数理解析研究所講究録 1204 巻 2001 年 77-84

77

Analytic Smoothing Effect for the Benjamin-Ono
Equations加藤圭一 (東京理科大・理)(Keiichi Kato)
小川 卓克 (九州大・数理) (Tabyoshi Ogawa)Elena I. Kaikina(Instituto Tecnologico de Morelia)
Pavel I. Naumkin(Universidad Michoacana)
1. INTRODUCTION
We study smoothing effect for the following nonlinear dispersive equation of the BenjaminOno type:(1.1)$\partial_{t}u+H_{x}\partial_{x}^{2}u+\partial_{x}u^{2}=0$$\{$,$t\in(-T,T)$ ,$x$$\in \mathbb{R}$,$v(0,x)=\phi(x)$ ,denotes the Hilbert transform
is aunknown function andwhere $u(t, x)$ :. This equation arises in the water wave theory and describes
defined bylong internal gravity wave in deep stratified fluid (see [2], [31]). Our problem here is toinvestigate asufficient condition of the initial data on which the solution has regularizing
property up to analyticity.
The existence and well-posedness problem of this equation is studied by many authors.
We refer to T. Kato [21], Iorio Jr. [14], Ponce [32], Kenig, Ponce and Vega [26] and reference therein. In the recent studies for the nonlinear dispersive equations, large amounts-posedness ofof studies are devoted to the smoothing effect. When we consider thebased (Sobolev) space is usually considered and the same orderthose type of equation,of the regularity for solutions is derived as the initial data . Concerning the dispersive, nonlinear Schr\"odinger and the Benjamin-Ono type equations, 10equation such as
cal or somewhat restricted version (in terms of weighted norm) of smoothing effect wasobserved. As the most well understood example, we would refer to the case of nonlinear
Schrodinger equations in [3], [4], [6], [9], [10], [11], [12],[18], [20], [30] and case of linear
Schrodinger equations in [16] and [33]. Since the Benjamin-Ono equation has asimilaras the Schr\"odinger equations, we would
dispersive structure in its linear part
expect that an analogous result holds for the nonlinear problem (1.1).
Concerning the analytic smoothing effect, we know that adrastic smoothing effectequation and nonlinear Schrodinger equations. Especially for theholds for theequation, it is shown that for aweak initial data including the Dirac delta measure, thecorresponding weak solution gains the regularity up to analytic in both space and time
variable by virtue of the conormal vector fields (see K.Kato and Ogawa [17] and also for$H_{x}$$\mathbb{R}\cross \mathbb{R}arrow \mathbb{R}$$u$$H_{ax}v=F \dot{.}\frac{\epsilon}{|\xi|}\hat{v}$$\phi$$\mathrm{w}\mathrm{e}\mathbb{I}$$L^{2}$$\phi$$\mathrm{K}\mathrm{d}\mathrm{V}$$\partial_{t}u+H_{x}\partial_{x}^{2}u$$\mathrm{K}\mathrm{d}\mathrm{V}$$\mathrm{K}\mathrm{d}\mathrm{V}$78
the Schrodinger cases [18] and K.Kato and Taniguchi [20] . In this paper, we would extend
these results to the Benjamin-Ono case (1.1). Our method is based on an operator method
which is common to the cases of theequation or nonlinear Schrodinger equations: We
introduce the generator of space-time dilationthat plays acompensatingrole where the main linear operator
can not gain the regularity. As aconsequence, we observe analytic smoothing effect for the solution to (1.1) with an initial
data having asingularity at one point. We state this more specifically as follows. Let$H$
be the Sobolev space of order defined by$)$$\mathrm{K}\mathrm{d}\mathrm{V}$$P=2t\partial_{t}+x\partial_{x}$$L=\partial_{t}+H_{x}\partial_{x}^{2}$.

$=H.(\mathbb{R})$$s$$||f||_{H}\cdot\equiv||\langle\xi\rangle.\hat{f}||_{2}$where$f\wedge=\mathcal{F}f$denotes the Fourier transform of

$f$

,

and

$\langle\cdot\rangle=(1+|\cdot|^{2})^{1/2}$.

