Partial differential equations and their applications 9781470439262, 1470439263

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Selected Title s i n Thi s Serie s Volume 12 P e t e r C . Greiner , Victo r Ivrii , Lui s A . Seco , an d Catherin e Sulem , Editors Partial differentia l equation s an d thei r application s 1997 11 Lu c V i n e t , Edito r Advances i n mathematica l sciences : CRM' s 2 5 year s 1997 10 Donal d E . Knut h Stable marriag e an d it s relatio n t o othe r combinatoria l problems : A n introductio n to th e mathematica l analysi s o f algorithm s 1997 9 D . Levi , L . Vinet , an d P . Winternitz , Editor s Symmetries an d integrabilit y o f differenc e equation s 1996 8 J . Feldman , R . Froese , an d L . M . R o s e n , Editor s Mathematical quantu m theor y II : Schrodinge r operator s 1995 7 J . Feldman , R . Froese , an d L . M . R o s e n , Editor s Mathematical quantu m theor y I : Fiel d theor y an d many-bod y theor y 1994 6 Guid o Mislin , Edito r The Hilto n Symposiu m 1 99 3 Topics i n topolog y an d grou p theor y 1994 5 D . A . Dawso n Measure-valued processes , stochasti c partia l differentia l equations , an d interactin g systems 1994 4 Hersh y Kisilevsk y an d M . R a m M u r t y , Editor s Elliptic curve s an d relate d topic s 1994 3 R e m i Vaillancour t an d Andre i L . Smirnov , Editor s Asymptotic method s i n mechanic s 1993 2 Phili p D . Loewe n Optimal contro l vi a nonsmoot h analysi s 1993 1 M . R a m Murty , Edito r Theta function s 1993

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Partial Differentia l Equations an d Thei r Applications

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https://doi.org/10.1090/crmp/012

Volume 1 2

CR M R PROCEEDING S & M LECTURE NOTE S c

Centre d e Reeherehe s Mathematique s Universite d e Montrea l

Partial Differentia l Equations an d Thei r Applications Peter C . Greine r Victor Ivri i Luis A . Sec o Catherine Sule m Editors The Centr e d e Reeherehe s Mathematique s (CRM ) o f th e Universite d e Montrea l wa s create d i n 1 96 8 t o promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Amon g it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l programs , an d publishing. Th e CR M i s supporte d b y th e Universit e d e Montreal, th e Provinc e o f Quebe c (FCAR) , an d th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h th e Institu t de s Science s Mathematiques (ISM ) o f Montreal , whos e constituen t members ar e Concordi a University , McGil l University , th e Universite d e Montreal , th e Universit e d u Quebe c a Montreal, an d th e Ecol e Polytechnique .

g America n Mathematical Societ y '! Providence , Rhode Island US A ^?VDED

T h e p r o d u c t i o n o f t h i s volum e wa s s u p p o r t e d i n p a r t b y t h e Fond s p o u r l a F o r m a t i o n de Chercheur s e t l'Aid e a l a Recherch e (Fond s F C A R ) a n d t h e N a t u r a l Science s a n d Engineering Researc h Counci l o f C a n a d a ( N S E R C ) . 1991 Mathematics Subject

Classification.

Primar

y 35Pxx , 35Qxx .

L i b r a r y o f C o n g r e s s Cataloging-in-Publicatio n D a t a Partial differentia l equation s an d thei r application s / Pete r C . Greiner.. . [e t al.] , editors , p. cm . — (CR M proceeding s & lectur e notes , ISS N 1 056-858 0 : v . 1 2 ) Includes bibliographica l reference s (p . - ) . ISBN 0-821 8-0687- 4 (sof t : alk . paper ) 1. Differentia l equations , Partial—Congresses . I . Greiner , P . C . (Pete r Charles) , 1 938 - . II. Canadia n Mathematica l Society . Semina r (1 99 5 : Universit y o f Toronto ) III . Series . QA377.P29731 99 7 515'.353— R n is

L

(6)

^)=JQ1]ITM

ds ds

and d(x,y) i s obtaine d b y minimizin g L(j) over curve s wit h 7(0 ) = y, 7(1 ) = x. In th e Hamiltonia n approac h on e take s a s Hamiltonia n functio n 1n

(7)

j,fc=i

looks fo r curve s (#(s),£(s) ) tha t satisf y Hamilton' s equation s (8)

dxj _ dH d^j ds d^jd£j' ds ds

_ dH dxj '

x(0) = y, x(r)

= x

and obtain s th e distanc e functio n d(x, y) fro m th e action (9)

S(x,y,r) =

/ Jo

^jdxj-Hixis),^))

0?5

d{x,y)2 2r

L?'=i

The actio n satisfie s th e Hamilton-Jacob i equatio n

— fffa; '— dr \ dx

(10)

= 0.

2. Th e subellipti c cas e In th e subellipti c cas e th e operato r ma y hav e th e for m (ii)

* = £*?.

m < dimension,

so C is not elliptic . A well-known theore m o f Hormander [8 ] state s tha t C i s hypoelliptic i f the commutator s [Xj,Xk], [Xj, [Xk,Xi]\ , . . . spa n th e tangen t spac e at eac h point . I f commutator s o f order k suffice a t al l point s (an d ar e necessar y a t some points ) th e operato r i s said t o b e o f step k. Such operator s aris e i n connection wit h som e problem s i n physics (quantu m anharmonic oscillators ) an d als o in connectio n wit h comple x analysis . I n th e latte r context suppos e tha t M 2 n + 2 i s a real codimensio n on e submanifol d o f C n + 1 . The n there ar e n independent holomorphi c vecto r field s Zj = Xj + iYj tangen t t o M and a natura l operato r (12)

