University LECTURE. Series. Why th e Boundar y of a Roun d Dro p Becomes a Curv e of Orde r Fou r

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2 University LECTURE Series Volume 3 Why th e Boundar y of a Roun d Dro p Becomes a Curv e of Orde r Fou r A. N. Varchenk o P. I. Etingo f American Mathematical Societ y Providence, Rhode Islan d

3 1991 Mathematics Subject Classification. Primary 30C99, 31A25, 58F07, 76S05. Library of Congress Cataloging-in-Publication Dat a Varchenko, A. N. (Aleksand r Nikolaevich ) [Pochemu granitsa krugloi kapli prevrashchaetsia v inversnyl obraz ellipsa? English] Why the boundary o f a round drop becomes a curve of order four/a. N. Varchenko and P. I. Etingof. p. cm. (University lecture series, ISSN ; 3) Includes bibliographical references. ISBN Boundary value problems. 2. Curves, Elliptic. 3. Fluid dynamics-mathematical models. I. Etingof, P. I. (Pave l I.), II. Title. III. Series: Universit y lecture series (Providence, R.I.). QA379.V '.35 dc20 CI P Copying an d reprinting. Individua l reader s o f thi s publication, an d nonprofi t librarie s actin g fo r them, ar e permitte d t o mak e fai r us e o f th e material, suc h a s to cop y a n articl e fo r us e i n teachin g o r research. Permissio n i s granted t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source is given. Republication, systemati c copying, or multiple reproduction o f any material in this publication (in - cluding abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society. Request s for suc h permissio n shoul d b e addresse d t o th e Manage r o f Editorial Services, America n Mathematica l Society, P.O. Box 6248, Providence, Rhode Island The owner consents to copying beyond that permitted by Sections 10 7 or 10 8 of the U.S. Copyright Law, provide d tha t a fe e o f $1.0 0 plu s $.2 5 pe r pag e fo r eac h cop y b e pai d directl y t o th e Copyrigh t Clearance Center, Inc., 2 7 Congres s Street, Salem, Massachusett s Whe n payin g thi s fe e pleas e use the cod e /9 2 t o refe r t o this publication. Thi s consen t doe s no t exten d t o othe r kirjd s o f copying, suc h a s copying fo r genera l distribution, fo r advertisin g o r promotiona l purposes, fo r creatin g new collective works, or for resale. Copyright by the American Mathematica l Society. Al l rights reserved. Printed i n the United State s of America The American Mathematica l Societ y retains all rights except those granted to the United State s Government. The paper used in this book i s acid-free an d fall s within the guidelines established to ensure permanence and This publication wa s typeset usin g A\^-J^, the American Mathematica l Society' s TgX macro system

4 Contents Preface vi i 1. Mathematica l mode l Filtration flo w o f an incompressible fluid The moving boundary proble m 2 2. First integrals of boundary motion Reconstruction o f a domain fro m th e values of its moments, and th e inverse problem of two-dimensional potentia l theory Results on the uniqueness of a domain with given moments (potential) The resul t o f injectio n doe s not depen d o n th e orde r o f wor k o f th e sources and sink s 9 Problems Algebraic solutions Algebraic and abelian domains Algebraic solutions The Cauchy transform an d its properties Singularity correspondence theore m Proof o f the theorem o n algebraic solutions Construction o f algebraic solutions 1 4 iii

5 iv CONTENT S 3.7. Examples 1 6 Problems Contraction of a gas bubble Formulation o f the problem Th e inclusion property Contraction o f a convex domain Contraction points An analogue of Richardson's theorem Dynamics of the gravity potential The gravity potential as the solution o f a boundary value problem Proof o f the main theorem Self-similar solution s Asymptotics of contraction Severa l sources 3 1 Problems Evolution of a multiply connected domain Statement o f the problem Integrals of motion Algebraic solutions Riemann's theorem Singularity correspondence, Proof o f the theorem o n algebraic solutions Construction o f solution s Reconstruction o f an annular domain fro m it s moments 4 0 Problems 4 2

6 CONTENTS v 6. Evolution with topological transformations Weak solutions of the evolution problem The simplest algebrai c solutions Weak solutions of the contraction proble m A sufficient conditio n o f a breakup o f a symmetric domain 5 1 Problems Contraction problem on surfaces Physical motivation Potential and contraction point s Calculation o f the generalized gravity potential o n the surface Breakup of the boundary o n symmetric surface s o f revolution Problems 6 0 Answers and clues to the problems 6 1 A few open questions 6 9 References. * 7 1

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8 Preface In th e fortie s P. Ya. Polubarinova-Kochin a an d P. P. Kufare v studie d the problem o f evolutio n o f a round oi l spot surrounde d b y water when oi l is extracte d fro m a wel l insid e th e spo t (Figur e 1). I t turne d ou t tha t th e boundary o f the spot remains an algebraic curve of degree four i n the course of evolution. Thi s curv e i s the imag e o f a n ellips e unde r a reflectio n wit h respect to a circle. I n Kufarev manage d to generalize this property: i f initially th e oi l spo t i s th e imag e o f th e uni t dis k unde r a conforma l ma p given b y a rational functio n o f th e comple x coordinat e i n th e disk, the n i t retains this property i n the course of evolution. In S. Richardson foun d a n infinite serie s of first integrals of motio n of the spot. H e proved that th e integra l o f an y harmonic functio n ove r th e oil domain change s linearly i n time. Thi s allowed Richardson t o give a new proof o f th e invarianc e o f rationalit y an d a n effectiv e metho d t o construc t explicit solutions. FIGURE 1 vii

