数理解析研究所講究録1102 巻 1999 年 1-8

1Time local well-posedness for the KP II equation

東京大学大学院数理科学研究科
高岡秀夫 Hideo TAKAOKA)

Introduction
We consider the well-posedness for the Cauchy problem of the KadomtsevPetviashvili 1I (KP II) equation:(1)(2)$\partial_{\dot{x}}(\partial_{t}u+\partial 3ux+\partial_{x}(u)2)+\partial_{y}2u=0,$$u(0, x, y)=u0(_{X,y)},$$(t,x, y)\in[-T,T]\cross \mathbb{R}^{2}$$(X, y)\in \mathbb{R}^{2}$,,

where the unknown function is a real valued and gives a time interval
to be determined later.
While the equation (1) is known as the KP II equation, the following
equation corresponds to the KP I:
$u$

$T$

$\partial_{x}(\partial_{t}u+\partial^{3}u+x\partial x(u^{2}))-\partial 2\mathrm{o}yu=$.

These KP equations describe a propagation of a weakly nonlinear dispersive
long wave which is essentially one dimensional with a weak transverse effect[3].
Our purpose ofnote is to consider the well-posedness for the Cauchy
problem of the KP 1I equation in a weak class. As usual, we rewrite theCauchy problem (1) $-(2)$ as the integral equation and use the contraction
argument. The usual contraction argument seems meet with the difficultyof the derivative loss which stems from the derivative nonlinearity of theequation (1).
$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{S}$Known results

2
There are alarge amount of work for the Cauchy problem (1) $-(2)$ (see e.g.,[2,5,6,7] . Recently J. Bourgain [2] showed the time local well-posedness innorm, the time global well. By the conservation law of theposedness was also obtained in the same space.
this note we consider the equation (1) in the integral form. On the other
hand, N. Tzvetkov [6] considered the following equation in the distribution
sense instead of the equation (1):$)$$L^{2}$$L^{2}(\mathbb{R}^{2})$

$\ln$

(3)

$\partial_{t}u+\partial_{x}^{3}u+\partial_{x}(u^{2})+\partial_{x}^{-12}\partial_{y^{u}}=0$.

For the Cauchy problem (3) $-(2)$ , the time local well-posedness was shown in,for $s_{1}>-1/4$ and $s_{2}\geq 0[6]$ , whereis the anisotropic Sobolev
space with the following norm:$\tilde{H}_{x,y}^{s_{1^{S_{2}}}}$$\tilde{H}_{x^{1}y}^{s},’ S_{2}$$||f||_{\overline{H}_{x^{1^{s}2}}}s,\nu’=||(1-\partial_{x}^{2})^{s}1/2(1-\partial_{y}2)s_{2}/2f||_{L}2x,\nu+||(-\partial_{x}2)-1/2(1-\partial_{y}^{2})s_{2}/2f||_{L^{2}}x,y$.

is necessary. ThisIn [6], the conditionhomogeneous Sobolev space of index-l in variable seems natural for theequation (3). But even if the equation is considered in the integral form (1)instead of the differential form (3), the proof of [6] seems to require the useof the homogeneous Sobolev space of negative index. We remark [6] does
not cover, nor be covered by [2].
$(-\partial_{x}^{2})^{-}1/2(1-\partial^{2})^{S}y2/2u0\in L^{2}(\mathbb{R}^{2})$$x$Results of this note
To state our theorems, we give some notations.

Definition 1. Let
denote the Fourier transformation of $g(t, x, y)$in the time and the space variables, i.e.,$\sim g(\tau,\xi, \eta)$$\sim g(\tau,\xi, \eta)=\int\int\int_{\mathrm{R}^{3}}\exp(-it\tau-ix\xi-iy\eta)g(t,X,y)dtdxdy$

Let
and

$\psi\in C_{0}^{\infty}(\mathbb{R})$.

denote a smooth cut off function such that $\psi=1$ on [-1, 1]. For $\delta>0$ , we put. Let$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\psi\subseteq(-2,2)$$\psi_{\delta}(t)=\psi(t/\delta)$$k( \tau,\xi,\eta)=\tau-\xi 3+\frac{\eta^{2}}{\xi}$.