£=5»+^]=i;w+j?] 3=1

3=1

SUBELLIPTIC GEOMETR Y AN D FUNDAMENTA L SOLUTION S

3

which i s hypoellipti c provide d tha t M satisfie s a pseudoconvexit y condition . W e consider her e th e mode l example s (13) C

1 n+

D Ma 2 n + 1 = ( l m z 2

(14) C (15) C +

1

n+1

=JT taj\zj\2\, a

D Ml = {lmz 2 = \zi\ 2k), k

D M f +1 = | l m , n + 1 =

5

> 0;

e Z +;

( | > f )'} •

For eac h o f thes e w e tak e (z, t) = (z\, ..., z n, R e 2 n + i ) s o a s t o coordinatiz e M a s Cn x R . Various author s hav e considere d a n associate d subelliptic geometry. Fro m th e Lagrangian standpoin t on e look s a t piecewis e smoot h curve s 7 : [0,1 ] — • M tha t satisfy th e constraint s (16) ^

G

span{Xi,... X m } , 7(0

) = y, 7(1

) = x.

The commutato r conditio n implie s tha t suc h curve s alway s exist . Introduc e a Rie mannian metri c fo r whic h X\, . . . , X m ar e orthonormal . Th e lengt h o f a curv e (16) i s independent o f the choic e o f such a metric . Minimizin g th e lengt h give s th e so-called Carnot-Caratheodory distance dc{x,y). In th e 2-ste p cas e (1 3) , M 2 n + 1 , on e ha s (17) d c ( M ; 0 , o ) ~ | *

l + l*l

1 /2

-

For th e associate d operato r C th e parametri x an d th e hea t kerne l fo r d/du — C satisfy 2n

(18) G ( ^ ; 0 , 0 ) ~ d c M ; 0 , 0 ) (

(19) i o g P U ^ ^ ; Q . Q ) - - ^

;

"2yQ)2, ti-o

.

Thus the Carnot-Caratheodor y distanc e look s promising a s a means to understand ing th e operators . One can test thi s hypothesis in more precise fashion, becaus e exact fundamenta l solutions ar e known for th e models above. Fo r the 2-ste p example (1 3 ) the manifol d has a Heisenber g grou p structur e wit h respec t t o whic h C i s lef t invariant , s o on e needs onl y th e fundamenta l solutio n a t th e origin . Whe n al l aj ar e equal, say = 1 , the fundamenta l solutio n wa s foun d b y Follan d [5] ; it i s (20)

G(*,*;0,0) =

|2|4+j2)n/2-

When th e aj ar e distinct n o simple expression i s available. On e does have a formul a [4, 6] : +

(21) G(z,t;0,0)

=

cn J

°° voir) gn

dr,

4

RICHARD BEAL S ET AL.

where th e auxiliar y function s vo and g ar e n

t, r) = Y ^ a,j \ZJ |2 coth 2ajT — it,

(22) g(z,

i=i

*|; ZZ).

The fundamenta l solutio n i s (27) G(z,t;w,s)

=

—p — p — i\l — p2\

—^~ log

l + |p| 2

the auxiliar y variable s ar e (28) a /on\ _ {}

P

= | \z\4 + \w\ 4 - 2 Im z2w2 - i(t - s) |, V2zw ~ [\z\± + \w\±-i{t-s)Y/

2

*

At first sight , thi s resul t simpl y add s t o th e mystery . 3. Comple x Hamiltonia n mechanic s Let u s retur n t o (21 ) an d se t Zj = Xj + ix n+j, s o tha t th e Green' s functio n G can b e writte n a s

(30)

G

^f¥r^

where th e functio n tha t occur s i n th e denominato r i s n

(31) g(x,

t,r) = ^2 aj(x* +

x2

n+j) coth2ajT - it.

3 =1

Note the analog y wit h th e Hadamar d parametri x (3) . Thi s analog y i s much deepe r than simply a choice of notation. Indee d if we define a Hamiltonian H = H(x,t,£,0)

SUBELLIPTIC GEOMETR Y AND FUNDAMENTA L SOLUTION S

5

from th e principa l symbo l o f £ a s i n (7 ) the n th e functio n (31 ) satisfie s th e corre sponding Hamilton-Jacob i equatio n

Now g o f (31 ) canno t b e th e actio n S fro m (8) , (9) , becaus e g i s no t real valued. Howeve r g is th e actio n (9 ) associate d t o a varian t o f (8) . Instea d o f denning Hamiltonia n trajectorie s b y standar d boundar y condition s a s i n (8 ) on e takes curve s tha t satisf y Hamilton' s equation s dx_ dt

_ d£

H

(33) T

$~

He

^ dS-

98

_ Hx

' dS--

Ht s-~ -°

' d~

but wit h modifie d boundar y condition s (34) x(0)

= y,

= x, t(r)

X{T)

= 6 0 = -i.

=t, 0

The integratio n i n (30 ) ca n b e considere d a s a summatio n ove r th e fiber o f th e characteristic variet y o f C a t th e bas e point . No w th e characteristi c variet y o f an ellipti c operato r i s trivial—th e fiber ove r a poin t i s th e singl e poin t 0—s o th e analogy betwee n (30 ) an d (3 ) i s complete . 4. Application s If th e precedin g viewpoin t i s correc t i t shoul d b e usefu l i n a t leas t tw o ways . First, i n the "variabl e coefficient " versio n o f the 2-ste p cas e (1 3 ) on e should b e abl e to obtai n a ful l parametri x analogou s t o (3) , wit h eac h ter m a n integra l alon g a fiber, wit h integrand s o f th e for m o f the term s i n (3) : (35)

G

(

,

I

(

;

M

^

+

j

^

. . . .

