# Dispersive decay bound of small data solutions to Kawahara equation in a
finite time scale

Jongwon Lee  Department of Mathematical Sciences, Korea Advaced Institute of
Science and Technology, 291 Daehak-ro Yuseong-gu, Daejeon 34141, South Korea
<EMAIL_ADDRESS>
###### Abstract.

In this article, we prove that small localized data yield solutions to
Kawahara type equations which have linear dispersive decay on a finite time,
depending on the size of the initial data. We use the similar method used to
derive the dispersive decay bound of the solutions to the KdV equation, with
some steps being simpler. This result is expected to be the first result of
the small data global bounds of the fifth-order dispersive equations with
quadratic nonlinearity.

Keywords. Dispersive decay bound; Finite time scale; Kawahara equation;
Modified Kawahara equation

##  1\. Introduction

A global behavior of a small data solution to a nonlinear dispersive equation
has drawn attentions of many authors. It might result in the finite or
infinite time linear decay bound, or the linear or modified scattering. The
goal of this paper is to find a dispersive decay bound of a small data
solution to the Kawahara equation:

(KW) $u_{t}-u_{5x}=uu_{x}.$

We also consider the case of the modified Kawahara equation

(mKW) $u_{t}-u_{5x}=cu^{2}u_{x},$

with $c=\pm 1,$ which is the special case of the fifth-order KdV equation

(5KdV) $u_{t}-u_{5x}=c_{1}uu_{3x}+c_{2}u_{x}u_{2x}+c_{3}u^{2}u_{x}.$

The equation (KW) arises from the study of capillary waves on a shallow layer
and magneto-sound propagation in plasmas; see [19].

The local and global Cauchy problems of the fifth-order nonlinear dispersive
equations such as (KW), (mKW), (5KdV) have been extensively studied. Kenig,
Ponce, and Vega [21, 22] proved the local well-posedness (LWP) of the general
class of dispersive equations

$\partial_{t}u+\partial_{x}^{2j+1}u+P(u,\partial_{x}u,\cdots,\partial_{x}^{2j}u)=0$

where $P$ is a polynomial without constant or linear terms, on the space
$H^{s}(\mathbb{R})\cap L^{2}(|x|^{m}dx)$ with $s,m$ being a sufficiently large
positive integer. Kwon [24] proved LWP of (5KdV) in $H^{s}$ with $s>5/2$
without the weighted $L^{2}$ structure. Kenig and Pilod [20] proved the global
well-posedness (GWP) of the equations (5KdV) with $c_{2}=2c_{1}$ and (KW) in
$H^{s}$ with $s\geq 2.$ Very recently, Bringmann et al. [1] proved GWP in
$H^{-1}(\mathbb{R})$ of (5KdV) with $c_{1}=10,\;c_{2}=20,\;c_{3}=30$.
When it comes to (KW), Cui, Deng, and Tao [5] proved LWP in $H^{s}$ with
$s>-1$ and GWP in $L^{2}$. This result is later refined by Wang, Cui, and Deng
[29] down to the regularity $s\geq-\frac{7}{5}$ for LWP and $s>-\frac{1}{2}$
for GWP. Chen et al. [4] proved LWP of both (KW) and (mKW), with the
regularity being $s>-\frac{7}{4}$ for (KW) and $s\geq-\frac{1}{4}$ for (mKW).
Chen and Guo [3] extended the LWP result to $s\geq-\frac{7}{4}$ and also
proved GWP for the same regularity. Kato [18] proved LWP for $s\geq-2$ and GWP
for $s\geq-\frac{38}{21}.$ He also proved that the LWP threshold is optimal by
proving ill-posedness for $s<-2.$

Also, the small data global analysis of nonlinear dispersive equations has
been spotlighted for a long time. For instance, Deift and Zhou [6] proved the
small data asymptotics of mKdV first, taking advantage of the complete
integrability of the equation. Hayashi and Naumkin [11, 12, 13] proved the
similar result without using the complete integrability, which helped them
extend the result for nonlinearities with time-dependent coefficients.
However, their results required an initial data to have zero mean. [2]
introduces some classical scattering results for defocusing NLS, together with
asymptotic completeness. It also mentions that the one-dimensional cubic NLS

$iu_{t}+u_{xx}=\lambda|u|^{2}u,\quad\lambda=\pm 1$

does not show a linear scattering. Rather, the modified scattering results for
(1) and similar types of equations are proved by many authors with different
settings. Ifrim and Tataru [16] proved the global dynamics of (1) by testing
the wave packets. Adopting this idea of Ifrim and Tataru, Harrop-Griffiths
[10] proved the modified scattering and asymptotic completeness results for
the mKdV equation. The work of Harrop-Griffiths is meaningful in that it
requires neither the integrability of the equation, nor the zero mean of the
initial datum. Okamoto [27] later proved the global behavior of solutions to
the fifth-order mKdV type equation, without proving asymptotic completeness,
using the same idea as above. Okamoto later proved [26] the asymptotic
behavior of solutions to the higher order KdV-type equation with critical
nonliearity. Here the nonlinearity is said to be critical in the sense that
the small data solution shows the modified scattering behavior at that order
of nonlinearity.

Unlike the mKdV-type equations, the global dynamics of KdV-type equations have
not been studied so much. The main difficulty of the global analysis of KdV-
type equations is that the nonlinearities are less perturbative than those of
the equations on the mKdV hierarchy; the nonlinearities of mKdV-type equations
are cubic, whereas those of KdV-type equations are quadratic. The first result
regarding the nonlinear dispersive equation with a quadratic nonlinearity is
due to [17], which discovered that the solution to the Benjamin-Ono equation

$u_{t}+Hu_{xx}=uu_{x}$

has linear dispersive decay bound in an almost global time scale $|t|\sim
e^{\frac{c}{\epsilon}}$ where $\epsilon$ is the size of an initial datum. A
global result for KdV equation is discovered fairly recently, by Ifrim et al.,
[15] which stated that the linear dispersive decay bound may break down beyond
the quartic time scale $|t|\sim\epsilon^{-3}$ where $\epsilon$ is the size of
an initial datum. They proved that the result is optimal, i.e. the time scale
cannot be extended further, using the inverse scattering theory. No other
meaningful results are known about the global dynamics of KdV-type equations.
The main goal of this paper is to show the first result regarding the global
behavior of the fifth-order dispersive equations with quadratic
nonlinearities, whose byproduct is the small data global behavior of (mKW).

###  1.1. Main result

First we state the main result of this paper:

######  Theorem 1.1.

Consider the Cauchy problem

$\begin{cases}u_{t}-u_{5x}=\pm u^{m}u_{x},\\\ u(0,x)=u_{0}(x),\end{cases}\quad
m=1,2.$

Suppose that the initial datum $u_{0}$ satisfies

$\|u_{0}\|_{\dot{B}^{-\frac{1}{2}}_{2,\infty}}+\|xu_{0}\|_{\dot{H}^{\frac{1}{2}}}\leq\epsilon\ll
1,$

then we have the following pointwise bound

$|\partial_{x}^{k}u(t,x)|\lesssim\epsilon t^{-\frac{1}{8}-\frac{k}{4}}\langle
x\rangle^{-\frac{3}{8}+\frac{k}{4}},\quad k=0,1,2,3$

provided $|t|\ll\epsilon^{-\frac{5}{5-2m}}.$
Moreover, in the elliptic region $\\{x\gg t^{\frac{1}{5}}\\},$ we have a
better bound

$|\partial_{x}^{k}u(t,x)|\lesssim\epsilon t^{-\frac{k}{4}}\langle
x\rangle^{-1+\frac{k}{4}}\log(t^{-\frac{1}{5}}\langle x\rangle),\quad
k=0,1,2,3$

when $|t|\ll\epsilon^{-\frac{5}{5-2m}}.$

######  Remark 1.2.

Note that the case $m=1$ corresponds to (KW), and $m=2$ to (mKW).

######  Remark 1.3.

The time scale in (KW) is worse than that in the KdV equation, which is even
worse than that in the Benjamin-Ono equation. This can be deduced
heuristically as follows: The group velocity of the fifth-order linear
equation is given as $\omega(\xi)=-5\xi^{4}$, so the high frequency part of
the solution travels much faster towards to the negative $x$-axis than in the
second or third-order equations. However, the possible emergence of nonlinear
“bulk”, such as solitons and dispersive shocks, inturrupts such linear
dispersive decay. The faster the linear part travels, the earlier the
nonlinear bulk emerges.
The time scale in (mKW) is better than that of the KdV and (KW). This is
because the nonlinearity becomes cubic, making it more perturbative. However,
such time scale is worse than that of the mKdV equation, as the mKdV equation
has modified scattering and global dispersive decay bound; see [10]. Such
discrepancy arises from the same heuristics used in the comparision between
(KW), KdV, and Benjamin-Ono equations. One may also compare this result with
[27] or [26], in which case the nonlinearities become more perturbative than
(mKW).

######  Remark 1.4.

The two norms used in the main result are inspired from [15]. As seen in the
later analysis, the $\dot{B}^{-\frac{1}{2}}_{2,\infty}$ norm helps properly
control the low frequency, and the growth of the high frequency norms is slow
enough that it does not disrupt our analysis. The $\dot{H}^{\frac{1}{2}}$ norm
exactly does the opposite. It does the control of the high frequency, and
guarantees the mild growth of the low frequency part.

##  2\. Preliminaries

###  2.1. Definitions and Notations

Let $1\leq p<\infty.$ $L^{p}(X)$ be the set of measurable function on $X$ with
the norm $\|f\|_{L^{p}(X)}=\left(\int_{X}|f|^{p}d\mu\right)^{1/p}$ on the
measure space $(X,\mu).$

If $p=\infty,$ then $\|f\|_{L^{\infty}(X)}=\inf\\{M>0:|f(x)|\leq
M\;\;\text{a.e.}\\}$. If $X=\mathbb{R}$ and $\mu$ is the Lebesgue measure,
then we write $L^{p}(X)=L^{p}$ for simplicity, in either case $p<\infty$ or
$p=\infty.$

We use the time-dependent Japanese bracket $\langle
x\rangle=(x^{2}+t^{\frac{2}{5}})^{\frac{1}{2}}.$ In some of the following
sections, we adapt the time scale to $t=1$ by a proper scaling, and in this
case $\langle x\rangle=(x^{2}+1)^{\frac{1}{2}}.$

We define the spatial Fourier transform of $f\in\mathcal{S}(\mathbb{R})$, the
space of all Schwarz functions on $\mathbb{R}$, by

$\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(x)e^{-ix\xi}dx,$

and its inverse transform by

${f}^{\vee}(x)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(\xi)e^{ix\xi}d\xi.$

Both transforms can be extended to $\mathcal{S}^{\prime}(\mathbb{R})$, the
space of tempered distributions, by duality.

Let
$u_{kx}(x):=\partial_{x}^{k}u(x)=\left((i\xi)^{k}\hat{u}(\xi)\right)^{\vee}$
be the $k$-th order partial derivative of $u$ with respect to the spatial
variable $x$. Also let $|D|^{s}$ be the homogeneous fractional derivative of
order $s\in\mathbb{R}$ whose symbol is $|\xi|^{s}$. Note that
$|D|^{s}=H\partial_{x}$, where $H$ is the Hilbert transform defined by the
Fourier symbol $-i\mathrm{sgn}(\xi).$

Let $A,B$ be two nonnegative quantities. If there exists $C>0$ such that
$A\leq CB,$ then we denote by $A\lesssim B.$ If $A$ need not be nonnegative
and $|A|\lesssim B,$ then we denote $A=O(B).$ If $A\lesssim B$ and $B\lesssim
A,$ then we denote $A\sim B.$ If the implied constant $C$ depends on some
parameters $a,b,c,\cdots,$ then we denote $A\lesssim_{a,b,c,\cdots}B,$
$A=O_{a,b,c,\cdots}(B),$ $A\sim_{a,b,c,\cdots}B$ respectively.

