APRIL 30 1983(U) STANFORD UNIV CA DEPT OF MATHEMATICSJ B KELLER 38 APR 83 AFOSR-TR-84-0034 AFOSR-79-@±34

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REPORT DOCUMENTATION PAGEla REPORT SECURITY CLASSIFICA ION 1b. RSTRICTIVE MARKINGS

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Stanford University Air Force Office of Scientific Research

* 6c. ADDRESS (City. State and ZIP Code) 7b. ADDRESS (City. State and ZIP Code)Department of Mathematics Directorate of Mathematical & InformationStanford CA 94305 Sciences, Boilling AFB DC 20332

Be. NAME OF FUNDING/SPONSORING Sb. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER

AFOSR NM AFOSR-79-0134Sc. ADDRESS (City. State and ZIP Code) 10. SOURCE OF FUNDING NOS.

PROGRAM PROJECT TASK WORK UNITBoilling AFB DC' 20332 ELEMENT NO. NO. NO. NO.

j 11. TITLE tlnclude Security Ckawification)

: SEE REMARKS Ca l&,k) 61102F 2304 A412. PERSONAL AUTHOR(S)

Joseph B. Keller13. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT 1Yr. Mo.. Daby) 15. PAGE COUNII

Interim FROM 1/9/82 T030/4/83 30 APR 83 1316. SUPPLEMENTARY NOTATIONI

17 COSATI CODES 1B. SUBJECT TERMS (Continue an reverse if necessary and identify by mbe~ jn

-FIELD GROUP sue. GR

19. ABSTRACT eContinue on reuerse if necessary and identify by block number)

This is a progress report of the Applied Mathematics Group in the Mathematics Department,

Stanford University. This group began functioning officially on 1 September 1979, and is,. supported by the Office of Naval Research, the National Science Foundation, the Army

Research Office, the Air Force Office of Scientific Research, and Stanford University. The

personnel comprising this group during all or part of the reporting period are Joseph B.v4] Keller, Professor of Mathematics, Stanford University; Russel E. Caflisch, Assistant-' m- Professor, Stanford University; John H. Maddocks, Acting Assistant Professor, Stanford

University; Meira S. Falkovitz, Postdoctoral Fellow, Stanford University; Victor Twersky,Professor of Mathematics, University of Illinois, Chicago Circle; John G. Watson, Assistant

' D Professor, University of Miami; Bernard A. Lippmann, Professor of Physics, Emeritus, New

York University; Kevin C. Nunan, Graduate Student; Michael I. Weinstein, Postdoctoral--J Fellow, Stanford University; Stephanos Venakides, Assistant Professor, Stanford (CONTINUED)

LA. 20 OISTRIBUTION/AVAILASILITY OF ABSTRACT 21 ABSTRACT SECURITY CLASSIFICATION

UNCLASSIFIED/UNLIMITED SAME AS APT DTIC USERS

22a. NAME OF RESPONSIBLE INDIVIDUAL 22t, TELEPHONE NUMBER f22c OFFICE SYMBOLI lnlciudr A 'a L odi,

Dr. Robert N. Buchal 7(,7- 4939 ,

DO FORM 1473, 83 APR EDITION OF I JAN 73 IS OBSOLETE

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ITEM #11, TITLE: PROGRESS REPORT, GRANT AFOSR-79-0134, SEPTEMBER 1, 1982 - APRIL 30, 1983.

ITEM #19, ABSTRACT, CONTINUED: University; Margaret Cheney, Research Associate, StanfordUniversity; Graham Eatwell, Postdoctoral Fellow, Stanford University; Luis Lopez-Bonilla,Research Associate, Stanford University; James Geer, Professor of Mathematics, School ofAdvanced Studies, SUNY, Center at Binghamton, New York; Michael Reeken, Professor ofMathematics, University of Wuppertal, Germany; Erik Hansen, Professor of Mathematics,Technical University of Denmark, Lyngby. The various research activities of the members ofthis group are indicated by the list of publications contained in Section II. Abstractsor introductions from the papers produced during the report period are contained in SectionIII. In addition to the manuscripts already submitted, a number of projects are nearingcompletion. Thus Professor Keller, while at the Woods Hole Oceanographic Institute duringthe summer of 1982, studied the turbulent diffusion of a chemically reacting substance.Professor Keller and Dr. Falkovitz are completing a paper on Liesegang rings to account forthe recent results of John Ross and his co-worker in the Chemistry Department at Stanford.Their theory uses the fourth order Cahn-Hilliard equation to describe diffusion. Dr. Eatwelis studying the propagation of an acoustic wave through a bubbly fluid, or through a mediumcontaining resonant scatterers. Dr. Bonilla is studying the effective behavior of an elasticsolid composed of randomly oriented cry-tallites. Mr. Nunan is calculating the effectiveviscosity of a suspension of rigid spherical particles in a viscous fluid and the effectiveelasticity of a lattiqe of rigid spheres in an elastic solid. Professors Geer and Kellerare studying scattering cf waves by thin bodies. In December 1982 Professor Keller wasinvited to deliver the Weizmann Lectures at the Weizmann Institute of Science in Rohovot,Israel. In March 1983 he presented an invited lecture at the British Theoretical MechanicsCollt quium in Manchester, England. In April 1983 he spent a week as Distinguished VisitingProfEssor of Mechanics and Applied Mathematics at Cornell University.

