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Invent. math. 98, 511-547 (1989) I l l l}e l l f io~ le$matbematicae9 Spr i ng e r -Ve r l ag 1989

O rdinary differential equ ations, transport theoryand Sob olev spacesR . J . D i P e r n a 1 a n d P . L . L i o n s 2i D e p a r t m e n t o f M a t h e m a t i c s , U n i v e r s i t y o f B e r k e le y , B e r k e le y , C A 9 47 20 , U S A2 C e r e m a d e , U n i v e r s it 6 d e P a r i s - D a u p h i n e , P l a c e d e L a t t r e d e T a s si g n y , F -7 5 7 7 5 P a r i s C e d e x 1 6,F r a n c e

S u m m a r y . W e o b t a i n s o m e n e w e x i s te n c e , u n i q u e n e s s a n d s t a b i l i t y re s u l t s f o ro r d i n a r y d i f f e r e n t ia l e q u a t i o n s w i t h c o e f f ic i en t s i n S o b o l e v s p a c es . T h e s e r e su l t s a r ed e d u c e d f r o m c o r r e s p o n d i n g r e s u l ts o n l i n e a r t r a n s p o r t e q u a t i o n s w h i c h a r ea n a l y z e d b y th e m e t h o d o f r e n o r m a l i z e d s o l u t i o n s .C o n t e n t sI . I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111I. L in e ar t ra n sp o rt e q u at io n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514II.1 E x i st en c e a n d r e gu l ar iz a ti o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514II .2 U n iq u en e ss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51711 .3 E x i st e nc e o f r e n o r m a l i z e d s o l u ti o n s a n d s t ab i li ty . . . . . . . . . . . . . . . . . . . . 5 20I1.4 D u a li ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52911.5 S t ab il it y a n d t im e c o m p a c tn e ss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 30I II . A p p l i c a ti o n s t o o r d i n a r y d i ff er e nt ia l e q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . 5 32III .1 T h e d iv e rg e n ce free a u t o n o m o u s c as e . . . . . . . . . . . . . . . . . . . . . . . . . . 532II I.2 T h e g en era l a u t o n o m o u s c ase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5371II.3 T i m e -d e p en d e n t t he o ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539IV . C o u n te re x a m p le s a n d r em a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540IV .1 W l ' v v e c to r -f ie ld s w i t h u n b o u n d e d d i v e r g en c e . . . . . . . . . . . . . . . . . . . . . 5 40I V .2 D i v e r g e n c e f re e v e c t o r- f ie l d s w i t h o u t i n t e g r a b l e firs t d e r i v a t i v e s . . . . . . . . . . . 5411V .3 S m a l l n o is e a p p r o x im a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543IV .4 R e m ark s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

I . Introduct ionT h e f a m o u s C a u c h y - L i p s c h i t z t h e o r e m ( i n i t s g l o b a l v e r s io n ) p r o v i d e s g l o b a ls o l u t i o n s o f o r d i n a r y d i f f e r en t ia l e q u a t i o n s

f ( = b ( X ) f o r t E R , X ( 0 ) = x ~ N (1 )w h e r e b , s a y , i s L i p s c h i t z o n ~ N ( N > 1). T o s i m p l i f y m a t t e r s i n t h i s i n t r o d u c t i o n ,w e r e s t r i c t t e m p o r a r i l y o u r a t t e n t i o n t o s u c h a u t o n o m o u s c a se s . I n f ac t, t h e


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512 R.J. DiP erna and P .L . LionsC a u c h y - L i p s c h i t z t h e o r e m p r o v i d e s m u c h m o r e i n f o r m a t i o n t h a n t h e m e r e ex is t-e n c e a n d u n i q u e n e s s o f a s o l u t i o n o f (1 ) s i n ce it p r o v i d e s a u n i q u e c o n t i n u o u s f l o wX ( t , x) i.e. a u n i q u e c o n t i n u o u s f u n c t i o n X o n N x RN s a t is f y i n g ( 1 ) - - i n i n t e g r a lf o rm - - a n d X ( t + s ," ) = X ( t , X ( s , ' )) on A N, fo r a l l t , s~ A . (2)A n d t h e c o n t i n u i t y i n x o f X r e fl e c ts th e c o n t i n u i t y o f t h e s o l u t i o n u p o n i n it ia lc o n d i t i o n s , w h i c h i n f a c t c a n b e s t r e n g t h e n e d t o

]X (t , x l ) - X ( t , x 2 ) l< e C ~ f or t e A , x l , x 2 ~ A N (3)w h e r e C O is t h e L i p s c h i t z c o n s t a n t o f b. T h e s t a b i l it y o f X w i t h r e s p e c t t op e r t u r b a t i o n s o n i n i t i a l c o n d i t i o n s c a n b e a l s o m o d i f i e d t o t a k e i n t o a c c o u n ts t a b i l i t y w i t h r e s p e c t t o p e r t u r b a t i o n s o n b : f o r in s t a n c e , if b , c o n v e r g e s u n i f o r m l yo n c o m p a c t s e ts t o b, X , s o lv e s (1) w i t h b r e p l a c e d b y b , a n d X . is b o u n d e d o nc o m p a c t s e ts o f A x A N u n i f o r m l y i n n , t h e n X , ( t , x ) c o n v e r g e s t o X ( t , x) u n i f o r m l yo n c o m p a c t s e t s o f ~ x A N - n o t i c e t h a t b . d o e s n o t n e e d t o b e L i p s c h i tz . I n a llt h e s e s t a n d a r d r e s u l t s , m e a s u r e t h e o r y p l a y s n o r o l e . H o w e v e r , s i n c e o u r g o a l i s t oe x t e n d a ll th i s t h e o r y t o v e c t o r - f ie l d s ly i n g in S o b o l e v s p a c e s i n s t e a d o f b e i n gL i p s c h i t z , i t i s t h e n n a t u r a l t o a d d t h e f o l l o w i n g ( e a s y b u t n o s o s t a n d a r d )i n f o r m a t i o n a l s o d e r i v e d f r o m t h e C a u c h y - L i p s c h i t z t h e o r e m :

e - C l t 2 < ~ o X ( t ) < e C 1 ~ 2 for a l l t > 0 (4)f o r s o m e C 1 > 0 , w h e r e 2 i s t h e L e b e s g u e m e a s u r e o n N N a n d 2 o X ( t ) d e n o t e s t h ei m a g e m e a s u r e o f 2 b y th e m a p X ( t ) f ro m A N in to RN i.e .

c b d ( 2 o X ( t ) )= ~ ( a ( X ( t , x ) ) d x , V q b E ~ ( ~ u ) .~N ~u

S e v er a l p r o o f s o f (4) a r e p os si bl e: th e s i m p l e s t - - b u t t h e w r o n g o n e - - u s e s (2) a n d( 3 ) t o d e d u c e

[X (t, x i ) - X ( t , x 2 ) l > e - C ~ f o r a l l t > O , x l , x 2 e ~ N (5 )a n d t h u s X ( t ) is a L i p s c h i t z h o m e o m o r p h i s m f r o m ~ N o n t o ~ N sa t is f y in g (4) w i t hC o = C 1. A b e t t e r p r o o f - - b e t t e r s in c e it y i el d s a s h a r p e r e s t i m a t e a n d t h e c o r r e c te x p l a n a t i o n o f ( 4 ) - - i s b a s e d u p o n t h e f o ll o w i n g (s t a n d a r d ) o b s e r v a t i o n : le t Z(t)d e n o t e 2 o X( t ) , t h e n o n e c a n s h o w t h a t ) ~ (t) s a t is f i es in t h e s e n s e o f d i s t r i b u t i o n s

0 so ~ - d i v ( b s 0 o n (0, oo) x ~ u , s = 2a n d 2 - a d m i t s a d e n s i t y r w i t h r e s p e c t t o 2 w h i c h s a t i s f i e s

~ t - d i v ( b r ) = 0 o n ( 0 , ~ ) x ~ N , r l e = o - 1 o n ~ N . (6 )A n d o n e d e d u c e s e a s i l y

e - C ~ t < r ( t , x ) < e c~e o n ( O , o o) x ~ N (7)


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Ordina ry differential equations, transport theory and Sobo lev spaces 513w h e r e

C 1 = I l d i v b l t L ~ . 18)R o u g h l y s p e a k i n g , th e d i v e r g e n c e o f b g o v e r n s t h e e x p o n e n t i a l r a t e o f c o m p r e s s i o no r d i l a t io n o f L e b e s g u e ' s m e a s u r e t r a n s p o r t e d b y t h e f lo w .I t h as b e e n a p e r m a n e n t q u e s t i o n t o e x t e n d a n y p a r t o f t hi s e le m e n t a r y th e o r yt o l es s r e g u l a r v e c t o r f ie ld s b - - q u e s t i o n p e r t i n e n t t o a w i d e v a r i e t y o f a p p l i c a t i o n sr a n g i n g f r o m F l u i d M e c h a n i c s t o C o n t r o l T h e o r y . V a r i o u s ( s o m e w h a t l i m i t e d )e x t e n s i o n s h a v e b e e n p r o p o s e d b u t s e e m e d t o b e o f r e s tr i c te d a p p l i c a b i l it y i n v ie wo f s t a n d a r d e x a m p l es .

I t is o u r g o a l h e r e to p r o v i d e a q u i t e g e n e r a l (a n d n a t u r a l ) e x t e n s i o n t o v e c t o r -f ie ld s b h a v i n g b o u n d e d d i v e r g e n c e a n d s o m e S o b o l e v t y p e r e g u la r it y . O u rm o t i v a t i o n s t e m s f r o m k i n e t ic t h e o r y a n d f lu i d m e c h a n i c s ( se e f o r i n s t a n c e [ 5 ] , [ 6 ] )w h e r e s u ch q u e s t i o n s a r e f u n d a m e n t a l t o u n d e r s t a n d t h e " c h a r a c t e r i s t ic s " o f t h ep h y s i c a l s y s t e m a n d w h e r e o n l y l i m i t e d S o b o l e v r e g u l a r i t y s e e m s t o b e a v a i l a b l e .M o r e p r ec ise l y , w e w i ll sh o w t h a t i f b ~ W llo c l ~ N ) , di v b e L~'~(Nu ) an d

b = b l +bz, bleLP(RN), b e ( l + l x l ) L e L ~ 1 7 6N) ( fo r so m e 1 =


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514 R.J. DiPe rna and P. L . Lionst h e e v id e n c e o f s i n gu l a r p h e n o m e n a in 3 - D E u l e r e q u a t i o n s ( [ 7 ] ) o r e x is t en c er e s u l t s f o r d e n s i t y - d e p e n d e n t m o d e l s ( [ 6 ] ) .

I I . L in e ar t ran sp or t e q u at ion sH.1 Ex i s t ence and regular iza t ionW e b e g i n w i th a s i m p l e e x is t e n c e r e s u lt f o r t h e fo l l o w i n g l i n e a r t r a n s p o r t e q u a t i o n

u- - - b ' V x u + c u = O i n ( 0 , T ) (11)&w h er e T > 0 i s g i v en an d w e w i ll a l w ay s a s su m e t h a t b , c s a ti s fy a t l e a s t

b ~ L l ( 0 , T ; (L ~o c(~ N))N ), c ~ L l ( 0 , T ; L~oc(~N)). (12)G i v e n a n i n it ia l c o n d i t i o n u ~ i n L P ( ~ N ) f o r s o m e p e [ 1 , ~ ] , w e w i sh t o b u i ld aso lu t io n o f (11) in L~(O, T; LP(~N)) . O f c o u r se , th e e q u a t i o n w ill b e u n d e r s t o o d i nd i s t r i b u t i o n s s en se t h a t i s ( f o r i n s t an ce )

- ! d t [. d x u - [ . u ~ + d t [ . d x u { d i v ( b c ) ) + c c b } = 0 ( 1 3 )~ " N N 0 N N

f o r a ll t e s t f u n c t io n s 0 6 C ~ ( [ 0 , T ] x ~ N) w i t h c o m p a c t s u p p o r t in [ 0, T ) ~ N - - w e w ill d e n o t e th is s p a c e b y 9 ( [ 0 , T ) ~N ).

O b s e r v e h o w e v e r t h a t t h i s d e f i n i t i o n m a k e s s e n s e p r o v i d e d w e a s s u m ec + d i v b ~ L a( 0 , T ; L ~o o(~ u)), b~ L~ ( O , T ; (L~or ~) (14)

w h e r e q is t h e c o n j u g a t e e x p o n e n t o f p + - = 1 .PW i t h t h e s e n o t a t i o n s , w e h a v e t h e

P r o p o s i t i o n I L l . L e t p c [1 , oo] , u ~ e Lr(~ u) , assume (12), (14) andc + - 1 d iv b ~ L l ( 0 , T ; L ~([R N ))p i f p > 1 (15)c, d i v b ~ L a ( 0 , T ; L~(ff~N)) i f p = 1 .

