Advances in nonlinear partial differential equations and stochastics

• J ^ M W*W*tf)

X'aXl3X^Vs=0

(i = l

m)

(1.7)

3u-0

for any real vector Xi = (X^X{:X^)

satisfying

(X|) a -cf£(XJ) 2 = 0.

(1.8)

Thus we can interpret this as the null condition for the system (1.1) with different speeds. If

d2cf

di&yU^didiU1)

=0 Su=o

UA7,i«U*J

(L9)

45

then the null condition (1.7) is automatically satisfied. The main aim of this paper is to prove the global existence of solution to (1.1) with small data under the assumption (1.9). It is still open whether the null condition (1.7) guarantees the global existence of solution to (1.1). The null condition is also considered with respect to nonlinear elastic wave equations. For this problem, see T. C. Sideris' work.13 In section 3, we introduce some notations and in section 4 we state the main result. In section 5, using the representation of solution to (1.1) of Kovalyov,10 we estimate the first order derivatives of solution. Finally we prove the main results in section 7 using the estimates and energy inequalities in section 6. 2

The Null Condition.

In this section we introduce the null condition for the system (1.1) with different speeds stated in Introduction. We consider the system in the form m

£

2

Y.

a*?{du)daa0u> = 0

(i = 1,..., m),

(2.1)

where du stands for space-time gradient of u t

du = a

(du\...,dum), d

x

8

d

TU

d

We assume that

a?f{0) = -<$6ij6aa

for

c, > 0, c, ^ Cj for

i^ j

(a,0) * (0,0),

(2.2)

and C${du) - 1, a°f(du) + c*6uSai) vanish at du = 0 at least of second order, that is, a°°(du) = \+0(\duf) = - c ? M a J + 0(|9u| 2 ) a°f(du) near du = 0. Set v v'

= =

V,-.-.""1) (u0.w!
e

(24)

46

Then we find from (2.1) and (2.4) that the vector v satisfies a system of first order which is hyperbolic near v = 0: 2

J2aa(v)dav = 0

(2.5)

Here the 3m x 3m matrices aa are defined by aa = {A°j : i,j = l , . . . , m ) ,

(2.6)

where 4,

=

/ «p 0

0 6i}

0 \ 0 ,^

\ o

o %/

/ 2aJ? -*y

=

e$ 0

V °

ajf \ 0 ,

o o / (2.7)

^,

=

0

V -*&

0

0

o

oy

.

We next consider the plane wave solution v(t, x) of the equation (2.5): v{t,x) =w{t,s),

s = ^2&xi,

(2.8)

t=l

where C = (Ci, 6 ) £ R 2 and C / 0. Then we ffnd from (2.5) and (2.8) that the vector w satisfies a system in one space dimension. dtw + a(w)dsw - 0

(2.9)

where 2

a(iu) = o°(u)) _I 53CiO*(«i)-

(2.10)

1= 1

We take the initial values for the solution w of (2.9) in the form w(Q,s) = E
(2.11)

where

00 We shall seek the eigenvalues A = A(tu) of the matrix a(w) and the corre­ sponding eigenvector £ = £(tu). By the definition, A satisfies an equation det(Aa°(u>)-£Cia>) ] - 0 .

(2.12)

47 We can verify by induction that det (\a°{w) - ^2Qa^w))

= Y"det( P l J : ij - 1, . . . , m )

(2.13)

where py = a°°A2 -

2 A £ < $ 0

+ £

agOC.

(2-14)

Therefore we find from (2.2), (2.3), (2.12), (2.13) and (2.14) that the eigen­ values A±(0)(i = 1,...,m) of the matrix a(0), aside from the trivial multiple eigenvalue A = 0, become

A?
(2.15)

According to (2.4) we arrange the components of a vector £ G R 3 m as follows: (2.16) Then we find from (2.2), (2.3), (2.7) and (2.12) that the eigenvector £±(0) corresponding to Xf (0) becomes (^(0))J

(fiW

-

'(0,0,0)

=

t

for

j*i

(=Fi,CiMlCU2/ci|CI)

and the eigenvectors & {« = I,,.. .m) corresponding to the trivial eigenvalue 0 become (&}J

=

'(0,0,0)

for j / i ,

f

{&)'

= (0,C2,-CO-

Since ^ ( 0 ) , ^ (t = l , . . , ,m) are linearly independent, we see that the system (2.9) is hyperbolic near w = 0. We now require that a solution tu(t,«) to the initial value problem (2.9), (2.11) has a lifespan Ts which is at least of order e - 3 for any ( £ R 2 . This requirement is equivalent to the following facts 12 : 2

dXf

2-, 2—i a 3

(#W)i=*0 u/=0

(2.18)

48

and 32A±

£ £

(«?(o)):(«?(o));=o

(2.19)

tu=0

for i = 1 , . . . , m a n d C £ R 2 Set P(A) =det(py(A) : i,j = l , . . . , m ) . Differentiating the equations P(Af(iu))=0

(2.20)

(t = l t . . . , m )

in a variable < and evaluating the results at w = 0, we get

2c-icr- i n^- c ?)^

= 0 UJ=0

which implies d\*

(2.21) = 0 dwi m = 0 for all ij,a. Therefore it follows from (2.2), (2.3), (2.14) and (2.21) that the condition (2.18) holds trivially and

9fa{Xt))

(2.22)

=0

dw'

u'=0

for all i,j,kj,a. Next differentiating twice the equations (2.20) in variables mi and wk0 and evaluating the results at IU = 0, we get

ICI2 m - 2

!!<<*-*

pfavt)) dwidwg

t*i

for all i,j,k,a,0.

By the definition (2.14) of pu(X), we have

2

d tpu(Xf)) dwidwg

(2.23)

= 0 ui=0

=

2Xf(Q)

w=0

d22\± \ dwadw%

±

-2A1 (0)£ ,=1

+z ,yi

+ Xf{0)2

w-0 d2.2„I0 a'

dwidwg

?i2n00

0 ait

dwadw'k

ur=0

m=0

Baftf

a^iaw*

li! = 0

(2.24)

49

Then it follows from (2.15),(2.17), (2.23) and (2.24) that

d2\\ ±

TC.KI

dwadw$

2 liS=0

ii=0

dw{dwks

(tfW)iUf(O))!

(2.25)

u>=0

Therefore we find from (2.17) and (2.25) that the condition (2.19) is equivalent to d21„-rS a „ dwlndw\

, .

(4f(0))lfe*(Q))J(4f (0)); (^(0))i = 0

(2.26)

u>=0

fori = 1 , . . . , m a n d ( e R 2 By the definition (2.17) of (^(O))'\ we have

{ftfTO^-^Et^wtf-o

(2.27)

J=I

for all ( £ R 2 Consequently we have proved the following Proposition 2. 1 The fc/espan Te of a unique solution w{t,s) of the initial value problem (2.9), (2.11) is at least of order e - 3 for any < e R 2 if and only if it holds that

£

2 3,2„1<(S a

i,8.~,.i=0

a

X>aX<0X\X>s=O

(f = l

m)

(2.28)

& u/=0

for all real vector X1 — (XQ,XX,Xl2)

satisfying (2.29)

Setting

C$(du) C?f(du)

= l-af(du)> = -cfSi^-affidu)

(a,0)jMO,O),

that the null condition (1.7), (1.8) follows from Proposition 2 . 1 .

50

3

Notations.

To begin with, we introduce some notations that are used throughout the paper. Partial derivatives are denoted by dt

dx\

ox2

We also use the angular derivative: SI — x\di — x-idi. We set

and define VA = V?'V^V$\\A\

= Ai + A2 + A$,

where A.= (AU A3,A3) is a multi-index. Let u = ^ ( u 1 . . . , um) be an unknown vector and set Wi(t,r)

= ( r + l ) 1 ' 2 " 1 ^ + r + l) 7 (|r - fcjt| + i ) ^ a

(3.1)

for 0 < 7 < 1/2. Then we define, for a non-negative integer fc, \du(t,x)\

=

£^|30uU*)l m

\du(t)\k =

Y.

[&*«]*

=

SU

E E

P !*>***«'(M)|

|A|<* i=l a=o^eR = m

IWOIU -

2

(3.2)

2

U,

E EE1I^« "MUR., E

SU

E E

P |^(*,kl)^9Q«'(i,x)|.

Moreover, we define |du|*(t)

=

sup

\du(s)\k,

[du]k{t)

=

sup [M*)]k-

(i A}

-

0
In what follows, M denotes various constant depending on Fitf\ Cj.

gi and

51

4

Statement of the Main Result.

The initial value problem to be considered is f dfu'-ct&u'^FtO^dhi)

in

ul(0,)=ef1,dtul(Q,)

= Eg> in

[0,oo)xR2 (i = l , . . . , m )

R2

(4.1)

where a are positive constants and e > 0 is small parameter. Moreover, / ' and a1 are C0"-functions with compact support. We describe some assumptions on the initial value problem (4.1) and state the main theorem. First, we assume that F* are of first degree with respect to the second derivatives of u: Fi(du,d2u)

m

2

= J2

Y,

^f(du)dad$uJ

+ Ei(du).

(4.2)

Here, Off and Et are C00-functions of du in {\du\ < 1} that satisfy C?f = C?; = C?f,

(4.3) 2

| c , f (3u)| < M\du\ ,

(4.4)

|£,(3u)| < M\duf

(4.5)

Assuming (4.2)-(4.5), M. Kovalyov10 proved the almost global existence of the solution to (4.1). Second, we assume the null condition (1.9) for different speeds introduced in Introduction: c, f Cj for

i^ j

2

8 C?f d{d^ul)d{dsui

du=0

(4.6)

- U^-TJ- ™

The conditions for E{ are d*E,

8)

in accordance with (4.7). Theorem Z,ef iu assume (4.2)-(l8). Then there exists a positive con­ stant ea depending on given functions such that the initial value problem (11)

52 has a unique C°°- solution in [0, oo) x R 2 for all e with 0 < e < e0. M. Kovalyov showed in his paper n1 that the theorem holds when Cfj = 0 and Ei (i = 1 m) satisfy the condition d3E,

(U =

=0

h-,m)

instead of (4.8). 5

Estimate of the First Derivatives of the Solutions to Initial Value Problems.

The aim of this section is to estimate the first derivatives of the solution to the initial value problem: f Sfti -&u=

F(t,x)

[ u(0, ■) = &««>, •)=<)

in

[0,D x R 2

in

R2

(5.1)

Here, F is a C~ function m [0,T) x & For this purpose, we use the represervation formula of the solution to (5.1) which has proved by M. Kovalyov 10.

Proposition 5.1 Let u £ C°°([0, T) x R 2 ) be the solution of the initial value problem (5.1). Then, u has the following representation: u(i,x)

=

— 11

rdrds I

+ Y-\(t-a)

f[

KiF(.%re'/^ie+'i'))di> rdrds f

A',F(s, r e ^ ^ 1 " ' ) ^

where x if

f 1 X'(s)

— (a cos 0, a sin #) = a e * ^ 9
=

t

(

(s,r) e D

{ { f - s ) 2 - a 2 - r 2 + 2arcosV} l <' 2 (s >> 0) 0) / 1 1 {5 | 0 (s < 0)

53 Moreover,

the domains D' and D" are. defined as follows.

&

-

{ ( » , r ) ] 0 < »<■*, rj < r < r 2 }

{

{(s, r) I 0 < s < ( - a, 0 < r < n } for

t >a

0

K a

for

where n = \a ~ t + s\, r2=a

+ t - 8.

(5.2)

Next, we derive representation formulae for the first derivatives of the solution of the initial value problem (5.1) from Proposition 5. 1 . In order to Dresent the formulae, we set 6

=

min{l/2,a} (a = |*|)

6

=

(5.3)

min{l/2,(t-a)/2}

and split the domains D' and D" as follows: D'

=

blue U white

blue

=

{(s,r)€D'\r1

=

f D'\blue I

for

white

y 0

foT


D"

=

black U red

black

=

{(s,T)eD"\n-i
red

=

f D"\black I {
or

5-1/2 I5 = Q

&c

for

6=1/2

for

6 = (t -

a)/2

We set hiue(F)(t,x)

=

It J Jblue

r2-S
rdrds I < —4?

KxF(i

0
54

We(F)
-

ff

W*(F)(f,i)

-

ff

J—\p

=

K,F{s,re^~^+^)di>

rdrds f

J Jblack

h,d{F)(Ux)

K,F{s,re^~^e+^)d^

rdrds T

JJ white

ff

J -n

K]F(s,re"/^IiS+,t,])d^

rdrds f

J J red

J —n

Then, by Proposition 5. 1 , d^u {/i = 0,1,2) is represented as

+X(t - o J W t t ^ f ) + x(t - a 10

Following M. Kovalyov, the map ip = * , where

l)I„d{duF)}.

we change the variable of integration from $ to r by $

=

arccos[l + PT — r], a2 + r2 -(t-s)2 2ar

Then we have the following Proposition 5. 2 IwhitAdpF){t,x) m

X) I / /

*

JJ

j.Q=0 [ S(whtU)

- ff

rh'2a°(8 + * > / ( « , r e ^ T , , + * ' l ) d r

/ Jo

drds f Va{rK2a°(9

JJaikife

+

VJ)}F(s,re^(e+'t>))dT

JO

- ff rdrds f K2a^(e + * J ) ( n F ) ( S , r e ^ I f e + + ^ ) V a * J d r l JJwhite Jo J + 11

K1a2l(e +

drds j

J J white

iJ})(QF)(s,re^~HB+'l'))diP

J "¥?

U
Jd(red)

'Jiff + ff

9 tre^ +^)di}>

J-n drds

drds f

I

V

o { f " ^ i } < ( ^ + V')F(s,re^ T T ( s + ^)d0

K1Ql($ +

$)(nF)is>re*/=Il6+W)dj>

55

where Ko

=

*j

=

{2arr{l - r)(2 + PT - r ) } 1 / 2 ' (-l)J*,

f 0 fa = 0) <(*)=$!.

<(*)={

cos* I sin<9 ^ V Q

(JI = 1 ) , (/i = 2)

(0 <£(«)={

-sin* I casff

(,x = 0) (p = l) , (|t = 2)

(a = 00) (a = 1)

_ / 9 s " \ dT

n = W m ) w the unit outer normal vector field on d(white) U fi(red), anrf do- is the line element on d(white)Ud(red). M. Kovalyov used these formulae in his work,10 but he has omitted the terms containing the first derivatives of *j in the above formulae. So we show the proof for completeness. Proof. We denote F^re^))

= G(s,r66), Then,

a =0

Therefore,

U«(8/)

= Jiff

rfrdB f Kta*(0+i>)(VaG)i*,r,9 + \t>)
Q _ Q J J white

+ //"

J — \fi

drds I

J Jwhite

Kial(8

+ i}>){deG)(s,r,8 + ip)di>.

J-,f

Changing variable from $ to r by the map ii> = * , we have //

rdrd.5 /" «Tia*(# + ^)(V a G)(*,r,* + ^ ) #

J J white 1

-Jill

J—ip

f/

rdrds

r1

/ ^ < ( ^ + *;)(V o G)(s,r.0 + *j)d7-.

f

{5o

.

>

56

Notice that (VaG)(s,r,8 + *j) = Va{G(s,r,$ + #,-)} - (deG)(s,r,8 + * j ) V a * j . (5.6) Substituting (5.6) into (5.5) and integrating by parts give I white (dp F)

= Y

do

I //

- //

rK

f

KW

+ *j)naG(s,r,8

+ *j)dr

drtfe / V o {rtf 2 <(0 + *,-)}
JJwhite

Jo

- ff rdrds f K2a°(9 + Vj)(d$G)(s,r,9 + V})VaVjdT\ J Jwhite Jo i + ff drds f K^Ke + tpKOeG^s^r^ + ^dtp. J Jwhite

J —&

(5-7) Similarly, htdid^F)

= Y

rdrds

ff

a=0J

+ ff =

Y

f

Jred

Kia"(* + ^)(V a G)(«,r,0 + ifr)d0 J-x

drds f ff

~Y + //

ff

Kml{6 + rp){deG){s,r,e + ip)dip

rda

f

ff

i<(e

drds

f

Vo{r/C,}a°(0 + ^)G(s,r,0 + tf)(ty>

+ ^KC(s,r,« + ii)#

(5.8)

drds I tfia*(0 + tf)(flfcG)(a,r,0 + 0)«ty.

JJred

J— ir

Thus we get the representation formula from (5.7) and (5.8). I The following proposition is used to estimate the terms appearing in Propo­ sition 5. 2 . This was shown in M. Kovalyov,10 except the estimates containing the derivatives of $. For the sake of completeness, we give the proof of all. Proposition 5. 3 I. Let (s,r) € D'. Then the following estimates hold: (i)

/ tf,d0 = 2 / J -^

Jo

K2dr

57

^ ( ^

H)

2+

l o g

fir

(f - ri)(r + r 5 )

/ {\d3K,\ + \drK,\)dr<

\(( - 5 - a)

g

H) Jo / ' ^ { | f l . » l + l^*|}rfr< f {«r(r . 2, -^ rf)(r| , V ]- 722 )} U 11//22 II. Let (s,r) G D". Thera (Ac following estimates hold: M

/ * « • , ■ • ■

# <

1/ — JT

ii)

/

log

{15,^1 +|ft.A-i|}d0<

«f — ir

(n - r ) ( r 2 +r)

U (rj - r ) { ( f + n ) ( r 2 - r ) > ^

Proof. The following identity can be easily verified by simple computation. (r + r a ){r + a - t + s) 2ar

_

_ (r2 ~ r ) ( t - s - a + r) 2ar

(5.9)

I-(i). Changing variable by the map $ = * , we have

^

far)*

j

{r{\ - r)<2 + P7 - r ) } - ^ 2 d r .

(5.10)

First, we notice that in the domain D', \P\<1 2 + Pr-r

= (P + 1)7 + 2(1 - 7 )

>2{1-T)

for

7 > 0.

Thus, splitting the interval of integration into two pieces, we have / Jo

{T{l-T)(2

+ pT-r)}-1/2dr<21/2

r~1/2dT = 2, Jo

f {7(l-7)(2+P7-7)}-l/2d7 Jl/2 <21/2/

{(l-7)(2 + Pr-7}}-l''2d7

(5.11)

58

< 2 1 ' 2 f 3 T {-2(1 - r)^2}{2 Jo = 2 + 2^2{\-P)

l T)- ^dr

+ PT-

f (l-T)l/2(2 Jo

Pr-T}-3/2dT

+

< 2 + ( l - P ) / .(2 + P T - T ) - ' d r <Mlog[2+r^p].

(5.12)

Since P + 1 > 1/2 for t - a < a, the estimate I-(i) follows from (5.10)-(5.12) and (5.9). I-(ii). Since (2ar)3^(l-T) 1 /2(2 + P T - T ) 3 / 2 9r

*

2 =

1 r'/2 K2 ~2~r ~ 23/2( ar )V2(i _ T )i/2(2 + PT-

/l P\ \ a ~ *■/ '

3 2

T) /

then we have

/ {|aSJftr-2| + |a r ^ 2 |}dr _1_ f1 ~ 2rJ0

K

. 2 T+

t-s + a + r f] T^_ (2ar)3/2 JQ (l - T ) i / 2 ( 2 +

Pj

_ T )3/2 d T (5.13)

By I-(i), we have 1 Z"1 — I K2dT

<

Ma(ar)-'i/2

log[2 + (1 + P)" 1 ]

<

A/a(arr3'3(l + P r ' -

(5.14)

On the other hand, since

we have by the method from which (5.12) was derived, fl J0

r1^ 2

(1-T)^ (2

+ PT-T)V2

3 -

2 1 /M

1 + P-

( 5 1 5 )

59 Therefore it follows from (5.13), (5.14) and (5.15) that

(ary/1 M {ary/2{r + s +

Jo

1+ P a-t)'

I-(iii). We can easily verify that r i/a

*• ■ (H)

(-5 W2 ar { ( l - F ) ( 2 + P T - r ) } V 2

-

a5*

{(1-P)(2 + PT-T)>V2

We use the same method as we used in I-(i) and obtain

Jo iry/2 (2or

M (ar)1/2

1 I P t -s \ a T
1

/'

"L_ (l-r)>/»{2 +

PT-T)

Thus we get the estimate I-(iii). H-(i). In the domain D", P < -1 and t - s> a + r. Therefore,

LKl4*

=

{{t-s?-a*-r>yi*L

(T

■fl2-r»}'/»{;0

P-'cos^)1''2 (1

+

2-1/2)1/2

(5.16)

Further,

r/4 /„

# {1 - cos^ + (1 + P - ' J c o s ^ } 1 / 2

60 r*/* ~ <

J0 M{\

dtp 2

{27r-^ + (l + J P - I ) 2 M / 2 } 1 / 2 -log(14-P-1)}

= A/ | l + log(l - r ^ p ) }

£«•")

From (5.17), (5.9) and ( t - s ) 2 -a2

-r2

= -2arP>ar(l

- P) = - ( r + n){r 2 - r),

we get H-(i). Il-(ii). We can easily see that ((( - s) 2 - a2 - r 2 + 2urcoST/>)3/2 r - a cos ip a r ^i = ((( - s) 2 - a 2 - r 2 + 2ar cos i/>)3/2 Thus,

/

{|a,K-,| + |ft.tf,|}rf0

■/ - I T

< 2 < « - . ) / ; { ( J - s ) 2 - a 2 - r 2 + 2arcos^) 3 / 2 4(t - s) f 3 2 (-2arP) / 70

dip (1-P-'cos^)3/2'

(5.18)

L (i

We get by the same way as the proof of II-(i) that

L (i

P-'cos^)3/2 <»/3 A* -y0 (i_2-i/2)3/2

f»/" +

jQ


^ M ( 1 + rfp) < M n~p-

(519)

Therefore it follows from (5.18) and (5.19) that

J — rr

A/(t - s) P 3 2 (-2arP) / 1 + P

61

<

M(t - s) -1 (ar)3/2{i_p)i/M + P M (n - r J t o r J ^ t l - P ) 1 / 2

and we get the estimate Il-(ii).