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r Setemor , 1982 - April1 30, 198i

-App) I jed i1;itllc :.i t i C. Group

1)epa -riel t 01 a theiua ticsStanford Ilnivers i t

1PricipaL l nv est igator: Joseph B~. Keller

1. In troduLc t ioni

LThis is a progress rcport of the Applied Mathemaitics (roupJ in the0

,liemat ics DepartmuIent, c)Ldford Univers ity. TiJs 01-01.I) began functioning

officially on September 1, 1979, and is supported by: DTI TAB

Unannounced [

1. Office of Naval Research;Jutfcio

2. National Science Foundation;Distribut ion/

3. Army 1,esea-rch, Office; Availability Codes

Avail and/or

4. Air Force Office of Scientific Research; Diet Special

5. Stanford University.

'1110 personnle I coui11pr i.s ing this group during all or pairt of the reporting

periLod ;ire

1. Josph B eller Profsso of Mathemtics, Stanford University;

2. Russe F I. Ca f i scli, Ass islant Prof-essor, stanfor01d tlin ivrs tv-

3.John 11. M~hkfock5 , Actim! A\ssistant Profe1'ssoI, 11InoII liVerS i Ly;

4o,.1 Mc~ i 1,; S. Fa I kov i t Z , Pos iOC. Loat L e low, Stiiford Uin i vers- i t\

2S. Victor Twersky, Professor of Mathemtics, Universit, of InoS,

(;hica11o Circle;

0. John G. 11a tsoi , Ass is tan t Pro essor, lhiitvers .[ y of U iu i

8'~ 02 0 13. A -

09.

* n.J.2

477

JlosepTh 11. Keller

Progress ReportP'age2

7. Brnadrd A. iLipIIiani, Pro lessor of hscEirt:,NwYr

University;

R. Kevin C. Nunan, G1,aduaIZte Student;

9. Michacl1 1. Weinsteini, Postdoctoral Fellow, Stanf'ord Un ivcrsit;

10. Stepnhanos Venakides, Assistant Professor, Stanford Univers ity;

11. Margaret Cheney, Research Associate, Stanford Universi ty;

12. Grahiam Fativell, Postdoctoral Fellow, Stan ford Uniivers ity';

13 . Luis Lopez-Bonilla, Reseirch Associate, Stanford University;

14. Jaes (Geer, Professor of lathema tics, School of Advanced Studies,

SUNY, (:enter at Binghamton, Ncw York.

15. .Michael Reeken, Professor of Mahthemalltic.s, Univers-ity Of WupJpertal,

16. Lrik Hansen, Professor of' Nathem'.atics, Tecluiical Ulnivers ity of

cnimark, Lyiighy.

Thei variouis research activities of the members of this group are

indicated by thle list of puiblications contained in Section 11. Albstracts

or introduIctions fromi the papers produced dUring' thle report period are

contained in Section 111.

Ini add it ion to tile malnusc ripts ad submtted, a Inlulbe r of proj ects

a reC ine' ing ComiplIe tion. '11lls Pro Lessor IKce1.[r, wh ile at the Woods IHole

'Jccanol'raphic Inst itut o dur 1ini, the simer of' 1982, StIUd i~ edth turbuhLenCt

di ffuision of a Chem icallyv reactill nUI)SUbsace. Pro fessor Keller and Dr.