Theorem 1.1. Let $s>3/2$ . Suppose that for some $A_{0}>0$ , the initial dataand satisfies$\sum_{k=0}^{\infty}\frac{A_{0}^{k}}{h!}||(x\partial_{x})^{k}\phi||_{H}\cdot<\infty$$\phi$$\in H.(\mathbb{R})$,then there exists a unique solution $u\in C(\mathbb{R}, H.)$ to the nonlinear dispersive equation (1.1)
and for any, we have for some $A>0$$(t,x)\in(\mathbb{R}\backslash \{0\})\cross \mathbb{R}$$|\partial_{t}^{j}\partial_{x}^{l}u(t, x)|\leq C\langle t^{-1}\rangle^{j+l}\langle x\rangle^{2l+3j}A^{j+l}(j+l)!$

for any.
for$j,l\in \mathrm{N}$. Namely$u(t$,$\cdot$$)$is a real analytic function in both space and time variables$t\neq 0$

Remark 1. The assumption on the initial data implies that the data have to be analytic
except $x=0$ . On this point the data is assumed to have only H. regularity. Hence theabove theorem states that this singularity disappears after time passed. The weaknessof this singularity on the data is depending on the space where we may establish thewell-posedness of the equation.
The existence and uniqueness result of the Benjamin-Ono equation can be found in thearticles by Iorio Jr. [14], Ponce [32]. The global well-posedness in time is also discussed
in Kenig, Ponce and Vega [26]. Our result is based on those well-posedness results in the
Sobolev spacewith $s>3/2$ . It seems that the well-posedness in aweaker spaces
thanis not well established so far as the authors know. If this is improved into
the lower regularity classes like H. with, we may extend our result into such a
weak space even negative exponent Sobolev spaces. See [17] for this direction for theequation case.
$H.(\mathbb{R})$$H^{\theta/2}$$s\leq\cdot 3/2$$\mathrm{K}\mathrm{d}\mathrm{V}$79
Remark 2. It is well-known that the global in time solution has been obtained to
Benjamin-Ono type equations by both the inverse scattering and analytical (continuing)
methods. Since our result shows that the solution reaches analytic in space time variables,
one can show that the result is valid globally in time through the result by T.Kato and
Masuda [23].
By the similar argument as in Theorem 1.1, one can also show the following weaker
theorem in the analytic and Gevrey regularity.

Theorem 1.2. Let $s>3/2$ . Suppose that for some $A_{0}>0$ , the initial dataand satisfies$\sum_{k=0}^{\infty}\frac{A_{0}^{k}}{(k!)^{2}}||(x\partial_{l})^{k}\phi||H$.

$<\infty$$\phi\in H^{\cdot}(\mathbb{R})$,to the nonlinear dispersive equation (L1)then there exists a unique solution, $u(t$ , is an analytic function in space variable and for
and for any, $u(t, x)$ is of Gevrey 2as a time variable function for $t\neq 0$ .
$u\in C(\mathbb{R}, H^{\cdot})$$(t,x)\in(\mathbb{R}\backslash \{0\})\cross \mathbb{R}$$\cdot$$)$$x\in \mathbb{R}$Remark 3. In both Theorems, the assumption on the initial data implies the analyticity
and Gevrey 2regularity except the origin respectively. In this sense, these results statethat the singularity at the origin immediately disappears after $t>0$ or $t<0$ up to
analyticity.

Some related results are obtained for linear and nonlinear Schr\"odinger equations. For
linear variable coefficient case, see Kajitani and Wakabayashi [16], Robbiano and Zuily[33] and for nonlinear case, Chihara [3]. They give aglobal weighted uniform estimatesof the solution with arbitrary order derivative in space variable.
The essential difference in proving the above type results from the case for the nonlinear. Since ourequation is the appearance of the nonlocal operator
Schr\"odinger or
method uses some localization technique, it is required to treat the non local term carefully
to show the higher regularity. We then introduce aweight function which has an explicitin the. This enables us to handle the nonlocal term
commutation estimate withlinear part of the equation. We explain this part in the following sections.
$H_{x}$$\mathrm{K}\mathrm{d}\mathrm{V}$$H_{ax}$$H_{x}$.
Here we summarize some notation that we would use in what follows.
be the linear part of the Benjaminis the Sobolev space of order . Let
be the dilation operator associated with . For operators
Ono equation andand $B$ , $[A,B]$ stands for the commutator $AB-BA$ . The ffee propagator group for$\langle\cdot\rangle=(1+|\cdot|^{2})^{1/2}$$s$$H^{s}$$P=2t\partial_{t}+x\partial_{x}$$A$$L=\partial_{t}+H_{x}\partial_{x}^{2}$

