A-A128 995 ON THE SOLUTIONS IN THVER 1 IKFLOW OF A NON-VISCOU..CU) MISCON51N UNIV-MADISONMATHEMATICS RESEARCH CENTER H BEIRAO DA VEIGA SEP 92

UNCLASSIFIED RC-TSR-2424 DAAO29-8S-C-04i F/G 29/4 N

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I~I~IIMCROCOPY RESOLUTION TESI CHART

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MC Technical Sumary Rqozt 2424

( ON THE SOLUJTIONS IN THE 1ARGZ OfTHE Two-Din=WNAL rLow or A

NON-VSCOWUS INCO NRESS I FLUID

H. Delrio da Veiga

* Mathematics Research CenterUniversity of Wisconsin-Madison610 Walnut StreetMadison, Wisconsin 53706

September 1982 f(Received April 20, 1982)

Approved for public releaseiistributlou ulimited

Sponsored byU. S. Army Research OfficeP. 0. Box 12211Research Triangle ParkNorth Carolina 27709

82 11 02 063

MMITX'U Y o summiUOZ-MUWISO.

OM TIM sowUs Iw m a or rIm "M -AOZfinuMzOaILOU OF A NOW-V COU ZII3N3ROSISl VIAD

UN. DeA. d VeLga,

1echnical SeMAry Report 02424September 1962

We study the ftler equatloms (1.1) for the motion of a non-vLsooms

comessible fluid in a plan. domain Q. Let a be the Saneeb speo

defi ed is (1.4)o Ift the initial data v bel"n to Be and let the external

foroee fit) belong to Lo ). ts eorm 1 we prove the tr te"

oetinuity and the glob"l bmddm of the (muque) solution v(t), end in

theorem 102 we proe the stoug-otnum depends of v on the data WO

and t. In particular the vortiLty rot v(t) is a cotinummus fwxo in

5, for every t 6 t, if and oaly it thL property bolds for one value of

t. In theorem 1.3 we stAte ei propertis for the aa.ooLated group of

nonlinear operators S(t). Finally, in tborem 1.4 we give a quite general

sufficient condition on the data in order to got classical eolutions.

M (MG) SubJect ClassLfioatLons, 35330, 35125, 3SQ20

Key Words, non-viscoms iaaepresible fluids, nonlinear evolutionequations, continuous dependence on the data

Work Uit musber I (Applied Analysis)

'Department of Hathematics, University of Trento (Italy).

Sponsored by the United States Army under Contract wo. OMG29-00-C-0041.

W

XSZrcaucin aD inmiauauOU

in this paper we study the Muer equations (0. 1) for the motion of a hoa-

viscous I oampressbl. flUid in a plane omn I.

Let a be the Damnef space consisting of all divergence free vetor

flws il 0, tanential to the bodezy r end having a oontinuoms courl in

As let the initial data v0 belong to And the external foroes f(t) be

integrable In time with values in Be Under these a -stions the (unique)

solution v(t) with values in a of the Soler equattone is globally bounded

And ontinmeo in time (theon 1,*1)* Nreover. we prove the strong

costinutimedpeed of the solution V with respect to the data V0 and

f mtho"M 1.2). in particalr, curl v(t) is a contiauous function La I,iP

for eveoy t a t if and coly if this property bolda for oe value of t.

In thevoem 1.,3 it is ithat if rot f l0# the nonlinear

operators Sit), mapping the initial data v0 to the solution at time t,

form a strongly ocntinuum gromp of Lsmetrisa. Irinally, a general eufficient

condition guaranteming the existenci of classical solutions is given in

theorem 1.4.Aeesston For"tS ORA&I

DTIC TAB

OD, ,_Distribution/

Aveila!lIty CodesAv.111 anI /or

The responsibility for the wording and vim exp resse in this descriptivesumry lies with MC, and not with the author of this report.