9 viii PREFACE It wa s realized recentl y tha t thi s approac h ca n be extende d t o multipl y connected domains. I n thi s case, th e uni t circl e i s replace d b y a hal f o f a comple x algebrai c curv e wit h a specifie d M-structure, an d the conforma l map i s given by a path integra l o f a meromorphic differentia l o n the curve that has no zeros or poles on the chosen half. Below we discuss thes e and other interestin g mathematica l subject s tha t arose recently in the theory of fluid flows with a moving boundary. This tex t i s an extende d versio n o f th e first author' s tal k a t a Mosco w Mathematical Societ y meetin g abou t th e results o f the second author. Th e authors gratefull y acknowledg e th e crucia l influenc e o f Prof. V. M. Ento v who introduced them to the circle of physical problems under consideration. The author s would als o like to thank Prof. V. I. Arnold, D. Ya. Kleinbock, Prof. I. M. Krichever, an d Dr. A. I. Shnirelman fo r very useful discussions, and Dr. Y. Peres for reading the English translatio n o f the manuscript and making important remarks.

10 References 1. R. Courant, Dirichlet Principle, Conformal Mapping, and Minimal Surfaces, Interscience, New York, Stanle y Richardson, Some Hele-Shaw flowswithtime-dependent free boundaries, J. Fluid Mech. 102(1981), P. S. Novikov, On the uniqueness for the inverse problem of potential theory, Dokl. Akad. Nauk SSSR 1 8 (1938), (Russian ) 4. Makot o Sakai, A moment problem on Jordan domains, Proc. Amer. Math. Soc. 70 (1978), P. I. Etingof, Integrability of the problem of filtration with a moving boundary, Dokl. Akad. Nauk SSSR 313 (1990), 42-47; English transl. in Soviet Phys. Dokl. 35 (1990). 6. B. Gustafsson, Quadrature identities and the Schottky double, Acta Appl. Math. 1 (1983), Stanle y Richardson, Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech. 56 (1972), A. I. Markushevich, Theory of functions of a complex variable, vol. 1, GITTL, Moscow, 1950; English transl., Prentice Hall, Englewood Cliffs, NJ, L. A. Galin, Unsteady seepage with a free surface, Dokl. Akad. Nauk SSS R 47 (1945), (Russian ) 10. B. Gustafsson, On the differential equation arising in a Hele-Shaw flow moving boundary problem, Ark. Mat. 22 (1984), P. Ya. Polubarinova-Kochina, On the motion of the oil contour, Dokl. Akad. Nauk SSSR 47 (1945), (Russian ) 12. P. P. Kufarev, The oil contour problem for the circle with any number of wells, Dokl. Akad. Nauk SSS R 75 (1950), (Russian ) 13. Makot o Sakai, Quadrature domains, Lecture Notes in Math., vol. 934, Springer-Verlag, Berlin and New York, V. M. Entov an d P. I. Etingof, Bubble contraction in Hele-Shaw cells, Quart. J. Mech. Appl. Math. 44 (1991), Murra y H. Protter an d Han s F. Weinberger, Maximum principles in differential equations, Springer-Verlag, Berli n and New York, J. W. Milnor, Singular points of complex hyper surfaces, Princeton Universit y Press, Princeton, NJ, Makot o Sakai, Null quadrature domains, J. Analyse Math. 40 (1981), S. D. Howison, Bubble growth in porous media and Hele-Shaw cells, Proc. Royal Soc. of Edinburgh Sect. A 102 (1986), A. I. Markushevich, Theory of functions of a complex variable, vol. 2, GITTL, Moscow, 1950; English transl., Prentice Hall, Englewood Cliffs, NJ, H. Cartan, Theorie elementaire des fonctions analytiques d'une ou plusieurs variables complexes, Hermann, Paris, 1961 ; English transl., Hermann, Paris, and Addison-Wesley, Reading, MA, B. A. Dubrovin, S. P. Novikov, an d A. T. Fomenko, Modern geometry: methods and applications, 2nd ed., "Nauka", Moscow, 1986 ; Englis h transl. o f 1s t ed., Springer - Verlag, Berlin and New York,

11 72 REFERENCES 22. L. Zalcman, Some inverse problems of potential theory, Contemp. Math. 6 3 (1987), P. I. Etingof, Inverse problems of potential theory and flows in porous media with timedependent free boundary, Comp. Math. Appl. 22 (1991), I. M. Kricheve r an d S. P. Novikov, Algebras of Virasoro type, Riemann surfaces, and structures of the theory of solitons, Funktsional. Anal, i Prilozhen. 2 1 (1987), no. 2, 46-63; English transl., Functional Anal. Appl. 21 (1987), no. 2, V. Isakov, On the uniqueness of a solution in the inverse problem of potential theory, Dokl. Akad. Nauk SSS R 245 (1979), ; English transl. in Soviet Math. Dokl., 20(1979). 26. D. Aharonov and H. S. Shapiro, Domains on which analytic functions satisfy quadrature identities, J. Analyse Math. 30 (1976), C. Ullemar, A uniqueness theorem for domains satisfying quadrature identity for analytic functions, Preprint TRITA-MAT , Mathematics, Royal Inst, of Technology, S , Stockholm, Sweden. 28. V. N. Strakho v an d M. A. Brodsky, On the uniqueness of the solution of the inverse logarithmic potential problem, SIAM J. Appl. Math. 46 (1986), L. Karp, Construction of quadrature domains in R n from quadrature domains in R, Complex Variables: Theor y Appl. 1 7 (1992),

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