$L$

80
the linear Benjamin-Ono type evolution is denoted by
ftom
to
.
$L^{2}(\mathrm{R})$$e^{-tH.iJj\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}}$

which is aunitary operator$L^{2}(\mathrm{R})$2. METHOD
In this section we give an overview of the proof and present some difference fromthe proof of the former cases in [17] and [18]. The results are based on the following
observation.
Noting the commutation relation between the generator of the dilationand the linear dispersive operator:$P=2t\partial_{t}+x\partial_{x}$$L\equiv\partial_{t}+H_{x}\partial_{x}^{2}$$[L, P]=2L$ ,we have$LP^{k}=(P+2)^{k}L$ ,(2.1)$(P+2)^{k}\partial_{x}=\partial_{l}(P+1)^{k}$

Applying(2.2)$P=2t\partial_{x}+x\partial_{x}$,$k$$=1,2$ ,$\cdots$

to the equation (1.1) iteratively, we have.

$\partial_{t}(P^{k}u)+H_{ax}\partial_{ox}^{2}(P^{k}u)=(P+2)^{k}Lu=-(P+2)^{k}\partial_{ax}(u^{2})$Setting $u_{k}=P^{k}u$ and $B_{k}(u,u)=-(P+2)^{k}\partial_{x}u^{2}$ , we have
(2.3)$\partial_{t}u_{k}+H_{x}\partial_{x}^{2}u_{k}=B_{k}(u,u)=-\partial_{x}\sum_{k=k_{0}+k_{1}+k_{2}}\frac{k!}{k_{0}!k_{1}!k_{2}!}u_{k_{1}}u_{k_{2}}$.

An important point here is that the nonlinear terms $B_{k}(u,u)$ maintain the bilinear
structure similar to the original Benjamin-Ono equation. This is due to the fact that the
Leibniz law can be applicable for an operation of . Thus each of satisfies the following
system of equations;$P$$\partial_{t}v_{k}+H_{l}\partial_{ae}^{2}u_{k}=B_{k}(u,u)$

(2.4)

$\{$

$u_{k}(0,x)=(x\partial_{x})^{k}\phi(x)$.

$u_{k}$,$t,x\in \mathbb{R}$,Firstly we establish the local-posedness of the solution to the following infinitely
coupled system of dispersive equation in aproper Sobolev space:$\mathrm{w}\mathrm{e}\mathrm{u}$$\partial_{t}u_{k}+H_{ae}\partial_{l}^{2}u_{k}=B_{k}(u,u)$

(2.5)

$\{$

$u_{k}(0,x)=\phi_{k}(x)$.

,$t,x\in \mathbb{R}$,, the uniqueness and local well-posedness alow us to say $u_{k}=P^{k}u$Takingfor aU $k=0,1$ , .
Through showing the existence and uniqueness process, we obtain the estimate$\phi_{k}=(x\partial_{l})^{k}\phi(x)$$\cdots$$||P^{k}u||_{H}\cdot\leq CA^{k}k!$.

Until this step, there is no effect ffom the appearance of the non local operator$H_{x}$.

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Next we would derive the pointwise derivative estimate by using the equation:(2.6)$H_{x} \partial_{x}^{2}P^{k}u=-\frac{1}{2t}P^{k+1}u+\frac{1}{2t}x\partial_{x}P^{k}u+B_{k}(u,u)$.

