数理解析研究所講究録 1201 巻 2001 年 83-95

83

On global well-posedness of some nonlinear
dispersive equations for rough data
東北大学大学院 理学研究科

高岡 秀夫 (Hideo Takaoka)

In this note, Iconsider the global well-posedness of dispersive equations forrough initial data. Most part of note concerns the results on the Cauchy problemof the$\mathrm{K}\mathrm{d}\mathrm{V}$equation:$u_{t}+u_{xxx}+(u^{2})_{x}=0$$\{$$u(0)=u_{0}$,.

The argument below has an application to another dispersive wave equations,but we shall not deal so here.

Anyway, my goal is to present the sharp global well-posedness for the$\mathrm{K}\mathrm{d}\mathrm{V}$equation. To be clear Iwould like to construct the content of note with three
sections. Section one recalls the work of the local well-posedness through theFourier restriction norm method. Next two sections are devoted to the extentionof the local solution obtained in section one to global one. In traditionally, it
is well-tried that the global well-posedness in the finite energy space. Recently,
J. Bourgain introduced quite interesting story for the global well-posedness formuch rough data. Section two will recall his method and try to apply thistechnique to the$\mathrm{K}\mathrm{d}\mathrm{V}$equation. In finally, Iwant to develop the argument ofBourgain to lead the sharp global well-posedness of$\mathrm{K}\mathrm{d}\mathrm{V}$equation, which is themain result in this note. The work in section three is ajoint study with J.
Colliander, M. Keel, G. Staffilani and T. Tao.

1Local well-posedness by the Fourier restriction norm method
For the well-posedness for dispersive wave equations, such as the$\mathrm{K}\mathrm{d}\mathrm{V}$equationand the nonlinear Schrodinger equation, there has been aremarkable progre ss

84
in recent year. This section will recall the refined performance of the local

well-posedness for the$\mathrm{K}\mathrm{d}\mathrm{V}$equation induced by the Fourier restriction norm

method. The Fourier restriction norm for the

linear$\mathrm{K}\mathrm{d}\mathrm{V}$equation is tailored with$\mathrm{K}\mathrm{d}\mathrm{V}$equation as follows [1]:,$||u||_{X_{*,b}}=( \int\langle\xi)^{2s}\langle\tau-\xi^{3}\rangle^{2b}|\hat{u}(\tau, \xi)|^{2}d\tau d\xi)\frac{1}{2}$where we denote the Fourier transform in

$t$

and

of

$x$

$u$

by \^u. The interest aims

not only the line case, but also the periodic boundary condition case throughthis note. Then the problem on the line case refers the measure

as the usual
Lebesgue measure on R. On the other hand, the counting measure on integer
is taken to the periodic boundary condition case. Through this note, we often
abbreviateas.
In ordinary way, by Duhamel formula, we attempt to solve the integral$||\cdot||_{X}..b$$d\xi$$||\cdot||_{s,b}$equation associated with$\mathrm{K}\mathrm{d}\mathrm{V}$equation:.

(1)$u(t)=e^{-t\theta_{xx\approx}}u_{0}- \int_{0}^{t}e^{-(t-t’)\partial_{x\mathrm{r}\mathrm{a}}}(u^{2})_{x}(t’)dt’$

Now, we start by making aplan for the estimate of the right hand side of thisintegral equation (1). The point of the proof is how handle the nonlinear term.

The standard computation estimates the Duhamel term of (1) byfor, $[13, 15]$ . Once the following bilinear estimate holds, which leads$||(u^{2})_{x}||_{s,b-1}$$b> \frac{1}{2}[1]$the local well-posedness of the Cauchy problem for the$\mathrm{K}\mathrm{d}\mathrm{V}$equation by theusual contraction argument:$||(uv)_{x}||_{s,b-1}\sim<||u||_{s,b}||v||_{s,b}$.

Problem 1Can one have the bilinear estimate (2)

(2)
$q$

Show when the estimate(2) holds.

Problems of this kind has been introduced first by Bourgain [1] and by C. E.
Kenig, G. Ponce and L. Vega $[13, 15]$ .
Proposition 1.1 ([1], [13, 15]) For

for the line case holds.

