数理解析研究所講究録 1261 巻 2002 年 18-23

18

Necessary Conditions for theGevrey-Well-Posednessof Schr\"odinger Type Equations
Michael Dreher, University of Tsukuba *1Introduction
We study necessary conditions under which the following Cauchy problem ofSchr\"odinger type,$Lu=(i \partial_{t}+\triangle+\sum_{j=1}^{n}b_{j}(x)\partial_{x_{j}}+c(x))u=f(\mathrm{t},x)$is well-posed in Gevrey spaces $G8$ , $1<s<\infty$ . Herethe Hilbert space$\langle\xi\rangle=(1+|\xi|^{2})^{1/2}$,$u(0,x)=\varphi(x)$$G^{s}=1\dot{k}^{\rho>0}G_{\rho}^{f}$,, and$G_{\rho}^{s}=\{v\in L^{2}(\mathrm{R}^{n}):||v||_{\epsilon,\rho}=||\exp(\rho\langle\xi\rangle^{1/s})\hat{v}(\xi)||_{p}<\infty\}$and$\hat{v}$(1.1)

is, where$G_{\rho}^{f}$is the usual Fourier transform of .
$v$

Definition 1.1. We say that the Cauchy problem for the operator is forward
well-posed if for every $T>0$ and every
there are constants $C=C(T, \rho_{0})$and $\rho>0$ such that for every,
there is aunique solution$u\in C([0, T], G_{\rho}^{s})$ to (1.1) with$L$$G^{s}$$\rho_{0}>0$$\varphi\in G_{\rho 0}^{s}$$f\in C([0, T], G_{\rho}^{s_{0}})$$||u(t, \cdot)||_{s,\rho}\leq C||\varphi||_{s,\rho 0}+C\int_{0}^{t}||f(\tau, \cdot)||_{s,\rho 0}d\tau$,.

$0\leq t\leq T$

If the coefficientsare purely imaginary valued, then a priori estimates of asolution to (1.1) in the spaces ,, andcan be easily derived, and the wellposedness of this Cauchy problem follows by standard arguments. The situation
is more delicate when
. For example, the Cauchy problem for the operator$b_{j}$$L^{2}$$u$$H^{\infty}$$G_{\rho}^{l}$$\Re b_{j}\not\equiv 0$

Faculty of Mathematics and Computer Science, Technical University Bergakademie Freiberg,
Agricolastrasse 1, 09596 Freiberg, German

191INTROD UCTION
is neither well-posed innor in , $1<s<\infty$ , as canbe shown by an explicit representation of the solution, see also [12]. Generally,
well-posedness requires acertain decay of $lbj (x) at infinity.
Therefore, we propose the following condition:$L=i\partial_{t}+\partial_{x}^{2}+\partial_{x}$$L^{2}$$G^{s}$

Condition 1. There is aconstant $M=M(d_{0})$ such that$\sup_{x\in \mathrm{R}^{n},\omega\in S^{n-1}}|\int_{0}^{\sigma}\sum_{j=1}^{n}\Re b_{j}(x+2\theta\omega)\omega_{j}d\theta|\leq M(1+|\sigma|)^{d_{0}}$We assume that the coefficients$||\partial_{x}^{\alpha}b_{j}(\cdot)||_{L\infty}\leq C^{1+|\alpha|}\alpha!^{s_{b}}$$||\partial_{x}^{\alpha}c(\cdot)||_{L^{\infty}}\leq C^{1+|\alpha|}\alpha!^{s}$

,

,

and

$b_{j}$$\forall\alpha$$\forall\alpha$,$\forall\sigma\in \mathrm{R}$belong to Gevrey spaces$c$$G_{p\circ}^{s_{b}}$.
,$G_{D^{-}}^{s}:$

,

(1.2)

.