To treat the second term of the right hand side of (2.6), we employ localzation argument.
With asuitable decaying weight function $a=a(x)$ , we can show that$||a\partial_{x}^{l}P^{k}u(t)||_{H^{1}(\mathbb{R})}\leq C\langle t^{-1}\rangle^{l}A^{k+l}(k +l)!$$k,l=0,1,2$ ,,$\cdots$

and then by iterative argument, we can shift from the estimate with the operator
and conclude
the one with

$P$

to

$t\partial_{t}$

(2.7)

,

$||(t\partial_{t})^{l_{1}}\partial_{x}^{l_{2}}u(t)||_{L^{\infty}(x0-\delta,x\mathrm{o}+\delta)}\leq C\langle t^{-1}\rangle^{l_{1}+l_{2}}\langle x_{0}\rangle^{3l_{1}+2l_{2}}A^{l_{1}+l_{2}}(l_{1}+l_{2})!$. Acrucial step for obtaining the above derivative estimates is tofor , $l_{2}=0,1,2$ ,equation or
which is an essential difference from thetreat the nonlocal operator
nonlinear Schrodinger equations. In order to handle this term, it is required to show an
explicit dependence of the iteration of the commutator estimate$\cdots$$l_{1}$$\mathrm{K}\mathrm{d}\mathrm{V}$$H_{x}$$||[H_{x}, a^{k}]||_{L(L^{2}arrow L^{2})}\leq C_{k}$,where $a=a(x)$ is acut-0ff function and $a^{k}=a(x)^{k}$ . We then choose aparticular weight, where $\langle x\rangle=(1+|x|^{2})^{1/2}$ and derive an explicit commutation
function
estimate with the Hilbert transform and . By this step, we may use the equation (2.6)
to gain the regularity and to show the analyticity (2.7). Here we only exhibit the following
and .
lemma which treats the commutator of$a(x)=\langle x\rangle^{-2}$$a^{k}$$H_{x}$

Lemma 2.1.

$If||a^{l}\theta_{x}f||_{2}\leq CA^{l}l!||f||_{2}$$a^{k}$

for $0\leq l\leq N-1$ , then we have.

$||[H_{x}, a^{N}]\partial_{x}^{N}f||_{2}\leq CA^{N}N!||f||2$

Proof of Lemma 2.1. The result is obtained by using the explicit expression of the. Sincecommutator $[H_{x},a]$ . 1 Let f$\in S$(2.8)$||[H_{x},a^{N}] \partial_{x}^{N}f||_{2}\leq\sum_{j=0}^{N-1}||a^{j}[H,a]a^{N-1-j}\partial_{x}^{N}f||_{2}$,it suffices to show that$||a^{j}[H_{x}, a]a^{N-1-j}\partial_{x}^{N}f||_{2}\leq CA^{N}(N-1)!||f||_{2}$.

An elementary computation gives$[H_{x},a]f=p.v.

\int_{\mathbb{R}}\frac{a(y)-a(x)}{x-y}f(y)dy=\int_{\mathbb{R}}\frac{x+y}{\langle x\rangle^{2}\langle y\rangle^{2}}f(y)dy$

is also possible to show the $N$ dependence of the operator norm ofFourier transform$\mathrm{x}\mathrm{I}\mathrm{t}$$[H_{l}, a^{N}]$.

directly by passing the82
By integration by parts, we have$||a^{j}[H_{x},a]a^{N-1-j} \partial_{ax}^{N}f||_{2}^{2}=\int_{\mathbb{R}}|\langle x\rangle^{-2j}\int_{\mathbb{R}}\frac{x+y}{\langle x\rangle^{2}\langle y\rangle^{2}}\langle y\rangle^{-2(N-1-j)}\partial_{x}^{N}f(y)dy|^{2}dx$$= \int\int\int\frac{(x+y)(x+z)}{\langle x\rangle^{4(j+1)}}\langle y\rangle^{-2(N-j)}\langle z\rangle^{-2(N-j)}\partial_{y}^{N}f(y)\partial_{z}^{N}\overline{f}(z)dydzdx$(2.9)$= \int\int\partial_{y}^{j+1}\partial_{z}^{j+1}\{(\int_{\mathbb{R}}\frac{(x+y)(x+z)}{\langle x\rangle^{4(\mathrm{j}+1)}}dx)\langle y\rangle^{-2(N-j)}\langle z\rangle^{-2(N-j)}\}$$\cross\partial_{y}^{N-j-1}f(y)\partial_{z}^{N-j-1}\overline{f}(z)dydz$

If we set$\sigma(y,z)=\int_{\mathbb{R}}\frac{(x+y)(x+z)}{\langle x\rangle^{4(j+1)}}dx=\sigma_{1}+\sigma_{0}yz$$(i=0,1)$ are constants of order$j^{1/2}$

and.