+

This i s indee d th e case : se e [1 ] . Th e denominator s involv e th e comple x actio n function g an d th e numerator s satisf y first orde r transpor t equation s r\ m

(36) i=

Tv0 = -7- ^ + Y^XjgXjV 0 = i

(Cg)v Q,

T»j = ^ 4 « * - i ) , . . . . Note tha t T i s the tota l derivativ e alon g th e Hamiltonia n trajector y (33) , (34) . The associate d hea t kerne l ha s a simila r expansio n (37) P

- ~ 7 ^ 1 / e~ * JR

t

T9/u

wo dr + ^ /

e'

T9 u

' w1 dr

+ •••.

JR

As a second test of this viewpoint, the auxiliary functions tha t occu r in solutions like (27) should be understandable a s geometric invariants associated to the comple x Hamiltonian mechanic s (33) , (34 ) an d thi s shoul d ai d i n finding result s i n othe r higher ste p models . Thi s i s als o th e case : se e [3] . I n fac t w e ca n obtai n th e fundamental solutio n i n th e genera l 2k ste p exampl e (1 4) : (38) C

2

D M f c = {lmz 2 = |2i|

2fc

}.

6

RICHARD BEAL S ET AL.

This fundamenta l solutio n ca n b e writte n i n th e for m F (39) G(z^s)= 4far2|il| > where th e numerato r F = F(p, p) i s (40) F Jo Jo

TJS+S-(1

-

s+)( l - s_)( l - s XlkV){\ - s]/ kp)(l - s+s-\p\

2k

)

and th e auxiliar y function s A, p ar e 2k

(41) A=\z\

+

\w\

2k

-i(t-s),

1k

2 ' zw (42)

p = _ _

.

The comple x actio n functio n g fo r thi s proble m ha s th e asymptotic s (43) er=

lim g(z,t;w,s;T) =

\z\ 2k + \w\ 2k - 2z kwk - i(t - s).

T—•+0O

The function CI = 2z kwk i s an angular momentu m invariant , constan t o n the Hamiltonian trajectories . Th e function s a and ft account fo r al l th e auxiliar y variable s (28), (29 ) an d (41 ) , (42) . Indee d (44) A

= a + fi, P=[j) •

The fundamenta l solutio n (39 ) ha s a n interpretatio n lik e (35) :

(45) G=-

[vd(logg) = - f-^-dr

9 or where th e numerato r v satisfie s a secon d orde r transpor t equatio n JR JR

(46) £:[Tv

drl

+ {Cg)v]-c(v^=0.

Similar result s ca n be obtained fo r highe r dimensiona l versions of these mode l problems, suc h a s (1 5) . References 1. R . Beals , B . Gaveau , B. , an d P . C . Greiner , Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians, Bull . Sci . Math . (2 ) (t o appear) . 2. , Subelliptic geometry, Proc . Internat . Conf . o n Comple x Hypercomple x Analysi s (Mexico City , 1 994 ) (t o appear) . 3. , On a geometric formula for the fundamental solution of subelliptic Laplacians, Math . Nachr. 1 81 , 81 -1 63 . 4. R . Beal s an d P . C . Greiner , Calculus on Heisenberg manifolds, Ann . o f Math . Stud. , vol . 1 1 9 , Princeton Univ . Press , Princeton , 1 988 . 5. G . B . Folland , A fundamental solution for a subelliptic operator, Bull . Amer . Math . Soc . 7 9 (1973), 373-376 . 6. B . Gaveau, Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents, Act a Math . 1 3 9 (1 977) , no . 1 -2 , 95-1 53 . 7. P . C . Greiner , A fundamental solution for a nonelliptic partial differential operator, Canad . J. Mat h 3 1 (1 979) , no . 5 , 1 1 07-1 1 20. .

7

SUBELLIPTIC GEOMETR Y AN D FUNDAMENTA L SOLUTION S

8. L . Hormander , Hypoelliptic second order differential equations, Act a Math . 1 1 9 (1 967) , 1 41 171. D E P A R T M E N T O F MATHEMATICS , Y A L E UNIVERSITY ,1

0 HILLHOUS E AVENUE , P . O . Bo

x

208283, N E W HAVEN , C T 06250 , U.S.A . E-mail address: bealsQmath.yale.ed u LABORATOIRE EQUATION S AU X DERIVEE S PARTIELLES , C A S E 247 , U N I V E R S I T E P I E R R E E T M A R I E C U R I E , 4 , PLAC E JUSSIEU , 7525 2 PARI S C E D E X 05 , F R A N C E

E-mail address: gaveauQmathp6.jussieu.f r D E P A R T M E N T O F M A T H E M A T I C S , UNIVERSIT Y O F T O R O N T O , T O R O N T O , O N M5 CANADA

E-mail address: greinerQmath.utoronto.c a

S 3G3

,

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https://doi.org/10.1090/crmp/012/02 Centre d e Recherche s Mathematique s CRM Proceeding s an d Lectur e Note s Volume 1 2 , 1 99 7

"Non-Standard" Spectra l Asymptotic s fo r a Two-Dimensiona l Schrodinge r Operato r Mikhail Sh . Birman an d Ari Laptev

1*. Fo r a Schrodinge r operato r (1) H(a)

= - A - aV(x), a

> 0 , x< E R d ,

we denot e b y N(a, — 72 , F ) th e numbe r o f eigenvalue s lyin g o n th e lef t o f poin t A = -7 2, 7 > 0, (2) N(a,V)

:=

N(a,0,V).

If d > 3 then th e asymptoti c formula e fo r N(a, —7 2, V) a s a — * o o hav e bee n wel l studied. Ther e ar e particularl y complet e result s i n th e Wey l (semiclassical ) case , i.e. unde r th e conditio n (3) VeL

d/2(R

d

), d > 3

.