$s>- \frac{3}{4}$, there exists$b> \frac{1}{2}$such that (2)85Remark 1.1 Precisely, the paper of [1] focused in the case of $s=0$ . After this,the proof was improved by the paper of, which reaches the value$[\mathit{1}\mathit{3}, 15]$$s>- \frac{3}{4}$of proposition 1.1.
Periodic boundary condition case. Proposition 1.1 refers the evaluation ofbilinear estimate (2) for the line case. It is also emphasized that the similar
problem is developed by the same papers [1], $[13, 15]$ for the periodic boundary

condition problem.
Proposition 1.2 ([1], [13, 15]) Let$b= \frac{1}{2}$. For$s \geq-\frac{1}{2}$, the estimate (2) forthe periodic boundary condition case holds.

As stated before, the bilinear estimate (2) leads the results of local well-posedness.
However, in contrast to the line case, the proof for the periodic boundary condi-tion case based on the estimate (2) needs the argument on some variant bilinear

estimate together with (2), because, for instance,
though, as far as we work for$s>- \frac{1}{2}$$H_{t}^{b}\not\subset L_{t}^{\infty}$$b= \frac{1}{2}$. Al-, not including the equal case, the paper[15] could show the local well-posedness in$H^{s}(\mathrm{T})$, whereRemark 1.2 The result of the local well-posedness in$s=- \frac{1}{2}$for$\mathrm{T}$$=\mathbb{R}/\mathbb{Z}$$H^{s}(\mathrm{T})$.

for the end point

was recently solved by J. Colliander, M. Keel, G. Staffilani, T. $Tao$ andthe author [9].
Now, we turn out attention to the opposite view on the failure of estimate(2). The same paper of [15] proved the following results, where they constructed

the counterexample.
Proposition 1.3 ([15]) For any$s<- \frac{3}{4}$(resp.

$s<- \frac{1}{2}$) and any$b\in \mathbb{R}$, theestimate (2) for the line case (resp. periodic boundary condition case) breaks

down.

It is remarked that their examples do not cover the end point
problem whether$s=- \frac{3}{4}$$s=- \frac{3}{4}$. The

was left open. Recently, this problem is fixed by K.

Nakanishi, Y. Tsutsumi and the author [23].

Proposition 1.4 ([23]) The bilinear estimate (2) of the line case fails for $s=$$- \frac{3}{4}$and any$b\in \mathbb{R}$.

86
Of cause, it is not enough to conclude the ill-posedness results ofin$H^{s}(\mathbb{R})$for$s<- \frac{3}{4}$, also for$s=- \frac{3}{4}$$\mathrm{K}\mathrm{d}\mathrm{V}$equationby the aid of Propositions 1.3 and 1.4.

We can find the results not only the failure of (2) but also the construction of
the exact example for the ill-posedness of$\mathrm{K}\mathrm{d}\mathrm{V}$equation.

Proposition 1.5 ([17]) The Cauchy problem of $KdV$ equation for the line caseis ill-posed in$H^{s}$

for$s<- \frac{3}{4}$.

For this proposition, it is open whether the case of$s=- \frac{3}{4}$is well-posed or

ill-posed. It is noted that the paper [2] gave the explicit ill-posedness results of$\mathrm{K}\mathrm{d}\mathrm{V}$equation for the periodic boundary condition case. Moreover, the papers$[1, 2]$and [14, 15, 17] covered the results for the modified$u_{t}+u_{xxx}\pm(u^{3})_{x}=0$$\mathrm{K}\mathrm{d}\mathrm{V}$equation:.

(3)

As one guesses, the trilinear estimate is treated there; for example,$||(uvw)_{x}||_{s,b-1}\sim<||u||_{s,b}||v||_{s,b}||w||_{s,b}$.

(4)

Their results are sketched in the following table:line
periodicline mKdV
periodic mKdV$\mathrm{K}\mathrm{d}\mathrm{V}$well-posedness$s>--$$s\geqq--$$\mathrm{K}\mathrm{d}\mathrm{V}$ill-posedness$s<--$$s<--$$s<-$$s<-$$s\geqq-$$s\geqq-$(2) fails for $s\leqq--$(2) fails for $s<--$(4) fails for $s<-$(4) fails for $s<-$

Icomplete this section by denoting the best known results for the local wellposedness of$\mathrm{K}\mathrm{d}\mathrm{V}$equation by Kenig-Ponce-Vega [15].

Theorem 1.1 ([15]) The Cauchy problem of $KdV$ equation for the line case islocally well-posed in$H^{s}$

for$s>- \frac{3}{4}$.