Theorem 1. Let (1.2) be satisfied, and let
be a number with $d_{0}>3/(s+1)$
and $d_{0}>2/(s+1-s_{b})$ . If Condition 1is $violated_{f}$ then the Cauchy problem forthe operator is not
well-posed.
$d_{0}$$L$$G^{s}$

Sufficient conditions for thewell-posedness of the Cauchy problem for thewere given in [2], namely $\Re b_{j}(t, x)=$. In case of the model operator
with,and $0<d<1$ , the Cauchy problem is therefore well-posed if $d<1/s$ . On theother hand, Theorem 1implies ill-posedness for $d>3/(s+1)$ .
This gap can be closed if we suppose that the coefficients decay not too rapidly:
operator$G^{s}$$L=i \partial_{t}+\triangle+\sum_{j=1}^{n}b_{j}(t, x)\partial_{x_{\mathrm{j}}}+c(t, x)$$o(\langle x\rangle^{1/s-1})$$L=i\partial_{t}+\triangle+\langle x\rangle^{d-1}\partial_{x}$$x\in \mathrm{R}^{1}$$b_{j}$

Condition 2. There are$x_{0}\in \mathrm{R}^{n}$,$\omega_{0}\in S^{n-1}$$- \sum_{j=1}^{n}\Re b_{j}(x+\tau\omega’)\omega_{j}\geq 2c_{0}\langle\tau\rangle^{d_{0}-1}$for all$\tau\geq 0$, $|x-x_{0}|<\epsilon_{0}$ , and all$\omega$,, and$\epsilon_{0}>0$, $c_{0}>0$ such that,$\omega’\in S^{n-1}$

with$|\omega-\omega_{0}|<\epsilon_{0}$Theorem 2. Suppose (1.2) with $s_{b}<s$ and Condition 2.
necessary for thewell-posedness.

,$|\omega’-\omega_{0}|<\epsilon_{0}$Then $d_{0}\leq 1/s$ is$G^{s}$

Anecessary condition for$H^{\infty}$.

well-posedness was given in [7]:,$\sup_{x\in \mathrm{R}^{n},\omega\in S^{n-1}}|\int_{0}^{\sigma}\sum_{j=1}^{n}\Re b_{j}(x+2\theta\omega)\omega_{j}d\theta|\leq M\log(1+|\sigma|)+N$$\forall\sigma\in \mathrm{R}$.

20This condition is sufficient in the case of one space dimension; and it is sufficientin the cases of two or more space dimensions if one supposes certain relations onderivatives of the coefficients , see .
The investigation of an operator with variable coefficients in the principal part,, where $a(x, \xi)=\sum_{j,k}a_{jk}(x)\xi_{j}\xi_{k}\geq$,, requires the introduction of the bicharacteristic strip $(X, P)=$$(X, P)(t,x,p)$ , which is the solution to the Hamilton-Jacobi equations,$b_{j}$$[8\mathrm{j}$$L=i \partial t+\sum_{j,k}ajk(x)\partial_{x_{\mathrm{j}}}\partial_{x_{\mathrm{k}}}+\sum_{jj}b(x)\partial_{x_{\mathrm{j}}}+c(x)$$c_{0}|\xi|^{2}$$\mathrm{C}0>0$$\partial_{t}X_{j}=\partial_{P_{\mathrm{j}}}a(X, P)$,$\partial_{t}P_{j}=-\partial_{X_{\mathrm{j}}}a(X, P)$

Then anecessary condition for the$H^{\infty}$,$(X, P)(0,x,p)=(x,p)$ .

well-posedness is,$\sup_{x\mu}|\int_{0}^{\sigma}\sum_{j=1}^{n}\Re b_{j}(X(\theta,x,\omega))P_{j}(\theta,x,\omega)d\theta|\leq M\log(1+|\sigma|)+N$$\forall\sigma\in \mathrm{R}$,under some additional condition. For details, see [6].
Sufficient and necessary conditions forwell-posedness were discussed in [3], [4]and [13]. These conditions are similar to the conditions forwell-posedness
if aloss of regularity is allowed, otherwise similar to the conditions ofwill$H^{s}$$H^{\infty}$$L^{2}$

posedness.
In [9] and [11], the following necessary condition for$L^{2}$

well-posedness was shown:$\sup_{x\in \mathrm{R}^{n}\mu\in S^{n-1}}|\int_{0}^{\sigma}\sum_{j=1}^{n}\Re b_{j}(X(\theta,x,\omega))P_{j}(\theta,x,\omega)d\theta|\leq M$,$\forall\sigma\in \mathrm{R}$.

