TSUKUBA
J. MATH.
Vol. 16 No. 1 (1992), 235-256
THE
L2-WELLPQSED
CAUCHY
SCHRODINGER
TYPE
FOR
PROBLEM
EQUATIONS
By
Akio Bab a
§1. Introduction.
We study the Cauchy problem for a Schrodinger type operator
P=P(x,
where
t,Dx, Dt)=D2
a{x, t)={al{x,
t),･･･ , an{x, t)),c(x, f)eC?([O,
―id/dt, Dj=―id/dxj
valued
functions
whose
derivatives
and
in
a/%,
t)=aRj{x, t)+iaj(x,
R"xRt).
of
any
(k = 0, 1, 2, ･･･) means
tinuously
+ Tfl(Dj-aj(x,
Z 1=1＼
Hence
order
are
{ P(x,
we
give
t)=f(x,
t)and
￡￡=
aj(x, t)are real
$°°(Rn) denotes
the set of C°°-functions
g(x, t)<ECkt(＼0,To] ; X)
and
[0, T^-it-^gix,
t)e^X
is
^-times
con-
of X.
a sufficient condition
t, Dx, Dt)u(x,
To] ; $°°(Rn)),(T0>0),
t)(a%x,
all bounded
that the mapping:
differentiable in the topology
In this paper
t)f+c(x, t)
/
t)
for the Cauchy
(x, 0ei?nx[0,
T]
problem
(T>0),
(*)
I U(X, O)-Mo(x)
to be
L2-wellposed.
We
such
XGi?"
say
that
that for any
exists a unique
the
Cauchy
initial data
solution
problem
uo^H2
(*) is L2-wellposed
and
for any
where
L2=L2(Rn),
(Dxy2v^L2(Rn)＼.
S, and (Dxy
When
T]
defined by Hz={v^S';
Here, S' is the dual space of the rapidly decreasing functions
is the pseudo-differentialoperator of symbol
<￡>2≪￡>―Vl+I^H.
the coefficientsaj(x)=aIj(x)+iaIj(x) (/=1, ･･･, n) are independent
t, Mizohata
(Co)
Received
which
there
T] ; H2) satisfying
s)＼＼Llds＼ fe[O,
and H2 is the Sobolev space
exists T>0
f(x, ;￡)eC?([0, T] ; H2)
m(x, t)^C＼([O, T] ; L2)nC?([0,
Hu(-, 0IL2^C(0{￨￨mo￨!l2+Jj￨/(-,
if there
[3], [4] proved that the condition
pso, xefl", weSn-l
May
14, 1991.
CP
n
＼ S
aIj(x ― sw)(ojds
Jo j=i
Revised October 8, 1991.
< oo
of
236
Akio
Baba
is necessary for the Cauchy problem (*) to be L2-wellposed. S11"1denotes the
unit sphere in Rn
He also proved (Co) and the following condition
(Ca)
i
max
are
sufficient for
proved
the
sup
Cauchy
I ＼DSaj(x
problem
that if there is ￡0>0 such
Daxa'j(x)＼^
(*)
+ s<o)＼ds＼<o°
to be
Z/2-wellposed.
Takeuchi
[5]
that ai(x)=aIj(x)-＼-iaIj(x) (j = l, ･･･ , n) satisfy
ca
<X>1
{all
+ ￡°+￨Q'1
a)
CO
D%a%x)＼fZ-f±
for zei?R,
then the Cauchy
(kl^l),
problem (*) is L2-wellposed.
In the present paper we give a sufficientcondition for the Cauchy
(*
problem
)t o he /^-weilnosed.
Theorem.
Suppose that there are elf s2>0
Co
such that
(kl=O)
<x>1+￡
Dlaftx, t)＼<
Ca
(5)
(kl^l),
<x>2+￡
ca
D*a*j(x,t)＼^ <X>1
for (x, t)(ERnX[Q
Tnl
and
;'=1, ･･･ , n.
(Ia￨^2),
+ ￡2
Then
the
Cauchy
problem
(*) is L2
wellposed.
We
can see from
removed
To
the above Theorem
that the condition for aj(x) in (T) is
for ￨a ￨ = 1.
prove our Theorem,
imaginary
we reduce the operator P to an operator of which
part of firstorder term vanishes identically,by use of pseudo-differ-
ential operator of type So.o. To do so, we need some
of a
characteristic equation for P. In §2
we
properties of a solution
investigate the characteristic
equations, in § 3 conjugate P by a pseudo-differentialoperator eA{x, Dx) of type
So,o and
in §4 prove
that the Cauchy
problem
welloosed.
§2.
We
Properties of characteristic curve.
consider a characteristic equation for P
(*) for P with
<^=0
is L2-
337
The L2-wellposedCauchy problem
(2.1)
At(x, ￡;t)+$-Ax(x,￡; t)+aRx(x, t,￡)･A(x, ￨; O+a'U≫
^ ^)=0
where
aR(x, t,$)=
S
a%x,
taf.^x,^,
aRx{x, t, $)=(
sn
X'$=
The
characteristic
curve
a＼x, t,￡)=
t)h,
2 a'j(x,Mj
-, 2 ≪?.*≫(*,
06)
X&,
(x,$<=Rn).