I

Now we can show the estimates for the first derivatives of the solution to the initial value problem (5.1). Proposition 5. 4 Let u £ £7°°([0,T) x R 2 ) be the solution of the initial value problem (5.1). And let w(s,r) be a positive function that satisfies

(i)

fc-l
+ l=5. T7tu{s, it) < w(s, r) < Mw{s, k)

[k — 1,2,• •';M

w{s,r) ~

y^ +

(5.20)

is independent of s, k,r)

(r + s + l)(\r - cts\ + 1)

1 (r + s + l) 1 + 2 ^(r + l ) 1 - 2 ^

+

(r + s + l ) > + * ( | r - s | + l ) 1 " ' / (d j i l (i = 1,2, •■-,£), 0 < 7 < l / 2 , 0 < e < l )

(5 21)

-

Then the following estimate holds: M |5 (f,a:)1

"

"

(|x| + l J ^ - H M + * + 1WIM - *| + l) 1 ' 2 "

£

»p|IM-.i-i)»^(-.-)||0

JA|<3

+ V

sup

|,4|<20<S<(

||™(s,[-n^Ffs

>>llo}

{522)

62 Proof. By (5.4) and Proposition 5. 2 , we have

{

sup sup l^l11/2 w(ff,|*l)l^{*»*)l < \ sup Bap|*i (

\du(t,x)\

°<*< *eR*

+ sup sup |x| 1/2 w(,s, 0<«<'reR=

\x\)\dF(s,x)\ (5.23)

]/ 2

+ sup sup o<*<'isR=

\x\ ' w{s,\x\)\QF(s7x)\ )

. { / ; + . . - + !>.+!»+ ... + I") where/,' f( = l , . . . , 5 ) a n d /," (i = 1

4) are defined as follows.

/; = / / 4—r<*«fe /" K^ r2 = // -4—/h I K>dT

/; = ,;

drd

IL,^ 'iy^ '

= /jLw3f M, j£' JMM ' +IMD *

I'l = ff

I'l

+ia mdT

= If

—,—-dnfo / K\M

—,2

-drds /

/ . / r e d UAS.fJ

k\dTp

./_,,

Here, /,' (i = 1,...,5) are integrals that are related to the domain D', and /," (i = 1 4) to the domain D". We show in the following that 1

"

(a+l)'/a->(a-M+i)M|a_t| + i ) i / 2

„

1

(» = 1..-..5), (5.24)

M ~

(a+l)i/2-^(a

+

(4-i).(|a_(|

+ 1 )i/2

( i - l . — »4), (5-25)

63

where a = |ar|. M. Kovalylov showed in his paper 10 that \x\\f(x)\2w(s,

\x\)2 < M Y, IM». I • \WAf\\l

{5.26}

|.4|<2

for / € C£°(R2). Then we get the estimate (5.22) from (5.23), (5.24), (5.25) and (5.26). First, we prove (5.24). To prove this, we introduce some notations. Set

f

,

.... y 1 h{ Cr + s + l ) ( | r - c , s | + l)

+

1 (r + s + l j ' + ^ r + l ) 1 - ^ '

(r + s + l V + ' d r - s l + l ) 1 - ' Then by the assumption (5.21) on tu(.s,r), 1

w(s,r)

<M$(s,r).

(5.27)

Moreover, set r)(s,r)

= r)i(s,r) +r? 2 (s,r), £

VL

(s,r) - J^ ] (r + s +

i)z(s,r)

=

1

1

l)*(|r - c,.s| + 1)

A 2

{T + s + l) + T(r + l ) 1 ^

= , (r + s + D ^ + ' d r - s l + l ) 1 " '

where 0 < A < min{ 7 ,1/2 - 7}, A = 1/2 - 7 - A.

(5.28)

Since r + s > \a-t\ for (s,r) G £>', we have M 6(s

'r)

"

(ja-f|

tu(a,r)

< ~

„ . „ M,„nMMs,r) ( | a - t | + l) 1 / 2 + T + A

+

l)i/3+^(fi'r)

(, = 1 2 )

'

'

(5 29)

'

(5.30)

for (*,r) £ £>'. But in the estimate of /,' and J|, &(*,r) is treated in another way. (i) Estimate of l[

64 By Proposition 5. 3.1-(i),

2+

[T

nr X(t -s- r\){r + r-x

a) drds.

(5.31)

Therefore it follows from (5.31) and (5.20) that

ds ds

r < 2L\ /

[

^+6

ar X(t- s - a) dr [T ri)(r + r2) 1 z - a '/ y 0 w{a,ri)JTl ar a) dr > -X(t-s J0 w(s,r2) fri—6 ]r2(r - n){r + r2)' (5.32) Let us consider the integrals of log[2 + ar/(r - r, )(r + r22 ) •(t ~ s - a)]. FoF 0 < * < t-a and ra - * < r < r a , it folfows from (5.2) ana (5.3) that log 2 +

r - ri > r 2 - 6 - n — 2a - d > a. Then we have log 2 +

ar (r - ri){r + r2)



(5.33)

r — ri

For 0 < s < i - a and n < r < n + 8, we have fi+'S

log 2

/

ar dr (r - n ) ( r + r2)

log 2 +

< /

r — fj

dr

- <S [{log(35 + n ) - log(5} + ^ log(l + M / r , ) ] < <Slog(3/2 + ( - a) +■ ( S ' ^ e " 1 + 6 <M51/2log[2+|a-t|].

(5.34)

Therefore it follows from (5.32), (5.33) and (5.34) that Mb1/2,

„

,

.„ f f*

ds

r log[2 4-1* - t\] { f - *

+ /" - *

I.

(5.35)

We next show ds M '„ w{a,r,) ~ < | a - * | + l)i/a+T+A

I

(i=l,2).

(5.36)

65

We use (5.30) for l/w(s,r2)

and obtain

ds

f

/'

M

Moreover, 1

I V(s,r2)ds < M f [y1 +

1

< Mrl± + <

1 ^ } ds

- +

i i

r i

- -\

l , - > ds

M.

(5.38)

Therefore, from (5.37) and (5.38) we have (5.36) for i = 2. The treatment for t = 1 is slightly different. We remark that | n -s| = \a - tj ffo rr - o)+ < s < t by the definition (5.2), where x+ = max{0,x}. Then we see from (5.28) that &(3,ri) <

(5.39)

~ (jfl - t\ + l)V2+^+A( r i + 3 + l)l+mta{..X>

for
f(

-

ds

1

/(t-a)+ ( r i + a + l)1*™1"*'.*} J M (\a-t\ + iy/2+-<+*'

Combining (5.35) and (5.36), we have

l

'

66

< < "

1 ( a + l ) 1 / 2 ( | a - ( | + l)i/2+7 * (a + l)'/ 2 ->(a +■ t + l)-r(\a - t\ + l)1/2

(5.40)

Here we use the fact that a + t +1 < 4 for ( a + l ) ( | o - * | + i;

a,t> 0.

(5.41)

(ii) Estimate of V% By Proposition 5. 3 .I.(i), 1

~ a1/2 jJs(white)

w(s,r)

°S

2+

ar X{t-s(r -rt){r + r2)

a) da. (5.42)

Let 0 < s < t - a and (s,r) E white. Then or < -rJL-- n < (r - r L )(r +L r-T 2) So we have

or

x(t

-

TJ

4^ 1/2

8

< 2(* " « + 1/2)-

- a ) < 2(2 + |o - i|)

(5.43)

for (5,r) € wftlte. Hence from (5.42) and (5.43) we get M , r. , „ // -log[2 + | a - « | ] / /

ll/2

tfo its) w ( s , r - ) '

(5.44)

We have already computed the integral of l/w(s, r) in the estimate of l[, except the one on {0} x (|a - t\ + S,a + t - S). Applying (5.30) for s = 0, we have a+t-S

rr a+t-°

dr

M__

L/2+7+A' J\a~t\+6 w(0,r) - ( | a - t | + l)W

(5.45)

Therefore it follows from (5.44), (5.36) and (5.45) that ji

2

M < log[2 + [a - (fl. " a 1 / a ( | a - t | + l)'/a+i+*

(5.46)

Since a > 5 = 1/2 when the domain wUte is not empty, we have (5 24) for i = 2 by the way from which (5.40) was derived. (in) Estimate of J's

67

By Proposition 5. 3.I.(i), 7

3<^//

^-Tlog[2 +7

£

x{t-s-a)

drds. (5.47)

Further, by (5.30) and (5.43), ■-

—

al/2(|a-t|

M : ,7757^71°S[2+I""'I]// ^drds. + l)l/2+T+A T ■/ J white

(5.48)

Since r > 5 = 1/2 in the domain white, we have

2tLI> <

M

l

i ^

1)I+X/2(|T. _ CiS\ + i)i+x/a

—

(r + l) 2 + Ar +(r

^ 1 (s-

TT^TT: ' ....!(5.49) + l ) i + V 2 ( | r - * | + l)»+V3 J for far) € «?i»te. Concerning the right-hand side of (5.49), the integral of the first and the third term are shown to be bounded by a constant M in the same way as (5.38). As for the second term, we see that ft JLhite

drds (r + 1}2+A

"

f Jo

<

_J_ f

<

"

n dr Jri (r + 1)2+* ^

l + AJo { n + l }

r 1+

<

M

.

(5.50)

* "

Therefore it follows from (5.49) and (5.50) that T}{S,T

drds < M.

(5.51)

J Jtii white

Hence from (5.48) and (5.51) we have (5.24) for i = 3. (iv) Estimate of l\ By Proposition5. 3.I.(ii), / ' < M ff 4 -a^JLhitew(s,r)(r

1

-drds. + s + a-t)

(5.52)

Applying (5.30) to the right-hand side of (5.52), we obtain I> < -

M w

,

, / /

-J&^drds.

(5.53)

68

Since r + s + a- t > S = 1/2 in the domain white, we have

r+s+a~t

<

^

(v

n

r + s + a-t

+l | ^ (

r + s

.

1 (r + s + l)V2( r

^

+ i)V«(| r - Cis\ + 1)' + V2

1 l)i+V2

+

+

1

(r + s + i J V ^ I r - s l + i J i + V z j

for(s,r) G tohiie. Hence we find by the change of variables (a, 0) = that

{s+r,s-r)

?

-
(5.54)

y7nrj /«*«t< r

(v) Estimate of JJ By Proposition 5. 3.I.(iii), M

ff

1

a+ r

We notice that for (s,r) G white, r2 + r > a, r2 + r > r; T + n > a, r + n > r for r2 - r > a, r 2 - r > r for

r > (r2 - fj )/2; r < {r2 - ^ )/2.

Hence we have a+ r

{(r 2

f

1

_ r ? ) ( r 2 _ r 2 ) } i / 2 < 2 { {r2 _ r ? ) i / 2 +

1 {(r

_ ri)(ra _ r)}1/2 }

1

(5-56)

for (s,r) G white. Therefore it follows from (5.55), (5.27) and (5.56) that

* " ^ ff^^W^fP**

{(r - r,)(Ja - r)}i/» } **■

(5.57)

We show in the following that

69 We use (5.29) for £i(s,r) and obtain

Since r±{s

+ a-t)

t?i(s,r) _ r aji/a

{r 2

>S = 1/2 for (s,r) e white, we have A/ _ t + i)i/3 ( r . _

<

-

(r + s + a

s

_

fl + ( + 1} i/2

r

, , M s < r>

M

< ~

r+s+a-t+1 l

{£

frj (f + 8 + l)V*(|r - c iS | + 1)»+V2

_)

J

+

M r — s — a + (+1 (

^

1

1

^ ( | r - 5 | + l)V 2 (r + l ) I + V 2 J for {s,r) £ white. Hence by the change of variables (a,#j = (s + r,« - r) we have // - ^ l f * ' r ^ , . .(frda < M. (5.60) Therefore it follows from (5.59) and (5.60) that

II

^S,I\ „drds < 12

JJwhitt {^ - r\Y

,

w

^ ^-

~ (|a " *l + l) 1 ' , 2 + -' + x

On the other hand, we see that, for (s,r) e white,

ft(»,r) (r2_r2)l/2

A/ < (r + s + a - f + l)'/ 2 (r -s-a

M

+ t + l)1'2

(5.61)

70

•[{1 - X(\r -s\-\a-

t\{2)}U*,r)

+ K(\r - s\ - |a-t|/2)&(*,r}]

tf (|o- - t\ + I)»/*+7+A (r + 5 + a - t + l ) 1 ' 2 1

(

"+ s +

to

1) l/2+A + ( ( | r

- s j + 1)1"'

1 i {r + s + l ) 1 + m i r ( r — s —2+t+iy i 1 M (|o -t\ + l)i/a+->+* (r + s + a - t + l ) 1 ' 2 +

(

(r

2

+

1)1/2+V2(|r_sj

S +

'{ r _ s -

a

+

l ) l + A/2

1 + t + l)i/2(|r - «| + l)i/a+mio{..*>/3

(r + S + l)l/2+min{(.A}/2 J Here, we have used that ( - a > 0 on the support of 1 - \{\r - s\ - \a - (j/2). Therefore it follows that

fr<^>

// J

dTds ;drds

<<

■

^ —j _

r.

(5.62)

Jwhite

Combining (5.61) and (5.62), we obtain (5.58). To estimate the second term in the right hand side of (5.57), we note that 1 {(r-r,)(r2-r)}'/» *

{(a +

+-

, . , .

r

+ 1)(' +

,

+

{{a + t - s - r+ l)(r~s-a

fl-f

+ 1 ) } ,/2

+t+

(5.63)

l)}'/2'

Using (5.29) for the first term of (5.63) and the method above for the second term of (5.63), we get

71

Therefore from (5.57), (5.58) and (5.64) we obtain (5.24) for i = 5. Conse­ quently, we have proved (5.24). Next, we prove (5.25). £(s,r) and jj(s,r) are used again, but we do not consider £i{s,r) and f 2 (s,r) separately. Since t > a when D" is not empty, ri=t—a-s. (vi) Estimate of I[' By Proposition 5.3.II.(i),

1

"

M

JLck

w(*,r){(r + r i ) f o - r ) } V a

lo

2+

S

or drds. ()"! -r)(r + r2).

(5.65) In the domain D", we use the following facts: 1 M {{r + n ) ( r 2 - r ) } i / 2 - {(* - a )(a + i ) } 1 ' 2

for

r + s<{t-

a)/2, (5.66)

- r ^ - r < T; , «1Mi^iHf«.f) w(s,r) ~ ( | a - t | + l)V2+T+^' v ' r 2 — r > 2o + <S for r < r\ — 5, or (ri - r ) ( r + 7-2)

^

r + *>(t-a)/2,

(5.67) (5.68)

< M { ( t - a ) x ( 2 r - r ! ) + l}

for

r < n - 5.

(5.69)

ri-r It follows from (5.66)-(5.69) and (5.3) that

n < {(t-a)(* + t)V/* i i «- )/a «(«,*)** M "(2a + J ) V * ( | o - t | + l)Va+i+A 7>(s,r)

// / /

(r

+

1/2 n )l/2

drds

black

iff ( | a - t | + l) 1 /2+7+* ■j

,ek

^ ' ^ { ( r + r O t r j - r ) } ' /2

•log 2 + - — ': (ri - r ) ( r

r drds. +r2)

(5.70)

72 Moreover, ff jj

rV2

a

drds < M f w(s,r) Jn

tuck

f r 1 * * < M&&, Jo

£(s,Q)ds

(5.71)

rl/2

s

TTrt r f r r f s

V( > )-,

/ /

-

r

JD M

7

7o

(1 + r - a - *)Va \ ^

0

+

(l+n)1'2

(s

+

1 -= + (8 1 )A+2 7

+

( s + 1)*{ C ] S + 1)

1 1 , r > ds l)l+A J

(5.72)

<M

because A + 27 > 1/2 by (5.28). It remains to estimate the third term of the right-hand side of (5.70). We show t h a t Mi

)(r2-r)}V2 M

^(^fl)^

IOg[2 +

log 2 +

(ri — r)(r + ra)

A* (5.73)

^^

from which we obtain -1/2 ^'r,{(r + r 1 ) , ( r 4 - r ) } . / " t o 8 [ 2

/(M,

or drds {ri - r){r + r 2 )

ft —a

< M I Jo p

J](s,ri)ds ri/2

r (ri - r ) ( 7 - - t - r 2 )

M

^^Tij^la6{2+*"4

cfr

(5.74)

To prove (5.73), we consider the following two cases separately: (a) 1 < a and (b) 0 < a < 1.

73

(a) 1 < o Since r 2 - r > r2 - n > a + 1, we have

1

p

-(«T7jW

>

ri — r

Hence by the way from which we derive (5.34), we have (5.73) for a > 1 (b) 0 < o < 1 Since log[2 + ar/(ri - r)(r + r2)] < 1 + a^/(ri - r)1** and r2 - r > 2a, we have log 2 +

(T-J

ar dr - r ) ( r + r2)

< 4<51/2 < 2
Thus we obtain (5.73) for 0 < a < 1. Therefore it follows from (5.70)-(5.72) and (5.74) that

n<

{(t-a){a

MS + t)}1/2

M ( 2 o + l / 2 ) 1 ^ ( | o - * | + l)1/44-H-A

(0 + l ) , / 2 ( | o - ( | + l) 1 / a +'H-*

M 1

(t-a+l) /2{(

+ a+l)V2

tQg[2 + i - a]

A/ ( a + l ) i / 2 { | a - t | + l)i/2+-r

Af (a + l)»/ -^(a + * + l)">(|a - t| + l ) 1 ' 2 '

(5.75)

2

(vii) O r n a t e o/ I'i By Proposition 5.3.II.(i), r

l/2

2+

ar da. (n - r ) ( r + r 2 )

Moreover, by (5.66)-(5.69), we have J" 2

< ~

M {(

ft

r1/2

74

M

^T1)1/,(|a_t| + ■/ / JJ , +

1 ) ./, + ^

n(s,r) r > ( fl(r.J) t-a)/2

1

°g[ 2 + f - a l -

T-do. ' ( r + r , )i/2

(5.76)

Here we notice that £* - a)/2 > 6 = 1/2 when the domain red is not empty. We further see that //

—

//

8<»=
(5.77)

rda < M,

w(s.r) T

/"/" n(«,r) JJ s + i -8(rcd| '(r >((~n>/2

2

'

(5.78)

, da < M.

ri)l/2

+

Hence from (5.76), (5.77) and (5.78) we obtain (5.25) for i = 2. (viii) Estimate of Ig By Proposition5. 3.II.(i), I

"-MjLlirV*w(s,r){(r

r1)(r2-r)y/i[0g

+

2+

(ri - r)(r + r2)

drds.

Further, by (5.66)-(5.69), we have M

j» < H

~

{{t-a+l){a

1

ff +t+ M

,,^8,r\,.vdrd8.

II jJ

(5.79)

r'/^ r + r,}1/2

rt(I-a)/2

»

'

Since r > 6 = 1/2 in the domain red, we have 1 r^w^r)

"

"

M£(s,r) (r + 1) 1 / 2 M +

l ^ t Cr + l) 5 / 4 (|r - CJ<| + l) 5 /* ^

!