The followin g Cwikel-Lieb-Rozenblu m estimat e i s well known (se e [R , L , Cw ] an d also [RS , BS] ) - 7 2 , V) < C(d)a d/2 f

(4) N(a,

V+ /2 dx, 7

> 0, d> 3

and i t i s accompanied b y th e Wey l typ e asymptotic s (5) li

m a~

2

d/2

N(a, -

7

, V) = (27r)" da;d / V?

/2

dx, 7

> 0 , d > 3,

ujd : = vol{ x 6 R d : \x\ < 1}. 1991 Mathematics Subject Classification. 35P1 5 , 35P20 . The firs t autho r wa s supporte d b y th e Roya l Swedis h Academ y o f Scienc e (projec t numbe r 1400) an d Gran t INTAS-93-1 8-1 5 . The secon d autho r wa s supporte d b y th e Swedis h Natura l Science s Researc h Council , Gran t M-AA/MA 09364-320 . This i s th e final for m o f th e paper . *This i s a shor t versio n o f the pape r whic h ha s bee n accepte d fo r publicatio n i n Comm . Pur e Appl. Math . © 1 99 7 America n Mathematica l Societ y

9

MIKHAIL SH . B I R M A N A N D AR I L A P T E V

10

Let u s assum e fo r a momen t tha t V(x) > 0 . I t i s know n tha t 1. Asymptotic s (5 ) i s valid unde r onl y on e conditio n (3 ) whic h i s equivalent t o the finiteness o f th e asymptoti c coefficient . 2. Asymptotic s (5 ) i s valid simultaneousl y fo r al l 7 > 0 . Moreover, th e followin g propert y i s pointed ou t i n [R ] (se e als o [BS] ) 3. I f fo r som e 7 > 0 2

(6) N(a,-y

,V) =

0(a

d 2

/ ),

then (3 ) i s fulfilled an d likewis e (4 ) an d (5) . Therefor e th e Wey l orde r ot dl2 in th e estimat e (6 ) implie s th e Wey l asymptoti c formul a (5) . 2. Th e mai n purpos e o f thi s pape r i s t o sho w tha t th e spectra l propertie s o f the operato r (1 ) which were described abov e are not tru e i n the cas e d = 2 . Namel y let u s assum e agai n tha t V > 0. The n fo r d = 2 we hav e 1. Th e conditio n V £ Li(R 2 ) i s not enoug h t o justif y th e Wey l asymptotic s m a- lN(a,V) =

(7) li

-? - [Vdx.

a—>oo 47

TJ

2. Wey l asymptotic s lim a- lN(a,-i2,V) =

— Vdx,

a—KX> 47

7

> 0,

TJ

might b e valid . However , a t th e sam e time , N(a,V) ~ are example s fo r a n arbitrar y p > 1 . ) It migh t happe n tha t (8)

lim a- 1 N(a,V) = a—»oo 47

^- [vdx+ /3,

p

ca p , p > 1 . (Ther e

> 0,

TJ

i.e. Weyl' s orde r o f asymptotic s doe s no t guarante e th e Wey l formul a (7) . All the phenomena described abov e depend on the behaviour o f the operator (1 ) on function s u(x) = u(\x\). Weyl' s asymptotic s i s alway s define d b y th e behaviou r of V o n compac t sets . I t "competes " wit h th e "non-Weyl " contributio n define d b y an auxiliar y proble m o n semiaxi s wit h th e potentia l

(9) V(r)

= ±J*V(r,6)d0,

where V i s no longe r assume d t o b e a nonnegativ e potential . Moreover , everythin g is determined b y the behaviour o f V(r) a t infinity . Th e contribution o f the potentia l (9) t o formul a (8 ) i s additive . However , i t ca n hav e th e orde r a p , p > 1 , and the n it dominate s th e Wey l term . Already whil e studyin g spectra l asymptotic s fo r th e Dira c operato r (se e [BL] ) the author s me t wit h th e phenomeno n wher e th e Wey l (hig h energy ) asymptotic s "competed" wit h th e non-Wey l (threshol d wit h respec t t o energy ) asymptotics . There th e caus e o f thi s competitio n wa s mor e explici t an d wel l observe d i n th e momentum representation . I n th e presen t pape r i t i s mor e convenien t t o us e a variational techniqu e i n coordinat e representation . Our mai n result s ar e formulate d i n term s o f uppe r an d lowe r limit s fo r powe r type asymptotic s fo r th e countin g functio n o f a n auxiliar y operato r o n a semiaxis .

NON-STANDARD SPECTR 1 A L ASYMPTOTIC S

1

We coul d hav e give n mor e genera l (nonpowe r type ) statement s usin g th e result s of [Wl ] an d [W2 ] concernin g correspondin g operato r ideal s an d functio n ideals . Such generalization can be obtained automaticall y an d we do not give them avoidin g cumbersome formulations . The approac h o f thi s pape r wa s recentl y use d fo r th e stud y o f th e negativ e discrete spectru m o f th e operato r — A — aV i n unbounde d domain s i n R d wit h Neumann boundar y condition s (se e [S3]) . 3. I n orde r t o formulat e ou r mai n resul t w e introduc e som e notation s an d restrict th e clas s o f potential s t o th e clas s satisfyin g th e followin g condition . The necessar y an d sufficien t condition s fo r th e Weyl asymptotics (5 ) t o b e tru e are no t known . W e shal l introduc e her e a sufficien t conditio n whic h i n particula r provides (5 ) fo r 7 > 0 . Let ^a,6 : = {x G R 2 : a < \x\ < 6}, 0 ttk : = fio.b, a

k 1

< a < b < 00, k

= e ~ ,b = e ,k G Z.