Let $\Psi\in\mathcal{S}(\mathbb{R})$ with $\psi=\widehat{\Psi}$ is even,
supported on $[-2,2]$, and $\psi=1$ on $[-1,1].$ Also let $\psi_{\leq
N}(\xi)=\psi(\frac{\xi}{N})$ and $\Psi_{\leq N}=(\psi_{\leq N})^{\vee}$ for
$N>0.$ Then define the Littlewood-Paley projection of
$f\in\mathcal{S}^{\prime}(\mathbb{R})$ by $P_{\leq N}f(x)=(\Psi_{\leq N}\ast
f)(x),$ $P_{N}=P_{\leq N}-P_{\leq N/2}.$ Also define $P_{<N}:=P_{\leq
N}-P_{N}$, $P_{\geq N}=1-P_{<N},$ $P_{>N}=1-P_{\leq N}.$ By a slight abuse of
notation, we denote a Fourier multiplier whose symbol is supported on
$\\{|\xi|\sim N\\}$ by $P_{N}.$ In particular, if $N=2^{j}\in 2^{\mathbb{Z}},$
then we denote $P_{N}:=P_{j}$ and similarly for $P_{\leq N},$ $P_{<N},$
$P_{>N},$ $P_{\geq N}.$

Now define Sobolev and Besov spaces. For $s\in\mathbb{R}$ and $1<p<\infty$,
the inhomogeneous Sobolev space $W^{s,p}$ and its homogeneous counterpart
$\dot{W}^{s,p}$ are defined as the subspace of
$\mathcal{S}^{\prime}(\mathbb{R})$ under the norm

$\|f\|_{W^{s,p}}=\left\|\left((1+\xi^{2})^{s/2}\hat{f}\right)^{\vee}\right\|_{L^{p}},\quad\|f\|_{\dot{W}^{s,p}}=\left\|\left(|\xi|^{s}\hat{f}\right)^{\vee}\right\|_{L^{p}},$

respectively. Recall the Littlewood-Paley inequality

$\|f\|_{W^{s,p}}\sim_{s,p}\|P_{\leq 0}f\|_{L^{p}}+\bigg{\|}\big{(}\sum_{j\geq
1}2^{2js}|P_{j}f|^{2}\big{)}^{1/2}\bigg{\|}_{L^{p}}$

and its homogeneous counterpart

$\|f\|_{\dot{W}^{s,p}}\sim_{s,p}\bigg{\|}\big{(}\sum_{j\in\mathbb{Z}}2^{2js}|P_{j}f|^{2}\big{)}^{1/2}\bigg{\|}_{L^{p}}.$

Motivated by the above Littlewood-Paley inequalities, we define the
inhomogeneous and homogeneous Besov spaces, denoted as $B_{s,q}^{p}$ and
$\dot{B}_{s,q}^{p}$, respectively, is defined as the subspace of
$\mathcal{S}^{\prime}(\mathbb{R})$ under the norm

$\|f\|_{B_{p,q}^{s}}=\big{(}\|P_{\leq 1}f\|_{L^{p}}^{q}+\sum_{j\geq
1}2^{qjs}\|P_{j}f\|_{L^{p}}^{q}\big{)}^{1/q},$
$\|f\|_{\dot{B}_{p,q}^{s}}=\big{(}\sum_{j\in\mathbb{Z}}2^{qjs}\|P_{j}f\|_{L^{p}}^{q}\big{)}^{1/q},$

with $1\leq q<\infty$ and the natural modification for $q=\infty.$

###  2.2. Useful theorems

In this subsection, some of the theorems are presented for the later use. I
omit the proofs here, since the proofs can be found in many literatures.

######  Theorem 2.1 (Bernstein’s inequality).

Let $1\leq p\leq q\leq\infty$ and $s\in\mathbb{R}.$ Then one has

(2.1)
$\displaystyle\|P_{N}f\|_{L^{q}}\lesssim_{p,q}N^{\frac{1}{p}-\frac{1}{q}}\|P_{N}f\|_{L^{p}},$
(2.2) $\displaystyle\||D|^{s}P_{N}f\|_{L^{p}}\lesssim
N^{s}\|P_{N}f\|_{L^{p}}.$

(2.1) still holds when $P_{N}$ is replaced by $P_{\leq N}.$ Also, if $s>0,$
then (2.2) holds when $P_{N}$ is replaced by $P_{\leq N},$ and if $s<0,$ then
(2.2) holds when $P_{N}$ is replaced by $P_{\geq N}.$

In particular, a notable case of Bernstein’s inequality is (2.1) with
$q=\infty$ and $p=2$, namely $\|P_{N}f\|_{L^{\infty}}\lesssim
N^{\frac{1}{2}}\|P_{N}f\|_{L^{2}}.$
Also we have the Gagliardo-Nirenberg inequality, a refined version of the
Sobolev embedding:

######  Theorem 2.2 (Gagliardo-Nirenberg inequality).

Let $1<p<q\leq\infty$ and $s>0$ be such that

$\frac{1}{q}=\frac{1}{p}-\frac{\theta s}{d}$

for some $0<\theta<1.$ Then for any $u\in W^{s,p}(\mathbb{R}^{d})$ we have

$\|u\|_{L^{q}(\mathbb{R}^{d})}\lesssim_{d,p,q,s}\|u\|_{L^{p}(\mathbb{R}^{d})}^{1-\theta}\|u\|_{\dot{W}^{s,p}(\mathbb{R}^{d})}^{\theta}.$

For the proof, see for example [28].

Throughout the paper, a special case of Theorem 2.2, namely

$\|u\|_{L^{\infty}(\mathbb{R})}\lesssim\|u\|_{L^{2}(\mathbb{R})}^{1/2}\|u\|_{\dot{H}^{1}(\mathbb{R})}^{1/2},$

will be used frequently.

Let $M$ be the one-dimensional Hardy-Littlewood maximal operator defined as

$Mf(x)=\sup_{r>0}\frac{1}{2r}\int_{x-r}^{x+r}|f(y)|dy.$

Then we have following pointwise bound by [9] :

######  Lemma 2.3.

Let $f$ be a function whose Fourier support is in the annulus $|\xi|\sim R.$
Then for any $x\in\mathbb{R}$ and $s\in\mathbb{R},$

$\left||D|^{s}f(x)\right|\lesssim_{s}R^{s}Mf(x).$

Moreover, if $f$ is not necessarily frequency-localized, then

$\left||D|^{s}P_{N}f(x)\right|\lesssim N^{s}Mf(x).$

In both cases, if $s$ is an integer, then replacing $|D|$ by $\partial_{x}$ is
acceptable. Also, if $s\geq 0$, then the annulus $|\xi|\sim R$ can be replaced
by the disk $|\xi|\lesssim R.$

######  Remark 2.4.

The above lemma implies that a polynomially decaying pointwise bound is stable
under the frequency restriction, since if $g_{s}(x)=\langle x\rangle^{-s}$ for
$s\geq 0,$ then $Mg_{s}(x)\lesssim_{s}g_{s}(x).$ Thus, we can apply the
bootstrap bounds of $u$ appearing in the later sections freely to the
frequency-localized pieces of $u$.

######  Lemma 2.5 (Interpolating pointwise bounds).

Let $0<s<1$. Then one has

$\left||D|^{s}u(x)\right|\lesssim(Mu(x))^{1-s}(M\partial_{x}u(x))^{s}.$

###### Proof.

This is a special case of the theorem in Section 12.3.2 of [25]. ∎

Note that, combining Lemma 2.5 and Remark 2.4, if $|u(x)|\leq A\langle
x\rangle^{-\alpha}$ and $|\partial_{x}u(x)|\leq\langle x\rangle^{-\beta}$ for
some $\alpha,\beta\geq 0$ and $A,B>0$, then
$||D|^{\frac{1}{2}}u(x)|\lesssim\sqrt{AB}\langle
x\rangle^{-\frac{\alpha+\beta}{2}}$. This kind of estimate will be used in the
proof of Proposition 4.1.

##  3\. Linear Analysis

First we consider the decay estimate of the solution of linear fifth-order KdV
equation

(Lin) $u_{t}-u_{5x}=0.$

We introduce the time-dependent linear operator $L(t):=x+5t\partial_{x}^{4},$
so that if $u$ solves (Lin) with initial datum $u_{0}$ then $Lu$ also solves
the same equation with initial datum $xu_{0}.$
Now we state the basic decay bound for the solution of (Lin), which is given
as below:

######  Proposition 3.1.

Let $u$ be a solution of (Lin) with initial datum $u_{0}\in L^{1}$. Then $u$
satisfies the uniform decay bound

$\|u(t)\|_{L^{\infty}}\lesssim t^{-1/5}\|u_{0}\|_{L^{1}}.$

###### Proof.

Solving the equation via spatial Fourier transform gives

$u(t,x)=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\xi^{5}t+i\xi(x-y)}u_{0}(y)dyd\xi.$

We change coordinates $\xi\mapsto t^{-1/5}\eta$ so that

$u(t,x)=t^{-1/5}\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\eta^{5}+i\eta
t^{-1/5}(x-y)}u_{0}(y)dyd\eta=t^{-1/5}A(t^{-1/5}x)\ast u_{0}(x),$

where $A(x)$ is an oscillatory integral

$A(x):=\int_{\mathbb{R}}e^{i\eta^{5}+i\eta x}d\eta.$

Then $A$ is bounded by standard stationary phase argument. Now the result
follows from Young’s inequality. ∎

To obtain a more refined version of the above estimate, we need the following
lemma due to [7] and [8]:

######  Lemma 3.2.

$A(x)$ in the proof of Proposition 3.1 satisfy the following asymptotic bound:

$A(x)\lesssim\langle x\rangle^{-3/8}e^{-Cx_{+}^{5/4}},$

where $C$ is an appropriate positive constant.

From the above lemma, it follows that solutions with initial datum $u_{0}$
satisfying $\|u_{0}\|_{L^{1}}\leq 1$ and $\mathrm{supp}u_{0}$ bounded has the
following decay bound for $t\gtrsim 1:$

$|u(t,x)|\lesssim t^{-1/8}\langle
x\rangle^{-3/8}e^{-Cx_{+}^{5/4}t^{-1/4}},\quad|u_{x}(t,x)|\lesssim
t^{-3/8}\langle x\rangle^{-1/8}e^{-C^{\prime}x_{+}^{5/4}t^{-1/4}}$

Now we aim to relax the compact support assumption to a decay estimate, in the
similar way with [15]. Our goal of this section is to prove the following
proposition:

######  Proposition 3.3.

Let $t>0$. Assume that a function $u$ satisfies:

(3.1) $\|u\|_{\dot{B}_{2,\infty}^{-1/2}}+\|L(t)u\|_{\dot{H}^{1/2}}\leq 1.$

Then it also satisfies the bound

(3.2) $t^{\frac{1}{8}+\frac{k}{4}}\langle
x\rangle^{\frac{3}{8}-\frac{k}{4}}|\partial_{x}^{k}u|\lesssim 1$

for $k=0,1,2,3.$ Moreover, in the elliptic region $E=\\{x\gg
t^{\frac{1}{5}}\\}$ we have the better bound

(3.3) $t^{\frac{k}{4}}\langle
x\rangle^{1-\frac{k}{4}}|\partial_{x}^{k}u(x)|\lesssim\log(t^{-\frac{1}{5}}\langle
x\rangle).$

This is an analogous result of the Lemma 2.2 in [15]. The only difference with
it is the linear operator $L(t),$ where $L(t)=x-3t\partial_{x}^{2}$ in [15]
and $L(t)=x+5t\partial_{x}^{4}$ here. The steps of the proof will basically
follow the idea of [15]. First we may rescale to $t=1$ since
$\|u\|_{\dot{B}_{2,\infty}^{-1/2}}+\|L(t)u\|_{\dot{H}^{1/2}}$ is invariant
under the scaling $u(\lambda^{5},x)\mapsto\lambda u(\lambda^{5},\lambda x).$
Also we will split the real line into the self-similar region
$S=\\{|x|\lesssim 1\\}$ (would be $=\\{|x|\lesssim t^{1/5}\\}$ in general),
the elliptic region $E=\\{x\gg 1\\}$ (would be $=\\{x\gg t^{1/5}\\}$ in
general), and the hyperbolic region $H=\\{-x\gg 1\\}.$ Also we split the
elliptic and hyperbolic regions into dyadic components, namely
$A_{R}=\\{\langle x\rangle\sim R\gg 1\\}$ and $A_{1}=\\{\langle
x\rangle\lesssim 1\\}=S$, $A_{R}^{H}=A_{R}\cap H$, $A_{R}^{E}=A_{R}\cap E$ for
$R\gg 1.$ Therefore the proof is reduced to showing that

(3.4) $\|\partial_{x}^{k}u\|_{L^{\infty}(A_{R})}\lesssim
R^{-\frac{3}{8}+\frac{k}{4}}$

for each $R\geq 1$ and $k=0,1,2,3$ (and the corresponding counterpart for the
elliptic region).