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pper we -tf the motor equationo

11.15p a1w. Ian

ubor the Velocity flow4 wit..) asi as am 1(ton) ft* tshesms. Ia (1.11 the

QNSWOa1 hem filW Men~) 410 the laitil 90eleet w*Iu) &M gILVW 0MMU,

Sulecomme of hemia eslatke of (1.11 was prees by I. Uteitels. Otl.st oeaiest

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bald ter equation Is. 1) La the spae 3().ams fel bm woe an am thathe I. 1 and 1.3 2 SUMPrve the afteiteso or

abometer lbs owl of the velocity field# o poesselye Eft wit) isa conetainus

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In - x I 1 31-tI(6aft

is the solution of *;() a oS(0)1(0 (e)). a 9 (0.?). with 01(t) -Is - x 1. news

51. 1 (a) fo b t. for a 4 t a omrtapedla awquost btds. 1%40

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amo .e ath oets luffatoa) - off1.t'aiI 4 aie - a I I sad

0see - afsla 4 of j5It - t 1I (aft (411# estimate (3.6) telirns.

Datiftewthesep COOP6) by

t

4 II)Am Em 13.61 and ee ane usaw essmaSA r ois. List a prove the

I t

M111 0(soa

1211 19 a sur tIf 16e,71d -0 48tha1" 1

Ume EISA) 4 241) sad L~As)S t~ or Menet &al a@ 16,1n, it 611amw

ftr- %me Labeso e isidum~, them t~ to ON& v * thete moroeedtes

0 isf taen

12.12) w (eiO)da ( '0

Vur re to e 0 > 0 there corrsponds a A 2 0 such that

( 2 13 ) m a ( t - t SI Olz - l 1) < 13 ) IU ( S~ t x ) - U (s ,t i #X I) I < C o0

uniformly with respect to I thls follows from (2.6)- IOO

(if) 1 *le,49Utt))ds - jt I(s.g(st,.x,))|M < 2V0 0

if MAclit - t11I Oi X1I -mia(A IA . fbs equicatinuity of the family O(K) i

proved. Tbs last statmnt falloen fro hecoli-rsea's oaoiaety theorem. 0

Thoorw 29 . fte m 0 1 K K bee a f ine Point .

Pof It ONaMSe ptoroVe tbOe ontinity of the Oa 0. lst S0 6 eS *

mfo m y m as 4.DOW" by W. the solutin of (1.3) with data $ it is clear thatS0 v unifory asa?.Lt £ 0 b given and % be ah that Iv- %IlQ < 9

* 2 Be. 90% Ws) - U(Stu). VD() a U(setex), and p(s) - IX(o) - a (s)I,

Awe % demotes theesolutioai 12.S) with v replaced by V. Por ) w one has

Ipelal 41 Iue) - X6()l 4 € * o1 IaX() - %(*d)l xlliis) - xW1)I. Bance

1014)1 4 a o + xo l Q%) because 2X(r) I an Lo Lag fusotn an (0,). Noreoer

*It) * S. OeMsqeu tly |O4o0t.n) - 1m6o t*x)| 4 ?(a * o1U9x(61)1 V a 9 (0,T1, and

0s(to,,x) 1 0LtnIon0ml moms to D(st) an (O,.]2 x 6l, wben 0 - 4-. It follows

"will from MG)0 and 42.12) theat Ca 0 9 uniformly in %~, Where Ca a(On). hotually.

It Duffl ee" to vow te poistwiss moverl e of t to Co uniform convergence follows

then Irm t"e C mpactes of swhme,, of $fl). 0

jL&Ta. bs "a aebmh of prowl etron continuity of C(t) in C() seems

net to work I& Older speces, evs if f a 0. ta tat if C0 9 C0 (6) we cannot prove

tet C*U(t'll) * C(.bCOA )) by asiag (only) regularity results for U(t.x) (other

arusamt mest eventually be addeud in fact, it C(l a '7 and u(t,x) a t - x the

twocics 41toa) a COiUlte)) verfies it) - C(tal - ;C(t.y) M :.y)I - Ix - Y1 1/2

if got. yt.

-9

iI

9 4+

fte situation become woere with respeo t to the strong continuous dependence on the

data.

Now V verify that the functoe v oorrespoeding to the f ixed point C is a

solution of 01.1h see alao (41.