, we define the anisotropic Sobolev spaceForfollowing norm:$s_{1},$ $s_{2}\in \mathbb{R}$$||f||_{H}s,’ sx^{1}y2=||(1-\partial_{x}2)s1/2(1-\partial_{y}^{2})s_{2}/2f||_{L_{x,y}^{2}}$$H_{x^{1}y}^{S,s},2$.

with the3, we define the space
to be the completion of the
Schwartz function space onwith the following norm:For$s_{1},$ $s_{2},$$b\in \mathbb{R}$$X_{s_{1},S_{2}},b$$\mathbb{R}^{3}$,$||g||xs_{1^{s}2^{b}},,=||g||_{S_{1},S_{2}},b+||g||_{-}3/4,S_{2},3/4$

where$||g||s_{1},S2)=b||\langle\xi\rangle^{s}1\langle\eta\rangle^{S_{2}}\langle k(\mathcal{T},\xi, \eta)\rangle^{b\sim}g(\mathcal{T},\xi,\eta)||_{L_{r,\xi,\eta}^{2}}$Theorem 1. Let $s_{1}>-1/4$ andfollowing properties. For any
solution of the Cauchy problem (1)that$s_{2}\geq 0$. Then there exists $b>1/2$ with the)$u_{0}\in H_{x,y^{S}}^{s_{12}}$$u$$-(\mathit{2})$, there exist $T>0$ and a uniquein the time interval $[-T,T]$ such$u\in C([-\tau,\tau]:H^{Ss}1_{),y}2)x,’\psi\tau(t)u\in xs_{1},S_{2},b$

For any $T’\in(0, T)$ , there exists

from$\{v_{0} :||v_{0}-u_{0}||H_{x,y}s1’ s_{2}<\epsilon\}$

In $addition_{f}$ if
holds withandinterval $[-T,T]$ .

$\epsilon>0$

to

with
replaced by$u_{0}\in H_{x^{1}y}^{s’,s_{2}’}$$s_{1}$,$s_{2}$.

such that the map.

$v_{0}\mapsto v$

is Lipschitz$||v-u||L\infty(\tau’ y’)H_{x^{1^{s}2}}^{\theta},+||\psi\tau’(v-u)||X_{\theta}1^{s_{2^{b}}}$”.

andthen the above resultand , respectively, in the same time$s_{1}’\geq s_{1}$$s_{1}’$$s_{2}’\geq s_{2;}$$s_{2}’$

Remark 1. The admissible values ofachieved by Theorem 1 are thesame as in [6]. In Theorem 1, we do not require the use of the homogeneous
space of negative order (see [6]). Therefore, Theorem 1 recovers thetime
global well-posedness [2].
$(S_{1}, s_{2})$$L^{2}$

Remark 2. Scaling argument. If
does
(4)

$u$

solves the Cauchy problem (1) $-(2)$ , so$u_{\lambda}(t,x,y)=\lambda 2(u\lambda 3t, \lambda X, \lambda^{2}y)$,for any
with the data
. Then the order difference
on betweenandin (4) suggests that the anisotropic Sobolev space,
may be natural for the Cauchy problem (1) $-(2)$ , rather than the usual
Sobolev space. Taking the homogeneous norm, we have$\lambda\in \mathbb{R}$$\lambda$$u_{\lambda}(\mathrm{O})=u_{\lambda}(0, X, y)$$\lambda x$$\lambda^{2}y$$H_{x,y}^{s_{1^{S_{2}}}}$$H^{s}(\mathbb{R}^{2})$$||u_{\lambda}(\mathrm{o})||_{\dot{H}_{x^{1}\nu}}s,’ s_{2}=\lambda^{S_{1}+2}S_{2}+1/2||u_{0}||_{\dot{H}}x^{1^{s_{2}}}s,y’$.

