section we prove the scattering estimate \eqref{est:ScatteringBound}.
We will use what we have already proved so far, namely  that we have a global solution $u$ which satisfies the bounds \eqref{sln-energy} and \eqref{sln-point}.

For fixed \(\alpha>0\), let
\[
w=P_{t^{-\frac\alpha2}\leq\cdot\leq t^{\frac\alpha2}}u,
\]
where the projection acts on \(x\)-frequencies. If \(\Omega\) is defined as in Section~\ref{sect:WP}, then for fixed \(\mathbf{v}\), the ray \(\Gamma_{\mathbf{v}}\) will eventually lie in \(\Omega\) and hence \(w^{\mathrm{hyp}}\) will capture the hyperbolic part of \(u\) at infinity. For concreteness we take \(\alpha=\frac16\) although any sufficiently small \(\alpha>0\) will suffice.

\medskip

\begin{lemma}
Let \(t\geq1\). We then have the estimate
\begin{equation}
\|uu_x-2\Re(w^+w^+_x)\|_{L^2}\lesssim \epsilon^2t^{-\frac{49}{48}+C\epsilon}\label{est:ScatHelper}.
\end{equation}
\end{lemma}
\begin{proof}
Let \(\chi\) be a bump function, identically \(1\) on the region \(\Omega\).
From \eqref{est:Hyperbolic} and \eqref{est:Elliptic} we have the estimate
\[
\|(1-\chi)uu_x\|_{L^2}+\|(1-\chi)w\uj_x\|_{L^2}\lesssim \epsilon^2t^{-\frac{49}{48}+C\epsilon}.
\]
On the other hand, from the localization of \(u^{\mathrm{hyp}}\), we have \(\chi(u^{\mathrm{hyp}}-w^{\mathrm{hyp}})\), \(\chi(u^{\mathrm{hyp}}-w^{\mathrm{hyp}})_x=0\), and from the elliptic estimate \eqref{est:Elliptic},
\[
\|\chi(u^{\mathrm{ell}}-w^{\mathrm{ell}})\|_{L^\infty}+\|\chi(u^{\mathrm{ell}}_x-w^{\mathrm{ell}}_x)\|_{L^\infty}\lesssim\|\chi u^{\mathrm{ell}}\|_{L^\infty}+\|\chi u^{\mathrm{ell}}_x\|_{L^\infty}\lesssim \epsilon t^{-\frac98+C\epsilon}(1+\log t).
\]
Thus we are left with estimating the difference arising for the hyperbolic
parts, for which, using $u^{hyp} = 2 \Re u^{hyp,+}$, we write
\[
\chi(u^{hyp}u^{hyp}_x-2\Re(w^{hyp,+}w^{hyp,+}_x))
= \chi(u^{hyp}u^{hyp}_x-2\Re(u^{hyp,+} u^{hyp,+}_x))
= \chi \partial_x |u^{hyp,+}|^2
\]
To bound this last term, we use \eqref{est:LzHyp} to obtain
\[
\|\partial_x(|w^{{\mathrm{hyp}},+}|^2)\|_{L^2}\lesssim t^{-\frac12}\|w^{{\mathrm{hyp}},+}\overline{L_z^+w^{{\mathrm{hyp}},+}}\|_{L^2}\lesssim\epsilon^2 t^{-\frac74+C\epsilon}.
\]
\end{proof}

\medskip

As \(2\Re(w^+w^+_x)\not\in L^1L^2\), we look to find an approximate solution to the equation
\[
\mathcal{L}u_{\mathrm {mod}}\approx 2\Re(w^+w^+_x)
\]
with error terms in \(L^1L^2\). As the bulk of $u^{{\mathrm{hyp}},\pm}$, and thus of $w$, is localized at frequency $\pm ( \omega(\xi_v,\eta_v), \xi_v,\eta_v)$, a standard frequency/modulation analysis leads to the choice of a quadratic correction term
\begin{equation}\label{defn:umod}
u_{\mathrm {mod}}=\frac83\partial_x^{-3}\Re(w^+w^+_x).
\end{equation}