2

For V G Laj i oc (M ), a > 1 we denot e l/a

f7fc(V,a):= ( f \x\ m(V,cr):=( [

2

^-^\Vf dx

V'dx)

a > 1 ,A

; = 1 ,2,... ,

.

!\x\
1 . It i s clea r tha t (1 0 ) i s fulfille d fo r an y a G (l, 1 . Besides, i t follow s fro m (1 0 ) tha t 2

(11) F G L i ( R

).

Now th e precis e definitio n o f th e operato r H(a) unde r Conditio n A ca n b e give n via it s quadrati c form , i.e . th e operato r i n (1 ) i s understood a s form-sum . Assuming tha t Conditio n A i s satisfie d an d usin g (9 ) w e introduc e (12) F

v{t)

:=e

2

*V(e*), t>l,

and th e quotien t o f quadrati c form s

ritt r*V(*)M')l (13) f?\w(twdt

a

'

*

wm 1 w{ )

_0 -°-

Let / = (l,oo ) an d

HHi) •= W e HUi) •"' e L 2(i)Mi) = o} .

12 MIKHAI

L SH . BIRMA N A N D AR I L A P T E V

Let u s assum e tha t th e quadrati c for m J x Fy(t)\uj(t)\ 2 dt generate s i n W 1 (/) a compact selfadjoin t operato r Ty. Th e sequenc e o f it s s-number s (eigenvalue s o f the operato r \Ty\ = {FyFv) 1 !2) i s denote d b y Sk(Fy)> Furthermor e n(s, JV) : = card{/ c : Sfc(^V) > s} > Ap(^V) : = limsups p n(s,Ty), 8

P(fy)

s-^0

p • = liminfs n(s,jFy), p> s^o v ' " "r

~ 2

r

Considering th e negativ e an d positiv e part s o f Ty, 2(J y)± = |JV | ± JV , w e als o need th e functional s n±(s,Jrv) :

= n(s , (^V)±) ,

A ^ ( ^ ) : = Ap((JV)±) , ^ C

M : = *p((JV)±) .

Obviously n(«, JV) = n+(s , JV) + n_(s , J V ). 4. Ou r mai n resul t i s contained i n th e followin g theorem . 1 . Let a potential V of the operator (1 ) satisfy Condition A and let the number of negative eigenvalues of the operator (1 ) .

THEOREM

N(a,V) be (a) / /

Ai(%|)l, then | limsu

Pe^00a-'W(a,

+ p \fv),

V)= A

(p>l)

(c) J/Ai(^"|v| ) < o o and +)

(14) A[

(FV) =

0,

then the Weyl asymptotic formula holds m a- lNd(a, V)

(15) li

= ^- ( V+ dx.

a—»oo 47

IfV>0, \x\

> R for some R>0, then

TJ

(1 5 ) implies (1 4) .

REMARK 1 . Conditio n A in Theorem 1 can be weakened. Namely , the sequenc e of weighte d L^-norm s o f V ove r ring s i n (1 0 ) ca n b e replace d b y th e sequenc e o f Orlicz norm s correspondin g t o th e clas s Llog + L . Th e estimate s fo r N(a, V) whic h are require d fo r thi s purpose , wer e obtaine d i n [SI] . Fo r V > 0 th e statemen t o f sufficiency i n th e par t (c ) o f Theore m 1 is contained i n [S2] .

NON-STANDARD SPECTRA 1 L ASYMPTOTIC S REMARK

2 . Le t

wher

\F

e k

1e

Co(Fv)= [

3

v(t)\dt,

(k(F

v)=

I

t\F(t)\dt,

A

r = l,2,... .

It ca n be proven that i f G(Fy) £ / Pi0 o, p > | , the n A P (JV) < oo. I n the case Fy > 0 the invers e statemen t als o hold s true , namel y A P (JV) < o o implie s G{Fv) ^ i p o o 5. Le t u s giv e her e a n exampl e whic h cover s al l th e statement s o f Theore m 1 . We conside r th e potentia l j V*(x) = 2 1 /P- 2 $(0)r- 2 (logr)- 2 (loglogr)- 1 /P, r (16)

\

> e 2,2p > 1 , e 2,

V*(x) = 0 , r
1.

a

The potentia l (1 6 ) satisfie s Conditio n A wit h th e sam e a a s i n (1 7) . W e denot e

$o := -^ f

*(0) l,2 p > 1 .

Substituting t = exp(2r) , u) = (expr)v w e transfe r (1 3 ) t o 18

A

i /1 2J 1 2 w '

Jo (M 2 + M 2 )dr

^ ° =0 .

For (1 8 ) w e impos e th e additiona l conditio n V(TQ) = 0 , r 0 = s _ p . Th e proble m splits an d th e respectiv e quotien t o n th e semiaxi s (TO , 00) doe s no t contribut e t o the asymptotic s o f n(s,Ty) i f n(s,^V ) ~ 0 ( s _ p ) , p > i . Finall y w e hav e s—+0 z

C r-

1 /p

2

\v\ dr

S 2 2 - > ' r°n„M2 ...m-i !o°(\v'\ + . \v\ )dr

w

(°) = U (T °) = °'

and afte r substitutin g r = Toy we obtai n

Iftr^-DW*^„ /„VI2*

(0) = „ (1 ) = 0.

The las t proble m ha s th e standar d Wey l asymptotic s o f th e orde r — ^, bu t wit h respect t o th e paramete r s 2p. Thi s implie s th e followin g statement . PROPOSITION

(19) h W n

2 . Let V is defined according to (1 6) . Then

(s,^) =

I jf V1/ p _ 1)1/2d2/ = I^m =

: M(p).

The latte r resul t allow s u s t o formulat e th e followin g statement .