(x(t), t-(t))=(x(y, v, t), %(y, in, t)) for (2.1) is defined
by
f x=$,
x(O)=y
＼e=a%x,t,e),
(2.2)
m=v
where
x=dx(t)/dt, ￡=d￡(t)/dt.
Put
A(y, r)＼O=-jV(x(s),
Then
s, $(s))ds
we can prove the following
Lemma
2.1. Assume
that the conditions(S) in Theorem
are valid. Then
the
solution of (2.2) satisfiesthe following properties.
(i)
(ii)
x(y, rj, t),
There
is a constant
${y, 7], 0eC
C
such
(i?2reX[0,
To]).
thai
C-l＼t7)＼^＼x(y, i), t)-y＼^C＼trj＼,
C-l＼T)＼￡＼￡(y,
y ,t)＼<C＼T)＼,
(iii)
There
are T>0
(iv)
There
is T>0
x＼%(y, v> t),
w;/jere x[%(y,
(T^T0)
(T^T0)
Wy,
rj, t)=d°D§x(y,
the set of symbols
of which
and
such
C>0
(y, y, t)^R2nx
such
v> 0e^S([0,
rj, t), ?$(y,
k-th derivatives
T];
0 and
1.
S8.0)n^J([0,
v, t)=d≪D^(y,
are bounded
j is the symbol
class of
{p(x, ?)￡EC (/?27i);＼plf)(x,%)＼^Ca,p(Om-pu*[+Slh
between
that
that for ￨a + j8￨^l
pseudo-differential
S
[0, To] .
in
rj, t), ^{([0,
in
operator
T] ; SJ.o),
X for
which
R2n),
T] ; Z) ≪
te [0, T]
defined by
for
some
p
and
S
and
a=
8
238
Akio Baba
(v)
There exist T>0
d(x(y,
c-1^
(vi)
(T￡T0)
rgC,
v)
(T￡T0)
fiD
(2.2) we
T] .
T] ; S00.o)M^K[0, T] ; SJ,0).
(i) This fact is well known
From
(y, V, t)^R2!tX[0,
such that
A{y, V> Oe^?([O,
Proof,
such that
0)
to, t), ￡(y
d(y,
There is T>0
and C>0
(see [1], Chapter 1, Theorem
4.2).
have
2＼m＼ft＼m＼=ft＼m2
^21 $(011 ￡(01
^2＼m＼z＼a≪(x(t), 01,
which shows
16(01^16(0)1
Similarly, we
exp
b:
a%(x(s), s)＼ds＼
obtain
dt
i￡(0l'^-2￨￡(0ll&0l
2>-2￨￡(OI2￨a≪(x(O,
01
and
1^(01^11(0)1 exp [-jWx(s),
s)￨rfs]
>C'＼r)＼.
Therefore, we
can get
c＼v＼^＼^t)＼^c-l＼v＼.
From (2.2)and the mean value theorem, there existsa 8 (O<0<1)
xj(t)=yj+＼
t
t-j(y,
0
=yj+t$j(y,
for
j
1, ･･･, n.
Therefore
we
rj, s)ds
v> Ot)>
get
C-1＼trj＼^＼x(t)-y＼^C＼trj＼.
(iii) From
(ii) we
have
such that
The L2-wellposedCauchy problem
sup
V&Rn,
^
y<=Rn
sup
yeRn,
<n
When
f'l&y, y, s)＼ ,
Jo
<x(S)>1+￡3
dS+VG*≫"P,,siJo
＼rj
121Jo
max
lsisra
￨ f]i＼/＼
7 ]＼^l/Vn
<x(s)>1+￡*
<x(s)>1+￡*
I
sup
, we
have
C'JIt^jgL^l+c.
i<
(x(s),
s, ￡(s))ds
1
^CT^
for T￡1/2VnC.
2Vn
Hence
we obtain ￨￡,￨^￨>?￨/2Vh.
r*
Since x(y, -n,s)=v+＼ ￡(t)(It,
w e have
Jo
f_!!Osii)!_ds<r'
Put er=(^t(r)dr
ci^
<f"
<:
L
C
＼f]＼
c
<3>*+<?>1+￡2
da^C"
This completes the proof of (iii).
(iv)
Taking
the derivative of (2.2) with respect to y,
xv
Gy
(2.3)
qy― ux
where
*≫=
a§x(x,
i
and
t $)XXy
x(x, t,&Xxy+a*(x,
t)X$v
239
240
Akio
/a?Xl(x,
a%x,t)X$y=＼
Baba
t) ･･■ a%Xl(x,
!
■■.
t) ＼
(t)/dy
%i(t)/dy :)
!
I
WXn(x,
t) ■■■
a*Xn(x,
t)
t)/dyx
■■■
d ￡n{t)/dy
nl
I.pf
p(t)=V＼xv(t)＼2+＼$y(t)＼＼
where
＼xy(t)＼
= {^lj=1＼dxi(t)/dyj＼*y<＼
Then
we
^(OI^IS?,^!^)/^!2}1'2.
have
^2ujiiy(oi+2i^iiey(oi
<2＼xv＼＼Sy{t)＼+2＼$v＼(＼aRxx{x,tt?)＼＼xv＼
+ ＼a≪{x,t)xZy＼)
^2p{ty{l+＼aRxx{x,t,%)＼
+ ＼aRx(x>t)＼＼
￡2Cp(t)2{l+a(t)}.
where
a(t)=＼a*x(x(t),
t,￡(t))＼.