We observe that \(w^+w^+_x\) is localized at \(x\)-frequencies \(t^{-\frac{1}{12}}\lesssim\xi\lesssim t^{\frac{1}{12}}\), so \eqref{defn:umod} is well-defined and satisfies the \(L^2\) bound
\begin{equation}\label{est:umodBound}
\|u_{\mathrm {mod}}\|_{L^2}\lesssim\epsilon^2 t^{-\frac34}.
\end{equation}

The scattering bound \eqref{est:ScatteringBound} is then a consequence of the following Lemma.

\medskip

\begin{lemma}
For \(t\geq1\) and \(\epsilon>0\) sufficiently small, we have the estimate
\begin{equation}\label{est:ModScat}
\left\|2\Re(w^+w^+_x)-\mathcal{L}u_{\mathrm {mod}}\right\|_{L^2}\lesssim \epsilon^2 t^{-\frac74+C\epsilon}.
\end{equation}
\end{lemma}

\begin{proof}
We calculate
\[
\begin{split}
\mathcal{L}\partial_x(fg)&=\partial_x\left((\mathcal Lf)g+f(\mathcal Lg)+3f_{xx}g_x+3f_xg_{xx}-\frac{1}{2t}fg\right)\\
&\quad+\frac{1}{4t^2}(L_y^2\partial_xf)g_x+\frac{1}{4t^2}f_x(L_y^2\partial_xg)-\frac{1}{2t^2}(L_y\partial_xf)(L_y\partial_xg),
\end{split}
\]
which gives
\[
\begin{split}
\frac43\mathcal L\partial_x^{-3}(w^+w^+_x)&=\partial_x^{-2}\left(\frac43w^+[\partial_t,P^+_{t^{-\frac{1}{12}}\leq\cdot\leq t^{\frac{1}{12}}}]u+\frac23w^+ P^+_{t^{-\frac{1}{12}}\leq\cdot\leq t^{\frac{1}{12}}}(u^2)_x+4w^+_{xx}w^+_x-\frac{1}{3t}(w^+)^2\right)\\
&\quad+\partial_x^{-3}\left(\frac{1}{3t^2}(L_y^2\partial_xw^+)w^+_x-\frac{1}{3t^2}(L_y\partial_xw^+)^2\right).
\end{split}
\]
We observe that
\[
[P^+_{t^{-\frac{1}{12}}\leq\cdot\leq t^{\frac{1}{12}}},\partial_t]=t^{-1}P^+_{t^{\frac{1}{12}}}+t^{-1}P^+_{t^{-\frac{1}{12}}}, \qquad [P^+_{t^{-\frac{1}{12}}\leq\cdot\leq t^{\frac{1}{12}}},L_y]=0.
\]
For sufficiently small \(\epsilon>0\), the frequency localization, \eqref{sln-energy} and \eqref{sln-point} then yield the estimate
\[
\left\|\mathcal Lu_{\mathrm {mod}}-4\partial_x^{-2}\left(w^+_xw^+_{xx}\right)\right\|_{L^2}\lesssim\epsilon^2(1+\epsilon)t^{-\frac74}.
\]

We may write
\[
\partial_x^2(w^+w^+_x)=4w^+_xw^+_{xx}+\frac{1}{3t}w^+_xL_zw^+-\frac{1}{3t}w^+ L_z\partial_xw^+.
\]
As
\[
[P^+_{t^{-\frac{1}{12}}\leq\cdot\leq t^{\frac{1}{12}}},L_z]=t^{-\frac{1}{12}}P^+_{t^{\frac{1}{12}}}+t^{\frac{1}{12}}P^+_{t^{-\frac{1}{12}}},
\]
we may commute the frequency localization with \(L_z\) and estimate as before to get
\[
\left\|w^+w^+_x-4\partial_x^{-2}\left(w^+_xw^+_{xx}\right)\right\|_{L^2}\lesssim\epsilon^2 t^{-\frac74+C\epsilon}.
\]
\end{proof}