###  3.1. Localized bounds of a linear solution

Now we present the following low frequency bound, which is the same as in
[15]:

######  Lemma 3.4.

If (3.1) holds, we have

(3.5) $\|Lu\|_{L^{2}(A_{R})}\lesssim R^{1/2}.$

######  Remark 3.5.

The reason why we call (3.5) as a low frequency bound is that the suppressed
low frequency factors in $\dot{H}^{\frac{1}{2}}$ norm become dominant in the
$L^{2}$ norm. Such low frequency bound turns out to be worse than the high
frequency bound, but it is still acceptable thanks to the good low frequency
bound of $u$ due to the $\dot{B}^{-\frac{1}{2}}_{2,\infty}$ norm.

###### Proof.

The proof is almost identical to the case of [15], but I present the details
here to clarify the ideas.

First we split $u$ at some frequency cut-off $M$, where $M$ is to be chosen
later:

$u=u_{<M}+u_{\geq M}.$

Then the Besov bound from (3.1) tells that
$\|u_{M}\|_{L^{2}(\mathbb{R})}\lesssim M^{1/2}.$ Hence

$\|u_{<M}\|_{L^{2}(\mathbb{R})}\leq\sum_{N<M}\|u_{N}\|_{L^{2}(\mathbb{R})}\lesssim\sum_{N<M}N^{1/2}\sim
M^{1/2}$

where the sum runs over all dyadic numbers $<M.$ Therefore one has

(3.6)
$\|Lu_{<M}\|_{L^{2}(A_{R})}\leq\|xu_{<M}\|_{L^{2}(A_{R})}+\|\partial_{x}^{4}u_{<M}\|_{L^{2}(\mathbb{R})}\lesssim
RM^{1/2}+M^{9/2}.$

On the other hand,

$Lu_{\geq M}=P_{\geq M}Lu+[L,P_{\geq M}]u=P_{\geq M}Lu+[x,P_{\geq M}]u,$

where the last equality follows since $[\partial_{x}^{4},P_{\geq M}]=0$ (note
that any two Fourier multipliers commute). Now we can estimate

$\|P_{\geq
M}Lu\|_{L^{2}(A_{R})}\leq\|(1-\psi(\xi/M))\widehat{Lu}\|_{L^{2}(\mathbb{R})}\lesssim
M^{-1/2}\|Lu\|_{\dot{H}^{1/2}(\mathbb{R})}\leq M^{-1/2},$

with $\psi$ being a bump function supported in $|\xi|\lesssim 1.$ Also we have

$\|[x,P_{\geq
M}]u\|_{L^{2}(A_{R})}\leq\|M^{-1}\psi^{\prime}(\xi/M)\hat{u}\|_{L^{2}(\mathbb{R})}\lesssim
M^{-1/2}$

by (3.1). This gives

(3.7) $\|Lu_{\geq M}\|_{L^{2}(A_{R})}\lesssim M^{-1/2}.$

Now we are going to optimize the bound, compromising between (3.6) and (3.7)
by choosing the appropriate $M$. Note that the bound in (3.6) becomes smaller
if $M$ is small and the same holds for the bound in (3.7) if $M$ is large.
Hence the bound is optimized when the two bounds are comparable. If
$RM^{1/2}\sim M^{-1/2},$ then $M\sim R^{-1},$ and the resulting bound becomes
$R^{1/2}.$ Also if $M^{9/2}\sim M^{-1/2}$, then $M\sim 1,$ so the resulting
bound becomes $R$, and the former is better, so we choose $M=R^{-1}.$ Hence
(3.5) follows. ∎

######  Remark 3.6.

The above argument works in both cases $R=1$ and $R>1.$

######  Lemma 3.7.

If (3.1) holds, we have

(3.8) $\|\partial_{x}^{k}u\|_{L^{2}(A_{R})}\lesssim
R^{\frac{1}{8}+\frac{k}{4}},\quad k=0,1,2,3,4.$

######  Remark 3.8.

In (3.8), one more derivative corresponds to the additional $R^{\frac{1}{4}}$
factor. This can be deduced heuristically from the fact that the fifth-order
Airy function

$A(t,x)=\int_{\mathbb{R}}e^{itx+i\xi^{5}t}d\xi$

is concentrated near $(|x|/t)^{\frac{1}{4}}$ with $x<0,$ as the principle of
stationary phase implies.

###### Proof.

Again we split $u=u_{<M}+\sum_{\lambda\geq M}u_{\lambda}$ and choose $M$ later
as before. The bound on $u_{<M}$ follows by:

(3.9) $\|\partial_{x}^{k}u_{<M}\|_{L^{2}(A_{R})}\lesssim
M^{k}\|u_{<M}\|_{L^{2}(\mathbb{R})}\lesssim M^{k+\frac{1}{2}}$

where the last inequality follows from (3.1). Now consider the high frequency
bound. The bound of $Lu_{\lambda}$ can be computed in the same way with
$Lu_{\geq M}$ in Lemma 3.4, and the bound of $u_{\lambda}$ directly follows
from (3.1) so that

(3.10a) $\|Lu_{\lambda}\|_{L^{2}}\lesssim\lambda^{-1/2},$ (3.10b)
$\|u_{\lambda}\|_{L^{2}}\lesssim\lambda^{1/2},$ (3.10c)
$\|u_{\lambda,x}\|_{L^{2}}\lesssim\lambda^{3/2}.$

Now integrating by parts gives

$\displaystyle\int\chi_{R}u_{\lambda,2x}^{2}dx=\frac{1}{5}\int\chi_{R}u_{\lambda}Lu_{\lambda}-\frac{1}{5}\int\chi_{R}xu_{\lambda}^{2}+\int(\chi_{R}^{\prime\prime}u_{\lambda,2x}+2\chi_{R}^{\prime}u_{\lambda,3x})u_{\lambda}.$

The first integral can be estimated via Cauchy-Schwarz inequality, which
yields the bound 1 by (3.10a) and (3.10b), and the bound of the second
integral follows directly from (3.10b), which yields $R\lambda.$ To bound the
third integral, we integrate by parts again to obtain:

$\displaystyle\int(\chi_{R}^{\prime\prime}u_{\lambda,2x}+2\chi_{R}^{\prime}u_{\lambda,3x})u_{\lambda}=-\frac{1}{2}\int\chi_{R}^{(4)}u_{\lambda}^{2}+2\int\chi_{R}^{\prime\prime}u_{\lambda,x}^{2}.$

Hence the third integral is bounded by $R^{-4}\lambda+R^{-2}\lambda^{3}$ due
to (3.10b), (3.10c), and the fact
$\|\partial_{x}^{k}\chi_{R}\|_{L^{\infty}}=O_{k,\chi}(R^{-k}).$ This gives the
preliminary bound

(3.11) $\|u_{\lambda,2x}\|_{L^{2}(A_{R})}\lesssim
1+R^{1/2}\lambda^{1/2}+R^{-1}\lambda^{3/2}:=N.$

Repeating the similar steps for the integral
$\int\chi_{R}u_{\lambda,3x}^{2}dx$ one gets the similar bound:

(3.12) $\|u_{\lambda,3x}\|_{L^{2}(A_{R})}\lesssim
R^{-1}N+\lambda^{1/4}N^{1/2}+R^{1/2}\lambda^{1/4}N^{1/2}:=N^{\prime}.$

Now we are going to refine (3.10b) and (3.10c) using (3.11) and (3.12) so that
the bound becomes summable with respect to $\lambda.$ To do so we decompose

$\chi_{R}u_{\lambda}=\partial_{x,\lambda}^{-2}(\chi_{R}u_{\lambda,2x})-[\partial_{x,\lambda}^{-2},\chi_{R}]u_{\lambda,2x},$

where the antiderivative $\partial_{x,\lambda}^{-2}$ is localized at frequency
$\lambda,$ namely having the symbol $-\xi^{-2}\varphi(\xi/\lambda)$ with
$\varphi$ being a bump function supported on the annulus $|\xi|\sim 1.$ Let
$\psi$ be the Fourier inversion of $-\xi^{-2}\varphi(\xi),$ so that
$\psi\in\mathcal{S}(\mathbb{R}).$ Then the inversion of
$-\xi^{-2}\varphi(\xi/\lambda)$ becomes $\lambda^{-1}\psi(\lambda x).$ Then
one has

$\displaystyle\|[\partial_{x,\lambda}^{-2},\chi_{R}]u_{\lambda,2x}\|_{L^{2}(\mathbb{R})}$
$\displaystyle=\lambda^{-1}\left\|\int\psi(\lambda(x-y))(\chi_{R}(y)-\chi_{R}(x))u_{\lambda,2x}(y)dy\right\|_{L^{2}_{x}(\mathbb{R})}$
$\displaystyle\lesssim\lambda^{-1}\||\psi(\lambda\cdot)|\ast|u_{\lambda,2x}|\|_{L^{2}_{x}}\leq\lambda^{-2}\|\psi\|_{L_{x}^{1}}\|u_{\lambda,2x}\|_{L_{x}^{2}}$
$\displaystyle\lesssim\lambda^{-2}N.$

Also

$\|\partial_{x,\lambda}^{-2}(\chi_{R}u_{\lambda,2x})\|_{L^{2}(\mathbb{R})}\lesssim\lambda^{-2}\|u_{\lambda,2x}\|_{L^{2}(A_{R})}\lesssim\lambda^{-2}N.$

This yields a local bound for $u_{\lambda}$:

(3.13) $\|u_{\lambda}\|_{L^{2}(A_{R})}\lesssim\lambda^{-2}N.$

Now summing up (3.13) on the frequency range $\lambda\geq M$ gives

(3.14) $\|u_{\geq M}\|_{L^{2}(A_{R})}\lesssim
M^{-2}+R^{1/2}M^{-3/2}+R^{-1}M^{-1/2}.$

Repeating the same argument above to the decomposition

(3.15)
$\chi_{R}u_{\lambda,x}=\partial^{-2}_{x,\lambda}(\chi_{R}u_{\lambda,3x})-[\partial^{-2}_{x,\lambda},\chi_{R}]u_{\lambda,3x}$

we also get the refined local bound for $u_{\lambda,x}$:

(3.16) $\|u_{\lambda,x}\|_{L^{2}(A_{R})}\lesssim\lambda^{-2}N^{\prime},$

so that

(3.17) $\|\partial_{x}u_{\geq M}\|_{L^{2}(A_{R})}\lesssim
R^{-1}M^{-2}+R^{-2}M^{-\frac{1}{2}}+R^{\frac{1}{2}}M^{-\frac{7}{4}}+R^{\frac{3}{4}}M^{-\frac{3}{2}}+M^{-1}.$

Now we want to optimize the bounds (3.9), (3.14), (3.17) by making them
comparable. It turns out that $M=R^{1/4}$ is the case with the best bound, so
that $M^{-2}N\sim R^{1/8}.$ This gives

(3.18a) $\|u\|_{L^{2}(A_{R})}\lesssim R^{1/8},$ (3.18b)
$\quad\|u_{x}\|_{L^{2}(A_{R})}\lesssim R^{3/8},$ (3.18c)
$\quad\|u_{2x}\|_{L^{2}(A_{R})}\lesssim R^{5/8}.$

Here the bound (3.18c) follows from integration by parts below with (3.5),
(3.18a), and (3.18b):

$\displaystyle\int\chi_{R}u_{2x}^{2}dx=\frac{1}{5}\int\chi_{R}uLu-\frac{1}{5}\int\chi_{R}xu^{2}+\int(\chi_{R}^{\prime\prime}u_{2x}+2\chi_{R}^{\prime}u_{3x})u.$

Using the bound (3.5) and (3.18a) easily gives that

(3.19) $\|u_{4x}\|_{L^{2}(A_{R})}\lesssim R^{9/8}.$

Finally, it remains to find the bound of $\|u_{3x}\|_{L^{2}(A_{R})}.$ This can
be done easily by integration by parts:

$\int\chi_{R}u_{3x}^{2}=-\int\chi_{R}^{\prime}u_{3x}u_{2x}-\int\chi_{R}u_{4x}u_{2x}=\frac{1}{2}\int\chi_{R}^{\prime\prime}u_{2x}^{2}-\int\chi_{R}u_{4x}u_{2x}$

so the bound follows from (3.18c) and (3.19). ∎

###  3.2. Pointwise estimation on each region

Let $v:=\chi_{R}u.$ For notational convenience, we are going to restrict the
norm to the region $A_{R},$ so we will write $L^{2}(A_{R})$ simply as $L^{2}.$
Then $v$ solves an equation of the form

$(x+5\partial_{x}^{4})v=f$

in $A_{R},$ where we control

(3.20) $\|\partial_{x}^{k}v\|_{L^{2}}\lesssim
R^{\frac{1}{8}+\frac{k}{4}},\quad 0\leq k\leq 4$

by (3.8), and

(3.21) $\|f\|_{\dot{H}^{1/2}}\lesssim 1,\quad\|f\|_{L^{2}}\lesssim
R^{\frac{1}{2}}$

which follows by (3.1), (3.5).