Therefore
jp{t)￡Cp(!)a+a(t)).
Since
o(0) =(l*,(0)r+l￡,,(0)r)1'2=Vn,
p(t)<p(O) exp
By (iii)and (S) we
we get
＼c[ta(s)ds]
[cTaCs^/sl
= Vn exp
L Jo
J
L Jo
I
have
＼a(s)ds=＼ ' C^y'^s)ids<C
Thus
we
For
have
/^l
p(t)￡C
we
for I^￨a+j8￨^/.
which
yields
Then
put
we
f(y,
have
Similarly,
＼xv＼,￨￡,￨5SC.
suppose
It follows
from
dyd'D$x(y,
We
＼xy＼,￨￡J<:C.
-q,t)=a*x(x{y,
(2.2)
V> t)=dyd≫D^{y,
7], t), t,$(y, 7],t)) and
rj, t).
g(y,
7],t)=a%(x(y,
7],t), t).
241
The L2-wellposedCauchy problem
dydM&y,
7],t)=d!D?(fXxv+gx￡v)
S
C%＼$f＼fi](y
i), t)xdfD§2xy(y,
ij, t)
/32+i32=i8
s
+
a3+a4=a
C%l%＼g＼%l]{y,
7],OxdfD^yiy,
V, t)
where
l+lll
＼fl%{y,y,t)＼<Caii
<x(0>1+￡2
'
1
<x(0>1+￡2
Let
P(t)={ ＼d*x＼Wy> v> OI2+
＼dym](y,
rj, oi≪}1/f
Then we have
^2＼dyx＼f){y,r]J)＼＼dvx＼f){y,r]J)＼+2＼dy^f){y,r]J)＼＼dyk{%(y,r],t)＼
￡2p(t)2+2p(t){Ca(t)+C'a(t)p(t)}.
Consequently,
￨5(0^ ^＼Ca(s))ds+＼＼l+C'a(s))p(s)ds
<C+＼＼l+C'a(s))p(s)ds
Therefore, by Gronwall's inequality we
b:＼l+C'a(s))ds＼^C
,o(0.<Xexp
which
have
o
J
implies
＼dyxl%(y, V,t)＼, ＼dytt$(y, 7j,t)＼^C.
Similarly,
we
obtain
＼dvx[ap)(y,rj,t)＼,＼dv&%(y,
Thus
x＼%{y, 7],t) and
x＼ap](y,V, t) and
%{%{y,
￡{$&,
y,t)＼<C.
jj, t) are in ^?([0, T] ; Sg.o) and (2.2) implies
y, t) are in
^K[0,
T] ; SJ,0). This
of (iv).
(v)
Let
(dx(t)/dy
dx(t)/dr)＼
＼d&t)/dy
dm/dv)
completes
the
that
proof
242
Akio
Baba
where
(
*(0)=
Then
taking the derivative of (2.2) with respect to y and rj we
/
°
have
)X(t)=A(t)X(t),
a≪{x{t),
t)
where X(t)=dX(t)/dt.
On the other hand,
(' ＼A(s)＼ds^c[＼l+a(s))ds^C1,
Jo
Jo
where
￨4(OI = {2!.Way(O)T/8
and
aw(0 (i,/=1, ･･･, 2n) are
components
Consequently,
^-2￨i4(0ll-Y(0lf,
and
￨*(OI^￨*(O)￨
exp[-jjyl(s)￨ds]
>C＼X(Q)＼ = C
.
Similarly,
jf＼X{t)＼*=2X{t)X{t)
￡2＼A(t)＼＼X(t)＼*
and
X(t)＼ <＼X(0)＼ exp[J
o
￡C＼X(O)＼
=
J
C//
This completes the proof of (v).
(iv)
We
A＼%(y,
have
?,0
=
2)
S
c￡if^dfD^a'jixiy,
V> s), s)$A$(y,
where
＼avuy
a (X{y,
V, s), s)＼= <x{y>
s)>1+M-
V> s)ds
of
243
The L2-wellposedCauchy problem
Then
＼A{fl(y>V,t)＼^Ca^
＼a(s)+l)ds^C
.
0
It implies
Let
(vi).
y(x, $, t), 7](x, $, t) be the inverse
functions
of x(y, rj, t), $(y, rj, t) and
out
(2.4)
A(x, ￡; t)=A(y(x,
Lemma
is T>0
2.2.
(T^To)
Assume
such
that the condition (S) in Theorem
is valid.
Then
there
that
( i )
y(x, f, t), rj(x, ￡, OeC-(i?2"X[0,
(ii)
^{?1U,
(iii)
IdfflgxO'C*,
(v)
￡, 0, ?(*, S, 0; 0-
T])
￡,t), r}＼f>(x,6, 0e^?([0,T]
; Sg.0)n^K[0,r]
￡,0, i?(^^0,s)I^CaU-s￨,
; SJ.o),(k
(￨a￨^l,
A{%, ￨, f)ei≫?([O, T] ; S20)n^K[0,
+ ^￨^l)
fl//j8)
T] ; SJ,0)
f Cafit, (￨a￨^l)
[Cop,
i.e.
A(x,￡;
Proof,
(￨a￨=0)
t)^S°0,0
(i) It is well known
the proof of the differentiabilityof solutions
with respect to parameters (see [1], Chapter 1, Theorem
4.2).