Then, there e xis ts a solut ion u of (11 ) in L~(O, T; LP(~N ) ) corresponding to the ini tialcondi t ion u ~R e m a r k . T h e s a m e r e s u l t h o l d s i f w e r e p l a c e 0 i n t h e r i g h t - h a n d s id e o f ( 11 ) b y

f 6 L 1 0 , T ; LP(~N)).P r o o f T h e p r o o f c o n si s ts o n l y i n a j u s t if i c a ti o n b y a p p r o x i m a t i o n a n d r e g u la r iz -a t i o n o f t h e f o l l o w i n g f o r m a l e s t i m a t e s . F i r s t o f a ll , i f p = o o, o n e h a s f o r m a l l y b ys t a n d a r d a r g u m e n t s

I l u ( t ) l l~ _-< I l u ~ + i I I c u l l ~ d s0


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Ordina ry differential equations, transport theory and Sobo lev spaces 515h en ce i n v i ew o f (1 2)

Ilu(t)[l~ < Co[lU~ a.e. o n (0, T ) (16)w h e r e C o d e p e n d s o n l y o n t h e n o r m o f c in L I ( 0 , T ; L ~ ( ~ N ) ) . N e x t , i f p < c~ , o n eo b s e r v e s t h a t f o r m a l l y

0~ t lu lP - b ' V ~ l u f + p c J u f = 0a n d t h u s i n t e g r a t i n g t h i s e q u a t i o n o v e r ~ N o n e d e d u c e s

d ~ lu lPdx< l u f d x { llpc+divb ll~( t)} .T h e r e f o r e , u s i n g ( i 5 )

Ilu(t) G < Collu~ a.e. o n (O, T) (17)w h e r e C o d e p e n d s o n l y o n t h e n o r m o f c + - 1 d i v b i n L ~ (0 , T ; L ~ ( ~ N ) ) .P

N o w , t o p r o v e e x i s t e n c e , w e r e g u l a r i z e b , c , u ~ b y co n v o l u t i o n i n x i . e . w e~ u ~ p~ c ~ P ( i : ) P C ~ -@ + ([~ N ),o n si de r b ~ = b * p ~ , c ~ = c , p ~ , u~ = = ,

~ , , p d x = 1. S i n ce w e a s su m ed o n l y L~o~ i n t eg r ab i l i t y i n (1 2), a f u r t h e r a p p r o x i m a-t i o n b y t r u n c a t i o n is n e c e s s a r y th a t w e le a v e t o t h e r e a d e r a n d w e th u s a s s u m e t h a tb ~ L ~ ( 0 , T ; C ~ (~ N ) ) , c ~ L ~ ( 0 , T ;C ~ (E N ) ). T h e n , b y s t a n d a r d c o n s id e r a ti o n s ,t h e r e e x is ts a u n i q u e s o l u t i o n u ~E C ( [ 0 , T ] ; C ~ ( ~ ' ) ) o f

~u~ b . V ~ u ~ + G u = 0 i n (0, T ) x l /~ N, u~ l ,=o =U ~_ _ o in ~rT h e n , i n vi ew o f (1 6) a n d ( 1 7 ) - - e s t i m a t e s w h i c h c a n n o w b e p r o v e d r i g o r o u s l y - - , u ~is b o u n d e d i n L ~ ~ T ; L P (~ N )) u n i f o r m l y i n e . E x t r ac t i n g su b seq u en c es i f n eces s -a r y , w e m ay a s su m e w h en p > 1 t h a t u~ co n v e r g es w eak l y i n L ~ ' (0 , T ; L P ( ~ N ) ) an dw e ak l y * i f p = ~ t o so m e u . C h eck i n g t h a t (1 3) h o l d s i s n o w a s i m p l e ex e r c is e t h a tw e s k i p : r e m a r k o n l y t h a t

c~ + di v b~, b~ ---, c + d iv b, b in L 1 0, T; LqorW h e n p = l , t h e s a m e p r o o f a p p l ie s p r o v i d e d w e s h o w t h a t us is w e a k l y r e l a ti v e l yco m p a c t in L ~ ( 0 , T ; L ~o c(~ N )). I n o r d e r t o d o so, w e co n s i d e r u ~ u ) co n -v e r g in g i n L I ( ~ N) t o u ~ a n d w e d e n o t e b y u ,,~ t h e c o r r e s p o n d i n g a p p r o x i m a t e ds o l u t i o n s a s a b o v e .

B y t h e p r eced i n g a r g u m en t s , u s i n g ( 1 5 ) , w e see t h a t[lu,,~ IL~(0,T.L,(~')) < C ( n , p ) ( i n d ' o f e ) f or a ll p > 1

w h i l eN u _ u n tI IL r ) < ColIUo o o= - u , , , ~ l l ~ < C o l l U ~ l l ~ .


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516 R.J. DiP erna and P.L. Lions

A n d t h is y i e ld s t h e d e s i r e d w e a k c o m p a c t n e s s . AW e n o w t u r n t o t h e m a i n r e s u l t o f t h is s e c t io n : th i s r e su l t w ill s h o w t h a t , u n d e r

a p p r o p r i a t e c o n d i t i o n s o n b , ( w e ak ) s o l u t io n s o f ( l l ) c a n b e a p p r o x i m a t e d b ys m o o t h (in x ) s o l u t i o n s o f ( 1 1 ) w i t h s m a ll e r r o r t e rm s . T h i s r e s u l t w il l b e o n e o f t h ef u n d a m e n t a l t e c h n ic a l t o o l s r e q u i r e d t h r o u g h o u t t h e p a p e r. L e t Pc b e a r e g u l a ri z in gkerne l i . e .

1 ( - )p ~ = ~ p ~ w i th p e .@ + ( N ~ ), S p d x = l , e > 0 .T h e o r e m I L l . L e t 1 q . T h e n

(B" Vw) * p~ - B " V (w * p~) --* 0 in L~ o~ (~ N)s

w h e r e f l i s g i v e n in T h e o r e m I I . l .ii ) L e t B ~ L X ( O , T ; (W~o~(~N))N) , w ~ L ~( O , T; L~o~(~n)) ; t h e n

( B ' V w ) * p ~ - B ' V ( w * p ~ ) - - ~ O in LI(O, T ; L ~ o c ( ~ N ) ) .e

P r o o f P a r t i ) o f L e m m a II .1 s e e m s t o b e l o n g t o t h e f o l k l o r e o f r e a l a n a l y s is a n dt h u s w e w ill p r e s e n t a r a t h e r s k e t c h y p r o o f o f it. A n d w e w i ll e n t i r e l y sk i p t h e p r o o f


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O rdina ry d i fferent ia l equa t ions , t ran spo r t theory and Sob olev spaces 517o f i i) s in c e it r e q u i r e s o n l y t o r e p r o d u c e c a r e f u l ly t h e p r o o f o f i), k e e p i n g t r a c k o f t h et i m e d e p e n d e n c e . I n o r d e r t o p r o v e i ), w e fi rs t o b s e r v e t h a t( B - V w ) 9 P c - B " V ( w 9 Pc ) = - ~ w ( y ) [ d i v y { B ( y ) p c ( x - y ) } + B ( x ) ' V p ~ ( x - y ) ] d y

= ~ w ( y ) { ( B ( y ) - - B ( x ) ) " V p c ( x - y ) } d y - ( w d i v B ) * P c .B y s t a n d a r d r e s u l ts o n c o n v o l u t i o n s , t h e s e c o n d t e r m c o n v e r g e s i n L~or a s e g o e s t o0 t o w d i v B .

N e x t , w e e s t i m a t e t h e f ir s t t e r m a s fo l l o w s f o r e s m a l l e n o u g h

w ( y ) { ( B ( y ) - B ( x ) ) " V p c ( x - y )} d y L~IB ,) N C I I w IIL~r

B R + I i X y l < C c 13w h e r e B M d e n o t e s t h e b a ll o f r a d i u s M , R is f ix e d , a n d C d e n o t e v a r i o u s c o n s t a n t si n d e p e n d e n t o f e , R , w , B . T h e n , w e r e m a r k t h a t{ . , Ix ~ I B ( y ) - B ( x ) I d y =

B ~ Ix -y l < Ce '~d x ~ d z d t r V B ( x + t ~ z )l

B 1 I z l < C< C I IV B I[ Lo IB . . . . ~ -

I n o r d e r t o co n c l u d e , w e j u s t n e e d t o o b s e r v e t h a t i t i s n o w e n o u g h t o s h o w t h a tw ( y ) { B ( y ) - B ( x ) } 9 V p c ( x - y ) d y ~ w d i v B i n L~or

C

w h e n w a n d B a re s m o o t h . I n d e e d , t h e g e n e r a l c a se fo l lo w s b y d e n s i ty u s i n g t h ea b o v e b o u n d s . B u t , th i s c o n v e r g e n c e i s c l ea r if w a n d B a r e s m o o t h s in c e

I w ( y ) { B ( y ) - - B ( x ) } ' V p c ( x - - y ) d y ~ . - -w ( x ), .i = l~ Bj(x)'j'z~.,~jp(z)dz~-a n d

- - i ,J =1~ ~ i B j ( x ) " ~ z i ~ j p ( z ) d z = d i v B . A

H . 2 U n i q u e n e s sT h e o r e m I I . 2 . L e t 1 < p < ~ , l et u e L ~ ( 0 , T ; L P ( R s ) ) b e a so l u ti o n o f ( l l ) f o r t h ei n i ti a l c o n d i t i o n u ~ 0 (i .e . u s a t i s f i e s ( 1 3 ) w i t h u ~ 0 ). W e a s s u m e t h a t c ,d i v b ~ L l ( 0 , T ; L ~ ( ~ N ) ) , b ~ L l ( 0 , T ; W l~o~q(~N )) a n d

b- - e L l ( 0 , T ; L I ( ~ N ) ) + L I ( 0 , T ; L ~ ( ~ N ) ) . (2 0)1 + Ix [T h e n , u - O .


8/37

518 R. J . D iPe rn a and P .L . L ionsR e m a r k . I t w i ll b e c l e a r f r o m t h e p r o o f b e l o w t h a t ( 20 ) m a y b e s o m e w h a tr e l a x e d . . .

C o m b i n i n g P r o p o s i t i o n I I . 1 a n d T h e o r e m I I. 2 , w e i m m e d i a t e l y d e d u c e t h eC o r o l l a r y I L l . L e t 1 I w h e r eq~ e ~ + ( N N ) , S u p p q5 c B 2 , q~ - 1 o n B 1 . T h e n , w e m u l t i p l y ( 21 ) b y ~bR a n d w e f in d

d s f f ( U ) O R d X + ~ { c u f f ' ( u ) + d i v b f f ( u ) } c h R = - ~ f f ( u ) b ' V q ~ R . (2 2)L e t M e ( 0, o c ), w e w o u l d l ik e t o c h o o s e f f (t ) = ( Itl A M ) p w h i c h is L i p s c h i t z o n N b u tn o t C 1 : t h is p o i n t m a y b e o v e r s o m e b y te d i o u s a p p r o x i m a t i o n a r g u m e n t s t h a t w es k i p a n d w e d e d u c e f r o m ( 22 )

d C S ( /u [ /x M ) p Ib (t , x )l d x .d t S ( l u l A M ) P d p R d X < C ~ ( [ u l/ ', M ) P ~ ) a d X + ~ R < = I ~ I < = 2 RN e x t , w e o b s e r v e t h a t ( lu l /x M ) e e L ~ ~ T ; L ~ ~ L ~ 1 7 6 h i l e

Ib ( t , x ) l I R < I b ( t , x ) l 1R


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Ordina ry differential equations, transport theory and Sobo lev spaces 519W h e n p = o% s o m e f u r t h e r a r g u m e n t s s e e m t o b e n e c e s s a ry . F i r s t o f a ll, if

u 6 L ~ ( O , T ; L ~ n L ~) t h e p r o o f a b o v e a p p l ie s a n d y ie ld s th e u n i q u e n e s s . I n t h eg e n e r a l c a se , w e w i ll u se a d u a l i t y a r g u m e n t t h a t w e o n l y s k e t c h b e l o w ( si nc e w ewi ll dea l w i th m uc h m ore gen era l d ua l i ty r e su l t s l a t e r on) : l e t ~b ~ ~ ( ( 0 , T ) EN) , i ti s e n o u g h t o s h o w t h a t

TS u d x d t = o .0 0~

I n o r d e r t o d o s o, o n e c o n s i d e rs t h e s o l u ti o n o f th e f o l lo w i n g b a c k w a r d s p r o b l e mo ~ - b ' V r b - ( c + d i v b ) c b = O in (0, T ) x ~ N, 4 ' l , = r - - 0 o n ~ N .

B y P r o p o s i t i o n I I. 1, su ch a so l u t i o n 4~ ex i s ts an d is in f ac t u n i q u e b y t h e ab o v ep r o o f . F u r t h e r m o r e , ~b ~ L ~ ( 0 , T ; L ~ ~ L ~ ) .