2Global well-posedness of KdV equation for theline case
It is natural to extend the local solution to global one, once the local solution is

obtained. One may expect the global solution via the iteration of the proof of
local well-posedness. However, iteration methods can not by themselves yiel$\mathrm{d}$87the global solution. In general, the proof of global well-posedness relies on the
combination of the proof of local well-posedness and the apriori estimate ofsolution. It is known that the conserved quantities are available for providing
equationconservation law for
the apriori estimate. Indeed, the use of$L^{2}$$\mathrm{K}\mathrm{d}\mathrm{V}$leads the following theorem.
Theorem 2.1 ([1]) The Cauchy problem of $KdV$ equation for the line case isglobally well-posed in$H^{s}$

for$s\geq 0$.

But, there is none that the global well-posedness below$L^{2}$, because of the lack

, we haveequation. As for the results below
of conservation law for thethe following theorem, which is joint study with J. Colliander and G. Staffilani.
$L^{2}$$\mathrm{K}\mathrm{d}\mathrm{V}$Theorem 2.2 ([7]) Let $\frac{a}{12}<s<0$ and$a \geq-\frac{3}{4}$. Then the Cauchy problem, where theof $KdV$ equation for the line case is globally well-posed in
space
denotes the usual homogeneous Sobolev space of order .
$H^{s}\cap\dot{H}^{a}$$a$$\dot{H}^{a}$

The proof of theorem is based on the new argument introduced by Bourgain [3].
He showed that the Cauchy problem of two dimensional nonlinear Schrodinger

equation with the$L^{2}$

critical nonlinearity:$i\partial_{t}u+\triangle u=u|u|^{2}$$V_{t}$$(t, x)\in \mathbb{R}^{1+2}$,(5)

determined by the conservaand $V(t)$ denote the flow maps of the nonlinear and the linear

was globally well-posed intion laws. Let,$H^{s}$

between$L^{2}$

and$H^{1}$

Schr\"odinger equations, respectively. Let $X$ and$\mathrm{Y}$be Sobolev spaces such that

$X\subset 1^{\Gamma}arrow$’

A:conservation law class,$Y$: initial data space.

His strategy is that if $(V_{t}-V(t))u_{0}\in X$ for$\forall t\in \mathbb{R}$, then we have the global

well-posedness in Y. Roughly speaking, his argument aims to estimate the highSobolev norm of solution by the low Sobolev norm, which tells us the informationof aspread of energy between the low frequency and the high frequency. He. We follow his strategy
applies this argument to (5) with $X=H^{1}$ and$\mathrm{Y}--H^{s}$

88
and try to have$(S_{t}-S(t))u_{0}\in L^{2}$

for$t\in \mathbb{R}$, where$S_{t}$

denotes the flow map of

because of $X=L^{2}$ and$\mathrm{Y}=H^{s}$,$\mathrm{K}\mathrm{d}\mathrm{V}$(6)equation and $S(t)=e^{-t\partial_{xxx}}$ ,in our case.

There are some differences between [3] and our problem. In [3], the nonlinearity without the derivative was considered in the usual Sobolev space

the positive index. For the case of the$\mathrm{K}\mathrm{d}\mathrm{V}$$H^{s}$

withequation, the difficulty of deriva-

tive loss stems from the derivative nonlinearity of$\mathrm{K}\mathrm{d}\mathrm{V}$equation. Moreover in[7], we consider the well-posedness for the negative index

of H8. To show
the well-posedness, we use the Fourier restriction norm method, which was also
applied to the local well-posedness for. When we follow one and the same
argument as Bourgain’s with the Fourier restriction norm method, it is required
to show the following bilinear estimate for some $\gamma<0$ :$s$$\mathrm{K}\mathrm{d}\mathrm{V}$$||(vw)_{x}||_{0,b-1}\leq c||v||_{0,b}||w||_{\gamma,b}$.

(7)

A glance of (7) recalls us the bilinear estimate (2) in section one. Unfortunately,unless $7\geq 0$ , the above estimate (7) fails for any $c>0$ and any$b\in \mathbb{R}$, wherethere is acounter example. One derivative can be regained by the smoothing
effect of$\mathrm{K}\mathrm{d}\mathrm{V}$equation, but we have to gain the derivative of order more than one

for the above bilinear estimate (7). In [3], the smoothing effect of Schrodinger
equation was available, where there is no derivative nonlinearity in (5). But
the above bilinear estimate (7) holds when we restrict the frequency of both
$v$

and

$w$

to high frequency. The failure occurs in the interaction between the

low frequency and the high frequency. This observation motivates us to adopt

the framework of homogeneous Sobolev space. Then we use the homogeneous

Sobolev space$\dot{H}^{\epsilon}$of negative index and we try to have
$S_{t}u_{0}\in\dot{H}^{a}$for$t\in \mathbb{R}$,(8). We note that the framework of homogeneous space is convenient to thegain of more regularity than in the inhomogeneous space. But the homogeneous