(ii) Set
/x(y, rj)-x＼
Then,
dF
d(x,
8(y,V)
dF
￡)
d{y,ri)'
d{x, $)
=-/
By the implicit function theorem,
1
d(x,
Thus
f)
＼d(
/
dF
＼
y, w
the conclusions of (ii)follow from (iv) and (v) of Lemma
(iii) Since x(y(x, ￡,t), 7](x,￡,t),t)=x, we have for ￨a￨^l,
2.1.
244
Akio
Baba
＼d$Dlx(y(x,$,t), y(x,t,t),s)＼
d$D§{x+^x{y{x,
d$D§{＼'&y(x,
t
(iv)
The
&
￨, t),V(x, $, t),T)dr}
0, V(x, $, t), t)dc)
estimate (iii)and (S) imply
＼D&fA(x,S,t)＼
= ＼D§d2A(y(x,e,t), T)(x,$,t),t)＼
=
Dgd^a'ixWx,
=
￡,0, V(x> S> *), s), s)-￡(y(x, ￡,t), v(x, $, t), s)ds
ds+
tf/JUo"<jc(s)>i+*i
implies
if lam>
if
=Can)o(x(s)y^ds+)
This
J!oc(S)>^c*>'
o<x(s)>1+￡iJ
=
0(3
＼a＼ =0
(iv)
§3. Transform
by eA.
Let a(K)(x,￡;t)=eMx-^
and a(K)(x, $; t)=e'A'x'^
Where
A(x, ￡;f)is
given in (2.4). Then, we have
a(K°K)(x,$; t)= Os―ffe-'≫"'<H<x-*+':￡V1<*+Vlf:Od:y<fy
1+
r
S Os-[[e-iv-i{-DlA(xt
Jo 17-1
=1
JJ
'
xd rxA(x
￡+7); tye-'11*-**1'-"
+ dy, ￡; t)eA<x+f>y-^dy3Vd0
= l+o(RYx,￡;t)
where
o(R)(x,￡:t)
Jo
XdrxA{x + dy, ￡; t)eA<x+0y-^dyd
7}dd
Here
3r]^(27r)-ndrj=(27:)-ndv1
･■･dnn ,
and
Os-＼＼e-iy^a(y, rj)dy3r]= lim
[[e-iVmiX(ey,
ey)a{y, r))dyd
r)
The
L2-wellposed Cauchy
for X<bS in RZn such that Z(0, 0)=l.
tion 3.1 from Lemma
2.2 (iv).
Proposition
Assume
3.1.
Then, for any a, (2 (￨a ￨^1)
(3.1)
Then
we
problem
245
can get the following Proposi-
that the same condition(S) as in Theorem is valid.
we have
＼rltl(x,$;t)＼￡Cafit,
(fe[0, T]),
u^/ierer(x, f; t)=a{R){x, ￡;0From
<1
(3.1) and
for fe[0,
the Calderon-Vaillancourt
T], if we
(3.2)
take
T>0
theorem,
sufficiently small.
Q(x, Dx;t)=f](-R(x>
3=0
which
converges in the sense of L2 norm.
the symbols
we
obtain
Hence
＼＼R(x,
D x;
we
can
t)＼＼Lz
define
Dx;t)Y
Moreover
by virtue of estimates of
of multiple products of pseudo-differential operators we can show
that (3.2) is convergent in the symbol
Proposition
3.2 (Kumano-go
class So 0.
[2]). Let <?/x,￡)(/=1, ･･･, v+1)
be in S°o,o.
Define for v^Q
(3.3) pv+1(x,$)
= 0s-＼＼exp(-i%}yi'T)>)-liqJ(x+yi-1,
where
$°=0, yj=yl+
is 00
independent
(3.4)
■■■
+yj,
of v such
(/=1,
%+f]j)dyl ･･･dy^rf
･■■
, v), )f+1 = 0 and
■■■
3yv
yj, rf^R71.
Then
there
that
no=2[n/2+l] ,
＼pv+l(x,$)＼￡C"+1n.＼qj＼%,
.7= 1
for x,^Rn.
Define pv+l(x, Dx ; f)=(―R(x, Dx ; t))v+l. Then
given by (3.3) with ^=-r(x,
￡;0 (y=l, ･･･, v+1).
given by (3.3) with qj=ritfj](x,$; t)(S^=≪,
it's symbol
Moreover
S/3;'=i3). Therefore
virtue of (3.4)
f C+1(￨r￨f2?+/s,+I,orl
if ＼a+p＼>v,
pv+iWx,e;t)＼￡＼
for x, ￡e/?B.
From
(3.1) we have
pv+i(x, ￡;t)is
pv+l＼$(x,￡; t)is
＼r＼$>^Cot(fe[0, T]).
Hence
we have by
246
Akio
if T
is sufficiently small, which
Q(x, ￡; 0=
Now
we
can
S
implies
pv+i(x, ￡; 0+leiB?([0,
construct
K{%,
where
Baba
the inverse
i^"1 of K(x,
Dx;t)-'={l+R{x,
￡=*-**･≪:'>.
The
symbols
T] ; Sg ≫)n^i([0,
Dx;
of
K
t))
Dx;
T] ; SJ.o).
t) as follows
lK{x, Dx;t),
＼x,Dx;t)
is in
^?([0, T];
^K[O,T];SJ..).