N e x t , w e i n v o k e t h e r e g u l a r i z a t i o n r e s u l t T h e o r e m I I . 1 t o d e d u c ec3u~t 3 ~ - - b ' V u ~ + c u ~ = r ~ i n ( O , T ) v, u ~ l t= o = O o n WvOq~~ 3 - t - b ' V c b ~ - ( c + d i v b ) c b = 4 9 + ~ i n ( O , T ) u , r o n ~ u

w here r~, ~b~--*0 in LI (0 , T ;L~oc(~N)). M ul t ip ly in g the f i rs t e qu a t io n by t / l e ~ b R ,i n t e g r a t i n g b y p a r t s a n d u s i n g t h e s e c o n d e q u a t i o n w e f i n d

T T- - I ~ u ~ (q 5 + O ~ ) d p R d x d t + r ~ c I ) j a R d x d t + ~ ~ u~ 4 ) , b . V 4 ) R d x d t = O .0 NN 0 ~

L e t t i n g e g o t o O , w e d ed u ce]b] 1g N , r < p - 1 i f q = N , r = ~

(23)( 2 4 )

( 2 5 )


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520 R . J. D iPe rna and P .L . L ionsPP r o o f ( 2 4 ) is a n e a s y c o n s e q u e n c e o f ( 21 ) o b s e r v i n g t h a t f l ( u ) ~ L ~ ( 0, T ; ~locI~ 1 ( ~ N ) )

a n d u s i n g S o b o l e v i n e q u a li t i e s t o d e d u c e t h a t b " f l ( u ) ~L l ( O , T ; L ~o~( ~N)) i f (25)h o ld s . N e x t , t h e p r o o f o f T h e o r e m I I . 2 sh o w s t h a t

d ~ [ u f d x + ~ { p c + d i v b } [ u f d x 0 a . e . o n ( 0 , T ) ( 2 6 )T h e r e f o r e , N u ( t ) ] l p 6 C ( [ O , T ] ) a n d t h is i m p l i e s e a s i l y , i n v i e w o f ( 11 ), t h a tu ~ C ( [ 0 , T ] ; L P ( ~ U ) ) i f p > 1. T h e c a s e p = 1 i s s l i g h t l y m o r e d e l i c a t e : f i rs t o f al l,a p p r o x i m a t i n g u ~ b y u ~ ~ L ~ c~ L p ( f o r s o m e p > 1) a n d u s i n g ( 26 ) t o d e d u c e t h a t t h ec o r r e s p o n d i n g s o l u t i o n u , o f (1 1) c o n v e r g e s t o u in L 1( ~ u ) u n i f o r m l y o n [ 0 , T ] , w ea l r e a d y o b t a i n t h a t u ~ C ( [ 0 , T ] ; L ~ o ,( ~u )) a n d t h a t

s u p e s s S lu( t) l l lu( t) l__>Mdx~O a s M ~ . ( 2 7 )t~[O, T] ~uN e x t , w e c o n s i d e r ~ C ~ ( N N ) , 0 < { < 1, ~ --= 0 o n B 1 / 2 , ~ =- 1 i f Ix l > 1 a n d w ei n t r o d u c e ~R = { ( R x ) f o r R > 1. T h e n , c o p y i n g t h e p r o o f o f T h e o r e m I I.1 w e f i ndf o r a l l M > 0

_d ~ [ u I A M ~ R d x < C ~ ] u I A M ~ R d X + C ~ ]u [A M --]b [dt ~N n ~ R/2 < Ix[ -< R 1 + [X[T h i s y i e l d s

s u p e s s S l u lA M ~ R d x - ' O a s R ~ , f o r a l l M > 0 . (2 8 )r E [O , T ] ~ 'A n d w e c o n c l u d e c o m b i n i n g (2 7) , (2 8) a n d t h e f a c t t h a t u ~ C ( [ 0 , T ] ; L~oc). A

H .3 E x i s t e n c e o f r e n o rm a l i z e d so lu ti on s a n d s ta b i l i tyI n th i s s e c t io n , w e e x t e n d t h e r a n g e o f t h e e x i s t e n c e a n d u n i q u e n e s s r e s u l ts p r o v e ni n t h e p r e c e d i n g s e c t i o n s b y r e q u i r i n g l e ss i n t e g r a b i l i t y c o n d i t i o n s o n t h e d e r i v a -t iv e s o f b a n d t h e i n it ia l c o n d i t i o n s a n d w e p r o v e a f u n d a m e n t a l s t a b i l i t y r e su l t . I no r d e r t o s t a t e p r e c i s e l y o u r r e s u lt s , w e n e e d t o i n t r o d u c e a f ew n o t i o n s a n dn o t a t i o n s .

F i r s t o f a ll , t h e c o n d i t i o n s o n b , c w e w il l a s s u m e t h r o u g h o u t t h is s e c t io n ( a n dt h e f o l l o w i n g o n e s ) a r e

{ b ~ L (O , T ; W ~or d i v b ~ L l ( O , T ; L ~ ( ~ N ) )c ~ L ~ ( O , T ; L ~ ( ~ N ) ) , ' (* )]b(t, x )[ E L I ( 0 , T ; L I ( N N ) ) + L I ( 0 , T ; L ~ ( N N ) ) . ( ** )

1 + I x lN e x t , w e n e e d t o i n t r o d u c e a s e t o f f u n c t i o n s t h a t w e w il l d e n o t e b y L ~ ~ is t h e s eto f a l l m e a s u r a b l e f u n c t i o n s u o n N u w i t h v a l u e s i n ~ s u c h t h a t

m e a s { [ u [ > 2 } < ~ , f o r a ll ) . > 0 . (2 9 )


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O rdina ry d if fe ren ti al equa t ions , t r ansp or t t heory and Sobo lev spaces 521O b s e r v e t h a t w h e n e v e r f l ~ C ( ~ ) is b o u n d e d a n d v a n i sh e s n e a r 0 th e nf f ( u ) ~ L 1 c~ L ~ ( ~ N ) . W e w i l l s a y t h a t u " ~ u in L ~ i f f f( u~) --. f f (u ) i n L 1 f o r a l l s u c h f l

n n

a n d t h a t u " is b o u n d e d i n L ~ i f f f( u ") is b o u n d e d i n L ~ f o r a l l s u c h ft. I n t h is w a y , t h es e ts L ~ ( O , T ; L ~ C ( [ O , T ] ; L ~ a r e w e l l - d e f in e d . F i n a l l y , L ~ w i ll s t a n d f o r t h ec o r r e s p o n d i n g l o c a l v e r s i o n s ( i n f a c t L ~ 1 6 2s n o t h i n g b u t t h e s e t o f a ll m e a s u r a b l ef u n c t i o n s f r o m ~ N i n t o ~ ) .

W e n o w t u r n t o t h e n o t i o n o f r e n o r m a l i z e d s o l u t i o n s o f (1 1) . W e w i ll s a y th a tu 6 L ~ ( O , T ; L ~ is a r e n o r m a l i z e d s o l u t i o n o f (1 l ) i f t h e f o l l o w i n g h o l d s

O t f l (u ) - b " V f f ( u ) + c u f f ' ( u ) = 0 i n ( 0 , T ) ~ N ( 3 0 )f o r al l f f e C ~ ( ~ ) , f f a n d f l'( 1 + I tl) - x a r e b o u n d e d o n ~ a n d fl v a n i s h e s n e a r 0 . W ew i ll c a ll s u c h f u n c t i o n s f f a d m i s s ib l e f u n c t io n s . O b s e r v e t h a t t h e s e c o n d i t i o n s i m p l yt h a t

f l (u ) a n d u f l ' ( u ) ~ L ~ ( O , T ; L ~ ( ~ N ) ) .A n d , o f c o u r s e , u E L ~ (0 , T ; L ~ w i ll b e a r e n o r m a l i z e d s o l u t i o n o f ( l l ) c o r r e s p o n d -i n g t o t h e i n i t i a l c o n d i t i o n u ~ ( g i v e n ) in L ~ i f f f ( u ) s o l v e s ( 3 0 ) w i t h f l ( u ~ a s i n i t i a lc o n d i t io n f o r a l l/ 3 a s a b o v e . W e m a y n o w s t a t e o u r m a i n r es u lt s.T h e o r e m 1 1 . 3 . W e a s s u m e ( * ) a n d (**) .

1 ) ( C o n s i s t e n c y ) . L e t u 6 L ~ ( 0 , T ; L V ( ~ N ) ) a n d l et b ~ L l ( 0 , T ; L p ( ~ N ) ) w i t h1 < p < ~ . I f u is a r e n o r m a l i z e d s o l u t i o n o f ( 1 1 ) , t h e n u is a s o l u t io n o f ( l 1). I f u is as o l u t i o n o f ( 1 1 ) a n d b ~ L 1(0 , T ; W ~ J ( ~ N ) ) , t h e n u is a r e n o r m a l i z e d s o lu t io n .

2 ) ( E x i s t en c e a n d u n i q u e n e s s ) . L e t u ~ 1 7 6 t h e n t h e re e x i s ts a u n iq u er e n o r m a l i z e d s o l u t i o n u o f ( 1 1 ) i n L ~ ( O , T ; L ~ c o r r e s p o n d i n g t o t h e i n i ti a lc o n d i ti o n u ~ F u r t h e r m o r e , u 6 C ( [ O , T ] ; L ~ u 6 C ( [ 0 , T ] ;L P ( R N ) ) i fu ~ ~ L P ( ~ N ) f o r s o m e 1 < p < ~ a n d

u e L ~ ( O , T ; L ~ ( R N ) ) c~ C ( [ 0 , T ] ; L f o c ( ~ N ) ) (V p < ~ ) i f u ~F i n a l l y , t h e f o l l o w i n g i d e n t i t y h o l d s f o r a l l f l 6 C ( ~ ) b o u n d e d a n d v a n i s h in g n e a r 0

d ~ f f ( u ) d x + ~ c u f f ' ( u ) + d i v b f f ( u ) d x 0 , a .e . o n ( 0 , T ) . (3 1)d t ~= ~=T h e n e x t r e s u l t is a s t a b i l i ty r e s u lt w h i c h c o r r e s p o n d s t o th e c a s e w h e n c = 0 .

W e w i ll i n d i c a t e b r i e f ly a f t e r t h e p r o o f o f a ll th e s e r e s u l t s h o w s t a b i l i t y r e s u l ts m a yb e o b t a i n e d i n t h e g e n e r a l c a se b y a s im p l e t r ic k ( r e d u c i n g t h e g e n e r a l c a s e to t h ec a s e w h e n e = 0 ).T h e o r e m 1 1 . 4 . ( S t a b i l i t y ) . L e t b , , c , ~ L ~ ( 0 , T ; L ~oc) b e s u c h t h a td i v b . E L l ( 0 , T ; L lo ~ ) a n d b , , c , , d i v b , c o n v e r g e s a s n g o e s t o ~ t o b , 0 , d i v b( r e s p e c t i v e l y ) i n L I ( 0 , T ; L l t o ~ ) w h e r e b s a t i s f i e s ( * ) a n d ( * * ) ( w i t h e = 0 ). L e t u ~ b e ab o u n d e d s e q u e n c e i n L ~ ( 0 , T ; L ~ s u c h t h a t u = i s a r e n o r m a l i z e d s o l u t i o n o f ( 1 1 ) w i t h

0(b , c ) r e p l a c e d b y ( b . , c . ) c o r r e s p o n d i n g t o a n i n i t ia l c o n d i t i o n u ~ E L ~ A s s u m e t h a t u ,c o n v e r g e s i n L ~ a s n g o e s t o o o to s o m e u ~ ~ L ~


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522 R.J. DiPerna and P.L . Lions1) ( L o c a l co n v e r g en c e ) . T h e n , u . c o n v e r g e s a s n g o e s t o ~ i n C ( [ 0 , T ] ; L ~ to

t h e r e n o r m a l i z e d s o l u t i o n u o f ( 1 1 ) ( w i t h c = O ) c o r r e s p o n d i n g t o t h e i n it ia l c o n d i t i o no c onv e rge s t o u ~ i n L~ oc for so m e p E [1, oo), t h a t~ I n a d d i t io n , w e a s s u m e n o w t h a t u .

u" i s bo un de d i n L~ T ; L foc) , t ha t b . , c . , d i v b . are bo un de d i n L I (O , T ; L l~ oc) or{ lu"(t)lP /t ~ [0 , T ] , n > 1 } i s r e l a t i v e l y w e a k l y c o m p a c t i n L ~ o~ . T h e n , u . c o n v e r g e s t ou in C([0 , T ] ; Lfoc).

2 ) ( G l o b a l c o n v e r g e n c e ) . A s s u m e t h a t c . c o n v e r g e s t o 0 i n L a O , T ; L 1 + U ) ( f o rs o m e r < o o ) , t h a t d i v b . = fl~ . + f12 w he re f12 i s b ou n de d in L~ (O, T; L ~ an d f l lc on v e r ge s i n L ~ 0 , T ; L ~ , t ha t u ~ c onv e rg e s t o u ~ i n L ~ and t ha t u" sa t i s f ie s (31) w i t h( b, c ) r e p l a c e d b y ( b . , c . ) . T h e n , u . c o n v e r g e s t o u i n C ( [ 0 , T ] ; L ~ I n a d d i t i o n , w e

0 U oa s s u m e n o w th a t u . c o n v e r g e s to i n L P f o r s o m e 1 < p < o o, t h a t u" is b o u n d e d i nL ~ ( O , T ; L p ) o r c . = 0 , t h a t c , , d i v b . a r e b o u n d e d in L I (O , T ; L ~ o r { [ u " (t )l P /r e [ 0 , T ] , n > 1} is r e l a t i v e l y w e a k l y c o m p a c t i n L 1 . T h e n , u . c o n v e r g e s t o u i nc ( [ o , T ] ; L ~).R e m a r k s . 1) N o t i ce t h a t w e a r e n o t a s su m i n g i n t h e s t ab i l i ty r e su l t t h a t b . - ~ b i n

nL 1 0, T ; wlgc 1 ) .2 ) S i m i l a r r e s u l ts h o l d f o r e q u a t i o n s w i t h a r i g h t - h a n d s id e . AW e w i ll p r o v e T h e o r e m s I I .3 a n d T h e o r e m I I. 4 in s e v e r a l s te p s: fi rs t o f a ll , w e

p r o v e p a r t 1) o f T h e o r e m I I.3 a n d t h e u n i q u e n e s s s t a t e m e n t o f p a r t 2 ) i n th e c a s ew h e n c = 0 . T h e n , w e w i l l p r o v e T h e o r e m I I . 4 i n t w o s t e p s . N e x t , w e p r o v e t h ee x i s t e n c e s t a t e m e n t o f p a r t 2 ) i n T h e o r e m I I.3 . F i n a l ly , w e w i ll e x p l a i n h o w t or e c o v e r t h e g e n e r a l c a s e f r o m t h e c a s e w h e n c = 0 .