Then the norm of
is invariant for $s_{1}+2s_{2}+1/2=0$ under the scalingtransformation (4). This argument suggests that the critical values for the
Cauchy problem (1) $-(2)$ in
may be $s_{1}+2s_{2}+1/2=0$ . Note that theadmissible values
achieved in Theorem 1 are far from the criticalvalues.
To state our second theorem of this note, we introduce some notations.
$\dot{H}_{x^{1}y^{S}}^{S},’ 2$$H_{x^{1}y^{S}}^{s},’ 2$$(s_{1}, s_{2})$4, we define the anisotropic homoDefinition 2. For $s_{1}>-1/2$ and,as follows:
geneous Sobolev space$s_{2}\in \mathbb{R}$$\overline{H}_{xy)}^{s_{1^{S}2}}$$\overline{H}_{x^{1}y}^{s},’ S_{2}=\{f\in s’(\mathbb{R}2):||f||\overline{H}_{x}^{s,s_{2}},1y<\infty\}$

,

where
$||f||\overline{H}_{x,y}^{SS}1’ 2=||(-\partial_{x}^{2})s_{1}/2(1-\partial^{2})^{s}y2/2f||L_{x,y}2$

For$s_{1},s_{2},b\in \mathbb{R}$and $a>-1/2$ , we define the space.

$Y_{S_{1},S_{2},a,b}$

as follows:$Y_{S_{1},S_{2}},a,b=\{g\in S’(\mathbb{R}^{3}):||g||_{\mathrm{Y}}\epsilon 1,s_{2^{a,b}},=||g||_{s_{1}},S_{2},b+|||g|||a,s2,a+1<\infty\}$where$|||g|||_{a}\}S_{2},a+1=|||\xi|a\langle\eta\rangle^{s_{2}}\langle k(\mathcal{T},\xi,\eta)\rangle^{a}+1g\sim(\mathcal{T},\xi,\eta)||L^{2}r,\xi,\eta$,.

Theorem 2. Let $s_{1}>3a+1>-1/2,$ $-1/2<a<-1/4$ and $s_{2}\geq 0$ .
Then there exists $b>1/2$ with the following properties. For any
there exist $T>0$ and a unique solution of the Cauchyin the time interval $[-T,T]$ such thatproblem (1)$u_{0}\in$$u$$H_{x,y}^{\theta_{1_{)}}}s_{2}\cap\overline{H}_{x,y\prime}^{a,s_{2}}$$-(\mathit{2})$$u\in C([-T,\tau]:H_{x,y}s1,s2\mathrm{n}\overline{H}a,S)x,y^{2}’\psi\tau(t)u\in \mathrm{Y}_{Ssa}1,2,,b$

For any $T’\in(0,T)_{f}$ there exists

from$\{v_{0} :$\epsilon>0$.

is Lipschitz
such that the map
to $||v-u||L^{\infty}T’(H_{x},1’)ss_{2}y+||v-$$v_{0}\mapsto v$||v_{0}-u0||H_{x^{1}y}s,’ s2+||v_{00}-u||\overline{H}_{x,y}a,s_{2}<\epsilon\}$$u||_{L_{T’}(\overline{H}^{\mathrm{O}}’}\infty x,ys2)+||\psi_{\tau}’(v-u)||_{\mathrm{Y}}s1,S2,’\circ,b$.

$-1/4>a’\geq a,$ $s_{1}’>3a’+1$
with
In addition; if, then the above result holds with , and replaced byandrespectively, in the same time interval $[-T,T]$ .
and$u_{0}\in H^{s^{l}}x^{1},y’\cap\overline{H}^{a’,S}x,ys_{2}/2$$s_{1}’\geq s_{1},$$s_{2}’\geq s_{2}$$s_{1}$$a$$s_{2}$$s_{1}’,$$a’$$s_{2f}’$Remark 3. When $s_{2}=0$ , the scaling argument of Remark 2 suggests that
Theorem 2 may be the best possible result, except for the limiting casein Theorems$s_{1}=-1/2$ which remains an open problem. We assume
1 and 2. In the case of $s_{2}<0$ , the well-posedness result remains to be an
open problem.
$s_{2}\geq 0$