Put
m(jc, t)=Kv(x,
Fm(x,
t). We
t)=P°Kv(x,
have
t)
= ＼e^eA≫'^＼DtA+±UDjA-ia^+Dt+^±^j-a^
+ c(x)+＼ ±(DjA-a,y+＼
tDjtDjA-aJ―tiaJ)*}
Xf>(￡,t)3$
On
the other
hand,
K*{Dt
+
4
ibiDj-atyUx,
u j=1
=
K° {D.
+
t)
>
i￨l(flj-2a^+(fl?)l-/)jfl?)kx,
0
4
S tf≫(a?(x,0￡>>(x, 0+4
and
a(K.aRtDt)(,x, l;0
=%Os-＼V
n
=
2
.7 = 1
where
eA (*.!;￡) aR
Uj
i=i
ir 1=1
2 K°((aj)2-Djaj)v(x, t)
S8,0)n
The L2-weliposedCauchy problem
e
b(x
247
0
n
fl ＼―f)
j=i iri=2Jo
I
T!
o(%K'(a^y＼x,
＼＼e-iv-HeA<x-Z+^yr>afm(x
+ dy,t)%jdy8r}＼de
I
￡; 0=
=
?
Os-＼＼e-iy-*eA<x-t+>>;t＼a?(x+y,
t )fdydr]
S
eA<-x'**＼a%x, t)Y+Cl(x,
￡; t)
and
Ci(*, ￡;0=
Therefore
2
(10s-[[e-iI'-'(^<x'≪+':O)<'PJ(a^(A:
2
+ ^3>, t))＼vdydf]d6.
we have
K-^P-Kvix, f)={/)t+i jt(Dj- a^f)v{x, t)+c(x,Dx; t)v(x,t)
where
(3.5)
c(x,
Dx ; t)v(x,
t)
= K-1o＼e*'^^-^＼-i){At(x,
･A^x, ^;t)+aI(x,
￡;t)+t-Ax(x, ￡;t)+aRx{x, t,￡)
t,$)}${$,t)3$
+ K-1ob(x, Dx; t)v(x,t)
+
Dja%x,
t)v(x,t)-jfK-x°Cl{x, Dx; t)v(x,t)
^(DjA-ajf+jDjiDjA-aj)
+
Lemma
3.3.
Assume
that the condition (S) in
c(x, ￡;0 is in ^?([0, T] ; S°o
o )H^K[O,
Proof.
of (2.1)
Note
and
it is
that the first term
evident
i￡?([0,T] ; Sg,0)n^K[0,
in j≫?([0,T] ; Sg.0)ni≫K[0,
and
e
that
T] ; SJ,0).
have
is valid.
Then
T] ; SJ 0).
of the right
the
terms
Hence
T] ; SJ.o).
y^^<yy2l<Dvy2le-y^=<7]y2l<Dyy2le-y^,
1―A,,, we
Theorem
4w}
By
side of (3.5) vanishes
in (3.5)
except
it suffices to prove
Lemma
b(x,t-;
that
2.1 (iii),Lemma
where
because
t) are
in
b(x, ￡; t) is
2.2 (iii),(iv)
<Z)^>2=1-A,,
</>,>"=
Akio
248
Baba
＼b＼%Kx,$;t)＼
7
x$>4b>-!I<D<>
Xd?D%'a?n(x+0y,
^ca/9
rf[<v>-2'<r7>wr(-
61+1
(x+dyy+s*
X
t)-%dydr^dd
s)ll(s)+^(s)i^+r
ds
＼
dydydd
^'.^-"^y^'dyavW^L
dsX
<X>1
+￡2
Here we take /=[≪/2+2] and we use the inequality
(f-s)(￨g￨+l)
<x(s)><^>1+^
<(t-s)O
%
+ fe(:V' v>
T)dT＼<x>
C<(t-s)￡>
<
<(t-sMy, v, d(t-s)+s)>
^c,
1
1
<x+0v>1+￡Kv>2
1
^
<x+dyy+s*<dy>1+E*
l
^
<X>1
+ ￡2 '
and
iW,.*oisc.,{j:j￨gpu+s:
for any
By
then
we
a (￨≪￨ ^1)
(see
Lemma
3.3
can
transform
we
the proof
have
of Lemma
I
This
a(c)(x, ￡; t)<=S°OiO. Finally
the Cauchy
problem
j K-'oPoKvix,
(**)
2.2 (iv)).
ds
v(x. O)=vo(x)
completes
put
(*) to the problem
t)=K-xf,
(=uo(x))
}
<x(s)>1+￡i
the proof.
u(x, f)―Kv{%,
t),
The L2-wellposedCauchy problem
249
where
K-ioPoK(x,Dx;t)=Dt
+
￡(Dj-a%x,
t)f+c(x, Dx; t)
and
c(x, ￡; Oe^?([O,
§ 4.
The
existence
real
valued
T] ; SU)r＼<B＼(＼S),Tl;
theorem
for Schrodinger
type
SJ.O)
operators
of
coefficients.
Denote
ib(Dj-aj(x,
P(t)=
Assume
^K[0,
that
T];
aj(x,t)
are
real
SS.o), then
P(0
satisfies
(4.1)
where
t)f+c{x,
valued
and
t).
c(x,l-;t)
is in
adjoint
(u, P*(t)v)Lo for any
u, v^S.