\bibliographystyle{abbrv}


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A study released by Kimberly-Clark this past week revealed that one in three American mothers have trouble providing diapers for their babies. Often these mothers end up cutting back on other basics, such as food or paying for utilities, so they can buy diapers.
Bolling said when a family can't afford to buy diapers, it can impact both their physical and emotional health. The Atlanta Community Food Bank has received diapers in the past, but usually they've been surplus or damaged in some way.
The Food Bank plans to distribute diapers to where the need is the greatest — such as homeless shelters for women and children.
"Diapers are a huge expense for them," Bolling said. A total of 10 communities received diapers from Kimberly-Clark this past week. The company also has committed to give a total of 20 million diapers in the next eight months as part of its "Every Little Bottom" initiative.
Because more communities will be included if future donations, ACFB spokeswoman Angie Clawson said she didn't know how many more diapers Atlanta will receive from Kimberly-Clark in the coming months.


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EV Legislative Update, Feb. 4, 2020
February 4, 2020 February 4, 2020 Lanny
Maryland Right-to-Charge Hearing
Today, I testified before the House Environment and Transportation Committee in support of "Right to Charge" bill HB 111. This bill would prevent HOAs or condo boards from unreasonably restricting a homeowner from installing an EV charging station in their designated parking space.
The hearing went well. No opposition to the bill. In fact, the group that represents the interests of HOAs and condominiums in Maryland testified that they've worked out a few tweaks to the bill with the sponsor, Delegate Marc Korman, and are supporting the bill with those amendments.
Right-to-Charge Senate Bill Filed
The Senate cross-file for the Right to Charge legislation is SB 734 sponsored by Senators Lam, Peters, Rosapepe, and Young. The hearing date hasn't been scheduled yet. Keep an eye on the Electric Vehicle Legislation Tracker for the hearing date on SB 734.
Bill to Extend ZEEVIC
HB 232, sponsored by Del. Fraser-Hidalgo is being heard in the House Environment and Transportation Committee on Thursday, Feb. 6 at 1:00 PM. This bill would extend the term of the Maryland Zero Emission Electric Vehicle Infrastructure Council (ZEEVIC) until June 30, 2026. The term of the council sunsets this year.
Clean Cars Act (EV Rebate) Hearings
Governor Hogan's Clean Cars Act of 2020 has hearings in the House and Senate. SB 277 is on Feb. 11 in Senate Finance and HB 359 is on Feb. 13 in the House Environment and Transportation Committee.
Here is an update for EV-related legislation in Richmond:
Virginia Anti-ICEing Fine Cut From $500 to $100-$250
SB 911 which proposed a $500 fine for ICEing was amended in the Committee on Transportation to cut the fine to $100-$250. I spoke to Senator Hashimi's Legislative Aide about this bill. The reasoning behind the amended fine is to bring it in line with the penalty for parking in handicapped spots.
Legislative Services also clarified for me that the traffic infraction/penalty would apply to blocking charging stations at publicly accessible, private property.
I asked what does "clearly marked mean? They responded that the Code is not specific as to what is required to be "clearly marked." It would likely include signage and/or painting of the parking spot (similar to a handicap spot). It's not sufficient for enforcement to just have a charging station located next to the spot.
Critical Typo Caught – Being Fixed
I also informed Virginia Legislative Services of a typo in the Committee substitute printed 20107318D-S1 text. They intend to fix the typo, otherwise, the way it is worded it will be illegal to park an EV in an EV charging spot. It shall constitute a traffic infraction for any person to park a plug-in electric motor vehicle, as defined in § 56-1, in a parking space clearly marked as reserved for charging plug-in electric motor vehicles.
Virginia EV Rebate Probably Not Happening
The Virginia EV Rebate bill, HB 717, is probably not going to produce incentive money this year. The bill was changed to study the feasibility of an electric vehicle rebate program instead.
The electric school bus grant bill, HB 1140, is in the Committee on Labor and Commerce, Subcommittee 3.
For daily updates see the PlugInSites Electric Vehicle Legislation Tracker
Follow @PlugInSites on Twitter and subscribe below to get updates via email.
No, You Can't Charge Here – Taxi, Uber, Lyft Drivers Banned
Ocean City, Maryland Supercharger is Open