Now we turn to the pointwise estimate in each region. In the case of self-
similar region, the result follows directly from Sobolev embedding (note that
in this region just $\lesssim 1$ bound is enough, since $R=1$). Hence we
concentrate on elliptic and hyperbolic regions.

First we see the elliptic region, namely $x\sim R,$ which seems to be easier
to deal with than the hyperbolic region. Split $f=f_{lo}+f_{hi},$ where the
frequency scale is $R^{1/4}.$ Here the operator $L=x+5\partial_{x}^{4}$ is
elliptic, so the leading part will then be $x^{-1}f_{lo}.$ Then let
$v_{1}:=v-x^{-1}f_{lo}$ be the remainder, which solves

$Lv_{1}=f_{1}:=f_{hi}-5\partial_{x}^{4}(x^{-1}f_{lo}).$

By (2) one has

$\|f_{hi}\|_{L^{2}(\mathbb{R})}\lesssim R^{-1/8}\|f\|_{\dot{H}^{1/2}}\lesssim
R^{-1/8},$
$\displaystyle\|\partial_{x}^{4}(x^{-1}f_{lo})\|_{L^{2}(\mathbb{R})}\lesssim$
$\displaystyle
R^{-5}\|f_{lo}\|_{L^{2}(\mathbb{R})}+R^{-4}\|\partial_{x}f_{lo}\|_{L^{2}(\mathbb{R})}+R^{-3}\|\partial_{x}^{2}f_{lo}\|_{L^{2}(\mathbb{R})}$
$\displaystyle+$ $\displaystyle
R^{-2}\|\partial_{x}^{3}f_{lo}\|_{L^{2}(\mathbb{R})}+R^{-1}\|\partial_{x}^{4}f_{lo}\|_{L^{2}(\mathbb{R})}$
$\displaystyle\lesssim$ $\displaystyle
R^{-9/2}+R^{-31/8}+R^{-21/8}+R^{-11/8}+R^{-1/8}$ $\displaystyle\lesssim$
$\displaystyle R^{-1/8},$

where for $k\geq 1$,

$\|\partial_{x}^{k}f_{lo}\|_{L^{2}(\mathbb{R})}\lesssim
R^{\frac{k}{4}-\frac{1}{8}}\|f_{lo}\|_{\dot{H}^{1/2}}\lesssim
R^{\frac{k}{4}-\frac{1}{8}},$

which yield

$\|f_{1}\|_{L^{2}(\mathbb{R})}\lesssim R^{\frac{k}{4}-\frac{1}{8}}.$

Hence we may integrate by parts in the following identity

$\int_{\mathbb{R}}v_{1}Lv_{1}dx=\int_{\mathbb{R}}f_{1}v_{1}dx$

and arrive at

$\int_{\mathbb{R}}x|v_{1}|^{2}dx+5\int_{\mathbb{R}}|\partial_{x}^{2}v_{1}|^{2}dx=\int_{\mathbb{R}}f_{1}v_{1}dx.$

Then by Cauchy-Schwarz inequality,

$R\|v_{1}\|_{L^{2}(\mathbb{R})}^{2}+5\|\partial_{x}^{2}v_{1}\|_{L^{2}(\mathbb{R})}^{2}\lesssim\|f_{1}\|_{L^{2}(\mathbb{R})}\|v_{1}\|_{L^{2}(\mathbb{R})}\leq\delta
R\|v_{1}\|_{L^{2}(\mathbb{R})}^{2}+\frac{4}{\delta
R}\|f_{1}\|_{L^{2}(\mathbb{R})}^{2}$

for any $\delta>0,$ so letting $\delta>0$ sufficiently small gives

$R\|v_{1}\|_{L^{2}(\mathbb{R})}^{2}+5\|\partial_{x}^{2}v_{1}\|_{L^{2}(\mathbb{R})}^{2}\lesssim
R^{-1}\|f_{1}\|_{L^{2}(\mathbb{R})}^{2}.$

Thus, using the bound on $f_{1},$ we arrive at

$\|\partial_{x}^{k}v_{1}\|_{L^{2}(\mathbb{R})}\lesssim
R^{-\frac{9}{8}+\frac{k}{4}},\quad k=0,2,$

and using the $v_{1}$ equation shows that this bound also holds for $k=4,$ and
by interpolation it still holds for $k=1,3.$ To sum up, we have

$\|\partial_{x}^{k}v_{1}\|_{L^{2}(\mathbb{R})}\lesssim
R^{-\frac{9}{8}+\frac{k}{4}},\quad k=0,1,2,3,4.$

Finally, by Gagliardo-Nirenberg inequality, one has

$|\partial_{x}^{k}v_{1}|\lesssim R^{-1+\frac{k}{4}},\quad k=0,1,2,3,$

which is exactly as needed.

Now we need to get the bound of $x^{-1}f_{lo}.$ Proceeding as above, one has

$\|\partial_{x}^{k}(x^{-1}f_{lo})\|_{L^{2}(\mathbb{R})}\lesssim
R^{-\frac{9}{8}+\frac{k}{4}}$

for $k=1,2,3,4.$ Hence again Gagliardo-Nirenberg inequality gives

$\|x^{-1}f_{lo}\|_{L^{2}(\mathbb{R})}\lesssim R^{-1+\frac{k}{4}},\quad
k=1,2,3.$

For $k=0$, we decompose $f_{lo}$ further as

$f_{lo}=f_{<R^{-1}}+\sum_{R^{-1}\leq\lambda\leq R^{1/4}}f_{\lambda}.$

Then Bernstein’s inequality and Littlewood-Paley characterization of Sobolev
norms give

$\displaystyle\|f_{<R^{-1}}\|_{L^{\infty}(\mathbb{R})}$ $\displaystyle\lesssim
R^{-1/2}\|f_{<R^{-1}}\|_{L^{2}(\mathbb{R})}\lesssim 1,$
$\displaystyle\sum_{R^{-1}\leq\lambda\leq
R^{1/4}}\|f_{\lambda}\|_{L^{\infty}(\mathbb{R})}$
$\displaystyle\lesssim\sum_{R^{-1}\leq\lambda\leq
R^{1/4}}\lambda^{1/2}\|f_{\lambda}\|_{L^{2}(\mathbb{R})}$
$\displaystyle\lesssim(\log R)^{1/2}\Big{(}\sum_{R^{-1}\leq\lambda\leq
R^{1/4}}\lambda\|f_{\lambda}\|_{L^{2}(\mathbb{R})}^{2}\Big{)}^{1/2}$
$\displaystyle\lesssim(\log R)^{1/2}\|f\|_{\dot{H}^{1/2}}\lesssim(\log
R)^{1/2}.$

where the inequality used in the penultimate line is just an elementary
inequality

$\sum_{k=1}^{n}a_{k}\leq(n\sum_{k=1}^{n}a_{k}^{2})^{1/2}.$

Hence one has

(3.22) $\|x^{-1}f_{lo}\|_{L^{\infty}}\lesssim R^{-1}(\log R)^{1/2},$

which is exactly needed.

Finally we turn to the hyperbolic region. As in the third-order case, $f$
might be worse at lower frequency (below $R^{\frac{1}{4}}$) than higher
frequencies, due to the weight $|\xi|^{1/2}.$ Hence we proceed by splitting
$f$ to low and high frequencies:

$f=\widetilde{\chi}_{R}f_{\leq
R^{1/4}}+\widetilde{\chi}_{R}f_{>R^{1/4}}:=f_{lo}+f_{hi},$

where $\tilde{\chi}_{R}$ is smooth, supported on $|x|\sim R,$ and
$\tilde{\chi}_{R}\chi_{R}=\chi_{R}$.

Then we may use an energy estimate,

$\frac{d}{dx}(-x|v|^{2}+5|v_{xx}|^{2}-10v_{x}v_{3x}+2f_{lo}v)=-|v|^{2}-2f_{hi}v_{x}+2f_{lo,x}v\leq-2fv_{x}+2f_{lo,x},$

and integrating both sides gives

(3.23)
$-x|v|^{2}+5|v_{xx}|^{2}-10v_{x}v_{3x}\lesssim\|f_{hi}\|_{L^{2}}\|v_{x}\|_{L^{2}}+\|f_{lo,x}\|_{L^{2}}\|v\|_{L^{2}}+|f_{lo}v|$

whenever $x\in A_{R}^{H}$, so the term $|f_{lo}v|$ can be controlled via
(3.22) and the Gagliardo-Nirenberg inequality

$\|v\|_{L^{\infty}}\lesssim\|v\|_{L^{2}}^{\frac{1}{2}}\|v_{x}\|_{L^{2}}^{\frac{1}{2}}\lesssim
R^{\frac{1}{4}},$

and the other terms can be bounded by (3.20) and (3.21). This finally gives

(3.24) $-x|v|^{2}+5|v_{xx}|^{2}-10v_{x}v_{3x}\lesssim R^{\frac{1}{4}}(\log
R)^{\frac{1}{2}}.$

Now we have a task to deal with the extra $v_{x}v_{3x}$ term, which was absent
in the third-order case. The problem is that, we should control the pointwise
bound of $v_{x}v_{3x}$, either.