Fa lkov itz are conpiet ing I paper Oil Li esegang rings to account for thle

copy available to DTIC dOes riOt

por~i kuiW legible septoductiOl

.. e, %

Joseph B. Kellerh' Proiress Report

Page 3

recent results of Johin Ross and his co-worker in the Chemistry 1)cpartimnlt

at Stanford. Their theory uses the foujrthi order Cdli-Iilliard equation

to describe diffusion. Di-. Eatwell is studying the propagation of an

* acoustic wave through a bubbly fluid, or through a mediui containing

resonant scatterers. D)r. Bonilla is stud-,Lag the effective behavior of

an1 elastic solid Composed of randomlIy oriented crysta .1~i tes. Mr. Nunan

J. ~is calculating tile effective viscosity Of Za suspeRs ion of rigid spherical

particles in a viscous fluid and the effective elastic ity of a lattice of

rigid spheres in an elastic solid. Professors Geer and Keller are studying

scattering of waves by thin bodies.

In D~ecember, 1982 Professor Kller was invited to deliver the 11eizin

Lectures at the Weiviiain Institute of Science in Rohlovot, Israel. Ini March,'

19183 hie presented mi invited lecture at the British Theoretical Mechanics

* (:ollojuiLBii in Mancester, Eingland. In April, 1983 hie sp ent a week as

Dis tingzuished Visiting Pro fessor of Mechianics and App lied Mathematics at

Cornell University.

]. RE!CIN!Y H []BI, IC.ATIO)NS 01I: ;''1: Al)[',1l:l) G.'IIL\ I 5 (ROUPII

50. J. B. KelLer Op I ,ir inaspec Lion pol ic ics

I_ Management Sc i . , 28, 447-450, 1982.

o.5L. P. S. Hlagan Tr:,'el.iig n .;Lc'k(.d travel] ilg wave

soItLiolls of parabolic U(jat tiOlns

- Pub: SIAM J. Math. Anal., 13, 717-738, 1982.

52. J.-M. Vanden-Broeck Conact problems involving the flow past an" inf lated aerofoil

P 1ub: J. AppI. Mech. , 49, 263-265, 1982.

53. J.-M. Vanden-Broeck Non inar Lwo-dirnensionaL sail theory

Pub: Phys. Fluids, 25, 420-423, 1982.

55. P. S. lagan Spiral waves in reaction diffusion equations

Pub: SIAM J. Appl. Math., 42, 762-786, 1982.

,56. J..-HI. Vandeia-Broeck Parabolic approximations for ship waves andJ. B. Keller wav,, resistance

Pub: Proccedings of the Third InternationalConference on Numerical Ship Hydrodynamics,Paris, France, June 16-19, 1981.

-S

57. A. Jeffrey Asymptotic Methods in Nonlinear Wave ProblemsT. Kawahara Pub : Pitman Publishing, Ltd., London, 1982.

58. 1l. .1. Miksis Rising Bubbles

J.-I. Vaden-lroeck Pub: I. Fluid Mech., 123, 31-42, 1982.J. B1. Keller

J. . C. Noll l':;IItiI I V illLi' rl,' ilW} \¢,.vt'S

hi ll S1 :1 .I. AppI . t-h., 4i_ , 4 , 1.4L- 150, 1983.

60. ,J. B. Ko I Icr Surface tension driven f lows

ri. .1. tliksis Pib: SIAM .. AppIl. math., 4_, It, 268-277, 1983.

U 61. A. .e oI I:l (IS ill Sl( (t-iol (,f two parameter systemsA. Spence and their ca lcu[ation: Part I

I'i1.: Sta lford Univ. Numer. Anal. Report,82-02.

I . .. :, .... . ' v .- .. .

62. J. B. Keller Tim dependent Clue1es

Pub : SIMI Rev., 24, 401-412, 1982.

63. R. E. Cal isc h Shock profile solutions of the Boltzmann equationB. iolenko Pub: Coilun. Math. i'hys., 86, 161-194, 1982.

64. P. S. lagan Arrow's model of optimal pricing, use and

R. E. Cafl[sch exploration of underlain natural resourcesJ. B. Keller Sub: Econometrica

65. R. E. CafIisch Radiation transport in a hot plasma

Ace: SIAM J. Appl . Ma Lh. , in press.

66. J. B. KelLer Jet,4 rising and falling under gravityJ.-M. Vanden-Broeck Pub: J. Fluid Mech., 124, 335-345, 1982.

67. R. E. Caflisch Fluid dynamics and the Boltzmann equation

Pub: Stud. Stat. ech., 5, 194-223, 1983.

68.1 N. S. Falkovitz Theory of periodic structures in lipid"" Soul bilayer membranes11~I. K. Frisebc,. r- Fcn Pub: Proc. Nat. Acad. Sci., 79, 3918, 1982.

,11. 9. McConnell -

69. R. F. Caflisch The fluid-dynamic limit of a model Boltzmannequation in the presence of a shock

Pub: Inst itte Nationfl1 de Recherche enInformatiquie et en Automiatique, INRIANo. 81, June 1981, 1-34.