We start by eowiAg that for fixed (set) the mp a* U(st.s) is mare

preserving in 0. Lot 0 e S C1 to 1tjC1 (6)), on 0 u lo&ly a t . it v i

the solution of (.3) with data s mee bag v a (O(TIOC I ) I I and div va - 0. soe

x* ist.,x) Is sear preseeving. On the other band we kno from the pro of theoren

2.1 that U. - U fonmly on 0,-]2 x 6. it followe that U is easure preeerving.

Vor, define 1% - U(s~tex). Tmx - 0,(@Ot#u), x 0 and let 9 be a compect subeet of

A1 and A an arbitrary open et verifying ?(a) C a C 0. Meslg that Tax* In

uniformly and that tIM) Is o op e ow that there exists an ite e each that

I (a) C As bene IT (a)J- 9 19 4 A onequently 191 4 S~i, -ee |*g demotes

Imbeaque measure. An analogou property bolds for the no 1y - O(t,.y). beo the

measure preserving property bolds.

Line 2.3 . € C 0 be t find osatrusted above. 'bhi

Proof We show that

(2.161 L (got) * (;vY) * (9,?)s v Y a 1).

D ng by C 2(tx) the second toer in the rimt bend side of (2.4) and taking nteo

ac ount the easure preeerving propert one get, by the shnge of variable y - U(st., ),

t

Nonce

d t(C201) f(tr.yydy do I *(e~ylv.t~yl*y

0 0 ABonaa

and returning to the variable x - (toy) in the last Integral one get (42.16) for C2.

One argues similarly vith the first tom on the rijbt hand aide of (2.0).

II -10-

1"

L w 2.4. v 4v 1 2 (o) d01v,& & 0 ad _V=0 o 0 .

rot v . ram rotl(V)v3 a div(vc) 12 the son" of diatributioe in 0# i.e.

((WI)v~rot V) - (M.O1Y) V Y C"(01.

po i. a direct omtaton am that for a regalar v, say v 0 C (0), the above

equatio bolds polatwis. For a gemeral v osider a equee of regular C Eu 8d that

C In C Is L a2CO) Deaoting by 9 the solution of (1.2) with data C and" deani""

v- not 9 t followstbtat v *v la 1' 2(a). Ibis allows to pass to the limit

whem a* # in Atheab" Week fogm. 0

now we verify that v is a solstie of (1.1). Coearly %v 9 W(OMIsLP(M)),

9 p 4 00 rWOVOWS C 91Sj 2 CO)) hescefrnlmml2.3 amsgts

3; 1t a (0.YeV 1 '2 0)). aosaiUlig that 0 a C euatios (1.2) yilds

-&1mf4i1 a sl t is as atfit a * as r. ammammay wflt e L is,.,,2l'(a)) an4

*widt a IPst(&$it) 6 Ll.(0Y,1 LA)). is a 11/&W + (ve)v - f a LI I0,5L2()).

Moreover, t($Vwit) + (vDD) - f- a 0 in the rLdbttlOSJS oaee I lame 2.4 and

2.3. Comeeqetly there Mj*WA , 41 L I vowl1,2()) that (1.1)1 bolds. 0 the other

ban4 i41* i i.e. o a C L-et co La O, daV fm-diWv0-s Ro lA, and

wieo -a ve -a em on '. easev - v* 0 .finall the emiquemes afth Owolution

v Conom a S mrdoe 111 th m 2 slae fee emvery p 12 + a1 the eetimate

lVl)li 4 e ,It)iP) bolde, this follow from (1.2) and frem well n eset teem

for I*&Upt partial Wforential equsotoseI LO iP epeses for inetance (31. theorm

2.1).

3. too fteoe .. Z this setion we write C a 0IC,* instead of

C a 9O() since Ce e # ae variable. For convenience we demoto by 01, 3

respectively the gqp v - 9I0I detimed by (1.3), - 2( l deftni b, (2)S) and

C * 3 00 o#) dfined i M2O). Ieaee *(O.C0*) - #3 (902 (9 1 (l),C 0 ,t). the mp

Is defined for every 0 A,. 0) a Cas) a aff) I. i(6.18c(5i. vote that V is the

M0Sotlom of Problem (1.I) if and cal - 1(C) for a C verifying C -I*C,00.