Now try a similar estimate as follows:

$\displaystyle\frac{d}{dx}$
$\displaystyle(-x|v_{x}|^{2}+5|v_{3x}|^{2}+2xvv_{xx}+2f_{lo}v_{2x})\leq
2vv_{xx}+2f_{hi}v_{3x}-2f_{lo,x}v_{2x}$ $\displaystyle\lesssim$
$\displaystyle|x|^{-\frac{1}{2}}(-x|v|^{2}+5|v_{xx}|^{2})+|f_{hi}v_{3x}|+|f_{lo,x}v_{2x}|$
$\displaystyle\lesssim$ $\displaystyle R^{-\frac{1}{2}}(R^{\frac{1}{4}}(\log
R)^{\frac{1}{2}}+10v_{x}v_{3x})+|f_{hi}v_{3x}|+|f_{lo,x}v_{2x}|,$

and integrating both sides, together with Cauchy-Schwarz inequality gives

$\displaystyle-x|v_{x}|^{2}+5|v_{3x}|^{2}+2xvv_{xx}$ $\displaystyle\lesssim$
$\displaystyle R^{\frac{3}{4}}(\log
R)^{\frac{1}{2}}+R^{-\frac{1}{2}}\|v_{x}\|_{L^{2}}\cdot\|v_{3x}\|_{L^{2}}+\|f_{hi}\|_{L^{2}}\|v_{3x}\|_{L^{2}}+\|f_{lo,x}\|_{L^{2}}\|v_{2x}\|_{L^{2}}+|f_{lo}v_{2x}|$
$\displaystyle\lesssim$ $\displaystyle R^{\frac{3}{4}}(\log
R)^{\frac{1}{2}}+|f_{lo}v_{2x}|$

for $x\in A_{R}^{H}$. Here the term $|f_{lo}v_{2x}|$ can be treated in the
same way as above, so one has

(3.25) $-x|v_{x}|^{2}+5|v_{3x}|^{2}+2xvv_{xx}\lesssim R^{\frac{3}{4}}(\log
R)^{\frac{1}{2}}.$

Now we obtained the system of two inequalities:

(3.26a) $-x|v|^{2}+5|v_{xx}|^{2}-10v_{x}v_{3x}\leq CR^{\frac{1}{4}}(\log
R)^{\frac{1}{2}},$ (3.26b) $-x|v_{x}|^{2}+5|v_{3x}|^{2}+2xvv_{xx}\leq
CR^{\frac{3}{4}}(\log R)^{\frac{1}{2}}.$

From (3.26a) and (3.26b) we have

$\displaystyle-x|v|^{2}+5|v_{xx}|^{2}\leq$ $\displaystyle
CR^{\frac{1}{4}}(\log R)^{\frac{1}{2}}+10v_{x}v_{3x}\leq CR^{\frac{1}{4}}(\log
R)^{\frac{1}{2}}+\sqrt{\frac{5}{-x}}(-xv_{x}^{2}+5v_{3x}^{2})$
$\displaystyle\leq$ $\displaystyle CR^{\frac{1}{4}}(\log
R)^{\frac{1}{2}}+2\sqrt{-5x}vv_{2x},$

so that

(3.27) $|\sqrt{-x}v-\sqrt{5}v_{2x}|\lesssim R^{\frac{1}{8}}(\log
R)^{\frac{1}{4}}$

and in the same way one can show

(3.28) $|\sqrt{-x}v_{x}-\sqrt{5}v_{3x}|\lesssim R^{\frac{3}{8}}(\log
R)^{\frac{1}{4}}.$

Now it remains to show that

(3.29) $\|\partial_{x}^{k}v\|_{L^{\infty}(A_{R})}\lesssim
R^{-\frac{3}{8}+\frac{k}{4}}(\log R)^{\frac{1}{4}},\quad k=0,1,2,3.$

To deduce a contradiction, suppose that $\|v\|_{L^{\infty}(A_{R})}\not\lesssim
R^{-\frac{3}{8}}(\log R)^{\frac{1}{4}}$. Then there exists a sequence
$\\{R_{n}\\}\uparrow\infty$ as $n\to\infty$ such that
$\|v\|_{L^{\infty}(A_{R_{n}})}\geq 2^{n}R_{n}^{-\frac{3}{8}}(\log
R_{n})^{\frac{1}{4}}.$ Then by (3.27) it turns out that
$\|v_{2x}\|_{L^{\infty}(A_{R_{n}})}\gtrsim 2^{n}R_{n}^{\frac{1}{8}}(\log
R_{n})^{\frac{1}{4}}$ whenever $n$ is large enough. Also by (3.26a) and (3.28)
one may show that $\|v_{x}\|_{L^{\infty}(A_{R_{n}})}\gtrsim
2^{n}R_{n}^{-\frac{1}{8}}(\log R_{n})^{\frac{1}{4}}$ and
$\|v_{3x}\|_{L^{\infty}(A_{R_{n}})}\gtrsim 2^{n}R_{n}^{\frac{3}{8}}(\log
R_{n})^{\frac{1}{4}}$ again for $n$ large enough. Now define the set
$B_{n}:=\\{x:|v(x)|\geq 2^{n-1}R_{n}^{-\frac{3}{8}}(\log
R_{n})^{\frac{1}{4}}\\}.$ Here $v$ is assumed to be differentiable
sufficiently many times, so by taking a connected component we may assume that
$B_{n}=[a_{n},b_{n}]$, where $a_{n}$ and $b_{n}$ are the first left and right
points where $v(a_{n})=v(b_{n})=2^{n-1}R_{n}^{-\frac{3}{8}}(\log
R_{n})^{\frac{1}{4}}.$ Note that $-\infty<a_{n}$ and $b_{n}<\infty$ because
$v$ is compactly supported.

Now observe by (3.26a) one has

(3.30)
$v_{x}v_{3x}\geq\frac{1}{10}\left(-x|v|^{2}+5|v_{2x}|^{2}-CR_{n}^{\frac{1}{4}}(\log
R_{n})^{\frac{1}{2}}\right)\gtrsim 2^{2(n-1)}R_{n}^{\frac{1}{4}}(\log
R_{n})^{\frac{1}{2}}$

whenever $x\in B_{n}$ with $n$ large enough. Similarly, by (3.26b) one also
has

(3.31)
$\begin{split}vv_{2x}&\geq\frac{1}{-2x}\left(-x|v_{x}|^{2}+5|v_{3x}|^{2}-CR_{n}^{\frac{3}{4}}(\log
R_{n})^{\frac{1}{2}}\right)\\\
&\geq\frac{1}{-2x}\left(2\sqrt{-5x}v_{x}v_{3x}-CR_{n}^{\frac{3}{4}}(\log
R_{n})^{\frac{1}{2}}\right)\gtrsim 2^{2(n-1)}R_{n}^{-\frac{1}{4}}(\log
R_{n})^{\frac{1}{2}}\end{split}$

under the same assumption above. (3.31) shows that for sufficiently large $n$,
$v$ and $v_{2x}$ always have the same sign on $B_{n}$, and the similar
statement holds for $v_{x}$ and $v_{3x}.$

First we may assume that $v>0$ on $B_{n}$ (otherwise we may replace $v$ by
$-v$). Then $v\geq 2^{n-1}R_{n}^{-\frac{3}{8}}(\log R_{n})^{\frac{1}{4}}$ and
it is strictly convex on $B_{n}$ since $v_{2x}>0$. Also by definition
$v(a_{n})=v(b_{n})=2^{n-1}R_{n}^{-\frac{3}{8}}(\log R_{n})^{\frac{1}{4}}$.
However, this is a contradiction because strict convexity means
$v(ta_{n}+(1-t)b_{n})<tv(a_{n})+(1-t)v(b_{n})=2^{n-1}R_{n}^{-\frac{3}{8}}(\log
R_{n})^{\frac{1}{4}}$ on $B_{n}.$ Hence it follows that
$\|v\|_{L^{\infty}}\lesssim R^{-\frac{3}{8}}(\log R)^{\frac{1}{4}}$. This
proves (3.29), and plugging (3.29) into the right hand side of the inequality
(3.23) gives

(3.32a) $-x|v|^{2}+5|v_{xx}|^{2}-10v_{x}v_{3x}\lesssim R^{\frac{1}{4}},$
(3.32b) $-x|v_{x}|^{2}+5|v_{3x}|^{2}+2xvv_{xx}\lesssim R^{\frac{3}{4}}.$

and repeating the same argument removes the logarithm from (3.29), and the
Proposition 3.3 is proved.

######  Remark 3.9.

  1. (1)

The overview of the linear analysis is similar to that of [15]. However, since
we have to deal with the higher order equation, there are some additional
steps which was not in [15]. More specifically, in (3.26a) and (3.26b) we had
to control the cross terms which did not appear in [15]. This is one of the
main obstacles we have to pay attention, since such cross terms may become
more and more complicated as the order of the linear equation becomes higher,
making the linear analysis more and more difficult.

  2. (2)

A notable advantage of the argument is that it does not depend on the fact
that $u$ is a solution of (Lin). Thus, as one can see in the last section, the
same argument can be applied to the nonlinear case after a slight
modification.

##  4\. Outline of the nonlinear analysis

Let $u$ be a solution to (KW) with the smallness assumption of the initial
datum

(4.1)
$\|u_{0}\|_{\dot{B}_{2,\infty}^{-\frac{1}{2}}}+\|xu_{0}\|_{\dot{H}^{\frac{1}{2}}}\leq\epsilon\ll
1.$

Motivated from Proposition 3.3, we are going to make bootstrap assumptions

(4.2) $|u_{kx}(x,t)|\leq M\epsilon t^{-\frac{1}{8}-\frac{k}{4}}\langle
x\rangle^{\frac{k}{4}-\frac{3}{8}},\quad k=0,1,2,3,$

where $M$ is a large universal constant, independent of $\epsilon,$ to be
chosen later, and $\epsilon\ll 1$ is small depending on $M$. The overall
process for nonlinear estimates is similar with that in [15], sometimes being
simpler. More precisely, the process may include energy estimates for both
nonlinear and linearized equations, uniform Sobolev bounds for $L^{NL}u$ for
some nonlinear operator $L^{NL},$ and finally the nonlinear dispersive bounds.

Let us explain the overall strategies of this paper. Mainly we focus on the
case of (KW), since the case of (mKW) is entirely similar. The case of (mKW)
is briefly discussed in the appendix.
Basically we are going to follow the argument of [15], but the case becomes
much simpler, thanks to the higher dispersive effect on high frequencies.
To prove our main result, Theorem 1.1, we are going to split the proof into
few steps. Below is the list of the propositions to be proved.

######  Proposition 4.1.

Let $u$ be a solution to (KW) with the smallness assumption of the initial
datum (4.1) together with the bootstrap bound (4.2). Then one has

$\|u(t)\|_{\dot{B}_{2,\infty}^{-\frac{1}{2}}}\lesssim\epsilon$

for the time scale $|t|\ll_{M}\epsilon^{-\frac{5}{3}}.$

Since the proof of Proposition 4.1 is very short, we give the proof right
here.
A direct energy estimate gives

$\displaystyle\frac{d}{dt}\int(P_{\lambda}u)^{2}=\frac{1}{2}\int
P_{\lambda}u\cdot\partial_{x}P_{\lambda}(u^{2})\lesssim\|P_{\lambda}u\|_{L^{2}}\lambda^{\frac{1}{2}}\||D|^{\frac{1}{2}}(u^{2})\|_{L^{2}}.$

Also we may find the bound of $\||D|^{\frac{1}{2}}(u^{2})\|_{L^{2}}$ as
follows: From $|u^{2}|\leq M^{2}\epsilon^{2}t^{-\frac{1}{4}}\langle
x\rangle^{-\frac{3}{4}}$, $|\partial_{x}(u^{2})|\leq
M^{2}\epsilon^{2}t^{-\frac{1}{2}}\langle x\rangle^{-\frac{1}{2}}$, and Lemma
2.5 one has $||D|^{\frac{1}{2}}(u^{2})|\lesssim
M^{2}\epsilon^{2}t^{-\frac{3}{8}}\langle x\rangle^{-\frac{5}{8}}$, so that

$\displaystyle\||D|^{\frac{1}{2}}(u^{2})\|_{L^{2}}\lesssim$ $\displaystyle
M^{2}\epsilon^{2}t^{-\frac{3}{8}}\left(\int(x^{2}+t^{\frac{2}{5}})^{-\frac{5}{8}}dx\right)^{\frac{1}{2}}=M^{2}\epsilon^{2}t^{-\frac{3}{8}}\left(\int(y^{2}+1)^{-\frac{5}{4}}t^{-\frac{1}{4}}t^{\frac{1}{5}}dy\right)^{\frac{1}{2}}$
$\displaystyle\lesssim$ $\displaystyle M^{2}\epsilon^{2}t^{-\frac{2}{5}}$

where the change of variables $x:=t^{\frac{1}{5}}y$ is used. Hence it should
be the case that

$\|P_{\lambda}u\|_{L^{2}}\lesssim\|P_{\lambda}u_{0}\|_{L^{2}}+\lambda^{\frac{1}{2}}M^{2}\epsilon^{2}t^{\frac{3}{5}}\lesssim\epsilon\lambda^{\frac{1}{2}}$

under the assumption of the time bound $M^{2}\epsilon t^{\frac{3}{5}}\leq
1$.111This is why the time scale for (KW) in Theorem 1.1 is restricted to
$|t|\ll\epsilon^{-\frac{5}{3}}$. Therefore, we get the desired bound

$\|u(t)\|_{\dot{B}^{-\frac{1}{2}}_{2,\infty}}\lesssim\epsilon,\quad
t\ll_{M}\epsilon^{-\frac{5}{3}}.$

######  Remark 4.2.

The reason why the direct energy estimate of the Besov bound for (KW) is
possible is that it has a better spatial decay of the solution that it is
directly square integrable. One may see that the above argument does not work
for (5KdV) unless $c_{1}=c_{2}=0$222The case $c_{1}=c_{2}=0$ is merely (mKW),
where we can apply the same argument as discussed in the appendix. due to more
derivatives in the nonlinearity and thus having worse spatial decay. It is
also not applicable to the KdV equation for the same reason, which is why
Ifrim et al. [15] used the small data $H^{-1}$ conservation law of KdV
equation, which is aided from the complete integrability.