Step 1 . I n o r d e r t o p r o v e p a r t 1) o f T h e o r e m I I. 3, w e f ir st r e c a l l t h a t s o l u t i o n s o f(1 1) ( in d i s t r ib u t i o n s s e ns e ) a r e r e n o r m a l i z e d s o l u t i o n s o f (1 1) w h e n b ~ L l ( 0 , T ;W 1 j ) , a f a c t w h i c h h a s b e e n s h o w n a l r e a d y ( s e e C o r o l l a r y I I . 2 a n d ( 2 4 ) i np a r t i cu l a r ) . N ex t , i f u is a r en o r m a l i z ed so l u t i o n o f ( 1 1 ) an d u E L ~ (0 , T ; L p ) t h e n u isa s o l u t i o n o f (1 1): i n d e e d , o n e j u s t n e e d s t o c h o o s e a s e q u e n c e o f a d m i s s i b l ef u n c t i o n s f t , s u c h t h a t

[f l,( t)l ~ It] an d f t , ~ t un i fo rm ly on co m pa ct se t s o f ~ .n

T h e n , ( 11 ) f o l lo w s f r o m (3 0) b y e a s y m e a s u r e t h e o r y c o n s i d e r a t i o n s .N e x t , t h e u n i q u e n e s s a s s e r t i o n i n p a r t 2 ) o f T h e o r e m I I. 3 a l so f o ll o w s fr o m

T h e o r e m I I. 2 w h e n c -- 0 s in c e f l (u) i s the n a so lu t i on o f (11) in L~ (0 , T ; L 1 c~ L~176T h e r e f o r e , f l (u) i s u n i q u e an d s i n ce th i s h o l d s f o r a ll ad m i s s i b l e f l w e d ed u ce ea s i l yt h a t

u l o o > lu l > o = v l | a . e . , 1 ,= o = l v = o a . e . ,l u = = l v =

i.e . u = v a.e ., if u , v a r e t w o r en o r m a l i z ed so l u t i o n s . O b se r v e a l so t h a t a l l t h ec o n t i n u i t y i n t im e s t a te m e n t s a n d t h e i d e n t it y (3 1) c o n t a i n e d i n p a r t 2) o f T h e o r e m1 1.3 f o l l o w in t h e s am e w a y f r o m C o r o l l a r i e s I I.1 an d 11 .2 .


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O rdi na ry differential equations, tran spo rt theo ry an d Sobolev spaces 523Step 2 . Po intwise s tab i l i t yW e n o w w i s h t o sh o w , u n d e r t h e a s s u m p t i o n s o f p a r t 1) o f T h e o r e m I I. 4, t h a t u .c o n v e r g e s a . e . o n ( 0 , T ) x ~ N t o u o r t h a t f o r a n y a d m i s s i b l e f l , f l (u.) c o n v e r g e s a .e .o n (0, T) ~ to fl(u). W e t h u s f ix s u c h a / 3 a n d d e n o t e b y v . = f l (u . ) . O b s e r v e t h a tv . is b o u n d e d i n L ~ ( 0 , T ; L 1 c~ L ~ ) a n d s o l v e s

0 v .& b . " V x v . + c . u . f l '( u . ) = 0 in (0, T ) x ~N (32)o f l ( u~ R e m a r k i n g t h a t f 1 2 i s s ti ll a d m i s s i b l e , u s i n g t h e d e f i n i t i o nhi le v. It = o = v. =

o f r e n o r m a l i z e d s o l u t i o n s , w e s ee t h a t w . = v 2 ~ L ~ ( 0 , T ; L 1 ~ L ~ ) s o l v e s~ w . b . " V x w . + 2 c . u . v . f l '( u . ) = 0 in (0, T) ~N (33)c~t

a n d % 1 , = o = ( v ~ 2.W i t h o u t l o ss o f g e n e r al i t y , w e m a y a s s u m e t h a t v . a n d w . c o n v e r g e w e a k l y ( s a y

in L~ ' ( (O, T ) x ~ N ) * ) t o v a n d w e L l ' ( O , T ; L 1 ~ L ~ ( ~ ) ) w h i c h a r e s o l u t i o n s o f( 11 ) ( in d i s t r i b u t i o n s s e n s e) i n v i e w o f t h e a s s u m e d c o n v e r g e n c e s o f b ", c", d i v b ". I na d d i t i o n , v a n d w c o r r e s p o n d r e s p e c ti v e l y to t h e in i ti a l c o n d i t i o n s f l (u ~ a n d / 3 ( u ~ 2

o c o n v e r g e s t o u ~ i n L ~i n c e u .T h e n , i n v i e w o f p a r t 1) o f T h e o r e m I I. 3 , v is a r e n o r m a l i z e d s o l u t i o n o f ( 1 I ) a n dt h u s v 2 is a s o l u t i o n o f (1 1) c o r r e s p o n d i n g t o t h e i n it ia l c o n d i t i o n (v ~ 2 = / 3 ( u ~

i n d e e d , r e c a ll t h a t c - - 0 h e re a n d t h a t o n e j u s t h a s t o l et t h e a d m i s s i b l e n o n l i n e a r i t yi n (3 0 ) g o t o t 2. T h e r e f o r e , b y t h e u n i q u e n e s s r e s u l t T h e o r e m I I .2 , v 2 - w . B u t t h i sm e a n s t h a t

v , --* v 2 w e a k l y i n L ~ 1 7 6 T ) x ~ N ) _ .h e n c e v . c o n v e r g e s i n L 2 ( 0 , T ; L 2 oc ) t o v, t h e r e f o r e i n m e a s u r e . R e c a l l i n g t h a tv . = / 3 ( u . ) a n d /3 is a n a r b i t r a r y a d m i s s i b l e f u n c t i o n , w e se e e a s i ly b y v a r y i n g / 3a m o n g a c o u n t a b l e c o l l e c t io n o f s u c h a d m i s s i b le f u n c t i o n s fl k s u c h t h a t

/ 3 k = O if Itl _ - < ~ , 0 < B ~ , ( t) i f I t [ > ~/3; , (0(1 + I t l) -1, /3k a r e b o u n d e d o n R

t h a t u , h a s t o c o n v e r g e i n m e a s u r e t o s o m e u . B u t t h e n v , h a s t o c o n v e r g e t o fl(u).H e n c e v = / 3 ( u ) a n d u is a r e n o r m a l i z e d s o l u t i o n o f ( 1 1 ) c o r r e s p o n d i n g t o t h e in i ti alc o n d i t i o n u ~ ( a n d t h u s is u n i q u e ).

S t e p 3 . C o n c l u s i o n o f t h e p r o o f o f T h e o r e m 1 1.4T h e r e o n l y r e m a i n s t o s h o w t h a t c o n v e r g e n c e s a r e u n i f o r m i n t a n d g l o b a l w h e n t h ed a t a c o n v e r g e g l o b a l l y .

T h e u n i f o r m c o n v e r g e n c e s i n t f o ll o w s f r o m A s c o li ty p e a r g u m e n t s . I n d e e d , ifw e f i rs t f ix a n a d m i s s i b l e f u n c t i o n /3, w e k n o w b y s t e p 2 t h a t / 3( u,) a n d 7 (U n)


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524 R .J . D i P e r n a and P . L . L i on sc o n v e r g e r e s p e c t i v e l y t o f l(u) and 7(u) in LP (O, T; L for fo r a l l 1 ~ p < ~ w he re7 = / 1 2 . F u r t h e r m o r e , c h o o s i n g qSg a s in t h e p r o o f o f T h e o r e m I I.2 , w e se e th a t

d ~ 2 (u ,)(O R dX + ~ { c . u , y , ( u , ) + d i v b , y ( u , ) } 4 o R + y ( u , ) b . V ( a R d x O .d t ~ ~A n d o n e d e d u c e s e a s il y th a t

d [ 7 ( u " )O R d x - -* ~ { div b~ gR + b ' V O R } ) ' ( u ) d x i n L ~ (0 , T ) .n ~ NT h er e f o r e , s i n ce u is a r e n o r m a l i z ed s o l u t i o n o f (1 1),

f l ( u . ) ~ ( o , , d x - - , ~ f l ( u f c b R a x ,nNr," NN u n i f o r m l y i n [ 0 , T ]

a n df l ( u . ( t .) ) 2 ( O R d X + ~ f l( u (t) )2 d P R d X , i f t , ~ t i n [ 0 , T ] . (3 4)NN NN

O n t h e o th e r h a n d , f o r a n y b o u n d e d b a ll B R , o n e c h eck s ea s i l y u s i n g ( 30 ) t h a t f l ( u , )is r e l a ti v e l y c o m p a c t i n C ( [ 0 , T ] ; H - S ( B g ) ) f o r s o m e l a rg e s > 0 ( i n d e p e n d e n t o f n).T h e r e f o r e , i f t , V t i n [ 0 , T ] , f l ( u , ( t , ) ) 7 f l ( u ( t ) ) in H - S ( B R ) f o r a ll R < o % an d t h u sw e a k l y i n L 2 ( B R ) . Th en , in v iew o f (34), f l ( u . ( t . ) ) ~ f l ( u ( t ) ) in L 2 ( B R ) . S i n cef l (u)eC([O, T ] ; L P ) ( v 1 __< p < oo), this im pl ies th a t f l (u . ) ~ f l(u) i n C( [0 , T ] ; L2oc) .

O n e p r o v e s i n a s i m i la r f a s h i o n t h e r e m a i n i n g a s s e r t i o n s o f T h e o r e m I I .4c o n c e r n i n g t h e L p co n v e r g en c es a t l e a s t w h en p > 1 r ep l ac i n g f l (u , ) 2 b y [u,[ p, o r t h eg l o b a l c o n v e r g e n c e s ( a g a in w h e n p > 1). O n e j u s t h a s t o n o t i c e t h a t w h e n c , = 0 , L pb o u n d s m a y b e o b t a i n e d u s i n g ( 3 1 ) t o d e d u c e

d I { ( l u , , L - - 2 ) + A M } p d x 0 , M < o o. B u t t h e n t h e a s s u m p t i o n o n w e a k L ]o c c o m p a c t n e s s i m p l i estha t , fo r a l l R < oo , we have

s u p ~ l u . l l I . . I > > _ M d x ~ O a s M ~ + ~ . ( 3 6 )t, n B R

C o m b i n i n g (3 5) an d (3 6), i t is e a sy t o d ed u ce t h e c o n v e r g en ce i n C ( [ 0 , T ] ; L~o~).I n t h e g l o b a l s i t u a t i o n , w e w a n t t o s h o w t h a t , f o r a n y s e q u e n c e t . i n [ 0 , T ]

c o n v e r g i n g t o s o m e t , u . ( t . ) c o n v e r g e s i n L 1 t o u(t) . T h i s i s c l e a r l y e n o u g h s i n c eu e C ( [ - 0 , T ] ; L 1 ). S i n c e u . c o n v e r g e s t o u i n C ( [ 0 , T ] ; L ~ w e a l r e a d y k n o w t h a t


15/37

O rdina ry d i fferent ia l equa t ions , t ransp or t theory and Sobo lev spaces 525u , ( t . ) c o n v e r g e s t o u ( t ) i n m e a s u r e o r t o s i m p l i f y t h e p r e s e n t a t i o n a l m o s t e v e r y -w h e r e ( e x t r a c t a s u b s e q u e n c e i f n e c e s sa r y ) . O n t h e o t h e r h a n d , b e c a u s e o f (3 1) , w eh a v e

tne - " " " ) u . ( t , ) d x + ~ ~ { c . + d i v b . + a ' ( t ) } e - a ~ ~ d t d x = O ( 3 7 )

e - ~ d x + i ~ { d i v b + a ' } e - ~ d t d x = 0 t 3 S )[~N 0 ~N

f o r a n y f u n c t i o n a ~ W L ~ 0 , T ) . A n d w e c h o o s e a s u c h t h a ta ( O ) = O , a ' ( t ) + c , + d i v b , > O a .e . o n ( 0, T ) R N .

T h e n , b y F a t o u ' s l e m m a ( re c a l li n g t h a t u , , u > 0 ), w e d e d u c e t h a tu . ( t . ) d x ~ ~ u ( t ) d x .

~N ~NA n d r e c al l i n g a s t a n d a r d e x e rc i se i n m e a s u r e t h e o r y w e c o n c lu d e :

( 3 9 )

l u . ( t . ) - u ( t )l d x = ~ u . ( t . ) - u ( t ) d x + 2 ~ ( u . ( t . ) - u ( t ) ) - d xa n d ( u , ( t , ) - u ( t ) ) - ~ 0 in L 1 b y L e b e s g u e ' s l e m m a .

S t e p 4 . E x i s t e n c e w h e n e = 0I /-\W e

T ; W k ' ( ~ N ) ) f o r a l l k > 1 b e -P e @ + ( ~ N ) , ~ N p d x = 1 . T h e n , ( 1 + i x i 2 ) 1 / ~ E L I ( O ' ~c a u s e o f ( *) a n d ( ** ).

N e x t , c o n s i d e r fl k a s i n S t e p 2 a b o v e : w e w i l l i m p o s e i n a d d i t i o n t h a ti l k ' = Yk , , k ~ f l k f o r s o m e 7k ' , k e C I (J~) , f o r a l l k ' _ > k > 1. T h e n , w e d e n o t e b y u ~= f l k (UO) , Uk ,O = f l k (U O ) * p ~ f o r J => 0 ( w i t h t h e c o n v e n t i o n P o = 5 0 ). B y s t a n d a r dr e su l ts , t h e r e e x i s t s a u n i q u e s o l u t i o n u ~,~ i n L ~ ( 0 , T ; L ~ ( N N ) ) o f th e f o l l o w i n gp r o b l e m

, ~ u ~ , ~c3t b ~ ' V u ~ , ~ = 0 i n ( 0 , T ) x ~ N , u ~ k ,~ lt =o = u o k, ~ o n W v ( 40 )a n d u k , ~ W 1 ' ~ ( ( 0 , T ) x B R ) ( V R < ~ ) f o r 6 > O , u ok,~ c o n v e r g e s t o u ~ t o 0 ( s a y in

0 C l e a r l y , -~ o( [ 0 , T ] ; L~o~) a n d w e d e n o t e b y Uk . ~ = Uk , ~ . U k '~ = 7 k ' , k ( Uk , ~ ) s o l v e s( 0 ~ , ~ ) - b ~ ' V 0 ~ , ~ = 0 i n ( O , T ) U k , ~ = T k , k (U R,a ) o n

f o r a ll k ' > k , 6 > 0 . T h e r e f o r e , l e t t i n g 6 g o t o 0 + a n d c o m p a r i n g w i t h ( 4 0 ) w e s e et h a t

U k ' ~ = y k ' , k ( U k , ~ ) o n ( 0 , T ) f o r a l l k ' > k _ > 1 .