(r + s + l ) ' + i ( r + l ) i + i / H

+

drds

l)}WjJm+r~^}ftriftm{*,r)

1

s

1

(r + l ) / < ( | r - * | + 1 ) 5 / 4 /

75

for (s,r) G red. Therefore it follows that //

-rrri

r * d * < M.

(5.80)

Moreover, 17(3, r )

ri/a^ + ^ji/a 7j(s,r) "

(r

l)l/3( r

+

+

r

i

(t

+

1)1/3

^

< Jtf ■

+A/2

1 (T-S

+

+ t - a + l ) » / a ( | r - S| + l)l/2+min{T,-A,A/2} ( f . + !)l+A/2

^

(r

^1

for (s,r) e red. Therefore it follows that 7](s,r)

drds < M.

/ /

(5.81)

Hence from (5.79), (5.80) and (5.81) we obtain (5.25) for i = 3. (ix) Estimate of l'l Bv ProDosition 5. 3 .Il.fii).

Further, by (5.66), (5.67) and (5.68) we have

r4

< -

M

r

.//

2!l drds

{ ( ( - o + l)(o + t + l ) } i / a y y j + r S « - _ B j / i i « ( * , r ) ( r 1 - r ) M +

( a + l ) 1 / 2 ( | a - ( | + l)i/2+7+A

■ If

JJ

n(s,r)

"d

—

j-drds.

(ri - r)(r + n)1'2

(5.82)

76

Both rl/JM«,r)(ri - r) and r,( S ,r)r 1 / 2 /(n - r)(r + n ) 1 ' 2 are bounded by T)(s,r)/(n - r). And since n - r > * = 1/2 in the domain red, we have

II r)(s,r)-drds JJred

ri-r

1

<M[[ ~

(f-

^

//«* n - r + 1 | £ j ( r + s + l)V2(| r 1 ^

1

=

i

>

■

-

■

- ,

-.,

— s\i Ci

+ l)1

■ = =— > drrfs ( r + s + l)V2(| r - s| + l)i+V2 J < AT.

+V 2

H

(5.83)

Therefore from (5.82) and (5.83) we have (5.25) for i = 4. Consequently we have proved the estimate. I 6

Energy Estimates.

In this section we prove Proposition 6. 1 Let u = ( u \ . . . ,u™) e C°°([0,T) x R 2 ; R"1) be a solution of the following system of wave equations with ti(0, ■) € Cg°(R 2 ; R m ) . m

#«*-<$*«' = £

2

XI C^{du)dad0ui + Ei(du)

(6.1)

Here, CfftEi(i,j = 1,2,.. , m ; a,/3 = 0,1,2) are C°°-functions in { \8u\ < 1 }, uAicA satutfe tAe etmdffiow (4-3)-(4.8). Moreover we assume that C?/(&0|<^min{l,c2,c2}

for

\du\ < Si

(6.2)

and that there exists a positive number T{ such that [du\0(T1)
and

|0u| o (Ti) <

Then,we have the following energy estimates for 0 < t < Tj:

II «**(*) H3v <

MN{\\du(o)\\%

fc.

(6.3)

77

(6.4)

l|9«(0lfir

<

MN \\ du{0) \\% (t + l)^USu]t(«t)a_

(6 5)

Proof. Since ft commutes 0? - c'f A, m

S J V I I ' - C ^ I I

1

2

J£ Pj4{C5d(fti)aoflflu-'}

^ j= l

+P- 4 £,(5U}.

(6.6)

a,p-0

We set aa -tf

=

f 1 - Cf { _ C j C j i ^ _ C,f m

(a = 0 = Q,j = i) (otherwise)

^7)

2

+r> A £ i (5u).

(6.8)

Then from (6.6), (6.7) and (6.8) we have m

2

H X! af/^Rc^'V = w'A

(6.9)

j = l a,0=0

Multiplying both sides of (6.9) by dtVAuJ and using (4.3), we get m

1

t.j = 1 a,0=0

- ft {aJ'/Ou)0oZ>V -3^11')] m

1=1

+£

J ] {dtCff(du)daVAul-d3VAuj

-23 Q C;f (9u)9(£>V - c ^ P V }

(6.10)

78 Integrating (6.10) over [0,t] x R 2 , we have

$

in +

JJR?

m

2

H

E

{ftc5tf(«u)fl|[ri>*iil*fls»©V

i,j = l a , / J = 0

- 2daCf{du)dtVAu'

■ dsVAu>} dx,

(6.11)

where lla^^Oll^

=

Y,\
/ /

- £ c J

dtVAui{t,x)

s§(^)0*X*M-&1>M(t,s)W 1

(6.12)

J

Notice that by (6.2) and (6.7) we have ~\\dVAu{t)\\0<\\dVAu{t)\\E<

M\\dVAu{t)\\0

Therefore it follows that

B«^«(0|i:<w(||OT-u(0)||: + jV , +jJ. ai )

(6.13)

where m

■#' =

ft

£ldsJjRJd,I>Au'WA\dx<

J ]

A = E l fds//D kr/Hisp-vna^vid*.

We first prove the estimate (6.4). Since \du(s)\D < 1, then by the assump­ tion (4.4)-(4.8), we get

(=2

j,,. .j , = l lB,|,....|B,|<|e|

*=1

(6.14)

79 \A\+4

A

\v E,{du)\ <

(

m

MAY,

E

< = 3 Ji

E

^^ni^^l-

Ji = l Mi|,-...|.4,|<1>1|

*=]

(6.15) Here we set

Since det (a°°(du))™ =i I > l / 2 m from (6.2), we can solve the following simul­ taneous linear equation with respect to dfu>:

E ag(0u)a ( V = £<{&/) - E

rfidujdadeu3

E

(i = 1,2,.,.,m).

Moreover, by (4.4), (4.5), (6.2) and Cramer's formula, we see dt2u( = - {det(a™(9u))™ = 1 }

cf&u1 h higher order terms.

(6.16)

Therefore it follows from (6.8), (6.14), (6.15) and (6.16) that

4" < M . E E E 1=1

'=

3

E

>'j-J' = »|Ail

rfs

t

«**-*■

Mil
(6.17)

jfc

■/ // nip'* 9u iip'*A'i*BNext, we consider JA2). By (4.4) and (4.7), m

\dc?f(du)\ < M E ^A' M l52u'l ■ k.l = \

Therefore it follows from (6.16) that

jf < M\ E + fdsff

J2fdsJJJ^\8uk\\dhdu'\\dvAu,\\di:>AuJ\dx \d*\3\8VAu\2dx\.

(6.18)

80

Hence we find from (6.17) and (6.18) that

J P + J? 1 < MAT, E /=3

£

J O , . . . , J , = I |,4 0 [

***-* (6.19)

]4,|<|/t| + l

fafhur*^*By Holder's inequality,

/j^ni^*^fe*)i
|^'a^'( s ,o[| L J ((it R :2 )

i,2(R=)

L°°(R*)

2

L (R )

t=0 1-1

n

w

2

l»"'3«J'(».-)IL.lR.)

('-!)/' («,j-I)

Jt=0

z.~(R2)

(-1

*=o

(6.20) Without loss of generality we may suppose that jo = ji = j a does not hold for / = 3. Therefore, it follows from (3.1) that

n-^'^i-D

k-0

< M(s + l)- 1 - m i n ^ 1 / 3 ' 2 ■ , '. 2

L~(R )

In order to estimate (6.20), we need Gagliardo-Nirenberg inequality: Lemma 6. 1 Let f € C^(R2), \A\ = i < k. Then, i/k

\|B|<*

>

(6.21)

81

where X = K2 or {x | x £ B2, n < }x\ < n + 1}

(n = 0 , 1 , 2 , . . .).

F. John and S. Klainerman proved this lemma in their paper 6 when X = R 3 We modify the proof of them and obtain the above lemma. Since (6.22)

rjUujfon) < u!j(s,r) < MWj(s,n) for n < r < n + 1, we find from Lemma 6. 1 that

00

n=0 2

E

9

K ^,-)iu { „ < | I | < n + 1 } )

|B|
< M^[au(s)]o ( '" 1) ||9u(s)||, 2 Mt|

(6.23)

Therefore, from (6.3), (6.19)-(6.23) and (6.13), we get (6.4). Next, we prove (6.5). For the proof, we use the following two lemmas: Lemma 6. 2 Let f,g € C0°°(R2). Then, \\VA(fg)

- fVAg\\Q

< M([Ms\\lA\-i

+ \ff\a\\f\\\A\)-

Lemma 6. 3 Let f = f/j,.. . , / r ) € C 0 °°(R 2 2R r ) and let u) = w{/) be a C°°function that satisfies

N/)| < M|/|«

|B|<M|

See F. John and S. Klainerman's paper 6 for the proof of Lemma 6. 2 , and M. Kovalyov's paper 1 0 for Lemma 6. 3 .

82

By these lemmas, (6.25)

K I L * ( R = ) ^ MA\du(s)\0\du(s)U\\du(S)\\lAl Therefore from (6.18) and (6.25) we get JA1] + JA2) < MA f \du{S)\o\du{s)\A\du{s)\\\Mds. Jo Further, it follows from (6.13) and (6.26) that \\du(t)\\% < MN Uduiml

+j

\du(8)\o\du(3)\i\\du(8)\\%daj

(6.26)

.

Hence by Gronwall's lemma we find \\du(t)\\% < M\\du(0)\\%cxV (MN

f

|0u(*)| o |fti(s)|i
(6.27)

Since \du(s)\0\du(s)\i

< MN(s +

l)-'[du(s)]2l,

we obtain (6.5) from (6.27). 7

I

Proof of the Theorem.

Making use of the method by R. Agemi' and F. John,5 we find that a solution u(t,x) to (4.1) is unique and u{t,•)(t > 0) has compact support. The local existence theorem of a solution to (4.1) has proved by F. John 4 and T. Kato.7 Let u(t,x) be a C°°-solution to (4.1) in [0.T) x R 2 We write u as « = «o + u l !

(7.1)

where «o is the solution of the initial value problem

1

dfui - c'jAu}, = 0 (7.2)

4 ( 0 , ■) - e/*,ftuj(0, ■) =89'

(1 = 1,2,..., m),

and m is the solution of the initial value problem d2u\ -c2&u[

=Fi(du,d2u) (7-3)

u[(0,-) = dtu\{0,-) = 0

(i = l , 2 , . . . , m ) .

83

R. T. Glassey has proved in his paper 2 by the method of W. von Wahl 14 that | U o ( ' - * ) l < 77T1 ov

777^,

;

7777^-

(7.4) v

' " - {(N+cit + i)(|W-*i«l + l»>^

'

Here M depends on L'-norm of / \ &P and g*. We set 1 i{s,r)

"

^ 1 1 2-f, (r + s + l)(\r - Cjs] + 1) + {r + s + l)l+2i(r

+ l) 1 " 2 ^

1 (r + s + l ) / 3 ( | r - c , s | +1)2/3-

+-

4

Then u>i satisfies (5.20) and (5.21). By Proposition 5. 4 , we get

[ft»i(0W

<

M

N{Y, Yi sup ||uii(*,[-|)2^/l(ftt,9»u)(« t .)|Ix««« [-l|^|<JV+3° < S < t m

4- £ £ sup ||iu,(«,| ■ MT^Fiidut&vlfaUppp) ,=1 | J 4|<W+2° < S < (

\. J (7.5)

Since [9u(*)]o is continuous, we can take for 0 < £ <
m

\VAFt{du,d2u)\<MA E (=3

I

E

E

*W«-J. 1 1 1 ^ ^ * 1 ' (7.6)

J i , . - J " i = l |A h tit+i Ci = 1.2 ') |J4|+8

(=3 j i

m

J'I=1 Mnl<MI+=

t=I

I

k-\

(7.7)

84

Hence from (7.5), (7.6) and (7.7) we get JV+10

[dUl(t)\N

m

< MN^2

Y,

(=3

■ sup 0<s<(

j ,

H

j, = l

**-*

|Ahl<«+4 ifc = i . s i)

(7.8)

"iW-Dn^^'^-)

LHB?)

k=i

We notice that M provided h =j2=j3

= i does not hold. Thus we get (7.9)

By (7.9) and Holder's inequality,

hh-ji

Ms,\-\)Y[vA*du^(s,-) k=i

<M

\]wf~1)/t(s,\-\)i>A>>dtjn»,-) L = (R=)

-)

„

(7.10)

Hence by (7.8), (7.10) and (6.23) we have [dui(t)]N < MN sup

[du(s)}l\\du{s)\\Nl,

(7.11)

0<s<(

where Ni = (N + 10)(JV + 4). Therefore it follows from (7.1), (7.4) and (7.11) that [du{t)]N <MN\e+ L

sup [du(s)]l\\du(s)\\Ni\ 0<s
.

(7.12)

J

By Proposition 6. 1 , \\MB)\\%

<MN{l

+ [du]&8) ■

||9u(T)|| (A r ]+i)(/Vi+5 j < M N ( r +■ i)M*[5w]fW

(7.13) (7 H )

85 We fix the constant MN in (7.12), (7.13) and (7.14) so that >max{8,2/mm{l/3,27},(2/
MN We take £0 to be

0 < £o < min{l/4M^,,ei}. Moreover, we suppose that eo is small enough to define the following T0 for 0 < e < en:

io — sup {t | [du]N(t) <4eA/jv}. Suppose that 0 < t < T0. Then, [du{t)]0 < AeMN < ljM% < 1/2 \du{t)\0 < l / A ^ < 5 ] / 2 . Therefore T 0 < ft. Then for 0 < E < £0 and 0 < { < T0,

[Bu]l(t) < {teMNf < l/Mf,, - m i n { l / 3 , 2 7 } + M N [au]J(r) < - m i n { l / 3 , 2 7 } / 2 .

(7.15)

Here from (7.13), (7.14) and (7.15) we get

\\Ms)\\2Nl < MN{\+~-MNJ\T <

+ l)-l-™W3^/2dT}

2MN.

(7.16)

Therefore it follows from (7.12) and (7.16) that [du{t)]N

<

MNL+

sup [du(s)}0 ■ AeMN -

\

<

0<s<(

(2MN)^2\ J

MNe + - sup \du(s)]o 2 0o<(

<

MN e + -

sup [du(s)]iv,

6 00
which implies sup [du(t)]n < 2MNe. 0
Therefore T0 cannot be finite, and we complete the proof of the theorem.

86

References 1. R. Agemi, Blow-up of solutions to nonlinear wave equation in two space dimensions, Mamiscripta Math. 73 (1991), 153-162. 2. R. T. Glassey, Existence in the large for au = F(u) in two space dimen­ sions, Math. Z. 178 (1981), 233-261. 3. F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Cornm. Pure Appl. Math. 27 (1974), 377-405. 4. F. John, Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, Comm. Pure Appl. Math. 29 (1976), 649-682. 5. F. John, Nonlinear wave equations, Formation of smgurahties, Pitcher lectures in the Math. Sci., Lehigh Univ., American Math. Soc, 1990 6. F. John and S. Klainerman, Almost global existence to nonlin­ ear wave equations in the three space dimensions, Comm. Pure Appl. Math. 37 (1984), 443-455. 7. T. Kato, The Cauchtj problem for quasilmear symmetric hyperbolic sys­ tems, Arch. Rational Mech. Anal.58 (1975), 181-205. 8. S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), 43-101. 9. S. Klainerman, The null condition and global existence to nonlinear wave equations, Lectures in Appl. Math. 23 (1986), Amer. Math. Soc, 293326. 10. M. Kovalyov, Long-time behaviour of solutions of a system on non-linear wave equations, Comm. in PDE., 12 (1987), 471-501. 11. M. Kovalyov, Resonance-type behaviour in a system of nonlinear wave equations, J. Differential Equations 77 (1989), 73-83. 12. Li Ta-tsien, Kong De-xing and Zhou Yi, Global classical solutions for quasilinear non-strictly hyperbolic systems, preprint. 13. T. C. Sideris, The null condition and global existence of nonlinear elastic waves, Invent, math. 123, (1996), 323-342. 14. W. von Wahl, IS-decay rates for homogeneous wave equations, Math. Z. 120 (1971), 93-106.

87

SCALING LIMITS FOR LARGE SYSTEMS OF INTERACTING PARTICLES KOHEI UCHIYAMA Department

of Applied Physics, Tokyo Institute of

Technology, Meguro-ku, Tokyo J5S, Japan. E-mail:

tichiyama^neptune.ap.titech.acjp

A system of a large number of classical particles moving on the real line is stud­ ied. The particles interact with one another through repulsive pair-potential forces and are subject to resistance proportional to their velocities. Because of the latter it is only the number of particles that is conserved under the evolu­ tion of the system. It is proved that under suitable scaling of space and time the normalized counting measure of particle locations converges and its limit is governed by a non-linear evolution equation, which is local (diffusion equa­ tion) or non-local (integral equation) according as the tail of the potential is bounded by l/|i| or obeys the power law l / | i p with - 1 < 7 < 1. The scaling also depends on the tail of the potential.

0

Introduction

This paper concerns the problem of deriving a macroscopic equation from a microscopic description of a large system of interacting particles. The system studied in this paper consists of a large number of particles that move on the whole real line R according to a classical equation of motion. Particles inter­ act by exerting upon one another repulsive potential forces given by a common pair-potential function and are subject to a damping force, 'resistance', pro­ portional to the velocity of each particle. In this system it is only the number of particles that is conserved under the evolution of the system. We take a cer­ tain scaling limit for the normalized counting measure of particle locations in the system and prove, under mild regularity conditions for the pair potential, that its limit is governed by a non-linear evolution equation, which is local (dif­ fusion equation) if the tail of the potential either is integrable or behaves like 1/|arj, and non-local (integro-differential equation) if its first derivative obeys the power law l/x"'+1 a s i - f D o with - 1 < 7 < 1; the scaling also depends on the tail of the potential. In [U2] we investigated the problem in the case when the pair potential U(x),x € R, has an integrable tail (short range). In that case we take the par-

88

abolic scaling and the limit density p(9,t) solves a non-linear diffusion equation Pi = {P{p))ee, which is obtained as a result of local equilibrium structure in the microscopic description of the system; the function P accordingly depends on the the whole shape of U. In the long range case the local equilibrium is irrelevant for the macroscopic description of the density and the limit equation for it is determined only by the tail of U unlike the short range case as we shall see in this paper. The proof accordingly differs from that employed in the short range case: we do not need to prove ( and can not utilize) the local equilibrium. When the tail of U behaves like 1/1 x\i with 0 < 7 < 1, the mathematical structure of the problem is similar to that for typical mean-field model of McKean-Vlasov type (cf.[Sp]). The interaction kernel is virtually given by {x - y)!\x - y\2+i, which is singular rlong the eiagonal and dll the eifficulty comes from this singularity. The limit equation may read

provided that a limit density p is nice enough. We shall verify uniqueness of a weak solution to the Cauchy problem of this non-linear evolution equation. The uniqueness is important since in our approach the limit is characterized only as a weak solution to it. We include the case - 1 < 7 < 0, in which estimation of the kinetic energy becomes somewhat non-trivial, although the singularity along the diagonal disappears. For the critical case U{x) ~ l/|x| (i.e.,7 = 1) we encounter a new type of problem concerning a singular limit that is to lead to the non-linear diffusion Pt = (p2)eeFor the proof we shall use the Young measure technique, which is applied by Varadhan [V] to derive a non-linear diffusion equation for a stochastic model of short range interaction. (Cf. also [OV],[SU], [U2], and [FIS], where it is applied in similar contexts.) A stochastic version to the present system has been studied by OllaVaradhan [OV] in which independent white noises are added as random forces

89

acting on particles and the pair potential is supposed smooth and to have a compact support. If the potential has a long tail, we choose a temporal scale under which we look at the system in shorter times compared with that un­ der parabolic scaling, so that influence of the noises diminishes as the particle number becomes large , making no difference to the macroscopic description of the density. Our dynamics is described by a system of the second order equations. It turns out that the collective behaviour of its solution is well approximated by that for an associated system of the first order equations. Miirmann [M] inves­ tigates the same scaling limit as ours for the latter system but by restricting the interaction to nearest neighbour particles. An infinite particle system of similar nearest-neighbour interaction is studied in [F] [PS]. 1

The Model and The Main Results

Consider a system of N Newtonian particles of unit mass moving on the onedimensional real line R such that each particle is subject to the resistance equal to its momentum and interacts with the other particles through a pair potential. The equation of motion for the system is written as follows: (1.1a)

-T-fc(0=R(0 at

(1.1b)

jpt(t)

= -Pi{t)

- Y,

u

'(l'(t)

~ Qjit))

where t = 1,...,JV, 9i(t) and Pi(t) denote, respectively, the location and the momentum of the i-th particle at time t > 0 and V the derivative of a function U. The pair potential function U(x), defined for x e R - {0}, is supposed to be continuously differentiable and satisfy (1.2)

U(x) = U(-x);

(1.3)

liminf-—

s

'

HO

[/(0+) = oo;

inf Mx) > 0,

lp[d) 0 < i « 5

where tf(i) := -xU'{x),

x^O.