70. I'. I'. Lide:-RoLui sso Oil the ,;hort- tira'e wusves d,,e to haltf-immersedc ireu I" cvi i er osciIlatiug on water of

if inite depth

u,b): Proc. Royal Soc. London A, 384, 333-357, 1982.

.4

,71. P . F. Rhodes-Robin sol Nti e oil the refl xion Of water waves at a

• .wll in the presLece of surface tension

ull]b: Ma th. l'roc. C;.mmbridge lhilosophical Soc.,

92, 309-373, 1.982.

ANot supported by AI"OSR or ONR.

4.

72. P. V. R1hodes-Robinson On lie, )eneraL ion of wa Ler wavts ;it anj

C. C. Papanico taoti of f c Ls,

AL Ac: SIAN J . Appi . Math. , n press.

74. R. F. Caflisch Instability in settling of suspensions dueG. C. Papanicolaou to P-rownian effects

Pub:- Proceedings of conference Two-Phase Flow.

75. 11. E. CaffI.i sch Shiock waves and the Bol tzmann equation

11.NiclackoAcc: Proceed ings NSF-M-IS conference non-linear

76. J.* It. Maddocks Resltric ted quiadrat ic forms and their appl~i cationLo .fklrca tio n and stLabil ity in cons tinmedvariational principles

Sub: SIAI4 J. Appi. Math.

77. M. S. Fa [ovitz Spat iall~y .tinuogencous polymerization inL. A. Segel unstirred bulk

Pub: SIAM J. Appl. Math., 43, 386-416, 1983.

78. M. S. Falkovitz Thc scalc of non-liomiogeneity as defined byJ. L. Frisch diffusion measurements

Pub): Journal of Membran Scjnce, 10, 61,192

79. PI. S. Fa L W vi t Op tiina.1 catal ys't diStribution in a mombraneIt. L. F'L isich --

A cc: Chemii. Enig. Scli

10 . I. S. FIi I :ov i I Iri~ ih or nic

J. It. K\I C 1 i .c c L.Iior.Bo

81. A. Jeffrey Terandomn rho ire jeoled and the f -ce- suirfaceJ. MN1j.1 xngi wLer profi ii ic fLr the collaIpse of a dain wall

71 Pt t:1 Wave Hot ion, -s,~ Mh-389, 1982.

*Not supported by AFOSR or OINR.

82. .1. C. W.Ison A : t i.st i.cal. theory fur iiiperfct 1 hi fuzrcaL ionE. 1L. Reiss Ri'b' S[AN J. Appl. th. , 42, 135-148, 1982.

83. .. G. Nat son Rel !ect[on, scati-crin),, ilid absorpLpiOaofacousticJB. Ke lleC waves by rough surfaces

e.To be subuLtted.

84. t. 1. Weinstcin Clol,al cxi srcc for a gencra ized Kurteweg -

de Vries equation

Sub: SIAMI J. MIath. Anal.

85. H. I. Weinstein Nonllnear Schr6din gkr cquations and sha rp[nt,.rpohit iou estimltes

11111C: Commun. ia LiI. Pliys., 87, 567-576, 1983.

V.

86. 11. Cheney Two-dimensional scattering: the number of

bound states from scattering data

Sub: J. of tath. Plys.

87. . I,. Bonjilla Reltxation oscillations, pulses, nad travellingA. l LiIn1l w;Ives ill the diffusive Volterra delay-differential

equi Lion

Actc: SIA J . of Appi . Math. , in press.

88. P. F. Rhodes-Robiuson Not( on the effec t of surface tension on waterwaves at an inertial surface

Pul: J. Fluid Mech., 125, 375-377, 1982.

89. J. B. Keller Weak shock diffractionJ. Hunter Ac: Wave otion, in press.

•.O .,1. k. IK&I Ic'r I '.f v I n11( i' ,, r Ihi;lk fretilut.Cv wa'vke2;Ac'c: Comm , Plunl' t d A1)I. ItLh., ill

91. . B. ll I cr A:;yuiultoti.c analys.is of viscous Cochlear model.T. C. N'u Sulb: .J A.S.A.

92. .1. r. .11I.r I';,r;Ibi I(- ipproxhit.t ijl for ship waves and wave

' Pu : !','.. Ird Iit I l. lf . ,,Ii NI,1 1l-ur ic.ll Sli.. J 9 1

A.. 111 *.. r - c, I I co-ti ,Itio l tLiI1 i, p t o

lkCb Ie tiClIt oC 1l1t CY )-;I I i 1 Mt ion 01w Ntir ica1'Ictliods, 1(1-183U, londlon, 1982.