P*(t)
is
operator
We
of P(t) which
consider
u(x,O)=Uo(x)
we
obtain
Theorem
is defined
the Cauchy
(j-iP{t))u{x, t)=f(x, t)
(4.2)
4.1.
Sg,0)r≫
x^Rn,
as (P(t)u, v)l2=
problem
fe[O, T]
xGi?"
Assume
C?([0, 7] ; <B°°(Rn))a nd
that
c{x, ￡; t) in
an^
aj(x, t)
^?([0,
/(x, OeC?([0,
are
real
(4.3)
[Ua)l!//^c(￨￨Wo￨kft +
for fe [0, T] and k
=0,
valued
T] ; S8.o)n^i([O,
functions
T] ; SJ,0).
T] ; J72) ^/zere ex/s?s a unique
u(x, t) of (4.2) u;fe"c/z6e/onffs /o C?([0, T] ; //2)nCK[0,
in
T/ze≪
sulution
T] ; L2) anrf safeyzes
jj￨/(r)￨U/r)
1, 2
Following the idea of Kumano-go
We need severallemmas.
[1], [2], we shall prove the above theo-
First,we define{￡,(￡)}"=!
as
(4.4)
＼
v
v/
and Pv(f)=Pv(x,Dx; t)as
(4.5)
T];
the following.
/or an}' M0G//2(i?ra) and
rem.
^?([0,
o(P*(t))-o(P{f))<B<B＼(＼Q, T] ; S%,,)f＼B＼{＼0,T] ; SJ.o),
the
Then
Dx;
pv(x,?; t)=p1(x,US); t)+c(x,%; t)
250
Akio
where
p^x, Dx; t)=^Ef=1{Dj-aj(x,
Then,
Bab a
t))＼
we consider the following Cauchy
problem
f Lvuv=dtuv-iPy{t)uv=f{t)
(f E[O, T]),
(4.6)
I Uv j=o―Mo .
We
define the series of weight
function U,/￡M?_1 as
;u0=<w$)>={i+
(4.7)
s(vsin^)2}1/2
then we have
'
i)
ii)
l^v(￡)^min(<￡>, Vl+ny2),
＼d$u&＼izAM$y-iai,
(4.8)
iii)
*..(￡)-―><￡>(v->oo) on R$ ,
(uniform convergence in a compact set),
In
fact
￡,(￡) satisfies
i)
IC(S)l^min(￨￡l, Vnv),
ii)
(4.9)
iii)
(uniform convergence in a compact set).
Denote by STv the set of symbols
p(x, f)eC°°(i?2re)
satisfying
i^K*,￡)i^ca/5;u￡r-'≪'
for any multi-index a, B.
Lemma
4.1.
Then
For p(x, f)eSTO
we get the following lemma.
put
and for any a, /3 there is constant Aa^
/>,(*≫￡)=/>(*,￡,(￡))■
Then
which is independent
p(x, $)(eSTv>
of v and p, we
have
i ＼P.n＼ix,
^)＼<{Aa^＼p＼＼r+^MT-]a＼
(4.10)
I pv(x, $) ―> p(x, f) (uniformly)
(v->oo) at RnxxK^,
where K$ is an arbitrary compact set of Rf.
Denote Hiv.t={us=S';
Lemma
4.2.
UOsu^Lz}.
P―p(x, Dx)ESfv
is a continuous mapping
from
space Hi s+m to Hx s and for a constant Cs m and l=l(s, m) we have
the Sobolev
The
(4.11)
L2-wellposed
(4.12)
251
problem
＼＼Puhv.s^(Ct,m＼P＼lim>)＼＼uhv,t+m,
Especially for m―0 and s=0
u<=Hiv.l+m.
we have
II^Pmlk2^(CI/≫￨i<0>)￨￨Miiz:8,
u^L,{Rn).
Lemma
^0
Cauchy
4.3. For
c(x, ￡;Oe.0?([O, T] ; Somo),^(l)eS^
(J=l, 2) anrf /^/^
we have
o(g1,ocoq2v)(x,
￡ ;t)<B<B%[0, T] ; So f'≫+'*).
Proof.
We
take 2l>h+n
^C<^>^v(f)± we
(/"i=max (/lf0)),then by Lemma
4.1 and A^+i))*
have
＼(qi≫°c°qtv)[aB)(x,￡;
t)＼
=
Dldf＼Os-fy-iv-iqlvG+y)c(x
<L^cf-a2-a3
+ y, ￡;t)q^)dydf]]
Os-^<Tiyal<Dyyil{<y>-sl<Dvyte-tv'i>}
XqiulalK$+y)c$Kx
+ y, I; t)q^a3＼$)dydV
^c＼%)mu%yi+l2＼＼<yy2l<vyu+lidy3r)
<C//<6>m+il+*2
where
f
al-＼-a2jraz―a.
Lemma
4.4.