90

The assumption (1.3) implies that $(x) either diverges to infinity or is bounded off both zero and infinity as x -*■ 0: in particular the pair interaction works repulsively in a neighbourhood of the origin. We shall assume additional conditions on U which imply that it is repulsive for large values of \x\. The initial configuration % = qt(0),i = 1,2,..., N, is always supposed to be such that qi/qjifi^ j for every t- The assumption 1/(0+) = oo guarantees that this condition for the initial configuration is preserved for all later times. The conditions (1.2) and (1.3) are assumed throughout the paper. In our previous paper [U2] we studied the case when the potential has short range: (1.4)

f

\U(x)\dx
In the present paper we mainly concern the long range case:

(1.5)

tf(x)~-L

as

|Z[-K»

\x\ where 7 is a constant such that - 1 < 7 < 1 and ~ means that the ratio of two sides approaches unity. In the short range case U will be assumed to be bounded below. We are interested in a macroscopic picture for the particle configuration (fc(*))tei1 which i i s 0 be viewed over large spatial and temporal lcales. Let us introduce the macroscopic position variables (1.6)

Xi{t) =

jjqi(X^t)

and the normalized counting measure 1

N

1=1

for an open subset ACRit

takes the value

a? (A) = -J= the number of* such that x&) £ A.

91

The temporal scale parameter XN is taken to be N2 in the short range case (1.4). In the long range case (1.5) it is chosen as follows (1.7)

A* - - ^ logW

if 7 = 1

= NX+~*

if

7 < l.

We need the following bounds for the initial phase (p^qi)^

£ t / - ( ? , ) = o(yV2)

(1.8a)

1

(1.8b)

: as N -> oo

N

1

LYJpf + - Y,

U

^' -«>) =o(N3),

where U~ stands for the negative part of U and the second sum extends to all ordered pairs (ij) with i ^j,1 a{J) as N -» oo for every bounded continuous function J on R, where a(J) stands for the integral of J by a. In the following theorems it is supposed that the sequence of initial phases fe,*&j = ( f t ( 0 ) , * ( ( ) ) ) £ i , * = 1,2,..., satisfies (1.8) and that a$ —> p0

as

N -» oo

where pe is a probability measure on R. Clearly the point mass Jo, eg., is admitted as \ia under the restriction (1.8). T h e o r e m 1. Suppose that U has the short range (1.4) and is convex on (0, oo), and let X^ — N2 in (1.6). Then o f converges uniformly for t
irP(0.O = ~ w W M ) . ot off*

0 6R.OO

92 with the initial condition that (1.10)

p{8,t)d6

converges to (i0 as l | O ,

where CO

foru 0 0 and P(0) = 0. Theorem 2. Suppose tfiat 1/ fcas the long range (1.5) and that either (1.11)

^(z)>0

/or x<ER

or (1.12)

#c)|*|—*©Q

a* se-*G.

7Tien a^ converges, as N -* oo, to a prob 0. / / 7 = 1, fit is absolutely continuous relative to the. Lebesgue measure dO and the density p(8, t) is a unique solution of the non-linear diffusion equation

(1.13)

JWt) = 4^2(M,

9eR,f>o

ot av that satisfies the initial condition (1.10). If -1 < 7 < 1, then fit is a unique solution, starting at the initial measure fa = fin.to the non-linear integro-differential equatton

(1.14)

J {e J m Jl {d ] de T^J) = \II ' \8f—^8'y '\ a'i at 2 JJn? ' l~ 6—8

t>0,

which is to be valid for every J e C0°°(R) (the set of smooth functions that vanishes outside a finite interval) . Though solutions might not be different]able (we do not know even whether solutions are absolutely continuous), the equation (1.14) is formally a weak form of the integro-differential equation (1.15)

~p{8,t) =}~-§g (/>(*.0^ f°

W^8%d&') ■

93 In the short range case certain analytic properties of empirical measures af (such as Z,p-bounds for its regularization) that are essential for the deriva­ tion of the limit equation are obtained by exploiting the non-linear nature of the problem. That this works is largely due to locality of the interaction. In long range case we shall encounter a different situation. Since the interaction is non-local, the non-linearity also is of non-local nature so that any direct ana­ logue of the previous approach is inadequate. Owing to the one dimensionality of our model, however, the difficulty arising thereupon will be coped with by introducing the transformation that maps a function of one variable p() into that of two variables p(; •) defined by

M<9") := e^Te I

P^dr<

which will make it possible to adapt the method that was devised in the short range case. We shall give a self-contained proof of Theorem 2 under (1.11) in Sections 2 through 7. In Section 3, after preparing an energy estimate in Section 2, we shall expose how the limit, equations arise from the underlying dynamics in formal level. We shall prepare a few lemmas concerning the transform fi{ ■) i-). p(-, •) in Section 4 and various moment estimates related to the empirical distribution a f in Section 5. The proof of Theorem 2 will be given in Sections 6 (in case 7 = 1) and 7 (in case 7 < I}The proof of Theorem 1 is substantially different from that of Theorem 2: the local equilibrium plays a key role for the proof of Theorem 1, while it is irrelevant for Theorem 2. In Section 8 we shall outline the proof of Theorem 1 with the emphasis laid upon the explanation of how the local equilibrium is formulated and utilized in the proof. We shall discuss the case when the potential V has a hard core in Section 9. The same macroscopic equations as above will be derived at least if they are local or the solutions are nice enough, while for the legitimate derivation we need to change the variable as is carried out by Rost [R]. The results on the associated first order equation and stochastic model will be briefly mentioned in Sections 10 and 11, respectively.

94 2

Bounds of Kinetic Energy

Put (Recall xi(t) =

N--qx(XNt).)

Then the equation of motion (1.1) becomes (2.1a) (2.1b)

^-Xi{t) = A/vAT't^i) at jvi(t) = -XNvt(t) - V £

f/'(N(x,(() - Xj(t))).

For the total energy

we calculate its derivative to obtain N

d

(2.2)

eN

_

T M = * « £ ■?W ^ °-

Proposition. The condition (1.8) implies that N

1

(2.3)

lim

sup —

Yvf(t)=0

and

(2.4) where eN = \NIN2, (2.5)

lim £N /

4V^(()di-0,

namely e* - 1 1 logiV = N~1+^

if

J™\U\dx
if -y = i «/ 7 < 1-

95

Proof. By (2.2) we obtain

(2.6) t„ j

i£i|(t)
53 U-{Ntpt{T)-Xi(T))),

where (7" denotes the negative part of V. Thus (2.4) clearly follows from (1.8) if U~ is bounded. Suppose that V has the long range (1.5) with - 1 < 7 < 0 so that U(x) ~ — b~lxb

as

X-KXI

if

6 :— - 7 > 0.

Then T V - ^ ^ j ET-fW&ei(r) - Xj(T))) is dominated by a constant multiple of

JJ-2 £ \Nx,(T)\" < ,4* + ^ £ ( / |Ui(()|d(J where A* - A ^ E k . f and nee equation (2.1) is appiied for obtaining the inequality. By Holder's inequality the right-hand side is bounded by

|(|gfN('>l*)>sf(sgiTW')i»*)"! Put v := ife2 - \b + 1 > 0. Then v > 0 for r < 1 and we may conccude a/2

This combined with (2.6) yields (2.4) under the condition (1.8). By the in­ equality £N{t) < £N(0) and the last inequality given above, (2.3) also follows from (1.8) as asserted. In the case 7 = 0 we have U(z) ~ - logi and the proof may proceed in a similar way.

3

Derivation of Limit Equations in Formal Level

Let J be a smooth function on the whole real line R. By (2.1) we obtain N

d

(3.1)

0 1

di * ^

= fN

£ - J'M*)M(*)

96 and

(3.2)

^J]/{^

as well. Here the time parameter is suppressed and we put CN = XN/N2 in (2.5)). Since U' is odd, the last term in (3.2) may be rewritten as

(as

Taking this into account we rewrite (3.2) as

ttf^2

J'(xi)vi

^E^^-^E'W, >.j(y)

Substitute this expression into (3.1) and then integrate both sides of the re­ sulting equality on a time interval [0,(] and you will see (3.3)

a?(J)-ag{J)

s=0

+

; ^

£

J0 2 N ^ iJM

^ ) - ^ ) J

xt-Xj

))ds.

In view of (2.4) the first term on the right-hand side converges to zero as N ->■ ooo Since (3.4)

3

^EMOI^[*~ E^>

1/2

97

the second term also vanishes in view of (2.3). Thus (3.3) is reduced to

(3.5)

a?(J)-ag{J)

=f | j ;

J,{Xi

}-_J'{Xi)^{*i-*i))d*+<m>

where o(l) is locally uniform in t. In the short range case (1.4) the ratio (J'fo) - J*&)}/(A - *i) is shown to be replaced by J"(ij) up to a negligible error term. Since ew = 1, (3.5) therefore turns into (3.6)

a?{J)-c$(J)

For the identification of the inner sum on the right-hand side we turn back to the microscopic variables qi(N2s) = Nxt(s). Equilibrium states for the infinite system corresponding to (1.1) are steady states in which each particle stands still in complete balance of forces exerted on it by the other parti­ cles; any steady state must be a configuration of equal spacing, of which the common span is necessarily equal to the reciprocal of its density. By the damping effect due to the resistance working in (1.1) velocities would get very small after so long time as of order iV2, so that the equilibrium state must be locally built up in the limit under our scaling. If there exists a limiting density p(8,s), this would allow one to replace T.j^irp{N{x,{s) - xj(s))) by 2 Y,k>! iKfc/pfoto. «s)) which results in

and by passing to the limit in (3.6) and noticing P(p) = p£jt>i tf(Jt/p)i ac­ cordingly yields a weak form of (1.9). When U has a long tail in the sense of (1.5), i.e., ip{x) ~ \x\~7 (7 < 1), it will be shown that for positive constants T and I

(37)

/

J*F E ^i{^(xi{t)-Xj(t)))dt-*0

as JV-»co,

98 where ^(x) = i>(x)x(\x\ < l), with x(\x\ < I) denoting the indicator function of the predicate | - | < 1 1i.e., x(P(x)) = 1 or 0 according as she statement P(x) is true or false). Suppose that 7 < 1. Then eN = W~1 and (3.7) will allow us to rewrite (3.5) as

(3.8) /•' 1 ^rJ'(xi)-J'(xi)

J0 2N* jf

ds

xi-xj

Vh

(\Xi -Xj\ + l/Np

where o(l) vanishes as TV -> 00. If it is not for the singularity of the kernel (in the case 0 < 7 < 1), it will be easy to show that the limit of a^ solves the integro-differential equation (1.14). When 7 = 1 the problem is subtler. Letting ft be a smooth, non-negative function on R such that h(x)=0

for |*| >1

and

/h(x)dx

= 1,

we introduce the empirical density p?{8) := hN * < ( 0 ) - £ h(N(8 - Xi)), 1

where hN * eg denotes the convolution of o f and the mollifier kN{x) := Nk(Nx). This time, e^ = 1/logiV, and we may rewrite (3.5) as follows

(3.9)

a?{J)-a(f{J) [* ds ftJ'm-J'ifi) JQ logN jj e>-e

p?(8)p?(8>)d0d8> 8>-9 + 1/JV

oy

'"

If $(0) had converged to pt(9) sufficiently strongly (convergence in L 2 -norm is certainly sufficient), then this integral would converge to ft

/»oo

/ ds / J0

J-00

J"(8)pl{8)d8,

99 yielding the weak form of the desired equation (1.13). However, we shall not prove the strong convergence of ${8) itself, which seems difficult to verify. Instead of it we shall consider the 'average'

(3.10)

${d,e')~-

— f p?{r)dr

and prove that p?(8,8 + N*-1) converges to p{8,t) strongly enough as a se­ quence of functions of three variables (0, t,A) eRx [0,T] x [0,1]. The proof then will be accomplished by further rewriting the integral on the right-hand side of (3.9) in terms of p?{0,8 + A^" 1 ). 4

Auxiliary Lemmas

Let p(dx) be a finite continuous measure on R and a and 7 constants such that a > 0 and - 1 < 7 < 1. Lemma 1. Letp>0.

Suppose that p + 7 > 0. Then

x
_ ( p + 7)(p + l + 7 ) [f dxdy f (• (p + 2)(p+l) JJ {y-x + a)PW\Jt

W

\p+2 ')

x
where Jfx
Proof. Suppose that a > 0 and that fi is absolutely continuous and has a continuous density / , say. With the help of the formula

MM (/><*>)' = - w ' ^ , i a > {L""idr)Y~ the integration by parts turns the left-hand side of the equality of the lemma into __.

P+7

ff

(j> + 2)(p + l)JJ y>x

dxdy (y-x

®-(r

+ a)r+^dy\Jx

f
> )

*

100

Integrating by parts once more we obtain the equality of the lemma. If ft is a continuous measure, it is a weak limit of smooth measures fi( for which ttt{x,)) uniformly converges to £(*>#)> provided a > 0. The case a = 0 is obtained by letting a I0 since both sides of the equality are non-negative and non-decreasing relative to a (the two sides may be infinite simultaneously) . A similar computation shows the following lemma. Lemma 2. (i) Let v be another finite continuous measure on R. Then for a>0

_ i l l ff ( f

idr{]' K*)* + *»(*) 2+ (y - ar 4- a) i

This is valid also for a = 0 with all the terms appearing in it finite if ff jijdxMdy) f* ff v{dx)v{dy) f» II 1 II l+yTTTT / M(ar) < oo and 1+-—— I i/(dr) < oo.

JJ (y-x) Jx

JJ (v-x) t

Jz

(ii) For each K <E R ff dxfijdy)

JJ

(y-x + a)-<

K < K j: rj r<
In particular, z / 7 ^ 0 , then K\(a)

-

x
where

Ky(a)

= [(I + o ) 1 - 1 ' - * 1 ~ 7 ] / { 1 - 7 ) = log(o

_l

A*(R)

(l + o)i »/ 7 < 1;

+ 1) j / 7 = 1.

The next lemma will verify Theorem 2 in the case 0 < 7 < 1.

101

Lemma 3. Let 0 < 7 < 1 and fin,n = 1,2,... fee a sequence of continuous, finite measures on R. Suppose that fin weakly converge to a finite measure ft and that for some p>0 (4.1)

sup ff r.Hj

ggfokjggfrjgj ( P (y - x + i/n)*-f U *

tiJdr)] < 7

00.

Tfcen JJ fi(dx)fi{dy){y - x)-< < oo and, as n -> 00, x
Hn(dx)fin{dy)

^i{dx)^{dy)

uAere tfie convergence taJtej place in the dual space of the set of bounded con­ tinuous functions on R x R. Proof. It suffices to show that JJn(dx)fi{dy)(y

- x)~* < 00 and

Ky

ff *!&&te^M __» / / fi(dx)n(dy)

(4.2)

JJ (y-x + l/nr

(y-x)"'

JJ

since it then follows that (\y - x| + 1 / n ) - 1 is uniformly integrable with respect to iin(dx)(in(dy). An application of Lemma 1 with p = 0 and with /x„ and 1/n in place of p and a, respectively, ,educes our rask to showing

//ir^^CMW/^GM

I<JI

2

I
Owing to the hypothesis (4.1), Holder's inequality combined with Lemma 1 controls the integral over \y - x\ < t. On the other hand jin(x,y) converges to $x,y) point-wise and boundedly whenever \z - y\> e for each positive e. These conclude that (4.2) holds.

5

Bounds for $(B,P)

and £ $ % A T f o -

x

i))

In this and succeeding three sections we shall assume ffr(x) > 0 for

x € R \ {0}

102

(instead of (1.11) for simplicity). We divide this section into two parts. The first part is adapted from Section 3 of [U2]. 5.1. Let h(x),x e R, be a smooth, non-negative even function such that h{0) > 1, /h(x)dx = 1 and h{x) = 0 for |*| > 1. Recall that where *{|x| < a} = 1 or 0 according as |x| < a or |x[ > a. The main purpose of this subsection is to prove the following lemma, which is fundamental for the discussions in the next subsection. L e m m a 4. There exists an even function w(x), x € R - {0} suck that u > 0; w(x) is positive and continuous for 0 < |x| < 1; w(x) converges to a posi­ tive number or diverges to infinity as x -)• 0 according as ^ is bounded or unbounded; and such that for each I > 1 (5.1)

§

/

E

( * - 0 % - i j ) ) —

r^(N(y-xk))dydt

U *-" i)dt


*w(xj:= ^

W^di-*;))£>(*(*(-**)) ,

C, and C2 ore constants independent of N, aV b stands for the maximum of two numbers a and b, and the triple sum on the left-hand side extends over all ordered triplets with three components different from one another. Let g = g(x),x # 0, be an even, continuous function such that g is nonincreasing on (Coo), 9(x)=

-4=7,

v\x\

= 0,

0<|x|<2

lil > 3

103

where o is a positive constant (the precise form of g as given above is not important) and eo

g{x)dx = 1. / ■CO We write gN{x) = Ng(Nx) and define a function G(N) on R by x

/ so that

ry

dy / -oo

G

'I'N)(X)

(9N{U)

-h(u))du

J—oo =SN(X)

-h(x).

Both |G(N)| and |G\N)\ are bounded by 1, and vanish for |x| > 1 if N > 3. Let G = G(N). T h e equation (2.1) yields dt

Y

G'{xi-Wj)vi

ij&)

= €NN

G

"(X* - xj)vdvi

Y tjm

- ^ ^ ^ G ' ( z , - ^ i

~ vj) ~ f N ^ 2 Y

G X

'( i ~

X V

i) i

urn £W(2V(s*-*i))

k^i

^i& (the time variable is suppressed from functions Zi{t),Vi{t) etc. as before). In the third term on the right-hand side above we may replace G'(xi - xk) by %{G'(xi-Xs)-G'(xj -xk)] if j + k, since U' is odd. Noticing G[N) is also odd, we see that the first term on the right side equals 5eivNY,(vi-vj)2G"{x, -Xj) and the second -\N{d/dt) J G ( i r i j ) . Integrating both sides of the resulting equality on [0, T] and then dividing them by Nz, we obtain after rearrangement of terms

/

2/V2

Y

+ TC5 /

[ - W ( ^ f o - xj))][G'(xi - **) - G'(ij - xk)]dt

Z [-W(JV(ii - ^))]G'(x, - I J ) * ij(^)

1

1 «
*J<*)

-$§*£ T,te-vA26"te-*i>%«j(^)

t=o

104

By the inequality (2.3) and (3.4) the first term on the right-hand side of this inequality vanishes in the limit. The second term is bounded by ! | | G | U Since -G'(N)(x) < h{x), the third term is bounded above by 2j|h|Uejv Jo JV"1 E i L i w.?W which also vanishes in the limit. it follows that 5§L /

T

[-NU'iNixi

+s r

- xMG'izi

E HTOW*

- xk) - G'(Xj -

Therefore

xk)\dt

- «f)no'(*j - XJW

Substituting the following two inequalities; G'(Xi-xk)-G'(xj-xk)

1 -

= >

]

Xi — Xj

— — // b« [9N ~-fe](y-x h\{y - xt k))dy /

— > 9N(Xi - Xj)

9N{y-xk)dy-

-—'

lw1

;

jXj — Xj\ V 1 '

we see that for a constant Co (5.2)

^

/

+

~N*

dt £

dt

^ ( j v ^ - a r j ) ) — — [*'gN(y

Y,

$(N(xi

- xk)dy

- Xj))9N(Xi - Xj)

•j(*)

L e m m a 5 . TAere ezists an even function u>{x), x e R \ {0} weft tAat w > 0; w(x) is jwrittve and continuous for 0 < |*| < I ; w(ar) converges to a posi­ tive number or dwerges to infinity as x -> 0 according as 0* is bounded or

105

unbounded; and Y,

$'i(N{xi - Xj)) -——

j

9N(V - Xk)dy

Froe/. If we define «{x),0 < x < 2, by *<x)=

inf g(u),

g{x) = ^(x)

inf

4T4

(recall we have supposed that ^(x) > 0), then w is non-increasing; (x) ap­ proaches infinity or a positive number as x j 0 according as ^J is unbounded or bounded by virtue of the assumption (1.3); moreover if 0 < 8 < t < 1, ip(s)g(s +t) > - = max{^(s + t)w(s), 0(t)d>{s),^(s)w(t)}. v2 This inequality implies, as a little reflection shows, that if we define u{x) = (6\^)-MNk{|x|<1},then (5.3)

TGS(3)

where 6(3) is the permutation group of three letters i,j,k and ® , $ and qk are arbitrary three real numbers different from one another. The inequality required in the lemma follows from (5.3) since rv iy-x)

j

9N(U-z}du>min{gN(x

- z),gN{y - z)}.