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5 26 R J . D i P e r n a a n d P . L . L i o n sF r o m t h is , w e d e d u c e t h a t t h e r e e x i s ts u , e L | ( 0, T ; L ~ s u c h t h a t Uk. e = f t k ( U e ) a n dt h u s u~ i s a r e n o r m a l i z e d s o l u t i o n o f (1 1 ) w i t h ( b , c ) r e p l a c e d b y ( b~ , 0 ) c o r r e s p o n d -i n g t o t h e i n i ti a l c o n d i t i o n u o . I n o r d e r t o c o n c l u d e t h e e x i s t e n c e p r o o f o n e j u s tn e e d s t o c h e c k t h a t u , is b o u n d e d i n L ~ ( 0 , T ; L ~ a n d t o u s e th e s t a b i l i t y r e s u l tl e t t i n g e g o t o 0 .

S t e p 5 . R e d u c t i o n t o th e c a s e w h e n c = 0L e t a e L l ( 0 , T ; L ~ ( ~ N ) ) . I n v i e w o f (* ) a n d ( ** ), t h e r e e x i s ts a u n i q u e s o l u t i o n

- - - b ' V q ~ = a o n (0 , T ) x ~ N , 4 ~ ] t = o = 0 O n ~ u (4 1 )&( se e C o r o l l a r y I I . 1 a n d I I.2 ).

T h e n , t h e r e d u c t i o n o f t h e g e n e r a l c a s e t o t h e c a s e w h e n c = 0 fo l l o w s. i m m e d i a t e l y f r o m t h e f o ll o w i ng .L e m m a I I . 2 . L e t ( b , c ) s a t i s f y (* ) a n d ( * * ) . L e t u ~ ~ T h en , u ~ L ~ ( O , T ; L ~is a r e n o r m a l i z e d s o l u t i o n o f ( 1 1 ) f o r t h e i n i t ia l c o n d i t i o n u ~ i f a n d o n l y i f e - ~ u isa r e n o r m a l i z e d s o l u t io n o f ( 1 1 ) w i t h ( b , c ) r e p la c e d b y (b , a + c ) f o r t h e i n it ia lc o n d i t i o n U o .P r o o f F o r m a l l y , t h i s is n o t h i n g b u t t h e c h a i n r u le a n d w e h a v e t o j u s t i f y th eo b v i o u s f o r m a l m a n i p u l a t i o n s . O f c o u r se , b y s y m m e t r y , i t i s e n o u g h t o s h o w o n ed i r e c t i o n o f t h e a b o v e e q u i v a l e n c e . H e n c e , l e t u ~ L~ 1 76 T ; L ~ b e a r e n o r m a l i z e ds o l u t i o n o f ( 1 l ) f o r t h e i n i ti a l c o n d i t i o n u ~ a n d l e t 7 , f l b e a d m i s s i b l e f u n c t i o n s . W ew a n t t o s h o w t h a t ~ ( e - ~f l (u ) ) = co s o l v e s

0o9- - - b " V c o + 7 ' ( e - ~ f t ( u ) ) e - ~ { c f t' ( u ) u + a f t ( u ) } = 0S t o n ( 0 , T ) x [ RN, col, = o = 7 ~ f t ( u ~ o n E N . ( 4 2 )

T o t h is e n d , w e a p p l y t he r e g u l a r i z i n g r e su l t T h e o r e m I I .1 a n d w e f in d th a t{ t ~ - b - V q ~ = a + t ) ~ in (0 , T ) x N ~ , 4 5 ~ l t = o = 0 O n N s (4 3 )o ~ N- b ' V v ~ + c f l ' ( u ) u = r , i n ( 0, T ) x I R N, v ~ l , = o = V ~ o n~ --_ 0 1 3 0w h e r e v = f t ( u ) , v ~ f t ( u ~ = * p ~ , a n d r

L I ( 0 , T ; L ~o ~).W e n e x t s e t r ~ = y ( e - ~,v ~) a n d w e m a y n o w u s e t h e c h a i n r u l e t o d e d u c e f r o m( 4 3 )

~tn_~ _ b " V ~ ~ = y ' ( e - ~ v ~ ) e - ~ {r~ - - c f t' ( u ) u - - a f t ( u ) - - ~b~ft(u)}Oti n ( O , T ) ~ N , o g * l , = o = ? ( v ~ ~ N .


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O rdina ry d i fferent ia l equa t ions , t ransp or t theory and Sobo lev spaces 527A n d ( 4 2) f o l lo w s u p o n l e t t i n g e g o t o 0 i n t h e p r e c e d i n g e q u a t i o n , c o n c l u d i n g t h u st h e p r o o f o f T h e o r e m s I 1.3 a n d I1 .4 . AR e m a r k s . 1) T h e o r e m I I .4 w i t h L e m m a I I .2 y i e l d s c o r r e s p o n d i n g s ta b i l i ty r e s u l tsf o r g e n e r a l e q u a t i o n s ( w i t h o u t a s s u m i n g c = 0 ).

2 ) I t i s p o s s i b l e t o e x t e n d s o m e o f t h e r e s u l ts a b o v e t o m o r e g e n e r a l i n i t ia lc o n d i t i o n s n a m e l y u ~ = { v m e a s u r a b l e f r o m R N i n to ~ } ( = L~ T h e n , w e m a yd e f i n e a d m i s s i b l e f u n c t i o n s fl a s fo l l o w s : f l ~ C l ( R ) , f l a n d f l ' ( l + I t [ ) a r e b o u n d e d .A n d w e t h e n u s e th e s a m e d e f i n i t io n a s b e f o r e f o r r e n o r m a l i z e d s o l u t io n s o b s e r v i n gt h a t i f u ~ L ~ ( 0 , T ; / 2 ) t h e n f l ( u ) e L ~ ( ( O , T ) x ~ i:~ ). L e t u s r e m a r k a t t h i s p o i n t t h a ti f u ~ e L ~ t h e n b o t h d e f i n i t i o n s a r e e a s il y s h o w n t o b e e q u i v a l e n t .

T h e n , t h e p r o o f s a b o v e s h o w t h a t t h e r e e x is ts a u n i q u e r e n o r m a l i z e d s o l u t i o n uo f (1 1) i n L ~ 1 76 T ; / 2 ) f o r a n y i n i ti a l c o n d i t i o n u ~ a n d u ~ C ( [ 0 , T ] ; s P a r t 1)( c o n c e r n i n g t h e L ~ = s c o n v e r g e n c e ) i s t h e n s ti ll t r u e .

o c o n v e r g e s in m e a s u r e l o c a l ly t o u ~ t h e nu r t h e r m o r e , i f lu ~ < o o a .e ., a n d u ,c h e c k i n g t h a t

s u p I { l u I ~ M } c ~ B R I ~ O a s M - - , o o ( f o r a l l R < o o )t 6 [ O , T ]

a n d d e d u c i n g f r o m t h a t f o r a ll R < ~ a n d e > O, t h e r e e x is t s M l a r g e e n o u g h s u c ht h a t f o r a l l n l a r g e

su p I{}u ,I > M } m B R I


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528 R . J. D iPe rna and P .L . L ionsS i n c e a l a r g e p a r t o f t h e p r o o f o f t h is r e s u l t is a n a l o g o u s t o t h e p r o o f o f0T h e o r e m I I .4 , w e s k e t c h it a n d c o n s i d e r o n l y t h e c a s e o f a n i n it ia l c o n d i t i o n u ,

w h i c h is b o u n d e d i n L 1 c~ L ~ a n d c o n v e r g e s i n L I , t h e r e f o r e u " is b o u n d e d i nL ~ ( 0 , T ; L ~ c~ L ~ ) . N e x t , b e c a u s e o f ( . ) a n d ( * *) , w i t h t h e n o t a t i o n s o f t h e p r o o f sa b o v e , w e d e d u c e t h e e x i s t e n c e o f u~ s o l u t i o n o f

O u ~ R - b ' V u ~ = f ~ i n ( 0 , T ) ~ NOtw it h u~ ], = o = q~Ru ~ w h e r e u ~ is s m o o t h in x ( u n i f o r m l y i n t), c o m p a c t l y s u p p o r t e di n x ( u n i f o r m l y i n t ) a n d f d ~ 0, u~ , u a s R ~ + 0o a n d t h e n e ~ 0 . I n

L I C ( L 1 )f a c t , u ~ i s n o t h i n g b u t ~ bR u~ w h e r e u~ is o b t a i n e d u s i n g t h e r e g u l a r i z i n g r e s u l tT h e o r e m I I . l . N e x t , w e w r i t e

~ ( u " - u ~ ) - b , " V ( u " - u ~ ) = (b - b , ) V u ~ + f d .S i n c e u n i s a r e n o r m a l i z e d s o l u t i o n s a t is f y i n g (3 1) , o n e d e d u c e s f r o m t h e a b o v ee q u a t io n a n d a t e d i o u s a p p r o x i m a t i o n a r g u m e n t

d~ ~ [ u " - u ~ l d x < E~ [ b - b . ] ] V u ~ l d x + E~ I fd ]d x + ][divb. (t )]JL~ ~,'~ ]u "- u~ ]d x .H e n c e , s e t t i n g A n ( t ) = i IL div b , (s )[ IL ~ ds, w e d e d u c e

0

[0, T ] ~N ~;,'T+ ~ e - A " ( ~ I ] b -- b .] I V u ~ l + I f ~ l d x0 ~ N

o r , in v i e w o f t h e b o u n d s o n ( d iv b , ) , ,

s u p ( d t u " - u ~ l d x ) ( ~ ) < =C o ~ lU ~ -d pR U ~[0, T] Ez~T+ C o S ~ d t d x { [ b - b . l t V u ~ ] + f f ~ ] }0 j~N

f o r s o m e c o n s t a n t C o i n d e p e n d e n t o f n , e , R .H e n c e , w e h a v e

, i r a C S C S[0, T] \ Er [0, T] KENT+ C O~ ~ d t d x I [ ~],0 ~ N

a n d w e c o n c l u d e l e t t i n g f i rs t R g o t o + 0o a n d t h e n e g o t o 0 . A


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O rdina ry d i fferent ia l equa t ions , t rans po r t theory and Sobo lev spaces 529H . 4 D u a l i t yT h e o r e m I I . 6 . L e t ( b , c ) s a t i s fi e s ( * ) a n d ( * * ) , l e t u ~ L ~ ( O , T ; L P ( ~ N ) ) ,

- P 1 ' b e r e s p e c t i v e l y r e n o r m a l i z e d s o l u -e L ~ ( O , T ; L q ( N N ) ) w i t h 1 < p


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530 R.J. DiPe rna and P. L. LionsF i n a l ly , in o r d e r t o c o n c l u d e w e j u s t h a v e t o l e t fl c o n v e r g e t o i l l( t ) = t ( i m p o s i n gth a t ]fl(t)[ < t, Ifl'(t)[ _-< 1 o n R). A

W e c o n c l u d e t h i s s e c t io n w i t h a n e x a m p l e o f t h e p o s s i b le a p p l i c a t i o n s o f t h ed u a l i t y f o r m u l a t o w e a k c o n v e r g e n c e a n d s t a b i li t y r e su l ts . It c a n a l s o b e a p p l i e d t or e c o v e r th e " s t r o n g " s t a b i li ty r e s u l ts o f t h e p r e c e d i n g s e c ti o n .C o r o l l a r y I I .3 . L e t b , c o n v e r g e t o b in L ~ 0 , T ; L~ o r b e s u c h t h a t d iv b , i s b o u n d e d inL~(0 , T ; L ~ ) . L e t u . b e a r e n o r m a l i z e d s o l u t i o n o f ( 1 1 ) w i t h ( b, c ) r e p l a c e d b y (b . , 0)

o W e a s s u m e in a d d i t io n t h a t b , s a t i sf ie s (**) a n d t h a t u ~o r a n i n i t i a l c o n d i t i o n u , .c o n v e r g e s w e a k l y in L P ( ~ U ) f o r s o m e p ~ ( 1 , + c o ] t o s o m e u ~ T h e n , u , c o n v e r g e sw e a k l y i n L ~ ( 0 , T ; L p ) t o t h e r e n o r m a l i z e d s o l u t i o n o f ( 1 l ) ( w i t h c = O ) b r t h e i n i t i a lc o n d i t io n u ~P r o o f O n e f ir s t o b s e r v es t h a t , b y t h e r e su l t s o f th e p r e ced i n g sec t i o n s , u , i sb o u n d e d i n L ~ T ; L P ( ~ N ) ) . N e x t , w e c o n s i d e r t h e s o l u t io n o f

8 c l ) , b , ' V q ) , - d i v b . 4 ) , 4) i n ( O , T ) N, ~ , [ t = r 0 o n ~ N&w h er e 0 is g i v en i n @((0 , T ) x R N ). B y t h e s t ab i l i t y r e su l t T h eo r e m I I .4 ( an d t h erem ark s fo l lo w ing it s p ro o f ) 4 ), co nv erg es in C ( [0 , T ] ; U ) fo r a l l 1 _


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Ord i na r y d i f f e ren t i a l equa t i ons , t r anspo r t t heory and Sob o l ev spaces 5312 ) I n g e n e ra l , T h e o r e m I I . 7 i s m o r e g e n e r a l t h a n T h e o r e m I I .4 a n d , e v e n i f i t

s e e m s a r a t h e r t e c h n i c a l e x t e n s i o n o f T h e o r e m I1 .4 , t h e g a i n i n g e n e r a l i t y w i ll b eq u i t e i m p o r t a n t f o r a p p l i c a t i o n s . O b s e r v e i n p a r t ic u l a r t h a t ( ** * ) h o l d s a s s o o n a s~b, i s b o u n d e d i n U ( 0 , T ; X ) w h e r e :~ > 1 , a n d X is a n y s p a c e w i t h a c o m p a c te m b e d d i n g i n L ~ o r i n s t a n c e a n y S o b o l e v s p a c e W " ' p w i t h p > 1, m > 0 h a s t h isp r o p e r t y . &P ro o f o f Th eorem I I . 7. S i nc e i ts p r o o f i s v e r y m u c h s i m i la r t o t h e p r o o f o f T h e o r e mI I .4 , w e o n l y e x p l a i n t h e n e w i n g r e d ie n t : t o t h i s e n d , w e t a k e c , = 0 a n d w e c o n s i d e rv " b o u n d e d i n L ~ s o l u t i o n o ft , x

(~Vn- - - d i v ~ ( b , v ' ) = 0 i n ~ ' ( ( 0 , T ) x ~ u )O tw r i t in g t h e e q u a t i o n i n d i v e r g e n c e f o r m a l l o w s t o s i m p l i f y a b it t h e p r e s e n t a t i o n ,a v o i d i n g t o k e e p t r a c k o f t h e e x t r a " d i v e r g e n c e " t e r m p r e s e n t in t h e ca s e o f t h e n o n -d i v e r g e n c e f o r m e q u a t i o n . O f c o u r se , w e m a y a s s u m e t h a t v" c o n v e r g e s w e a k l y inL , t o s o m e v a n d w e w a n t t o p r o v e t h a t v s a t is f i es

- - - d i v ~ ( b y ) = 0 i n ~ ' ( ( 0 , T ) E N ) ;o r in o t h e r w o r d s t h a t b,v" c o n v e r g e s w e a k l y ( in @ ' o r i n L 1) t o by.