redpajama_set_name: RedPajamaCommonCrawl


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NEW: What Ningxia Jiaze Renewables Corporation Limited (601619) Investors Ought To Know (2019-07-05)
First of all, if you are trading Ningxia Jiaze Renewables Corporation Limited as of the time of writing (2019-07-05), we strongly advise subscribing to Wall Street Journal AT THIS LINK:
Reporting for 2019-07-05 | by tallahasseescene.com: Ningxia Jiaze Renewables Corporation Limited (601619) opened at 4.54, reaching a high of 4.55 and a low of 4.47 before closing at a price of 4.51. There was a total volume of 5113820.0. We saw an accumulation-distribution index of 126.54, an on-balance volume of 4.53, chaikin money flow of 15.0 and a force index of 0.0028. There was an ease of movement rating of -4e-05, a volume-price trend of -0.5566 and a negative volume index of 1000.0.
We calculated a Moving Average Convergence Divergence (MACD) of 0.00067, a MACD signal of 0.00037, a MACD difference of 0.0003, a fast Exponential Moving Average (EMA) indicator of 4.47, a slow Exponential Moving Average (EMA) indicator of 4.47, an Average Directional Movement Index (ADX) of unknown, an ADX positive of 20.0, an ADX negative of 20.0, a positive Vortex Indicator (VI) of 1.0, a negative VI of 1.0, a trend vortex difference of 0.15519, a trix of -9.1456, a Mass Index (MI) of 1.0, a Commodity Channel Index (CCI) of 66.66667, a Detrended Price Oscillator (DPO) of 0.6327, a KST Oscillator (KST) of -123.99318 and a KST Oscillator (KST Signal) of -123.99318 (leaving a KST difference of 1.46981). We also found an Ichimoku rating of 4.545, an Ichimoku B rating of 4.545, a Ichimoku visual trend A of 5.2206, an Ichimoku visual trend B of 5.0339, an Aroon Indicator (AI) up of 4.0 and an AI indicator down of 4.0. That left a difference of 4.0.
We found a Relative Strength Index (RSI) of 50.0, a Money Flow Index (MFI) of 50.23934, a True Strength Index (TSI) of -100.0, an ultimate oscillator of -14.4744, a stochastic oscillator of 800.0, a stochastic oscillator signal of 800.0, a Williams %R rating of 700.0 and an awesome oscillator of 0.0175.
There was a daily return of -12.39932, a daily log return of 0.6689 and a cumulative return of 0.67114.
BREAKING: Important News On SHANGHAI CHINAFORTUNE CO., LTD (600621) (2019-07-16) - July 16, 2019
Crucial Technical Analysis Released on Zhejiang Dingli Machinery Co.,Ltd (603338) (2019-07-16) - July 16, 2019
A New Analysis of Fushun Special Steel Co.,LTD (600399) (2019-07-16) - July 16, 2019
BREAKING: Important News On Bohai Ferry GROUP Co.,Ltd. (603167) (2019-07-10)
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What Is Happening With Beijing Qianjing Landscape Co., Ltd. (603778)? (2019-07-03)
On 2019-07-03 Beijing Qianjing Landscape Co., Ltd. (603778) opened at 4.28, reaching a high of 4.37 and a low of 4.23 before closing at a price of 4.33. There was a total volume of 4351446. We saw an accumulation-distribution index of 14.00311, an on-balance volume of -4.29, chaikin money flow of 2.11111 and a force […]
Thoughts on Jangho Group Co., Ltd. (601886) (2019-07-05)
New Report Released on SHAN DONG JIANG QUAN INDUSTRY CO.,LTD. (600212) (2019-07-05)