Lemma 6. Let tp{x) be a positive, integrate continuous function of x > 0 such that /)■ Then forp>\ £[$>i+1(tf(si-**)>]


f^h(N(xt-Xk))

106

where C is a constant independent of N and I > 0. Proof. Choose b > 2 so that h{x) > 0 if |x| < 1/6 and define p{8) = Y,i h(bN{xi-6)). The left-hand side of the asserted inequality may be written / E l 7 y,
- xk\)htN(xk

- 9)d»]

hbN(xi-8')d$',

which, employing Holder's inequality, we dominate by a constant times N1+p

f \f

W (iV|0|-l)de]

< N\I j


p{8')d8'

/ \f
It is easy to see that N / R ^ I + P W ^ < const E = i E * * H^t Thus Lemma 6 has been proved.

p(9')d8'

- **»]'■

Proof of Lemma 4. We may suppose u> < g. Substituting this inequality and that of Lemma 5 into the left-hand side of (5.2) we have fT 1


1

(*'

'

-LIIIMI

~c))dydt

eN

fT

V"

*P(N(xi-xi))^

We decompose the right-hand side according to i> = {$ - V,') + $f. Lemmm 4 follows if we show that for each 6 > 0 and / > 0 there exists a constant C (independent of N) such that for any configuration x = (xu...>stf) Cm. R

(5.4)

^ ^(N{xi-xj))


107

since then the contribution of V,* can be absorbed into the second term on the left-hand side. The inequality (5.4) is proved as follows. By Lemma 6 there exists a constant M = Mt such that £ ( j{:jt) ^,*(JV{*j - */)) ^ M E i j ( ^ ) ^(N(xi - xj)). We can then choose a constant C = Cs so that \x

%

uji :^22u(N{xi-xj))

> -j

ft

whenever

Consequently £ { j m t y { N ( x i -x3)) The proof of Lemma 4 is complete. 5.2. Let p?{9){= £ih(N(9 denned in Section 3.

Q ■- ^ ^ ( / V f x , - xj)) > —.

< M£Q


aa sesired.

- x*(*)))) be the empirical density

as

Lemma 7. Suppose that U has the long range (1.5). Then

*NN L Mn p-i+i/w** Ipt{T)dT 6<»'

; sck+aww-i> ■■ •IS -■•

MS*'

a<e'<#+i

(Recall that exN1-!

= 1/]ogNtf-y

= 1 andeNNl->

= 1

if-r
Proof. The lemma is essentially a corollary of Lemma 4: it is obtained by rewriting the sum in the inequality of Lemma 4 by means of the density func­ tion pf. Our task is to ascertain that the error terms arising therein are really negligible. We insert the following expression of unity (5.5)

/ hN(xi - $)d$ f hN(xj - e')dB' JR.

JR

under the summation symbol on the right-hand side of the inequality (5.1). Taking summation first as in the proof of Lemma 6 and replacing the function

108

(0 - ty)(x) bb yas + l)-> (for x > 0), we eotain (5.6)

ty-WKNjxt-Xj))

f

™b J^, ~ *NN

\xt-Xj\V\ L Jj (e>-6 + l/Nn\8>-8\Vl)dt{1

+

° (1}) '

9<9'

where o(l)-> 0 a s I -Kx> uniformly in N. Next we look at the first term on the left-hand side of (5.1). Whenever \xi -8\< 1/iV, \XJ -ff\< 1/N and xj >xt+ l/N, it holds that (5.7)

^N (y

—z— / Xj

X\ j

*V-d

-xk)dy

X i

hNiy Xk )dy

+ llNf9 I

-

I" rXi + l/N

ylj

hN{y Xj — X{ \JXi

xk)dy.

Jxj—1 /N

If we put h±(x) =

/ h{y±

x)dy,

Jo

the last term may be written as (XJ - Xi^lh+iNixi

- xk)) + h-{N{xj

-

xk))].

Now we proceed as we did for the right-hand side of (5.1), namely we in­ sert the expression (5.5) under the triple sum on the left-hand side of (5.1). Substituting the lower bound (5.7) we see that for m > 1 (5.8)

U.Z

0-ifrD(iv(n-*j)) fZ' HN(y - xk))dydt JXi

hAftfet)

9<9<

p(l)

p&)

109 and d(^m

where o(l) -■ 0 as / -► oo uniformly in N Vnd d^

and

*is

stand for

X ( ^ l » j - g i l < 0 ) . x W x j - a t | < 0) r f (

respectively. The function tp,(x) := (V - ^,*)(x)/|x| is integrable and ft± van­ ishes outside [-2,2]. Therefore, by Lemma 6 (applied with p = 1), for any positive number 6 there exists a constant L such that (5.9a)

f

RMIK&^L

"

9N(x)dt

for

I > L.

Jo

Lemma 6 also deduces (5.9b)

R^m

<

l

\C+l

^

/

*jv(x)d(,

where c = [ i n f ^ o ^ K i n f | ^ 1 0(a)]. The inequality of Lemma 7 now follows from (5.6), (5.8), (5.9) and Lemma 4. Lemma 8. For each l>1

(5.10)

and each T > 0 there exists a constant C such that

~ f

Yl

I*'-SJIV1

IM+)

-dt < C;

T1

(5.11)

^ I * N {x ( 0 ) * < C; i* Jo

and in the long range case fT

(5.12)

ff

oN(9\nN (8!\d6dS' ; f

fe'

\@

if y < 1

/ dt / / 7 ' / ; \ ' ..„ / /)f(r)dr < { _, Rr ., 7o J . / (O'-e + i/Ny+i j9 " w lciogjv tf 7 = 1.

no Proof. Let U has long range with 0 < 7 < 1. Apply Lemma 1 (in Section 4) with ti{d8) = p?(Q)d6 and with p = 0,1 and a = 1/JV to the inner integrals on both sides of the inequality of Lemma 7. Let p? be defined by (3.10). Then one on the right-hand side is bounded by a constant multiple of ft

fflWtfdBdff

JJ

(0'-0 + l)->

8 + KS'

ff

\fl{8,8'))*MdB'

Jj

{ff-e+l/Np

9<9'<S+\

/

\

\e<9'

1

•*< / 1 y / 2 ( ff[^{e,8')fdem' < 214 + I JJ {B'-8+\INY> " 7 ( 1 + 7) \i{\-i)J

1/2

where the inequality is obtained by applying Schwarz inequality and Lemma 2 (ii) to the second integral on the left side (also notice p^(M') < W - 6\~l). The last double integral also arises from the right-hand side (without the power of 1/2. Comparing these we deduce (5.11) and (5.12), and, recalling (5.6) together with (5.4), also the inequality

(5.10')

~ [ JQ

Y,

&(N{x,(t)-Xj(t)))dt
urn

which obviously implies (1.10). The proofs in the other cases of long range are similarly (when 7 = 1) or rather immediately (when 7 < 0) deduced from Lemma 7 and (5.4). In the short range case use an inequality as given in Lemma 6 (cf. [U2]). Lemma 9. Suppose that U has the long range (1.5). Then JOT each I > 0 ^v" / Jo

H

&{N{xi(t)

- xj(t)))dt —> 0

as

N -> 00.

urn

Proof. This follows from (5.4) and (5.11).

6

Proof of Theorem 2 in Case 7 = 1

Let 7 = 1. Recall that eN = 1/logiV. Let p? be defined by (3.10), namely

We will apply (5.12) in the following form

111

Lemma 10. 1 / 7 = 1 , then for each T > 0

/ ds If &*(8'e'^ dM9' = Q(logN).

(6.1)

e<8'

Proof. Writing down the inequality (5.12) in terms of p^{9,9') Lemma 1 we find (6.1) to hold.

according to

The same argument as deriving (5.6) together with Lemma 9 shows that the relation (3.5) can be written as (6.2)

1

ft

ffJ'(P)-J'(6)

p^{8)p^{B!)d0d9'

,

^

where o(l) -» 0 as JV -— oo ana J is ana smsoth function on R having a compact support. Since the contribution from outside a neighbourhood of the diagonal 8 = $' to the integral is negligible owing to the factor 1/logTV, [J'{9) - J'{8)]/{8' - 8) may be replaced by J"(9). After this replacement we carry out the same integration by parts as in the proof of Lemma 1 to rewrite the inner double integral on the right-hand side of (6.2) in terms of pf(9,9'). It then becomes (6.3)

(j J"{9) ^

f

y ^ y

+ \ [I J'"{8)[p™{9,9')fd9d8'.

The second term of (6.3) being bounded (use Lemma 1 if necessary), we obtain (6.4)

a?(J)-ag(J) 1

/ ' r f , ff e<6'

r,m\p»{e,8>)fd8d8>

112 Lemma 11. lim sup sup Q(V((—oo, —L] U [£,oo)) = 0. L-HX

N

Q
Proof. Let g be a smooth non-negative function such that g = 1 fat \x\ > 1 and g = 0 for \x\ < 1/2. If we substitute J(x) = g(x/L) in (3.3) and apply the bound (5.10), then sup a f {(-co, - L ] U [L, oo)) < o f ( ( - o o , - L / 2 ] U [L/2, oo)) + C/L, 0
where C is a constant independent of N. The assertion of the lemma is now obvious. Lemma 12. The family of empirical measures a?, N=.1,2,..., i/ considered as a sequence of functions from a time interval [0,T] into the space of the probability measures on R, is relatively compact. Proof. According to Lemma 11 it suffices to verify the equi-contimiity of a f (J) for each testing function J <E Cg°(R), which follows from (6.4) if it is true that (6.5)

lim lim sup

1 _ fl

sup

^4-0 Af->oo 0
[f[p»{e,e>)YdBd6<

iNJJJ

log

e'-e + i/N

ds

-°-

#<$

Apply Holder's inequality to the integration relative to s in (6.5) and then, applying it once more together with Lemma 10, you will see (6.5) holds. Owing to (6.1) as well as Lemma 11 the range of the inner integral on the right-hand side of (6.4) may be restricted to 8 + 1/N < 6' < 8 +1. Introducing new variable A, instead of
(6.6)

a?(J)-a£(J) = f ds f M f J"(9)[p^(9,e Jo JK JO

+ N^-1)]2dX

+ o(l).

113

(We have neglected the contribution of the integral over A > 1, while it turns out that every interval included in 0 < A < 1 remains significant.) The family {Q^ : N > 2} is relatively compact according to Lemma 12. Let af converge to a measure-valued continuous function & along a subsequence JVV Then [Jo* $($,9+ Nx-l)d\]d9 also converges to &(<&) along N* as is easily seen. From (6.1) it follows that (6.7)

sup f dt f d$ f {p?(6,8 + N Jo JR JO

Nx-l)fd\ < oo.

By standard regularization argument fit is shown to have a density, p($, t) say, such that T

(6.8)

/ Jo

J / p.3/ (9j)d&dt


JR

Since a weak solution of the Cauchy problem for the non-linear diffusion equa­ tion (1.13) satisfying this integrability condition is unique (see Theorem 9 in Section 8), Theorem 2 follows from (6.6) if we can show that the measures (£M'(M + N^fdXdt) dB weakly converge to (/ Q T P 2 (6,t)dt) d&. Ac­ tually we prove the following somewhat stronger result. Theorem 3. Let p{8}t) be a weak solution to the equation (1.13) having initial measure p.Q and satisfying the moment condition (6.7). Then lim

f

dt [ dB [ \p?(016 +

Nx-l)-p{e,t)\1ld\=0.

Theorem 3 is to be proved by means of the Young measure. Let M be the set of Radon measures n on I : = [ 0 , T ] x H x [O,l]x[0,oo) such that Jxnm
TT

+ u2)n(dtd$d\du)

< oo for every K > 0.

6 M by f Fd*N = [ dt f dB f F ( r , M , P ? ( M + Jx JO JR JO

Nx'1))dX.

Define

114

where F = F{t,8,Xyu)

ranges over all continuous functions on X such that \F(t,8,X,u)\ ' \ ' ' ' ' < oo

sup (t.M,u)e.v

'

+

u

and that for some K > 0 f = 0 whenever

|0| > K.

We equip the space M with the topology according to which a sequence JT„ in M converges to n if and only if 7n{F) -4 ^(F) for every continuous F subject to the condition above. By (6.7) as well as Lemma 11 the family (nN (dtd8dXdu)\ N = 1,2 ) constitutes a relatively compact set of measures. Theorem 4. Let N' be a subsequence of N along which both o f and wN converge. Let p{8,t)d$ and Tt(dtdddXdu) be respective limit points along N* Then the Young measure n(dtd8d\du) is degenerate at p(9,t) , namely K{dtd6d\du) = dtdOdXSp{f)J]{du). The deduction of Theorem 6 from Theorem 7 is straightforward since the function \u-p{8, t)\2f{6) ii admiited as F if / ii conttnuous and has a compact support owing to the moment bound (6.8) and since as A' - J oo sup /

u2dTTN < s u p , / /

N J\8\>K

N V ■'0

f

p?d8dt . / fu^di!*

J\G\>K

—> 0

\J

in view of (6.7), Lemma 11 and Lemma 2 (ii). The proof of Theorem 4 is similar to that given in [V: Theorem 7.6], [OV: Theorem ] or [U2: Theorem 7] except that we define the measure nN by means of p?{9,9 + Nx~1) instead of

Proof of Theorem 4- As in [V] we define a function G(Nfi] on R by G
/

dy

(hN{s) - hb(s))ds

so that G'('Nb){x) = hN{x) - hb(x). Here b is a positive constant less than one and h the same function that is used to define />* It is immediate to see that (6.9)

|IGiK.ftjlloo-K)

as

N -> oo and

b ^ oo in this order.

115 We put G = G{Ntb) in (5.2). If we substitute the equality

G'jXi-x^-G'jXj-x*)

_

z fr-

i1

— - — / [hN - hb](y *• XJ Ji,

xk)dy,

the first term on the left-hand side of (5.2) becomes

(6.10)

^- [ N

Y,

Jo

^(Nixi-xj))—^— Xi

,JM*)

/*'[ft*->»&](*-«*)]<&. Xj

Jx

'

By (6.9) this is non-positive in the limit since the second term on the left-hand side of (5.2) is non-positive. In view of Lemma 9 the function ip in (6.10) may be replaced by (ip - -0,** so that we have the following L e m m a 13. For each I > 0 (6.11)

liminf* IN(N) N-too

< limsup limsup* IN(b), b-too

N-too

where * indicates that the limit is taken along the subsequence N* specified in Theorem 4 and

/„(*) = ££ f

V ty-$*)(X(Xl-Xj))—L_ f hK(s-xk}ds.

TakingIlargeenoughwemayreplace(^-^;){x)by (|i|+m)-1x{]xl We apply (5.8) and (5.9) with 7 = 1 to deduce

r ,.« IN{N)>

2

-^NL

fT , ffp?(8)p?{e')d6d8'

dt

!SV e
f

dr ._, + o{l)„.

Nl

elmiNyh ^

>/)■

>

where o(l) - > 0 a s N -» 00,Z-> 00 and m -■ 00 in this order. Also, by Lemma 9 and (5.6),

2

fT . ff p?(9)p?($')d9d$' f

Nt

..

M

116

where o(l) -» 0 as AT -> oo.

By expressing ftft(fl - *( 0 (6.12)

V?(0,t) —► /"/iftCff

-8'}p(8')d8'

a s J V - K » uniformly in (#,i). Rewrite the inner integrals appearing in the above bounds of IN(N) and /w(&) in terms of $-m(8,B + A'*"1) and p?(8,8 + Nx~l% respectively, according to Lemmas 1 and 2, where

*""<'■'> = ,-,'+./*

£*<■»■

Then observing pf>m{6,Q + N^)

+ l

= ^ ^

Nx-1).

p?{8,S +

and applying Lemma 10 we deduce from (6.11) liminf / dt. f d8 f [p? (9,8 + *-** Jo JK JO -n»

N-tou

dt j ffl j [p* (9, 6 + AT*-1}]3 x

Jo

x {^{8,8

Nx~l)}3d\

JR

Jo

x l

+ N - )-^(S)-^(8

+

Nx-})}d\.

To the left-hand side we apply Fatou's lemma; as to the right-hand side we apply the bound (6.7) as well as the uniform convergence (6.12). As a general fact the Young measure TT is automatically of the form ir(dtd8dXdu) = dtd8d\-Ktj,x{du)\ in particular we can write p(8,t) = /n°° uvt.B.\{du). Therefore

j dt j d8 j d\ j Jo

JR

JO

< limsup I u2\

u^t,e,s(du)

JO

kb{8 - 8')p(8\t)d8'\

= I dt d8 d\ I Jo JR. JO JO

u2xtigiX(du)

n{dtd8dXdu) / JO

untf,\(du).

117

But the inequality in the opposite direction is always true; hence we have the equality. This is possible only if the Young measure jr,,s,*(du} concentrates on a point, which necessarily equals p(9,t). Thus Theorem 4 has been proved. 7

Uniqueness for (1.14) and Proof of Theorem 2 in Case 7 < 1

Let 7 < 1. In place of (6.2) we have (7.1)

a?(J)-ag(J)

The same compactness of the empirical measures a f as in Lemmas 11 and 12 holds. The proof is similar. We let JV -> >0 0i n7.1) )long g aubsequence for which « f has a limit. We can take the limit under the integral symbol if - 1 < 7 < 0. Thii si allowed even in the case 0 < 7 < 1 according to Lemma 3 of Section 3 and in view of the bound (5.12) and leads to the equation (1.14) as required. It remains to show uniqueness of a solution to the initial value problem for (1.14). Owing to (5.12) any limit point pt of af satisfies (7.2)

/ h

dt [f JJB?

^(
< oo.

h

Theorem 5. Let-1 < 7 < 1. For T > 0 and a (positive) finite measure p0 on R there exists one and only one solution, under the integrability condition (7.2), of the non-linear problem

(7.3)

v,(J) - fi„(J)

•\C'L

J'(y) - J'{x) y-x

fit(dx)pt{dy) \y - x\->

which is to be valid for every 0 < s
fy

118

Let Pr be the semigroup of the heat equation (d/dt)u = {d2/dx2)u Prfi(x) =. I

on R, i.e.,

gr(x,y)y{dy).

JR.

where

Lemma 14. For 0 < a < 1, p > 1 and i > 0 ft [{PanT{x,y)\vdxdy < ft

\y-x\y

JJ\y-x\«

P(x^dxdy

\y-Ay

JJ\y-x\<,

Proof. By Holder's inequality (T

Pa(i(r)dSj

< (f*J

<j

ff.(sMr+ds))

U\{dr-s)\

ga(s)ds.

By substituting this bound and changing the variable of integration according to x = c - A, y = c + A, ,he llft-hand ssde of fhe inequality of the lemma ii dominated by

r*^2 dx fx

ff

JJ\v-x\«.

f™ ( \

fx

fipp(x,y)dxdy £i (x, y)dx

^-A"1

This finishes the proof of Lemma 14. Put rvyiu) „. ff Kdx)(i{dy)v{x,y) JJR* \y - ^ where v(dx) is another finite measure on R.