T o t h i s e n d , w e i n t r o d u c e a r e g u la r i z in g k e r n e l a s i n t h e p r o o f o f T h e o r e m I I.1a n d w e o b s e r v e t h a t ( * ** ) y i e l d s i m m e d i a t e l y

b , ( v " . p ~ ) - ( b , v ' ) * p ~ - - * O i n L L ( 0 , T ; L ~ o ~ ) ,a s e ~ 0 + , u n i f o r m l y i n n .

~ ( v ' * p ~ ) = ( d i v x ( b , v " ) ) * p ~ i s c l e a r l y u n i f o r m l y ( i n n )i n c e b o u n d e d i nL 1(0 , T ; L~oc), w e d e d u c e e a s il y f r o m t h e c o m p a c t n e s s o f S o b o l e v e m b e d d i n g s t h a tf o r e a c h f i x e d ~ > 0

v ' . p ~ v * p ~ a . e . i n ( 0 , T ) x ~ N( e x t r ac t in g a s u b s e q u e n c e i f n e c es sa r y ) . T h i s a l m o s t e v e r y w h e r e c o n v e r g e n c e c o m -b i n e d w i th t h e u n i f o r m b o u n d s o n v ' * p ~ a n d t h e w e a k * c o n v e r g e n c e o f b , inL 1(0 , T ; L ~ oc ) y i e ld s t h e d e s i r e d c o n v e r g e n c e n a m e l y

b , ( v " . p ~ ) ~ b ( v * p ~ ) w e a k l y in L l(O , T ; L ~oc ), f o r a ll ~ > 0 .I n d e e d , c o l l e c ti n g a ll t h e a b o v e c o n v e r g e n c e s a n d t h e o b v i o u s o n e

b ( v . p ~ ) ~ b v i n L l ( O , T ; L ~ o c ) , a s ~ O + ,w e d e d u c e e a s il y t h a t w e h a v e

b,v" ~ by w e a k l y i n L I ( 0 , T ; L ~ or A


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532 R.J. DiP erna and P, L Lions

I I I . A p p l i c a t i o n s t o o r d i n a r y d i f f e r e n t i a l e q u a t i o n s

I I I .1 T h e div e rg e n ce f r e e a u t o n o m o u s c a seW e c o n s i d e r i n th i s s e c t i o n t h e c a s e w h e n b d e p e n d s o n l y o n x a n d s a ti sf ie s

b ~ wlio~1 ( ~ N ) , d i v b = 0 a . e . o n ~ N , ( 4 8 )b- - ~ L ~ + L ~ 1 7 6 ( 4 9 )

1 + I x lI n s o m e o f t h e e s t i m a t e s b e l o w w e w il l s t r e n g t h e n (4 9) a s f o l lo w s

b ~ L P + ( 1 + I x l) L ~ f o r s o m e p c [ l , o o ] . (5 0)W e a r e g o i n g t o s h o w t h e e x i s t e n c e a n d u n i q u e n e s s o f s o l u t i o n s o f (1). S t a b i l i t y

r e s u l t s w i ll b e g i v e n i n t h e n e x t s e c t i o n i n a m o r e g e n e r a l s i t u a t i o n . B u t , if w ea s s u m e o n l y (4 9) , t h e s o l u t io n m a p X ( t , x ) w e w i ll o b t a i n w i ll n o t b e i n L ~ oc ( f o r af ix e d t ) s o w e h a v e t o d e f i n e s o l u t i o n s o f (1) i n a m a n n e r s i m i l a r t o r e n o r m a l i z e ds o l u t i o n s o f (1 ): w e w i ll s h o w t h a t X ( t ) e C ( ~ ; L ) N w h e r e L = {05 m e a s u r a b l e f r o m~ N i n t o ~ a n d ]051 < o o a.e .} ( f o r e x a m p l e ) e n d o w e d w i t h t h e d i s t a n c e

1d ( 0 5 , ~ ) = ~ > 2 - - ; t I I 0 5 - - 0 t A l I I L , ( B . )n = l

w h i c h c o r r e s p o n d s t o t h e c o n v e r g e n c e i n m e a s u r e o n a r b i t r a r y b a ll s. In a d d i ti o n ,b e c a u s e o f ( 4 8 ), X w i l l s a ti s f y

2 o X ( t ) = 2 , f o r a l l t e R ( 5 1 )( re c a ll t h a t 2 is t h e L e b e s g u e m e a s u r e a n d t h a t 2 o X ( t ) is t h e i m a g e m e a s u r e o f 2 b yX (t) i.e.:

4 d ( 2 o X ( t ) ) - - ~ 0 5 ( x ( t ) ) d x .~N ~N

B e c a u s e o f ( 5 1 ), 05 o X ( t ) m a k e s s e n s e i n L f o r a ll 05 E L . T h e O D E ( 1) w i ll h o l d i n t h ef o l l o w i n g s e n s e : f o r al l f l ~ C l ( ~ N, ~ N ) s u c h t h a t f l a n d I D f l ( z )l ( l + I z l ) a r e b o u n d e do n ~ N , f l ( X ) E L ~ ( R ; L ~ o c a n d w e h a v e

~ f l ( X ) = o n x f l ( X ) l ,= o = f l( x ) o n (5 2)f f ( X ) ' b ( X ) R u , R u

w h e r e t h e e q u a t i o n h o l d s i n d i s t r i b u t i o n s s e n se . W e w i ll a l s o c al l a d m i s s i b l ef u n c t i o n s s u c h f u n c t i o n s ft. N o t i c e t h a t b e c a u s e o f ( 49 ) a n d (5 1) ,b ( X )- - ~ L ~ L 1 + L ~1 76l + lX lF i n a l l y , th e g r o u p p r o p e r t y w i ll n o w h o l d i n t h e fo l lo w i n g s e n s e

X ( t + s , ' ) = X ( t , X ( s , ' ) ) a .e . o n ~ N , f o r a l l t, s e R . ( 5 3 )W e m a y n o w s t a te o u r m a i n e x i s te n c e a n d u n i q u e n e s s r e s u lt .


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O rdina ry d if fe ren ti al equa t ions , t r anspo r t t heory and Sobo lev spaces 533Theorem I I I . 1 . W e a s s u m e ( 4 8 ) a n d (49). T h e n , t h e r e e x i s t s a u n i q u e X ~ C ( ~ ; L )~s a t i s f y i n g (51) , (52) , (53) . I n a d d i t i o n , X s a t i s f i e s

ri(X )c Lloc(~u; C([~)), fo r a l l a d m i s s i b l e r i ( 5 4 )O xf o r a lm o s t a l l x 6 ~ N , X e C l ( ~ ) , b ( X ) e C ( R ) a n d ~ t = b ( X ) o n ~ . ( 5 5 )

F u r t h e r m o r e , i f U o ~ L ~ ( o r s ), u ( t , x ) = u o ( X ( t, x ) ) i s t h e u n i q u e r e n o r m a l i z e ds o l u t i o n i n C ( R ; L ~ o f ( l 1) w i t h c = O , o r t h e i n i ti a l c o n d i t i o n u ~ ( f o r a l l T ) . F i n a l l y ,i f b s a t i s f i e s (50) , t h e n X ~ L Po r C ( R ) ) .

P r o o f . S t e p 1 ( E x i s t e n c e )W e r e g u l a r i z e b a s u s u a l : s e t b~ = b 9 P c. B y C a u c h y - L i p s c h i t z t h e o r e m , t h e r e e x i s tsa u n i q u e s m o o t h m a p X ~ o n R R u s a t i s fy i n g

0 X ~ -b(X~) o n R x ~ N , X ~ l , : o = X o n ~ N . (56)?,tI n a d d i t i o n , (5 1) a n d (5 3) h o l d f o r X ~ ( o f c o u r s e ( 53 ) h o l d s n o w e v e r y w h e r e ) a n d f o re a c h u ~ ~ ( o r i n s u ~ is t h e u n i q u e ( r e n o r m a l i z e d ) s o l u t i o n o f

~u~ - b , ' V u ~ in ~ x ~ N , u , l , = o = U ~ o n ~N . (5 7 )0 tI n p a r t i c u l a r , X ~ s o l v e s

0 X ~ - b ~ 'V X ~ in ~ x ~ N , X ~ l , = o = X o n ~ N . (58)a tN e x t , c h o o s i n g r io(Z) = z (1 + I z I 2 ) - 1 / 2 L o g ( 1 + ]z] 2 ) f o r z e ~ ? e w e d e d u c e t h a t

~ r i o ( X ~ ) = b ~ ' V ( r i o ( X ~ ) ) = V r i o ( X , ) ' b ( X ~ ) i n IR x R N . ( 5 9 )

S i n c e ] V r i o ( z ) ' b ( z ) ] < C ]b (z )J w e d e d u c e f r o m ( 49 ) a n d (5 1 ) t h a t: 1 + ]z l 'c~~ ( r i o ( X ~ ) ) is b o u n d e d i n L ~ ~ L * + L ~ 176 n d b e l o n g s

t o a r e la t iv e l y c o m p a c t s e t o f L ~ 1 7 6 T , T ; L ~ ( B R ) ) ( V R , T < o o ) ( 60 )I n p a r t i c u l a r , r i o ( X ~ ) is b o u n d e d in L ~ 1 7 6 L~ orW e m a y t h e n u s e t h e s t a b i l i ty r e s u l ts t o d e d u c e t h a t X ~ c o n v e r g e s a s e g o e s t o 0 ,i n C ( [ - T , T ] ; L f f (V T e (0 , m ) ) t o X w h i c h s a t is f ie s (5 2 ). I n a d d i t i o n , c h o o s i n g f ir s tu ~ i n ~ ( ~ N ) , u s i n g t h e s t a b i li ty r e s u lt s a n d t h e n a p p r o x i m a t i n g g e n e r a l u ~ in L ~ w es e e t h a t f o r a l l u ~ e L ~ ( o r / 7 ) , u ~ is t h e u n i q u e r e n o r m a l i z e d s o l u t i o n o f ( l 1 ) w i t hc _= 0 f o r t h e i n i ti a l c o n d i t i o n u ~ I n p a r t i c u l a r , w e d e d u c e t h a t f o r a ll u ~ e ~ ( ~ N )

S u ~ t , x ) ) d x = ~ u ~ V t ~


24/37

534 R.J. DiPe rna and P .L . Lionst h e r e f o r e (5 1) h o l d s . T h e u n i q u e n e s s o f r e n o r m a l i z e d s o l u t i o n s a l s o y i e ld s th e g r o u pp r o p e r t y (5 3).

Because of (51) , (52) y ields0O - ~ f l ( X ) e L ~ ( N ; L 1 + L ~ ) . (61)

Sinc e L~ (I~; L 1 + L ~) ~ L~oc(Nu; L~oc(N)), we d ed uc e tha tf l ( X ) e L ] o c ( ~ u ; W ~ g c '( ~ ) ) ~ L~oc(~u ; C (~ ) ) .

Then , (60) a l so y ie lds0

L ~f l o ( X ) e L '~ (N ; + L ~ ) (6 2)L 1 t~N.f r o m w h i ch w e d ed u ce a s ab o v e t h a t f l o ( X ) e ~oc~ , C ( R ) ). I n p a r t i cu l a r , fo r

a l m o s t a l l x s N N , p o ( X ) ~ C ( N ) a n d s in c e t ~ ~ L o g ( 1 + t 2) is s t ri c tl yx / l + t ~i n c r ea s i n g o n [ 0 , o o ) w e d ed u ce t h a t X 9 C(N) .

N e x t , w e s h o w t h e c la i m c o n t a i n e d i n (5 5) a b o u t t h e t i m e c o n t i n u i t y o f b ( X ) f o ra l m o s t a ll x e N N. I n o r d e r t o d o so , w e f ir s t ch o o se t ) s C ~ N ) su ch t h a t 0 > 0 o n R ,0 i s ev en an dt p (l zl )l D b = ( z ) [


25/37

O rdin ary differential equations, trans po rt theo ry and Sobolev spaces 535a n d e x a c t l y a s b e fo r e w e d e d u c e t h a t , f o r a l m o s t a l l x e ~ N , t p ( X ) f l o ( b ( X ) ) isc o n t i n u o u s o n R a n d t h u s, in v ie w o f t h e p r e v i o u s p r o o f s , b ( X ) is c o n t i n u o u s o n R .

A t t h i s s t a g e, p r o v i n g t h a t t h e O D E h o l d s f o r a l m o s t a l l x ~ ~ N is e a s y : u s e (5 2 )i n i n t e g r a l f o r m , le t fl g o t o t h e i d e n t i t y m a p p i n g , u s e th e a .e . i n x t e m p o r a lc o n t i n u i t y t o d e d u c e t he i n te g ra l f o r m o f t h e e q u a t i o n . . .