-11-

• ,jA

Ibeowem 3.1. b , e A relatively-aimot Ot An C(A), a2 " ultively copfat

se I LIRIOYCCU)) Ad aha o is C(67). Ibnt ft # Ki * 1 W 1 x 2 ) it

relatvelyomit lk)

Let 'I. % as" IL be Contained Is balls with cete is the Ocila ad

radim hki, ka a"i 91 mepeaively. fte got of ftnotIMe ,O(D(@,t,x))* for 1e 0 1

and C K.I. is heemd in 41) by k1. 91 the mesear esDFLtLmof .1 &oLi-&Arellae

theovem the tematlefe4 N eo eqletAWm I By 12.6) the femothas

u(Sptea) ewe eq iaain M emos the family M*W(Otax)) isa euosatimmes s

aaiby AaoiAaletbeore, 0eometutee a rotatively omact Oet inLa )

Amalogaely the familyr

t

Is bamed by h2 In C62). w e ova that every 1egeemee otalms a

o-- "I,gt etheaoe in C(§,). Ibla gevue comatsems for tdo fomlly (3.1).

Net I S41 ad #n 1 2 he arUr sameo m'omi

(3.2) ;U(tux) -J *(8.U (epton))d

Dy the omeates of % 12 hr inlte a subseeemoe of ad a funotion

9 L 1. (601104) ech that * JA* La I (GOVICt(E)I 14 ), forver a wel, Iau theorem

emeuces the ealetemoe of a euemam sa& that

(3.3) to (see) * #(so*) In ctal for alse" all 6 (0.to?)

Daeot by *(seg) the metltn of countinuity of #U(m,.) ina losie (3.11)) ad

*0. 1e~n u a a" a (14). ftem (3.4) aid ECM AeOoIL-Arsels'u theorem it follow that

(3.5) no j(.6) - a. for aast all a a (0.?)eg

(4)v Par cmemee we moe the auma iabs a for sems ana ftw subeeem.

Now let k=1 b e a sequefce of real pelttva nmbers such that % .

a a

&imoe ~ ~ inci L OTCd)taemnsasbeumo# uhta

0

40Defime b,(e)I is(a - e 6 (0) a ilI Clearly %o Le iategrabl. Over 10.?).

Moreover wk($A) k S (a)# 4 2$.l *a(e) I ab(s) heoce ;(*,g) 4 3b(m) bee

to eftae reepect to the eeheeqmem a k and b(s) Le iat.grabte. Sy *&Lag (3.5) and

Laees lwamated man aerwemo theorem Lt follow that to every w )- 0 thae oorrempoale

en a Blac edthat

ST

a 04,60 )lds cve W k v

Uqaatiom (3.6) gumeralee (3.12) Is the proof of theorem 2.1.

On the other heal, by the heendaebege of 16 the functions vk " U Wf 24

ad (2.6) inileumiy with respect to k. Neaf. (2.13) holds for every Uk with

A20 2(90) an~paat of k. We am proceel as Ia the proof of theorem, 2.1 =A we show

the m fst"aety of the met of famouise" 2 k t ni (mote that (2.10) holde

unkfozuly with reepeat to k, siamsI k() 4 b(s)). From the equicontlauity follows the

existaft of a subeequence com eg 9- a 0(i,).0

"Wce 10 2* no 0(1) X C(O) X L (0,71C(d)) + vi? ) Is ataus

IVW! tat (0 A (a) A 1 *(019 ##). Acquing sa La the proof of the contirnilty of

the amp 0 In theorem 2.2 meon e that we 5 #,(0a) '0 va VI(e) uniformly am 1

conseuentl us a v)*41 #2(v) snlfozuly 'a 10,?) 2 xl A- ov one easily verifiLs

that (% *, 1 1 4(m), A 0 c a 0 MAO,) poLstwia. La elacei

3 0

tt4 J5(sU~~ ) ()3e1 (, 3 at))-*.UetuId

0 a

0,0

NOW by miSU theorem 3.t 1With a O) K,(00) sad w (~ it folLow that the

oosvezgsmoe of C a to C is ""fors to (this as be ahown without reeowt to theem

3.1). 0

keel of theoren t .2. Asme the hypothesis of theorem 1. 2 and put C* rot

0 irotf to rotwe C 0 a Mot . *8 Mr tn, Ca lout. 98410. Dytbe

0~ 0

L fSe'votl)).