14

M I K H A I L SH . BIRMA N A N D A RI L A P T E V

THEOREM

3 . Let V e L 1 , and

(20) V{x)-V?(x)=o{vW(x)),

|s|-oo,2p>l

.

Then the following asymptotic formulae hold (a) If in (20 ) p = 1, tfien m a " 1 ^ , V ) = - i - [v+dx +

(21) li

a->oo 47

^ *+ .

TJ 2

(b) If p > 1, then (using notation (1 9) ) m a~ pAT(a, V) = $$ . M(p), p

(22) li

> 1.

a—xx)

(c) If p 0 , p > 1 . 2. Conditio n A implies th e "lattice" conditio n

(23)

W

/ \V\°dx) neZ2

1,

W +nJ

where Q 2 is a unit cub e i n E2. I t was shown in [BB] tha t i f (23) i s satisfied then 1 2 lim a " N(a, - 7 , V) = ^- fv+dx, 7

> 0.

at—»oo

However, fo r an arbitrar y p > 1 and V = V®, $+ > 0 , th e (non-Weyl ) asymptotics (22 ) of the orde r a p holds . 3. Fo r V = V*, $ + > 0 , th e asymptotic s (21 ) has the Weyl orde r a , bu t the asymptoti c coefficien t i s greater tha n th e Weyl coefficient . (I t is worth mentioning tha t i f $0 < 0 then th e asymptotic formul a (21 ) is of the Weyl type). The contributio n t oA T (a, V) o f the radial proble m (non-Wey l contribution ) in coordinat e representatio n i s generate d b y a neighborhoo d o f infinity . O n the contrary, the Weyl contribution i s defined b y boundary value problems on compacts. Things becom e differen t i f we take the Fourier transform . I t can be shown tha t the Weyl contributio n correspond s t o larg e momenta , bu t the contribution fro m th e radial proble m correspond s t o small moment a (threshol d effect) . When d > 3 the Weyl asymptotic formul a for N(a, V) als o corresponds to large momenta. A possible non-Wey l asymptotic s ha s a threshold origin . Howeve r whe n d > 3 the "addition " o f these effect s i s impossible a t leas t whe n V > 0. I f d == 2 then thi s additio n become s possible . Thi s i s described i n part (a ) of Theorem 1 and par t (a ) of Theorem 3.

NON-STANDARD SPECTRA 1 L ASYMPTOTIC S

5

7. Th e natura l analo g o f the operato r (1 ) fo r d > 3 (fro m th e poin t o f view of the behavio r o f th e negativ e spectrum ) i s the operato r (24) H

d(a)

2

= -A- x{d)\x\-

(25) x ( d )

-

aV(x), a

= ^ ~ ^,

d>3

> 0,

.

Here w e alway s assum e th e conditio n (3 ) t o b e satisfied . Th e precis e definitio n o f the operato r (24 ) i s given vi a th e close d quadrati c for m i n L 2 ( (|Vt;|2 - x(d)\x\' 2\v\2 /
3.

The last inequalit y implie s that Hd(0) i s a positive operator. W e denote the numbe r of it s negativ e eigenvalue s b y Nd(ot, V). Let r , 0 be th e pola r coordinate s i n R d1 0 e S d _ 1 . Similarl y a s i n (9 ) an d (1 2 ) we introduc e

v{r)=

w^Lv{r'd)de

and Fv(t) =

e

2t

V(et).

Apart fro m (3 ) w e assume her e tha t (26) /

a

dx < 00 , e

\V\

> 0,2a > d ,

J\x\ d,

MIKHAIL SH . BIRMA N AN D AR I LAPTE V

16

(c) / / (27 ) is fulfilled and (28) A$(JV)=0

,

then the Weyl asymptotic formula holds (29) li

m a- d/*Nd(a, V)

=

-^/

IfV > 0, \x\ > R for some R>0, then

vf 2 dx.

(29 ) implies (28) .

References [BB] M . S . Birma n an d V . V . Borzov , The asymptotics behaviour of the discrete spectrum for certain singular differential operators, Spectra l Theory , Problem s i n Mathematica l Physics , vol. 5, Izdat. Leningra d Univ. , Leningrad , 1 971 , pp. 24-38 (Russian) ; Englis h transl. , Topic s in Mathematica l Physics , vol . 5 , Consultant s Bureau , Ne w York-London , 1 972 , pp . 1 9-30 . [BL] M . Sh . Birma n an d A . Laptev , Discrete spectrum of the perturbed Dirac Operator., Ark . Mat. 3 2 (1 994) , no . 1 , 1 3-32 . [BS] M . S . Birma n an d M . Z . Solomjak , Quantitative analysis in Sobolev's embedding theorems and applications to spectral theory., Tent h Mathematica l Schoo l (Kaciveli/Nalchik , 1 972) , Izdanie Inst . Mat . Akad . Nau k Ukrain , SSSR , Kiev , 1 974 , pp . 5-1 8 9 (Russian) ; Englis h transl., Amer . Math . Soc . Transl . Ser . 2 , vol . 1 1 4 , Amer . Math . S o c , Providence , R.I. , 1980. [Cw] M . Cwikel , Weak type estimates for singular values and the number of bound states of Schrodinger operators, Ann . Math . (2 ) 1 0 6 (1 977) , no . 1 , 93-1 00 . [L] E . H . Lieb , Bounds on the eigenvalues of the Laplace and Schrodinger operators., Bull . Amer. Math . Soc . 8 2 (1 976) , no . 5 , 751 -753 . [RS] M . Ree d an d B . Simon , Methods of modern mathematical physics. IV . Analysis of operators, Academi c Press , Ne w York-London , 1 978 . [R] G . V . Rozenbljum , Distribution of the discrete spectrum of singular differential operators., Izv. Vyss . Ucebn . Zaved . Matematik a (1 976) , no . 1 (164), 75-8 6 (Russian) ; Englis h transl. , Soviet Math . (Izv . VUZ ) 2 0 (1 976) , 63-71 . [Si] M . Solomyak , Piecewise-polynomial approximation of functions from H l((0,l)d), 12 = d, and applications to the spectral theory of Schrodinger operator, Israe l J . Math . 8 6 (1 994) , no. 1 -3 , 253-276 . [S2] , Spectral problems related to the critical exponent in the Sobolev embedding theorem, Proc. Londo n Math . Soc . (3 ) 7 1 (1 995) , no . 1 , 53-75 . [S3] , On the negative discrete spectrum of the operator — AJV — OLV for a class of unbounded domains in R d (i n thi s volume) . [Wl] T . Weidl , General operator ideals of the weak type, Algebr a i Analiz 4 (1 992) , no. 3, 1 1 7-1 4 4 (Russian); Englis h transl. , St . Petersbur g Math . J . 4 (1 993) , no . 3 , 503-525 . [W2] , Estimates for the operators b(x)a(D) in nonpower type ideals, Algebr a i Anali z 5 (1993), no . 5 , 47-6 7 (Russian) ; Englis h transl. , St . Petersbur g Math . J . 5 (1 994) , 37-58 . D E P T . O F MATHEMATIC S AN D MATHEMATICA L P H Y S I C S , P H Y S I C S FACULTY , S T . P E T E R S BURG UNIVERSITY , ULYANOVSKAY A 1