KATHY GRIFFIN COMING TO GLEE? YOU BETCHA!
Home > TV Guide > KATHY GRIFFIN COMING TO GLEE? YOU BETCHA!
Comedian Kath Griffin confirmed Friday via Twitter that she will be appearing on an upcoming episode of Glee. Griffin will play a "recent Tea Party candidate and home schooler – a Sarah Palin type," according to Movieline.com. While Griffin will undoubtedly relish the role after taking numerous shots at Sarah and Bristol Palin in her stand-up act (and took the father of the younger Palin's child, Levi Johnston, on a date), Glee creator Ryan Murphy told TV Guide last June that the new character is all about inclusiveness. "I'm adding a Christian character," he told the magazine. "We've taken a couple jabs at the right wing this year, so what I want to do with this character is have someone who Christian kids and parents can recognize and say, 'Oh, look – I'm represented there, too!' If we're trying to form a world of inclusiveness, we've got to include that point of view as well." Griffin appeared last fall with the cast of Glee at Models of Pride, a benefit for LGBT youth, in Los Angeles.


This educational series is intended to give the Bay Area audience an opportunity to form a deeper understanding of Chinese music and performing arts, and to meet some accomplished artists "up close and personal". At each session, a musician or artist will give a talk on his or her area of specialty, which could be a Chinese musical instrument, a regional style of dance, or certain aspect of performing arts. A question and answer session follows to foster interactions between the artist and the audience.
Professor Ding Chengyun was from Dengzhou, Henan Province. He is a famous music scholar and is most noted in the realm of "Guqin". From a very early age, professor Ding developed his love for Chinese arts and culture and started playing qin and zheng under the tutorage of his elder sister Ding Boling. Later he became a pupil of guqin masters, Gu Meigeng and Zhang Zeqian. He graduated from the Department of Chinese Instruments in 1969, and taught in Henan University (lecturer, 1972, associate professor, 1986, professor / head of department, 1992, Director of Graduate School, 1998). Currently he is a professor of Wuhan College of Music, and Director of Changjiang Traditional Musical Culture Center. He is also a visiting professor of Taiwan Nanhua University, and Chinese College of Music. He is a member of Chinese Musician's Association, Association of Famous Chinese, and Director of Boya Qin Association.
Guqin, also known as qin or seven stringed instrument, is one of China's oldest string instruments. It was already popular among ancient gentlemen, scholars, and saints over 2500 years ago. It was said Confucius consoled himself singing and playing this instrument during his low and sad period roaming the kingdoms of Chen and Cai.
Se dates back to even before the guqin, and is considered an extinct instrument. Prof. Fu is one of only a handful of people in the world who know how to play the se. This will be a rare opportunity to experience the legendary harmony attributed to the guqin and the se.