Sobolev space of negative order is not conserved for the$\mathrm{K}\mathrm{d}\mathrm{V}$equation. We have89
to evaluate precisely the growth order of this norm. This is an essential difference

from [3]. Aconsequence of (6) and (8) is Theorem 2.2.
Roughly speaking, the advantage of [3] is related with the division of solution

into three potions:$S_{t}u_{0}=S_{t}(u_{0}^{1\mathrm{o}\mathrm{w}})+S(t)u_{0}^{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}+$where$u_{0}^{1\mathrm{o}\mathrm{w}}$and$u_{0}^{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}$(9)(error term),mean respectively the low frequency and the high fre-

. The evaluation of low frequency associated with the nonlin, thanks
equation can be controlled in the conserved space
ear flow of. On the other hand, the evolution of high frequency
to the smoothness ofquency parts of$u_{0}$$L^{2}$$\mathrm{K}\mathrm{d}\mathrm{V}$$u_{0}^{1\mathrm{o}\mathrm{w}}$concerning the linear flow of$\mathrm{K}\mathrm{d}\mathrm{V}$equation is globally well-posed. Bourgain has

demonstrated in his proof that the error term, which stems from the interaction between the low frequency and the high frequency, is very small. Then we
conclude that most part of solution is represented by the first two portions of(9), which dominates the behavior of solution. Note that the smallness of error
term is derived by the gap of estimate between the high Sobolev norm and thelow one of the estimate (6).

Remark 2.1 The above argument is also applicable to another kind nonlinear dispersive equations and nonlinear wave equations, see [4, 3D-NLS], [12,$mKdV]$, [16, semilinear $NLW$],$l^{\mathit{2}}\mathit{4}$, KP-2], [25, DNLS], [26, KP-2].

3Sharp global well-posedness for the KdV equathan
equation
We again come to the position for the global well-posedness ofspace, we find the. Concerning the local regularity weaker thanbelow$\mathrm{K}\mathrm{d}\mathrm{V}$$L^{2}$$L^{2}$

result of Theorem 2.2.
Problem 2In Theorem 2.2, to obtain a global solution, we have imposed an. The proof
additional assumption in the homogeneous Sobolev space$\dot{H}^{a}(\mathbb{R})$controls the high regularity by low one, so that it seems difficult to cover all
local solution in section one. Furthermore, from such a reason, it seems to bedifficult to adapt the proof of section two to the periodic boundary condition
problem, because this problem has no dispersive smoothing effect of solution.

90
The main result in this note is the following theorem, which is ajoint work with

J. Colliander, M. Keel, G. Staffilani and T. Tao.

Theorem 3.1 (Colliander, Keel, Staffilani, T. and Tao [9]) The Cauchy
problem of KdV for the line case is globally well-posed in.
for s$H^{s}$$>- \frac{3}{4}$This theorem allows us to succeed in solving the sharp global well-posedness

for the line case. In the rest of this note, Iwant to outline of the proof. Our
proof is also motivated by the argument of Bourgain in section two. However

we do not evaluate the high Sobolev norm of solution by the low one, which isasignificant difference from section two. Turn to the proof of theorem, my goalis the following:

Goal 1Give the a priori estimate forlinear operator from$m(\xi)=1$

for

to

$H^{s}$$|\xi|\sim<N$,$L^{2}$$E_{I}^{2}(t)=||Iu(t)||_{L^{2}}^{2}$, where I denotes thedefinedJ with the Fourier multiplier$m(\xi)=|_{N}^{\epsilon}|^{s}$for $|\xi|>>N$ ,$m(\xi)\in C^{2}$$m(\xi)$such that. We will mentionof $N>>1$ soon later.
It is noted that$||u(t)||_{L^{2}}$$E_{I}^{2}(t)$

”