Detumite a,(e) - (C~) %m teos (2.6) it toile" that a set

03 1 8) to bacmded wbaev 0 and 82 are bomadede Indspendently of the

pwar lar set S. Womeequemtly X Ise boanied because CS a 3(- (1(s)I 0)

Vs 0 4. Now theosem 3.1 shom that *1 (KO 1.K2 ) Is a relatively oapct set is C(OJv)

homce a C 0 1(K.K 1 9 2 ) verifies the sam property.

Let C, he mny convergent subeequeam of C a ad put for convaesem CUis C

Pro. the idestity C, a-, (C VC ( ) ad tros theorem 3.2 it follows that V4

C0. .) 0 eeusl isC ia aolutiosof (1.1) hesoe v v and

C- C *It follows that all the eusC a ossverse to C osifoguly is C(AV) i.e.

vaO i 0C(ISfu)). 0

WAS~ 3.1n1 theorem 1.2 Coosvergem Of f~a) to t iO 00t reqUested since V is

determined by system (4.2). Oeverqsnce of f~m to f in L iI (0)) would implythe additional convargso Yw * wt in 1 2

a LlO(L (11)).

4. &ro" Of teorem 1,4. We start by grOoming that coSPOeltoO Of C-fWotioft. With

Older catisuous functions yields Ce-fursatioma.

ww"m 4.1. a 0 C*(l) adU a C1(did). 0 CI 1. Tbsa .U 0 S A

0 % r

is articular

If RUa mi the aomd fma m tho rLdt be"d "do of (4.2) is -orngd Lt

PcoL ft C Ua * Uo (U1& 5 go One easily reati that

we

It= 4 L I ) (01C4 ad Mass)S

a atx fsoatuM

?ban4j c a. 0(QIlCC§)ao

2 , T 1e a foetiamern 1rn ,ariia

tt(4.4) ist~) I I *(e.W~ut sw 0

Ibi C6(tir,to )) m ea

bt VWith straigh1tforward oaloulatims one doAq that

(4.4 ((t), 4I del e t(s.u(*,tfx)) - (a.u(setoy)I0 0 9-(fx-v(t t

t

0i%,t

bere 0l) a is,*) and ua It Us't.*). my "uLg (4.2) oa gets

(4. t)(* . 1 (j. . f wel(l-)"(' s.

L.o. equation (4.5).

We am POW the contaInty statement. Asume for instance t o . oFrm dateinitiom

(4.4) ome gets

•t It

(4.8) (C(t) - C(to)* • I de I eu I$(mU(..t~u)) - $(eU(et,yl)) dr0 % 0 O(5a3-Y14r

+ do/ a* I$(so.u(et.)x) - $(eO(e.Ox)) -0 0 0(CIz1Ct

- ~~ ~ ! 91ftai% * *2~

As for (4.5) WO aho that 4 1 is bounded by the right band side of (4.7) vith the

interval (0,t) replaned by (tt) hne 51 goes to sr when It - tol goo to

Sero. we am prov tht a2 *0 a t t o oA tion (4.3) yields

V(tOter) 4 2r'%# (a) (K is) Where P(t%,to~sr) is the Lntegrand in 32. "e above

fwntion is Lnut ablo ovr• (0,) 10,R) ainie for anost all a 9 (0.T one has

0 3 0

as one shove by arguing as in the proof of leon 4.1. Moreover for ever e (OT for

Whiah #(a,*) 9 cdu), an for every r a 1oal, one has Li lto 0 totr) - 0. An

application of Lebequeo's dominated convergenc theorem proves that a 4 0 if t t 0 . 0

2o0

IM~e .3. l 1) if Z ULU ie .assmion s of the Orcedinae laws. lot Co 0COO

4aaeJ CI t(x) 1 0 (0Otx)). Then . C((O.eT|C*(I)) moreover

* 6(4.9) [Cl I , a 0o

it Rti te (4.9) follow from 1i 4.1. the oontLauLty statement follows as in

the preceding lam (vith wany slqlpftcstLows).