, S T . P E T E R S B U R G - P E T R O D V O R E T Z , 1 9890 4 RUSSI A

D E P A R T M E N T O F M A T H E M A T I C S , K U N G L TEKNISK A HOGSKOLAN ,1 0 0 4 SWEDEN

E-mail address: laptevQmath.kth.s e

4 STOCKHOLM

,

https://doi.org/10.1090/crmp/012/03

Centre d e Recherche s Mathematique s CRM Proceeding s an d Lectur e Note s Volume 1 2 , 1 99 7

Generic Singularitie s fo r th e Stead y Boussines q Equation s Russel E. Caflisch, Nichola s Ercolani, and Gregor y Steel e ABSTRACT. Fo r th e stead y Boussines q equation s describin g two-dimensional , stratified flow, w e analyz e generi c singularities . Th e result s sho w tha t fo r codimension 1 singularities, ther e ar e tw o generi c singularit y type s fo r genera l solutions, an d onl y on e generi c singularit y typ e i f ther e i s a certai n symmetr y present. Th e analysi s depend s o n a specia l choic e of coordinates, whic h greatl y simplifies th e equations . Th e solutio n i s viewe d a s a surfac e i n a n appropriat e compactified je t space .

1. Introductio n In thi s paper , singularitie s ar e analyze d fo r th e steady Boussinesq equations: u V p = 0 ; u - V C = -Ar 5 ^' ' Vx

u= (;V

• u = 0.

This syste m describe s steady , two-dimensional , stratifie d (i.e . variabl e density) , incompressible flo w i n whic h p i s density , u i s tw o dimensiona l velocit y an d £ i s vorticity. Th e buoyanc y ter m p x play s the rol e of the vorte x stretchin g i n the Eule r equations. Thi s syste m i s derived i n th e "Boussines q limit" , i n which th e variatio n of density i s important i n the buoyancy terms bu t insignifican t i n the inertia l terms . For the steady Boussinesq equations (1 .1 ) we will derive the generic (i.e . typical) form o f singularities , unde r certai n natura l restrictions . Numerica l computation s for stead y Boussines q [8 ] confirm thes e generi c singularit y types . The mai n resul t o f this pape r ca n b e informall y state d a s follows : Consider singularities for the steady Boussinesq system (1 .1 ) which have codimension 1 and for which the vorticity £ is infinite. The generic form of such a singularity is of two types: 1. «z 3 / 2 ; vnx 1 /2; C « z " 1 / 2 2. ^ « * 1 / 2 ; v « * - 1 / 2 ; C « * - 3 / 2 . 1991 Mathematics Subject Classification. Primar y 76C05 ; Secondar y 35B65 , 1 4B05 , 65N35 . The researc h o f th e firs t autho r i s supporte d i n par t b y th e ARP A unde r UR I gran t numbe r #N00014092-J-1890. The researc h o f th e secon d autho r i s supporte d i n par t b y th e NS F unde r gran t numbe r #DMS93-02013. The researc h o f the thir d autho r i s supported i n part b y the NS F unde r gran t #DMS-9306488 . This i s th e final for m o f th e paper . © 1 99 7 America n Mathematica l Societ y

17

RUSSEL E . C A F L I S C H ET AL.

18

If in addition the stream function ip is assumed to have an additional symmetry (T/>(X, 0) = tp(—x,Q)), then the singularity type is

3. v > ~ *

2/3

; ^ ~ z ~1

/3

;C~*~~4/3.