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Wednesday, Sep 14 2016
Patients With High Deductible Plans Sometimes Face EpiPen Cost Issues
News outlets report on stories related to pharmaceutical drug pricing.
Consumer Reports: EpiPen Costs Add To High-Deductible Insurance Woes
When Isa Traverso, of Morrison, Col., went to fill her son's EpiPen prescription in early September 2016, the pharmacy said it would cost her $1,500 for a pair of the life-saving devices—despite having health insurance coverage. That's because her family's Blue Cross Blue Shield health plan requires that they meet a $5,200 deductible before insurance coverage kicks in, a figure they haven't reached yet for the year. That left Traverso on the hook for the full price—and she couldn't afford to fill the prescription. "I'm angry. This is not something you want to have. It's something you have to have," says Traverso. She says she'll consider not using her insurance and switching to the cheaper EpiPen alternative, generic Adrenaclick ($140 at Walmart using a GoodRx.com coupon). (Skinner, 9/12)
Stat: Food Allergy Group Says No To New Donations From EpiPen Manufacturer
A major food allergy advocacy group said this week that it would stop accepting donations immediately from drug companies selling epinephrine auto-injectors until there is "meaningful competition" in the market. While not mentioning Mylan by name, the statement from Food Allergy Research and Education (FARE) implies that the EpiPen manufacturer is one of the targets. (Swetlitz and Silverman, 9/9)
FiercePharma: After Grassley Decries 'Incomplete Response,' Mylan Agrees To Pony Up EpiPen Docs
Congress is starting to get some answers on EpiPen pricing. And though, so far, Mylan's not offering much beyond what's readily available via Google search, that could change later this week. The House Committee on Oversight and Government Reform, which demanded answers earlier this month, is expecting documents late this week, sources close to the investigation said. The committee had set a deadline for initial word from the company Monday. (Staton, 9/13)
KPBS Public Media: Criticism Continues Over High-Priced EpiPen
Mylan, the company that makes the EpiPen, has been under fire for making the drug too expensive for many consumers. EpiPen is an auto-injector form of ephedrine that can stop a severe allergic reaction. It's only sold in pairs, retailing for about $600. But it's not the only prescription drug with an exorbitant price. (Goldberg, 9/12)
The Desert Sun: Woman Says She'll Go Without Her EpiPen Because Of Price
A bee sting while swimming one year ago prompted Donna Myrowto start carrying a life-saving EpiPen. When a pharmacy recently called to say her cost for a new pen had jumped $100, the Palm Springs retiree felt stung all over again...Myrow's pharmacy told her in August she would be responsible for roughly $300, or about half of the total price for her EpiPen renewal. Medicare and supplemental insurance would cover the rest. A year ago she paid $200. (Newkirk, 9/12)
Forbes: BIO's CEO On The Drug-Price Brouhaha And New Ads Touting Pharma Innovation
Anyone who has watched college football on TV or played the game Words with Friends on their phone in recent days may have seen the ad featuring Stephanie Haney, Bill Remak and Ryan Dant, all of whom were beneficiaries of biotechnology treatments. The ad, called "Innovation Saves," suggests that money spent by biotech companies on research and development ultimately saves healthcare costs, because drugs lessen the need for hospital stays, reduce the number of surgeries performed and, in some cases, cure costly diseases. The group behind the ad is the Biotechnology Innovation Organization (BIO), which is running it as part of a larger campaign to counter all the negative publicity surrounding drug companies of late, much of which has been fueled by pricing controversies surrounding drugs like Mylan's EpiPen and Turing Pharmaceuticals' Daraprim. (Weintraub, 9/13)
Stat: Pharma Suffers A Setback In Battle Over Ohio Drug Pricing Ballot Measure
An Ohio court has given a significant boost to a controversial ballot measure that is designed to lower the cost of medicines. In a ruling last Friday, the state Supreme Court decided that thousands of contested signatures on petitions submitted to the General Assembly were valid. And the court also ordered the Secretary of State to permit consumer activists to submit yet another petition that is required for final approval by county officials so the measure can go before voters in November 2017. (Silverman, 9/12)
Business Insider: Scrutiny Of PBMs, Express Scripts Grows With Drug-Price Fury
It's easy to see why EpiPen has become the focus of America's fury over drug prices. It treats potentially deadly allergic reactions — for example, in a child who is stung by a bee — and its price has spiked by over 500% in a few years. While it's easy to jump all over drugmakers, like EpiPen's maker, Mylan, other actors in the healthcare system ought to draw as much scrutiny.One group of companies, called pharmaceutical-benefit managers, or PBMs, serve as middlemen, and they touch every part of the purchase of a prescription drug. (Lopez, 9/12)
FiercePharma: 'Extraordinary' Generics Price Hikes Hit Medicare Part D Amid Big Reduction Overall
Generic drug prices in Medicare Part D decreased significantly in recent years, a new report from the U.S. Government Accountability Office (GAO) says. So why the worry about price hikes? Hundreds of products saw "extraordinary price increases," that's why. For a group of 2,378 generic drugs--including those that entered or exited the market from 2010 to 2015--Medicare Part D prices fell overall by 59%. But "established generics"--the 1,441 drugs that stayed on the market the entire time--fell by just 22%. More than 300 cases of "extraordinary price increases" kept prices from falling further, the GAO said. (Sagonowsky, 9/13)
California Healthline: How Generics Lower Drug Prices
The popular furor over rising drug prices has led to increased scrutiny of the patent protections that can allow drugmakers to sell medicines exclusively for many years. A new study helps quantify just how much a generic competitor can drive down the price of a brand name drug. The price of brand name drugs falls steeply in the first year after the loss of exclusivity and continues to fall over the next three years, according to a study published by the IMS Institute for Healthcare Informatics with numbers from IMS Health, a medical data wholesaler. (9/12)
Stat: Feds Probe Drug Maker For Prices Of Generic Dermatology Meds
Yet another drug maker is under scrutiny for its pricing. The US Department of Justice last Thursday sent grand jury subpoenas to Taro Pharmaceuticals and two of its senior commercial officers in its US unit seeking documents about generic pharmaceuticals and pricing, according to a filing that Taro made with the US Securities and Exchange Commission. (Silverman, 9/12)
Stat: Drug Maker Loses Appeal Of Antitrust Pay-To-Delay Case In Europe
European Commission court upheld an antitrust fine that was imposed three years ago against Lundbeck and four other drug makers for allegedly conspiring to delay the availability of a lower-cost generic version of an antidepressant. The ruling by the General Court of the European Union came in response to an appeal of a 2013 decision that found Lundbeck and the generic drug makers pursued a pay-to-delay deal that violated European Union anticompetition regulations. The European Commission had fined the companies a total of $165 million with Lundbeck ordered to pay the bulk of the fine, or about $105 million. (Silverman, 9/8)
CNBC: Government Involvement In Drug Pricing Is A Trojan Horse, Analyst Says
Drugmakers have been under scrutiny since Mylan's EpiPen price hike, but government involvement in controlling costs is not necessarily the answer, Wells Fargo Securities analyst David Maris said Thursday. Democratic presidential nominee Hillary Clinton has said if elected, she would create an oversight panel to monitor price increases and impose penalties for unjustified hikes. "I don't think that having government run that process is really the right way, because it's a Trojan horse for price controls. Once they start controlling a little bit of price, then the temptation is to control all of price, and if you do that, you really get fewer drugs invented," he said. (Fox, 9/8)
Stat: FDA Slams Drug Maker For Touting Unapproved Leukemia Treatment To Docs
Last June, Celator Pharmaceuticals proudly displayed a large poster touting its experimental Vyxeos medication as an effective salve for treating acute myeloid leukemia. The poster was, in fact, one of countless placards featured prominently on the exhibit floor at the American Society of Clinical Oncology meeting in Chicago, a sort of Woodstock for cancer doctors and eager investors. Despite the numerous displays, however, the Celator poster managed to stick out.That's because Vyxeos has not yet been approved to treat AML. (Silverman, 9/7)
This is part of the KHN Morning Briefing, a summary of health policy coverage from major news organizations.	Sign up for an email subscription.