######  Proposition 4.3.

Let $u$ be a solution to (KW) which satisfies the smallness assumption (4.1)
for the initial data, as well as the bootstrap assumption (4.2). Then the
linearized equation

(LinKW) $z_{t}-z_{5x}=uz_{x}$

is well-posed in $\dot{H}^{\frac{1}{2}}$ with uniform bounds

$\|z(t)\|_{\dot{H}^{\frac{1}{2}}}\sim\|z(0)\|_{\dot{H}^{\frac{1}{2}}},\quad
t\ll_{M}\epsilon^{-\frac{5}{3}}.$

The proof of Proposition 4.3 will be discussed in the Section 5. Using
Proposition 4.3, one may prove the following proposition:

######  Proposition 4.4.

Let $u$ be a solution to (KW) satisfying the smallness assumption (4.1) and
the bootstrap assumption (4.2). Then $L^{NL}u:=xu+5tu_{4x}+\frac{5}{2}tu^{2}$
satisfies the bound

$\|L^{NL}u(t)\|_{\dot{H}^{\frac{1}{2}}}\lesssim\epsilon,\quad|t|\ll_{M}\epsilon^{-\frac{5}{3}}.$

Proposition 4.4 will be proved in the Section 6.
Let me explain the relations between the above propositions and our main
theorem. First, gathering the results from the above propositions, we conclude
the following: Under the smallness assumption (4.1), we have the bound

(4.3)
$\|u(t)\|_{\dot{B}_{2,\infty}^{-\frac{1}{2}}}+\|L^{NL}u(t)\|_{\dot{H}^{\frac{1}{2}}}\lesssim\epsilon,\quad|t|\ll_{M}\epsilon^{-\frac{5}{3}}.$

Under the bound (4.3), we may apply the same argument as in the Section 3,
with some additional procedures to control the nonlinear terms. This will be
done in the following subsection. Finally, in the Appendix, we briefly sketch
the proof of our main result in the case of (mKW).

###  4.1. Proof of Theorem 1.1

In this section, we finish the proof of our main theorem, Theorem 1.1,
assuming that the Propositions 4.1, 4.3, and 4.4 are true. For the sake of
simplicity, we are going to rescale the problem to $t=1.$ Namely, given the
equation

$xu+5tu_{4x}+\frac{5}{2}tu^{2}=f,\quad f=L^{NL}u$

let
$\tilde{u}(x)=t^{\frac{4}{5}}u(t,xt^{\frac{1}{5}}),\quad\tilde{f}(x)=t^{\frac{3}{5}}f(t,xt^{\frac{1}{5}}).$
Then a direct calculation gives that $\tilde{u}$ and $\tilde{f}$ solve the
same equation with $t=1,$ so that

(4.4) $(x+5\partial_{x}^{4})\tilde{u}+\frac{5}{2}\tilde{u}^{2}=\tilde{f}.$

Such scaling also gives the new bounds as follows:

(4.5)
$\|\tilde{u}\|_{\dot{B}_{2,\infty}^{-\frac{1}{2}}}\lesssim\tilde{\epsilon}:=\epsilon
t^{\frac{3}{5}}\ll 1,\quad|\partial_{x}^{k}\tilde{u}(x)|\lesssim
M\tilde{\epsilon}\langle x\rangle^{-\frac{3}{8}+\frac{k}{4}},\;k=0,1,2,3,$

where $\langle x\rangle=(x^{2}+1)^{\frac{1}{2}}$ does not depend on $t$ and

(4.6) $\|\tilde{f}\|_{\dot{H}^{\frac{1}{2}}}\lesssim\tilde{\epsilon},$

and moreover, the rescaled time bound assumption

(4.7) $M^{2}\tilde{\epsilon}\leq 1.$

For notational convenience, we are going to drop the tilde notation. Then the
proof is reduced to showing that

$\|\partial_{x}^{k}u\|_{L^{\infty}(A_{R})}\lesssim\epsilon
R^{-\frac{3}{8}+\frac{k}{4}}$

and the corresponding counterpart for the elliptic region $A_{R}\cap E.$
Now most of the procedures are the same with the linear analysis, so I am
going to skip most of the details, except the necessary nonlinear analysis.

######  Lemma 4.5.

Under the assumptions of (4.4), (4.5), (4.6), and (4.7), we have

(4.8) $\|f\|_{L^{2}(A_{R})}\lesssim\epsilon R^{\frac{1}{2}}.$

###### Proof.

As in the proof of Lemma 3.4, we split $u$ at the frequency cutoff $R^{-1}$
and compute:

$f=P_{>R^{-1}}f+P_{<R^{-1}}Lu+P_{<R^{-1}}(u^{2}).$

The first two terms can be estimated in the same way as Lemma 3.4. For the
last term, the pointwise bootstrap bound (4.5) and Lemma 2.3 gives

$|P_{<R^{-1}}(u^{2})|\lesssim M^{2}\epsilon^{2}R^{-\frac{3}{4}},$

so that

$\|P_{<R^{-1}}(u^{2})\|_{L^{2}(A_{R})}\lesssim
M^{2}\epsilon^{2}R^{-\frac{1}{4}},$

and using the time bound (4.7), the desired bound (4.8) is obtained. ∎

######  Lemma 4.6.

Under the assumptions of (4.4), (4.5), (4.6), and (4.7), we have

(4.9) $\|\partial_{x}^{k}u\|_{L^{2}(A_{R})}\lesssim\epsilon
R^{\frac{1}{8}+\frac{k}{4}}.$

###### Proof.

The low frequency case $\lambda\leq R^{\frac{1}{4}}$ can be dealt with in the
same way in the linear case, so we focus on the high frequency case,
$\lambda>R^{\frac{1}{4}}$. Again we split as

$Lu_{\lambda}=P_{\lambda}L^{NL}u+\lambda^{-1}u_{\lambda}-P_{\lambda}(u^{2}),$

and the first two terms can be estimated in the same way as in the linear
case, so we again focus on the nonlinear terms. Using (4.5) and Lemma 2.3
gives

$\|P_{\lambda}(u^{2})\|_{L^{2}(A_{R})}\lesssim\epsilon R^{-\frac{1}{4}}$

and

$\|\partial_{x}P_{\lambda}(u^{2})\|_{L^{2}(A_{R})}\lesssim\epsilon
R^{\frac{1}{4}},$

so that

$\|P_{\lambda}(u^{2})\|_{L^{2}(A_{R})}\lesssim\epsilon\lambda^{-\frac{1}{2}}.$

Now the argument is completed as in the linear case. ∎

Now we may localize the solution $u$ to the dyadic regions $|x|\sim R.$ Let
$v:=\chi_{R}u,$ then $v$ solves the equation

(4.10) $(x+5\partial_{x}^{4})v+\frac{5}{2}uv=g,$

with the bounds

(4.11) $\|\partial_{x}^{k}v\|_{L^{2}(A_{R})}\lesssim\epsilon
R^{\frac{1}{8}+\frac{k}{4}},\quad k=0,1,2,3,4,$

and

(4.12)
$\|g\|_{\dot{H}^{\frac{1}{2}}}\lesssim\epsilon,\quad\|g\|_{L^{2}(A_{R})}\lesssim\epsilon
R^{\frac{1}{2}}.$

As in the linear case, we consider three different regions; self-similar,
elliptic, and hyperbolic.

The self-similar region is entirely similar to the linear case, so we omit the
details.

The elliptic region is also almost similar, but there is a subtle difference
due to the presence of nonlinearity. More precisely, we again split
$g=g_{lo}+g_{hi},$ where the frequency scale is $R^{\frac{1}{4}}.$ Moreover,
letting $v_{1}:=v-x^{-1}g_{lo}$ and

$\tilde{L}^{NL}v_{1}=(x+5\partial_{x}^{4})v_{1}+uv_{1}=g_{hi}-5\partial_{x}^{4}(x^{-1}g_{lo}):=g_{1}.$

The estimation of $g_{1}$ is the same as in the linear case, yielding the
$\epsilon R^{-\frac{1}{8}}$ bound. The only difference is the estimation of
integrals. To be more specific, we start from the equality

$\int_{\mathbb{R}}v_{1}\tilde{L}^{NL}v_{1}dx=\int_{\mathbb{R}}g_{1}v_{1}dx$

and integrating by parts gives

$\int_{\mathbb{R}}x|v_{1}|^{2}dx+5\int_{\mathbb{R}}|\partial_{x}^{2}v_{1}|^{2}dx=\int_{\mathbb{R}}g_{1}v_{1}dx+\int_{\mathbb{R}}uv_{1}^{2}dx$

The first integral on the right is routine, so we focus on the second integral
including nonlinearity. A direct calculation gives

$\int_{\mathbb{R}}uv_{1}^{2}dx=O(M\epsilon
R^{-\frac{3}{8}})\|v_{1}\|_{L^{2}}^{2},$

so this bound can be absorbed into the left hand side provided $R$ is large
enough. Note that by the time bound (4.7) and the assumption $M\gg 1,$ the
choice of $R$ is independent of $M$. Thus one may conclude

$\|v_{1}\|_{L^{2}}\lesssim\epsilon
R^{-\frac{9}{8}},\quad\|\partial_{x}^{2}v_{1}\|_{L^{2}}\lesssim\epsilon
R^{-\frac{5}{8}}.$

Now the same process as in the linear case gives the extra bounds

$\|\partial_{x}^{4}v_{1}\|_{L^{2}}\lesssim\epsilon
R^{-\frac{1}{8}},\quad\|\partial_{x}v_{1}\|_{L^{2}}\lesssim\epsilon
R^{-\frac{7}{8}},\quad\|\partial_{x}^{3}v_{1}\|_{L^{2}}\lesssim\epsilon
R^{-\frac{3}{8}}.$

Hence Gagliardo-Nirenberg inequality gives

$|\partial_{x}^{k}v_{1}|\lesssim\epsilon R^{-1+\frac{k}{4}},\quad k=0,1,2,3.$

The pointwise bound of $x^{-1}g_{lo}$ can be obtained in the same way as in
the linear analysis, so we omit the details.

Lastly, let us turn to the hyperbolic region. Again we do a spatial energy
estimate to obtain

$\frac{d}{dx}(-x|v|^{2}+5|v_{2x}|^{2}-10v_{x}v_{3x}+2g_{lo}v)=-|v|^{2}+5uvv_{x}-2v_{x}g_{hi}+2g_{lo,x}v.$

Now integrating both sides and using Hölder’s inequality gives

$\displaystyle-x|v|^{2}+5|v_{2x}|^{2}-10v_{x}v_{3x}$
$\displaystyle\lesssim\|v_{x}\|_{L^{2}}\|g_{hi}\|_{L^{2}}+\|g_{lo,x}\|_{L^{2}}\|v\|_{L^{2}}+\sup_{x\in
A_{R}^{H}}|g_{lo}v|+\int|uvv_{x}|dx$
$\displaystyle\lesssim\epsilon^{2}R^{\frac{1}{4}}+M\epsilon^{3}R^{\frac{1}{8}}\lesssim\epsilon^{2}R^{\frac{1}{4}}.$

A similar calculation gives

$-x|v_{x}|^{2}+5|v_{3x}|^{2}+2xvv_{2x}\lesssim\epsilon^{2}R^{\frac{3}{4}},$

and the rest is now the same with the linear case. This completes the proof of
the main theorem.