In thes e formula s x an d v denot e som e suitabl y chose n spac e an d velocit y coordinates. A precis e statemen t o f thi s resul t wil l b e presente d i n Sectio n 3 . The principa l ingredien t i n thi s stud y wil l b e a geometri c approac h t o differ ential equations , whic h ha s bee n develope d i n [7] , as wel l a s b y Bryant , Griffiths , and Hs u [3—5] . I n thi s approac h th e solutio n o f a differentia l equatio n i s viewed a s a surfac e i n a n appropriat e je t spac e (describe d i n Sectio n 3 ) an d th e PD E serve s as a constrain t o n th e possibl e for m o f thi s surface . Th e generi c singularit y type s for thi s surfac e ar e then analyzed , usin g singularit y theor y [1 ] ; they ar e exactl y th e generic singularitie s fo r a clas s o f Legendria n surfaces . An analyti c resul t showin g tha t th e relevan t surfac e i s smooth , whic h i s a necessary ingredien t i n th e singularit y theor y analysi s i s give n i n [8] . I t use s th e Cauchy-Kowalewski Theore m t o show that th e PD E solutio n i s smooth, whe n writ ten i n term s o f th e unfolde d variables . The proble m o f singularit y formatio n fro m smoot h initia l dat a fo r th e (time dependent) 3 D Eule r equation s i s the motivatio n fo r thi s work . A s pointe d ou t b y Childress et al [9 ] an d b y Pumi r an d Siggi a [1 0] , th e time-dependen t Boussines q equations ar e very similar t o the Eule r equation s fo r axi-symmetri c flo w with swirl . Moreover, we believe that th e form of steady complex singularities may be indicativ e of th e for m fo r dynami c rea l singularitie s o r fo r nearl y singula r flow . Finally , thi s problem serve s a s a vehicl e fo r developmen t o f method s tha t ma y b e applicabl e t o the singularit y formatio n problem . In addition , th e singularit y analysi s o f thi s stud y wa s motivate d b y a simila r analysis o f th e generi c singularit y typ e fo r firs t orde r system s wit h a t mos t tw o speeds b y Caflisch , Ercolani , Ho u an d Landi s [7] . Tha t analysi s use d a generaliza tion o f th e hodograp h transformatio n t o unfold th e differentia l equations . Whil e that unfolding transformation ha s been generalized [6 ] to a larger clas s of equations, including th e Boussines q equation s o r th e Eule r equation s fo r axisymmetri c flow with swirl , the analysi s i n the presen t stud y i s effected throug h a simple r an d mor e direct transformatio n o f th e equations . 2. Unfoldin g th e stead y Boussines q equatio n The Boussines q equation s fo r steady , 2D , stratified , incompressibl e fluid flow are (2.1) u - V (2.2) u (2.3) V x (2.4) V

p= 0 • V C = -p

x

u= ( •u=0

in whic h u = (u,v) i s velocity , p i s density , £ i s vorticity . Th e incompressibilit y condition (2.4 ) implie s th e existenc e o f a strea m functio n ip so tha t (2.5) u

= - V x j/> = (-ip y, ip x).

It i s illuminatin g t o recas t th e syste m (2.1 )-(2.4 ) int o a n exterio r differentia l system usin g differentia l forms . Sinc e dip = ip x dx + tp y dy = vdx — udy, the n

G E N E R I C S I N G U L A R I T I E S F O R T H E S T E A D Y B O U S S I N E S Q E Q U A T1 ION S

9

equation (2.5) , which i s equivalen t t o (2.4) , can b e writte n a s (2.6) dip

= v dx — u dy .

Similarly equation s (2.1 ) an d (2.2 ) ar e equivalen t t o (2.7) dtp

A dp = 0

(2.8) dtpAd(

= dyA dp.

This simple reformulation o f the equation is surprisingly potent: Equation s (2.7 ) and (2.8 ) sugges t tha t (y , ip) would b e convenient variables , i n terms o f which thes e equations becom e (2.9) p

y

= 0

(2.10) C

» = -Pi>

which ha s solutio n (2.11) p

= p{i>)

(2-12)
* M

in whic h p an d £ are arbitrar y functions . Next, conside r th e solutio n a s a surfac e i n (# , ?/, u, v, ip) spac e The n equa tion (2.6 ) o r th e rescale d for m 1u (2.13) dx = -dip+-dy vv provides a contact constraint o n the solution; i.e . a differential for m a = dip — vdx+ u dy tha t vanishe s everywher e o n the solutio n surface . Thi s surfac e i s a Legendria n surface du e t o th e contac t constraint , a s discusse d i n [8] . Finally, equation s (2.3 ) an d (2.4 ) ca n b e writte n i n term s o f th e ne w indepen dent variable s (y , ip) as f V (2.14) ^ vjip \ v

(2.15) ( u

2

/ y

2

+u )^ = 2 ^ + 2 C

Equation (2.1 4 ) i s equivalent t o the contac t constrain t (2.1 3) , while (2.1 5 ) provide s a furthe r constrain t o n th e system . Several furthe r manipulation s o f th e equation s ar e worthwhile . I f (u, v) ar e finite w e can writ e (2.1 4) , (2.1 5 ) a s a firs t orde r syste m

©/(?=:) CM? In thi s cas e th e previou s classificatio n o f [7 ] describe s th e generi c singularitie s fo r the system . Second, suppos e tha t (u,v) ma y becom e infinite . Followin g th e Beale-Kato Majda resul t [2] , singularities wit h infinit e vorticit y ar e o f most interest , s o we will assume that £ blows-up. Accordin g to the formul a (2.1 2) , blow-up of £ occurs alon g fixed value s o f ip. Therefor e w e unfol d th e singularit y throug h unfoldin g onl y th e ip variable b y a mappin g ip = ip(p). The n w e loo k fo r a solutio n whic h i s singl e valued i n y an d p. Furthermor e w e scale th e velocit y a s

(2.7)

- g , »= i

20

RUSSEL E . CAFLISC H ET AL.

in whic h (3 = (3(p) gives th e rat e o f blow-up . A s a consequenc e o f (2.1 2 ) w e ma y write ( a s (2.18) C

= fa - yph

in whic h 77 ^ = £ . Finally w e assume tha t £ is of the sam e orde r a s (u 2 + v2)^ i n (2.1 5 ) b y scalin g 77 an d p a s = f3- 2K(p) p

(2.19) V

= /T 2 7(p).

The resultin g pde' s fo r \x and 1 / are

, 2 ,o,

©,-