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can "flexible routes" solve the challenge of serving low density?
Posted on December 9, 2011 in General
Whenever I point out that low density areas tend to have low ridership potential regardless of the service offered — as I do at length in Chapter 10 of Human Transit — someone always asks: "But what if we used flex routes or 'demand-reponsive transport''?'
I've added a page — referenced in the book — that tries to answer that question. It's here.
countering the "empty buses" myth -- with video!
welcome, readers from portland's willamette week
more of a debate than i expected
3 Responses to can "flexible routes" solve the challenge of serving low density?
Miles Bader December 9, 2011 at 6:42 pm #
A little nit; your article says: "At the opposite extreme, a fully loaded rapid transit train may carry over 1000 passengers an hour with a single driver. A highly performing bus line can hit 100 passengers per hour in a very dense market."
I presume the bus line figure is "with a single driver" as well—presumably both train and bus can carry far, far, more passengers per hour once you factor in more frequent service allowed by multiple trains/buses.
[These figures seem a bit on the low side, even for a single driver… a single crush-loaded train can carry upwards of 3000 passengers, after all…]
Andrew December 10, 2011 at 11:45 pm #
My impression is that "dial a ride" services are kind of a joke. Basically it's sort of a shared taxi that you usually have to reserve an hour in advance or so. It's very inconvenient and choice riders are very unlikely to use it.
If an area is very low density it is probably best to have the taxi companies provide this sort of service rather than the transit system, because this sort of service is very inefficient. (The very wealthy and low density neighbourhoods of Bridle Path in Toronto and Lorne Park in Mississauga come to mind here. Both have very lightly used bus routes [TTC 162 and MT 14 respectively] that are often targeted for cutbacks by politicians, and would seem possible candidates for this sort of service). In moderate density areas, like the typical suburb in the Greater Toronto Area (houses on small lots and some apartment buildings), the existence of dial a ride services is a sign of underfunding of a transit system and inevitably produces low ridership. Oakville (a suburb west of Toronto) used to replace regular bus routes with this sort of service on Sundays.
anonymouse December 12, 2011 at 5:15 pm #
The low capacity of flex routes is just an extreme case of a general tradeoff between capacity and flexibility. What makes metro systems so efficient at serving very high numbers of passengers is precisely their inflexibility, and the less flexibility you have, the higher you can push your capacity. Moscow, for example, doesn't have branched lines, because with 39 tph, it just doesn't work out, while London's Central Line has only 30 tph, and works more or less fine with branches at both ends.
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Q: This slow MySQL Query needs improvement This query works and provides me with the information I need, but it is very slow: it takes 18 seconds to agregate a database of only 4,000 records.
I'm bringing it here to see if anyone has any advice on how to improve it.
SELECT COUNT( status ) AS quantity, status
FROM log_table
WHERE time_stamp
IN (SELECT MAX( time_stamp ) FROM log_table GROUP BY userid )
GROUP BY status

