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 Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/ulect/003

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 dro p becomes a curve of order four/A. N . Varchenko and P . I. Etingof .

p. cm. —(University lectur e series , ISSN 1047-3998 ; 3) Includes bibliographical references . ISBN 0-8218-7002- 5 1. Boundary value problems. 2 . Curves, Elliptic. 3 . Fluid

dynamics-Mathematical models . I . Etingof, P . I . (Pave l I.) , 1969 - . II . Title. III. Series : Universit y lecture serie s (Providence , R.I.) . QA379.V36 199 2 92-2098 5 515'.35—dc20 CI P

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Contents

Preface vi i

1. Mathematica l model 1

1.1. Filtration flo w of an incompressible fluid 1

1.2. Th e moving boundary proble m 2

2. First integrals of boundary motion 5 2.1. Richardson's integrability theorem

5

2.2. Reconstruction o f a domain fro m th e values of it s moments, and th e inverse problem o f two-dimensional potentia l theory 6

2.3. Results on the uniqueness of a domain with given moments (potential) 7

2.4. The resul t o f injectio n doe s no t depen d o n th e orde r o f wor k o f th e sources and sink s 9

Problems 1 0

3. Algebraic solutions 1 1

3.1. Algebraic and abelian domain s 1 1

3.2. Algebraic solutions 1 1

3.3. The Cauchy transform an d it s properties 1 2

3.4. Singularity correspondenc e theore m 1 2

3.5. Proof o f the theorem o n algebraic solutions 1 4

3.6. Construction o f algebraic solutions 1 4

iii

iv CONTENT S

3.7. Examples 1 6

Problems 1 8

4. Contraction of a gas bubble 2 1

4.1. Formulation o f the problem 2 1

4.2. Th e inclusion propert y 2 2

4.3. Contraction o f a convex domain 2 3

4.4. Contraction point s 2 4

4.5. An analogue of Richardson' s theorem 2 5

4.6. Dynamics of the gravity potentia l 2 7

4.7. The gravity potential a s the solution o f a boundary value problem . 2 7

4.8. Proof o f the main theorem 2 8

4.9. Self-similar solution s 2 9

4.10. Asymptotics of contraction 3 0

4.11. Severa l sources 3 1

Problems 3 2

5. Evolution of a multiply connected domain 3 5

5.1. Statement o f the problem 3 5

5.2. Integrals of motion 3 5

5.3. Algebraic solutions 3 6

5.4. Riemann's theorem 3 7

5.5. Singularity correspondenc e , 3 8

5.6. Proof o f the theorem o n algebraic solution s 3 9

5.7. Construction o f solution s 3 9

5.8. Reconstruction o f an annula r domain fro m it s moments 4 0

Problems 4 2

CONTENTS v

6. Evolution with topological transformations 4 3

6.1. Weak solutions of the evolution problem 4 3

6.2. The simples t algebrai c solutions 4 6

6.3. Weak solutions of the contraction proble m 5 0

6.4. A sufficient conditio n o f a breakup o f a symmetric domain 5 1

Problems 5 2

7. Contraction problem on surfaces 5 5

7.1. Physica l motivation 5 5

7.2. Potential an d contractio n point s 5 5

7.3. Calculation o f the generalized gravity potential on the surface . . . . 5 6

7.4. Breakup of the boundary o n symmetric surfaces o f revolution . . . . 5 8

Problems 6 0

Answers and clues to the problems 6 1

A few open questions 6 9

References . * 7 1

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Preface

In th e fortie s P . Ya . Polubarinova-Kochin a an d P . P . Kufare v studie d the proble m o f evolutio n o f a round oi l spo t 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 th e imag e o f a n ellips e unde r a reflectio n wit h respect t o a circle. I n 195 0 Kufarev manage d t o generalize thi s 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 rationa l 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 197 2 S. Richardson foun d a n infinite serie s of first integrals of motio n of th e spot . H e proved tha t 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

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 an d 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 fo r readin g the English translatio n o f the manuscript an d making important remarks .

References

1. R . Courant, Dirichlet Principle, Conformal Mapping, and Minimal Surfaces, Interscience, New York, 1950 .

2. Stanle y Richardson, Some Hele-Shaw flows with time-dependent free boundaries, J. Fluid Mech. 102(1981) , 263-278 .

3. P . S. Novikov, On the uniqueness for the inverse problem of potential theory, Dokl. Akad. Nauk SSSR 18 (1938), 165-168 . (Russian )

4. Makot o Sakai, A moment problem on Jordan domains, Proc. Amer. Math. Soc. 70 (1978), 35-38.

5. 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), 209-240.

7. 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), 609-618.

8. A . I. Markushevich, Theory of functions of a complex variable, vol . 1 , GITTL, Moscow, 1950; English transl., Prentice Hall , Englewood Cliffs , NJ , 196 7 .

9. L . A. Galin , Unsteady seepage with a free surface, Dokl. Akad. Nauk SSS R 47 (1945) , 250-253. (Russian )

10. B . Gustafsson, On the differential equation arising in a Hele-Shaw flow moving boundary problem, Ark. Mat. 22 (1984), 251-268.

11. P . Ya. Polubarinova-Kochina, On the motion of the oil contour, Dokl . Akad. Nauk SSSR 47 (1945) , 254-257. (Russian )

12. P . P . Kufarev , The oil contour problem for the circle with any number of wells, Dokl. Akad. Nauk SSSR 75 (1950), 507-51 0 (Russian )

13. Makot o Sakai , Quadrature domains, Lecture Notes in Math., vol. 934, Springer-Verlag, Berlin and New York, 1982 .

14. V . M. Entov an d P . I . Etingof, Bubble contraction in Hele-Shaw cells, Quart. J . Mech . Appl. Math. 44 (1991), 507-535.

15. Murra y H . Protte r an d Han s F . Weinberger, Maximum principles in differential equa-tions, Springer-Verlag, Berlin and New York, 1984 .

16. J . W . Milnor , Singular points of complex hyper surfaces, Princeton Universit y Press , Princeton, NJ , 1968 .

17. Makot o Sakai , Null quadrature domains, J. Analyse Math. 40 (1981), 144-154 . 18. S . D. Howison, Bubble growth in porous media and Hele-Shaw cells, Proc . Royal Soc. of

Edinburgh Sect . A 102 (1986), 141-148 . 19. A . I. Markushevich, Theory of functions of a complex variable, vol . 2, GITTL, Moscow,

1950; English transl., Prentice Hall , Englewood Cliffs , NJ , 196 7 . 20. 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, 196 3 .

21. 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, 198 4 .

71

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22. L . Zalcman , Some inverse problems of potential theory, Contemp . Math . 6 3 (1987) , 337-350.

23. P . I. Etingof, Inverse problems of potential theory and flows in porous media with time-dependent free boundary, Comp. Math. Appl. 22 (1991), 93-99.

24. 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, 126-142 .

25. V . Isakov , On the uniqueness of a solution in the inverse problem of potential theory, Dokl. Akad. Nauk SSSR 245 (1979), 1045-1047 ; 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), 39-73 .

27. C . Ullemar , A uniqueness theorem for domains satisfying quadrature identity for ana-lytic functions, Preprin t TRITA-MAT-1980-37, Mathematics, Royal Inst, of Technology, S-100-44, 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), 324-344.

29. L . Karp, Construction of quadrature domains in R n from quadrature domains in R , Complex Variables : Theor y Appl. 17 (1992), 179-188 .