\p

119

Lemma 15. Let ft and is be two finite measures such that r{fi\fi) < oo and T(v\v) < oo. Then (i)

limr(Pav\ri

=

r(i,\(i)
and m

B

ff

^y)PM^dy

=

ft .fi2^{x,y)v{dx)dy ^ ^

<

„

IJf-ip Proo/. Holder's inequality combined with the previous lemma shows (7.4)

sup o< a
ff yj

* (x>y)(p°v)^x)dxdy ly-ip

_^Q

M

c ; 0

We apply Lemma 2 (i) with o = 0 and with Pau{x)dx in place of u(dx) to see

II

(1 + 7 )(2 + 7) ff 2 JJ

[L2(x,y){Pav)A{x)dxdy \y-x\i

ff jx2{x,y){Pav{x)dxdy

JJ '

+ dxPav{y)dy}

lv-xp

"

+ i(^(R))V(R), and apply (7.4) and Fatou's lemma in taking the limit on the right-hand side and then Lemma 2 to go back to I>|f/). This yields l i m s u p a i 0 r ( / > | / 0 < r{v\n). The inequality in the other direction follows from Fatou's lemma. Now we have limQ4.0 r(Pav\{t) — r(i/|/i). Because of this equality we must also have the equality in the places where we have obtained the inequality by an appli­ cation of Fatou's lemma. This shows (ii). Proof of Theorem 5. Taking J such that J(x) = 1 for | i | < 1, replace J(x) by J(x/L) in (7.3) and let L tend to infinity. Then, using (7.2), one observes that fi,{R) = fi0(R) and that any bounded function with bounded derivatives is admitted as a test function J in (7.3).

120

Let fit and W be two solutions starting at the same initial measure. We put ut= fit- vt. Then (1.14) yields

d

ft *{**)*{*)-«{*)**)

dt

JJ

p

\y-xp

which hold outside a Lebesgue null set of t G [0, T). Applying this with J{x) = F,«t(x) and substituting [Prut)n = {d/dT)PTu>t we see that d

ID

\

„ f f (it(dx)fit(dy) ~ vt{dx)vt{dy)

fdPr^Y

y>x

Integrating both sides with respect to t and r over the square [0, sj x [a, b] we have (7.5)

f u;a(PTu>,)dr = /" /t(i)
JO

JO

where we put r

,>v

0

ff fit(dx)tit(dy)-vt(dx}v,{dy)

1

/•*

We let b -> oo. Then the term /Q5 /4(t)dt disappears in the limit. By Lemma 15 (and its proof) and (7.2) f* Ia(t)dt converges to f* IQ(t)dt as a \, 0. By the semigroup property PT = and since PT is symmetric we conclude that (7.6)

/ Jo

/ [PT/2vt(x)]2dzdT JR

= - [' Jo

I0(t)dt.

We write / 0 in the form TM

» f/* Mdx)ut(dy)

+ ui|(dr)i/K<^r)

1

\)~>X

We further rewrite it by integrating by parts by using vt(dy) I u!t{u)du = -dy ( I

u t (u)d«J

/»

121

and w((dx) / ut(u)du = ~~dx 1 I u)t(u)du)

.

This, which we can carry out owing to (ii) of Lemma 15, yields M , U k , n [[ fit{dx)dy + dxvt{dy)t_ 2 W) = h + 1) jj \y-x\y [ut{x,y)Y>0

a.e.

y>x

Thus, by (7.6), JK[Pr/^s(x)\2dx = 0 for all r > 0, implying that v, is identical to n„. The proof of uniqueness is complete.

8

Outline of Proof of Theorem 1

The proof of Theorem 1 differs from that of Theorem 2 in the respect that here the notion of local equilibrium plays decisive role as reflected in the form of P(p), the non-linear function in the diffusion equation, which depends on details of the potential U. We outline the proof, adapting from [U2], in which the problem is studied for the system defined on the unit interval with virtually reflecting boundary. Let U be a short range potential. What we need as the local equilibrium statement is formulated in Theorem 6 below. Given a (large) positive number L, let * JV - t {dq) be the probability measure on the space of configurations q = ( * } £ , ! on R defined by (8.1) where x3 = xS'N

$N-L(H)

= - i - / ds f H{Nxe(s))d0, *i*l Jo j-L stands for the configuration x(£) = (x.(O)^i viewed from $ :

x»(t) = (*,(«)-*)£, and H = tf (q) ranges over bounded, continuous local functions of locally finite configurations on R. (The idea of averaging not only in time but also in space is due to [GPV].) We continue to suppose (8.2)

#C)>0

for

zeR{{0}.

The local equilibrium means that any limit point of * ; v ' t as N -+ oo, which is necessarily translation invariant, concentrates on configurations of equal spac­ ing; to be precise it is stated as follows:

122

Theorem 6. Suppose that (8.2) as well as the same assumption as m Theorem 1 is fulfilled. Then for each L > 0 any limit point $ of $N'L as N -* oo must be of the form 1.00

* = / Jo

rl/p

ipX(dp),

^p{dq)-p

o~e{i/e,<,)(d
Heree(r,q)rr> 0, stands for (..., -r + q,q,r + q,2r + q,...), the configuration of particles on R with equal successive spacings of common span r and having a particle at q; \ = \L is a probability measure on [0, oo); and * 0 is understood to concentrate on the empty configuration. Theorem 6 follows from Lemma 16 and Theorem 7 below. Lemma 16. For every limit point $ of$NL

(8.3)

as N -> >o

$ | q = (
Theorem 7. Let ft be a probability measure on the space of locally finite con­ figurations ((<,),ez)) on R. If fi is translation invariant and (8.3) holds with a strictly convex U satisfying (1.2) and (8.2), then (qi) has equal successive spacings of a common span with fi -probability one. In formal level the result of Lemma 16 is inferred from an identity obtained by calculating the right-hand side of

Jo dt

dt

according to (2.1) and then passing to the limit as N -> >o. . I nhe identity however there arise singular terms that should be avoided for the latter proce­ dure of taking limit. Theorem 7 is proved by Lang [L] under a certain moment condition, which is shown to be dispensable in [U2]. The details of proofs of Lemma 16 and Theorem 7 are omitted. We have compactness results analogous to Lemmas 11 and 12, which implies that the family of continuous probability-measure-valued functions

123

(a?.0 < t < T),N = 1,2 is relatively compact. From (5.10) it follows that any limit point of af is absolutely continuous for each t. For the identifi­ cation of the limit density, p(0, t) say, we are going to show that the following convergence of the integral on the right-hand side of (3.7): as N* -> oo ft

<8'4>

i

^(Ar,(xI(s)-xJ(3)))J"(i,(.s)}rfs

/ 5XrT H Jo *N* *-'

—y f

[

JO

P(p(8,s))J"(8)d8ds

J-CX)

for every J e Q ° ( R ) , where {TV*} is a subsequence of {N} along which of converges to p{8,t)dS. Since p{8,t) is then revealed to be a weak solution of the Cauchy problem (1.8) and (1.10) , the uniqueness theorem will complete the proof of Theorem 1. Let h be the same smooth even function as introduced in Section 3. Write hK{&) := Kh(KO) for K > 0 as before. Suppose J{x) = 0 for |x| > L. Then for t > 0

Introducing the function ff

*(«i) = 2 ! C <Wft " ?j) / l i/'(9 i ),

q - fat)" i.

we can rewrite the inner integral on the right-hand side above in a concise form: /

H^(Nx9(s))J"(8)dO. -L

For the proof of (8.4) it suffices to show that for every L > 1 (8.5)

lim limsup* f /-K»

/V-K3C

JO

\H$,{Nxe{s)) - P(P(8,s))\d$

ds f J-L

= 0,

124

where * indicates that the limit is taken along the subsequence {AT*}. By virtue of the bound (5.11) and Lemma 5 the function * " ( i , i ) := £i ; i#itf(JVte($)-*i(*))) is uniformly summable with respect to (t,i). This allows us to replace //£(q) in (8.5) by

We also replace P(p) by

PM(P):=

and PM{p(0,s))

( f>(*/p)

by Pw(pf(Nxfl(s))),

AA/U,

where

The last two replacements are justified by Theorem 8 and Lemma 17 below. Theorem 8. For every L > 1 lim limsnp limsup" /

t-tea

,l0

ftr-»oo

Jo

ds /

J-L

\p((Nxe(s))

- p(0,s)\d8 = 0.

Lemma 17. Any limit density p(B,t) of a? satisfies (8.6)

fdtfJ-<

Jo

P{p{9j))d8

< oo.

Now (8.5) follows from the following Lemma 18. For each L > 1 and each M > 1 (8.7)

lim limsup /" ds f

\H& M(JVx*(s)) - PM{P1[Nx$(s)))\d8

= 0.

125 Proof. By means of * * ' L the limit supremum in (8.7) is expressed in the form limsup f\H^M[n) N-HJO

-iWtq))!*"'^*!).

I

According to the local equilibrium assertion (Theorem 6) this equals

dq,

The proof of Theorem 8 is done by means of the Young measure in a way the proof Theorem 3. (8.7) The is bound of Lemma 17 is obtained whichsimilar clearlytovanishes as of I -> oo. Thus proved. incidentally in that proof. The following uniqueness theorem completes the proof of Theorem 1. Theorem 9. Let Qo be a finite Borel measure on R. Then for every T > 0 there is at most one non-negative weak solution p = p{0, t) to the problem (1.8) on [0,T] that satisfies the mtegrability condition (8.6) and the initial condition (1.10), as well. Proof The proof is similar to that of Theorem 3 (but here we have not to introduce^). (See also [til].)

9

Hard Core Potentials

Suppose that the pair potential function U has a hard core: U(x) = oc

for

|J|


for some a > 0. It is also assumed that U is even, and continuously differentiatle for | i | > a and has either a short range / " U(x)dx < oo or a long range 1>(x)~ l/larp ( 0 < 7 < 1 ) ; t h a t U{x) -» oo as

\x\ i a

126

and U'{x) < 0 if \x\ - a (> 0) is sufficiently small; and that the energy bound (1.8) holds. We can then show (3.5), so that (3.6) is valid. The formal ar­ guments made after it seem flawless, but if we proceed as before there arises serious difficulty. What is wrong? While it is true that

we cannot show the uniform integrability of *^()

:=

^

E

\U'(N(xi-xi))\x(\xi-x}\<2a).

(Notice that \U'{x)| is equivalent to ip(x) for a < \x\ < 2a.) The legitimate derivation is given by introducing the new variables as is done by Rost [R] for a system of Brownian rods. Let N particles are numbered in the increasing order: ii

< x-i < ■ ■ ■ <

XN

and put - &i(t) ~ a — . Then the equation (2.1) becomes tfiVM

= -#Vi(t)

~ *« E

V'iNfoit)

- y}(t}} +

(i-j)a)

and in place of the equation (3.7) we accordingly have (9.1)

= / §f E -( J '^') - ftyWWfa + o(l), where a?(J) = N~l £ \

J{yi(t)).

- Vi) + (i - J)o-)ds

127

First consider the short range case / ~ U{x)dx < oo. In this case eN = 1 and the local equilibrium is valid, so that Xi - x, may be replaced by (i - j)/p where p = p(9,t) is the density of a limit point of df.. The same proof as outlined in Section 8 leads us to the equation

where

P[p) = Yi ~hU' (-- + ka)

(9.2) By (9.3)

^ = o f ((-00,1,(1)]) - afU-oo.Sitt)]),

the rule of transform, which depends on t, between variables y and x is given by (9.4)

x-y

= a/i,((-oc,xj) = a/i t ((-oo,y]),

where fit and £, are limits of eg and d/1', respectively. ^From these relations we deduce (1 + ap){\ -ap) - 1; or p—

and p 1 — ap 1 + ap where /> = p(0, t) is the density of ft,. According to (9.4) we write xt(y) = V +

ajit((-oo,y}).

Then

tk(J) = f JMy))p(y^)dy

(9.5) and we see d_ (J) m dt

= f J'ixtiy^piy^dy-

j

^J(xt(y))^P(p(y,t))dy,

128

^i

= a^P(p(y,t))

and

^

= l + «*»,i)s

and substituting the last two relations into the first one, ftMJ)

=- j

J'{xt{y))^P(p{y>t))dy

=- j

J'{x)dyP(p{xi(y),t))

=- J

J'(x)dxP(p(x,t))

=J

J"(x)P(p(x,t))dx,

where we have used the relation P{p) = P(p) and p = p{8J) denotes the limit density of af The equation (9.2) is therefore translated to

the same non-linear diffusion equation as in Theorem 1. In the long range case we put rj}{x) = -\x\U'{\x\+a) and write (9.1) in the form

= jf 2 £ B -,7^(T-?-'.)/ W ^ - » > * « - ' - ^ + o(l). By (9.3) we can rewrite it as Qf(J)-6 0 N (J)

129 This would give us

(9.6)

j-a=

a

p

„

,„,

if 7 = 1

and

(9.7) = ['#[[

J'{y')-J'{y)

MdVMdy')

if 0 < 7 < 1. The equation (9.6) is translated to (d/dt)p = (d^/dO^p2 as before. If pt has a sufficiently nice density p then from (9.5) and (9.7) we deduce ftMJ)

= f J'(xt)^^-p(y,t)dy

^ 1

+f

=-ap(y,t}A(p)(ytt)

and

J'(xt)^01p{yJ)A(p)(y,t)dy%

^ | M = 1 + a^fj,, *),

where A/-U

*\

f°°

signjy -

x)pt(x)dx

U*Vt{dr) := (sign(y-x))/*([xAy,iV»]). These relations verify ((.14). We notice that the equation (9.7) admits a solution that has point masses. For such solutions the argument above breaks down in its last step and (1.14) would be false. 10

The First Order Equation

The equation (1.1) or its scaled version (2.1) is regarded as the second order equation for the variable (**(*))£j. Its solution as a bulk is well approximated by that of the following first order equation (10.1)

jtXi(t)

= -eNN

£

U'(N{Xi(t) - Xj(r))),

130

where U is supposed to satisfy the same conditions as in Theorems 1 or 2 and eyv is given by (2.5). This time we have the identity

JO

ZJV

u {

^

X

*

X

3

without error term o(l) as in (3.6). The same (or rather simpler) proof as before can be applied to deduce the bounds parallel to those given in Section 5 and derive the same limit equations as given in Theorems 1 and 2. Here we have not to assume anything about the energy £*,;(*) U(N{Xi - Xj)), i.e.. nothing corresponding to (1.8) is needed. It is worth noting that the functions

1 £ U(N(Xi(t)-Xj(t)))

and

IW^X((())

are shown not to increase as time goes on if U" > 0. In Miirman [M], where the nearest neighbour interaction version of (10.1) is studied under the condition that these quantities are initially bounded, this fact is used for derivation of the corresponding non-linear diffusion. The result is stronger than ours in the sense that the empirical current field N

-E'WiW E t=l

converges to -tkP{p{9,t)) 11

u'(N[Xi(t)-Xj(t)})

j=i±l

uniformly in t as N -+ ooo

Stochastic Models

Adding the independent standard white noises tyt{t) to the right-hand side of (1.1b) we have the interacting Ornstein-Uhlenbeck process. Assuming that V has a compact support and is continuously differentiable on R and repulsive, i.e., -xU'(x) > 0, OUa-Varadhan [OV] proves that if the particles move on a circle instead of the whole real line and if the laws of initial configurations are subject to a certain entropy bound, the empirical measure af converges in probability and its density solves the non-linear diffusion equation (1.9) with the function P given as follows. Let $„,; be the finite volume Gibbs state

131

on the interval (-1,1) of parricle number n with empty boundary condition, namely it is the probability measure on the n-dimensional hyper-cube [-1,1]" defined by

*„j(dgi

dqn) = -=— exp

dgi - ■ ■ dg„

where Z„,, is the normalization. Then 2P{p) = p+

lim

/

\ \ ij>(qi -

qJ)h(qJ)'i>n,t{dq\---dqn),

i,i(#)

n/<2()-»/>

where h may be any non-negative smooth function on R having a compact support such that f hdx = 1. If the potential U has a long range (1.5) (with 7 < 1), then the corre­ sponding equation for the scaled variable x,(t) becomes rfxj(r) =

XiwN~]vt{t)di

dvi[t) = -XNVi{t)dt

- A/v £

0'{N(xi(t)

-Xj{t)))

+

^dWt(t).

where

Wt(t) = (^Ny'WdeNNh)

~ d Wx{t)

(~d means that the probability laws are identical ). Recalling our derivation of the scaling limit for the deterministic model we see that the effect of the white noise disappears in the limit so that there comes up the same equation in the limit. The scaling limit for an analogous stochastic model associated with the first order equation (10.1) is studied in [V], [Ul].

REFERENCES [FIS]

Feng, S.; Iscoe, I; Seppalainen, T., A microscopcc mechanism for the porous medium equation. Preprint.

132 [F]

Fritz, J., On the asymptotic behaviour of Spitzerss model for evolution dimenstonlt point systems, J. Stat. Phys. 38 (1985), 615-645.

of one-

[GPV] Guo, M.Z.; PapanicoLaou, G.C.; Varadhan, S.R.S., Non-linear diffusion limit for a system of nearest neighbor interaciion,, Comm. Math. Phys. 118 (1988), 31-59. [LI

Lang, R., On the asymptotic behaviour of infinite gradient system,, Phys. 6 5 (1979), 129-149.

Comm. Math.

[M]

Miirmann, M. G., The hydrodynamcc limit of a one-dimensional gradient system, J. Stat. Phys. 48 (1987), 769-788.

[OV]

Olla, S.; Varadhan, S.R.S., Scaling limit for interaciing cesses, Comm. Math. Phys. 135 (1991), 355-378.

[PS]

Presutti, E.; Scacciatelli, E., The evolution of a one-dimensional note on Fritz's paper, J. Stat. Phys. 38 (1985), 647-653.

[R]

Rost, H.; Diffusion de spheres dures dans la droite reelle comportement macroscopique et equilibire local, L.N.Math. 1059 (1981), 127-143.

[Sp]

Spohn, H.; Large scale dynamics of interacting particles, Springer (1991).

[SU]

Suzuki, Y.; Uchiyama, K-, Hydrodynamcc limit for a spin system on a sional lattice, Probab. Theory Relat. Fields, 9 5 (1993), 47-74.

[Ul]

Uchiyama, K., Scaling limits of interaciing diffusions with arbitrary initial tions, Probab. Theory Relat. Fields, 99 (1994), 97-110.

[U2]

Uchiyama, K., Scaling limit for a mechanical system of interacting particles, Comm. Math. Phys., 177 (1996), 103-128.

[V]

Varadhan, S.R.S., Scaling limit for interaciing diffusion,, (1991), 313-353.

nearest neighbor

Omstein-Uhtenbeck

pro­

point system:

u

multidimen­ distribu­

Comm. Math. Phys., 135

133

REGULARITY OF SOLUTIONS OF INITIAL B O U N D A R Y VALUE PROBLEMS FOR SYMMETRIC HYPERBOLIC SYSTEMS WITH B O U N D A R Y CHARACTERISTIC OF CONSTANT MULTIPLICITY YOSHITAKA YAMAMOTO Department of Applied Physics, Osaka University, Yamada-oka 2-1, Suita, Osaka 565, Japan E-mail: [email protected] We discuss the higher order regularity of solutions to initial boundary value prob­ lems for linear symmetric hyperbolic systems with boundary characteristic of con­ stant multiplicity. By means of the standard energy method it is shown that the solutions and their derivatives with respect to the time variable lie in certain weighted Sobolev spaces under a suitable compatibility condition between the data.