Here's what it does/what it needs to do in plain text:
I have a table full of logs, each log contains a "userid", "status" (integer between 1-12) and "time_stamp" (a time stamp of when the log was created). There may be many entries for a particular userid, but with a different time stamp and status. I'm trying to get the most recent status (based on time_stamp) for each userid, then count the occurrences of each most-recent status among all the users.
My initial idea was to use a sub query with GROUP BY userid, that worked fast - but that always returned the first entry for each userid, not the most recent. If I could do GROUP BY userid using time_stamp DESC to Identify which row should be the representative for the group, that would be great. But of course ORDER BY inside of group does not work.
Any suggestions?

A: The first thing to try is to make this an explicit join:
SELECT COUNT(status) AS quantity, status
FROM log_table join
     (select lg.userid, MAX( time_stamp ) as maxts
      from log_table lg
      GROUP BY userid
     ) lgu
     on lgu.userid = lg.userid and lgu.maxts = lg.time_stamp
GROUP BY status;

Another approach is to use a different where clause.  This will work best if you have an index on log_table(userid, time_stamp).  This approach is doing the filtering by saying "there is no timestamp bigger than this one for a given user":
SELECT COUNT(status) AS quantity, status
FROM log_table
WHERE not exists (select 1
                  from log_table lg2
                  where lgu.userid = lg.userid  and lg2.time_stamp > lg.time_stamp
                 )
GROUP BY status;

redpajama_set_name: RedPajamaStackExchange


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