For
any
uo^L2
and
any
/(OeC°([O, 7] ; L2) there existsa
solution mv(OgC￨([0, T] ; L2) of (4.6) which satisfiesthe energy ineqalities
(4.13)
(4.14)
(4.15)
(4.16)
where
llMpCOII^^IlMoll+
fV^'-^H/CONr
Jo
MWOII^^M^oll+fVi^-^M^/Wllrfr
Jo
T])
(fe[0, T] ; ; = l,2,3, ･･･)
^CT{＼＼Ai+2u0＼＼+max＼＼Ai+2f(T)＼＼}
co,n
at
ae[0,
(*e[0,
T] ; ;=0,1,2,
･･･)
＼＼Ai(uv(t)-uu(n)＼＼^C'T＼t-t'＼
{＼＼Ai+2u0＼＼+
mox ＼＼Ai+2f(T)＼＼}
[0'r]
(t, *' =[0, T]; /=0, 1,2, ･･･),
y, yl} CT,
C'r are
constants
which
are independent
of v, and
AV=XJDX)
252
Akio
II-II=
Baba
11-Ik
Proof
of Lemma
4.4. I) If we
M([0, T];
&
{R*≫e))
(k = l, 2).
We
4.2. Then
Pv{t)is an L2-bounded
fix v arbitrarily, we
note ^"(/?^f)=SS
have
pv(x, t
t;)(E
tf=l), and use Lemma
operator uniformly with respect to t. There-
fore uJt) to be a unique solution of the integral equation
(4.17)
uv(t)=u0+i＼
which
can be solved as follows;
u
fT1...
r*"1^^)
Jojo
Jo
p(O=Mo+S*"*
k=i
Ct r
n
Si*
+
Then
we
r*
-
JO Jo
k=2
Pv{T)uv(z)dv+＼
'iVrO
Jo
･･･Pv(rk)u,dtk
■■■
drr+i1
･･･ Pv(Xk-i)f(Tk)dTk
･■･dr1
f(t)dt
Jo
(n=t)
have uv(t)(EC＼([O,
T ] ; L2)
II) From
(4.6) we have
llMvll1
=2Re(-ruv, uvj
dt
= Re(i(Pv-P*)uv,
By the way,
from
uv)+2Re(f,
uv).
the definitions of (4.1) and (4.5) we have
(4.18)
Thus
'/(,)dr
J≫= a(Plv)-o(PS)eC0t([0,
by Lemma
T] ; 5^).
4.2 and the Calderon-Vaillancourt theorem, for a constant f>0
which independent
of v, t we have
^￨￨M,(0ll2^2rlkv(0il2+2￨￨/(0IIN,(0!l
(t(E[0, T];
v=l, 2, ･･■)
and
^IWOII^rllttvCOII+
Therefore
ll/COII-
we get (4.13). Moreover from (4.6) we have
d
.
A
dt
and PlViJ=Plv+[Al,
4.3 AlcAzJ^&K[Q,
iuu=i(Plv+ ＼_A{,
Plv
PXv
KHere, [A{, P1v
T]; Sg.o)n#K[O,
Plvj satisfies(4.18).
hand, noting
＼A
Hence
]A-3+A{cA-j)A{uv+Aif
we
]A;j(eCK10,T
T] ; SJ>0) uniformly
＼;
S%) and by Lemma
with respect to v and
get (4.14) similarly to (4.13). On the other
253
The L2-wellposed Cauchy problem
jAiuv=i(AiPlvA?-2+AicA;J-2)Al+2uv+Aif
where <y(v4'JVf;7>-a)<EC2([O,
T ] ; S%) and JM^2e
<8?([Q,T] ; Sg.o)n^K[0,
T] ;
So o)(uniformly on v), we have
jAiuMW^dWAl+'uiW
Then
we
from
this and (4.14), we
get (4.16) from
Proof
of
(4.15).
Theorem
C?([0, T] ; H^.
Then
of (4.6).
from
Then,
completes
4.1.
1)
it follows
(4.9) we
(4.19)
ft d
Put uv(t)―uv(t')=＼
r-uv{r)dT.
get (4.15).
This
the proof
First
from
we
where
Therefore
of Lemma
assume
Lemma
Then
4.4.
that
uo^HA,
/(t)E
4.4 that there is a solution
ujf)
have
Mfroll^M'uJ,
for y^4,
+ UlfW.
＼＼Alf＼＼<＼＼Aif＼＼
A ―{Dxy.
by (4.13), (4.14), (4.16) we
(4.20)
have
＼＼Aiuv(t)＼＼C1{UiUo＼＼+
max M>/￨￨},
[O.T]
(4.21)
for j￡A.
Therefore
if we
fix to<^[0, T]
{uv(to)}=i is a bounded
sequence in
arbitrarily, then
L2.
subsequence such that {uVm}m=i of -^2 we
uVm―>
and we
(4.22)
Then
Then
there exists u(to)^L2
and
a
Then
(m->oo).
by the diagonal method
there exists
have
uvjtk)―>
u(tk)(weakly) in L2
(m-^oo).
from (4.21) with j―0, for any t,t'^{tk}f=1 we have
(4.23)
For
(4.20) with ;―0,
have
u(t0)(weakly) in L2
Let {tk}f=i be a dense set in [0, T].
u(tk)^L2
Hence
from
＼＼u(t)-u(t')＼＼^C2＼t-t'＼{＼＼A2u0＼＼+max＼＼A2f＼＼}.
any
^0e[0, T]
we
choose
subsequence
by (4.23), {M(^s)}f=i becomes
exists u(to)^L2
and
we
a Cauchy
U*s}r=i °^ {**} sucn tnat tks-^t0.
sequence in L2(Rn).