This condition is also sufficient, see [10].
Schr\"odinger type equations with alower order term of order strictly less than 1wereinvestigated in [1]; and sufficient conditions forwell-posedness were proved.
Theorem 1and Theorem 2will be discussed simultaneously; and the both cases
will be called Case Iand Case , respectively. The following lemma, which gives
us an integrated estimate offrom below, is quite essential.
$G^{s}$$\mathrm{I}\mathrm{I}$$\Re b_{j}$Lemma 1.1. Assume that $0<h$ $<1$ and that Condition 1is violated. Then, for, there areeach,,with the property that$k\in \mathrm{N}$$x_{k}\in \mathrm{R}^{n}$$\sigma_{k}\in \mathrm{R}_{+}$$\omega_{k}\in S^{n-1}$$- \int_{0}^{\sigma_{k}}\sum_{j=1}^{n}\Re b_{j}(x_{k}+2\theta\omega_{k})\omega_{k_{\dot{\beta}}}d\theta=k(1+\sigma_{k})^{d_{0}}$,$- \int_{0}^{\sigma}\sum_{j=1}^{n}\Re b_{j}(x_{k}+2\theta\omega_{k})\omega_{ki}d\theta\geq kd_{0}\sigma(1+\sigma_{k})^{d_{0}-1}$

where$\sigma_{k}$tends to infinity for $karrow\infty$ .

,$0\leq\sigma\leq\sigma_{k}$,211INTRODUCTION
tending to infinity in Case I. In Case , weThis lemma gives us asequencechoose this sequence arbitrarily, but still tending to infinity. Now we fix special
initial data, $\varphi_{k}(x)=\varphi(x-x_{k})$ (in Case ), and $\varphi_{k}(x)=\varphi(x-x_{0})$ (in Case ),. Assuming that (1.1)
is given by:(4)whereofwell-posed, there is aunique solutionis$\mathrm{I}\mathrm{I}$$\{\sigma_{k}\}_{k}$$\mathrm{I}\mathrm{I}$$\mathrm{I}$$=\langle\xi\rangle^{-(n+1)/2}\exp(-\rho_{0}\langle\xi\rangle^{1/s})$$\varphi\in G_{\rho 0}^{s}$$u_{k}\in C^{1}([0, T], G_{\rho}^{s})$$G^{s}$$Lu_{k}=0$,$u_{k}(0, x)=\varphi_{k}(x)$

(1.3)

.

, .
for the function
Next we define aseminorm
Let $h=h(x)\in G^{s0}$ (with $s_{0}>1$ very close to 1) be afunction with$u_{k}(t$$E_{k}(t)$$h(x)=\{$

0: $|x|\geq 1$ ,1: $|x|\leq 1/2$ ,$0\leq h(x)\leq 1$$\cdot$$)$.

We choose the symbols$Ek(t)x,$$\xi)=h(\frac{x-x_{k}-2t\sigma_{k}^{\delta_{3}}\omega_{k}}{\sigma_{k}^{-\delta_{1}}})h(\frac{\xi-\sigma_{k}^{\delta_{3}}\omega_{k}}{\sigma_{k}^{\delta_{2}}})$$Ek(t)x,$

where
multiindizes$0<\epsilon$$\xi)=h(\frac{x-x_{0}-2t\sigma_{k}\omega_{0}}{\epsilon\langle 2t\sigma_{k}\rangle})h(\frac{\xi-\sigma_{k}\omega_{0}}{\sigma_{k}^{\delta_{2}}})$$\ll\epsilon_{0}$$\alpha$,, $\delta_{1}=1-d_{0}$ , and, we specify$\delta_{2}$,$\delta_{3}$(Case ),,$\mathrm{I}$

,

(Case

$\mathrm{I}\mathrm{I}$),are certain positive constants. For$\beta\in \mathrm{N}^{n}$’$w_{k}^{(\alpha\beta)}(t, x, \xi)=\partial_{y}^{\alpha}h(y)\partial_{\eta}^{\beta}(\eta)|y=\sigma_{k}^{\delta_{1}}(x-x_{k}-2t\sigma_{k}^{\delta_{3}}\omega_{k})$,$\eta=\sigma_{k}^{-\delta_{2}}(\xi-\sigma_{k}^{\delta_{3}}\omega_{k})$$w_{k}^{(\alpha\beta)}(t, x, \xi)=\partial_{y}^{\alpha}h(\epsilon^{-1}y)\partial_{\eta}^{\beta}(\eta)|y=\langle 2t\sigma_{k}\rangle^{-1}(x-x0-2t\sigma_{k}\omega 0),\eta=\sigma_{k}^{-\delta_{2}}(\xi-\sigma_{k}\omega 0)$’symbols are supported near thein Case , Case , respectively. These,bicharacteristic strip. With some positive constant , we set$s_{1}>s_{0}$, and define the seminorm
choose$\mathrm{I}$$\mathrm{I}\mathrm{I}$$\mathrm{c}\mathrm{u}\mathrm{t}-\mathrm{o}\mathrm{f}\mathrm{f}$$\mathrm{N}\ni N_{0}=\lfloor\sigma_{k}^{\kappa/s_{1}}\rfloor$$\kappa$$E_{k}(t)$$= \sum_{|\alpha|\leq N_{0},|\beta|\leq N_{0}-2}(\alpha!\beta!)^{-}"||W_{k}^{(\alpha\beta)}(\mathrm{t},x, D_{x})u_{k}(t, x)||_{D(\mathrm{R}_{ae}^{\mathfrak{n}})}$.