3quations (2.4). (2.7), (2.0) dsfinition of 4 and the two preceding l s give the

following result I

Lm.4.~ Ase got kweOths"i 09 theorem 1.4 hold Wn lot C a rot v,* 9 c 6 t g rot v 0 T. " 4 C(3C.E)), moreover for every t a

(4.10) 1C(tl , e 0) ( 0 ( I) 0 O ci ) * 31 g08 + 31 01L (o.t Ic())

wher a • 01 + 1 t(Ott (i))

The followl theorem is cruiaRL for our proof.

fteorem 4.5. ge O4 C, (5) an lo h, e the solution of urobe Q(-2). Then

a C 2(g), moeoe

(4.11) 1" 2 4 cool. V O a C(I)

this result seem well known even if an exact reterance is not available to us (see

(2)# chapter 4, problem 4.2); ve are able to prove it for a uniformly elliptic second order

equation Lo - in a, s a 0 on r, at least it L has smooth coefficients and the

boundary operator % is regular (for Instanoe Diriohiet or Neumann boundary value

problem). ?his result doeenot depend an the dimolon a v 2.

The win statement in theorem 1.4 follows lmediately from v I Eot 0 and from

(4.10). (4.11)o recall that 0 - C. Moreover if g and VI are continuous in it

follows from (5.3) that V2 is continuous, from Yv - Tr that V8 Ls continuous

and from (1.1) 1 or (5.2)1 that 3v/St is continuous. 0

-17-

A.

£AmeadiU 1. e recall sone well known tats about vector fields detLed Ln sam

shaply-coaaed domanls. lot 0 be an (W + 1)-tla onnected bounded regioe, the

boundary e come a to of ouAo c.losed mrve r 0vres...r. the oune r

oetakala the otrs. In that os the keg%l of the Linar yestm rot 0 Ia o

day v - 0 in 9, voa 0 an r bha finite dimsion U. lt m in a bas ele... 0%

and a&saw for mocaveolea that (uIa.u) - iah k - tos.. 1. ay taagtt" flow

(veetor fiel verifyLg Ulv v - 0 in Go von - 0 on r) Is uniqu ly deterLmod by the

field "t v La B1 m4 by the real mers (Vwluk), k I 1... to. n. quantity

ae - Iat V + I(v'up)l t a noa L 3(A). equivalent to the morakh,

I rot v + I v 2 0lt no f be an arbitrary vector fid La A. Solve the rblm - - rot f L

a, %o- a ar u Uoe g st 0 . clearly o -t%- ,rt t ivoo-o and

g a a on r. if f m go + IAftu%, A I k a (took) .t oll~owthat is a

tmagential flow, Moreover "t f - g) o 0 Ln a# (5 - gook) a s, k a I,....,on. Nw

there enists a scala: field amb that I - g -, I i .e. the vector fild g Is

the tangential flow In the camonical decomoition

(5.t) f 1- q 1 .

Uote that g 4Pn. only on rotru f aud 0 tbe U reel be (ftuk).

1

- Is

Apo J a.lot w doomae the emtegel totes f insequatio (1-*1), es adicated

US. 51) mAolRt an omil the auxiliary pwobJlm

IVm

i V - 0L

Ihm geltins .1 (1.1) osasieta an the velooity MIoA v as Is (5.2) and on the

Pessur. tern Yo a Yt I YVI. Nmwver. Iro (5. 2) It follows that

IVI

.1

Aser that the rey"laity of Yv(t) Is komm. Ihes the *siLliti bmlery vain

Pgaum (5.3) givs the reguLaty otft Sod e (5.2) Oives the regularity of IV/It. in

portioula varios regslarity results foe Iw/St (Oel ftm Yi) are trivially obtainel byp

awrnig litfoeret ooeLioes on f. Seams the regalarity of Yv(t) Le the basic ose.

was by the way that Ye is the only tar leps"Lng fully an f. the other torn

osidre abwoe ImPes oly arOft * as as O (fluk). k usooo

III C. ardoe. " tmes at malitS do I& soluton do I .quation 4lul '5 an disaloo

dsuae. J. Math. &a". Aggi. a 11972), 79-790.