W e c o n c l u d e S t e p 1 b y s h o w i n g t h a t X e L ~ 'o r C ( ~ ) ) i f b s at is fi es (5 0) i.e.b = b I + b 2 w he r e b~ eLP([~N) , b 2 (1 + I x l ) -* ~ L ~ ( ~ N ) . T h e n , w e h a v e i f p < o c ( t h ec a se p = c ~ i s e a s i e r )

~ X ~ _ b ~ ( X ~ ) + b ~ ( X ~ )&h e n c e

o r se t t i ng Y~ = e - o [X~lc~t =

~ Y~ < C e ~ 1 7 6L e t t i n g e, g o t o 0 , w e f i n d t h a t

~ 7 < C e - C ' + e C' lb~(X)L.- c , X e N . L fo ~(~ N ; C ( ~ ) ) a n d o u r c l a i m i s p r o v e n .n pa r t i c u l a r e I I L f o r , W ) o 'S ( ~ ))

O b s e r v e t h a t t h e a b o v e p r o o f a ls o y ie ld sX ~ Lfo~(~N; W ~od(R)) 9 (66)

S t e p 2 . ( U n i q u e n e s s )I n o r d e r t o p r o v e u n i q u e n e s s , w e j u s t h a v e t o p r o v e t h a t , i f X s a ti sf ie s t h ec o n d i t i o n s l i s te d i n th e u n i q u e n e s s s t a t e m e n t a n d i f Uo ~ ~ ( ~ u ) , t h e n u o ( X ( t , x ) ) ist h e s o l u t i o n o f (1 1 ) w i t h c = 0 c o r r e s p o n d i n g t o t h e i n i ti a l c o n d i t i o n u o . S i n c e U o i sa r b i t r a r y , t h i s y i el d s o f c o u r s e t h e u n i q u e n e s s . H e n c e , w e se t u ( t , x ) = u o ( X ( t , x ) )a n d w e w i s h t o s h o w t h a t u - - w h i c h b e l o n g s to C ( ~ ; L f o c ( ~ N ) ) fo r a l l 1 < p < ooa n d t o L ~ 1 7 6 L P ( ~ N ) ) f o r a ll 1 < p < c ~ - - s a t is f i es ( 1 1 ) i n d i s t r i b u t i o n s s e n s e .

I n o r d e r t o d o s o, w e w r it e f or a ll ~ ( ~ N ) , h > 0 , t ~1a h ( t ) = j" ~ { u ( t + h , x ) - u ( t , x ) } q , (x ) d x

~N1= ~ ~ { u o ( X ( t + h , x ) ) - u o ( X ( t , x) )} O ( x ) d x

~N

a n d s i n c e X s a t i s f i e s t h e g r o u p p r o p e r t y , w e d e d u c e1& ( t ) = ~ ~ { u o ( X ( t , X ( h , x ))) - u o ( X ( t , x ) ) } q , ( x ) d x .

~N


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536 R.J. DiP er na and P.L. Lion s

A n d u s i n g t h e g r o u p p r o p e r t y a n d t h e m e a s u r e i n v a r i a n c e o f X ( h ) , t h i s y ie ldsAh(t) = ~ u(t , z) {O (X ( - h , z ) ) - 0 ( z ) } d z . (67)

NNN e x t , w e o b s e r v e t h a t b ( X ) ' V t p ( X ) ~ L ~ ( R ; L 1) an d t h a t f o r a l l ad m i s s i b l e f u n c -t i o n s f l

~ O ( f l ( X ) ) = V t ) ( f l ( X ) ) ' D f l ( X ) ' b ( X ) on I~ x NNa n d l e tt in g f l c o n v e r g e t o t h e i d e n t i t y m a p (a s w e d i d s e v e r a l ti m e s b e f o r e ) w ed e d u c e

~ t O ( X ) = b ( X ) ' V t k ( X ) o n N x N N .I n p a r t i c u l a r , w e h a v e

hO ( X ( - h , z)) - t) (z ) = - y b ( X ( - a , z))" V O ( X ( - a , z ) ) d z .0

I n s e r t i n g t h i s e x p r e s s i o n i n ( 6 7 ) a n d u s i n g o n c e m o r e t h e g r o u p p r o p e r t y a n d t h em e a s u r e i n v a r i a n c e o f X ( a ) w e f i n a l l y o b t a i n

Ah(t) = -- y { b ( x ) 'V O ( x ) } " ~ ! u (t + a , x ) - u ( t , x ) d a d x .N N

S i n ce b ' V O e L ~, u i s b o u n d e d i n L ~ ( N ; L ~ ( N N ) ) a n d u e C ( N ; L f o r ( fo r1 =< p < o o), w e d ed u c e f r o m t h i s ex p r e s s i o n t h a t

A h(t) ,--* -- S b ( x ) ' V O ( x ) u ( t , x ) d x u n i fo r m l y f o r t b o u n d e d .NN

S i n c e , o n t h e o t h e r h a n d , w e h a v e o b v i o u s l yAh (t) , ~ S u (t , x ) O ( x ) d x in ~ ' ( N ) ,

~Nw e f i n a l l y o b t a i n t h e d e s i r ed e q u a t i o n (1 i ).R e m a r k s . l ) I n t h e u n i q u en ess s t a t em en t , i t is p o s s i b l e t o r ep l ace ( 52 ) b y ( 5 5) an d , i nf a c t, o n e m a y s h o w t h a t ( 55 ) i m p l i e s (5 2) ( u n d e r t h e a s s u m p t i o n s o f t h e T h e o r e mI I I . l ) . I n t h e ca se w h en ( 5 0 ) h o l d s , i t i s ev en p o ss i b l e t o r ep l ace ( 5 5 ) b y

~ X- - = b ( X ) in ~ ' ( R x N N ) , (6 8 )s ince in tha t case X e C(N; L~oc) .

2) U n d e r t h e a s s u m p t i o n s o f T h e o r e m I I I. 1, w e d o n o t k n o w o f a n y e s t i m a t e o nt h e d i s p e r s i o n D x X ( t , x ) e x c e p t f o r t h e f o r m a l f o l l o w i n g o n e : d i f f e r e n t i a t i n g t h eO D E w i t h r e s p e c t t o x , w e f i n d f o r m a l l y

~ D x X = D b ( X )" D x X


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Ordinary differential equations, transport theory and Sobolev spaces 537h e n c e

0~ t { g o g l D x X l } < I D b ( X ) l ,T h e r e f o r e , i f D b e L P ( N N) f o r s o m e p e [ 1 , o o ] , w e d e d u c e

I lLoglDxXl l lL , < Cl t l , f o r a ll t e n . (6 9)N o t i c e t h a t w h e n p = o o t h is y ie l d s t h e u s u a l e x p o n e n t i a l r a t e fo r d i s p e rs i o n . A

111.2 The general autonomous caseI n t h i s s ec t i o n , w e r ep l ace t h e co n d i t i o n ( 4 8 ) b y

b e W jlo ~t(E ~'), d i v b e L ~ ( E N ) . (7 0)O f co u r se , t h i s w i ll a f f ec t t h e p r o p e r t y s t a t ed i n (5 1) n a m e l y t h e i n v a r i an c e o f 2 b yX ( t ) a n d i n s t e a d w e w ill o b t a i n f o r s o m e C o ~ [ 0, ~ )

e-C~ 0 an d f o r al l t e e

e-Colt l

T h e n , w e h a v e t h ed p d x < ~ 4 ) ( X ( t , x ) ) d x < = e c~ ~ ~ ) d x .

T h e o r e m I I I . 2 . W e assume (70) and (49). Then, the same conclusions as in TheoremIII .1 hold provided condi t ion (51) i s replaced by (71).R e m a r k . T h e u n i q u e s o l u t i o n X ( t , x ) sat i sf ies in fact (71) w i th C o < N iv b l i t ~(w~Nj.Pr oo f o f T heorem 111 .2 . S t e p 1 o f t h e p r o o f o f T h e o r e m III.1 m a y b e r e p e a t e dw i t h o u t a n y c h a n g e s ; h o w e v e r , t h e u n i q u e n e s s p r o o f ( st e p 2) h a s t o b e m o d i f i e d ab i t. I f w e f o l l o w t h e p r o o f g i v en i n s t ep 2 ( k eep i n g t h e s am e n o t a t i o n s ) an d u se (7 1)ins tea d o f (51), we o b ta in fo r a l l t~ E , h > 0 , r e c~ (E N)

1 z ) ) d z- ~ u ( t , X ( h , x ) ) O ( x ) d x - ~ u ( t , x ) r= h ~ ~< C ( e c ~ 1)114'IIL, ~ N )

a n d f r o m t h i s , w e d e d u c e l e t t i n g h g o t o 0Ou_ _ _ d i v ( b u ) e L ~ ( N ; L ~ ( E N ) ) ;&


28/37

538 R . J. D iPe rn a and P .L . L ions1 i b ( X ( - - ~ r , z ) ) ' V O ( X ( - a , z))d~r i s b o u n d e d i n L 1, u n i -o b s e r v e i n d e e d t h a t ~ o

0 uf o r m l y i n t e g r a b l e a n d c o n v e r g e s i n L~oc t o b ( z ) . V t p ( z ) ) . T h e r e f o r e , i f w e s e t F = - -c~t- b . V u , w e a l r e a d y k n o w t h a t F E L ~ ( N ; L ~ ( N N ) ) a n d w e w a n t t o s h o w t h a t Fv a n i s h e s .

W e t h e n u s e t h e re g u l a r iz i n g r e s u l t ( T h e o r e m I L l ) t o d e d u c e t h a tc3u~0 t ~ - b ' V u ~ = F + r ~ in N x N N

w h e r e r~ V 0 i n L~ oc(N x N N ). T h e n , w e i n t r o d u c e ~bn a s in t h e p r o o f o f T h e o r e m I I . 2a n d w e o b s e r v e t h a t0~ ( u ~ 4 ) n ) - b ' V ( u ~ d ~ n ) = F + r ~ - b ' V 4 ) R i n N x N N .U s i n g t h e r e g u l a r i t y o f u~ ( a n d (5 2)) , i t i s n o w e a s y t o i n t e g r a t e t h is e q u a t i o n " a l o n gt h e c h a r a c t e r i s t i c s X " i n o r d e r t o f i n d

(qSau~)(t , X ( - t , x ) ) - ((a Ru ~)(x, 0) = i { qSR(F + r ~ ) - b 'V ( ~ R u ~ } (a , X ( - a , x ) ) d a0

a .e . x e N N , f o r a l l t e ~ .W e t h e n l e t e g o t o 0 , u s i n g (7 1), a n d w e o b t a i n

( c~ n u) (t, X ( - t , x ) ) - ( ( o , u ) ( x , 0 ) = i { ~ b R r - b V q ~ R u } ( a , X ( - - ~ r , x ) ) d c r .0

T h e n , l e t t i n g R g o t o o o , u s i n g ( 7 1 ) a n d ( 4 9 ) , t h i s y i e l d stu (t, X ( - t , x ) ) - u (x , O ) = ~ F ( a , X ( - a , x ) ) d a a . e. x e N N , f o r a l l t e N .0

B u t t h e l e f t - h a n d s i d e v a n i s h e s , t h e r e f o r e w e h a v eF ( t , X ( - t , x ) ) = O a .e . x e N u, f o r a ll t e N .

A n d u s i n g o n c e m o r e (7 1 ), w e f i n a ll y o b t a i n t h a t F v a n i s h e s a .e . o n ~ x N s ,c o n c l u d i n g t hu s t h e p r o o f o f T h e o r e m I II .2 . A

U s i n g t h e s t a b i li ty r e s u l t s p r o v e n in s e c t io n I I . 3 , w e im m e d i a t e l y d e d u c e t h eC o r o l l a r y I I I . 1 . L e t b , ~ L ~o b e s u c h t h a t d i v b , E L ~oc a n d b , , d i v b , c o n v e r y e a s ng o e s t o b , d i v b in L~o~ (resp ec t ive ly ) w he re b sa t i s f ie s ( 7 0 ) a n d (49) . A s s u m e t h a t t h e r ee x i s t s X , e C ( R ; L ) N s u ch t h a t ,. fo r a n y Uo ~ ( R N ) , u o ( X , ( t , x) ) i s a r e n o rm a l i z eds o l u t i o n o f

~ ? U . _ b . . V u , = O in ~ x ~ N , u ,[ ,= o = U o O n ~ N . ( 7 2 )O tT h e n , f o r a ll T ~ ( O , o o ) , X , c o n v e rg e s in C ( [ - T , + T ] ; L ) N t o t h e m a p p i n gX ~ C ( N , L ) N s a t i s fy i n g (71) , (52) , (53) . I n a d d it io n , X , c o n v e rg e s to X u n ! f o r m l y f o r tb o u n d e d , i n m e a s u r e f o r x b o u n d e d i n R N.


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O rdina ry d i fferent ia l equa t ions , t rans po r t theory and Sobolev spaces 539R e m a r k . U s i n g T h e o r e m I I . 5 i n s t e ad o f T h e o r e m I I .4 , w e s ee t h a t w e m a y a s s u m et h a t d i v b , i s b o u n d e d i n L ~ i n s t e a d o f a s s u m i n g i ts L~or c o n v e r g e n c e .

1 1 1 . 3 T i m e - d e p e n d e n t t h e o r yW e n o w c o n s i d e r g e n e r a l v e c t o r f i el ds b = b( t , x) w h i c h s a t i sf y ( ,) a n d ( ** ) f o r a l lT < o o . T h e n , w e w a n t t o s o l v e f o r a ll t > 0 , x e R N t he f o l l o w i n g o r d i n a r yd i f f e r e n t i a l e q u a t i o n

~ X - b ( s , X ) f o r s > t , X l s = ~ = x (7 3 )0 sa n d t h u s X i s a f u n c t i o n o f ( s , t , x ) : X = X ( s , t, x ). T h e m a p p i n g X w i l l b e l o n g t oC ( D ; L ) w h e r e D = [ 0 , o o ) x [ 0 , o o ) .