1

Introduction

Let fibea bounded open set in Rn, n > 2 with smooth boundary r . We consider the initial boundary value problem for the system of linear partial differential equations of first order n

Y^AjdjU + An+1u = F Qu = 0 L u(0) = /

in

[0,3*)xfi

on on

[0, T] x T O,

where x0 is the time variable, sometimes written ast,dj= d/dx-j, 0<j
134

The strong solution to the non-characteristic problem evolves continuously in the usual Sobolev space just like the solution to the Cauchy problem ([18], [27]). Some characteristic equations enjoy the same property thanks to their special structure ([7], [10], [11]). This is not always true of all the character­ istic problems, as illustrated by several equations including the one of ideal magneto-hydrodynamics ([10], [13], [26]). Hence, we are forced to introduce some other function spaces than the usual Sobolev spaces in handling the higher order regularity of solutions to the characteristic problem of a general form. A few spaces have been proposed when the boundary matrix is of constant rank. Rauch [16] proved that the strong solution and its derivatives in ( evolve continuously in the function spaces in which only the regularity of tangen­ tial derivatives in the Z,2-sense is taken into account. This result, referred to as the tangential regularity, is not available for solving quasilinear problems because the function space lacks several properties indispensable to nonlin­ ear analysis. Yanagisawa Matsumura [29] introduced some weighted Sobolev spaces in which the regularity of normal derivatives is appropriately consid­ ered and succeeded in solving the equation of ideal magneto-hydrodynamics. Ohno-Shizuta-Yanagisawa [15] handled the equation of a general form using the same function spaces. We note that the weighted Sobolev space, denoted by H?{Q), was first introduced by Chen Shuxing [4] in the study of a class of quasilinear hyperbolic systems. The continuation of solutions in the weighted spaces needs further im­ provements on the known results. Shizuta-Yabuta [22] presented a compati­ bility condition for the solution to lie in Hr,"(ii) but failed to find the solution in this class. A proof of this part was given by Secchi [20], [21]. His idea is raising the regularity of the strong solution one by one up to the desired order. To obtain the tangential regularity, for instance, he considered the equations for the tangential derivatives of the solution. With some equations added they form a system of first order. Secchi expected the derivatives BIS smooth cts the solution of the system and tried to solve it. The claim is that the solution is the fixed point of a contraction map sending an element of a rprtain rnptric space to the solution of the equation in which the unknown fuTction of the system is partially replaced by the element His plan howeve seems not to work well here for some other hypotheses on he stmctuTe of t h e S ^ r ^ matrices are required than the assumut on^"to solvethis e m l on S ^ r S S Ve th S eqU elements of the metric spaU ^ ^ ^ °' ' a t ' ° n f0r *° the In fact, the conclusion itself is true and the proof is straightforward as we will show in this paper. Unlike [20], [21] we pick up the system of equations for the tangential derivatives. By taking the degeneracy of the boundary matrix

135

into account carefully the system is just of the same form as (1). Hence, we have only to concentrate on the study of the first order regularity of strong solutions. The energy method suffices for our argument. It is also used to obtain the regularity of the normal derivatives of the solution. No space with negative norm is involved as compared with [20], [21]. We plan this paper as follows. In section 2 the definitions of several func­ tion spaces and their basic properties are given. In section 3 we present the assumptions and the statement of the main results. Section 4 is devoted to the proof of the existence of solutions of first order regularity. The next two sections treat the higher order regularity of solutions. All the technicalities are collected in Appendix. 2

Notation and function spaces

R and C denote the fields of real and complex numbers respectively. N is the set of natural numbers and Z+ the set of nonnegative integers. Let £ be a Banach space, m £ Z+ and 1 < q < oo. We set several function spaces with values in £ as follows. For a compact interval / we denote the space of m times continuously differentiable functions on / by Cm{I; E). C™(7; E) is the space of m times weakly continuously differentiable functions on I. Let I be an open interval. I/*(l) E) is the L?-space with respect to the Lebesgue measure on / . W™{I;E) is the Sobole v space in / of order m: {u 6 L«(/;£);distributional

derivatives &u S £ » ( / ; £ ) , 0< j < m}.

These spaces are equipped with the natural norms and are Banach spaces. Let ft be a bounded open set in Rn, n > 2 with smooth boundary T. # m ( f i ) , m € Z+ is the usual Sobolev space in ft of order m. We see ff°(ft) = L2(ft). We introduce the subspaces #, m (ft) and H£{Sl) of L2(ft) which play crucial roles in this paper. Also the space #£ n (ft) is given. We begin with the notion of tangential vector fields. Let A be a C°°-vector field on ft. A is said tangential if for any C 00 -function u on ft vanishing on T we have AM = 0 on

r. Definition1 W m 6 N . H? (f2) u the set of a function in L* (Q) « d that all the distributions which result from operating j tangential vector fields and k vector fields to the function lie in L2(ft) provided 0<j

(2)

+ 2k<m.

The spaces H£[Cl) and i/™n(ft) are defined by putting the conditions 0<j + 2k<m+l, 0<j<m, k =0

0<j

+ k<m

(3) (4)

136

in place of (2) respectively. We define H*?(Q) - H?m{fl) = H°an(U) = L2(Q). In the region apart from the boundary T elements of these spaces behave like functions in Hm(Q). For describing the behavior of the elements near T it is convenient to introduce some standard function spaces. Let RJ = {x; i „ > 0}. For a - ( Q ] , . . . , Q „ ) e Z£ we put

Definition 2 feme N. ff.m(R£) t$ 2 fl,°a„fl*« € L (R£), |o| + 2* <m. fl»(Bl) d«dknu E L 2 (R"), \o\ + 2k <m + l,\o\ of u I I?{Rl) satisfying df~u € L^K), H ? R ) , ffSCSi) and H*L(Rl)

the set of u € L 2 (R£) 5oturA»nff ft the set ofue L2(Rl) satisfying + k <m . fi2» (R£) is the set \a\ < m . We de/ine H?(H1) =

are Hilbert spaces with respective norms 1/2

l"l«J>{R») . . . 1/3 !"1HS„(R«J

It is noticed that we may replace the operator 8fon with

to obtain the same definitions of the spaces as Definition 2 and the equivalent norms to the original ones. We often make use of this observation. Returning to the case of the domain ft, we choose a finite open covering {Vf,0 < k < N} of n with the properties 1. V0 is a relatively compact and open subset of 0; 2- Kfc, 1
*k(Vk n ft) = Bk n R"+, *k(vk n r ) = B , n r - ' ; and then a partition of unity {-functions, u e L2(fi) belongs to H™(Q) if and only if tp0u € Hm(U) and

137

(pku) o * ; ' e / C ( R " ) , 1 < k < N. HZ(fy and H™(Q) are characterized similarly by means of //™(R") and Hg„(Kl) respectively. Thus, H?(Sl), H™W) and Hgn(n) me Hilbert spaces with respective norms

l " l " , ' : n ( « ) ^ { ^ 0

U

l « " ( n )

+

E r =

1

l ( ^ « )

O

* ^

l

| / / -

n

| R n

)

}

•

Let C m (H), m € Z + be the space of m times continuously differentiable functions on U. Using C°{f!} in place of Z*(«), we define the spaces C?{Ti), C%m and C- n {H) as in Definition 1. The spaces C T ( R l ) , C™(RT) and C™n(R") are given as in Definition 2. These norrned in the same way as above and become Banach spaces. It is well-known that a function in Hm (ft) has the trace on the boundary. The trace belongs to Hm~^2(r). This is also true of a function in ff™{fl). Let « € HS(fi)- Writing x = (i',x„), x' e R"" 1 , i „ t R ' , we regard Upku) o SC ! as an element of W ^ nlfy* . > ( R V ) ) and apply + ; H - \ R ^ ) ) the trace theorem of Lions < Lions-Magenes [9]). then, the'boundary value

fa*)»•r'k-Bexists

and |ies in

Thus, the trace operator 70 : u H> u | r is defined as a linear continuous map from H£{«3 to / / m " 1 ^ ( r ) . Similarly, when m > 2, u € J7,m(n) has the trace which belongs to H™"1^). For several results on the higher order traces and the characterization of the ranges of the trace operators we refer the reader to Ohno-Shizuta-Yanagisawa [14] and Shizuta-Yabuta [22]. We are concerned with solutions of the problem (1) some components of which lie in //™(fl) while the others in H?{Cl) after certain transformation of unknown functions. Such a structure of solutions is known as the extra regularity in the literature [15], [20], [21], [22] and realized in the following function space. If L € C°°(tt) vanishes on I\ we have_Lu G ff™(fi) for any u € H?(Sl). Moreover, 7o (Lu] = 0 holds since C°°{n) is dense in H?(tl). FVom this observation the subspace of H?{Sl) determined from P e C°°(fi) by

depends only on the boundary value P = 7o[P]- We denote this space by Hp(tl). This is a Hilbert space with the norm

138

For u € Hf(Q) the trace -ya\Pu] € H m " , / 2 ( r ) depends only on P, which is denoted by (Pjo)\u\. The boundary condition of the problem (1) is described by using the closed subspace of tf£(fi) given by ^ ( a ) = {ue^(ft);(PTD)[a]=0}. Finally, we introduce several function spaces on intervals. All the spaces are Banach spaces. Let / be a finite open interval. We define

rn

j=0 m

j=0

In this definition we replace the spaces HJ(fi} with tf^fl), tf^(fi) and // t J arl (0) to obtain A\ m (7 ; n), n m ( / ; f l ) , H™(J;fi); * £ ( / ; « ) , *??(/; D), K™ (I;ft) and X™n(7;£l), Yt™n(I;i1), W^JI;U) respectively. Corresponding function spaces in R" are defined in the same way. F o r d = (cto,c*i,...,G n ) £ Z" + 1 we denote by d?on and d? the differential operators d£°d?1 ■•■d*lll(xn&R)ti* and «S-6ff°aj" ■ •■S£*r§£" respectively. For P g C°°(r) we put A'J?(7 ; n)= f | C m - J ( 7 ; ^ ( a ) ) .

3

Assumptions and main results

We state the main results in two theorems. One deals with the existence of solutions of first order regularity. The other is concerned with the higher order regularity of solutions. We make use of the first theorem to show the latter. The statements are given in such a way as they are applied to the problem in which the coefficient matrices lie in the same type of function space as that of solutions, the linearized problem of quasilinear equations kept in mind. Let fi be a bounded open set in R \ n > 2 with smooth boundary T. v{x) = ( { ^ ( i ) , . . . , ^ ( a : ) ) denotes the unit outward normal to V. Supposing that Aj{L,x)i 0 < f < n + 1 and Q{x) are /„ *
139

H. 1 Aj{t,x), 0 < j < n_are hermitian and A0(t,x) is positive definite at each point (t,x) G [0,T] x Q. There exists a positive constant KQ such that A0(t,x)

>K&I,

( t , x ) e [0,T] xfi.

H. 2 The subspace kerQ(x) is maximal nonnegative at each point (t,x) G [0T] x T, that is, the boundary matrix Av{t,x) = £ " ^ ( ^ ^ ( t . i ) is nonnegative on the subspace kerQ(ar) and any suhspace which enjoys this property and contains kerQ(x) must coincide with kerQ(x). H . 3 There exists a function P on T with values in l0 x (0 matrices such that kerA,{t,x)=kerP(x)holdsat each point (t,x) G [0,T]xF. The rank of P(x) is a constant k G (0,Z0) everywhere on T. H. 4 The rank of (?(x) is a constant l2 everywhere on T. Remark 1 As was proved in [8], H.2 implies ker>U(M) C kerQ{*),

(i,r) G [0,T] x T.

(5)

Remark 2 In the treatment of the equation of ideal magneto-hydrodynamics with a perfectly conducting wall condition under a certain constraint on the initial data the boundary matrix of the linearized equation is determined from the shape of fl only, and dose not depend on a parrtcular choice of functions about which the quasilinear equation is linearized (Yanagisawa-Matsumura [29]). Hence, the hypothesis H.3 and the assumption on the smoothness of P in the theorems below are not too restrictive in application, though the other types of hypotheses are possible if we confine ourselves to the linear equation (1) with smooth coefficients. Theorem 1 Assume that

tAseWUQiTi&mnL*(0)T-tC*.{Ti)),

\An+1 e H^(o,r;c°(n))nL~(o,:r;C!(f2})

0<j
w

and P,Q G C~(r). Then, for (f,F) G (H^il) n H^fii)) x W/.(0,T;O) the problem (1) has a unique solution m Xlp([0,T];ii). Theorem 2 Letm > 2 and put r = max{m,2[n/2] + 6}. We assume that Aj e y / I O J i f i ) , X

0 < j
(7)

and P,Q G C {T). Suppose that u G X$- ([0,T];ft) satisfies (I). Then, if F belongs to WR(0,T;fi) and fp = d?u(o) e ffjr^ft) n H£~p(fi), 0 < p < m - 1 , (8) wekaveueXp([0,Ty,Q).

140

It is worthwhile to mention the meaning of the boundary condition in (1). Let P(x) and Q(x) be the orthogonal projections to (kerPfx)^ and (kerQOr))1respectively. Since P(x) and Q(x) are of constant ranks on T and dependent on x smoothly, so are P(x) and Q(x). By (5) we have kerP(i) C kerQ(x) and hence Q(x) = Q(x)P(x). Therefore, Hp(Q) = Hf (Q) c Hg^(ft) = H | ( f t ) - H£{fl). This implies Xj?([0,T];n) C ^ ( | 0 , T ] ; 0). Thus, the condition "Qu = 0 on U T x T" for u € XW(IQ ZTfil makes sense by saying u(t) e H£(Q), 0 < t < T. By the continuity of the trace operator Q-ya it is also proved that a function u € Xp([0, T]; Q) with the boundary condition must satisfy (8). We may express fp in Theorem 2 as a linear combination of the derivatives of / and the values at t - 0 of the derivatives of F with coefficients in fo x (0 matrix-valued functions on SI. The relations between / and F given by (8) is called the compatibility condition of order m - 1. When m = 1 the compatibility condition is stated that / belongs to HlP{ii) f"l H^(ii). ShizutaYabuta [22] showed that if a function u € Ar.m([0,T];fi) satisfies the first equation in (1) with F e Wg{0,T;(l), it necessarily belongs to Xf(p,T];ii). Hence to solve the problem ("l) in the class X™([0,T\;ft) we must impose the compatibility condition on the data. The above theorems say that we can solve the problem (1) in the class Xff(%T\;f§ for any data satisfying the compatibility condition. In this paper, instead of proving the theorems themselves, we will present the ideas of the proofs using an equation with smooth coefficients in the half space. Let us consider the problem (1) in the half space R™. All the hypotheses H.l to H.4 are meaningful also in the case Q = R™. We write

A

Aj

(Af

A 2

\\

~ \A? Af)

with A)1 and Af, square matrices of order J, and /0 - {, respectively and Aj% = (AfY, an li x(lD-li) matrix. In addition to the hypotheses above the boundary matrix -An ii assumed to have the properries 1. A£ is not singular on [0,T] x R " " 1 ; 2. A™ = (Al'y

and Af vanish on [0,T] x R " " 1

We further assume that there exists a positive constant r<, such that VgYA$>&

[0,T]xRy.

(9)

141 The matrices P and Q are assumed to be of the forms

«"(*?)■ «-(*!)■ where Et is the identity matrix of order /. The relation (5) implies d > fe. As for the smoothness of the coefficients we put Ai£J5oo([0,3lxS|),

0<j
(10)

in place^of_(6) and (7), where Bm([0,r] x R£) is the space of functions on [0,T] x Rl whose derivatives with respect to the operators do,...,dn and xndn of order up to m are bounded and continuous on [0,T] x R™. We set # p ( R + ) = {u e fff (R£);P« 6 #.™(R+)} £Q(R+)

= {« 6 Hr(Rn+);Qu

X£([Q,T];R1) =

e Hr.(Rt),7o[Q«] - 0}

f)Cm-J([0,T];HUK))-

Then, all the statements in the theorems on the equation in Q = R™ make sense. In the sequel we write u 6 C'° as ' ( « / , u / / ) with it/ € C'1 and «n e C'o- i l . por tne sake 0f simplicity we assume that the support of the data (/, F) is compact, and so is the support of the solution by the finiteness of the speed of the propagation. 4

Existence of solutions of first order regularity

We solve the problem (1) by the method of non-characteristic regularization. Let i] be a positive parameter. We eonssder rhe approximattng problem to ((): n

^Ajdju

- i}dnu + An+lu-F

in

[0,T]xR£

Qu = 0

on

[0,T] x R " ' 1

u(0) = /

on R£.

The boundary matrix to the problem (11) i s ^ ( ( , x ) = -An{t,x)+r}I. As was proved by Schochet [19], A*{t,x) is regular and the subspace kerQ is maximal nonnegative at each point (t,x) € [0,T] x Rn_ 1 if JJ is small enough. Hence the problem (11) satisfies all the hypotheses in Theorem 1 but H.3, which is replaced by the hypothesis that the boundary matrix has full rank everywhere

142

on the lateral boundary. For such a problem the existence of solutions in the class X ^ f O T ] ; ^ ) is known. See Rauch Massey HI [18]. Making use of this fact, and the data (/,F) fixed in the space Hl(K) x ^ V . ( 0 , r ; R ^ ) , we first prove that the sequence of solutions to (11) remains bounded in A^([0,T];R£) as r} tends to 0. Next, by a sort of weak compactness method we find a solution to (1) in Xlp{[Q,T\;R1). Finally, by approximating the data the existence theorem in the general case is established. The uniqueness of solutions in the class X},([0,r];R^) follows from the standard energy estimate. The first step Surmose that the data if F) 6 Hl(TC\) x W} (0,T;R") satisfies Q~ro[f] - 0. If rj > 0 is small enough, (11) has a umque solution in Let us derive some uniform estimates of 3?u, a e Z" + 1 , \a\ < 1 and dnut with respect to the parameter r}. We first consider the case a = 0. By the hypothesis H.l the energy equality

dt{Mt)u(t),u(t))L2{RV n

-(A n (t)B(t),u(t)) £ a t R „_ l j + g(»(*li«(*))x«^-ij =

2%(u(t),F(t))L2{Rl)

holds. Since -An is nonnegative on kerQ, we have

<|^o(0)^2U(0)|L2{Rn)+ /

e^lJloM-^FWUnB.,*

Jo

with a constant A0 satisfying ^A0{t)-^{An+l(t)

+ An+}(ty

-f^djA^AoW-^

> X0I-

j=o

Henceforth we often make use of similar arguments to estimate solutions of various symmetric systems. In order to estimate 3?u, \a\ = 1 we use the mollifier Me in Appendix A. C h o o s e e 0 e ( 0 , T ) . F o r a e Z ^ 1 , \a\ < 1, 0 < e <e0 we put uE°

=d?{Mcu).

143

< , |a| = l belongs to X'([0,T-£o];R+) and satisfies the equation '

n

Y,AA<

+ > W i » ? - r,da.dnKUu = J °

,(?u f a - 0

in

[0,T - £„] x R^

on

[ 0 , r - £ o ] > Rn_1-

The forcing term Jf is expressed as Jf = Ja(u°,Fc),

where

n

and n

We derive the estimate of < as above and let e -> 0. Since u 6 ^ ( [ Q . T f c E ^ ) , we have M,u -» u in A M ([0,r - e 0 ] ; R + ) . By Lemma A 1 the commutators [Ajdj,Me]u, 0 < j < 7i and [ft,, ME]u tend to 0 in ^ ' . ( O . T - e 0 ; Rn+). Hence, {R} converges to F in IV,1.(0, T - E0\ R"). Consequently, we obtain

We have

< K^flfiMi 1 !*- + l«M"k-)lft.«/(*)k*(R;) + {\x;]d?Al?\L~ + +an(\xnlAlni\L-

\x-ld:A»\L~)\i;ndnul!(s)\LilK) \x;lA?\L-)\a?ull{,)\L>iK)

+

n —1

+\A0(s)-i/2d?F(s)\LiiRl).

(12)

To estimate the norm of dnu, on the right-hand side of (12) we use the equation n-1

i4Vft.li/ = ^ u , - £ j=0

n

y^ftu, - £ j=(t

A 2

] 9ju„

- A^u,

- A?+lu„

+ F,

144

together with (9) to obtain (co - r})\dnui{s)\LilRn+) T1-]

<X>"k™ia,"/(s)|^
+ Y,\A?\L~\djU,,{s)\LnRl)

+ Ix^

Axn2]L™}xndnu,i{s)\L,iRV

+Mn+lli»Kt/(s)liatR5.3 + l^n+jk»K/( s )li 2 (R^.) + l*/(s)| t = {B .,. Combining these estimates, then summing up those of 3 > for \a\ < 1, we get eAo' 1 ] |AoW 1/? 5?a{t)U> tR »j

< £ lAolOl'^tOJI^a,, WSJ

/fl

W<1

with constants M and M' independent of 77. Putting MO(01/23>(0IL=(R;),

E(f)= ^

F(0= E

|a[<]

l^otO-'^FWU^B-),

|„|<1

we obtain by Gronwall's inequality that E(0 < E(0)exp(-A,0 + M' j exp(-A1(i - s))F(s)ds

(13)

Jo

with Ai = AQ - M/K0. We have allo |3„U/(0IL= ( R;>

< A*"{ S

|3X0k= ( R';i + |F(()Ua(R«)}

|a|
with a constant A/" independent of n.

(14)

145

The second step Let u„ be the solution of (11) in Xl{[Q,T\;B$).

Since

n

J"=l 2

{5,u^{0)} converges in L (R£) as JJ tends to 0. Hence, from the estimates (13), (14) the sequence {«„} is bounded in H ^ ( 0 , r ; L 2 ( R ; ) ) n L « > ( 0 , T ; ^ ( R ; } n »Q(K))We apply Lemma B 1 in Appendix to {u„} and find a subsequence

{M^d"e^i(0,T;L2ro)n£»(OT;//MR+)ni%(R?)) lim um{t) = u(t)

weakly in

//j>(R") n

such that HQ(R+).