Remark 4. The algebraic relation (9) below plays an important role to overcome the difficulty of a loss of the derivative. In the case of the KP I
equation, our method seems useless to avoid this difficulty (see [2, \S 10]).
The proofs of Theorems 1 and 2 are based on the Fourier restriction norm
method $[2,4]$ . This method has been used for the dispersive equation withand the KP II equations.
the quadratic nonlinearity, such as the$\mathrm{K}\mathrm{d}\mathrm{V}$5
There are several differences between the Cauchy problem for theequation and that for the KP II equation. In the case of theequation,
the following smoothing effect is valid to overcome a loss of the derivative:$\mathrm{K}\mathrm{d}\mathrm{V}$$\mathrm{K}\mathrm{d}\mathrm{V}$$||\partial_{x}\exp(-t\partial_{x}^{3})u_{0||_{L_{x}^{\infty}}}L_{l}^{2}\leq c||u0||L_{x}^{2}$.

On the other hand, it seems difficult to see such a kind of smoothing effect for
the KP II equation case. Then using an argument similar to that of theequation case, we need to consider the integrability in variable, where is
the Fourier variable in . To get such the integrability, we define the spaces
for the proofs of Theorems 1 and 2, respectively. More
, andprecisely, the second terms that
andplay a role,
to get such an integrability for the proofs of Theorems 1 and 2, respectively.
These Fourier restriction norm spaces
are the essential, anddifference from $[2,6]$ .

$\mathrm{K}\mathrm{d}\mathrm{V}$$\eta$$\eta$$y$$X_{s_{1)}S_{2}b}$$\mathrm{Y}_{S_{1^{S_{2)}}}},a,b$$||\cdot||_{-3/4_{S_{2}},3/4}$$X_{S_{1},S_{2}b}$$|||\cdot|||_{a,s_{2},a+1}$$\mathrm{Y}_{s_{1},s_{2},a,b}$

Proofs of Theorems 1 and 2

We regard the Cauchy problem (1) $-(2)$ as the following integral equation:$u(t)=W(t)u0- \int^{t}0)W(t-S)\partial_{x}(u(S)2d_{S}$

,

where
.
First we state the lemma concerning the estimates of the linear and thenonlinear part of the KP II equation in the function spaces we consider.
Lemma 1. Let $1/2<b<b’<1$ . For $\delta\in(0,1)$ andwe have$W(t)=\exp(-t(\partial_{x}^{3}+\partial_{x}^{-1}\partial 2)y)$$s_{1},$ $s_{2}\in \mathbb{R}_{f}$(5)$||\psi_{\delta}(\cdot)W(\cdot)u0||s1,S_{2},b\leq C\delta^{1/}2-b||u_{0}||H_{x^{1^{\epsilon}2}}\epsilon,y$

”

.

(6)

(7)

$|| \psi_{\delta}(\cdot)\int_{0}$$W(\cdot-t’)F(t)/d\mathrm{t}’||_{s_{1,2}}S,b\leq c\delta^{1/}2-b||F||_{s}1^{S},2,b-1$$||\psi_{\delta}(\cdot)F||_{s_{1},s_{2}},b-1\leq c\delta^{b’-}b||F||S_{1},s_{2},b’-1$,.

See [4], for the proof of Lemma 1.
Remark 5. Lemma 1 holds with.
andreplaced byand, respectively.
The following two propositions play an important role in the proofs ofTheorems 1 and 2, respectively.
$||\cdot||_{s_{1},s_{2},b}$$|||_{S_{1)}S_{2},b}$$||\cdot||_{\overline{H}_{x,y}^{s_{1},s_{2}}}$$||\cdot||_{H_{x,\nu}^{s_{1},s}}2$

$|||$

6

$\mathrm{P}\mathrm{r}o$position 2. Let $s_{1}>-1/4$ andwith the following properties. For$s_{2}\geq 0$$\theta>0$(8). Then there exist $b>1/2$ and$T\in(\mathrm{O}, 1)_{f}$we have$||\psi_{\tau}\partial_{x}(u^{2})||s1,s_{2},b-1\leq C\tau\theta(Tb-1/2||u||S_{1^{S_{2}}},,b+T^{1/}4||u||-3/4,s_{2},3/4)$$\cross T^{b-1/2}||u||s_{12},sb1$’$||\psi_{\tau}\partial_{x}(u^{2})||-3/4_{S_{2}},,3/4-1\leq cT\theta(\tau b-1/2||u||S_{1},s_{2},b)2$.