fromThe ill-posedness of the Cauchy problem can be proved by estimates ofif we choose , , , , suitably.
above and below which contradict for large
For reasons of space, we omit the tedious calculations, which can be found in [5],and only sketch the proof$E_{k}(t)$$\delta_{1}$$\sigma_{k}$$\delta_{2}$$\delta_{3}$$\kappa$$\epsilon$

22REFERENCES
It is easy to estimate
ffom above: the symbols
belong to the H\"ormander, then the Calderon-Vaillancourt theorem and the presumed wellclass
posedness of the Cauchy problem give$w_{k}^{(\alpha\beta)}$$E_{k}$$S_{0,0}^{0}$$E_{k}(t)\leq C\sigma_{k}^{C}||\varphi||_{s,\rho 0}$.

To get an estimate ffom below, we write$v_{k}^{(\alpha\beta)}(t, x)=W_{k}^{(\alpha\beta)}(t,x,D_{x})u_{k}(t,x)$$B(x,D_{x})=- \sum_{j=1}^{n}\Re b_{\mathrm{j}}(x)D_{x_{\mathrm{j}}}$,,and can deduce that$||v_{k}^{(\alpha\beta)}||_{p}\partial_{t}||v_{k}^{(\alpha\beta)}||_{p}=\Re(\partial_{t}v_{k}^{(a\beta)},$$v_{k}^{(a\beta)})$$=\Re(-i[L, W_{k}^{(\alpha\beta)}]u_{k},v_{k}^{(a\beta)})+\Re(i$IS$v_{k}^{(a\beta)}$,$v_{k}^{(a\beta)})$$+ \sum_{j=1}^{n}\Re(ib_{j}\partial_{x_{\mathrm{j}}}v_{k}^{(a\beta)},v_{k}^{(a\beta)})+\Re(icv_{k}^{(a\beta)},v_{k}^{(a\beta)})$$\geq-||[L, W_{k}^{(a\beta)}]u_{k}||_{p}||v_{k}^{(a\beta)}||_{p}+\Re$

Now we need an estimate of
$\Re$

(

$B(x, D_{x})v_{k}^{(a\beta)}$,$v_{k}^{(a\beta)}$($B(x, D_{x})v_{k}^{(a\beta)}$$||[L, W_{k}^{(a\beta)}]u_{k}||_{p}$,$v_{k}^{(a\beta)}$)$-C||v_{k}^{(a\beta)}||_{p}^{2}$.

from above, and an estimate of) from below.

The symbol of
can be written as an asymptotic expansion, up to someremainder, andcan be estimated by certain norms
plus some remainder which becomes negligible for.
,The term
can be estimated using Condition 2and Garding’s inequality, or Lemma 1.1 and Gronwall’s Lemma.
$[L, W_{k}^{(a\beta)}]$$||[L, W_{k}^{(a\beta)}]u_{k}||_{p}$$||v_{k}^{(a+\gamma,\beta+\delta)}||_{p}$$\sigma_{k}arrow\infty$

$\Re$

(

$B(x, D_{x})v_{k}^{(a\beta)}$$v_{k}^{(a\beta)}$)

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equations of Schrodinger type. Math. Models Methods Appl. Sci., $6(3):295-$
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[2] A. Baba and K. Kajitani. The Cauchy problem for Schrodinger type equations. Bull. Sci. Math. (2), 119:459-473, 1995

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$H^{\infty}$

32(2):287-305, 1992.

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[8] W. Ichinose. Sufficient condition on

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$H_{\infty}$

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$H^{\infty}$