######  Remark 4.7.

Here one may see that when estimating on the hyperbolic region, one may
estimate the nonlinear interaction term $uvv_{x}$ directly, unlike in the KdV
case where one has to split the nonlinear interaction term and send some
pieces to the left hand side. This difference is again due to the better
spatial decay of our bootstrap bound (4.2) than in the case of KdV equation.

##  5\. Bounds for linearized equation

In this section, we give the proof of Proposition 4.3.
Note that (KW) enjoys scaling symmetry
$u(t,x)\mapsto\lambda^{4}u(\lambda^{5}t,\lambda x),$ which is critical in
$\dot{H}^{-\frac{7}{2}}$ in the sense that the scaling preserves
$\dot{H}^{-\frac{7}{2}}$ norm. The scaling vector field which generates the
symmetry is given as $\mathcal{S}=5t\partial_{t}+x\partial_{x}+4.$

Define $\Lambda=\partial_{x}^{-1}\mathcal{S}$ and
$\mathcal{L}=\partial_{t}-\partial_{x}^{5}.$ Then by the commutator relation

$[\mathcal{L},\mathcal{S}]=5\mathcal{L},\quad[\mathcal{S},\partial_{x}]=-\partial_{x}$

one has

$\displaystyle\mathcal{L}(\Lambda u)$
$\displaystyle=\partial_{x}^{-1}(\mathcal{S}+5)\mathcal{L}u=u\partial_{x}\Lambda
u.$

Hence, $z=\Lambda u$ solves the linearized equation

$z_{t}-z_{5x}=uz_{x}.$

However, the function $z=xu+u_{4x}+\frac{5}{2}u^{2}+3\partial_{x}^{-1}u$
contains the inverse derivative of $u$, which can cause problem if $u$ is not
mean-zero. Hence, we rather remove the $\partial_{x}^{-1}u$ term and work with
the function

$w:=L^{NL}u:=xu+u_{4x}+\frac{5}{2}u^{2},$

which solves the inhomogeneous linearized equation

$w_{t}+w_{5x}=uw_{x}-\frac{3}{2}u^{2}.$

Now we are going to prove Proposition 4.3.
Let $y=|D|^{\frac{1}{2}}z,$ then $y$ satisfies the equation

$y_{t}-y_{5x}=-|D|^{\frac{1}{2}}(uH|D|^{\frac{1}{2}}y).$

A direct calculation and frequency analysis as in [15] gives

$\frac{1}{2}\frac{d}{dt}\int y^{2}=-\int|D|^{\frac{1}{2}}y\cdot
uH|D|^{\frac{1}{2}}y=O(M\epsilon
t^{-\frac{2}{5}})\|y\|_{L^{2}}^{2}-\int|D|^{\frac{1}{2}}y_{hi}\cdot
u_{hi}H|D|^{\frac{1}{2}}y_{hi}$

in the time scale $M^{2}\epsilon t^{\frac{3}{5}}\leq 1.$ Here
$y_{hi}:=P_{\gtrsim t^{-\frac{1}{5}}}y$ means a piece of $y$ frequency-
localized in the scale $|\xi|\gtrsim t^{-\frac{1}{5}}.$

Hence, we shall try a normal form analysis to cancel the high frequency term,
following the idea in [14]. Let $\tilde{u}=u+B(u,u)$ where $B$ is a symmetric
bilinear Fourier multiplier. We want to find $B$ such that
$\tilde{u}_{t}-\tilde{u}_{5x}$ consists of cubic or higher order terms of $u$.
A direct calculation gives that

$\tilde{u}_{t}-\tilde{u}_{5x}=uu_{x}+B(u_{5x},u)+B(u,u_{5x})-\partial_{x}^{5}B(u,u)+h.o.t.,$

where $h.o.t.$ denotes the cubic or higher order terms of $u$. Here our goal
is to cancel out $uu_{x}+B(u_{5x},u)+B(u,u_{5x})-\partial_{x}^{5}B(u,u).$ It
turns out that the formal normal form is given as
$\frac{1}{10}B(\partial_{x}^{-1}u,\partial_{x}^{-1}u)$ with the symbol of $B$
being $\hat{B}(\xi,\eta)=\frac{1}{\xi^{2}+\xi\eta+\eta^{2}}.$ However, this
can cause problems on the low frequency region, so we truncate the low
frequency part and get
$\frac{1}{10}B(\partial_{x}^{-1}u_{hi},\partial_{x}^{-1}u_{hi}).$ Now
linearizing the normal form gives the normal form correction of (LinKW) as
$\frac{1}{10}B(\partial_{x}^{-1}u_{hi},\partial_{x}^{-1}z_{hi}).$ Then
plugging $y=|D|^{\frac{1}{2}}z$ gives
$\frac{1}{10}B(\partial_{x}^{-1}u_{hi},\partial_{x}^{-1}|D|^{-\frac{1}{2}}y_{hi}).$
Let
$\tilde{z}=z+\frac{1}{10}B(\partial_{x}^{-1}u_{hi},\partial_{x}^{-1}z_{hi}).$
Then observe that

$\displaystyle\|\tilde{z}\|_{\dot{H}^{\frac{1}{2}}}$
$\displaystyle=\|y+|D|^{\frac{1}{2}}\frac{1}{10}B(\partial_{x}^{-1}u_{hi},\partial_{x}^{-1}|D|^{-\frac{1}{2}}y_{hi})\|_{L^{2}}^{2}$
$\displaystyle=$
$\displaystyle\|y\|_{L^{2}}^{2}+\frac{1}{5}\langle|D|^{\frac{1}{2}}y,B(\partial_{x}^{-1}u_{hi},\partial_{x}^{-1}|D|^{-\frac{1}{2}}y_{hi})\rangle+\|\frac{1}{10}|D|^{\frac{1}{2}}B(\partial_{x}^{-1}u_{hi},\partial_{x}^{-1}|D|^{-\frac{1}{2}}y_{hi})\|_{L^{2}}^{2}.$

Here the last term is quartic, so it becomes more perturbative, and we do not
include this into the corrected energy term. Hence the energy correction term
is given as

$E^{\prime}[y]=\frac{1}{10}\int|D|^{\frac{1}{2}}y_{hi}B(\partial_{x}^{-1}u_{hi},\partial_{x}^{-1}|D|^{-\frac{1}{2}}y_{hi}).$

This corrected energy plays the desired role itself, but a more symmetric form
would be convenient, so shifting $|D|^{-1}$ from the rightmost $y_{hi}$ to the
leftmost one gives

$E^{\prime}[y]=\frac{1}{10}\int|D|^{-\frac{1}{2}}y_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})$

First check that
$E[y]:=\frac{1}{2}\|y\|_{L^{2}}^{2}+E^{\prime}[y]\sim\|y\|_{L^{2}}^{2}.$ This
directly follows from:

$|E^{\prime}[y]|\lesssim
t^{\frac{1}{10}}\|y\|_{L^{2}}t^{\frac{2}{5}}t^{\frac{1}{5}}\|u\|_{L^{\infty}}t^{\frac{1}{10}}\|y\|_{L^{2}}\leq
M\epsilon t^{\frac{3}{5}}\|y\|_{L^{2}}^{2}\leq\frac{1}{M}\|y\|_{L^{2}}^{2},$

so by letting $M$ sufficiently large, we conclude $E[y]\sim\|y\|_{L^{2}}^{2}.$
Now a direct calculation gives444Here I intentionally dropped any
insignificant constant multiplications.

$\displaystyle\frac{d}{dt}E^{\prime}[y]$ $\displaystyle=$ $\displaystyle
D_{1}+D_{2}+D_{3}+\int|D|^{-\frac{1}{2}}\partial_{x}^{5}y_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})$
$\displaystyle+$
$\displaystyle\int|D|^{-\frac{1}{2}}y_{hi}B(\partial_{x}^{4}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})+\int|D|^{-\frac{1}{2}}y_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}\partial_{x}^{5}y_{hi})$
$\displaystyle=$ $\displaystyle
D_{1}+D_{2}+D_{3}-\int|D|^{-\frac{1}{2}}y_{hi}\partial_{x}^{5}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})$
$\displaystyle+$
$\displaystyle\int|D|^{-\frac{1}{2}}y_{hi}B(\partial_{x}^{4}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})+\int|D|^{-\frac{1}{2}}y_{hi}B(\partial_{x}^{4}u_{hi},H|D|^{-\frac{1}{2}}\partial_{x}^{5}y_{hi})$
$\displaystyle=$ $\displaystyle
D_{1}+D_{2}+D_{3}+\int|D|^{-\frac{1}{2}}y_{hi}\partial_{x}(u_{hi}|D|^{\frac{1}{2}}y_{hi})=D_{1}+D_{2}+D_{3}+\int
H|D|^{\frac{1}{2}}y_{hi}\cdot u_{hi}|D|^{\frac{1}{2}}y_{hi}$

where the penultimate equality follows from the symbol calculation

$\frac{(\xi+\eta)^{5}-\xi^{5}-\eta^{5}}{\xi^{2}+\xi\eta+\eta^{2}}=5\xi\eta(\xi+\eta).$

Hence the last integral cancels out the “bad” term of
$\frac{1}{2}\frac{d}{dt}\int y^{2}$, and the other terms($D_{1}$, $D_{2}$,
$D_{3}$) are perturbative, as demonstrated below:

  1. (a)

$D_{1}$ arises from the time derivative of the frequency scale truncation,
i.e. $\partial_{t}P_{hi}.$ Namely,
$\partial_{t}(1-\varphi(t^{\frac{1}{5}}\xi))=-\frac{1}{5}t^{-\frac{4}{5}}\xi\varphi^{\prime}(t^{\frac{1}{5}}\xi)=-t^{-1}\psi(t^{\frac{1}{5}}\xi)$
where $\psi(\eta)=\frac{1}{5}\eta\varphi^{\prime}(\eta)$ is the Littlewood-
Paley projection on the frequency scale $\sim 1.$ Thus the bound for $D_{1}$
is given as

$\displaystyle\left|\int|D|^{-\frac{1}{2}}(\partial_{t}P_{hi})y\cdot
B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})\right|\lesssim
t^{-\frac{9}{10}}\|y\|_{L^{2}}\|B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})\|_{L^{2}}$
$\displaystyle\lesssim$ $\displaystyle
t^{-\frac{9}{10}}\|y\|_{L^{2}}t^{\frac{2}{5}}\|\partial_{x}^{-1}u_{hi}\|_{L^{\infty}}\|H|D|^{-\frac{1}{2}}y_{hi}\|_{L^{2}}\lesssim
M\epsilon t^{-\frac{2}{5}}\|y\|_{L^{2}}^{2}.$

The rest terms, i.e. $\int|D|^{-\frac{1}{2}}y_{hi}\cdot
B(\partial_{x}^{-1}(\partial_{t}P_{hi})u,H|D|^{-\frac{1}{2}}y_{hi})$ and
$\int|D|^{-\frac{1}{2}}y_{hi}\cdot
B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}(\partial_{t}P_{hi})y)$ can be
estimated in a similar fashion.