.
Proposition 3. Let $s_{1}>3a+1>-1/2,$ $-1/2<a<-1/4$ andThen there exist $b>1/2$ and $\theta>0$ with the following properties. For
we have$s_{2}\geq 0$$T\in(\mathrm{O}, 1)_{f}$$||\psi_{T}\partial_{x}(u2)||_{S}1,s2\}b-1+|||\psi_{\tau^{\partial}}x(u2)|||_{a},s_{2},a$$\leq cT\theta(\tau b-1/2||u||S_{1},S_{2},b+\tau^{a}+1/2|||u|||a,s2\prime a+1)\tau^{b}-1/2||u||_{sS_{2},b}1,\cdot$

For the proofs of Propositions 2 and 3, we use the following identity [2]:$k(\tau,\xi, \eta)-k(\mathcal{T}_{1},\xi_{1}, \eta_{1})-k(\tau-\mathcal{T}1,\xi-\xi_{1}, \eta-\eta 1)$$3|\xi(\xi-\xi 1)\xi_{1}|^{2}+|\xi\eta 1-\xi 1\eta|^{2}$$=-\overline{\xi(\xi-\xi 1)\xi_{1}}$.

This identity implies(9)$\max\{|k(\mathcal{T},\xi,\eta)|, |k(\mathcal{T}1,\xi_{1}, \eta 1)|, |k(\mathcal{T}-\mathcal{T}_{1},\xi-\xi 1,\eta-\eta_{1})|\}$$\geq|\xi(\xi-\xi 1)\xi 1|+\frac{|\xi\eta_{1}-\xi_{1\eta}|^{2}}{3|\xi(\xi-\xi 1)\xi_{1}|}$.

Using this inequality, we overcome the difficulty of the derivative loss andthe following, we only give a sketch of
get the integrability in variable.
the proof of (8) in Proposition 2.
$\eta$$\ln$

Proof. For simplicity we consider (8) for the case of $s_{2}=0$ . We first consider

the following:(10)$( \int\int\int_{\mathrm{R}^{3}}\frac{|\xi|^{2}\langle\xi\rangle 2s_{1}}{\langle k(\tau,\xi,\eta)\rangle 2(1-b’)}$$( \iiint_{\mathrm{R}^{3}}|u(\mathcal{T}_{1},\xi 1, \eta 1)||u(\mathcal{T}-\tau_{1},\xi-\xi 1,\eta-\eta 1)|d\mathcal{T}1d\xi 1d\eta 1)^{2}d_{\mathcal{T}d\xi)}d\eta 1/2$,

7
where $2b-1/4<b’<1$ . For briefly we only consider (10) in the following
case:
(11)

$|\xi_{1}|\leq 1$

and $|k( \tau_{1},\xi_{1}, \eta_{1})|\geq\max\{|k(\tau,\xi, \eta)|, |k(\tau-\tau 1, \xi-\xi_{1}, \eta-\eta_{1})|\}$ .

Using the Schwarz inequality, we have that (10) is bounded by(12)$|\xi|\langle\xi\rangle^{s_{1}}$$c \sup_{\tau,\xi,\eta}\overline{\langle k(\tau,\xi,\eta)\rangle^{1-}b’}$$-$$\cross(\int\int\int \mathrm{R}^{3},\frac{\langle\xi_{1}\rangle^{3/2}\langle\xi-\xi 1\rangle-2s_{1}}{\langle k(\mathcal{T}_{1}\xi_{1,\eta_{1}})\rangle 3/2\langle k(\tau-\tau_{1},\xi-\xi_{1},\eta-\eta 1)\rangle^{2}b}d_{\mathcal{T}}1d\xi_{1}d\eta 1)1/2$$\cross||u||-3/4,0,3/4||u||_{S_{1},0},b$.