91. .1 B. Ie I icP B tts por)oc' Ias t icit Vc(I Citiol OIL 1w ollioi',cn i z'ati onl

- l~. Brr id lb pi i iur Itt lure Notes, ill MIhCrOSCOp)ic

Propcrt ics of- li sorcicred Iedi a, NY, SI- 57, 1982.

()5 .. B. Kellecr Cap ii ' waves Onl ;I vortica I jet

Ac c: I. Fluid ?leci. , inl press.

9o B1 . I]e I ici' t-t'Irv 1-vzI I eS timn til ui IU- ceiisored dat aA. .ii ttinreSub Ji. Royal Stat Lst. Soc., Series 1B

9)7. J. B. Kelle r F igeniti bis of slender cay it jes and waves in'J. F'. Geer- S]lnder tUlbeS

lB To be sillitted..

9. J1. 11. Keller Vailuation of Stocks ;I11 opt ionsR. Vo ronika To be submlit tc I.

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oil aliber ofparailoUr -ic funlction:-; (f Ow p~tin conictra Lionl of thc

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Tiii ri.rnrn Cho ire (IUc) IIWLlIod khIe 10 CIiI. 1 1u .1 1 ,kiibe;,1qIzcIItI Y mod Lf it'd by

C (lorill t "1~ ;11( "n odl j, . i s j)1 ed U' tm slil Iow~ waer eqtaL ions of Stokerx [4,

ill o tduir Lii (Ict vvi ot tht, C 0' .ii' IWee(Or -u ilf Of r w1I cr, el! it fMin ' L I 0)) it

-~~~~~~ po i(iII .ili t i ni', iiri a d.iiI] w. Iil ornIJd en v Tii a s ;; lt' ii e r t It

A) .I4; i necd 1)' LIIii!; ;itLIit'd ;IrI ill p.rd ' i t e it 1;1 y L i t, 1 1 re n . L' dIle to Stoker

for ti his cI asst ia ("I 1p)robi) em I d i%(1 'i -li mit., r ica L r es ii Its obtLai ned b ;I f tIr t ter

niod i I ie Cit . on 0o10f Lu I cGLi imm-GCIit io ii-w citi III i iv() 1.v L i, i siiimit LI Ii Fig o le ra t LO I. The

uiriilt)rv il I )ro Ilt"" ;Ihu w I ILIvLhe !;moo I Iiii !, a lid Ltle )general ri Iv c Lo se .1 gr t-,iI('nt

W it Ii . 101 r L.r su L i L I i il 0 the t I j. ILd , i S ai fa tOrn e1 C o 0f LIIthi 11od itI i '

L* I-kI t 1

s. ~*9 La L I sLi- t Ie L heO ry Vo f0rIIu rif ec L )i f IrI r t, i 0 1, i) y J. G. IsatLso n and LL

An ''nstL'" stLat i Lti C 1 l eL Icun io (Ij re) 7senIt ed to ai lyr . ;.e Itue e ffctLs o f

imlrne tI '(-Liils0, ;and o Lieor (I is t-irtbatices on tHeL h)i ura i l 0 1~ SO 11 L ion 0 i) f 110 I-

I.u-i irobiceius. Fir-st, uni formliiy Validi asy'fLJtic apprlO;r iml it iois of tule

I I. s i I lils a re o01) 1 a i ned ftr a:ityI rea',l 7,;l L i on o f Llit! i vipe C I mis. '[h a1S TCpp rex-

~~rn~rnat ions arek v;i Id :Is tb', maininiide iof tue z;upertert imle; iioiiiszeo. Th

1:1,>L i nirr s i'~tLjtiai*nr~n1V si itnutiy It ervi~

% ei!;iI y ;iii I .izoii re,,iil t'; fior Uiii' expect ecl va tuies and var lances of Lte, so lotions.

(:oi f i deuice 1 ii i t s Or al t I so i

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A itr bit cruid Dt imi Fur J1"b.(I L',-.i ti~ I' 'Iti iitd Jbr OwI ;%rriorrii ized

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Thut coiid i t lo)1 1 i ' i r ill to (itL I t ' iI i I tI)-v 1 1 i~lVel 11W (I.lv SO(IuLIOt

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A dl.i rp ,;[ I ici 1en t conl it( ioll I 'I-v I OI I t'N i sl cii~c i s ob VII i Ied f 0 i t(,

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vir ;It L 11; 1 tuo 1) r ty )1on Lo 0 1 L ini L Ihw "12 st ol tn' for r iss ic il interpolat ion

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