  2. (b)

$D_{2}$ consists of the nonlinarity of $u$ from (KW):

$\displaystyle\left|\int|D|^{-\frac{1}{2}}y_{hi}B((u^{2})_{hi},H|D|^{-\frac{1}{2}}y_{hi})\right|\lesssim
t^{\frac{1}{10}}\|y\|_{L^{2}}t^{\frac{2}{5}}\|u^{2}\|_{L^{\infty}}t^{\frac{1}{10}}\|y\|_{L^{2}}\leq
M\epsilon t^{-\frac{2}{5}}.$

  3. (c)

$D_{3}$ consists of the linearized counterpart of $D_{2}$:

$\displaystyle\left|\int(uH|D|^{\frac{1}{2}}y)_{hi}\cdot
B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})\right|$ $\displaystyle=$
$\displaystyle\left|\int
Hy\cdot|D|^{\frac{1}{2}}\left(uP_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})\right)\right|\lesssim\|y\|_{L^{2}}\||D|^{\frac{1}{2}}\left(uP_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})\right)\|_{L^{2}}^{2}.$

Now the Kato-Ponce commutator estimate (see, for example, [23])

$\||D|^{s}(fg)-(|D|^{s}f)g\|_{L^{2}}\lesssim\|f\|_{L^{\infty}}\||D|^{s}g\|_{L^{2}},\quad
0<s<1,$

gives

$\||D|^{s}(fg)\|_{L^{2}}\lesssim\||D|^{s}(fg)-(|D|^{s}f)g\|_{L^{2}}+\|(|D|^{s}f)g\|_{L^{2}}\lesssim\|f\|_{L^{\infty}}\||D|^{s}g\|_{L^{2}}+\||D|^{s}f\|_{L^{\infty}}\|g\|_{L^{2}},$

so that

$\displaystyle\||D|^{\frac{1}{2}}\left(uP_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})\right)\|_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\||D|^{\frac{1}{2}}u\|_{L^{\infty}}\|P_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})\|_{L^{2}}+\|u\|_{L^{\infty}}\||D|^{\frac{1}{2}}P_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}y_{hi})\|_{L^{2}}$
$\displaystyle\lesssim$ $\displaystyle M\epsilon t^{-\frac{3}{10}}\cdot
t^{\frac{2}{5}}\|\partial_{x}^{-1}u_{hi}\|_{L^{\infty}}\|H|D|^{-\frac{1}{2}}y_{hi}\|_{L^{2}}+M\epsilon
t^{-\frac{1}{5}}\cdot
t^{\frac{3}{10}}\|\partial_{x}^{-1}u_{hi}\|_{L^{\infty}}\|H|D|^{-\frac{1}{2}}y_{hi}\|_{L^{2}}$
$\displaystyle\lesssim$ $\displaystyle M\epsilon
t^{-\frac{2}{5}}\|y\|_{L^{2}}.$

Hence we obtain the bound $M\epsilon t^{-\frac{2}{5}}\|y\|_{L^{2}}^{2}$. The
other term
$\int|D|^{-\frac{1}{2}}y_{hi}B(\partial_{x}^{-1}u_{hi},H(u|D|^{\frac{1}{2}}y)_{hi})$
can be treated similarly after decomposing
$(u|D|^{\frac{1}{2}}y)_{hi}=\partial_{x}(uH|D|^{-\frac{1}{2}}y)_{hi}-(u_{x}H|D|^{-\frac{1}{2}}y)_{hi}$.
One may handle the former term by noting that the bilinear map $(u,v)\mapsto
B(u,v_{x})$ has symbol $\frac{i\eta}{\xi^{2}+\xi\eta+\eta^{2}}$, so that
$\|B(u,v_{x})\|_{L^{2}}\lesssim
t^{\frac{1}{5}}\|u\|_{L^{\infty}}\|v\|_{L^{2}},$ provided $u$ and $v$ are
frequency supported at the scale $\gtrsim t^{-\frac{1}{5}}.$
Hence, we conclude that, if we let the corrected energy $E[y]=\frac{1}{2}\int
y^{2}+E^{\prime}[y],$ then

$\frac{d}{dt}E[y]\lesssim M\epsilon t^{-\frac{2}{5}}\|y\|_{L^{2}}^{2}\sim
M\epsilon t^{-\frac{2}{5}}E[y].$

Hence Gronwall’s inequality gives

$E[y]\leq e^{CM\epsilon t^{\frac{3}{5}}}E[y_{0}]\lesssim E[y_{0}]$

in the time scale $t\ll_{M}\epsilon^{-\frac{5}{3}}.$ This completes the proof
of Proposition 4.3.

######  Remark 5.1.

  1. (1)

The above analysis also works when the right hand side of (LinKW) is
$u^{k}w_{x}$ with $k\geq 1,$ with $u_{hi}$ being replaced by $(u^{k})_{hi}.$
Namely, we may consider $u^{k}$ as a single component, which makes no
difference of the details. The only difference is the valid time scale of the
analysis. For instance, if $k=2,$ which corresponds to the case of (5KdV) with
quadratic terms gone, the time scale becomes $\epsilon^{-5}.$

  2. (2)

The analysis in this section is almost identical to that in [15], since the
KdV and Kawahara equation share the same form of nonlinear terms and their
linearized counterparts, on which the overall analysis mainly depends. The
linear part only affects the form of the corrected energy.

##  6\. Sobolev bound of $L^{NL}u$

In this section, we are going to prove Proposition 4.4. Unlike the [15], we do
not have to dive into a complicated normal form analysis, since we have a
better bootstrap bound for $u$. Namely, we already proved that
$\|u^{2}\|_{\dot{H}^{\frac{1}{2}}}\lesssim M^{2}\epsilon^{2}t^{-\frac{2}{5}}$
in the Proposition 4.1. Then it remains to see that the
$\dot{H}^{\frac{1}{2}}$ bound of the inhomogeneous term is perturbative enough
that the result of Proposition 4.3 is still valid to $L^{NL}u.$ Of course, the
answer is affirmative, which is merely the Proposition 4.4.
Let $z:=|D|^{\frac{1}{2}}L^{NL}u.$ Then $z$ solves the equation

$z_{t}-z_{5x}=-|D|^{\frac{1}{2}}(uH|D|^{\frac{1}{2}}z-\frac{3}{2}u^{2}).$

Hence one has:

$\displaystyle\frac{d}{dt}E[z]=$ $\displaystyle-\frac{3}{2}\int
z|D|^{\frac{1}{2}}(u^{2})-\frac{3}{10}\int(u^{2})_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}z_{hi})$
$\displaystyle-$
$\displaystyle\frac{3}{10}\int|D|^{-\frac{1}{2}}z_{hi}B(\partial_{x}^{-1}u_{hi},H(u^{2})_{hi})+O(M\epsilon
t^{-\frac{2}{5}})\|z\|_{L^{2}}^{2},$

where $E[\cdot]$ is the corrected energy defined in the Section 5. Here most
of the steps are identical to the Section 5, so it only remains to estimate
the integrals including the inhomogeneous term. Namely,

$\displaystyle\left|\int z|D|^{\frac{1}{2}}(u^{2})\right|\lesssim
M^{2}\epsilon^{2}t^{-\frac{2}{5}}\|z\|_{L^{2}}\lesssim M^{2}\epsilon
t^{-\frac{2}{5}}\|z\|_{L^{2}}^{2}+M^{2}\epsilon^{3}t^{-\frac{2}{5}}\lesssim
M^{2}\epsilon t^{-\frac{2}{5}}\|z\|_{L^{2}}^{2}+\epsilon^{2}t^{-1},$
$\displaystyle\left|\int(u^{2})_{hi}B(\partial_{x}^{-1}u_{hi},H|D|^{-\frac{1}{2}}z_{hi})\right|$
$\displaystyle\lesssim$
$\displaystyle\|u^{2}\|_{\dot{H}^{\frac{1}{2}}}t^{\frac{2}{5}}\|\partial_{x}^{-1}u_{hi}\|_{L^{\infty}}\||D|^{-\frac{1}{2}}z_{hi}\|_{L^{2}}\lesssim
M^{3}\epsilon^{3}t^{\frac{1}{10}}\|z\|_{L^{2}}\lesssim M^{2}\epsilon
t^{-\frac{2}{5}}\|z\|_{L^{2}}^{2}+M^{4}\epsilon^{5}t^{\frac{3}{5}}$
$\displaystyle\lesssim$ $\displaystyle M^{2}\epsilon
t^{-\frac{2}{5}}\|z\|_{L^{2}}^{2}+\epsilon^{2}t^{-\frac{6}{5}},$

and the rest terms can be estimated similarly.

To sum up, one obtains the bound

$\frac{d}{dt}E[z]\lesssim M^{2}\epsilon
t^{-\frac{2}{5}}\|z\|_{L^{2}}^{2}+\epsilon^{2}t^{-1}.$

Hence Grönwall’s inequality gives

$\|z\|_{L^{2}}^{2}\lesssim\epsilon^{2}e^{CM^{2}\epsilon
t^{\frac{3}{5}}}\lesssim\epsilon^{2}$

in the time scale $|t|\ll\epsilon^{-\frac{5}{3}}.$ This completes the proof of
Proposition 4.4.

##  7\. Appendix: The case of modified Kawahara equation

In this appendix, we sketch the proof of the case $m=2$, namely (mKW), of the
Theorem 1.1.
First, when it comes to the Subsection 4.1, one may rescale the equation

$xu+5tu_{4x}+\frac{5}{3}tu^{3}=f,\quad f=L^{NL}u$

to

$x\tilde{u}+5\tilde{u}_{4x}+\frac{5}{3}\tilde{u}^{3}=\tilde{f}$

by letting $\tilde{u}(x)=t^{\frac{2}{5}}u(t,xt^{\frac{1}{5}})$ and
$\tilde{f}(x)=t^{\frac{1}{5}}f(t,xt^{\frac{1}{5}})$. Also the localized
equation becomes

$(x+5\partial_{x}^{4})v+\frac{5}{3}u^{2}v=g$

with $v$ and $g$ satisfying the same bounds as in the Subsection 4.1. In these
settings, all the same steps work properly.
Moreover, one may easily see that the argument in the Section 5 does work to
prove Proposition 4.1 in the case of (mKW), under the time bound assumption
$M^{\frac{3}{2}}\epsilon t^{\frac{1}{5}}\leq 1.$ Here is the step where the
time scale restriction $|t|\ll\epsilon^{-5}$ is needed.
Secondly, as mentioned in the remark at the end of the Section 5, one may
prove the same bound from Proposition 4.3 with the same argument. Here we have
a subtly different setting, and I state here for convenience.
The equation (mKW) enjoys the scaling symmetry
$u(t,x)\mapsto\lambda^{2}u(\lambda^{5}t,\lambda x)$, so the scaling vector
field is given as $\mathcal{S}=5t\partial_{t}+x\partial_{x}+2,$ and one may
see that if $u$ solves (mKW), then $\Lambda
u:=\partial_{x}^{-1}\mathcal{S}u=xu+5tu_{4x}+\frac{5}{3}tu^{3}+\partial_{x}^{-1}u$
solves the linearized equation

$w_{t}-w_{5x}=u^{2}w_{x}.$

Now the rest of the arguments are the same as in the Section 5.

The Section 6 is again similar. In this case
$L^{NL}u=xu+5tu_{4x}+\frac{5}{3}tu^{3}=\Lambda u-\partial_{x}^{-1}u$ solves
the inhomogeneous equation

$z_{t}-z_{5x}=u^{2}z_{x}+\frac{2}{3}u^{3},$

and the energy estimate of the inhomogeneous equation is now routine. Note
that the sign of $c$ of (mKW) does not play any role in the entire steps of
the proof.

##  8\. Conclusion

We have discussed how long does a small data solution to Kawahara and modified
Kawahara equation shows a linear dispersive decay bound. The result shows the
tendency of change of the time scale where the linear dispersive decay bound
holds depending on the linear and nonlinear part of the equation. More
specifically, for the Kawahara equation, which has the same nonlinearity as
KdV equation and a higher order of the linear part, the time scale of linear
dispersive decay bound is possibly shorter than that of KdV equation. On the
other hand, for the modified Kawahara equation, which has a higher order
nonlinearity than KdV and Kawahara equations, has a longer time scale of such
decay bound. However, unlike the KdV equation, we still do not know whether
the time bounds found in this article is optimal due to the non-integrability
of the equations, which should be found in a future work.

## Declarations

Acknowledgements
The author appreciate Soonsik Kwon for helpful discussions and encouragement
to this work.
Funding statement
The author is partially supported by NRF-2019R1A5A1028324 and
NRF-2018R1D1A1A0908335.
Availability of data and materials
Not applicable.
Competing interests
The author declares that he does not have any competing interests.
Author’s contributions
The single author contributed and reviewed the work solely.

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