We use (9) first, integrate invariable and use the change of the variable
as $\mu=(\xi\eta_{1}-\xi_{1\eta})2/\xi(\xi-\xi 1)\xi 1$ . Noting that for the points in (11),, we have that the contribution of the region (11) to (12) isbounded by$\tau_{1}$$\eta_{1}$$\langle\xi\rangle\sim\langle\xi-\xi_{1}\rangle$$c \frac{|\xi|}{\langle k(\tau,\xi,\eta)\rangle 1-b’}(\int_{1\xi 1}|\leq 1\int^{\infty}-\infty\frac{|\xi_{1}|^{1/2}}{\langle|\xi(\xi-\xi_{1})\xi_{1}|+|\mu|\rangle 3/2|\mu|^{1/2}}d\mu d\xi_{1})1/2$$\cross||u||_{-}3/4,0,3/4||u||_{s_{1},0},b$$\leq c\frac{|\xi|}{\langle k(\tau,\xi,\eta)\rangle 1-b’}(\int_{1\xi|\leq 1}1\frac{d\xi_{1}}{\langle\xi\rangle^{2}|\xi_{1}|^{1}/2})^{1}/2-||u||3/4,0,3/4||u||_{s_{1}},0,b$$\leq c||u||_{-3/4,0,\mathrm{s}}/4||u||S_{1},0,b$.

Combining the above argument with (7) of Lemma 1, we have that the left
hand side of (8) restricted to the region (11) is bounded by$cT^{b’-b}||u||_{-}3/4,0,3/4||u||_{s_{1},0},b$$\leq cT^{b^{r}-2}b+1/4(\tau^{1}/4||u||-3/4,0,3/4)(\tau b-1/2||u||s_{1},0,b)$.

$\square$

Proof of Theorem 1.
We putfor $s_{1}>-1/4$ and $s_{2}\geq 0$ . For $T\in(0,1)$ and, we put $||w||x\tau=\tau^{b-1}/2||w||s1\mathrm{J}2S,b+\tau^{1/4}||w||_{-}3/4S_{2}3/$ . Now for, we define$r=||u_{0}||H_{x^{1^{s}2}}s,y$’$b\in \mathbb{R}$}$T\in(\mathrm{O}, 1)$$\mathfrak{B}(r)=\{w\in Xs_{1},S_{2},b :||w||\mathrm{x}\tau\leq 2cr\}$

,

’

$4$

8
.

$\Phi(u)(t)=\psi\tau(t)W(t)u0-\psi\tau(t)\int_{0}^{t}W(t-S)\psi\tau(s)\partial x(u(S)^{2})ds$

We choose same as in Proposition 2. Thus by (5) $-(6)$ of Lemma 1 and
Proposition 2, we have$b$$||\Phi(u)||\mathrm{x}T$$\leq c||u_{0}||_{H_{x,y}^{s}}1’ s2+c||\psi\tau\partial x(u2)||s1^{S},2,b-1+C||\psi\tau\partial x(u2)||-3/4_{S},2,\mathrm{s}/4-1$$\leq cr+CT^{\theta}||u||_{X}^{2}T$$\leq cr+cT\theta r^{2}$for,, where $\theta>0$ is the same as in Proposition 2. Similarly, if, we obtain$u\in \mathfrak{B}(r)$$u,v\in \mathfrak{B}(r)$$||\Phi(U)-\Phi(V)||X^{T}\leq cT^{\theta}r||u-V||X^{\tau}$.

Then we conclude that if we choose $T>0$ sufficiently small, thenisa contraction map. Then we obtain the unique local existence result inof the Cauchy problem (1) $-(2)$ by the contraction argument.
Replacing Proposition 2 by Proposition 3 in the proof of Theorem 1, we
obtain Theorem 2.
$\Phi$$\square$$X_{s_{1},s_{2},b}$

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$Schor\tilde{\mathit{0}}dinger$.O