The convergence is uniform with respect to I 6 [0,T] and u(0) = / holds. u is a solution of (1) in Xj,([QtT\; R+). To show this we rely on some basic facts in functional analysis. Let E and F be normed spaces. B{E,F) denotes the space of bounded linear operators from E to F. We write B(£, E) = B{E). We define the linear operators A(t) and L(t), 0 < t < T by {^(()9)(x)-^0(t,x)5(i) (L(t)g)(x) = Y,AJ(t,x)dig{x)

+ .4 n+1 (t,;c)g(x).

Obviously, Ait) belongs to B(L 2 (R")) with bounded inverse and A(-), AM"1 e C O ( [ 0 , T ] ; B ( L 2 ( R : ) ) ) We express%)g as n-l

2

^ J ^ S + And

~ p)d»9 + A„dn(Pg) + An+1g

and notice that the operator S " ! , 1 A& + i4„(/ - P)3„ is tangential. Then we have L(t) € B(H},(Rl),^(Rl)) and L() 6 C°([0T];B(lfKR?),i 2 (H?))). WeshaUproveUeCi,([0,r];L2(R:))nC«([01Tl;HMR:)nf/^(R;))and A ( 0 a t < * ) + £(*)*(*) = **(*)

in

LHK)'

0
(15)

Pf*$ Let Q be a relatively compact and open subset of R^. For a func­ tion g on R^ the restriction of g onto ft is denoted by Rg. We have R 6 B{LHRl),LHQ)) n B(tf>(B^) t ff*(ft))- We define A(t) € B(JE?(fi)), 0 < ( < rby (i(t)g){2) = 4>(M)s(z).

146

A(t) is invertible and A(-), A{-)~1 6 C°([0,T];B(L 2 (n))). B(Hl(£l),L2{h)). From the equation (11) we have RdtUn (t) = RA(ty}(F(t)

We see 0 n €

- L(t)u,h (()) + rjjA-'WdnRu^

(<)■

The right-hand side converges to RA(t)-l{F(t) - L{t)u(t)) weakly in L2{f>) uniformly on [D,T]. Taking the weak limits of the both sides of R{ " w CO " / ) = / «ft«^ (T)*-, we obtain fl(u(() - / ) = / " R A ( r ) - l ( F ( r ) Jo and immediately

L(r)u(r))dr

« { u ( t ) - / - I A{T)-l{F{j)-L{r)u{T))dT}

=0.

Jo

Since fi is arbitrary, we get u{t) - f -

f

A{T)-1{F(T)

- L(T)u(T))dr

= 0.

Jo This shows that U e C&([0,r];L 2 (R^)) and (15) holds. D We can prove that u lies in XlP{[0,T]\Kl) by using the mollifier Mt. The detail of the proof will be given in [23]. The third step (1) has a unique solution u 6 X>([0,T];R^) for (f,F) & Hl(Rn+)x WU0,T;Rn+) with Q1(t[f] = 0. The estimates (13) and (14) are valid. Since $u{0) = Ao(0)-*(F(0) - t ( 0 ) / ) , the existence theorem in the general case is proved by approximating / 6 ff£(R£) n ff£(R£) by a se­ quence {/c;f > 0} in H 1 (R+) with -> u{x',xn + E). It is easy to see that ff = Pf + (I - P)Sef gives a desired sequence in tf'fR"). 5

Tangential regularity

We proceed with the proof of Theorem 2. In this section we show the tan­ gential regularity of order m of solutions. Let m > 2. Suppose that u 6 X p ^ t M i R " ) is a solution of (1) with F £ ^ ( O . T ; R£) and (8). For a € Z " + ! , |o| < m - 1 we put ua = S.ftu

147

By the assumption it is clear that ua e C°([0,T];L 2 (K n + )). We will show that a* |tt| = m-l belongs to X|,([0,ri;RJ). We first prove that ua is the strong solution to the equation Y,Aidi^a

+ An+lua

= Ja

Qua = 0 Lu°(0) = uQ(0)

in

[0,7] x RJ

on on

[0,r]xR"-1 R^

(16)

with the forcing term Ja given below in (18). Next, choosing suitable functions BnS, 0 6 Z^+1, \0\ = m - 1 and Ga on [0, T] x R£ with values in square matrices of order / 0 and Cio respectively, we show that Ja is of the form \0\ = m-\

By Theorem 1 the first order system for the unknown (w°; |a| - m - 1 ) n

Y,AAV° + An+lVa = Y, J=0

gwa = 0 lw a (0) = u°(0)

Ba3y

* + G°

in

[°'T1 X R +

on on

(0, T] x Rn - l R£

\0\ = m-

(17)

has a unique solution in the class XlP([0,7/]; RJ). This together with the energy estimate for the difference ua-va leads to the conclusion ua G A>([OtT];R£). In the sequel we let ej = (<5jJt) e Z f 1 , where Jjt is Kronecker's symbol. The first step Let Mc be the mollifier in Appendix A. Choosing e 0 e (0,T), we define for a £ Z^ + 1 , |Q| < m - 1, 0 < e < SQ u?

=dJ{Mru).

Then, u», |a| = m - 1 belongs to A'j?-'([0, J1 - £o];R+) and satisfies the equation ( " y * 4 j d i < + -A*+l< = J?

in

[0,r-eo]xRJ

,Qu? = 0

on

[0,T-£o]xR""'

with the forcing term given by JEQ = Ja{u°, Fe), where n

Ja(v,G)

= anAndr*"d„v

+ Yi[Ail8?]div 3=0

+ lA"+^d>

+d G

?

148

and Fs = Y}Aidi>

M u

^

+

M»+i, M r ]u + MCF

It is clear that u° converges to ua in Ca{[Q,T - f 0 ];R" ) as z -4 0. Putting Ja = Ja[u,F),

(18)

we shall prove that ua satisfies the equation (16) in the strong sense: KmJ?=Ja Proof: FOT(V,G) e W™~l(0,T » u . _ I ( 0 , r - e o ; R ^ ) we have

in

Ll(0,T-£Q;L2(K))-

(19)

- £0;R"+) x W,™"'^, T - £ 0 ;R£) with «/ E

|J'tt(u,G)|f,i(o.r-Eo;tI(R5.)) < a n ( K , U ~ + |^ n 1 | L -)|ar f "a„^l/.' ( o,T-, ( >;t=iR;)) + a n ( | i ^ I J 4 ^ | i o = + |i~ , A5 , | i -)|9f»//|£i ( o,r-io:t»(I^)l 11-1

+C 2 j l^lo^-'l^^lvv^-^o.r-co.Ri)

+ C ( | / 4 „ Ijgcn-nva + M n I B ' ™ - 1 ' " ) I W "IH'™-'(D,7'-iro;ft n )

+C , |i4„ +1 |a"-il l, lw™-"(o 1 r-t 0 iR5.) + l 9 rG|Li ( o,T-fo^=(fi;))We see Meu, -+ U / in W'™."»(0, T - £ 0 ;R^), Meuu -» »/; in V ^ " l ( 0 , r e 0 ; R+) as e -> 0. The commutators [ 4 ^ , M e ] u t 0 < j < n - 1, [A'nlan, Jtfe]uj, / = 1,2 and \A*?dn,Mc]uu, t = 1,2 tend to 0 in H T r ' ( 0 T - e n R " ) by Lemma A I (1), (2) and Lemma A 2 respectively Hence' ft -» F in »■/"-> ( 0 , r - E 0 : R ; ) . Combining these with the estimate of Ja(v G) we obt.Lin (19). □ The second step We shall derive the following expression of Ja: Ja =

Y.

Batiu» + Ga,

(20)

|(J|=m-l

where B^ are functions in S°°{[0, T] x FVJj taking the values in square matrices of order l0 and determined from Aj, 0 < j < n, and Ga is a C'°-valued function in WL{0,T',m.) determined from u and F.

149

To begin with we recall the definition (18) of Ja: n a

e

J = anAndr -dnu

+ £M;,0?]a,-u + [AH+U$*]lt + d?F.

In the first term of J° we rewrite the normal derivative dnu, by using the equation n-l

n

A^dnUr = - J 2 AyBjUi -^Afdjun j=0

Then, And°~^dnu

- A\}+tut

- A^uj,

+ Fj. (21)

j=0

is written as

- ii {*t IF) "a"en+ej+1*1 (4 1 4 ? ) * * + ( o ) (22) with

/<• - £^ 1 [(^ , )- 1 Aj 1 ,^- e -]a J -ii / + x:>i»[(^ 1 )- I ^ a ,fif- e -]^H +>l|113re"{«1)-,(F/-J4V+i«;-Aj12+lU//)}. [(^1)-1^1.a?--]*fU/ and [(AV)-Mja,fir«-ia,u//, 0 < j < n - 1 belong to JC!([0,TJ;R£), and so dose [(AJ,11"1 A\\d?-<"}dnus, because Aj,2 vanishes 1 1 2 o n [ 0 , T ] x R " - ' . Since ( A ^ ) - ( F / - A j l V i U / - A j 1 + 1 u / ) ) € A : r - I { [ 0 , r ] ; R ^ ,

wehave/«exif[o,r];R:)We express the next terms [Aj,d?]djU, 0 < j < n as

- ^ata^'Ajar^a^ + GJ. Furthermore, by virtue of (21) the term d:
(23) can be rewritten as

150

with T1-]

n

+

Y,9:iAili[(AiiriA}\drc'}djuI1

If, 0 < I < n are shown to belong to ^ { [ 0 , r | ; R J ) , as Ia is. G°, 0 < ) < n - 1 lie in Xl(\0 TVR1) We have also Ga € X'ffO T1;R") because a ^ S . " lie b J f F ( 0 T*R»1 by virtue of U / € V™*'([0 T h R ? ) and'so do d^'dlun by the f i t that both J& and A22 vanish on [0,71 x R V I . " A 1 S O M^.^^beJongsto^ffCr];^). All the matrices in (22), (23), 0 < j < n - 1, (24) operating to the tangential derivatives u", |0| = m - 1 lie in B°°([0,T] x R j ) because the matrices A|,2, J42,1 and A™ vanish on [0,7] x R n _ 1 . Thus we can express Ja like (20) with the function Ga <E 1*7,(0,7; R£) given by

G

° = a" Co) " £ Q ' ('o) + ^ G ? + ^»+i'3> + ^F.

The t h i r d step It is easy to see that the system (17) satisfies all the hypotheses in section 3. By (8), u°(0), |<*j = m - 1 belong to #]>(R+) n H i ( R ^ ) . We apply Theorem 1 to obtain the solution (va; \a\ = m - 1) of (17) in the class Aj,([0,T];R£). By the energy estimate it holds that ew\A0(t)]^(ua(t)-va(t))\L,{RV < fe*°'\M')-1/2(Ja(*)-

E

Bal3(s)v^s)-Ga{s))\

ds.

Substituting (20) into this, we obtain e^\A0(t)^(ua(t)-va(t))\L,iRV <

Yl \S\=m-i

\^1/2Ba"A~l/2\L«

fe^\A0(sy/H^(S)-vd(s))\LHRl)ds. J

°

151 Summing up the both sides for \a\=m\Aa(t)^2(ua(t) that is, ua(t) = va(t),0
1, we get by Gronwall's inequality

- va(t))\L,[RV

= 0,

\a\=m-

1,

This proves u« € A^([0, 7]; R j ) .

Regularity of normal derivatives

In the previous section we proved the tangential regularity of solutions, that is, u € -Y£n([0,r];R"). Since u, e J C r [ f l O . r ] . R l ) by *** assumption, we Prom these facts we derite the regularity of the normal derivatives of u. In this paper we only prove that fl?%«/€£"(0,7;!?($$))

(25)

for jorj = min{m + 1 - 2p,m - p}, 0 < p < {(m + l)/2j and

a^U//enoj;i2(Rt))

(26)

for |Q| = m - 2p, 0 < p < [m/2], which imply U / g K™(0,7(5+) and uu e r™{0,T;R") respectively. The strong continuity in L 2 of the derivatives will be shown in [23]. The following lemmata are crucial. L e m m a 6 . 1 Suppose that 1 < p < \{m + l)/2]. // flf^-'u//

E i-(0,T;L2(R:)),

|/J| = m - 2(p - 1),

we have d°d?ui a ° ° ( 0 , T ; L X ) ) '

\Q\

=m+\-2p.

L e m m a 6. 2 Suppose that 1 < p < [m/2]. If d?8*ui 6 £ « ( » , 7; L 2 T O ) ,

|/J| = m + 1 - 2p,

we have d?dpnun 6 £<*>(0,7;L*(R+)),

[«j - m - 2p.

We postpone the proofs of the lemmata and start the proof of (25) and (26). We proceed by induction with respect to the number p. When p = 0, (25) and (26) are nothing but the tangential regularity of u. Suppose that (25) and (26) are valid for p = q - 1 with 1 < q < [m/2]. By the hypothesis of inductton the assumption in Lemma 6.1 is satisfied with p = q. Hence (25) holds for p = q. This in turn implies the assumption in Lemma 6.2 with p = q and we have (26) for p = q. When m is even, the proof is complete. When m is odd, it follows

152

from Lemma 6.1 that fljfm+i>/a,«j £ L°°(0,r;£ 8 (R+)) and this completes the proof. Proof of Lemma 6.1 We operate d?d%~[ to (21) and express A^d^d^ui as thp sum of the following terms0<j
(27)

-Af&fdt^djUii,

0<j
(28)

[Af^tdr-^dju,,

0<j
(29)

-/Ij'a^-'^u/,

l

(30) (31) (32)

[j*j*,#ar i%««. o < j < n - i K2,^^-l]a««/i flf ajj-i ( F/ _ ^ + 1 u / - <2+1u/>)-

Since u/ e X. m ([0,r3;R^), (27) and (29) belong to C°([0,r]; Z. 2 (R^)). The fact that x~lAyz e S^ffO 71 x R i ) and the assumption imply that (28) belongs L - ( 0 , r ; t f < R J ) ) The term (30) lies in C([0,T];L 2 (R" + )), and so does (311 because A12 vanishes on fO 71 x R"" 1 It is easy to see that F] - W l B , - AH^/I e X - H I O V ] ; R : ) . Thus we conclude d^m € Proof of Lemma 6.2 Abbreviating u;a = d?dPuu, |a| = m - 2p, we prove wa € £.°°{0,T;L 2 (R^)) by three steps. Noting that |a| +p < m - 1 and hence the function wa is once differentiable, we first derive the equation n

Y^Af&jvf

+ A%+lW* =

j=0

51

C a i V + tfa in

[0,r]xRJ,

(33)

|/J|=m-2p

where C°^ are elements of B°°([0,7/] x R ^ ) with values in (fe - h) X (fo - h) matrices, and Ha is a C' 0 - ' 1 -valued function in L'fO,T; L 2 (R")). We remark that the matrix A22 vanishes on 10 Tl x R"" 1 Next multiolvine the eauation (33) by such a weight ff+> as the function p*V is sufficiently smooth up to the boundary, we derive the energy estimate for (f^w*. Finally, taking the limit along an appropriate sequence of p, we remove the weight from the estimate and then arrive at the conclusion wa £ L°°(0,T;L 2 {R n + )). The first step It is easily verified that wa satisfies the equation n

Y.Afdjw"

+ A22+,wa

=Ka

in

[0,T]xRJ

(34)

153

with

n

if" is expressed as n—1

n

n

-J2alx-1dft'A^wa-','+^

-pdnAfwa

+ Ha,

(35)

where Ha is the sum of the following terms: n

[Af.fifagjdj-u/i + ^Q,a, ( Mf9r e '^9jix//,

o < j < ™ -1

(36)

(=0

-Afdte&d&t,

0<j
(38)

[Af&dftdjU,, [^".flf^ft.u/

0<j
(39) (40)

-A^&d&i

Mn+i.WW. SZ&ZFn

(4i)

[>Ci. $?*£]*»

(«) (43)

M the matrices in (35) operating to u/3, \0\=m - 2p belong to B°°([0,r] x R£) since the matrix A™ vanishes on [0,T] x R"" 1 . The terms (36) to (42) belong to L°°(0,T;L 2 (R£)). As for (36) and (37) it follows from the fact that un £ Xr_I{[0,r]iR?) and Af vanishes on [0,7] x R"" 1 . Since x"1^1 G B°°([0 r l x RT) (38) belongs to L^(0,T:L 2 fR?)) by the assumption. Since a, e W l O n K), (39) belongs to C°([0,r];L* R£)), so dose (40) because ^ v a n ' i s h e s o n f O T l x R " - 1 Also (41) belongs to C°(|0 r i - i a ( R " ) ) Both the terms in (42) Thus wa satisfies the equation like (33).

154

The second step Let p be a smooth function from [0, oo) to [0, oo) satisfying 0 0 ,

p{0)=0,

(44)

0<rp'(r)
Multiplying both sides of (33) by the function p(x„)" , we have

3=0

=

{p+ p

y 'Al2(pP+1wa)+

Y,

P.

Ca0(pV+1wV) + p*+lHo

|j3|=m-2p

The tangential regularity of u implies (?+1wa €X1([0,Il;R"). Hence we are led to the energy estimate

Jo

2 1/2

Mo " ^4 ~ u»/'^ os K +1 ^( S ) 1/2 ^(s)u 2 ( R ^d 5

+ £

2 1/2

- 70

|3l=m-2p

+ /

AoI e

|p p+1 ^ 2 ( S )- 1 ^ J tf»( 5 )t L , (R .,d.s

with a constant AQ satisfying

Uf{t)-^{Af+l{t)

+ Al^itr

~ i^djAf(t))AlHt)-^

> X0I.

Here we use the fact that the matrix Af vanishes on [0,T] x Rn_1 and so dose the integration on the boundary. Summing up the above estimates for laI = m- 2p and putting Ep(t)=

£

|/> p+1 ^ 2 (t) 1/ V'(t)U*(R ! ,)

|o|=m—2p

we have

|a|=m—2p

EpW^E^OJ + Arifo"1 / e A o %(s)ds+ / ex°*Fp(s)ds Jo Jo

155 with a constant N independent of p. By Gronwall's inequality we get E,(*)<E / ,(0)exp(-A 1 t) + / e c p ( - A i ( t - a ) ) F , ( 8 ) d s

(45)

JO

with Ai = A0 - N/K0. The third step We choose a sequence of functions with the properties (44) monotone increasing and converging to 1 at each point r > 0. Since wa(0) e £ 2 (R") by (8), passing to the limit along the sequence of p in (45), we have wa{t)GL2{Rl)axid E(t) < E(0) exp(-Aj*) + / exp(-Ai(r. - s))F(s)ds, Jo

0
with

E(t> =

Y.

M2{t)l/2va(t)\L*mi)

\a\—m—2p

F(O =

Yl

\Ai2(ty1/2Ha(t)\L,{Rl).

This shows w a e I w ( 0 , r ; L 2 ( R ^ ) ) . D Appendix 4 . Mollifier Let 0 be areal valued C^-function on R n + 1 with support contained in {(x0, x)\ 0<xo <1,i*| < l . ^ n > 0} and /

„ 4>{y^y)dyody = 1,

0 > 0.

Let a, 6 and eo be constants with 0 < eo < 6 - a . Let t < p < oo. We deffne the linear operator Me, 0 < e < e0 from £,"(«,i>;L2(R")) to L P ( o , ^ £ 0 ; ; i ( R ^ ) ) by A/ f «(x 0l x',3; n ) = / / 0(yo,y',lfn)u(2:o + £3/o,z' +ey',x n e f V ")dy 0 dy'dy„. ./o J R ; The operator Af, was introduced by Rauch [16] in the study of first order systems with boundary characteristics. The operation of the mollifier has smoothing effects in the following sense.

156 Lemma A 0 (1) Let u g W™(a, b;R? ) (nap. W™(a, b; R£), W™ (a,fc;R£) ;, 1 < p < oo, m € Z+. Tfcerc, ^BI1Af,u € X m ( [ a , 6 - £ o ] ; R : ) (Vesp. *. r a ([o,6eo];R"), JC™([o. & - «o];R+) ; for any aeZf1. We Aaw lim Meu = u

in

W™ (a, 6 - e0; R+)

(resp. W ^ ( a , f r - c 0 ; R ^ ) , W£,(a,b-e0;K))

■

The assertions are valid when we replace the spaces W™(I;Rl), W™(I\R$) andWm f/-R") withXm(l-R") Xm(lRl) andXm(J-R"_) respectively (2) LeYu e W™(a b-Rl) }resp W™ (ab-Rl) ) 1 < p< oo m l N We Q m e thai I u] ~ 0 L & *n Via b-H™~^ R"""1)) " V ^ i » W e We tat several properties of commutators between first order differential operators and the mollifier. For the proofs see [23]. In wh&t follows ws assume 1