$\{$

$||u(t)||_{H^{s}}$,if support\^u\subset $\{|\xi|\sim<N\}$ ,, if support\^u\subset $\{|\xi|\gg N\}$ ,where we transfer the Fourier transform in of $u(t, x)$ by the same notion \^u as
before, for simplicity. Therefore, the quantity
seems to be looked like the$x$$E_{I}^{2}(t)$combination ofandnorms. The inverse operator
leads the apriori
estimate in H8, once the apriori estimate foris obtained. So the problem$L^{2}$$H^{s}$$I^{-1}$$E_{I}^{2}(t)$comes down to the time global estimate for. The standard calculation
with the use ofequation yields the estimate.
$E_{I}^{2}(t)$$\mathrm{K}\mathrm{d}\mathrm{V}$Lemma 3.1 Let$s>- \frac{3}{4}$. For the solution of $KdV$ equation $u(t)$ , we have$\frac{d}{dt}E_{I}^{2}(t)=\int M_{3}(\xi_{1},\xi_{2},\xi_{3})\hat{u}(t,\xi_{1})$\^u(t, )$\xi_{2}$$u\wedge(t,\xi_{3})\delta(\xi_{1}+\xi_{2}+\xi_{3})d\xi_{1}d\xi_{2}d\xi_{3}$where$\Lambda f_{3}(\xi_{1}, \xi_{2}, \xi_{3})=\xi_{1}m(\xi_{1})^{2}+\xi_{2}m(\xi_{2})^{2}+\xi_{3}m(\xi_{3})^{2}$

.

,

91
Remark 3.1 A glance of the above expression shows us the cancellation of theinteraction among the low frequencies, since$M_{3}(\xi_{1}, \xi_{2}, \xi_{3})=0$

under $\xi_{1}+\xi_{2}+\xi_{3}=0$ . Thus, the quantity$E_{I}^{2}(t)$for $|\xi_{i}|<<N$

acts like almost conserved

quantity.

The energy transportation is observed by the following lemma.
Lemma 3.2 Let s$>- \frac{3}{4}$. For the solution of KdV equation $u(t)$ , we have$| \int_{0}^{T_{\mathrm{O}}}\frac{d}{ds}E_{I}^{2}(s)ds|\leq N^{-1+}||Iu_{0}||_{L^{2}}^{3}$

where

,

denotes the existence time assured by the proof of local well-posedness.

$T_{0}$

We may assume thatscaling of$\mathrm{K}\mathrm{d}\mathrm{V}$$||Iu_{0}||_{L^{2}}$is very small. This request is carried out by theequation:$u_{\lambda}(t, x)= \frac{1}{\lambda^{2}}u(\frac{t}{\lambda^{3}}, \frac{x}{\lambda})$,which leads the desired condition:$||Iu_{\lambda}(0)||_{L^{2}}\sim<\lambda^{-s-\frac{3}{2}}N^{-s}||Iu_{0}||_{L^{2}}=O(1)$for$\lambda\sim N^{-\frac{2}{3+2s}}.$,. Moreover, by the proof of local well-posedness, it is all right totake $T_{0}=1$ in Lemma 3.2. Fix $T>0$ , the goal of this section will be replaced

as follows.
Goal 2Assume $||Iu_{0}||_{L^{2}}=O(1)$ . Give the a priori estimate up to time t$\lambda^{3}T$$=$.

By Lemma 3.2, we have
$E_{I}^{2}(1)-E_{I}^{2}(0)\leq N^{-1+}$

.

(10)

Some iteration method using Lemma 3.2 can controls the spread of energy as far
as the correspoinding right hand side of (10) never be greater than$||Iu_{0}||_{L^{2}}=$$O(1)$ .

In order to reach up toat least$\lambda^{3}T$

steps. Thus, if$- \frac{6s}{3+2s}-1<0$, that is,$t=\lambda^{3}T$, we have to iterate the above procedure$\frac{\lambda^{3}T}{N^{1-}}\sim<1$$s>- \frac{3}{8}$, we achieve the goal, which holds forand for large $N>0$ . Here we make asuccess92of the removal of the additional assumption on data in$\dot{H}^{a}$However the gap between the sharp local well-posedness in
the result for$s>- \frac{3}{8}$of Theorem 2.2.

$H^{s}$

for$s>- \frac{3}{4}$andremains yet.

Observation. If
can be divided into two portions,
and
. The
first term
dominates the quantity ofbutis controls by(not so big). Therefore the second term
will correspond to the the more$\frac{d}{dt}E_{I}^{2}(t)$$\mathrm{a}[(\mathrm{t})$$a_{1}’(t)$$b_{1}(t)$$a_{1}(t)$$\frac{d}{dt}E_{I}^{2}(t)$$E_{I}^{2}(t)$$b_{1}(t)$explicit estimate for$\frac{d}{dt}E_{I}^{2}(t)$.