$\max(\sigma_{0},\sigma_{1})=\tilde{\sigma}$, where$\sigma_{i}$, then$|\partial_{y}^{j+1}\partial_{z}^{j+1}(\sigma(y, z)\langle y\rangle^{-2(N-j)}\langle z\rangle^{-2(N-j)})|$$\leq\sigma_{0}j^{2}\partial_{y}^{j}\langle y\rangle^{-2(N-j)}\partial_{z}^{j}\langle z\rangle^{-2(N-j)}+\sigma_{0}jy\partial_{y}^{j+1}\langle y\rangle^{-2(N-j)}\partial_{z}^{j}\langle z\rangle^{-2(N-j)}$$+\sigma_{0}jz\partial_{y}^{j}\langle y\rangle^{-2(N-j)}\partial_{z}^{j+1}\langle z\rangle^{-2(N-j)}$$+(\sigma_{1}+\sigma_{0}yz)\partial_{y}^{j+1}\langle y\rangle^{-2(N-j)}\partial_{z}^{j+1}\langle z\rangle^{-2(N-j)}$$\leq C_{0}j^{2}A_{0}^{j+1}(\frac{2^{j}N!}{(N-j-1)!})^{2}\langle y\rangle^{-2N+j}\langle z\rangle^{-2N+j}$(2.10)$+ \sigma_{0}jA_{0}^{j+1}(\frac{2^{j}N!}{(N-j-1)!})(\frac{2^{\mathrm{j}}(N+1)!}{(N-j-1)!})$$\mathrm{x}\{\langle y\rangle\langle y\rangle^{-2N+j-1}\langle z\rangle^{-2N+j}+\langle z\rangle\langle y\rangle^{-2N+j}\langle z\rangle^{-2N+j-1}\}$$+ \tilde{\sigma}\langle y\rangle\langle z\rangle A_{0}^{j+1}(\frac{2^{j}(N+1)!}{(N-j-1)!})^{2}\langle y\rangle^{-2N+j-1}\langle z\rangle^{-2N+j-1}$$\leq C\tilde{\sigma}(j+1)^{2}A_{1}^{j+1}(\frac{N!}{(N-j-1)!})^{2}\langle y\rangle^{-2N+j}\langle z\rangle^{-2N+j}$.

Hence it follows by the assumption that
(2.11)$||a^{j}[H_{ax},a]a^{N-1-j} \partial_{ax}^{N}f||_{2}^{2}\leq C\tilde{\sigma}(j+1)^{2}A_{1}^{j+1}(\frac{N!}{(N-j-1)!})^{2}||a^{N-j/2}\partial_{ax}^{N-j-1}f||_{1}^{2}$$\leq C.\tilde{\sigma}(j+1)^{2}A_{1}^{j+1}(\frac{N!}{(N-j-1)!})^{2}||a^{\mathrm{j}/2+1}||_{2}^{2}||a^{N-j-1}\partial_{x}^{N-j-1}f||_{2}^{2}$$\leq C\tilde{\sigma}(j+1)^{2}A_{1}^{j+1}A_{1}^{2(N-j-1)}(N!)^{2}||f||_{2}^{2}$$\leq 4^{-(j+1)}C^{2}A^{2N}(N!)^{2}||f||_{2}^{2}$

83
and we conclude.

$||[H_{x},a^{N}]f||_{2} \leq CA^{N}N!\sum_{j=1}^{N-1}2^{-(j+1)}||f||_{2}\leq CA^{N}N!||f||_{2}$$\square$$\square$Based upon the above Lemma 2.1, we can show the analyticity.
Acknowledgment. The last author would like to thank Dr. Tatsuo Iguchi for avaluable comment.

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