have u(tk.)-*u(t0)in L2 (s―>oo). Moreover
Thus
there
for all m(￡)
254
we
Akio
Baba
obtain (4.23) and
uvjf)―>
Thus
for (p<BS
(4.24)
noting
(4.25)
in Lz
(m->oo).
have
(MmuVm{t),
Therefore
for
we
u(t) (weakly)
<p)=(uUm(t), A*m<p) -h> (M(0, Ak<p)
M*mw,m(0>
y)l ^＼＼A*muVm(t)＼＼＼＼<p＼＼,
we get from
￨(m(0, ^^l^diM'uoll+maxMVIIIIIyll
k^L
We
define
UR.kit, $)=<$>ku(t,
{^}7=i
°f ^
uRtk(t, x)(eL2
￨) (￨f￨^i?),
and
as supp<p(Z{＼{j＼￡R},
for
Therefore
from
(4.25) we
any
￡<ee
[0, T]
taking
i?―>oo, we
(4.26)
Z?>0
take
such
that
the sequence
we
have
<$y＼a(t, e＼2d$ (y-oo)
JlllSft
get
have
Aku^L2
(k￡4).
and
＼＼Aku＼＼<C1{＼＼Aku≫＼＼+
max ＼＼A"f＼＼) (k^i).
[0,?]
(4.21) we
(4.27)
Thus
have
similarly
to (4.26)
＼＼Ak(u(t)-u(n)＼＼^C'2＼t-t>＼{＼＼Ak+*u0＼＼+max＼＼Ak+*f＼＼}(k^2).
[0.T]
we
We
obtain
u(t), Au(f),
A2u(t)(EC°t(l0, T] ; L8).
take ^(f, x)=f,(0^(x)eCo((0,
L*m^
we
and
and
<Pj->ur k(t, f) in L2 (y->°°). Then
￨￨M≪.*(OII^C1{M*Mo￨￨+maxM*/ll}
[O.T]
Hence
(4.20) and (4.24),
(^e6>)
fifliJt(f,
￡)=0 (￨f!>i?)
(u(t),Ak(pj)=＼u(t,$)Q>k<pM)d$―>＼
J
From
(m->oo).
―>
T)xRn)
L>
arbitrarily.
Noting
in C?([0, T] ; Lt),
have
≫rr
Lu-<pdxdt=＼＼
u-L*<pdxdt
= lim＼＼ LVmuVm-<pdxdt
m-≫oo
J JU-p
m-≫oo
JJUf
=＼＼ f-*ixdtBT
Thus
Lu=f
(fe[O, T]).
Therefore
u is a solution of (4.2). Also noting u(0,
Au(t), A*u(t)^C＼(＼R,T]; L2) and dtu=iP(t)u+f,
we get w(0eCJ([0,
T] ; L2)n
C?([0, 7] ; //2).
II) If M(OeCK[0,
T] ; L2)nC?([0,
T] ; //2)is a solution of (4.2), we
obtain
The
L2-wellposed Cauchy
problem
255
the energy inequality (4.3) similarly to the proof of Lemma
Assume
uo(eH2
and
/(f)eC2([0, T] ; Ht).
Put
4.4.
uo,t=Xt(Dx)uo
le(Dx)f(t), where Z(￡)e<SfsatisfiesZ(0)=l and Z,(￡)=Z(e￡).Then
gC°([0, T] ; //4). Therefore from I), we
and
/￡. Then
by (4.3) we
and
/,(*)=
uo.t^Hit ft(t)
get the solution us(t) of (4.2) for uo,s
have
＼＼u￡(t)-uAt)＼＼2^e^T{＼＼uo,s-Uo,e'h+Tmax＼＼f,(T)-fAr)＼＼}.
[0,7*]
On
the other hand
when
e, s'―>0we have
f ￨￨Mo,.-Mo.e'￨￨2=￨￨(Z.(^)-^(Z)x))Mo￨￨2―*0,
I max
￨￨/.(r)-/,.(r)￨￨2=max ＼＼{UDx)
[0,7"]
[0,71]
Therefore,
noting
dtue―iPue + f$, we
can
lADx))f(r)h ―> 0 .
see {ms(O}o<s<i becomes
a Cauchy
sequence in C￨([0, T] ; L2)nC?([0, T] ; H2) and the limit of this series u{t) is z
solution of (4.2). This
Proof
of Theorem.
completes the proof of Theorem
By Theorem
4.1 we
4.1.
can see that there exists a uni-
que solution v(x, f)eC?([0, T] ; H2)r＼CK[0, T] ; L2) of the Cauchy
problem (**)
with initialdata uo(x) and we set the energy inequality
IK-, 0IU^C(T)(￨￨u.lU4+f
for ￡<s[0,T]
and
k=0, I, 2.
Hence,
ji/ir-'/O,r)＼＼Bkdr)
we
obtain the unique solution u(x, t)=
Kv(x, t) of the equation (*) with initial data uo(x) and
Vaillancourt theorem
we
get the energy inequality
≪≪(-,
OIU^C'(T)([￨Wo￨U,+
for fe[0, T] and k=0,
by using the Calderon-
1, 2. Thus
we
ril/(-,
T)＼＼Hkdr)
complete the proof of our Theorem
Acknowledgement
The
author
wishes
to express
his appreciation to his advisor, Prof. K
Kaiitani, for his helpful suggestions.
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