Following this observation, we have obtained the following lemma.
Lemma 3.3 Let$a_{1}(t)= \int\frac{M_{3}(\xi_{1},\xi_{2},\xi_{3})}{\xi_{1}^{3}+\xi_{2}^{3}+\xi_{3}^{3}}\hat{u}(t,\xi_{1})$\^u(t, )$\xi_{2}$$u\wedge(t,\xi_{3})\delta(\xi_{1}+\xi_{2}+\xi_{3})d\xi_{1}d\xi_{2}d\xi_{3}$$b(t)= \frac{d}{dt}(E_{I}^{2}(t)-a_{1}(t))$

Let$s>- \frac{3}{4}$,.

. Then for the solution of $KdV$ equation $u(t)$ , we have$|a_{1}(t)|\leq o(E_{I}^{2}(t))$,$| \int_{0}^{1}b(t)dt|\sim<N^{-\mathrm{g}_{+}}2||Iu_{0}||_{L^{2}}^{4}$.

This lemma brings us the more explicit information of the energy transportation
than Lemma 3.2. The analogous argument to above will reduce the relation$\frac{\lambda^{3}T}{N^{1-}}\sim<1$to$\frac{\lambda^{3}T}{N\S-}\leq 1$, which holds for$s>- \frac{1}{2}$. However, unfortunately, there

appears agap again. It is remarkable that the quantity $E_{I}^{2}(t)-a_{1}(t)$ in Lemma

3.3 corresponds to theI as$\partial_{x}$$H^{1}$

energy conservation law of$\mathrm{K}\mathrm{d}\mathrm{V}$equation by regarding. So we recognize that the above argument is based on the use of$H^{1}$

conservation law as compered to the proof in section two. As one guesses, the$\mathrm{K}\mathrm{d}\mathrm{V}$equation has an infinite conservation laws, then we surely develop morecomputation by means of$H^{2}$

conservation law according to above observation.

Lemma 3.4 Let$M_{4}(\xi_{1},\xi_{2},\xi_{3},\xi_{4})$$=$,$\frac{M_{3}(\xi_{1},\xi_{2},\xi_{3}+\xi_{4})(\xi_{3}+\xi_{4})+(permutationamong\xi_{1},\xi_{2},\xi_{3},\xi_{4})}{6(\xi_{1}^{3}+\xi_{2}^{3}+\xi_{3}^{3}+\xi_{4}^{3})}$93$a_{2}(t)$\^u(t, )$= \int M_{4}(\xi_{1},\xi_{2},\xi_{3},\xi_{4})\hat{u}(t,\xi_{1})$$d\xi_{1}d\xi_{2}d\xi_{3}d\xi_{4}$$s>- \frac{3}{4}$$u\wedge(t,\xi_{3})\hat{u}(t,\xi_{4})\delta(\xi_{1}+\xi_{2}+\xi_{3}+\xi_{4})$,$b_{2}(t)= \frac{d}{dt}$Let$\xi_{2}$($E_{I}^{2}(t)-a_{1}(t)$-a2(t)).

. For the solution of $KdV$ equation $u(t)$ , we have$|a_{2}(t)|\leq o(E_{I}^{2}(t))$,$| \int_{0}^{1}\frac{d}{dt}b_{2}(t)dt|\sim<N^{-\mathrm{s}+}[|Iu_{0}||_{L^{2}}^{5}$.

The order $N^{-3+}$ of Lemma 3.4 is sufficient to win the sharp global well-posednessof$\mathrm{K}\mathrm{d}\mathrm{V}$equation for$s>- \frac{3}{4}$, in accordance with the above scheme.

$\mathrm{O}.\mathrm{K}$.

Iwould like to complete this note by noting further application of our argument.

Further applications. It is noted that the proof of section three does not needthe dispersive smoothing effect of solution directly in contrast to section two.
In fact, we have the sharp global well-posedness for the periodic$\mathrm{K}\mathrm{d}\mathrm{V}$equation[9]. Moreover, the same paper knows the sharp results on the modified$\mathrm{K}\mathrm{d}\mathrm{V}$equation (3) for both line and periodic boundary condition cases. More de-tails and the application to another type equations are developed in the papers[8, 9, 10, 11].

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