B e c a u s e o f ( , ) , w e w i ll f i n d t h e f o l l o w i n g r e l a t i o ne x p ( - } A ( t ) - A ( s ) ) ) 2 < 2 o X < e x p ( ] A ( t ) - A ( s ) [ ) 2 , f o r a l l t , s > 0 ( 7 4 )

w h e r e A ( t ) e W 1" 1 (0 , R ) ( V R < o o ) , A ( 0 ) = 0 , A ' ( t ) > 0 f o r t > 0 . I n f a c t , t h e s o l -u t i o n w e w i ll b u i l d w i l l s a t i s f y ( 7 4 ) w i t h

A ( t ) = i [[ d iv x b l[L ' ( ~ ~) d s . ( 7 5 )0N e x t , t h e g r o u p p r o p e r t y w e u s e d in t h e a u t o n o m o u s c a s e b e c o m e s

X ( t 3 , t l , X ) = X ( t 3 , t 2 , X ( t 2 , t l , X ) ) a .e . x e ~ N , f o r a l l t 1 , t 2 , t 3 > 0 . ( 7 6)b ( s , X ) L I( O T ; L I + L ~ ) ( V T < o o) a n d t h u s w e w il ln v ie w o f (7 4 ) a n d (* * ) , 1 ~ ~

d e f in e s o l u t i o n s o f (7 3) i n a s i m i l a r w a y t h a n in th e p r e c e d i n g s e c t io n s n a m e l y t h ef o l lo w i n g s h o u l d h o l d f o r a ll a d m i s s i b l e f u n c t i o n s a n d f o r a ll t > 00~ s f l ( X ) = D f l ( X ) ' b ( s , X ) o n (0 , o o) x N U , f l ( X ) l s = , = f l ( x ) o n N u , (7 7)w h e r e t h e e q u a t i o n h o l d s in d i s t r i b u t i o n s s en s e.

W e m a y n o w s ta te o u r m a i n e x i s te n c e a n d u n i q u e n e s s r e s u lt . L e t u s p o i n t o u tt h a t w e w i ll n o t g i v e s t a b i li t y r e s u l t s w h i c h a r e e a s i l y d e d u c e d f r o m t h e s t a b i l i t yr e su l ts o f s e c t i o n I I . 3 e x a c t l y a s w e d i d in C o r o l l a r y I I k l .T h e o r e m I I I . 2 . W e a s s u m e t h a t b s a t i s f i e s ( * ) a n d ( * * ) . T h e n , t h e r e e x i s t s a u n i q u eX s C ( D ; L ) u s a t i s fy i n g (74) , (76) a n d (77) . I n a d d i t io n , i f u ~ ~ ( o r L ) , u (s , t , x )= u ~ x ) ) i s, f o r a l l s > O, t he un ique r en orm al i z ed so lu t i on i n C ( [ 0 , o o ); L ~ o f

? u~ t + b ' V x u = O in (O, oo) x ~ U , u l t = s = u ~ o n ~ u . ( 7 8 )R e m a r k s . 1) T h e a n a l o g u e o f ( 5 4 ) -( 5 5 ) is n o w

f l ( X ) e C ( [ O < t < o o ); L I~ o~ (N N ; C ( [ 0 < s < o o ) ) ) ) f o r a l l a d m i s s i b l e f l ( 7 9 )


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540 R . J. D iPe r na and P .L . L ions

f o r a l l t > 0, f o r a l m o s t a ll x ~ N , X ( s ) ~ W 1'~ a n d ~ - = b ( s , X ) o n ( 0 , ~ )(8O)

F i n a l l y , i f b s a ti s f ie s (f o r a l l T < ~ )b E L ~ ( O , T ; L P ) + ( I + I x l ) L I ( O , T ; L ~ ) f o r s o m e 1 < p < ~ (8 1 )

t h e n X s C ( [ 0 < t < o o ); L f o c( ~ u ); C ( [ 0 < s < ~ ) ) ) ) . I n a d d i t io n , i n t h e l a s t s t a t e -m e n t o r i n (7 9) o n e m a y p e r m u t e s a n d t . A

W e s k i p th e p r o o f o f t h i s r e su l t s in c e it m i m i c k s t h e p r o o f s m a d e i n th ep r e c e d i n g s e c t i o n s , k e e p i n g t r a c k c a r e f u l l y o f t h e t - d e p e n d e n c e ( o r s - d e p e n d e n c e )u s i n g t h e s t a b i l i t y r e s u l t .

IV. Counterexamples and remarksIV.1 W I'p vec tor - f i e lds w i th unbounded d ivergenceI n t h is s e c t i o n , w e c o n s t r u c t v e c t o r - f i e ld s b w h i c h a r e a u t o n o m o u s (i.e . b d e p e n d so n l y o n x ) in t w o d i m e n s i o n s ( x ~ Z ) , b e l o n g t o W l l o ' c P ( ~ Z ) ~ B U C ( ~ 2 ) f o r a na r b i t r a r y p < ~ a n d y e t y i e l d i n f i n it e ly m a n y s o l u t i o n s o f t h e O D E

) ( = b ( X ) , X [ , = 0 = x ( 8 2 )s u c h t h a t X ( t , x ) s a t is f i es t h e g r o u p p r o p e r t y a n d X is c o n t i n u o u s .

T h i s c o n s t r u c t i o n f o l lo w s i n f ac t d i r e c t ly f r o m t h e c o n s t r u c t i o n m a d e b yA . B e c k i n [ 1 ] t h a t w e r e c a l l n o w : l et K b e a C a n t o r s e t in [ 0 , 1 ] a n d l et 9 e C ~ 1 7 6be su c h th a t 0 ___


31/37

O rdina ry d if fe ren tial equa t ions , t rans po r t theory and Sobo lev spaces 541t h e r e f o r e fro i s d i f f e r e n t i a b l e a t t a n d f , ~ ( t ) = f ' ( x ) o r i n o t h e r w o r d s j ~ , ( f , ~ l ( f m ( t ) ) ) =

f ' ( f - l ( x ) ) ) = f ' ( f 1 (fr o( t) ) ) . N e x t , i f x ~ K a n d s i s c l o s e to t, d e n o t i n g b y x ( s ) t h eu n i q u e s o l u t i o n o f

w e o b s e r v e f i r s t t h a tx ( s ) + m ( K c ~ [ 0 , x ( s ) ] ) = t ,

I x ( s ) - x l < I s - t l .T h e n

l f , , ( s ) - f r o ( t ) ] = I f ( x ( s ) ) - f ( x ) [ < C l x ( s ) - x [ 2


32/37

542 R J . D i P e r n a a n d P .L . L i o n s

f o r w h i c h t h e r e ex is ts t w o m e a s u r e - p r e s e r v i n g f lo w s s o lv i n g t h e a s s o c ia t e d O D E .S i n ce w e a re in t w o d i m e n s i o n s , d i v e r g e n c e - f r e e v e c t o r - f i e ld s c o r r e s p o n d t o H a m i l -t o n i a n s y s t e m s a n d w e w il l i n f a c t b u i ld a s i n g u l a r H a m i l t o n i a n s y s t e m a s f o l lo w s

x l i f [ x l l < l x 2 [ , = - - ( x l - - l x 2 [ + l ) i f x l > [ x 2 1 ,H ( x ) = Ix2 1 =- - - ( x t + { x 2 ] + l ) i f X l < -- Ix 21 , f or a l l x = ( x l , x 2 ) e [ R 2

t h e n b w i l l b e g i v e n b y

( 89 )

I O H 1b z ( x ) = ~ X l = { l ~ 2 1 1 1 x l l ~ , x d + l l x , , > l x 2 1 } ( 9 O )

( b l ( x ) ~ ? H _ { ] X ~ 2 ll;,,l_< ,x21 + ll :,,> lx21i g n ( x 2 ) )~ / t~ 2 t /)A n d o n e c h e c k s e a s i l y t h a t ( 8 7 ) a n d ( 8 8 ) h o l d : n o t i c e i n f a c t t h a t bj , Oxic~x

( V l < i , j < 2) a r e b o u n d e d m e a s u r e s o n ~ 2 - B a f o r e a c h 3 > 0 a n d t h e t o t a lv a r i a t i o n o f t h e s e m e a s u r e s o n ~ 2 _ B ~ g r o w s l o g a r i t h m i c a l l y a s 6 g o e s t o 0 .

G i v e n a n i n i ti a l c o n d i t i o n x ~ o o( x ~ , x 2 ) , w e n e x t w i s h t o d e f i n e t w o d i f f e r e n tf l o w s X ~, X 2 ( = X ~ , X 2 ( t , x ~ S i n c e w e a r e d e a l i n g w i t h " L 1 f l o w s " ( i.e . d e f i n e da .e .) , w e o n l y n e e d t o d e f i n e t h e s e f l o w s o n I = { x ~ ~ + O , x ~ + O ,I x~ 4= I x ~ T h e n , b y s y m m e t r y c o n s i d e r a t i o n s , w e o n l y n e e d t o d e f in e X 1, X 2 o n

Q = { x ~ ~ ~ 2 / x ~ > o , x ~ > o , x ~ , ~ o } .I n t h e c a s e w h e n x ~ ~ w e d e f i n e X 1 a n d X 2 b y

X ~ = X 2 = x ~ X ~ = X 2 = x ~ - t i f t < x 2 ,X ~ = X 2 = x ~ - 2 x ~ + t i f t > x 2 f o r a l l x ~ ~ (9 1 )

I n t h e c a s e w h e n x ~ < x 2 w e d e f i n e X ~, X 2 a s f o l l o w sx ~ t (x2~ 2 - 2t[ ~/2 ( 92 )~ = I (x ~ 2 - 2 t 1 1 / 2 , X ~ = x ~

x o= 1 0 2X 2 = e l ( x ~ 2 - - 2 t l ' /2 , X 2 x O e } ( x 2 ) - - 2 t [ 1 /2 w h e r e ~ = 1l 0 2 1 0 2i f t < ~ ( x 2 ) , e - - - - 1 i f t > ~ ( x 2 ) (9 3 )

f o r a l l t E ~ , x ~ > x ~ > 0 .N o t i c e t h a t i n b o t h c a s es , X a a n d X z a r e c o n t i n u o u s i n t , b e l o n g t o

W I 'P ( - T , T ) (V T < ~ ) ( f o r a ll p < 2 ), a r e s m o o t h e x c e p t f o r o n e t , s o l v e t h e O D Ew i t h b f o r a ll t e x c e p t f o r o n e v a l u e a n d s u c h t h a t b ( X ) is c o n t i n u o u s in t e x c e p t f o ro n e v a l u e , b ( X ( t ) ) e L P ( - T , T ) ( u T < ~ ) ( f o r a ll p < 2). F u r t h e r m o r e , f o r i = l , 2,w e h a v e

s u p I X i ( t , x ) l < = C r ( l + x ) f o r a l l T < o o , t e ~ , x e l ( 9 4 )-T< - - t


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O r d i n a r y d i ff e re n t ia l e q u a t i o n s , t r a n s p o r t t h e o r y a n d S o b o l e v s p a c e s 5 43a n d X i oo 2 . C (N ; L~ 'oc(N2)) (Vp < o o ) . (95)L , o c ( [ ~ , C ( ~ ) ) nO n e c a n a l s o c h e c k t h a t X a a n d X 2 a r e m e a s u r e p r e s e r v i n g i.e.

2 o X i ( t ) = 2 f o r a l l t E ~ , i = 1 , 2a n d s a t i s f y t h e g r o u p p r o p e r t y ( 5 3 ) .

F i n a ll y , l et u s a l s o r e m a r k t h a t t h e p r o o f o f T h e o r e m I I I . l a n d t h e a b o v ep r o p e r t i e s o f X 1, X z s h o w t h a t , f o r a n y u ~ ~ , @ ( ~ 2 ) ( o r L p , L ~ L , L . . . ) , u i ( t , x )= u ~ x ) ) is, f o r e a c h i = 1, 2 , a r e n o r m a l i z e d s o l u t i o n ( a n d t h u s a s o l u t i o n i nd i s t r i b u t i o n s s en s e) in C ( ~ ; L " ( ~ : ) ) ( V l < p < o e ) o f

c3u- - = - b ' V u i o n ~ x ~ 2 b l i l t 0 : uO o n [1~2 .0 t

N o t i c e t h a t t h i s a ls o s h o w s t h a t : i) t h e r e g u l a r i z i n g r e s u l t T h e o r e m I I.1 d o e s n o th o l d h e r e - - o t h e r w i s e , w e w o u l d d e d u c e u ~ -= u z , a c o n t r a d i c t i o n - - , ii) r e n o r m a l -i ze d s o l u t i o n s c a n n o t b e c o m p a r e d w i t h d i s tr i b u t i o n a l s o l u t i o n s i n t h e s e n se t h a tt h e n o t i o n s a r e n o t c o m p a r a b l e . I n d e e d , w e h a v e s ee n in t h e p r e c e d i n g s e c ti o n s t h a tr e n o r m a l i z e d s o l u t i o n s ar e m o r e g e n e r al t h a n d i s t r i b u t i o n a l s o l u t i o n s u n d e r s o m ec o n d i t i o n s o n b. H o w e v e r , t h e a b o v e e x a m p l e s h o w s t h a t t h is i s n o t a l w a y s t h e c a se :ind e e d , i f u ~ i s a n in i t i a l c o n d i t io n su c h th a t u t ~ u 2 ( i.e . u ~ ~ 0 ! ), t h e n v = u t - u 2s o l v e s t h e e q u a t i o n i n d i s t r i b u t i o n s s e n s e a n d s a ti sf ie s

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