(2) D. Gilbsr. U. 5. Yfuler. "B11pti* patial diffteatial equsatlome of emood

oudsr. ftrLa~s-Vezlag DIli. Uaidelb"r New Tort (1977).

131 T. 1. juieviobs "Moa-Ottloaery flow of a SIsal Lacosee lLiqmid Uur. Wyea.

Not. A Mt. 11,. 3(1962). 1032-1066.

14) 1. nato. 0 oaeiocal oautiose oil the 4-4--ON RM-UUtOaary WeArW

equatimap A"&b. rate Nsdcb. AMal. A (196?). 166-200.

15) p. 0", "gut lllu= d=e fGOmtIOuG de fta Ot do Ua demW W woisinae ft

CoMaau" he"a Math. 42 (193), M7-2679

(6 J.. C. W. Naguto, Emltmso ad =miqumm. theorms for Awaod Sampre"ibl,. foW,

4 Iheela U"*loIe (Vobruagy. 1906).

17) a. C. lobmeff at *xmiame theorem foc the fi., of sa ideal Isaapueeelle fluid in

two inaiOftg% lvi. am.e Raft. S. ai (193?). 497-513.

I W. weiibmer "me thsoinem O- l..iuteae do Mu m plai dove eluids part.t

haimogi. eupea . Iedn m tow Laialagt lougurn". math. 2. 3? (1933),

-D/ed

-20-

.. . ... .

ISUMTY CIAMPICAww" OF Tla PIaE f w' - awo)J

REOT DOC ENTATION PAGE _________________raw5. EPOUTNUMUN 55 ACCUSIO 3. EPEOr C€O.~G irNSE

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. DZN8ZIISZML FLOW OF A NOVIS8 ZNIMWP3ZSIMZJ a. pw6o M Ione. won MuuRO

. MjTWOR4 . GAUTNACM T ON GANT iWUM5E

H. Dirgo a veiga DAAG29-80-C-00 41

Mathematics Research Center, University of U UNI

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I. CNTMLUO OPPIC8 1AllE mIN A@.S IL EPORT DATE

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UNCLASSIFIEDM~a& ~IWCATIODWNGR0 bU

M UITUUUTMN ITAVrl T (0 W MOW

Approved for public release; distribution unlimited.

1?.W8TIU StAtEME"t (oSf Me~ aw=oein we Usia, w wft~ am Ra

W UPPIMESSTAYM NOV3

Is. ItEY WORNS (Cenobweso 80" ow 0890M MI Pei- 94 8wm ~~ own"*ad

non-viscous incempressible fluids, nonlinear evolution equations,continuous dependence on the data

IL. AMSTRACT fC Qft MWN i 0e- N9 000N AM. d~ OFut ~p -NO :.bel0A

We study the iEler eqpations (le1) for the notion of a non-viscous

inco pressible fluid in a plane domain ' Let 9 be the Banach space(V -,r 0)

defined in (1.4), let the initial data W0 belong to i, and let the external

forces f(t) belong to i4(Msg). Zn theorem 1.1 we prove the strong

D- 1? , 147 ca or I UNCLASSIFIED

SECUIftY CL"S VICAVON OF Twa, PA42 f. eft" MiM

20. A3WRA-r (coat.)

continuity and the global boundedness of the (unique) solution v(t). and in

theresm t.2 we prove the strong-contnuous dependence of v on the data v-

and f. Zn particular the vwrtioity rot v(t) is a continuous function inI

J. for every t 38t. if and only If this property holds for one value of

t. In thoorm 1.3 we state m properties for the associated group of

nonlinear operators S(t). finally, in theorm 1.4 we give a quite general

sufficient condition on the data in order to get classical solutions.

e.

0

00 i iiii" m i|I l